CAMBRIDGE BRIDGE COURSE
For SEE Appeared Students!
MATHEMATICS
CAMBRIDGE INSTITURE
Published by:
CAMBRIDGE INSTITURE
Corporate office:
New Baneshwor
Branch office:
Chabahil Chwok
Copyright© Cambridge institute
Revised and updated edition 2018
About the Book
This book is aimed to develop the competitive abilities of
the students for the succeeding the upcoming entrance
examination.
The topics are compiled to cover-up all fundamental topics
of Mathematics which are sufficient for the entrance
examination and makes easier your further study after SEE
examination.
The book contains 12 units, including multiple choice
questions(MCQ) on last section. Which have been so graded
that they will help student to developed their knowledge
levels and related skills systematically.
This is a modest attempt at making you all attain
proficiency for the competitive entrance examinations. We
hope that students would benefit much from this book.
Best wishes!
Cambridge Institute
New Baneshwor
Chabahil Chwok
Table of contents
Chapter
page
1.Quadratic Equations
1
2.Complex Numbers
6
3.Matrices and Determinants
14
4.System of Linear Equations
25
5.Sequences and Series
29
6.Sets and Logics
32
7.Properties of Triangle
37
8.Pair of line
45
9.Circle
51
10.Limit and continuity
55
11.Derivatives
64
12.Antiderivatives
73
13. Objective Questions -I
76
14. Objective Questions -II
94
15. Try yourself -I
102
16. Try yourself -II
116
Useful Formula
134
Chapter
1
Quadratic equation
The equation of the form ax2 + bx + c = 0 (a0) is called quadratic equation
•
The equation of the form ax2 + bx + c = 0 where a0, b0 is called
complete quadratic equation.
For example, 2x2 – 3x + 1 = 0
•
The equation of the form ax2 + c = 0, where a0 is called a pure
(incomplete) quadratic equation.
For example, 2x2 + 5 = 0
Theorem 1: The two roots of a quadratic equation
ax2 + bx + c = 0 (a0) are
–𝑏+√𝑏 2 −4𝑎𝑐
2𝑎
and
–𝑏−√𝑏 2 −4𝑎𝑐
2𝑎
Proof: Let the given equation be
ax2 + bx + c = 0 where a,b,c, R, a0
Then, ax2 + bx = -c
𝑏
−𝑐
or, x2 + 𝑎 x = 𝑎
𝑏
2
Adding (2𝑎) on both sides
𝑏 2
2𝑎
𝑏
𝑎
𝑥2 + 𝑥 + ( ) =
or, (𝑥 +
𝑏 2
)
2𝑎
𝑏
=
or, 𝑥 + 2𝑎 = ±√
−𝑐
𝑎
𝑏 2
2𝑎
+( )
−4𝑎𝑐+ 𝑏2
4𝑎 2
𝑏2 −4𝑎𝑐
4𝑎2
=±
or, 𝑥 =
−𝑏±√𝑏2 −4𝑎𝑐
2𝑎
𝑜𝑟 𝑥 =
−𝑏 ± √𝑏 2 − 4𝑎𝑐
2𝑎
√𝑏2 −4𝑎𝑐
2𝑎
Two roots of equation (i) are
−𝑏+√𝑏 2 −4𝑎𝑐
2𝑎
&
−𝑏−√𝑏2 −4𝑎𝑐
2𝑎
Cambridge institute/ Mathematics | 1
Nature of roots of quadratic equation
Let ax2 + bx + c = 0, (a0) be a quadratic equation and Let and be two roots of
given equation, then
𝛼=
−𝑏 + √𝑏 2 − 4𝑎𝑐
−𝑏 − √𝑏 2 − 4𝑎𝑐
&𝛽 =
2𝑎
2𝑎
The nature of roots of a quadratic equation is determined by the expression under
radical sign (ie b2 – 4ac) which is called discriminant of the quadratic equation.
Therefore there are following cases
1) If b2 – 4ac > 0, than the roots and are real and unequal.
2) If b2 – 4ac > 0 and perfect square then the roots and are rational & unequal
provided that a,b,c are rational
−𝑏
3) If b2 – 4ac = 0 then the two roots and are real and equal & = = 2𝑎
4) If b2 – 4ac < 0, then the two roots and are imaginary & unequal.
Relation between roots and coefficient.
Let and be the roots of quadratic equation ax2 + bx + c = 0 (a0).
So that
−𝑏 + √𝑏 2 − 4𝑎𝑐
−𝑏 − √𝑏 2 − 4𝑎𝑐
𝛼=
&𝛽 =
2𝑎
2𝑎
−𝑏+√𝑏2 −4𝑎𝑐
2𝑎
Now, + =
+
−𝑏−√𝑏2 −4𝑎𝑐
2𝑎
−𝑏+√𝑏2 −4𝑎𝑐− 𝑏−√𝑏 2 −4𝑎𝑐
2𝑎
−𝑏
+= 𝑎
=
−𝑏+√𝑏 2 −4𝑎𝑐
−𝑏−√𝑏2 −4𝑎𝑐
)(
)
2𝑎
2𝑎
Also, . = (
=
𝑏2 −𝑏2 +4𝑎𝑐
4𝑎 2
.=
𝑐
=𝑎
𝑐
𝑎
Formation of quadratic equation
2 | Cambridge institute/ Mathematics
Let ax2 + bx + c = 0 (a0) be quadratic equation
𝑏
𝑐
Then, 𝑥 2 + 𝑎 𝑥+𝑎 = 0
or, x2 – ( + ) x + . = 0
(∴ 𝛼 + 𝛽 =
−𝑏
, 𝛼. 𝛽
𝑎
𝑐
= 𝑎)
Hence the equation may be written as
x2 – (sum of roots ) x + product of the roots = 0
i.e. x – sx + p = 0.
Worked out examples
Example 1 : Determine the nature of the roots of 2x2 – 3x – 2 = 0
Solution, since a = 2, b = -3 & c = -2
b2 – 4ac = 25 > 0, the roots are real, rational & unequal.
Example 2: Prove that the roots of 2x2 – 6x + 7 = 0 are imaginary
Solution, Here a= 2, b = -6 & c = 7 so that b2 – 4ac = -20 < 0, Hence the
result
Example 3: If the equation x2 + (K+2) x + 2K = 0 has equal roots, find K.
Solution, Here, a=1, b=K+2 & c = 2k. since roots are equal, b2 – 4ac = 0 or
2
(K+2) – 4.1.2 K = 0
or k2 -4k -3 = 0
K=2
Example 4: Form a quadrates equation where roots and 2, - 3
Solution, + = -1, . = -6
Hence the equations.
X2 – (+) x + = 0
X2 + x – 6 = 0
Example 5: Find the value of K so that the equation
Cambridge institute/ Mathematics | 3
(3K+1) x2 + 2 (K+l) x + K = 0 many have reciprocal roots.
Solution: Here,
a= 3K+l, b=2(K+l), c=K
+ =
. =
−2(𝐾+𝐼)
3𝐾+𝐼
𝐾
3𝐾+𝐼
Since roots are reciprocals
Then . = 1
𝐾
3𝐾+𝐼
=1
K=3K + I
K=
−1
2
Exercise:
1.
2.
3.
4.
5.
Determine the nature of the roots of each of the following equations:
a) x2 – 6x + 5 = 0
b) x2 – 4x – 3 = 0
c) x2 – 6x + 9 = 0
d) 4x2 – 4x + 1 = 0
e) 2x2 – 9x + 35= 0
f) 4x2 + 8x – 5 = 0
For what value of P will the equation 5x2 – Px + 45 = 0 have equal roots?
If the equation x2 + 2(K+2)x+9K = 0 has equal roots final K.
For what value of a will the equation x2 – (3a-1) x + 2 (a2 -1) = 0 have equal roots
𝑎
If the roots of the equation (a2+b2)x2 – 2(ac+bd) x +(c2 + d2)=0 are equal the 𝑏 =
𝑐
𝑑
6.
7.
8.
9.
Show that the roots of the equation (a2-bc)x2+2(b2-ca) x+ (c2-ab)=0 will be equal
if either b=0 or a3+ b3 +c3 -3abc =0
Show that the roots of the equation x2-4abx+(a2+2b2)2 =0 are imaginary.
Form the equation whose roots are
a) 3, -2
b) -5, 4
c) √3,− √3
d) -3+5i, -3-5i
e) a+ib, a-ib
Find a quadratic equation where roots are twice the roots of 4x2 + 8x -5 = 0
4 | Cambridge institute/ Mathematics
10. Find the quadratic equation where roots are the reciprocals of the roots of 3x2 –
5x – 2 = 0
11. Form a quadratic equation where roots are the squares of the roots of 3x2 – 5x –
2=0
12. Find the value of K so that
a) 2x2 + Kx – 15 = 0 has one root = 3
b) 3x2 + Kx – 2 = 0 has roots where whose sum is equal to 6.
c) 2x2 + (4-K)x – 17 = 0 has roots equal but opposite in sign.
d) 3x2 –(5+K)x +8 =0 has roots numerically equal but opposite in sign.
e) 3x2 + 7x + 6 – K = 0 has one root equal to zero
f) 4x2 – 17x + K 0 has the reciprocal roots.
Answer
1) a) real, rational & unequal
b) real, irrational & unequal
c) real, rational & equal
d) real, rational & equal
e) imaginary & unequal
f) real rational & unequal
2)
3)
4)
8)
P = I 30
K = I or 4
a=3
a) x2- x – 5 = 0
b) x2 + x – 10 = 0
c) x2 – 3 = 0
d) x2 + x -1 = 0
e) x2 + 6x + 34 =0
f) x2 -2ax + (a2 + b2) = 0
9)
10)
11)
12)
x2 + 4x – 5 = 0
2x2 + 5x -3 = 0
9x2 – 37x + 4 = 0
a) K = -1
b) K = - 18
c) K = 4
d) K = -5
e) K = 6
f) K = 4
Cambridge institute/ Mathematics | 5
Chapter
Complex number
2
Introduction:
However the real number system is not sufficient in mathematics to solve all
algebraic problems for example, the equation of the types x2 + 1 = 0, x2 + 9 = 0 cannot
be solved into real numbers. In order to obtain the solution of such equation it
becomes necessary to extend the real number system. Euler was the first
mathematician to introduce the symbol i for √−1 with the property that i2 + 1 = 0 i.e.
i2 = -1 and called i = √−1 as the imaginary number.
"Integral power of i"
The basic definition that i2 = - 1 lead us
i3 = i2 x i = - I x i = -i
i4 = (i2)2 = (-1)2 = 1
i5 = i4 i = (I) I = i
i6 = (i2)3 = (i2)3 = (-i)3 = -1 and so on
Definition: An order pair (a, b) of real number express in the form a + ib is called
complex number, where a & b are real numbers and i is imaginary unit. We call a as
the real part of complex number and b as the imaginary part of complex number.
Generally the complex numbers are denoted by Z and W.
Equality of complex numbers:
Two complex numbers z1 = x1 + iy1 and z2 = x2 + iy2 are said to be equal if x1 = x2 and y1
= y2
Algebra of complex numbers
(i) Addition.
The sum of two complex number
z1 = x1 + iy1 and z2 = x2 + iy2 denoted by z1 + z2 is defined by z1 + z2 = (x1+iy1) + (x2 +
iy2)
= (x1 + x2) + i (y1 + y2)
(ii) Subtraction.
The difference of two complex number z1 = x1 + iy1 and z2 = x2 + iy2 denoted by z1
– z2 is defined by z1 – z2 = (x1 + iy1) – (x2 +iy2) = (x1 – x2) +i (y1 – y2)
(iii) Multiplication
a) The product of two complex numbers z1 = x1 + iy1 and z2 = x2 + iy2 denoted by
z1z2 is defined by
6 | Cambridge institute/ Mathematics
a) z1z2 = (x1 + iy1) (x2 + iy2)
= (x1x2 – y1y2) +i (x1y2 + y1x2)
(iv) Division
If Z1 = x1 + iy1 and z2 = x2 + iy2 be the two complex number
then,
=
=
𝑧1
𝑧2
=
𝑥1 +𝑖𝑦1
𝑥2 +𝑖𝑦2
=
𝑥1+𝑖𝑦1
𝑥2 +𝑖𝑦2
𝑥
𝑥2 −𝑖𝑦2
𝑥2 −𝑖𝑦2
(𝑥1 𝑥2 +𝑦1 𝑦2) + 𝑖(𝑥2𝑦1 −𝑥2 𝑦1−𝑥1 𝑦2)
𝑥22 +𝑦22
(𝑥1 𝑥2 +𝑦1 𝑦2)
𝑥22 +𝑦22
+
𝑖(𝑥2 𝑦1 −𝑦1 𝑦2 )
𝑥22 +𝑦22
The imaginary unit:
The complex number (0, 1) is denoted by i is called imaginary unit.
Conjugate of complex number.
If Z = a + ib be a given complex number, then its conjugate is denoted by 𝑧̅ and is
defined by 𝑧̅ = a-ib.
Properties of conjugates
If z = a + ib and w = c + id be any two complex numbers then
1
(i) (z+𝑧̅) = a = Re (z)
2
1
(ii) (z-𝑧̅) = b = Im (z)
2𝑖
(iii)𝑧̅̅̅̅̅̅̅̅
+ 𝑤 = 𝑧̅ + 𝑤
̅
̅̅̅̅
(iv) (𝑧̅) = z
(v)𝑧̅ 2 = (𝑧̅)2
Proof (i) 𝑧 + 𝑧̅ = a+ib + a – ib = 2a
1
= (𝑧 + 𝑧̅ ) = a = Re (z)
2
(iii) 𝑧̅̅̅̅̅̅̅̅
+ 𝑤 = ̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅
𝑎 + 𝑖𝑏 + 𝑐 + 𝑖𝑑
(𝑎 + 𝑐) + 𝑖(𝑏 + 𝑑)
= ̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅
= (a+c) – I (b+d)
= a-b + c-id = 𝑧̅+𝑤
̅
other results follow similarly.
Cambridge institute/ Mathematics | 7
Absolute value of complex number
If z = a + ib be any complex number, then its absolute value (or modulus) is nonnegative number denoted by |z| and is defined by |𝑧| = √𝑎2 + 𝑏 2 .
Properties of absolute values of complex number
If z and w are two complex numbers then
(i) |z| = |𝑧̅|
(ii)|zw| = |z||w|
𝑧
|𝑧|
(iii)| | = |𝑤|
𝑤
(iv)|z+w| |z| + |w|
Proof of (i) :
Let z = a + ib and 𝑧̅ = a-ib
by defn : |z| = √𝑎2 + 𝑏 2
| 𝑧̅ | = √𝑎2 + (−𝑏)2 = √𝑎2 + 𝑏 2
|z| = |𝑧̅|
remaining results follow similarly.
Square root of complex number
Let x + iy be the square of a complex number z = a + ib, then
√𝑎 + 𝑖𝑏 = 𝑥 + 𝑖𝑦
Squaring an both sides
a+ ib = x2 + 2ixy + i2y2
=(x2 – y2) + (2xy) i
Equating real and imaginary parts
x2 – y2 = a – (i) and 2xy = b – (ii)
We know
x2 + y2 = √(𝑥 2 − 𝑦 2 )2 + 4𝑥 2 𝑦 2
2x2 = √𝑎2 + 𝑏 2 - (iii)
Adding (i) & (iii)
2x2 = √𝑎2 + 𝑏 2 + a
8 | Cambridge institute/ Mathematics
x2 =
√𝑎 2 +𝑏2 +𝑎
2
= (
1/2
√𝑎 2 +𝑏2 +𝑎
)
2
Similarly, subtracting (i) from (iii)
1/2
√𝑎 2 +𝑏2 −𝑎
)
2
y = (
Note : (1) If b is +ive, both x & y must have same sign.
(2) If b is –ive, x & y have opposite sign.
The cube root of unity.
Let the cube root of unity be z
i.e. z = 11/3
Then z3 = 1
or z3 – 1 = 0
or (z-1) (z2 + z +1) = 0
either z – 1 = 0 or z2 + z + 1 = 0
z = 1 or z =
−1±√1−4
2
−1±√−3
2
−1±√3i
= 2
−1±√3i
= 2
−1±√3i −1−√3i
z = 1,
,
2
2
=
The first is a real number and other two are imaginary roots, any one of which is
denoted by w (omega). Then
𝑤=
−1 + √3i
−1 − √3i
& 𝑤2 =
2
2
Properties of cube roots of unity
1.Each imaginary cube root of unity is square of each other
i.e. (w)2 = w2 , (w2)2 = w
2.The sum of cube roots of unity is zero i.e. 1+w + w2 = 0
3.The product of cube roots of unity is one
i.e. 1.w.w2 = 1 or w3 = 1
Worked out examples
Example 1: Evaluate:
Cambridge institute/ Mathematics | 9
(a) (1,0)6 (b) (0,1)7
Solution:
(a) (1,0)6 = (1+oi)6 = (1)6 = 1
(b) (0,1)7 = (0+i1)7 = i7 = -i
Example 2: Find the value of x & y if
(x, y) = (1,3) +(2,3)
Solution (x, y) = (1+2, 2+3)
(x, y) = (3, 5)
x= 3, y = 5
Example 3: Simplify, 3√−4 + 5√−9 − 4√−25
Solution: 3√4𝑖 2 + 5√9𝑖 2 − 4√25𝑖 2
= 6i + 15i – 20i = i.
Example 4: Express (2+5i) + (1-i) in the form a+ib
Solution : (2+5i)+(1-i)
= 2+5i+1-i
= 3-4i
which is in the form a+ib.
Example 5: Find the absolute value of
Solution : |
1−2𝑖
1−2𝑖
√(1)2 +(−2)2
| = | 2+𝑖 | =
2+𝑖
√(2)2 +(1)2
1−2𝑖
2+𝑖
=1
Example 6:
Find the square roots of complex numbers 5-12i
Solution: Let x+iy be square roots of 5-12i
i.e. √5 − 12𝑖 = x+iy
or, 5-12i = (x+iy)2
or, 5-12i = x2 – y2 + 2ixy
Equating real & imaginary parts
x2-y2 = 5 (i) & 2xy = -12 – (ii)
Also (x2+y2)2 = (x2-y2)2 + 4x2y2
10 | Cambridge institute/ Mathematics
= 52 + 4 (-6)2 = 16g
x2 + y2 = 13 – (iii)
Adding (i) & (iii)
2x2 = 18 x+3
Again, subtracting (i) from (iii)
2y2 = 8, y2 = 4 y = 2
Since xy = -6, x& y have opposite sign
So that x=3 & y = -2 and x = -3 & y =2
The square roots are 3-2i & - 3 + 2i
The square roots of 5-12i are (3-2i)
To solve the above problems we can also use then following formulas,
1/2
√𝑎 2 +𝑏2 +𝑎
)
&
2
x = (
1/2
√𝑎 2 +𝑏2 −𝑎
)
2
y = (
Example 7 : Show that (1-w+w2)4 + (1+w-w2)4 = -16
1.H.S = (1-w+w2)4 + (1+w-w2)4 [1+w+w2 = 0]
= (-2w)4 + (-2w2)4
= 16w4 + 16w8
= 16 (w+w2) = 16(-1) = -16.
Exercise
1. Evaluate
a) (1,0)2
c) (0,1)5
2.
3.
b) (1,0)5
d) (0,1)11
Find the values of x and y if
a) (x,y) = (2,3) + (3,2)
b) (x,y) = (2,1) + (-2,-1)
c) (x,y) = (1,1). (2,3)
d) (x,y) =
e) x+iy = (2-3i)(3-2i)
f) (x-1) i+ (y+1) = (1+i) (4-3i)
Simplify
a) √−9 + √−25 − √−36
c) 3i2 + i3 + 9i4 – i7
b)(3— √4)(2 + √−9)
1
1
1
1
d) 2 + + 3 + 4
𝑖
𝑖
(1,1)
(3,4)
𝑖
𝑖
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4.
Express each of the following complex number in the form of a+ib
a) (2+5i)+(1-i)
b) (2+5i) – (4-i)
c) (2+3i) (3-2i)
d) 3+4i /4-3i
e) i/2+i
f) (1-i) / (1+i)2
g) 2 − √−25 1 − √−16
h) √1 + 𝑖/1 − 𝑖
5.
Complete the absolute value of the following
a) l + 2i
b) l + √3i
c) (1+2i) (2+i)
d) (3+4i) (3-4i)
e) (1+i)-1
f) 1 + i / 1 - i
6.
If z = 1 + 2i & w = 2i verify that
a) 𝑧𝑤
̅̅̅̅ = 𝑧̅ 𝑤
̅
c) |z+w| |z|+|w|
7.
b) |zw| = |z||w|
a) If (3-4i) (x+iy) = 3 √5, show that 5x2 + 5y2 = 9
𝑎−𝑖𝑏
b) If x+ iy = 𝑎+𝑖𝑏, show that x2 + y2 = 1
c) If
𝑙−𝑖𝑥
=
𝑙+𝑖𝑥
a-ib, prove that a2 + b2 = 1
𝑙−𝑖
d) If x-iy = √𝑙+𝑖, prove that x2 + y2 = 1
8.
Determine the square roots of the following complex number.
a) 5 + 12i
b) -5 + 12i
c) 8 + 6i
d) -8 + 6i
e) 7 – 24 i
f) -7 + 24i
g) i
h) 12 – 5i
9.
If w be a complex cube root of unity, show that
a) (1+w-w2) – (1-w+w2)3 = 0
b) (2+w+w2)3 + (1=w-w2)8 – (1-3w+w2)4 = 1
c) (1-w+w2)4 (1+w-w2)4 = 256
d) (1-w)(1-w2) (1-w4)(1-w8) = 9
𝑎+𝑏𝑤+𝑐𝑤 2
𝑎+𝑏𝑤+𝑐𝑤 2
e) 𝑎𝑤+𝑏𝑤 2 +𝑐 + 𝑎𝑤 2 +𝑏+𝑐𝑤 = -1
12 | Cambridge institute/ Mathematics
Answers
1.
a) 1
b) 1
c) i
d) –i
f) − 2 − 2 𝑖
2.a) x=5, y=5
b) x=0, y=0
c) x=-1, y=5
𝑥
−1
d) x=25 , 𝑦 = 25
e) x = 0, y = -13
f) x = 2, y = 6
5.a) √5
b) 2
c) 5
d) 25
1
e) 2
3.a) 2i
b) 12 = 5i
c) 6
d) 0
8.a) (3+2i)
b) (2+3i)
c) (3+i)
d) (1+3i)
e) (4-3i)
f) (3+4i)
1
g) 2 (1 + 𝑖)
4.a) 3+4i
b) -2+61
c) 12+5k
d) 0+i
1
2
e) 5 + 5 𝑖
1
22
1
3
g) 17 + 17 + 𝑖
h)
1
1
+ 2𝑖
√2
√
√
f) 1
h)
√
1
(5
√2
+ 𝑖)
f) (3 + 4𝑖)
Cambridge institute/ Mathematics | 13
Chapter
3
Matrices and Determinants
A rectangular arrangement of numbers in rows and columns enclosed in round or
square brackets is called a matrix. The numbers which form a matrix are called the
elements or the entries of the matrix.
𝑎 𝑑
1 2 3 𝑏 𝑒
For eg:[
][
]
4 5 6 𝑐 𝑓
Order of matrix.
If a matrix has m rows and n columns, we call it a matrix of order mxn or more briefly
m by n matrix.
For examples:
3 5
1) [
] is a matrix of order 2x2
−3 0
2)[
7 2 −3
] is a matrix of order 2 x3
1 4 2
1 4 6
3)[−2 0 8] is a matrix of order 3x3
5 3 6
Some different types of matrices.
1) Row matrix :- A matrix having only one row is called row matrix
For eg: (2 4 6) 1x3
2)
Column matrix:- A matrix having only are column is called column matrix.
1
For eg: (2)
3
3)
Square matrix:- A matrix having equals number of rows and columns is called
square matrix.
𝑎11 𝑎12 𝑎13
𝑎
For eg: A =( 21 𝑎22 𝑎23 )
𝑎31 𝑎32 𝑎33
14 | Cambridge institute/ Mathematics
4)
Diagonal matrix: A square matrix having all non diagonal elements zero is called
a diagonal matrix. In other words, A square matrix A = (aij) is said to be diagonal
matrix if aij = 0 for all ij.
1 0 0
For eg: (0 2 0 )
0 0 −3
5)
Scalar matrix : A square matrix A = (aij) is said to be a scalar matrix if
2 0 0
𝐾, 𝑓𝑜𝑟 𝑖 = 𝑗
aij = {
, where K is any number eg: (0 2 0)
0, 𝑓𝑜𝑟 𝑖 ≠ 𝑗
0 0 2
6)
Unit or identify matrix: A square matrix having diagonal elements unity and
1 0
non-diagonal elements equal to zero is called unit matrix examples:-[
]
0 1 2𝑥2
1 0 0
[0 1 0]
0 0 1 3𝑥3
7)
Null matrix or zero matrix:
A matrix having all of its elements zero is called a null matrix.
0 0
eg: (
)
0 0 2𝑥2
Equality of two matrices
Let A = (aij)mxn and B = (bij)mxn are two matrices of the same order mxn. We say that
matrices A and B are equal and we write A=B if aij = bij for all i & j.
Algebra of Matrices
1.Addition of matrices
2.Multiplication of a matrix by a scalar.
3.Subtraction of matrix from a matrix.
4.Multiplication of matrices
1.Addition of matrices
Let A = (aij)mxn and B = (bij)mxn be two matrics. Then their sum (i.e. A+B) is defined to
be the matrix [cij]mxn where cij = aij + bij for lim, ijn.
2.Multiplication of a matrix by a scalar
Let A be a matrix of order mxn and K be any scalar. Then the matrix obtained by
multiplying each elements of the matrix A by K is said to be scalar multiplication of A
by K and is denoted by KA.
In symbol: Let A = [aij]mxn then KA = [Kaij]mxn
Cambridge institute/ Mathematics | 15
Properties of multiplication by a scalar.
Let A and B be two matrix of same order then
1) K (A+B) = KA + KB, K is any scalar
2) (K1+K2) A = K1A + K2A
3) K2(K2A) = (K1K2) A
4) IA = A and (-I) A = -A
Subtraction of Matrices
Let A = (aij)mxn and B = (bij)mxn are two matrices of the same order mxn. Then their
difference denoted by A-B is defined by A-B = (aij-bij)mxn for all I & j
Product of two matrices
Two matrices, A of order mxn and B of order pxq are said to be conformable for
multiplication iff n=p (ie no. of columns of matrix A is equal to no. of roes of matrix
B). The product matrix AB of the matrices A& B is of the order mxq.
Working rule for product of two matrices
𝑔 ℎ
𝑎 𝑏 𝑐
Let A =[
] & B =[ 𝑖 𝑗 ]
𝑑 𝑑𝑒 𝑓 2𝑥3
𝑘 𝑙 3𝑥2
We see that the numbers of column of A is equal to the numbers of rows in B, so that
AB is exists.
𝑔 ℎ
𝑎 𝑏 𝑐
AB = [
] [𝑖 𝑗]
𝑑 𝑒 𝑓
𝑘 𝑙
=[
𝑎𝑔 + 𝑏𝑖 + 𝑐𝑘
𝑑𝑔 + 𝑒𝑖 + 𝑓𝑘
𝑎𝑏 + 𝑏𝑗 + 𝑐𝑙
]
𝑑ℎ + 𝑒𝑗 + 𝑓𝑙
Transpose of a matrix:
A new matrix formed by changing all rows into column (or column into rows) of a
given matrix A is called the transpose of matrix A. It is denoted by AI or 𝐴̅ or AT or
tan(A)
1 2
1 3 5
eg: Let A = [
] then AT = [3 0]
2 0 7
5 7
Properties of transpose of a matrix
Let A and B be the two matrices, then
(1) (AT)T = A
16 | Cambridge institute/ Mathematics
(2) (A+B)T = AT + BT
(3) (AB)T = BT.AT
(4) (KA)T = KAT. Where K is any scalar.
Worked out Examples
Example 1: Construct a 3x3 matrix A whose elements aij are given by aij = 3i+2j
Thus,
a11 = 3.1 + 2.1 = 5 a12 = 3.1 + 2.2 = 7 a13 = 3.1 + 2.3 = 9
a21 = 3.2 = 2.1 = 8 a22 = 3.2 + 2.2 = 10 a23 = 3.2 + 2.3 = 12
a31 = 3.3 + 2.1 = 11 a32 = 3.3 + 2.2 = 13 a33 = 3.3 + 2.3 = 15
Hence the required matrix A is given by
𝑎11 𝑎12 𝑎13
5
7
9
A = (𝑎21 𝑎22 𝑎23 )= ( 8 10 12)
𝑎31 𝑎32 𝑎33
11 13 15
Example 2: Find the product AB and BA and show that AB BA.
2 3
1 −2 3
A=(
) and B = (4 5)
−4 2 5
2 1
Solution : Here A is a 2x3 matrix and B is a 3x2 matrix. Then the product AB and BA
are defined.
2 3
1 −2 3
0 −4
AB = (
) (4 5) = (
)
−4 2 5
10 3
2 1
2 3
−10 2 21
1 −2 3
& BA = (4 5) (
)= (−16 2 37)
−4 2 5
2 1
−2 −2 11
Thus AB BA
Cambridge institute/ Mathematics | 17
Exercise:
1.
Construct a 3x3 matrix whose elements aij are given by
(i) aij = i + 2j
(ii) aij = 3j – 2i
(iii) aij = 2ij
(iv) aij = ij
3
If A = (
1
3
A -3x = (
8
2
); find the matrix x such that
5
5
)
2
3.
1
If A = (
2
−1
1 1
) and B = (
), show that AB BA.
−1
4 −1
4.
1
If A = (
3
2
), show that A2 – 2A – 5I = 0, where 0 is the 2x2 null matrix
1
5.
If A = [
6.
If A = [
7.
2 1 0
1 0
If A = [ 3 5 −4]and B = [−1 2
−1 2 6
3 0
2.
4 3
], show that A2 – 9A + 14I = 0, where I and 0 are identity and null
2 5
matrices respectively.
1 2
], find x and y such that A2 – 4A – 5I = 0
3 1
2
5] and K = 2
3
The compute AT and BT and verify,
(i) (AT)T = A
(ii) (A+B)T = AT BT
(iii) (A-B)T = AT - BT
(iv) (KA)T = KAT
(vi) (A2)T = (AT)2
Answers:
3 5
1) (i) (4 6
5 7
2)
(v) (AB)T = BTAT
7
8)
9
1 4 7
2
(ii) (−1 2 5) (iii) (4
−3 0 3
6
0 −1
(
)
3 1
18 | Cambridge institute/ Mathematics
4
6
1 1 1
8 12) (iv) (2 4 8 )
12 18
3 9 27
Determinants
A determinant is a number associated with square matrix. Corresponding to each
square matrix A, there is associated a number, called the determinant of A, denoted
by det (A) or |A|. a matrix is an arrangement of numbers and it has no fixed value
but a determinant is a number and it has a fixed value.
Determinant of a square matrix of order 1 Let A = [a11] be a square matrix of order
1x1 then the determinant of matrix A is denoted by det (A) or |A| or and is defined
by the no. a11
i.e. |A| = |a11| = a11
For examples:
Let A = [5], then |A| = |5| = 5
B = [-3] then |B| = |-3| = =3.
Determinant of a square matrix of order 2
𝑎11 𝑎12
Let A = [𝑎
] be a square matrix of order 2x2, then the determinant of A is
21 𝑎22
denoted by det (A)_ or |A| or and defined by the number a11 a22 – a21 a12
𝑎11
i.e. |A| = |𝑎
21
𝑎12
𝑎22 | = a11 a22 – a21 a12
Minors and Cofactors
Minor: Let A = (aij) be a square matrix. Then the minor of any element aij of the
matrix A is a determinant of the sub matrix obtained from A, by deleting ith row and
jth column of the matrix A. It is denoted by Mij.
Let us consider matrix A of order 3x3.
𝑎11 𝑎12 𝑎13
i.e.= [𝑎21 𝑎22 𝑎23 ]
𝑎31 𝑎32 𝑎33
𝑎22
Then the minor of a11 is M11 = |𝑎
32
𝑎21
The minor of a12 is M12 = |𝑎
32
𝑎23
𝑎33 | = a22 a33 – a32 a23
𝑎23
𝑎33 | = a21 a33 – a31 a23
Cambridge institute/ Mathematics | 19
𝑎21
The minor of a13 is M13 = |𝑎
𝑎22
𝑎32 | = a21a32 – a31a22
𝑎12
The minor of a21 is M21 = |𝑎
𝑎13
𝑎33 |= a12a33 – a32 a13 and so on.
31
32
1
4
3
−4
For example If A= [0
2
M11 = minor of a11 = | 3
−4
−2
−1]
5
−1
| = 15 – 4 = 11
5
M12 = minor of a12 = |
0 −1
|=0+2=2
2 5
M13 = minor of a13 = |
0 3
| = 0 – 6 = -6
2 −4
M21 = minor of a21 = |
4 −2
| = 20 – 8 = 12
−4 5
Similarly
M22 = 9, M23 = -12, M31 = 2, M32 = -1, M33 = 3
Cofactors: Let A = (aij) be a square matrix. Then the cofactor of any element aij is
denoted by Aij and defined by
Aij =(-1)i+j Mij, where Mij is minor of aij.
Consider a square matrix A of order 3x3
𝑎11 𝑎12 𝑎13
i.e. A = [𝑎21 𝑎22 𝑎23 ] then
𝑎31 𝑎32 𝑎33
A11 = Cofactor of a11 = (-1)1+1 M11
𝑎22 𝑎23
= (-1)2 |𝑎
|= (a22 a33 – a32 a23)
32 𝑎33
A12 = Cofactor of a12 = (-1)1+2 M12
𝑎21 𝑎23
= (-1)3|𝑎
| = -(a21 a33 – a31 a23)
31 𝑎33
A13 = Cofactor of a13 = (-1)1+3 M13
𝑎21 𝑎22
= (-1)4|𝑎
| = (a21a32 – a31a22) and so on.
31 𝑎32
20 | Cambridge institute/ Mathematics
1
Example Let A = [0
2
3 2
1 0]
1 5
1
A11 = Cofactor of a11 = (-1)1+1 M11 = (-1)2|
1
0
|= 5
5
0 0
A12 = Cofactor of a12 = (-1)3 |
|= 0
−2 5
Similarly
A13 = 2, A21 = -13
A23 = -7 A31 = -2
A22 = 9
A32 = 0 A33 = 1.
Determinant of a square matrix of order 3
𝑎11 𝑎12 𝑎13
𝑎
let A = [ 21 𝑎22 𝑎23 ] be a square matrix of order 3x3. Then
𝑎31 𝑎32 𝑎33
the determinant of matrix A is denoted by |A| or and is defined to be the number
a11A11 + a12A12, Where A11, A12 and A13 are the cofactors of a11, a12 and a13
respectively.
𝑎11 𝑎12 𝑎13
i.e. |A| = [𝑎21 𝑎22 𝑎23 ]
𝑎31 𝑎32 𝑎33
= a1A11 + a12A12 + a13A13.
= a11(-1)1+1 M11 + a12 (-1)1+2 M12 + a13 (-1)1+3 M13
𝑎22
= a11 |𝑎
32
𝑎23
𝑎21
𝑎33 |- a12 |𝑎31
𝑎23
𝑎21
𝑎33 |+ a13 |𝑎31
𝑎22
𝑎32 |
= a11 (a22a33 – a33a23) –a12 (a21a33-a31a23) + a12 (a21a32 – a31a22)
This process is called the expansion of the determinant along the first row.
Sarrus diagram for expansion of a determinant of order 3
Working rule:
𝑎11 𝑎12 𝑎13
let A = [𝑎21 𝑎22 𝑎23 ]
𝑎31 𝑎32 𝑎33
Cambridge institute/ Mathematics | 21
Write the elements as shown in following way
𝑎11
𝑎
| 21
𝑎31
𝑎12
𝑎22
𝑎32
𝑎13
𝑎23
𝑎33
𝑎11
𝑎21
𝑎31
𝑎12 −
𝑎22 |
𝑎32 +
|𝐴| = a11a22a33 + a12a23a31 + a13a21a32 – a31a22a13 – a32a23a11 – a33a21a12
Example :
−1 2
0
Let A =[ 1 −2 −1]then
2
1
0
−1 2
0
|A| =| 1 −2 −1|
2
1
0
−2 −1
1 −1
1 −2
= -1|
| -2 |
|+0|
|
1
0
2 0
2 1
= -1 (0+1) – 2(0+2) + 0(1+4)
= -1-4+0 = 5
−1 2
0
Example : Let A = [ 1 −2 −1]then
2
1
0
The value of determinant
−1 2
0 −1 2
1 −2 −1 1 −2
2
1
0
2
1
|A| = (-1) x (-2) x 0 + 2 (1) x 2 + 0 x1x1 – 2 x(-2) x 0 – 1 x (-1) x (-1) – 0 x 1 x 2
=0–4+0–0–1–0
= -5
22 | Cambridge institute/ Mathematics
Adjoint of a Matrix
The adjoint of a square matrix A is the transpose of the matrix obtained by replacing
each element of A by its cofactors and it is denoted by adj (A)
Inverse of a matrix
The inverse of a Non-singular matrix A is given by the formula
1
A-1 = |𝐴| 𝑎𝑑𝑗(𝐴)
Example : Find the adjoint and inverse of
1 0 −1
[3 4
5]
0 −6 −7
Solution :
1
Let, A = [3
0
0 −1
4
5]
−6 −7
Then,
1 0
|A| [3 4
0 6
−1
5]
7
3
4
5
=1|
|-9-|
0
−6 −7
4
|
−6
= (-28+30) – (-18-0)
= 2+18 = 20 0. Hence A-1 exists
4
5
A11 = Cofactor of a11 = |
| = (-28 + 30) = 2
−6 −7
A12 = 21
A13 = -18
A21 = 6
A22 = -7
A23 = 6
A31 = 4
A32 = -8
A33 = 4
2
The matrix of cofactor of A is[6
4
21 −18
−1
6 ]
−8
4
Cambridge institute/ Mathematics | 23
adj (A) =
2 21 −18 𝑇
2
6
4
[6 −1
6 ] = [ 21 −7 −8]
4 −8
4
−18 6
4
and A-1 =
(𝐴)𝑗
|𝐴|
adj(A)
2
6
4
1
= 20 [ 21 −7 −8]
−18 6
4
Exercise:
1) Evaluate the following determinants:
𝑎 0 0
1 2 3
(i) |0 𝑏 0|
(ii) |4 5 6|
0 0 𝑐
7 8 9
(iv) |
𝑎+𝑏
𝑎−𝑏
𝑎−𝑏
|
𝑎+𝑏
1
(v) |2
3
3
5
1
3|
−4 −6
𝑠𝑒𝑐𝜃
(iii) |
𝑡𝑎𝑛𝜃
𝑡𝑎𝑛𝜃
|
𝑠𝑒𝑐𝜃
1 −2 3
(vi) | 0 −1 4|
−2 2 1
2) Find the adjoint and inverses of the following matrices, if possible
3 0 −1
1 2 −1
3 0 −7
(i) [5 1 0 ]
(ii) [2 0 1 ]
(iii) [6 −5 2 ]
0 1 3
0 3 −1
1 0 −2
1 −1 1
(iv) [−1 1 −1]
−1 1
1
3) Find the transpose of a matrix given in example 2.
Answers
1)
(i) abc
(ii) 0
(iii) 1
(iv) 2b(a+b) (v) 14 (vi) 1
3
−1 1
3
−1 1
−3 −1 2
−3 −1 2
1
1
2) (i) [−15 9 −5], 4 [−15 9 −5] (ii) [ 2 −1 −3], 6 [ 2 −1 −3]
5
−3 3
5
−3 3
6 −3 −4
6 −3 4
24 | Cambridge institute/ Mathematics
Chapter
4
System of Linear equation
Cramer's rule:
Let us consider linear equations
a1x + b1y = c1 --- (i)_
a2x + b2y = c2 --- (ii)
Multiplying the first equation by b2 and second by b1 and subtracting (ii) from (i), we
set
(a1b2 – a2b1) x = b2c1 – b1c2
𝑏 𝑐 −𝑏 𝑐
x = 𝑎 1𝑏1 −𝑎1 𝑏2
1 2
=
𝑐 𝑏1
| 1
|
𝑐2 𝑏2
𝑎1 𝑏1 ,
|
|
𝑎2 𝑏2
&y=
2 1
provide that a1b2 – a2b1 0
𝑎1 𝑐1
|𝑎
|
2 𝑐2
𝑎
𝑏1 ,
| 1
|
𝑎2 𝑏2
We can write as follows
∆
∆
x= 1 , y = 2 , where
∆
𝑐
∆1 = | 1
𝑐2
∆
𝑎1
𝑏1
| , ∆2 = |𝑎
𝑏2
2
𝑐1
𝑎1
𝑐2 | & ∆ = |𝑎2
𝑏1
|
𝑏2
Similarly we can derive the formulae related to three variables.
Example :Solve : 3x + 4y = 14
5x + 6y = 22
3
Here, ∆ = |
5
4
| = 3x 6 -5 x 4 = 2
6
Cambridge institute/ Mathematics | 25
14 4
∆1 = |
|=-4
22 6
3
∆2 = |
5
Now x =
y=
∆2
∆
=
∆1
∆
−4
−2
14
|=-4
22
−4
= −2 = 2
=2
x=2&y=2
Row – equivalent matrix method:
Let us consider linear equations
a1x + b1y = c1
a2 x + b2y = c2
𝑎 𝑏1
From the augmented matrix [ 1
𝑎2 𝑏2
: 𝑐1
]
: 𝑐2
1 0 : 𝛼
]
0 1 : 𝛽
Then x = & y = so (,) is solution of given system.
by successive row operations, reduce it to the form [
Example: Using row-equivalent matrix method solve the system
4x + 5y = 2
2x + 3y = 0
Solution:
Here the augmented matrix of the system is
4
[
2
5 : 2
]
3 : 0
5
4
1
[1
2
: 2] R 1 R
1
1
4
3 : 0
1
[
0
5
4
1
2
:
1
2
: −1
] R1 (-2) R1 + R2
26 | Cambridge institute/ Mathematics
5
4
:
1
2
[1
0
1 : −2
1
[
0
−5
0 :3
] R 1 4 R2 + R 1
1 : −2
] R1 2 R2
Which gives x = 3, y = -2.
Exercise:
1) Solve the following system by using row equivalent matrices and Cramer's rule
(i)
x-y = 2
2x + 3y = 4
(v)
3x – 3y = 11
9x – 2y – 5 = 0
(ii)
6x + 4y – 18 = 0
2x + 3y = -6
(vi)
5x + 7y + 2 = 0
4x + 6y + 3 = 0
(iii)
2y – 3x = 0
x+y=5
(vii) 2x + 5y = 7
5x – 2y = - 3
(iv)
–x+y = - 9
x - 5 = 3y
(viii) 7x + 2y = 6
-5x + 4y = 3
Solve the following system of equations by using Cramer’s rule.
(i)
x+y+z= 7
x–y+z–2=0
x–y+z=3
x+y–z=3
(iv)
2x – y + 4z = - 3
x – 4z = 5
(ii)
2x + y + 3z = 7
6x – y + 2z = 10
3x + 5y + z = 0
5x + 2y – z = 1
(v)
9y – 5x = 3
x+z=1
(iii)
3y – 2z – 1 = 0
z + 2y = 3
5x + y – 5 = 0
Cambridge institute/ Mathematics | 27
Answers:
1. (i) x = 2, y = 0
(iii) x = 2, y = 3
1
(v) x = 3, y = -4
−1
(vii) x = 29 , 𝑦 =
2.
(i)
(iv)
(3,2,2)
(3,7, -1/2)
(ii)x = -2, y =0
(iv)x = 11, y = 2
9
−7
(iv)x = 2 y = 2
41
29
9
(viii)x = 19 , 𝑦 =
(ii) (1, -1, 2)
(v) (3,2, -2)
28 | Cambridge institute/ Mathematics
(iii)(0,5,7)
51
38
Chapter
5
Sequence and series
Definition: A sequence is a function whose domain is the set of natural number to
non-empty set X. i.e. f; N X
If X = R then the sequence is real sequence
If X = C then the sequence is complex sequence
Since, the domain of every sequence is the set of N of natural number, therefore a
sequence is represented by its range.
Example:
(1) 1,2,3,4 ………….
(2) t1, t2, t3, …………..
Series: If t1, t2, t3 ……. be a sequence then the expansion t1 + t2 + t3 …….. is series.
Examples:
(1) 1 +2+ 3 + 4 + ……….
(2) t2 + t2 +t3 + t4 ………..
Progressions
Those sequence whose terms follow certain patterns are called progressions.
Types of progression.
1) Arithmetic progression (A.P)
2) Geometric progression (G.P)
3) Harmonic progression (H.P)
1) Arithmetic progression: A sequence is called an arithmetic progression if the
difference of term and its proceeding term is always same i.e tn-1 – tn = constant, for
all nN
Example
(1) 1, 4, 7, 10 …….. is an A.P
(2) 11, 7, 3, -1, ……….. is an A.P.
2) Geometric progression. A sequence of non-zero numbers is called a geometric
progression if the ratio of a term to its preceding term is always a constant quantity.
Example:
(1) 4, 12, 36, 108 ………… is a G.P.
(2) a, ar, ar2 ……….. is a G.P.
Cambridge institute/ Mathematics | 29
3) Harmonic progression: A sequence t1, t2, t3, …… of non zero number is called a
Harmonic progression if the reciprocals of its terms form an A.P.
1 1 1
Example: The sequence 1, 3 , 5 , 7, …… is a H.P. because, 1, 3, 5, 7 …. is an A.P.
Formula for means
given a & b are two number, then
𝑎+𝑏
(i) A.M. = 2
(ii) G.M. = √𝑎𝑏
2𝑎𝑏
(iii) H.M. = 𝑎+𝑏
Sum to infinity of G.P.
𝑆∞ =
𝑎
,
𝑙−𝑟
if |r| < 1
Solved Examples:
Example : Show that a2, b2, c2 are in A.P.
If
1
1
,
1
,
𝑏+𝑐 𝑐+𝑎 𝑎+𝑏
Solution:
So,
or,
1
𝑐+𝑎
𝑏−𝑎
𝑏+𝑐
1
,
are in A.P.
1
,
1
𝑏+𝑐 𝑐+𝑎 𝑎+𝑏
1
1
−
=
𝑏+𝑐
=
𝑎+𝑏
are in A.P.
−
1
𝑐+𝑎
𝑐−𝑏
𝑎+𝑏
or, b2 – a2 = c2 – b2
a2, b2, c2 are in A.P
Example: Prove that 21/2. 21/4. 21/8 ………….. = 2
Solution: 21/2. 21/4.21/8 ………
1 1 1
= 22+4+8
30 | Cambridge institute/ Mathematics
=
1
2
1
1−
2 2
= 21
= 2.
Exercise:
1
1
1) If G is the geometric mean between a & b, show that 𝐺 2 −𝑎2 + 𝐺 2 −𝑏2 =
2) If H be the harmonic mean between a & b prove that
1
𝐻−𝑎
+
1
𝐻−𝑏
=
1
𝑎
1
𝐺2
+
1
𝑏
3) If A be the arithmetic mean & H, the H.M. between two quantities a & b, show
𝑎−𝐴 𝑏−𝐴
𝐴
that 𝑎−𝐻 x 𝑏−𝐻 = 𝐻
5) Determine k so that k+2, 4 k-6 and 3k-2 are three consecutive terms of A.P. [Ans:
k=3]
6) Find the three numbers in A.P. where sum is 21 and product is 315. [Ans 9, 7, 5 or
5, 7, 9]
7) Find the sum of the following series.
a) 16 + 8 + 4 + 2 + ………….
[Ans: 32]
5
5
5
1
1
1
b) 5 + 2 + 4 + 8 + … … … … [𝐴𝑛𝑠: 10]
c) 1 − 2 + 4 − 8 + … … … … [𝐴𝑛𝑠: 2/3]
d) 4-1 + 4-2 + 4-3 + ……………
[Ans: 1/3]
Cambridge institute/ Mathematics | 31
Chapter
6
Sets and Logics
The theory of sets is essential for the study in any area of mathematics and its
application and hence it is considered as the foundation of all branch of Modern
Mathematics.
The word "set" is an undefined term in mathematics. We shall use it as a primitive to
define other terms. As set is meant primitive to define other terms. A set is meant as
a collection of well define objects. A given collection of objects is said to be well
defined, if we can defiantly say whether a given particular objects belongs to the
collection or not.
Eg: 1) The no. of students in Xavier College.
2) A collection of beautiful girls in Pokhara is not set because the term "beauty"
is not well defined.
Types of sets:
1) Singleton set: A set having only one element is called singleton set.
2) Empty set: A set having no element. It is denoted by or { }
3) Equivalent set: Two finite sets A an B are said to be equivalent if n(A) = n(B).
4) Universal set: The set of all possible members under consideration is called the
universal set. It is denoted by U.
5) Subsets of a set: If A and B are any two sets, then A is called subset of B if every
element of A is also an element of B.
Union of two sets:
The union two sets A and B is defined as the set of all elements which belong to A or
B or both. In symbol
AB = {x: x A or x B}
Intersection of two sets:
The intersection of two set A & B is defined as the set of all elements belonging to
both A and B. In symbol.
AB = {x: x A or X B}
32 | Cambridge institute/ Mathematics
Difference between two sets.
The difference of two sets A and B is the set of all elements of A but not belonging to
B.
In symbol,
A-B = {x: x A and xB}
Complement of set.
The complement of set 𝐴̅ is denoted by A and define as
𝐴̅ = {x: x and xA }
= {x: xA }.
Algebra of sets:
Solve examples:
Example:
Let A and B are subset of a universal set U then
(i) A U A = {x: x A or X A}
= {x: x A}
(ii) A U U = {x: x A or X U}
= {x: x U}
=U
(iii) A U B = {x: x A or X B}
= {x: x B or X A}
= B U A.
Logical connectives:
The word used to combine two or more statement is called logical connectives. The
connectives used to form a compound statement are presented in Table.
Com pound statement
by the connectives
Conjunction
Disjunction
Negation
Conditional
Biconditional
Logical
connectives
And
Or
Not
If ……. Then
If and only if
Symbol
^
v
~
Cambridge institute/ Mathematics | 33
Conjunction:
Any two statements can be combined by word "and" to form a compound statement
is called the conjunction
Example :
P : Binod is an engineer
Q: Sovit is a doctor
Their conjunction is "Binod is an engineer and Sovit is a doctor: and this compound
statement is symbolized by P^q. The truth table of the conjunction is presented
below.
P
T
T
F
F
q
T
F
T
T
P^q
T
F
F
F
Disjunction:
Any two statements connected by the word "or" to form a compound statement is
called the disjunction of the original statements.
Let P and q be two prime statements. A disjunction is a statement of the form P or q"
is respectively symbolically by
P^q
Truth table of disjunction
P
q
Pvq
T
T
T
T
F
T
F
T
T
F
F
F
Conditional Statement
A condition statement or simply a conditional is a statement of the form "If P, then q"
and is represented symbolically by P q and is read as "p arrow q". The connective
if ….. then is called the conditional connective.
The statement P is called the hypothesis and the statement q is called conclusion.
The truth table of the conditional statement
P q is given by
34 | Cambridge institute/ Mathematics
Truth Table of conditional
P
q
Pq
T
T
T
T
F
F
F
T
T
F
F
T
Biconditional
A statement of the form "P if and only if q" is known as biconditional of the
statements P and q. It is denoted by P q
Truth table: Biconditional
P q P q q P (Pq)^(qP)
T
T
T
T
T
T
F
F
T
F
F
T
T
F
F
F
F
T
T
T
Negative: If P is given statement then it negation is given by ~ P
Tautology and contraction.
Tautology: A compound statement which is always true, whatever may be the truth
values of its components, is known as tautology.
Contradiction: A compound statement which is always false, whatever may be the
truth values of its components are known as a contradiction.
Consider the following example
(i)
P
q
PV ~ P
T
F
T
F
T
T
(ii)
P
~P
P^ ~ P
T
F
F
F
T
F
In example (i), the truth value of statement are true hence tautology.
In example (ii) the truth value of statement are false hence contradiction.
Cambridge institute/ Mathematics | 35
Exercise:
1) Let A, B and C are subsets of universal Set U, then
i) A U =
ix) A(BC) = (AB) (AC)
ii) A U B = A = and B
x) A(BC) = (AB) (AC)
iii) A A = A
xi)̅̅̅̅̅̅̅
𝐴𝑈𝐵 = 𝐴̅ ∩ 𝐵̅
iv) A =
xii) (𝐴 ∩ 𝐵) ̅ = 𝐴̅ ∪ 𝐵̅
v) A = A
xiii) A-(BC) = (A-B) (A-C)
vi) A B = BA
xiv) A- (BC) = (A-B) (A-C)
vii) (AB) C = A (BC)
xv) A(B-C) = (AB) – (AC)
viii) (AB) C = A (BC)
xvi) A – B = A 𝐵̅
2)Construct truth tables for the following compound statements
a)~ P q
d)~[P ~ q)
b)~ P ~ q
e)(P q) (q P)
c)~ (P q)
f)(~ P q) (P q)
36 | Cambridge institute/ Mathematics
7
Chapter
Properties of triangles
1.1 Introduction
A triangle ABC consists of three sides and three angles they are called six elements of
triangle. The angles of a triangle ABC is denoted by A, B and C, the sides BC, CA and
AB are denoted by small letters a, b and c. The semi perimeter of the triangle is
represented by s so that 2s = a+b+c. The various relations existing among them are
deducted in the following articles.
1.2 The sine law
Before proving this theorem, we have to define circum circle
A circle that passes through the vertices of a triangle is called circum – circle. The
centre of circum circle is called circum centre and radius is called circum – radius.
Statement: In any triangle ABC, the sides are proportional to the sine of the opposite
𝑎
𝑏
𝑐
angle, that is
=
=
𝑠𝑖𝑛𝐴
𝑠𝑖𝑛𝐵
𝑠𝑖𝑛𝐶
OR
𝑎
𝑏
𝑐
In any triangle ABC, 𝑠𝑖𝑛𝐴 = 𝑠𝑖𝑛𝐵 = 𝑠𝑖𝑛𝐶 = 2R, where R is circum –radius.
The proof of sine law
fig (i)
fig (ii)
fig (iii)
A
A
A
D
O
B
C
B
C
B
O
O
C
D
Let O be the circum –centre and R be the circum radius of a triangle ABC.
(i)The angle A is an acute in fig (1)
(ii)The angle A is obtuse in fig (2)
(iii)The angle A is right in fig (3)
Now, join BO and produce it which meets circumference at D as shown as in figures.
Join DC, then BO = R and BD = 2R
In figure 1
<BDC = <BAC = A and <BCD = 90º
Cambridge institute/ Mathematics | 37
Sin < BDC =
𝐵𝐶
𝐵𝐷
𝑎
2𝑅
∴ 𝑠𝑖𝑛𝐴 =
=
𝑎
2𝑅
In figure 2
<BCD = 90º and <BAC + <BDC = 180º
< BDC = 180º - A
Sin < BDC =
𝐵𝐶
𝐵𝐷
𝑎
or Sin (180º A) = 2𝑅
𝑎
sin A =
2𝑅
In figure 3
<BAC = A = 90º and BC = A = 2R
a = 2R
𝑎
or 2𝑅 = 1 = sin 90º
𝑎
2𝑅 = sin A [90º = A]
If we assume angles B and C are acute, obtuse and right, then we will get
𝑏
𝑐
= 2𝑅 𝑎𝑛𝑑
= 2𝑅
𝑠𝑖𝑛𝐵
𝑠𝑖𝑛𝑐
combining above result we get
𝑎
𝑏
𝑐
=
=
= 2𝑅
𝑠𝑖𝑛𝐴
𝑠𝑖𝑛𝐵 𝑠𝑖𝑛𝐶
Hence, this complete the proof of theorem
1.3 Cosine law
In any triangle,
(i) cosA =
𝑏2 +𝑐 2 −𝑎2
2𝑏𝑎
or a2 = b2 + c2 – 2bc cosA
(ii) cosB =
𝑐 2 +𝑎2 −𝑏2
2𝑐𝑎
or b2 = c2 + a2 – 2ca cosB
𝑎 2 +𝑏2 −𝑐 2
2𝑎𝑏
or c2 = a2 + b2 – 2ac cosC
(iii) cosC =
38 | Cambridge institute/ Mathematics
The proof of cosine – law.
Let ABC be any triangle. <C is acute angle in fig (i), <C is obtuse angle in fig. (ii) & <C is
right angle in fig (iii). Draw AD perpendicular to BC and produce BC if necessary.
A
A
A
c
b
b
b
B
a
CB
D
a
C
C B
a
C
In fig (i)
From right angled triangle ABD
AB2 = BD2 + AD2
or, c2 = (BC-DC)2 + AD2
or c2 = BC2 – 2BC. DC + DC2 + AD2
or c2 = a2 – 2a. AC cosC + AC2
[AD2 + DC2 = AC2]
or c2 = a2 -2ab cosc + b2
cos C =
𝑏2 +𝑎2 −𝑐 2
2𝑏𝑎
From ADC
cos C =
𝐷𝐶
𝐴𝐶
AC cosC = DC
In fig (ii)
AB2
or C2
or C2
cosc =
= AD2 + BD2
= AD2 + (BC+CD)2
=AD2 + BC2 + 2BC. CD + CD2
=AD2 + CD2 + a2 + 2a. - AC cosc
= AC2 + a2 – 2ab cosc
= b2 + a2 -2ab cosc
𝑏2 +𝑎2 −𝑐 2
2𝑏𝑎
In fig (iii)
AB2 = AC2 + BC2
or C2 = b2 + a2
or C2 = b2 + a2 – 2ab. cos 90º [ cos 990º = 0]
or c2 = b2 + a2 – 2ab cosc [< c = 90º]
Cambridge institute/ Mathematics | 39
or cosC =
𝑏 2 +𝑎2 −𝑐 2
2𝑏𝑎
Hence, for all values of C, we have
cosC =
𝑏 2 +𝑎2 −𝑐 2
2𝑏𝑎
Similarly, if we assume angles A and B are acute, obtuse and right, then we will get
cos A =
cos B =
𝑏 2 +𝑐 2 −𝑎2
2𝑏𝑐𝑎
𝑐 2 +𝑎2 −𝑏2
2𝑐𝑎
Hence, this complete the proof of theorem.
The projection law:
In any triangle,
(i)
a = b cos C + c cos B
(ii)
b = c cos A + a cos C
(iii)
c = a cos B + b cos A
Proof (i)
R.H.S. = b cos C + c cos B
= 2R sin B. cos C + 2R sin C cos B ( by sine law)
= 2R (sin B. cos C + sin C. cos B)
= 2R sin (B+C)
= 2 R sin A
=a
The other results follow similarly.
The above expression can also be proved by using cosine law.
40 | Cambridge institute/ Mathematics
The Half – angle formulae:
In any triangle ABC
𝐴
(i) 𝑠𝑖𝑛 2 = √
𝐵
2
𝑠(𝑠−𝑏)
𝑐𝑎
𝐶
2
𝑠(𝑠−𝑐)
𝑎𝑏
(v) 𝑐𝑜𝑠 = √
(𝑠−𝑏)(𝑠−𝑐)
𝑏𝑐
(vi)𝑐𝑜𝑠 = √
(𝑠−𝑐)(𝑠−𝑎)
𝐵
2
(ii) 𝑠𝑖𝑛 = √
𝐶
(iii)𝑠𝑖𝑛 2 = √
𝑐𝑎
𝐴
(vii) 𝑡𝑎𝑛 2 = √
(𝑠−𝑏)(𝑠−𝑐)
𝑠(𝑠−𝑎)
(𝑠−𝑎)(𝑠−𝑏)
𝐴
𝑎𝑏
(iv) 𝑐𝑜𝑠 2 = √
𝐵
𝑠(𝑠−𝑎)
𝑏𝑐
𝐴
𝐶
2
(ix) 𝑡𝑎𝑛 = √
𝑠(𝑠−𝑏)
(𝑠−𝑎)(𝑠−𝑏)
𝑠(𝑠−𝑐)
(𝑠−𝑏)(𝑠−𝑐)
To prove, 𝑠𝑖𝑛 2 = √
𝑏𝑐
We have, cos A = 1 – 2sin2
𝐴
(𝑠−𝑐)(𝑠−𝑎)
(viii) 𝑡𝑎𝑛 2 = √
𝐴
2
or, 2 sin2 2 = 1 – cos A
=1-
𝑏2 +𝑐 2 −𝑎 2
2𝑏𝑐
=
2𝑏𝑐− 𝑏2− 𝑐 2 −𝑎2
2𝑏𝑐
=
𝑎 2 −(𝑏2 −2𝑏𝑐+𝑐 2 )
2𝑏𝑐
=
𝑎 2 −(𝑏−𝑐)
2𝑏𝑐
=
=
2
(𝑎−𝑏+𝑐)(𝑎+𝑏−𝑐)
2𝑏𝑐
(2𝑠−𝑏−𝑏)(2𝑠−𝑐−𝑐)
2𝑏𝑐
𝐴 4(𝑠−𝑏)(𝑠−𝑐)
2
4𝑏𝑐
Sin2 =
∴ 𝑠𝑖𝑛
(𝑠 − 𝑏)(𝑠 − 𝑐)
𝐴
=√
2
𝑏𝑐
The remaining results follow similarly.
Cambridge institute/ Mathematics | 41
The tangent law:
In any triangle ABC
𝐵−𝐶
)
2
(i)𝑡𝑎𝑛 (
=
𝐶−𝐴
)
2
(ii) 𝑡𝑎𝑛 (
𝑏−𝑐
𝐴
𝑐𝑜𝑡
𝑏+𝑐
2
=
𝐴−𝐵
)
2
(iii) 𝑡𝑎𝑛 (
=
𝑐−𝑎
𝐵
𝑐𝑜𝑡 2
𝑐+𝑎
𝑎−𝑏
𝐶
𝑐𝑜𝑡 2
𝑎+𝑏
The area of triangle:
The area of triangle is denoted by (read as delta) and
1
1
1
(i) = 2 bc sin A = 2 ca sin B = 2 ab sin C.
(ii) = √𝑠(𝑠 − 𝑎)(𝑠 − 𝑏)(𝑠 − 𝑐)
1
(iii) = √2𝑎2 𝑏 2 + 2𝑏 2 𝑐 2 + 2𝑐 2 𝑎2 − 𝑎4 − 𝑏 4 − 𝑐 4
4
𝑎𝑏𝑐
(iv) = 4𝑅
Worked out examples:
Example 1: In any triangle ABC prove that,
a(bcosc – ccosB) = b2 -c2
Solution:
L.H.S. = a (b cosC- c cosB)
= ab cosC – ac cow B
𝑎2 +𝑏2 −𝑐 2
= 𝑎𝑏 (
=
2𝑎𝑏
𝑎2 +𝑐 2 −𝑏2
) − 𝑎𝑐 (
2𝑎𝑐
)
𝑎2 +𝑏 2 −𝑐 2 −𝑎2 −𝑐 2 +𝑏2
2
= b2 – c2 = R.H.S
Example 2: In any triangle ABC, prove that a2 (sin2B-sin2C) + b2 (sin2C – sin2A) + c2
(sin2A – sin2B) = 0
l.H.S: a2 (sin2B-sin2C) + b2 (sin2C – sin2A) + c2 (sin2A – sin2B)
42 | Cambridge institute/ Mathematics
𝑏2
4𝑅2
= 𝑎2 (
−
𝑐2
)
4𝑅2
𝑐2
4𝑅2
+ 𝑏2 (
−
𝑎2
)+
4𝑅2
𝑎2
4𝑅2
𝑐2 (
−
𝑏2
)
4𝑅2
1
= 4𝑅2 (𝑎2 𝑏 2 − 𝑎2 𝑐 2 + 𝑏 2 𝑐 2 − 𝑎2 𝑏 2 + 𝑐 2 𝑎2 − 𝑏 2 𝑐 2)
=
1
𝑥
4𝑅2
0
=0
= R.H.S
Exercise:
In any triangle ABC, prove that
1: i) a2 + b2 + c2 – 2 (bc cosA + ca cosB + ab cosC) = 0
𝑐𝑜𝑠𝐴
𝑎
𝑐𝑜𝑠𝐵
𝑏
𝑐𝑜𝑠𝐶
𝑐
ii) 𝑎 + 𝑏𝑐 = 𝑏 + 𝑐𝑎 = 𝑐 + 𝑎𝑏
1
iii) bc cosA + ca cosB + ab cosC = 2(a2+b2+c2)
𝑐
2
𝑐
2
iv) (a-b)2 cos2 + (a+b)2 sin2 = c2
2:
i)
𝑎 2 𝑠𝑖𝑛(𝐵−𝐶)
𝑠𝑖𝑛𝐴
ii)
3:
+
𝑎 𝑠𝑖𝑛(𝐵−𝐶)
𝑏2 −𝑐 2
𝑏2 𝑠𝑖𝑛(𝐶−𝐴) 𝑐 2 𝑠𝑖𝑛(𝐴−𝐵)
+ 𝑠𝑖𝑛𝐶
𝑠𝑖𝑛𝐵
+
=0
𝑏 𝑠𝑖𝑛(𝐶−𝐴) 𝑐 𝑠𝑖𝑛(𝐴−𝐵)
+ 2 2
𝑐 2 −𝑎2
𝑎 −𝑏
i) (b+c) cos A + (c+a) cosB + (a+b) cosC = a+b+c
ii) b2sin2C + c2sin2B = 2 absinC
1
iii) c2cos2B + b2 cos2C + bc cos(B-C) = 2 (a2 + b2 + c2)
iv) a cosB cosC + b cos C cos A + c cos A cosB =
4:
𝑎𝑏
4𝑅2
i) a2 cotA + b2 cotB + c2 cotC = 4
𝐴
2
𝐵
2
1
2
ii) b cos2 + a cos2 = (a+b+c)
iii)
𝑐𝑜𝑠𝐴
𝑎
+
𝑐𝑜𝑠𝐵
𝑏
+
𝑐𝑜𝑠𝐶
𝑐
=
𝑎 2 +𝑏2 +𝑐 2
2𝑎𝑏𝐶
iv) sin (A+B) : sin (A-B) = c2: a2-b2
𝑐−𝑏 𝑐𝑜𝑠𝐴
𝑐𝑜𝑠 𝐴
v) 𝑏−𝑐 𝑐𝑜𝑠 𝐴 = 𝑐𝑜𝑠 𝐶
vi)
𝑎−𝑏 𝑐𝑜𝑠𝐶
𝑐−𝑏 𝑐𝑜𝑠 𝐴
=
𝑠𝑖𝑛 𝐶
𝑠𝑖𝑛 𝐴
Cambridge institute/ Mathematics | 43
𝑏2 −𝑐 2
𝑠𝑖𝑛2𝐴
𝑎2
vii)
5:
+
𝑐 2 −𝑎 2
𝑏2
+
𝑎 2 −𝑏2
𝑠𝑖𝑛2𝐶
𝑐2
=0
i) If a4+b4+c4 = 2c2(a2+b2), prove that C = 45º or 135º
ii) If (a+b+c) (b+c-a) = 3bc, show that A = 60º
iii) (a sinA + b sinB + c sinC)2 = (a2+b2+c2) (sin2A + sin2B + sin2C)
iv) If 2 cos A = sinB : sinC, show that the triangle is isosceles.
6:
Solve the triangle ABC
i) c = 30º, b = √3 and a = 1
ii) a = 2, b = √6, A = 45º
iii) a = 2, b = √2, C = √3+1
iv) c = 30º, B = 45º, c = 6√2
v) a = 2, b = 4, c= 60º
Answer
i) A = 30º, B = 120º, C = 1
ii) B = 60º, C = 75º, C = √3 + 1, or B = 120º, C = 15º, C = √3 − 1,
iii) 45º, 30º, 105º
iv) 105º, 12, 6(√6 + 1),
v) 30º, 90º
44 | Cambridge institute/ Mathematics
Chapter
8
Pair of lines
Introduction:
Let us consider two st. lines be given by the equations.
a1 x + b1y + c1 = 0 ----- (i)
a2x + b2y + c2 = 0 ------ (ii)
Now, combine equations (i) and (ii)
(a1x + b1y + c1) (a2x + b2y + c2) = 0 ---- (iii)
We say that the set of points lying on (i) and (ii) will satisfy equation (iii). Similarly,
the set of points lying equations (iii) must be satisfied either equations (i) or (ii) or
both. Hence equations (iii) represents pair of lines given by (i) & (ii). If we simplify
equations (iii), we will get,
ax2 + 2bxy + by2 + 2gx + 2fy + c = 0
Every equation of pair of lines is second degree equation. But the converse is not
always true (why?)
Example: Find the single equation representing lines
2x – y + 1 = 0 & 3x + 4y – 8 = 0
Solution, we have given equations:
2x-y +1 = 0 --- (i)
3x+4y -8 = 0 ---- (ii)
combining equations (i) & (ii)
(2x-y+1) (3x+4y-8) = 0
6x2 + 5xy – 4y2 – 13x + 12y – 8 = 0
Homogeneous Equation:
The equation of the form ax2 + 2hxy + by2 = 0 where a, h& b are constants and they
are not zero at the same time is called homogeneous equation of second degree in x
and y.
Theorem 1: The homogeneous equation of second degree ax2 + 2hxy + by2 = 0,
represents a pair of lines through the origin.
Cambridge institute/ Mathematics | 45
Proof:
case I: If b = 0, then given equation can reduce to ax2 +2hxy = 0
or x (ax + 2hy) = 0
clearly, x =0 and ax+2hy = 0
Therefore x=0 and ax+2hy =0 are pair of lines through origin represented by given
equation
case II: If b 0, then dividing on both sides by b of given equation.
𝑎 2 2ℎ
𝑥 +
𝑥𝑦 + 𝑦 2 = 0
𝑏
𝑏
𝑦2
𝑥2 +
2ℎ𝑥𝑦
𝑏𝑥 2
𝑦 2
(𝑥 ) +
𝑎
+𝑏 =0
2ℎ 𝑦
( )
𝑏 𝑥
𝑎
+ 𝑏 = 0 --- (i)
𝑦
equation (i) is quadratic in 𝑥 , so that it have two roots or values say m1 and m2
𝑦
𝑦
𝑥
Then 𝑥 = 𝑚1 𝑎𝑛𝑑
= 𝑚2
y = m1 x and y = m2x are pair of lines through origin & represented by given
equation.
𝑦
Note: since m1 and m2 are two values of 𝑥 of
𝑦 2 2𝑏 𝑦
𝑎
( ) +
( )+ =0
𝑥
𝑏 𝑥
𝑏
then we have,
m1 + m2 =
−2𝑏
𝑏
and m1 . m2 =
𝑎
𝑏
Theorem 2: The angle between the pair of lines represented by ax2 + 2bxy+by2 = 0 is
tan = ±2
√ℎ2 −𝑎𝑏
𝑎+𝑏
, be the angle between the lines.
Proof: Let y = m1 x and y2 = m2x be pair of lines represented by ax2 + 2bxy + by2 =0
Where m2 + m2 =
−2𝑏
𝑏
𝑎
and m1m2 = .
𝑏
If be the angle between the lines then tan
𝑚1 −𝑚2
= 𝐼 1+𝑚
.𝑚
1
2
46 | Cambridge institute/ Mathematics
=±
√(𝑚1 +𝑚2 )2 −4𝑚1 𝑚2
1+𝑚1 𝑚2
(
= ±√
=±
−2ℎ 2
𝑎
) −4.
𝑏
𝑏
𝑎
1+
𝑏
√ℎ 2 −𝑎𝑏
𝑎+𝑏
∴= 𝑡𝑎𝑛−1 (
±2√ℎ2 − 𝑎𝑏
)
𝑎+𝑏
Condition of perpendicularity
Two tines will be to each other if = 90º
tan 90º = ±
2√ℎ 2 −𝑎𝑏
𝑎+𝑏
or, cot 90º = ±
or 0 = ±
𝑎+𝑏
2√ℎ 2 −𝑎𝑏
𝑎+𝑏
2√ℎ2 −𝑎𝑏
a+b = 0
i.e. coeff. of x2 + coeff. of y2 = 0
Condition for coincident (Parallelism)
Two lines will be coincidence if = 0
So, tan 0º = ±
2√ℎ2 −𝑎𝑏
𝑎+𝑏
or, 02 = h2 – ab
or, h2 = ab
Theorem 3: The general equation of second degree
ax2 + 2hxy+by2 = 2yx + 2fy + c = 0 represents a pair of lines if
abc + 2fgh – af2 – bg2 – ch2 = 0
Worked out examples:
Example 1: Find the separate equation of the lines represented by x2 – 5xy + 4y2 = 0
Cambridge institute/ Mathematics | 47
i.e. (x-4y)(x-y) = 0
x-4y = 0 & x – y = 0 are two lines represented by given equation
Example 2: Find the angle between the line pair 2x2+7xy+3y2 =0
Solution, we have, 2x2 + 7xy + 3y2 = 0 --- (i)
Comparing (i) with ax2 + 2hxy + by2 = 0
7
2
Where, a = 2, h = & b = 3.
If be the angle between lines then, tan =±
or tan = ±
=1
2
√ℎ 2 −𝑎𝑏
𝑎+𝑏
49
4
2√ −6
2+3
= 45º or 135º
Example 3: Prove that 2x2 + 7xy + 3y2 – 4x – 7y + 2 = 0
represents two straight lines.
Solution, we have, 2x2 + 7xy + 3y2 – 4x – 7y + 2= 0 ---(i)
Comparing equation (i) with,
ax2 – 2hxy + by2+ + 2gx + 2 fy + c = 0
7
−7
where, a =2 , h = , b=2. g = -2, f = , & c = 2
2
2
equation (i) represents pair of lines if abc +2 fyb–af2 –bg2 –ch2=0
l.H.S = 2.3.2 + 2+ 2(-2) (
7 −2
)( 2 )−
2
= 12 + 49 -
7 2
−7
2 ( 2 )2 - 3(-2)2 – 2(2)
49
2
12 -
49
=
2
0
The given equation represents two st. lines.
Example 4: Find the separate equations of two st. lines represented by 6x2 + 13xy +
6y2 + 8x + 7y + 2 = 0. also find point of inter section of the lines
Solution 6x2 + (13y + 8) x + (6y2 + 7y + 2) = 0 --- (i)
equation (i) is quadrate equation in x.
Solving for x
48 | Cambridge institute/ Mathematics
−(14𝑦+8)± √(13𝑦+8)2 −24 (6𝑦 2 +7𝑦+2)
12
12 x = (13y + 8) ±√25𝑦 2 + 40𝑦 + 16
x=
=
= 12 x = - 13y – 8 (5y + 4)
or, 3x + 2y + 1 = 0 & 2x + 3y + 2 = 0 are separate equations given by equation (i),
solving these two equation we get
x=
1
5
&𝑦 =
−4
5
Exercise
1) Find the single equation representating the line pair.
a)
x + y = 0, x + 2y = 0
b)ax by = 0, bx + ay = 0
c)
x+ + y + 2 = 0, x + 2y + 1 = 0
d)3x + y – 1 = 0, 2x – y + 3 = 0
e)
2y = 3x + 1, x – 5y = 2
2) Determine the lines represented by each of the following equations:
a)
x2 – 2xy = 0
b) x2 – 5xy + 4y2 = 0
c)
xy - 3x – 2y – 6 = 0
d) x2 + 2xy + y2 – 2x – 2y – 15 = 0
e)
x2 + 2xy + y2 + x + y = 0
f) 2x2 + 7xy + 3y2 – 4x + 7y + 2 = 0
g)
4x2 + 4xy + y2 + 2x + y – 12 = 0
3) Find the angle between the following pair of lines
a)
x2 + 9xy + 14y2 = 0
b) x2 – 2xy cot - y2 = 0
c)
x2 – 5xy + 4y2 + x + 2y – 2 = 0
d) 2x2 + 4xy + 2y2 + x + y – 3 = 0
e)
4x2 + 5xy + 4y2 + 2x + y – 12 = 0
4) Find the difference between the slopes of the lines represented by following
equations
a)
8x2 + 10xy + 3y2 = 0
b) 3x2 – 6xy + 2y2 = 0
5) Find the equation of two lines represented by x2 + 6xy + 9y2 + 4x + 12y – 5 = 0.
Prove that the two lines are parallel. Also find the distance between them
6) Find the equation of the two lines represented by the equation 2x2 + 3xy + y2 + 5x
+ 2y – 3 = 0. Find their points of intersection and also the angle between them.
7) Determine the two st. lines represented by 6x2 – xy – 12y2 – 8x + 29 y – 14 = 0.
Hence find the point of intersection for the lines.
Cambridge institute/ Mathematics | 49
8) Prove that each of the equation represents a pair of lines
a)
2x2 + 7xy + 3y2 – 4x – 7y + 2 = 0
b) 6x2 – xy – 12y2 – 8x + 2g – 14 = 0
c)
6xy – 8x + 9y – 12 = 0
d) 2x2 – xy – y2 + 5x + y + 2 = 0
Answers:
1)
a) x2 + 3xy + 2y2 = 0
b) abx2 + (a2 –b2) xy – aby2 = 0
c) x2 + 2y2 + 3xy + 3x + 5y + 2 = 0
d) 6x+2 – xy + y2 + 7x + 4y – 3 = 0
e) 3x2 – 17xy + 10y2 – 5x + y – 2 = 0
2)
a) x = 0, x – 2y = 0
b) x – ly = 0, x – y = 0
c) x + 2 = 0, y – 3 = 0
d) x + y – 5 = 0
e) x + y = 0, x + y + 1 = 0
f) x + 3y - 1 = 0, 2x + y – 2 = 0
g) 2x + 7 – 3 = 0, 2x + y + 4 = 0
3)
1
3
a) tan-1 (± )
b)
3
5
c) tan-1(± )
𝜋
2
d) 0
e) imaginary
4)
a) 2/3
b) √3
5) a) x + 3y + 5 = 0, x +3y-1 =0, 30/5
6) x + y + 3 = 0, 2x + y – 1 = 0,
11 20
, )
17 17
7) 3x + 4y – 7 = 0, x – 3y + 2 = 0 (
50 | Cambridge institute/ Mathematics
Chapter
9
Circle
A circle is a locus of point which moves so that it is always at a constant distance
from the fixed point. The fixed point is called centre and constant distance is called
radius of circle.
Equation of a circle centre at origin. Let O (0,0) be the centre and r be the radius of
the circle. Let P(x,y) be any pint on the circle.
Then OP = r
OP2 = r2
x 2 + y 2 = r2
Y
P(x,y)
r
X1
O
X
Y1
This relation is true for any point P(x,y) on the circle. Hence it is the equation of circle
in standard form.
Equation of a circle centre at any point.
Let c(h,k) be the centre and r the radius of the circle. Let P(x,y) be any point on the
circle so that
CP = r
or, CP2 = r2
Y
or (x-h)2 + (y-k)2 = r2
P(x,y)
r
c(h,k)
x1
x
0
Y1
Cambridge institute/ Mathematics | 51
This is the equation of circle
Equation of circle in general form.
p(x,y)
B(x2y2)
Let us consider the equation,
x2 + y2 + 2gx + 2fy + c = 0 – (i)
Equation (i) can be written as
(x2 + 2gx + g2) + (y2 + 2fy + f2) = g2 + f2 – c
A(x,y)
2
(x+y)2 + (y + f)2 = (√𝑔2 + 𝑓 2 − 𝑐)
Comparing this equations with
(x-h)2 + (y-k)2 = r2
h = -g, k = -f & r = √𝑔2 + 𝑓 2 − 𝑐
equation (i) represents a circle centre at (-g, -f) & radius √𝑔2 + 𝑓 2 − 𝑐
Equation of circle in diameter form,
Let A (x1, y1) and B (x2, y2) be the ends of diameter of a circle. Let P(x,y) be any point
on the circle. Join AP, BP and AB since AB is a diameter of the circle, < APB is a right
angle
𝑦−𝑦
Now slope of AP = 𝑥−𝑥1
1
& slope of BP
𝑦−𝑦
= 𝑥−𝑥2
2
Since AP is r to BP then
𝑦−𝑦1 𝑦−𝑦2
.
= −1
𝑥−𝑥1 𝑥−𝑥2
or, (x-x1) (x-x2) + (y-y1) (y-y2) =0
Which is true for all value of P (x,y) on circle. So it is required condition.
Worked out examples.
Example 1: Find the equation of the circle with the centre at (2,3) and radius 5.
Solution, The required equation is (x-2)2 + (y-3)2 = 52
or x2 + y2 – 4x – 6y – 12 = 0
Example 2: Determine the equation of the circle if the ends of a diameter be at
(3,0) and (7,-1)
Solution, The equation of a circle in diameter form is
(x-x1) (x-x2) + (y-y1) (y-y2) = 0
52 | Cambridge institute/ Mathematics
or (x-3)(x-7) + (y-0) (y+1) = 0
or x2 + y2 – 10x + y + 21 = 0
Exercise:
1)
Find the equation of a circle with
a) centre at (4,5) and radius 3
b) centre at (0,0) and diameter 8
c) centre at (p,q) and radius √𝑝2 + 𝑞 2
d) centre at (4,-1) and through the origin
e) two of the diameters are x+y = 6 and x+2y = 8 and radius 10.
f) centre at (-1,5) and through the point of intersection fo the lines 2x-y=5 &
3x+y = 10
g) (0,0) & (4,7) as the ends of a diameter.
h) concentric with the circle x2 + y2 + 8x – 6y + 1 = 0 & radius 3
i) concentric with the circle x2 + y2 – 8x + 12y + 15 = 0 and passing through
(5,4)
j) passing through the origin and making intercepts equal to 3 & 4 from the
+ive x and y-axis respectively.
k) centre at (3,4) and touching the x-axis.
l) centre at (a,b) and touching the y-axis.
m) centre at (4,5) and touching the line 3x+4y+18=0
2)
3)
Find the centre and radius of following circles
a)
x2 + y2 – 12x – 4y = 9
c)
4(x2 + y2) + 12 ax – 6ay – a2 = 0
b)x2 + y2 – 3x + 2y – 3 = 0
Find the equation of circle
a)
passing through the points (0,0), (a,0) and (0,b)
b)
passing through the points (1,2), (3,1) and (-1, -1)
Cambridge institute/ Mathematics | 53
Answers:
1) a) x2 + y2 – 8x – 10y + 32 = 0
b)x2 + y2 = 16
c)
x2 + y2 – 2px – 2ay = 0
d)x2 + y2 – 8x + 2y = 0
e)
x2 + y2 – 16x + 4y – 32 = 0
f)x2 + y2 + 2x + 10y -6 = 0
g)
x2 + y2 – 4x – 7y = 0
h)x2 + y2 + 8x – 6y + 16 = 0
i)
x2 + y2 – 8x + 12y – 49 = 0
j)x2 + y2 – 3x – 4y = 0
k)
x2 + y2 – 6x – 8y + 9 = 0
l)x2 + y2 – 2ax – 2by + b2 = 0
m) x2 + y2 – 8x – 10y + 37 = 0
2)
3
b)(2 , −1) , 2
a) (6,2), 7
−39 39 79
, 4 ), 4
2
c) (
3)
a) x2 + y2 – ax – bx = 0
b)
54 | Cambridge institute/ Mathematics
x2 + y2 – x + 3y – 10 = 0
Chapter
10
Limits and Continuity
Let us take a function f(x) = x+2
when x=3, then f(3) = 3+2 =5. Therefore 5 is called functional value of given function.
The value of function at fixed point is called functional value.
Consider a function y = f(x) =
𝑥 2 −1
𝑥−1
0
When x =1, the given function takes 0 form which is indeterminate. Now we shall see
the nature of given function in the neighborhood of x=1.
there are two cases (i) when x- approaches to 1 through the values greater than 1,
written as x 1+ and case (ii). When x approaches to 1 through the value less than 1,
written as (x 1-)
Case 1:
1.001
…………. 1.000001
x 1+ 1.1 1.01
1
y
2.1 2.01
2.001
…………. 2.000001
2
Case 2:
0.9998 …………. 0.99999
x 1- 1.9 0.99
1
y
1.9 1.99
1.998
…………. 1.99999
2
In table 1 if x approaches 1 from right hand side, then y approaches to 2.Then 2 is called
𝑙𝑖𝑚
right hand limit of given function. In symbol
𝑓(𝑥) = 2
𝑥 → 1+
In table 2 if x approaches 1 from left hand side, then y approaches to 2. Then 2 is called
𝑙𝑖𝑚
left hand limit of given function In symbol,
𝑓(𝑥) = 2
𝑥 → 1+
Now, In both cases,
We see that as x approaches to 1, y approaches to 2. In symbol
𝑙𝑖𝑚
𝑓(𝑥) = 2
𝑥 → 1+
Example
Cambridge institute/ Mathematics | 55
In above figure, the area of the regular polygon is increasing and the area of the
polygon cannot exceed the area of the circle. So the limit of the area of the polygon is
the area of the circle.
Example: f(x) = 1/x, for all x n
When x increases, the value of f(x) goes on decreasing and never becomes zero, no
matter how large x is chosen. In such case we say that f(x) approaches the limit o as x
increases indefinitely.
So in general,
A function y = f(x) is said to have limit 𝑙 at x=a if f(x) approaches to 𝑙 as x approaches
to a symbolically,
𝑙𝑖𝑚
𝑓(𝑥) = 𝑙
𝑥→𝑎
Rule of limits.
If f(x) and g(x) are the two functions of x such that
𝑙𝑖𝑚
𝑙𝑖𝑚
𝑓(𝑥) = 𝑚 and
𝑔(𝑥) = 𝑚
𝑥→𝑎
𝑥→𝑎
1)
𝑙𝑖𝑚
𝑙𝑖𝑚
𝑙𝑖𝑚
[𝑓(𝑥) 𝑔 (𝑥)] =
𝑓(𝑥)
𝑔(𝑥) = 𝑚 𝑛
𝑥→𝑎
𝑥→𝑎
𝑥→𝑎
2)
𝑙𝑖𝑚
𝑙𝑖𝑚
𝑙𝑖𝑚
[𝑓(𝑥) . 𝑔 (𝑥)] =
𝑓(𝑥) .
𝑔(𝑥) = 𝑚𝑛
𝑥→𝑎
𝑥→𝑎
𝑥→𝑎
3)
𝑙𝑖𝑚 𝑓(𝑥) 𝑥→𝑎 𝑓(𝑥) 𝑚
=
= (𝑛 0)
𝑥 → 𝑎 𝑔(𝑥) 𝑙𝑖𝑚 𝑔(𝑥) 𝑛
𝑙𝑖𝑚
𝑥→𝑎
4)
𝑙𝑖𝑚 𝑥 𝑛 −𝑎𝑛
=nan-1
𝑥 → 𝑎 𝑥−𝑎
Methods of finding limits
We generally evaluate algebraic limits by using the following methods.
1) Direct substitution
2) Factorization
3) Form:
𝑥 𝑛 −𝑎𝑛
𝑥−𝑎
4) Rationalization
5) Limit at infinity
1) Direct substitution:
Substitute the direct value of variable in the given expression yields the required
limiting value
56 | Cambridge institute/ Mathematics
Example: Evaluate the following
𝑙𝑖𝑚
(i)
(7x2 – 5x + 1)
𝑥→0
=
𝑙𝑖𝑚
𝑙𝑖𝑚
𝑙𝑖𝑚
7𝑥 2 −
5𝑥 +
1
𝑥→0
𝑥→0
𝑥→0
𝑙𝑖𝑚 2
𝑙𝑖𝑚
=7
𝑥 +5
𝑥+1
𝑥→0
𝑥→0
= 7.0 – 5.0 + 1 = 1
(ii)
𝑙𝑖𝑚 𝑥−1
𝑥 → 1 𝑥+1
1−1
= 1+1 =
0
2
=0
2) Factorization method
0
If the functional value takes the form 0 while substituting the value of variable then
factorize numerator or and denominator (if possible) and cancel the common factor
and put the value of variable in the determinate form to get the required limiting value.
Example:
𝑙𝑖𝑚 𝑥 2 − 1
(i)
𝑥 → −1 𝑥 + 1
0
When x= -1, the given function takes the form 0
Which is indeterminate
So,
𝑙𝑖𝑚 𝑥 2 − 1
𝑙𝑖𝑚 (𝑥−1)(𝑥+1)
𝑙𝑖𝑚 (𝑥
=
=
− 1)= -1 -1 = -2.
𝑥 → −1 𝑥 + 1 𝑥 → −1 (𝑥 + 1)
𝑥 → −1
3) Form
𝒙𝒏 −𝒂𝒏
𝒙−𝒂
Example:
𝑙𝑖𝑚 𝑥 5 − 32
(i)
𝑥 → 2 𝑥− 𝑎
=
𝑙𝑖𝑚 (𝑥)5 −(2)5
= 5.25-1 = 5.24 = 5x16 = 80
𝑥 → 2 𝑥−1
Cambridge institute/ Mathematics | 57
4) Rationalization
In function which involve square roots, rationalization of numerator or denominator
or both numerator and denominator is essential to find limit.
Example:
𝑥−1
𝑙𝑖𝑚
𝑥 → 1 √𝑥 2 + 3 − 2
0
When x =1, the given function takes 0 form.
𝑥−1
√𝑥 2 +3+2
𝑙𝑖𝑚
So,
x
2 +3−2 √𝑥 2 +3+2
√𝑥
𝑥→1
= 𝑙𝑖𝑚
(𝑥−1)(√𝑥 2 +3+2
𝑥→1
𝑥 2 +3−4
= 𝑙𝑖𝑚
√𝑥 2 +3−2
𝑥→1
𝑥 2 −1
= 𝑙𝑖𝑚
𝑥→1
√𝑥 2 +3−2
𝑥−1
2+2
4
= 1+1 = 2 = 2.
5) Limit at infinity
∞
Type 1: If functional value takes the form , when x = then divide the numerator and
∞
denominator of the fraction by the highest power of x present in the fraction
Example:
2
𝑙𝑖𝑚 3𝑥 + 5𝑥 + 3
𝑥 → 7𝑥 2 + 10𝑥 + 8
∞
It takes the form ∞ When x =
So,
5 3
𝑙𝑖𝑚 (3 + 𝑥 + 𝑥 2 )
𝑥 → (7 + 10 + 8 )
𝑥 𝑥2
=
3+0+0
7+0+0
3
=7
Type 2: If functional value takes the form ( - ) , when x = , then rationalize the
∞
numerator or denominator to convert into ∞ form and then use the process in type
(1).
𝑙𝑖𝑚
(√𝑥 − 𝑎 − (√𝑥 − 𝑏)
𝑥→
=
𝑙𝑖𝑚
𝑥→
(√𝑥−𝑎−√𝑥−𝑏)(√𝑥−𝑎)+√𝑥−𝑏)
(√𝑥−𝑎+√𝑥−𝑏)
58 | Cambridge institute/ Mathematics
𝑙𝑖𝑚
𝑥→
𝑙𝑖𝑚
=
𝑥→
𝑥−𝑎−𝑥+𝑏
𝑥−𝑎+
√𝑥−𝑏
√
=
=
𝑏−𝑎
√𝑥−𝑎+√𝑥−𝑏
𝑏−𝑎
∞
=0
Limits of trigonometric functions:
Two standard results
𝑙𝑖𝑚
→0
𝑙𝑖𝑚
(2)
→0
(1)
𝑠𝑖𝑛𝜃
=1, where
𝜃
is measured in radian.
𝑡𝑎𝑛 𝜃
=1
𝜃
Solved Examples:
Example 1:
𝑙𝑖𝑚
𝑥→0
𝑠𝑖𝑛 𝑎𝑥
𝑠𝑖𝑛 𝑏𝑥
𝑠𝑖𝑛 𝑎𝑥
=
𝑎
𝑎
𝑙𝑖𝑚 𝑎𝑥
𝑥 =
𝑥 → 0 𝑠𝑖𝑛𝑏𝑥𝑏𝑥 𝑏 𝑏
Example 2:
𝑙𝑖𝑚
𝑥→0
𝑠𝑖𝑛 𝑎𝑥
𝑥
𝑙𝑖𝑚
𝑥→0
𝑠𝑖𝑛 𝑎𝑥
𝑥
𝑎𝑥
𝑎 = 1𝑥𝑎 = 𝑎
Example:
𝑙𝑖𝑚
𝑥→0
1−𝑐𝑜𝑠 9𝑥
𝑥2
=
=
𝑙𝑖𝑚
𝑥→0
𝑙𝑖𝑚
=
𝑥→0
𝑙𝑖𝑚
=2
𝑥→0
1−𝑐𝑜𝑠 2
9𝑥
2
𝑥2
1−1+2𝑠𝑖𝑛2
𝑥2
9𝑥
2
𝑥2
𝑠𝑖𝑛2
9𝑥
=2
9𝑥
2
2
𝑠𝑖𝑛
𝑙𝑖𝑚
( 𝑥2 )
𝑥→0
Cambridge institute/ Mathematics | 59
9𝑥
𝑠𝑖𝑛
9
𝑙𝑖𝑚
=2
( 9𝑥2 𝑥 2)
𝑥→0
2
81
2
81
2
= 2x1x 4 =
𝑙𝑖𝑚 𝑥𝑠𝑖𝑛𝜃−𝜃𝑠𝑖𝑛𝑥
𝑥−𝜃
𝑥→0
𝑙𝑖𝑚 𝑥𝑠𝑖𝑛𝜃−𝜃𝑠𝑖𝑛𝜃+𝜃𝑠𝑖𝑛𝜃−𝜃𝑠𝑖𝑛𝑥
Solution:
𝑥−𝜃
𝑥→0
𝑙𝑖𝑚 𝑠𝑖𝑛𝜃(𝑥−𝜃)+𝜃(𝑠𝑖𝑛𝜃−𝑠𝑖𝑛𝑥)
=
𝑥−𝜃
𝑥→0
𝑙𝑖𝑚 𝑠𝑖𝑛𝜃 + 𝜃 𝑙𝑖𝑚 𝑠𝑖𝑛𝜃−𝑠𝑖𝑛𝑥
=
𝑥→0
𝑥 → 0 𝑥−𝜃
Example:
= 𝑠𝑖𝑛 +
𝑙𝑖𝑚
𝑥→0
2 𝑠𝑖𝑛
𝜃−𝑥
𝜃+𝑥
.𝑐𝑜𝑠
2
2
𝑥−𝜃
𝑥−𝜃
2 𝑠𝑖𝑛
.𝑐𝑜𝑠
𝑙𝑖𝑚
2
= 𝑠𝑖𝑛𝜃 + 𝜃
−
𝑥+𝜃
𝑥→0
2𝑥
2
𝜃+𝑥
2
= sin - cos
Exercise:
1) Find the following limits.
𝑙𝑖𝑚
(2x2 + 2x – 4)
𝑥→2
𝑙𝑖𝑚 3𝑥2 + 2𝑥 − 4
c)
𝑥 → 1 𝑥 2 + 5𝑥 − 4
𝑙𝑖𝑚
(2x2 + 2x – 9)
𝑥→5
𝑙𝑖𝑚 6𝑥 2 + 3𝑥 − 12
d)
𝑥 → 3 2𝑥 2 + 𝑥+ 1
a)
b)
2) Compute the following limits.
𝑙𝑖𝑚 4𝑥 3 − 𝑥2 + 2𝑥
𝑥 → 0 3𝑥 2 + 4𝑥
𝑙𝑖𝑚 𝑥 2/3 − 𝑎2/3
c)
𝑥 → 𝑎 𝑥−𝑎
a)
𝑙𝑖𝑚 𝑥 3 − 64
𝑥 → 4 𝑥 2 −16
𝑙𝑖𝑚 𝑥 2 + 3𝑥−4
d)
𝑥 → 1 𝑥−1
b)
e)
𝑙𝑖𝑚 𝑥 2 − 4𝑥 + 4
𝑥 → 2 𝑥 2 −7𝑥+10
f)
𝑙𝑖𝑚 √3𝑥 − √2𝑥+𝑎
𝑥 → 𝑎 2(𝑥−𝑎)
g)
𝑙𝑖𝑚 √2𝑥− √3𝑥−𝑎
𝑥 → 𝑎 √𝑥 − √𝑎
h)
𝑙𝑖𝑚 √2𝑥 − √3−𝑥2
𝑥−1
𝑥→1
60 | Cambridge institute/ Mathematics
i)
𝑙𝑖𝑚 √𝑥− √6−𝑥2
𝑥 → 2 𝑥−2
𝑙𝑖𝑚 √3𝑎−𝑥 − √𝑥+𝑎
4(𝑥−𝑎)
𝑥→𝑎
𝑙𝑖𝑚 5𝑥2 + 2𝑥−7
m)
𝑥 → ∞ 3𝑥2 +5𝑥+2
k)
o)
𝑙𝑖𝑚
(√3𝑥 − √𝑥 − 5)
𝑥→∞
q)
𝑙𝑖𝑚
𝑥→1
j)
𝑙𝑖𝑚 6√𝑥−2
𝑥 → 64 3√𝑥−4
𝑙𝑖𝑚 4𝑥 2 + 3𝑥 +2
𝑥 → ∞ 5𝑥 2 +4𝑥−3
𝑙𝑖𝑚
n)
(√𝑥 − √𝑥 − 3)
𝑥→∞
l)
p)
𝑙𝑖𝑚
𝑥→2
𝑥−√8−𝑥 2
√𝑥 2 +12−4
𝑥 − √2−𝑥 2
2𝑥 − √2+2𝑥 2
3) Evaluate the following
𝑙𝑖𝑚
𝑥→0
𝑙𝑖𝑚
3)
𝑥→0
1)
5)
𝑙𝑖𝑚
𝑥→𝑎
𝑡𝑎𝑛 𝑏𝑥
𝑥
𝑡𝑎𝑛 𝑎𝑥
𝑡𝑎𝑛 𝑏𝑥
𝑠𝑖𝑛(𝑥−𝑎)
𝑥 2 −𝑎 2
𝑙𝑖𝑚 𝑠𝑖𝑛 𝑎𝑥. 𝑐𝑜𝑠 𝑏𝑥
𝑠𝑖𝑛 𝑐𝑥
𝑥→0
𝑙𝑖𝑚 1−𝑐𝑜𝑠 6𝑥
9)
𝑥2
𝑥→0
𝑙𝑖𝑚 𝑠𝑖𝑛 𝑎𝑥−𝑠𝑖𝑛 𝑏𝑥
11)
𝑥
𝑥→0
7)
13)
𝑙𝑖𝑚
𝑥→0
𝑙𝑖𝑚
15) 𝑥 → 𝜋
4
17)
𝑙𝑖𝑚
𝑥→𝑦
𝑙𝑖𝑚
𝑥→𝜃
𝑙𝑖𝑚
21)
𝑥→1
𝑙𝑖𝑚
23) 𝑥 → 𝜋
4
19)
𝑡𝑎𝑛 2𝑥 − 𝑠𝑖𝑛 2𝑥
𝑥3
𝑙𝑖𝑚
𝑥→0
𝑙𝑖𝑚
4)
𝑥→0
𝑙𝑖𝑚
6)
𝑥→𝑝
2)
𝑠𝑖𝑛 𝑚𝑥
𝑠𝑖𝑛 𝑛𝑥
𝑠𝑖𝑛 𝑝𝑥
𝑡𝑎𝑛 𝑞𝑥
𝑥 2 −𝑝2
𝑡𝑎𝑛 (𝑥−𝑝)
𝑙𝑖𝑚 1−𝑐𝑜𝑠 𝑥
𝑥2
𝑥→0
𝑙𝑖𝑚 𝑐𝑜𝑠 𝑎𝑥−𝑐𝑜𝑠 𝑏𝑥
10)
𝑥2
𝑥→0
𝑙𝑖𝑚 𝑡𝑎𝑛 𝑥−𝑠𝑖𝑛 𝑥
12)
𝑥3
𝑥→0
𝑙𝑖𝑚
14) 𝑥 → 𝜋 (𝑠𝑒𝑒 𝑥 − 𝑡𝑎𝑛 𝑥)
2
8)
𝑠𝑒𝑒 2 𝑥−2
𝑡𝑎𝑛 𝑥−1
16)
𝑙𝑖𝑚
𝑥→𝑦
𝑡𝑎𝑛 𝑥−𝑠𝑖𝑛 𝑦
𝑥−𝑦
𝑠𝑖𝑛 𝑥−𝑠𝑖𝑛 𝑦
𝑥−𝑦
18)
𝑙𝑖𝑚
𝑥→0
𝑐𝑜𝑠 𝑥−𝑐𝑜𝑠 𝑦
𝑥−𝑦
𝑥 𝑐𝑜𝑡 𝜃− 𝜃 𝑐𝑜𝑡 𝑥
𝑥−𝜃
20)
1+𝑐𝑜𝑠 𝜋 𝑥
𝑡𝑎𝑛2 𝜋 𝑥
𝑐𝑜𝑠𝜃−𝑠𝑖𝑛𝜃
𝜋
𝜃−
4
𝑙𝑖𝑚
𝑥→𝜃
𝑙𝑖𝑚
22)
𝑥→𝜃
24)
𝑙𝑖𝑚
𝑥→𝑐
𝑥 𝑐𝑜𝑠 𝜃− 𝜃 𝑐𝑜𝑠 𝑥
𝑥−𝜃
𝑥 𝑡𝑎𝑛 𝜃−𝜃 𝑡𝑎𝑛 𝑥
𝑥−𝜃
√𝑥−√𝑐
𝑠𝑖𝑛 𝑥−𝑠𝑖𝑛 𝑐
Cambridge institute/ Mathematics | 61
Answers
1.
a) 0
b) 26
c) ½
a) ½
e) 0
i) 2
b) 61
f)
51
d) 52
2.
c)
1
4√2𝑎
2 +/3
𝑎
3
g) −√2
51
52
h) √2
d)
j)
5
m) - 4
√2
1
4
k)- 4
1
√2𝑎
n) 0
o) ∞
1
√2𝑎
p) 4
l) 4/5
q) 2
3.
a) b
i)
18
1
(𝑏 2
2
o)
2
t)
p)
see2y
sin
q)
cos y
u)
½
v)
tan -
b)
m/n
j)
c)
a/b
𝑑2 )
d)
p/q
k)
a-b
r)
–sin y
e)
1/2a
l)
½
s)
cot
f)
2p
m) 4
g)
a/c
n)
h)
½
−
+
𝜗
𝑠𝑖𝑛2 𝜃
cos
+
see2
w) −√2
0
x)
𝑠𝑒𝑒 𝑐
2√𝑐
Continuity of a function:
A function f(x) is said to be continuous at point x =a if
𝑙𝑖𝑚
𝑙𝑖𝑚
f(n) = l, l
+ f(x) =
𝑥→𝑎
𝑥 → 𝑎−
R. Otherwise, the function f(x) is said to be a discontinuous at x=a.
Example:
A function f(x) is defined as follows
2𝑥 + 3 𝑓𝑜𝑟 ≤ 1
𝑓(𝑛) = {
6𝑥 − 1 𝑓𝑜𝑟 𝑥 > 1
Is the function is continuous at x=1
62 | Cambridge institute/ Mathematics
Solution: Right hand limit at x=1
𝑙𝑖𝑚
𝑓(𝑥)
𝑥 → 1+
𝑙𝑖𝑚
(6𝑥 − 1)
= 𝑥 → 1+
=
= 6.1 – 1
=6–1
=5
Left hand limit at x=1
𝑙𝑖𝑚
𝑓(𝑥)
𝑥 → 1−
𝑙𝑖𝑚
(2𝑥 + 3)
= 𝑥 → 1−
= 2.1 + 3 = 5
Functional value at x=1
f(1) = 2.1 + 3 = 5
The given function is continuous at x =1
Exercise:
Discuss the continuity of functions at the points specified.
2
(i) f(x) = 2−𝑥
𝑥−4
𝑓𝑜𝑟 𝑥 ≤ 2
}
𝑓𝑜𝑟 𝑥 > 2
2
𝑎𝑡 𝑥 = 2
+1
(ii) f(x) = 2𝑥
4𝑥+1
𝑓𝑜𝑟 𝑥 ≤ 2
𝑓𝑜𝑟 𝑥 > 2
} 𝑎𝑡 𝑥 = 2
2𝑥
(iii)f(x) = 3𝑥+1
𝑓𝑜𝑟 𝑥 ≤ 3
𝑓𝑜𝑟 𝑥 > 3
} 𝑎𝑡 𝑥 = 3
2𝑥+1
(iv)f(x) = 2
3𝑥
2
+2
(v) f(x) = 𝑥3𝑥+12
Answers
(i) continuous
𝑓𝑜𝑟 𝑥 < 1
𝑓𝑜𝑟 𝑥 = 1
𝑓𝑜𝑟 𝑥 > 1
} 𝑎𝑡 𝑥 = 1
𝑓𝑜𝑟 𝑥 ≤ 5
}
𝑓𝑜𝑟 𝑥 > 5
𝑎𝑡 𝑥 = 5
(ii) continuous (iii) continuous (iv) discontinuous (v) continuous
Cambridge institute/ Mathematics | 63
11
Chapter
The Derivatives
Geometrical meaning of the derivative.
Let y = f(x) be a function. Let p(x, y) be any point on the curve. Let Q(x+x, y+y) be a
neighbouring point of P. Then draw PN OX, QM OX and PT QM
𝑟𝑖𝑠𝑒
𝑟𝑢𝑛
Now the slope of 𝑃𝑄 =
𝑄𝑇
∆𝑦
= 𝑃𝑇 = ∆𝑥 = 𝑡𝑎𝑛
If Q P, the secant PQ becomes a tangent at P, at that time
x 0, y 0. Let be the angle made by tangent with x-axis.
Then, as x 0
Q
Y
tan tan as x 0
The slope of tangent at P
= tan =
𝑙𝑖𝑚
tan
𝑥→𝜃
T
P
𝑙𝑖𝑚 ∆𝑦
=
O
X
N
M
∆𝑥 → 0 ∆𝑥
derivative is the slope of a tangent to the curve y = f(x) at a point P.
Definition: If y = f(x) is a function define on (a, b), then the derivative of f(x) with respect
to x is a number and defined as
𝑑𝑦
Therefore, 𝑑𝑥=
𝑙𝑖𝑚
∆𝑥 → 0
𝑙𝑖𝑚
∆𝑥 → 0
∆𝑦
∆𝑥
, it is denoted by
𝑑𝑦
𝑑𝑥
or f1 (x)
∆𝑦
.
∆𝑥
Working rules: to find the derivative using definition (i.e. from first principle)
1)Consider, the function y = f (x)
2)Let x and y are small change in x & y respectively, so that y+ y = f(x+x)
3)y = f(x+x) – y = f(x+x) – f(x)
∆𝑦
4)Find the ratio ∆𝑥,
∆𝑦
i.e. ∆𝑥 =
𝑓(𝑥 + ∆𝑥)−𝑓(𝑥)
∆𝑥
64 | Cambridge institute/ Mathematics
5)Taking the limits on both sides as x 0
𝑙𝑖𝑚 ∆𝑦
𝑙𝑖𝑚 𝑓(𝑥+∆𝑥)− 𝑓(𝑥)
=
.
∆𝑥
∆𝑥 → 0 ∆𝑥 ∆𝑥 → 0
∆𝑦
𝑙𝑖𝑚 𝑓(𝑥+∆𝑥)− 𝑓(𝑥)
=
.
∆𝑥 ∆𝑥 → 0
∆𝑥
Examples:
(1) Let y = f(x) = x
Let x and y are small increments in x and y respectively.
y + y = x + x
or, y = x + x –y
or, y = x + x – x
or, y = x
∆𝑦
or ∆𝑥 = 1
𝑑𝑦
𝑑𝑥
=
𝑙𝑖𝑚
∆𝑥 → 0
∆𝑦
∆𝑥
= 1.
(ii) Let y = f(x) = x2
y = y = (x+x)
or, y = (x+x)2 – y
or, y – (x+x)2 – x2
= x2 + 2x. x + (x)2 – x2
or,
𝑑𝑦
𝑑𝑥
=
𝑙𝑖𝑚
∆𝑥 → 0
∆𝑦
∆𝑥
=
𝑙𝑖𝑚 (2𝑥
+ ∆𝑥) = 2𝑥
∆𝑥 → 0
(iii)Let y = f(x) = x3
y = y = (x + x)3
or, y = (x+x)3 – y
or, y = (x + x)3 – x3
= x3 + 3x2. x + 3x. (x)2 + (x)3 – x3
𝑑𝑦
or, , 𝑑𝑥 =
𝑙𝑖𝑚
[3𝑥 2 + 3𝑥. ∆𝑥 + (∆𝑥)2 ]
∆𝑥 → 0
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= 3x2
From above we see that
𝑑(𝑥)
= 𝑥 1−1 = 1
𝑑𝑥
𝑑(𝑥 2 )
= 2. 𝑥 2−1 = 2𝑥
𝑑𝑥
𝑑(𝑥 3 )
= 3𝑥 3−1 = 3𝑥 2
𝑑𝑥
From these, results that we conclude that,
𝑑(𝑥 𝑛 )
𝑑𝑥
= 𝑛𝑥 𝑛−1
Derivative of constant function
Let y = f(x) = c
y + y = c
or, y = c-y
or, y – c-c
or,
𝑑𝑦
𝑑𝑥
𝑑𝑦
=0
𝑑𝑥 =
𝑙𝑖𝑚
∆𝑥 → 𝜃
∆𝑦
∆𝑥
= 0.
Examples: find from first principle the derivative of the following function.
(i) √𝑥
(ii)
1
√𝑥+2
(iii) 2x2 + 3x + 6.
i) Let y = √𝑥
ii) √2 − 3𝑥
Let x be small change in x, and y be the corresponding change in y then,
y + y = √𝑥 + ∆𝑥𝑦 = √𝑥 + ∆𝑥 − √𝑥 [∴ 𝑦 = √𝑥]
Dividing on both sides by x, we get
66 | Cambridge institute/ Mathematics
∆𝑦 √𝑥 + ∆𝑥 − √𝑥
=
∆𝑥
∆𝑥
Taking limit as x 0 on both sides, we get
𝑙𝑖𝑚 ∆𝑦
𝑙𝑖𝑚 √𝑥 + 𝑥 − √𝑥 0
=
( 𝑓𝑟𝑜𝑚)
∆𝑥 → 0 ∆𝑥 ∆𝑥 → 0
∆𝑥
0
1
𝑑𝑦
(𝑥 + ∆𝑥)2 − 𝑥 1/2
𝑙𝑖𝑚
=
𝑑𝑥 𝑥 + ∆𝑥 → 0 (𝑥 + ∆𝑥) − 𝑥
1
1
= 2 𝑥 2 − 1 (∴
∴
𝑙𝑖𝑚
𝑥→𝑎
𝑥 𝑛 −𝑎 𝑛
)
𝑥−𝑎
= 𝑛𝑎𝑛−1
𝑑𝑦
1
=
𝑑𝑥 2√𝑥
ii) Let 𝑦 = √2 − 3𝑥
Let x and y be the small in cerements in x and y respectively. Then
y +y = √2 − 3 (𝑥 + ∆𝑥)
or ∆𝑦 = √2 − 3𝑥 − ∆𝑥 − √2 − 3𝑥
∆𝑦
or ∆𝑥 =
=
(2−3𝑥−∆𝑥−2+3𝑥)
∆𝑥(√2−3𝑥−∆𝑥+√2−3𝑥)
−3
√2−3𝑥−∆𝑥+√2−3𝑥
𝑑𝑦
−3
𝑙𝑖𝑚
=
𝑑𝑥 ∆𝑥 → 0 √2 − 3𝑥 − ∆𝑥 + √2 − 3𝑥
=2
−3
√2−3𝑥
Derivatives of the trigonometric function
(i) Let (y = sinx)
Let x and y be the small increments in x and y respectively.
Then,
y + y = sin (x+x)
y = sin(x+x) – sinx
= 2 sin
𝑥+∆1−𝑥 𝑐𝑜𝑠 𝑥+ ∆𝑥+𝑥
2
2
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∆𝑥
2𝑥 + ∆𝑥
∆𝑦 2𝑠𝑖𝑛 2 . 𝑐𝑜𝑠
2
=
∆𝑥
∆2
∆𝑥
2𝑥 + ∆𝑥
𝑙𝑖𝑚 ∆𝑦
𝑙𝑖𝑚 2 𝑠𝑖𝑛 2 𝑐𝑜𝑠
2
=
∆𝑥
∆𝑥 → 0 ∆𝑥 ∆𝑥 → 0
2+ 2
𝑑𝑦
= 1 𝑐𝑜𝑠 𝑥 = 𝑐𝑜𝑠 𝑥
𝑑𝑥
(ii) Let y = tan x
y+ ∆𝑦 = 𝑡𝑎𝑛(𝑥 + ∆𝑥)
∆𝑦 = 𝑡𝑎𝑛(𝑥 + ∆𝑥) − 𝑦
= tan (x + ∆𝑥) − 𝑡𝑎𝑛 𝑥
=
𝑠𝑖𝑛( 𝑥+ ∆𝑥).𝑐𝑜𝑠 𝑥−𝑠𝑖𝑛 𝑥.𝑐𝑜𝑠( 𝑥+∆𝑥)
𝑐𝑜𝑠(𝑥+∆𝑥).𝑐𝑜𝑠 𝑥
𝑑𝑦
𝑠𝑖𝑛( 𝑥 + ∆𝑥 − 𝑥)
=
𝑑𝑥 ∆𝑥. 𝑐𝑜𝑠 (𝑥 + ∆𝑥). 𝑐𝑜𝑠 𝑥
𝑙𝑖𝑚 ∆𝑦
𝑙𝑖𝑚
=
∆𝑥 → 0 ∆𝑥 ∆𝑥 → 0
𝑑𝑦
1
=
= 𝑠𝑒𝑒 2 𝑥
𝑑𝑥 𝑐𝑜𝑠 2 𝑥
𝑠𝑖𝑛 ∆𝑥
∆𝑥.𝑐𝑜𝑠 (𝑥+∆𝑥) 𝑐𝑜𝑠 𝑥
Similarly we can show the following relation.
(1)
𝑑(𝑐𝑜𝑠𝑥)
𝑑𝑥
(2)
𝑑(𝑠𝑒𝑒𝑥)
= 𝑠𝑒𝑒𝑥. 𝑡𝑎𝑛𝑥
𝑑𝑥
(3)
𝑑(𝑐𝑜𝑡𝑥)
𝑑𝑥
= −𝑐𝑜𝑠 2 𝑥
(4)
𝑑(𝑐𝑜𝑠𝑥)
𝑑𝑥
= −𝑐𝑜𝑠𝑥 . 𝑐𝑜𝑡𝑥
= − 𝑠𝑖𝑛 𝑥
Some standard rules of differentiation (Derivatives)
(1) The sum rule
Let u and v are function of x then
𝑑(𝑢±𝑣) 𝑑𝑢
=𝑑𝑥
𝑑𝑥
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𝑑𝑢
± 𝑑𝑥
Examples:
Find
𝑑𝑦
𝑑𝑥
𝑜𝑓
a) 2x4 + 3x2 + 5
b) sin x+ 10 tan x
Solution
a) Let y = 2x4 + 3x2 + 5
𝑑𝑦 𝑑(2𝑥 4 + 3𝑥 2 + 5)
=
𝑑𝑥
𝑑𝑥
=
3𝑑(𝑥 2 )
2(𝑑(𝑥4)
+
𝑑𝑥
𝑑𝑥
+
𝑑(5)
𝑑𝑥
= 8x3 + 6x + 0
= 8x3 +6x
b) Let, y = sin x + 190 tan x
𝑑𝑦
𝑑(𝑠𝑖𝑛 𝑥) 10𝑑(𝑡𝑎𝑛 𝑥)
=
+
𝑑𝑥
𝑑𝑥
𝑑𝑥
= cos x+ 10 see2 x
(2) The product rule
𝑑(𝑢𝑣) 𝑢𝑑𝑣 𝑣𝑑𝑢
=
+
𝑑𝑥
𝑑𝑥
𝑑𝑥
Example: y = (2x2 + 3x)(4x-1)
𝑑𝑦
𝑑𝑥
= (2𝑥 2 + 3𝑥)
𝑑(4𝑥−1)
+
𝑑𝑥
(4𝑥 − 1)
(2𝑥 2 +3𝑥)
𝑑𝑥
= (2x2+3x) (4-0) + (4x-1)(4x+3)
=24x2 + 20x -3
3) The power rule
𝑑(𝑢𝑛 ) 𝑛𝑢𝑛−1 𝑑𝑦
=
𝑑𝑥
𝑑𝑥
𝑑𝑦
Example: Find of y = (3x3 + 3x) ½
Since
𝑑𝑥
𝑑(𝑢𝑛 )
𝑑𝑥
= 𝑛𝑢𝑛−1
𝑑𝑦
1
𝑑𝑦
𝑑𝑥
1
Then 𝑑𝑥 = 2 (4𝑥 3 +3x)2−1
𝑑(4𝑥^3+3𝑥)
𝑑𝑥
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1
=2 (4x3 + 3x)1/2 (12x2 + 3)
=
12𝑥 2 +3
2√4𝑥 3 +3𝑥
4) The quotient rule
𝑑𝑦
𝑑𝑦
𝑢
𝑑(𝑣 ) 𝑣
−𝑢
= 𝑑𝑥 2 𝑑𝑥
𝑑𝑥
𝑣
5) The chain rule
If y = f(u) and u = g(x) then
𝑑𝑦
𝑑𝑥
=
𝑑𝑦 𝑑𝑢
x
𝑑𝑢 𝑑𝑥
Exercise:
1)
Find from the first principle the derivative of the following:
(i)
2x2
(xiv) (2x+3) ½
(i)6x
(ii) x2 – 2
(xv) (1+x2)1/2
(ii)2x
(iii) x2 + 5x – 3
(xvi) 𝑥 1/2
(iv) 3x2 – 2x + 1
(v)
1
𝑥
(vi)
3
2𝑥 2
(vii)
1
𝑥−1
1
(iii)2x+5
1
(1−𝑥)1/2
(xvii)
1
(xviii)(1+x )
(xix) cos( ax-b)
(vi)− 𝑥 2
(xx) tan (3x – 4)
(vii) −
3
(xxi) sin
3𝑥
2
1
2𝑥+3
(xxii)tan 3
(x)
𝑎𝑥+𝑏
𝑥
(xxiii) cos2 x
(xii) x+√𝑥
(xiii) (1+x)1/2
1
(2−1)2
1
(viii)(5−𝑥)2
5𝑥
(ix)
(xi) x1/2
1
(v)− 𝑥 2
2 1/2
(viii) 5−𝑥
(iv)6x-2
2
(ix)− (2𝑥+3)2
−𝑏
(xxiv)sin 3x
(x) 𝑥 2
(xxv)√𝑠𝑖𝑛2𝑥
(xi)𝑥 𝑥 1/2
2
1
Answers
1)
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1
2
(xii)1 + 𝑥 −1/2
1
(xiii)2 (1 + 𝑥)−1/2
(xviii)4 cos 4x
(xxiii)-sin2x
(xiv)(2x+3)-1/2
(xix)- a sin (ax-b)
(xxiv)3sin 6x
(xx)3 sec2 (3x-y)
(xxv)
𝑥
(xv)
√1+𝑥 2
3
(xxi)2 𝑐𝑜𝑠
1
(xvi)− 3/2
2𝑥
3𝑥
2
5
(xxii)3 𝑠𝑒𝑐 2
1
(xvii)
2(1−𝑥)3/2
𝑐𝑜𝑠 𝑥
√𝑠𝑖𝑛2𝑥
3𝑥
3
2) Find the derivatives of the following
𝑥 2 −𝑎 2
𝑥 2 +𝑎 2
(i)x5
(ii)5x
√𝑥 2 +𝑎2 −√𝑥 2 −𝑎 2
3
2
(iii)3x – 5x + 7
(v) 2x3/4 – 3x1/2 – 5x1/4
3𝑥+3𝑥 3/4 +𝑥 1/2 +1
𝑥 1/4
(vi)
2
(vii)3x (2x-1)
(ix)(3x4+5)(4x5-3)
𝑥2
(xii)1−𝑥 2
(xvi)(2x+3)
(xxvii)tan(5x2 + 6)
2
(xxviii)cot √𝑥
(xvii)(3-2x)3
2
(xviii)(3x +2x-1)
4
(xx)√8 − 5𝑥
(xxi)(2x2-3x+1)3/4
2
(x)(3x +5x-1)(x +3)
𝑥
(xi)1+𝑥
(xxvi)cos (ax+b)
𝑥 2 −2𝑥
𝑥+1
(xiv)(2x2+3x-3)-6
(viii)(2x2+1)(3x2-2)
2
(xxv)sin (4x-5)
(xiv)𝑥 2
(xv)
3𝑥 3 +2𝑥−1
(iv)
2𝑥 2
1
(xxiv)
(xiii)
(xxix)see
1
𝑥
(xxx)sin5 (cx2-c)
(xxxi)tan (cos5x)
(xxxii)cos(sin(3x2+2)
(xxxiii)tan5(sin(px-q))
1
(xxii)
√𝑎𝑥 2 +𝑏𝑥+𝑐
(xxiii)
1
√𝑥+𝑎+√𝑥−𝑎
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Answers
(i)4x2
(xiv)-6(4x+3)(2x2+3x-3)-7
(ii)5
(xx)2√8−5𝑥
−5
(iii)6x - 5
3
(xxi)4 (4𝑥 − 3)(2𝑥 2 − 3𝑥 + 1)−1/4
(iv)6x - 2
1
(xxii)− 2 (2𝑎𝑥 + 𝑏)(𝑎𝑥 2 + 𝑏𝑥 +
6𝑥 1/2 −6𝑥 1/4 −5
(v)
4𝑥 3/4
3
(vi)2𝑥 1/4
+
3
2𝑥 1/2
+
1
4𝑥 3/4
−
1
4𝑥 5/4
(vii)6x (3x – 1)
𝑐)−3/2
1
1
√𝑥+𝑎
(xxiii)4𝑎 (
𝑥
−
1
(viii)2x (12x – 1)
(xxiv)2𝑎2 (
(ix)4x3(27x5+25x-9)
(xxv)4cos(4x-5)
(x)12x3 + 15x2 + 16x + 15
(xxvi)-9sin(ax+b)
2
1
(xi)(1+𝑥)2
2𝑥
(xii)
(1−𝑥 2 )2
4𝑎 2𝑥
(xiii)(𝑥 2 +𝑎2 )2
(xiv)-6/x3
𝑥 2 +2𝑥−2
(𝑥+1)2
√𝑥 2 +𝑎 2
1
)
√𝑥+𝑎
+
1
√𝑥 2 −𝑎2
)
(xxvii)10xsec2(5x2+6)
−1
𝑐𝑜𝑠𝑒𝑐 2 √𝑥
√𝑥
(xxviii)2
1
1
1
(xxix)− 𝑥 2 𝑠𝑒𝑐 𝑥 . 𝑡𝑎𝑛 𝑥
(xxx)10cxsin4(cx2-c).cos(cx2-c)
(xxxi)-5sec2(cos5x).sin5x
(xv)
(xxxii)-6xsin{sin(3x2+2)}cos(3x2 +2)
(xvi)4(2x+3)
(xxxiii)5Ptan4{sin(px-q).sec2(sin(px-
(xvii)-6(3-2x)2
q)}.cos(px-q).
(xviii)8(3x+1)(3x2+2x-1)3
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Chapter
12
Antiderivatives
Integration is the reverse process of differentiation. The process of the finding f(x),
when its derivative f1(x) is given is known as integration.
Constant of integration
We know that
(i)
𝑑(𝑥)
𝑑𝑥
= 1 ∫ 1 𝑑𝑥 = 𝑥
𝑑(𝑥+2)
𝑑𝑥
(ii)
(iii)
= 1 ∫ 1 𝑑𝑥 = 𝑥 + 2
𝑑(𝑥−10)
𝑑𝑥
𝑑(𝑥+𝑐)
𝑑𝑥
(iv)
= 1 ∫ 1 𝑑𝑥 = 𝑥 − 10
= 1 ∫ 1 𝑑𝑥 = 𝑥 + 𝑐
Integration of 1 may be x, x+2, x-10 or x+c, where c is any arbitrary constant.
𝑑(𝛷(𝑥)+𝑐)
If
= 𝑓(𝑛)
𝑑𝑥
Then, ∫ 𝑓(𝑥)𝑑𝑥 = 𝜙(𝑥) + 𝑐, 𝑐 𝑏𝑒𝑖𝑛𝑔 𝑎𝑛𝑦 𝑐𝑜𝑛𝑡𝑎𝑛𝑡
Thus , 𝜙(𝑥) + 𝑐 𝑖𝑠 𝑎𝑛𝑡𝑖𝑑𝑖𝑎𝑡𝑖𝑣𝑒 𝑜𝑓 𝑓.
Where, ∫ 𝑖𝑠 𝑠𝑖𝑔𝑛 𝑜𝑓 𝑖𝑛𝑡𝑒𝑔𝑟𝑎𝑡𝑖𝑜𝑛
The symbol, ∫ 𝑑𝑥 denotes, the antiderivative is performing w.r. to x.
Properties of integration.
(1)∫ 𝐾 𝑓(𝑥) 𝑑𝑘 = 𝑘 ∫ 𝑓(𝑥)𝑑𝑥, 𝐾 𝑖𝑠 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡𝑠
(2)∫[𝑓(𝑥) ± 𝑔(𝑥)]𝑑𝑥 = ∫ 𝑓(𝑥)𝑑𝑥 ± ∫ 𝑔(𝑥)𝑑𝑥.
Fundamental formulae
1) ∫ 𝑥 𝑛 𝑑𝑥 =
𝑥 𝑛+1
+c.
𝑛+1
2) ∫(𝑎𝑥 + 𝑏)𝑛 𝑑𝑥 =
n -1
(𝑎𝑥+𝑏)𝑛+1
𝑎(𝑛+1)
+ 𝑐 (𝑛 − 1)
1
3)∫ 𝑥 𝑑𝑥 = 𝑙𝑜𝑔𝑥 + 𝑐
1
4)∫ (𝑎𝑥+𝑏) 𝑑𝑥 =
𝑙𝑜𝑔(𝑎𝑥+𝑏)
𝑎
+𝑐
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5)∫ 𝑠𝑖𝑛 𝑥 𝑑𝑥 = − 𝑐𝑜𝑠 𝑥 + 𝑐
6)∫ 𝑐𝑜𝑠 𝑥 𝑑𝑥 = 𝑠𝑖𝑛 𝑥 + 𝑐
Solved examples
(1)∫ 𝑥 5 𝑑𝑥 =
𝑥 5+1
5+1
+𝑐=
𝑥6
6
+𝑐
1
(2)∫ 𝑥 4 𝑑𝑥 = ∫ 𝑥 −4 𝑑𝑥
=
𝑥 −1+1
−1+1
+𝑐 =
𝑥 −3
−3
+𝑐
2
(3)∫
𝑑𝑥 = 2 𝑙𝑜𝑔(𝑥 − 1) + 𝑐
𝑥−1
(4)∫(5𝑥 + 4)10 𝑑𝑥
=
(5𝑥+4)10+1
(5(10+1)
+𝑐 =
(5𝑥+4)11
55
+𝑐
(5)∫(𝑠𝑖𝑛 𝑥 + 𝑐𝑜𝑠 𝑥) 𝑑𝑥
= ∫ 𝑠𝑖𝑛 𝑥 𝑑𝑥 + ∫ 𝑐𝑜𝑠 𝑥 𝑑𝑥
= − 𝑐𝑜𝑠 𝑥 + 𝑠𝑖𝑛 𝑥 + 𝑐
(6)∫ 𝑠𝑖𝑛(4𝑥 + 3)𝑑𝑥 =
− 𝑐𝑜𝑠( 4𝑥+3)
+
4
𝑐.
Exercise:
Find the indefinite integrates.
𝑥+3
1)∫ 5𝑥 3 𝑑𝑥
10)∫ 𝑥−1 𝑑𝑥
2)∫ 7𝑥 5/2 𝑑𝑥
11)∫
3)∫(𝑥 2 + 2) 𝑑𝑥
4) ∫(2𝑥 + 1)(3𝑥 +
2)𝑑𝑥
13)∫
1
6)∫ (√𝑥 −
7)∫
1
) 𝑑𝑥
√𝑥
(3𝑥 2 −5𝑥+2)
𝑥
8) ∫
𝑎𝑥 2 +𝑏𝑥+𝑐
𝑥2
9)∫ 𝑥 +
𝑑𝑥
𝑑𝑥
1
𝑑𝑥
(𝑥+3)2
20)∫ 𝑠𝑒𝑐 2 (2𝑥 + 3)𝑑𝑥
3𝑥−1
𝑑𝑥
𝑥−2
𝑥 2 +3𝑥+3
12)∫ 𝑥+1
5)∫(𝑥 2 − 𝑥 2 ) 𝑑𝑥
19)∫ 𝑐𝑜𝑠(𝑎2 𝑥 + 𝑏) 𝑑𝑥
𝑥 2 +5
𝑥+2
21)∫ 𝑠𝑖𝑛 2 𝑎𝑥 𝑑𝑥
𝑑𝑥
𝑑𝑥
22)∫ 𝑐𝑜𝑠 2 bx dx
1
23)∫ 2
𝑑𝑥
𝑠𝑒𝑐 𝑥 𝑡𝑎𝑛2 𝑥
14)∫ 𝑥√𝑥1 𝑑𝑥
24)∫ √1 + 𝑐𝑜𝑠 𝑏𝑥 𝑑𝑥
15)∫ 2𝑥 √2𝑥 + 3𝑑𝑥
25)∫ √1 − 𝑐𝑜𝑠 𝑝𝑥 𝑑𝑥
16)∫(𝑥 +
26)∫ 𝑠𝑖𝑛 6𝑥 𝑐𝑜𝑠 2 𝑥 𝑑𝑥
2)√3𝑥 + 2 𝑑𝑥
27)
17)∫ 𝑠𝑖𝑛 5 𝑥 𝑑𝑥
∫ 𝑠𝑖𝑛 6 𝑥 𝑐𝑜𝑠 5 𝑥 𝑑𝑥
18)∫ 𝑠𝑖𝑛(𝑎𝑥 + 𝑏)𝑑𝑥
74 | Cambridge institute/ Mathematics
Answers
1)−
5𝑥 4
4
1
15)5 (2x+3)5/2 – (2x+3)3/2 + c
+𝑐
2 1
2)2x7/2+c
1
3)3 𝑥 2 + 2𝑥 + 𝑐
+c
7
4)2x3+ 2 x2 + 2x+c
1
18)- 𝑎 cos (ax+b) + c
2
6)3 x3/2-2x1/2 + c
3
7)2 x2 – 5x + 2 logx + c
𝑐
8)ax + b log x - 𝑥 + c
1
11)3x + 5 log (x-2) + c
2
sin(𝑎2 𝑥+𝑏)
+c
𝑎2
1
20)2 tan (2x+3) + c
1
21)2 [𝑥 −
22)2 [𝑥 −
10)x + 6 log (x-3) + c
1
13)2 x2
19)
1
9)2 x2 - 𝑥+3 + c
1
12)2 x2
1
17)-5 cos5x + c
1
1
5)3 x3 + 𝑥 + c
1
4
16)9 [5 (3x+2)5/2 + 3 (3x+2)3/2]
+ 2x + log (x+1) + c
– 2x + 9 log(x+2) + c
2
14)5 (x+1)5/2 - 3 (x+1)3/2 + c
𝑠𝑖𝑛2𝑎𝑥
2𝑎
𝑠𝑖𝑛2𝑏𝑥
2𝑏
]+𝑐
]+𝑐
3
1
23)− cot 𝑥 − 2 𝑥 − 4 𝑠𝑖𝑛2𝑥 + 𝑐
24)
2√2
𝑏
25)−
26)−
27)
1
𝑠𝑖𝑛 2 𝑏𝑥 + 𝑐
2√2
𝑃
𝑐𝑜𝑠8𝑥
16
𝑐𝑜𝑠2𝑥
4
1
𝑐𝑜𝑠 2 𝑝𝑥 + 𝑐
−
−
𝑐𝑜𝑠4𝑥
8
𝑐𝑜𝑠14𝑥
28
+𝑐
+𝑐
Cambridge institute/ Mathematics | 75
OBJECTIVE MATHEMATICS
1.
2.
3.
4.
5.
The median from the values: 31, 25, 20, 18, 35, 60, 27, 40, 20, 43 is
a) 29
b) 20
c) 24.5
d) 29.5
The L.C.M. and H.C.F of the two numbers are 840 and 14 respectively and
if one of the numbers is 42 then the other number is
a) 84
b) 280
c) 868
d) 42
20% of what number is equal to 2/3 of 90?
a) 30
b) 120
c) 600
d) 300
x-4
x-6
If 2 = 4 a , what is the value of a?
a) 2
b) 6
c) 6a
d) 2a
If 0 < < 1, which of the following lists the numbers in increasing order?
a) x , x, x2
b) x2, x, x
c) x2, x , x
d) x, x2, x
6. A checker is placed on a rectangular table 3 inches from one side of the
table and 4 inches from the adjacent side. How far, in inches, is the
checker from the nearest corner of the table?
a) 3
b) 5
c) 5
d) None
1
7. If the numbers 8 and 12 are increased by 25% and 33 /2% respectively,
then what will be the average increment?
a) 30%
b) 15%
c) 10%
d) None
8. A dealer ordinarily make a profit of 16%. If his cost goes down by 20% and
he decreases his price by 10%, what percent does he gain?
a) 28.2%
b) 30.50%
c) 15%
d) None
9. The principle value, which amounts to Rs 1200 at 8% p.a. S.I after 9
years will be:
a) 69729/43
b) 69711/43
c) 697
d) None
10. If (3x+1):( 5x+3) is the triplicate ratio of 3:4, then value of x will be:
a) 17
b) 17/57
c) 57
d) None
11. If x:y=2:3 and y:3=4:7 then x:y:3 will be:
a) –1 : 5 :7
b) 8 : 12 : 21
c) 3 : 5 : 1
d) None
12. Two quantities are in the ratio 7 : 4. If the greater quantity is 24.5, then
the smaller quantity will be:
76 | Cambridge institute/ Mathematics
13.
14.
15.
16.
17.
18.
a) 14
b) 15
c) 10
d) None
If a, b, c, d and e are in continued proportion, then a : e will be equal to:
a) a4 : b4
b) a3 : d
c) a : d
d) None
In a mixture of 35 liters, the ratio of milk to water is 4 : 1. Another 7
liters of water is added to the mixture. Then the ratio of milk to water in
the resulting mixture will be:
a) 2 : 1
b) 3 : 5
c) 10 : 13
d) None
The difference between C.I and S.I on sum of Rs 4800 for 2 years at 5%
per annum will be
a) Rs. 10
b) Rs. 30
c) Rs. 12
d) None
In what time will a sum of Rs 1562.50 produce Rs 195.10 at 4% per
annum compound interest?
a) Two years
b) Three years
c) Ten years
d) None
The compound interest per annum on Rs 50,000 for 2 years at 10% per
year, compound half yearly will be
a) 23205
b) 25000
c) 10000
d) None
At what rate percentage per annum compounded interest will Rs 2304
amount to Rs 2500 in 2 years
a) 25/6%
b) 30%
c) 17%
d) None
19. A function is defined by f(x)=
the value of
20.
21.
22.
23.
f(2)
f(3)
2
3x 2x 1
x 1
where x R and x 1. Then
1 will be:
a) -1/2
b) ½
c) 5/3
d) None
In a group of 50 students, 25 play hockey, 30 play football and 8 play
neither game. The number of students who play both games will be:
a) 10
b) 15
c) 13
d) None
The value of Tan /3. Sin /3 + Sin /4.Cos /2+Cos /2. Sin /3 will be
a) 3/2
b) -1/2
c) 3/2
d) None
o
o
The value of Cosec 35 -Sec 55 will be:
a) 0
b) -1
c) 10
d) None
The in centre of a triangle, the equation of whose sides are 3x + 4y = 0;
5x – 12y = 0 and y – 15 = 0 will be
Cambridge institute/ Mathematics | 77
24.
25.
26.
27.
28.
29.
30.
31.
32.
33.
34.
a) (-1, 2)
b) (8, 1)
c) (1, 8)
d) None
The orthocenter of the triangle formed by the lines whose equations are
x – y + 1 = 0, x -2y + 4 = 0 and 9x - 3y + 1 = 0 will be
a) (-1, 4)
b) (4, -1)
c) (0, 5)
d) None
If the lines 2x + 3ay – 1 = 0 and 3x + 4y + 1 = 0 are mutually
perpendicular, then the value of a is
a) -1/2
b) 3
c) 5
d) None
A rectangle with one side 4 cm inscribed in a circle of radius 2.5cm. The
area of the rectangle will be
a) 2cm2
b) 7cm2
c) 12cm2
d) None
Two right circular cones X and Y are made, X having three times the radius
of Y and Y having half the volume of X. Then the ratio of heights of X and
Y will be:
a) 1 : 9
b) 9 : 1
c) 2 : 9
d) None
o
A sector of a circle of radius 35 cm has an angle of 144 . It is folded so that
the two bounding radii are joined together to form a cone. Then the total
surface area of the cone will be
a) 2156 cm2
b) 2000 cm2
c) 1800 m2
d) None
The largest sphere is carved out of a cube of wood of side 21 cm. Then
the volume of the remaining wood will be
a) 4410 cm3
b) 4010 cm3
c) 4900 cm3
d) None
A circular hall has a hemispherical roof. The greatest height is equal to
the inner diameter. If the capacity of the hall is 48510 m3, then the area
of the floor will be
a) 1218 m2
b) 1386 m2
c) 1300 m2
d) None
The Range of the relation: R=(x, y) : x + 2y < 6 and x, y N, will be
a) {1,2}
b) {0, 2}
c) {1,5}
d) None
2
2
The Equation x + k1y + k2 xy = 0 represents a pair of perpendicular lines if
a) k1 = -1
b) k1 = 2k2
c) 2k1 = k2
d) None
2
2
If x - 10xy + 12y + 5x – 16y - 3 = 0, represents a pair of st. lines, then
the value of is
a) 4
b) 3
c) 2
d) None
Any four vertices of a regular pentagon line on a
78 | Cambridge institute/ Mathematics
35.
36.
37.
38.
39.
40.
41.
42.
43.
44.
45.
a) circle
b) square
c) parallelogramd) None
If two circles touch, the point of contact line on a:
a) St. line
b) quadrilateral
c) square
d) None
The domain of the Relation R where R = {(x,y) : y = x + 8/x ; x, y N and x
< 9} will be
a) {x, 2, 3}
b) {1, 2, 4, 8}
c) {1, 0, 4, 8} d) None
A sum of money is divided between Mary and David in the ratio 5:8. If
Mary's Share is Rs. 225, then the total amount of money will be:
a) 300
b) 400
c) 585
d) None
The angle between the vectors 2î + 3ĵ + k and 2î - 3ĵ - k is
a) /4
b) /3
c) /2
d) None
1 2
If A=
then the value of A2 - 2A – 5 I equals to
3 1
a) -1
b) 5
c) 0
d) None
2 3
The value of the determinant
equals to
2 3
a) 0
b) 2
c) 5
d) None
A dealer gains the selling price of 10 radio sets by selling 6 sets. His
percentage profit will be
a) 10%
b) 15%
c) 20%
d) None
A father is 2 times as old as his son. 16 years ago the age of the father was
three times the age of the son what is the present age of the father?
a) 64 years
b) 10 years
c) 80 years
d) None
A box contains 17 good oranges and 3 bad oranges. If 3 oranges were
drawn one after the other. Then, what is the probability that all the three
oranges are good?
a) ½
b) 20/27
c) 34/57
d) None
2
One of the factors of 2x + 5 x –3 = 0 is
a) x + 3
b) x + 5
c) x - 3
d) None
4
2
2
4
Factorized form of x +x y + y is:
a) (x2+xy+y2) ( x2- x y+ y2)
b) ( x2+xy+y2) ( x2+ xy+y2)
c) ( x2- x y + y2 )
d) (x+y) ( x –y) (x2+xy +y2)
Cambridge institute/ Mathematics | 79
46. Find the area of triangle in which base is 6 cm and height is 2 cm
a) 6
b) 12
c) 4
d) 2
47. The cost of carpenting a room at Rs. 50 per square meter is Rs. 1000. Find
the breath of the room, if it’s length is 5 m.
a) 2 m
b) 3 m
c) 2.5 m
d) 4 m
48. The market price of a watch is Rs, 640. What is the selling price if a
discount of 15 is allowed?
a) Rs. 736
b) Rs. 544
c) Rs. 625
d) Rs. 525
49. The angle of elevation of the top of a tree at a point 15 m from the tree
is 30. Find the foot of the tree
a) 8.25 m
b) 8.66 m
c) 9.2 m
d) 9.6 m
50. A garden is 25.50m wide and 35 m long. What is its perimeter?
a) 892.5 m
b) 121 m
c) 105m
d) 70 m
51. A quadrilateral with all sides equal is:
a) parallelogram
b) square
c) rectangle
d) rhombus
52. What is the probability of drawing a heart or an ace from a deck of 52
cards?
a) 26/ 52
b) 4/13
c) 1/52
d) 2/13
53. Find the value of k if a = 3i – 5j and b = 9i + kj are perpendicular.
a) 12/5
b) 23/5
c) 17/5
d) 32/5
54. The set (A-B) (B-A) (AB) is equal to the set
a) AB
b) A B
c) AB
d) none
2
55. The quadratic equation x -2x + 2 = 0 has
a) Rational roots. b) Irrational roots` c) Imaginary rootsd) none
56. The value of 2 . 3 is
a) 6
b) 6
c) 6
d) none
57. nth term of the sequence 2,4,6,8,10, ……….. is
a) 2n – 1
b) 2n +1
c) n2
d) 2n
58. Unit matrix is not a
a) Scalar matrix
b) Diagonal matrix
c) Square matrix
d) Null matrix
80 | Cambridge institute/ Mathematics
59. A 2x2 matrix A with Aij = ij is
1 2
2 4
4 2
2 1
a)
b)
c)
d)
2
1
2
4
2
1
4 2
60. If a matrix has 8 elements, which of the following is not the possible
order of the matrix?
a) 4 x 2
b) 2 x 4
c) 8 x 1
d) 4 x 4
61. O < a < b < 1. If x = a b and y = a + b , then
a) x > y
b) x < y
c) x = y
d) none
62. If A and B are square matrices of the same order, which of the following
statements may not be true?
a) AB is defined
b) BA is defined
c) Order of AB and BA are same
d) AB = BA
63. If one type of rice costing Rs. 20 per kg and another type of rice costing
Rs. 17 kg are mixed in the ratio 1:2, what is the cost of the mixture per
kg?
a) Rs. 17.50
b) Rs. 18
c) Rs. 18.50
d) Rs.19
64. If the point (x,2) is equidistant from (8,-2) and (2,-2), find the value of x.
a) 3
b) 4
c) 5
d) 6
65. If x + y = a, y + z=b and x + z = c, then the arithmetic mean of x, y and z is
abc
abc
abc
abc
a)
b)
c)
d)
6
2
3
4
66. The set ( A B ) is equal to
a) A B
b) A B
c) A B
d) A B
67. If the range and co-domain of a function are equal, then the function is
called
a) onto
b) one- to one
c) into
d) None
68. If 120% of 'a' is equal to 80% of 'b', which of the following is equal to a+
b?
a) 1.5a
b) 2a
c) 2.5a
d) 3a
69. If a-b=1, b-c=2 and c-a=d, what is the value of d
a) -3
b) -1
c) 1
d) 3
Cambridge institute/ Mathematics | 81
70. If 3x+5y= 14 and x-y=6, what is the average of x and y
a) 0
b) 3
c) 3.5
d) 2.5
71. The element(s) of power set of {1, 2, 3} is/are
a)
b) {1,2}
c) {1,2,3}
d) All
2
2
72. Sin A/2 Cos A/2 is equal to
a) Sin2A
b) ½ Sin2A
c) ¼ Sin2A
d) ½ Sin A
73. Janak had Rs. 1500. He used 85% of it to pay his electric bill and 5% of it
on a gift for his mother. How much did he have left?
a) Rs. 1350
b) Rs. 1275
c) Rs. 150
d) None
74. Which of the following is NOT equivalent to 3/5%
a) 24/40
b) 60%
c) 0.6
d)
3
7
7
5
75. If p painters can paint h houses in d days how many houses can 5
painters, working at the same rate, paint in 2 days?
5hp
dhp
2hp
10 h
a)
b)
c)
d)
2d
5d
10
dp
76. Nadia will be x years old y years hence. How old was she z years ago?
a) x+y-z
b) x-y-z
c) x+y+z
d) y-x-z
1
x
77. If x 9 , then the value of 2
is
x
x x 1
a) 1/10
b) 1/9
c) 1/8
d) 1/11
3
ab
78. a, b, c, d are in continued proportion ,
=
bc
a) a/b
b) a/c
c) a/d
x
x2
4 are
79. Roots of the equation
x2
x
a) Rational
b) Imaginary
c) Irrational
2
x xy
x
2
d) 1
d) Equal
x y
80. Value of 2
3
2 is:
x xy x x y
a) 1
b) x
82 | Cambridge institute/ Mathematics
c) x + y
d) x – y
1
81. The measure of satisfying Sinθ andCosθ
2
3
2
is
a) /6
b) 5/6
c) 7/6
d) 11/6
82. 2 Years ago, the population of a village was 16000. The rate of population
growth of that village is 5%, what is the population at present?
a) 17620
b) 17630
c) 1s7640
d) 17650
4 1 2 1 x 1
83. If
, then x and y are
=
7 3 1 3 11 y
a) 9 and 16
b) -9 and 16
c) -9 and -16
d) 9 and -16
84. If x+3 is a factor of x 3 (k 1)x2 kx 54, find k
a) 1
b) 2
c) 3
d) 4
2
2
85. Obtuse angle between the line pair x - 4xy + y = 0 is
a) 1350
b) 1200
c) 1500
d) 1750
86. Equation of the line passing through (-6, 4) and perpendicular to 3x - 4y
+ 9 = 0 is
a) 4x + 3y – 12 = 0
b) 3x + 4y + 12 = 0
c) 3x - 4y – 12 = 0
d) 4x + 3y + 12 = 0
87. When 2 cos2 = -3 cos, then the value of o0 θ 1800 is
a) 900
b) 1500
c) 900, 1500
d) 900,1200
1 1
2 3
and Q
88. P = 1 1
1 4
a) 0
b) 1
then PQ is
c) 2
d) 3
2
89. The set s {x : x 6 0 and x is real} is
a) Singleton set
b) a pair set
c) a null set
d) none
90. n(A) = 3 and n(B) = 6. Then the minimum number of elements in AB is:
a) 3
b) 6
c) 9
d) none
3
91. If f(x) = x +1 is defined in the closed interval - 1 x 2, what is the image
of -2?
a) – 7
b) 7
c) 7
d) none
Cambridge institute/ Mathematics | 83
92. If f(x) =
x 1
x 1
, x 1, then (ff) (x) =
x 1
1
b)
c) x
d) none
x 1
x
93. The quadratic equation having roots 2 and 3 is
a) x2 – 5x + 6
b) x2 + 5x + 6
c) x2 + 5x – 6 d) x2 – 5x – 6
94. Orders of matrices A and B are m x n and n x p. Which of the following
statements is not true?
a) AB is defined
b) BA is defined
c) order of AB is m x p
d) A+B is not defined)
95. Angle between the lines represented by x2 – 2xy Cot - y2 = 0 is
a) 30
b) 45°
c) 60°
d) 90°
96. Find the value of K in order that the point (K, 1), (5, 5) and (10, 7) may be
collinear.
a) -3
b) -4
c) -6
d) -5
97. If 80% of the adult population of a village is registered to vote, and 60% of
those registered actually voted in a particular election, what percent of
the adults in the village did not vote in that election?
a) 40%
b) 48%
c) 50%
d) 52%
1
5
3
98. If of a number is 7 more that of the number, what is of the
6
3
4
number?
a) 20
b) 24
c) 18
d) 15
99. A shopkeeper fixed the marked price of his television to make a profit of
40%. Allowing 20% discount on the marked price of the television was
sold, what percent profit will be make?
a) 10%
b) 12%
c) 15%
d) 11%
a)
100. Value of x in x-2 = x is
a) 1 or 4
b) 1 only
c) 4 only
d) none
101. There are 3 red, 2 blue and 5 white balls in a box. A ball is taken out
randomly. What is the probability of a ball being blue or white?
a) 7/10
b) 3/10
c) 2/7
d) 5/7
84 | Cambridge institute/ Mathematics
102.The sum of two numbers is 16 and sum of their squares is 130. Find the
numbers?
a) 9,7
b) 8,8
c) 10,6
d) 11,5
103.If √2 sin = 1, which of the followings is not the value of ?
a) 45º
b) 135º
c) 225º
d) 405º
104.Value of
2
x 3
2
x 2
2
is
x 2
a) 1
b) 2
c) 3
d) 4
105.The population of a village was 7200. 5% of the population was migrated
and 2% died due to different caused within a year. What would be the
population of the village after a year?
a) 6669
b) 6966
c) 6696
d) 9666
106.If x2 – y2 = 28 and x-y=8, what is the average of x and y?
a) 1.75
b) 3.5
c) 7
d) 8
107.O < a < b < 1. If x = a b and y = a + b , then
a) x > y
b) x < y
c) x = y
b) none
108.If one type of rice costing Rs.20 per kg and another type of rice costing
Rs.17 per kg are mixed in the ratio 1:2, what is the cost of the mixture per
kg?
a) Rs. 17.50
b) Rs. 18
c) R. 18.50
d) Rs. 19
109.The monthly salary of a civil servant is Rs. 7500. If 15% tax is levied on the
yearly income of more than Rs.60,000, how much tax should he pay?
a) Rs. 2500
b) Rs.3500
c) Rs. 4000
d) Rs.4500
110.A house of 1080 square meters in area was constructed in a land of 1800
square metres in area, what percent of land was covered by the house?
a) 40%
b) 50%
c) 60%
d) 70%
x+1
x
111.Value of x in 3 + 3 = 108 is
a) 3
b) 2
c) 4
d) -3
Cambridge institute/ Mathematics | 85
112.A natural number is chosen at a random from amongst the first 1000.
What is the probability that the number so chosen is divisible by 3.
a)
3
10
b)
33
100
c)
333
1000
d)
332
1000
113.The value of x in x 7 = 1 + x is
a) 8
b) 9
c) 10
d) none
114.A handkerchief is 20 cm long and 18cm broad. How much the breadth
must be decreased to cover a surface of 324 cm2?
a) 1.8 cm
b) 1.7 cm
c) 1.9 cm
d) 1.6 cm
115.The half plane y x + 1 contains the point
a) (3,3)
b) (1,3)
c) (0,0)
d) (2,2)
116.The difference between the compound interest and simple interest on
Rs.5120 for 3 years at 12.5% per annum is
a) Rs.150
b) Rs.200
c) Rs.250
d) Rs.300
117.2 years ago, the population of a village was 16000. The rate of population
growth of that village is 5%. Find the population at present.
a) 17640
b) 17460
c) 17064
d) 17046
118.If the height and radius of a cylindrical wood are equal and cured
surface area is 308 cm2, find the height.
a) 14 cm
b) 12 cm
c) 10 cm
d) 7 cm
2
119.A square garden has area 6400 m . If two paths of 2m widths are running
midway and intersecting each other inside the garden, find the area of
paths
a) 316 m2
b) 314 m2
c) 318 m2
d) 312 m2
120. The radius of a wheel is 35 cm. The distance it covers in 10 complete
revolution is?
a) 20 m
b) 22m c) 24 m
d) 4 m
121. In the triangle ABC, if A= 6B=3C, what will be the value of B?
a) 30
b) 20
c) 10
d) 15
122. If one angle of a parallelogram is 30, then it’s other angles are:
a) 30, 120 and 120
b) 30, 130 and 30
c) 30, 150 and 150
d) 150, 30 and 120
86 | Cambridge institute/ Mathematics
123. What is the value of cosec2/2. sec2 /2(sin3/6+4 cot 2-sec2/3)²
a) 2/3
b) ½
c) ¼
d) 1/3
124. Simplify 428 - 63
a) 8 7
b) 7 7
c) 6 7
d) 5 7
125. Find the compound interest on Rs. 50,000 invested for 2 years at rate of
4 per annum
a) Rs. 4050
b) Rs. 4080
c) Rs. 4025
d) Rs. 4045
126. In cos(90°-)= BC/CA, what is the ratio of cos
a) CA2-BC2/CA
b) BC2-AC2/BC
c) CA2-BC2 /bc d) BC2+CA2/CA
127. What is the relation between the central angle with the angle at the
circumference standing on the same arc?
a) equal
b) double
c) three times d) four times
128. Given that 176 dollars =£100 and £1= Rs. 119. Find in dollars for Rs.
8925
a) $ 122
b) $132
c) $134
d) $140
129. (xp)(xq)=
xr
xs
. Find s in terms of p, q & r.
a) r+p+q
b) rpq
c) r-q-p
d) none
130.The sum of two numbers is 40 and difference is 10. Find the ratio of two
numbers.
a) 5:4
b) 5:3
c) 3:2
d) 2:5
131. The two ends of a diameter are (-4,1) (2,1). Find area of the circle
a) 3
b) 9
c) 6
d) 36
132.Find (37+20 3 )1/2- 2 3
a) 2 3
b) 5
c) 5 3
d) 12
133.If 5x+13=31. Find 5x 31 .
a) 7
b) 173 / 5
b) 15
d) 13
134.The price of 50 books is Rs. 4000. If the price is increased by 25%. What
will be the price of 36 books?
a) 3500
b) 2500
c) 3600
d) 4000
135.12p+3q=1 and 7q-2p=9. What is the average of p & q?
a) 0.1
b) 1
c) 0.5
d) 2.5
Cambridge institute/ Mathematics | 87
136.If A & B are two sets. Then A(AB) is
a) B
b) A
c)
137.Which of the following is a singleton set?
a)
b) 0
c)
d) AB
d) 0
138.If f:RR and g: RR such that f(x)=x and g(x) = 1 x then
a) gf(x)fog(x)
b) gf(x)=fg(x)
c) gf(x)fg(x) d) all
2
139.Sum of n terms of a series is 2n+n then it's 10th term is
a) 31
b) 21
c) 131
d) 121
140.The amount according to the compound interest of 3 years is 79860 and
4 years is 87846. Find the rate of interest.
a) 10.5%
b) 10%
c) 9%
d) 11%
2
0
141.If x +4x+4=0 then x +6=?
a) x
b) 6
c) 0
d) 7
142.
2
2
(2.5) (1.5)
2.5 1.5
. Find value
a) 2
b) 1
c) 4
d) none
143.A Train travels at the rate of 58 miles/hr. Express it in m/s.
a) 25.78
b) 252.80
c) 25.28
d) 25
1 1
0 1
144.Find the inverse of matrix.
1
0
1 1
1 0
0 0
a)
b)
c)
d)
0 0
1 1
0 1
0 1
145. Find the perimeter of right angled triangle having two sides 12cm &
5cm
a) 12cm
b) 15cm
c) 17cm
d) 30cm
1 2
is
7
146.Find the additive inverse of matrix
3
1
2
1 0
1 2
0 0
a)
b)
c)
d)
0 1
0 0
3 7
3 7
147.Find the radius of hemisphere, whose total surface area is 27 cm2.
a) 4cm
b) 5cm
c) 3/2cm
d) 3cm
88 | Cambridge institute/ Mathematics
148.If x-y=5, and xy=6. Find the value of x3-y3
a) 219
b) 129
c) 215
1
1
1
d) 228
149.If ab+bc+ca=0, xa .xb .xc =?
a) 1
b) xa+b+c
c) 0
d) x
150.What percentage of 1 km 20m is 720m?
a)71%
b) 70.58%
c) 70%
d) 85%
151.x(x-3)=x, then x are
a) 0,-4
b) 0,4
c) -4,4
d) 0,3
2
2
152. ax+b =a +bx , then solve for x
a) a2-b2
b) a+b
c) a-b
d) 1
153.If the replacement set of the set of 5x-1 9 is 2,1,0,1,2,3,4. Find the
solution
a) 2,1,0
b) 2,1,0,1
c) 2
d) 2,1,0,1,2,
154. The locus of the point whose abscissa and ordinate are always equal is
a) x+y=0
b) x-y=0
c) x+y=1
d) x-y=1
155.The point of intersection of perpendicular bisectors of the sides of a
triangle is known as
a) centere
b) incentre
c) orthocentre d) circumcentre
156.The point (a,o) (o,b) and (1,1) are collinear if
a) a+b=ab
b) a-b=ab
c) b-a=ab
d) a+b+ab=0
157. The area of circle centred at (1,2) and passing through (4,6) is
a) 5
b) 25
c) 10
d) 15
158.Angle between the two lines x=0 and y=0 is
a) 45
b) 90
c) 180
d)
159.The area of a triangle whose sides are along the lines x=0,y=0 and
4x+5y= 20 is
a) 20 sq. unit
b) 10 sq. unit
c) 1/10 sq. unit d) 1/20 sq. unit
160.The set of male student in St. Mary's school is
a) singleton set
b) empty set
c) super set
d) sub set
161.The distance between the parallel lines 5x-12y+65=0 & 10x-24y-78=0 is
a) 2
b) 8
c) 16
d) 0
Cambridge institute/ Mathematics | 89
162.Find the value of A when cos3A=sin2A (A<90)
a) 90
b) 45
c) 18
d) 36
163. The greatest chord in a circle is
a) tangent
b) chord
c) secant
d) diameter
164. Two acute angle of right-angled triangle are
a) complementary b) supplementary c) equal
d) none
165.In a rhombus ABCD if AC = 5cm, BD = 6cm .Find area
a) 30cm2
b) 15 cm2
c)20 cm2
d) 60 cm2
166. What is the shape of the base of a cylinder?
a) square
b) circular
c) rectangular d) triangular
4
167.In the expansion of (p+q) , how many terms are there?
a) 4
b) 5
c) 6
d) 3
2
1 tan 15
168.The value of
2 is
1 tan 15
a) 1
b) 2
c) 3 / 2
d) 2/ 3
169.The points (0, - 1) (-2, 3) (6,7) (8,3) are
a) collinear
b) vertices of parallelogram
c) vertices of rectangle
d) vertices of square
170.The diameter when the area of circle is A is
a)
4A
b)
3A
c)
A
d)
2A
171.The circumference of the base of a cone is 44 cm and the sum of its
radius and slanting height is 32 cm. Find total surface area.
a) 32 cm2
b) 44 cm2
c) 102 cm2
d) 704 cm2
172.A 20 cm long stick cast the shadow of 20 3 cm in afternoon. What is the
altitude of sun
a) 60º
b) 30º
c) 45º
d) 0º
173.The center of circle is (5,2) and touches x-axis. Find the equation of
circle.
a) x2+y2-10x-4y=0
b) x2-2y2+5x+9y=0
c) x2+y2-10x-4y+25=0
d) x2+y2=4
90 | Cambridge institute/ Mathematics
174.Find the diameter of sphere whose volume is
a) 1 cm
b) 2 cm
4
cm3
3
4
cm
3
c) 7 cm
d)
c) 5 2
d) none
175.What is the square root of 9+4 5
a) 9+ 5
b) 2
176.When Tan = 1
a) sin - cos = tan
b) sin - cos =0
c) tan = sin
d) tan = cos
177. Which of the following is not a measure of dispersion?
a) variance
b) mode
c) standard deviation
d) mean deviation
178.State the transformation in which an object and it's image are similar
a) rotation
b) reflection
c) transformationd) enlargement
179. The sine of angle of inclination of line 3x- 3 y-2=0 to x-axis is
a) 1/ 2
b) 3 /2
c) ½
d) 2
180. The perimeter of triangle is 12 cm and ratio of sides are 3:4:5, find area
a) 12 cm2
b) 6 cm
c) 4 cm2
d) 6 cm2
181. What is the reminder of f(x)= x3+6x2-x-30 is divided by (X+1)?
a) 24
b) -24
c) 42
d) -42
182.If the interest of the loan is decreased by Rs.15 when the rate of interest
1
4
3
4
falls from 5 % to 4 %. What is the amount of money borrowed?
a) Rs.1000
b) Rs.1500
c) Rs.3000
d) none
183.A point (5,1) is reflected on a line y=x axis. The image thus obtained is
rotated about origin through +ve 90. The co-ordinate of final image is
a) (-5,1)
b) (5,1)
c) (-1,5)
d) (1,-5)
184. A post has
1
1
of it's length in mud, of it in water and 15m above the
4
3
water. What is the total length of the post?
a) 18m
b) 72m
c) 8 m
d) 36 m
Cambridge institute/ Mathematics | 91
185.The difference between CI and SI. of sum Rs.5,00,000 for 3 years at 6%
per annum will be
a) Rs.5500
b) Rs.5550
c) Rs.5508
d) Rs.5580
186.f (3m+1): (5m+3) is the triplicate ratio of 3:4 then the value of m will be
a) 57
b) 17
c) 17/57
d) 57/17
187.What is the difference between the arithmetic mean and geometric
mean between 3 and 27 is
a) 15
b) 6
c) 12
d) 0
188. A man bought an article for Rs.1 and sold it for Rs. 1.20. What is the
percentage gain?
a) 20
b) 12
c) 1.2
d) 10
92 | Cambridge institute/ Mathematics
Answer key:
1
2
3
4
5
6
7
8
9
10
11
12
13
14
a
b
d
a
b
b
a
b
a
b
b
a
a
a
15
16
17
18
19
20
21
22
23
24
25
26
27
28
c
b
d
a
b
c
a
a
d
a
a
c
c
a
29
30
31
32
33
34
35
36
37
38
39
40
41
42
a
b
a
a
c
a
a
b
c
d
c
a
c
a
43
44
45
46
47
48
49
50
51
52
53
54
55
56
c
a
a
a
d
b
b
b
b
b
c
a
c
b
57
58
59
60
61
62
63
64
65
66
67
68
69
70
d
d
a
d
b
d
b
c
d
b
a
c
a
d
71
72
73
74
75
76
77
78
79
80
81
82
83
84
d
c
c
d
a
b
a
c
c
a
c
c
d
c
85
86
87
88
89
90
91
92
93
94
95
96
97
98
b
d
c
a
c
b
d
c
a
b
d
d
d
a
99
100
101
102
103
104
105
106
107
108
109
110
111
112
b
c
a
a
c
a
c
a
b
b
d
c
a
c
113
114
115
116
117
118
119
120
121
122
123
124
125
126
b
a
b
c
a
d
a
b
b
c
d
d
b
a
127
128
129
130
131
132
133
134
135
136
137
138
139
140
b
b
c
b
b
b
a
c
c
b
d
b
b
b
141
142
143
144
145
146
147
148
149
150
151
152
153
154
d
b
a
c
d
a
d
c
a
b
b
b
d
b
155
156
157
158
159
160
161
162
163
164
165
166
167
168
c
a
b
b
b
b
b
c
d
a
b
b
b
d
169
170
171
172
173
174
175
176
177
178
179
180
181
182
c
a
d
b
c
b
c
b
b
d
c
d
b
c
183
184
185
186
187
188
a
d
c
c
b
a
Cambridge institute/ Mathematics | 93
OBJECTIVE MATHEMATICS_II
1.
2.
3.
4.
5.
The value of x in 2x-2 + 2x = 5 is
a) 0
b)1
c)2
d) none
The nth term of two AP's – 19, - 12, - 5, +2 ….. and 1+6+11+ …. Are equal.
The value of n is,
a)9
b)10
c)11
d)12
The numerator of a fraction is 4 less than its denominator. If the
numerator is decreased by 2 and the denominator is increased by 1, then
the denominator is eight times the numerator. Find the fraction.
a)7/3
b)3/7
c)1/5
d)5/9
If 2/5 of a pole is 3.60m, what will be the length of 5/9 of it?
a)5
b)6
c)7
d)8
If, are the roots of 4x2 + 5x – 21 = 0. Find
1
α
1
β
a)8/21
b)21/5
c)-5/21
d)5/21
6. If a = 3cm, b = 2.4 cm and c = 5.5 cm, you will construct
a) isosceles triangle
b) scalene triangle
c) right angled triangle
d) none
7. A line cannot intersect a circle more than
a) one point
b) two points
c) three points d) all
8. Each exterior angle of an equilateral triangle is
a)60
b)90
c)120
d)180
9.
The value of
3
3
(0.3) (0.7 )
0.09 0.21 0.49
is,
a)1
b)2
c)3
d)1.5
10. A number multiplied by two third of itself makes the product 10584, the
number is,
a)123
b)124
c)125
d)126
2
11. The area of square is 900 cm . The length of its diagonal is,
a)202
b)203
c)302
d)303
94 | Cambridge institute/ Mathematics
12. Roshan can see upto 14 km far. The area of land that he can see around
is,
a)612 sq km
b)614 sq km
c) 651 sq. km d)616 sq. km
13. In how many years will a sum of money double at 10% p.a. simple
interest
a)5
b)6
c)7
d)10
14. The value of Cos 15 is,
a)
31
2 3
b)
1 3
2 2
c)
3 1
2 2
d)none
15. If x and y are integers and xy = 5 then the value of (x+y) 2 is.
a)13
b)25
c)36
d)49
2
16. The roots of a quadratic equation x -x-30 = 0 are,
a)10,3
b) 5,6
c)-5, 6
d)-6,5
17. In a fourth proportion, the product of extremes is equal to
a) product of all the four
b) product of the means
c) product of the first and the fourth
d) product of the first and the third
18. After paying an income tax of 5%, a man has Rs.7600 left. What is his
income?
a) Rs. 800
b) Rs. 8000
c) Rs. 4000
d) Rs. 16000
19. A garrison of 960 men has food enough to last for 65 days. How many
men should be sent away so that the provision may last for 120 days.
a)440
b)
520
c)220d)630
20. If r is the radius, h the vertical height and l the slant height of a cone
then l2 is equal to ………
a)r2 – h2
b) h2 – r2
c) r + h2
d) 1/3(r2 + h2)
21. The roots of the quadratic equation x2 — 6x + 7 = 0 are __
a) 3, 2
b) 3+2, 3-2
c) 3,2
d) 3, 2
2
22. The product of roots of the quadratic equation x + 3 x – 6 = 0 is ……..
a) - 16
b)6
c)- 6
d)16
Cambridge institute/ Mathematics | 95
23. In a two digit number, the unit's digit is twice the ten's digit. If the digits
are reversed, the new number is 27 more than the original number. Find
the number.
a)63
b)18
c)3
d)72
24 Divide Rs. 81 among A, B and C so that, B may, get Rs. 7 more than A and
C gets Rs. 6 less than twice A's share
a) 20, 40, 20
b)30, 10, 40
c) 50, 10, 20
d)20, 27, 34
25. A metallic cylindrical pipe has an inside radius r and outside radius R and
the length I. Find the volume of the metal.
a) R2 – r2
b) (r2 – R2)
c) (R2 – r2) d) ½ Rl
26. The simplified form of (27)4/3 is …..
a)9
b) 999
c)88
d)81
1
27. If a3 =
, than a is
64
a)1/5
b) 1/6
c)1/3
d)¼
3
3
28. If SinA + CosecA = 2, then Sin A + Cosec A is …….
a)8
b)6
c)4
d)2
29. A boy is 3 years older than his sister. Two years ago the sum of their ages
was 19. How old is the boy now?
a)13 years
b) 12 years
c) 11 years
d) 10 years
30. If one angle of a triangle is equal to the sum of the other two, then the
triangle is
a) isosceles
b) equilateral
c) right angled d) none
31. If the equation of a straight line is 2x – 3y + 5 = 0, then the slope of the
line is
a)-3/2
b)-2/3
c)2/3
d)1/3
32. If CosA = 3/5 and A lies in the fourth quadrant, then tanA is
a) 4/3
b)-4/3
c)4/5
d)-4/2
2
2
33. If a-b = 10, a -b = 20, what is the value of b?
a) -6
b)-4
c)4
d)6
34. Tony drove 8 miles west, 6 miles north, 3 miles east and 6 more miles
north. How far was Tony from his starting point?
96 | Cambridge institute/ Mathematics
35.
36.
37.
38.
39.
40.
41.
42.
43.
a)13
b)17
c)19
d)21
At a speed of 48 miles per hour, how many minutes will be required to
drive 32 miles?
a) 40
b)45
c)50
d)2400
A store owner received a shipment of books. One Tuesday he sold half of
them, on Wednesday after two more were sold, he had 2/5 of the books
left. How many books were there in the shipment?
a)10
b)20
c)30
d)35
If a – b = 1; b – c = 2 and c – a = d, find the value of d.
a) -3
b)-1
c)1
d)3
4/7 of the 350 students of Alpha Beta Institute are girls, 7/8 of the girls
got admission in St. Xaviers College, how many girls did not get admission
in St. Xaviers?
a) 25
b)50
c)45
d)200
A cube has an edge of four inches long. If the edge is increased by 25%
then how much the volume will be increased approximately?
a) 25%
b)48%
c)73%
d)95%
An equilateral triangle of side 3" is cut into smaller equilateral triangle of
side one inch each. What is the maximum number of such triangles that
can be formed?
a) 3
b) 9
c)6
d)13
A pool is filled to ¾ of its capacity. 1/9 of the water get evaporated) If the
capacity of the pool is 24,000 gallons when it is full, how many gallons of
water is to be added to fill the pool?
a) 8,000
b)6,000
c)12,000
d)18,000
A bag contains 2 red marbles, 3 green marbles, and 4 orange marbles. If
a marble is picked at random, what is the probability of the marble not
having orange?
a) 9/5
b)5/9
c)1/9
d)2/9
The price of an imported car is 8,25,000, which includes a VAT of 10% of
the original cost. Find the price of the car before VAT.
a) 8,25,000
b) 7,50,000
c) 75,000
d)8,00,000
Cambridge institute/ Mathematics | 97
44. What is the compound interest on Rs. 8000 for 2 years at the rate of 6%
per annum.
a) 987
b)898
c)988.80
d)889.80
45. Find the value of x in 4x – 4x-1 = 192
a) 16
b)5
c)12
d)4
46. If the difference and product of two natural numbers are 7 and 78
respectively, find the two numbers.
a) -13, -16
b) -6, 13
c) 6, -13
d) 6, 13
47. In a triangle ABC, a=3, b=5, c=4; then CosC is
a) 10/3
b)9
c)10
d)3/5
48. In a triangle ABC, If a = 5, b = 12, c = 13; then CosA is
a) 13/2
b)12/13
c)5/12
d)13/5
49. A metallic sphere is melted into a solid right circular cylinder whose
height is twice the radius of its base. If the radius of the sphere and the
cylinder are 'r' and 'R' respectively, then R is ____
4
2
a) 3 r
b)3 r
b)r
d)2r
3
3
50. My salary was first increased by 10% and then decreased by 10%. What
is the total percentage change in my salary?
a) 2.2%
b)1.5%
c)1%
d)3%
51. When the rate of income tax is increased from 10% to 15%. I have to pay
Rs. 835 more. Find my income.
a) 16,700
b) 16,600
c) 10,000
d) 15,500
52. If A = {a,h,c,d}; B={e,f,g,h} find AUB
a) O
b) {a,h,c,d,e,f,g}
c) {e,f}
d){a,h}
53. If A = {a,b,c,}; B={1,2,3,4}, find n (AB).
a)2
b)3
c)0
d)7
54. The triangle formed by joining the points (a-a), (-a, a) and (a, 3 – a 3) is
a) scalene
b)right-angled
c) isosceles
d) none
55. Find the quadratic equation whose roots are –3+5i, -3 – 5i.
a) x2 – 9x + 34x + 1 = 0
b) x2 + 17x – 9 = 0
c) x2 + 6x + 34 = 0
d) 9x2 + 34x + 1 = 0
98 | Cambridge institute/ Mathematics
56. If 4Cos2x = 1, then the value of x is
a) 60, 120
b) 120, 240
c) 60, 240
d) 30, 60
57. In what ratio is the line joining the points (-3, 4) and (2, - 6) is divided by
the point (-1, 0) ?
a) ¾
b) ½
c)2/5
d)2/3
58. Two vertices of a triangle are at (5, 9) and (-4, 1). Find the third vertex if
the medians meet at (1, 1).
a) (7, 2)
b)(2,-7)
c)(1, 7)
d)(4, -2)
59. If the three vertices of a triangle are (2,0), (4,4) and (6,2), find the centroid
of the triangle.
a) (4, 1)
b)(1, 5)
c)(4,2)
d)(-2, 3)
60. If the two sides of a right-angled triangle are 5 and 12, then find the length
of the median that bisect the hypotenuse.
a) 13
b)6
c)2.5
d)6.5
61. If (2,6), (3,8) and (-1,y) lie on a straight line, find the value of y.
a) 10
b)-5
c)2
d)0
62. If the radius of a sphere is doubled, its volume becomes _____ the
original volume.
a) 16 times
b) 4 times
c)8 times
d) double
63. Find the angle between the lines 2x – 3y + 5 = 0 and 2x -3y — 7 = 0
a) 30
b)45
c)90
d)0
64. Find the angle between the lines 3x – y + 2 = 0 and x+ 3y + 4 = 0?
a) 90
b)45
c)30
d)60
65. Which of the following statement is true:
a) 42 + 52 = 52
b) 72 + 22 = 53
c) 52 + 32 = 16 d) 1 + 22 = 4
66. The value of Sin 75 is
a) (3 +1)/2 2
b) 1.5
c) -1
d)0
67. The two straight lines A1x + B1y +C1 = 0 and A2x + B2y + C2 = 0 will be
perpendicular if
a) A1A2 + B1B2 = 0 b)A2/B1 = A2/B2
c) A1/B1 = B2/A2
d)A1A2 – B1B2 = 0
68. The equation Ax + By + C = 0 always represent a
Cambridge institute/ Mathematics | 99
69.
70.
71.
72.
73.
74.
75.
76.
77.
78.
79.
80.
a) straight line
b)circle
c) parabola
d) square
The solution of the inequality 2x + 2 > x + 3 will be
a) x > 2
b)x < 2
c)x = 2
d)zero
The internal bisectors of a triangle meet at a point, the point is called
a) circumcentre
b) in-centre
c) centriod
d) rthocenter
2
The roots of a quadratic equation ax + bx + c = 0 will be unequal and
real if
a) b2 – 4ac > 0
b) b2 – 4ac < 0
c) b2 – 4ac = 0
d) none of the above
The value of Sin (n.360 + ) will be (n is +ve integer)
a) Sin
b) –Sin
c) Cos
d)Sec
2
2
The equation ax + 2hxy + by = 0 always represent two straight lines
a) passing through origin
b) not passing through origin
c) does not represents straight lines
d) represents two circles
The formula for finding the amount of simple depreciation on the
original cost is
a) 9/10 P
b) ½ P
c) PTR/100
d) A/100
Find the fourth proportion to 6, 18 and 5
a)18
b)6
c)5
d)15
In a race of 1km, A beats B by 20m and B beats C by 50m. By how much
does A beat C?
a) 70 m
b)35 m
c)69 m
d)80 m
Find the area of the triangle whose sides are 3 cm, 4 cm, and 5 cm
a) 25 cm2
b)6 cm2
c)8 cm2
d)10 cm2
If the radius of the sphere is made half, then the ratio of the volumes of
the bigger sphere to that to smaller sphere is equal to ______
a) 8:1
b)5:2
c)7:3
d)4:3
The total surface area of cuboid with dimension l, b and h is
a) 2(lb+bh+lh)
b) (lb+bh+lh)
c) ½(lb+bh+lh) d) lbh
If the length of sides of a cuboid are reduced to half, its surface area
becomes
a) ¼
b)1/3
c)½
d) double
100 | Cambridge institute/ Mathematics
81. The total surface area of a hemisphere of radius 'r' is
a) ¼r2
b) 2r2
c)3r2
d)4r2
Fill in the blanks:
82. If 99% of a number is 4.95, the number is ………..
83. If a + b + c = 0, then a3 + b3 + c3 – 3abc = ………..
84. If the sides of a triangle are 7cm, 24 cm and 25cm, then its area is ………
85. If the hypotenuse of a triangle is 13cm and one of its side is 5 cm, then
the area of the triangle is ………..
86. Three consecutive integers whose sum is equal to their product are ………
87. The L.C.M. of 4(a-b) and 6(b-a) is ……….
88. A triangle which is neither isosceles nor equiangular is said to be ………..
89. The square root of 0.01 is …………..
90. 90 is equal to ………….. grades.
91. The equilateral triangle is also called ……………
92. The figure formed by joining the mid points of the adjacent sides of a
quadrilateral is ………..
Answers: Mathematics
1.c
2.c
3.b
4.a
5.d
9.a
10.d 11.c
12.d
13.d
17.b 18.b 19.a
20.c
21.b
25.c 26.d 27.d
28.d
29.a
33.b 34.a 35.a
36.b
37.a
41.a 42.b 43.b
44.c
45.d
49.a 50.c 51.a
52.b
53.c
57.d 58.b 59.c
60.d
61.d
65.b 66.a 67.a
68.a
69.a
73.a 74.c 75.d
76.c
77.b
81. c 82. 5 83.0
84.84cm2 85.30cm2
87. 12(a-b)
88. scalene
89. 0.1
91. equiangular
92. parallelogram
6.d
14.c
22.c
30.c
38.a
46.d
54.d
62.c
70.b
78.a
7.b
8.c
15.c 16.c
23.c 24.d
31.c 32.b
39.d 40.b
47.d 48.b
55.c 56.a
63.d 64.a
71.a 72.a
79.a 80.a
86.1,2,3
90. 100g
Cambridge institute/ Mathematics | 101
TRY YOURSELF - I
1
5.
Find the two-digit number whose tens digit when multiplied by 3 equals
the sum of the digits, and the number that is obtained by reversing the
digits is 54 less than the product of 4 and the original number.
a) 42
b) 24
c) 33
d) 44
A Boat takes two trips on a rover. On the first trip it travels upstream for
5 hours and returns in 2 hours. On the second trip it goes down stream
for 3 hours, turns around and heads back upstream. After spending 7
hours on the return trip it is still 2 miles from its starting point. Which of
the following is the speed of the current in miles per hour?
a) 3
b) 4
c) 5
d) 7
If there are two containers of sugar solution; the first is 4 percent and the
second 8 percent. How much of each should we combine to get 40 gallons
of a 5 percent solution?
a) (20, 20)
b) (10, 30)
c) (30, 10)
d) (15, 25)
The angle between the lines 3x – y + 2 = 0 and x + 3y + 4 = 0 is
a) 0
b) 45o
c) 600
d) 90o
The shortest distance between the lines 3x + 5y – 1 = 0 and 3x + 5y + 23 is
6.
a) 43
b) 34
o
The value of sin 18 is
2.
3.
4.
a)
7.
8.
1 5
4
b)
5 2
2
c) 9 2
c)
5 1
4
d) 4
d)
1
5
Two vertices of a triangle are at (5, 9) and (-4, 1). Find the third vertex if
the medians meet at (1, 1)
a) (7, 2)
b) (2, -7)
c) (1, 7)
d) (4, -2)
My salary was first increased by 10% and then decreased by 10%. What
is the total percentage change in my salary?
a) 20%
b) 1%
c) 5%
d) 1.5%
102 | Cambridge institute/ Mathematics
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
A Pool is filled to ¾ of its capacity. 1/9 of the water get evaporated) If
the capacity of the pool is 24, 000 gallons when it full. How many
gallons of water is to be added to fill the poll?
a) 8,000
b) 6,000
c) 12,000
d) 18,000
The square root of 0.01 is ……….. to 0.01
a) less
b) equal
c) greater
d) twice
o
What is the least positive number x for which cos (30 – x) = sin (45o +
2x)?
a) 30
b) 20
c) 60
d) 15
How many points are in the graph of the solution set of the system
2x – y – 1 = 0
x2 + y2 = 9 ?
a) 0
b) 1
c) 2
d) 3
What is the minimum value that f ( x ) can attain if f ( x ) = 2x2 + 8x – 1 ?
a) -1
b) -8
c) 9
d) –9
9
f (a) f (b)
If f ( x ) = 3x - , then
=?
5
a b
a) -3
b) -1
c) 0
d) 3
A pyramid and a prism have equal altitudes and bases of equal area.
What is the ratio of their volumes?
a) ½
b) 1/3
c) 1/4
d) 1/6
If the sequence 5, x, y is proportional to the sequence x, 20, 32, which of
the following is y?
a) 16
b) 12
c) 14
d) 10
If the diameter of circle R is 30% of the diameter of circle S, the area of
circle R is what percent of the area of circle S?
a) 15%
b) 9%
c) 30%
d) 60%
For how many integer values of x will the value of the expression 3x – 4
be an integer greater that 4 and less than 250?
a) 82
b) 81
c) 83
d) 84
Set A consists of all multiples of 4 between 50 and 65. Set B consists of
all multiples of 3 between 50 and 65. What are possible number of the
element of the set A – B)
Cambridge institute/ Mathematics | 103
a) 2
b) 4
c) 3
d) 1
20. Rick is three times as old as Maria, and Maria is four year older than
Leah. If Leah is z years old, what is Rick’s age in terms of z?
z4
a) 3z + 4
b) 3z – 12
c) 3z + 12
d)
3
21. The base of an isosceles triangle exceeds each of the equal sides by 8
cm. If the perimeter is 89 cm, what is the length of the base?
2
a) 35
b) 27
c) 29
d) 70
3
22. What is the area of a rhombus with a perimeter of 49 and a diagonal of
10?
23.
24.
25.
26.
27.
28.
29.
a) 100
b) 50 3
c) 100 5
d) 200
George scored an average of 80% on three tests. What score must he
get on the fourth test to bring his average to 85%?
a) 85%
b) 90%
c) 95%
d) 100%
When the rate of income tax is increased from 10% to 15%, I have to
pay Rs. 835 more. Find my income.
a) 16700
b) 16,600
c) 10000
d) 83500
One tap gives 200 liters of water in 20 minutes; another tap throws all
the water in 25 minutes. If both the taps are open together, how much
water is collected n 20 minutes?
a) 10 lt
b) 2 lt
c) 20 lt
d) 40 lt
Equation of the line through origin and perpendicular to 2 x + 3y + 6 = 0
is
a) 2x + 3y + 6 = 0 b) 3x + 2y + 6 = 0 c) 3x + 4y = 0 d) 3x – 2y = 0
If (2, 6), (3, 8) and (-1, y) lie on a straight line, find the value of y.
a) 0
b) 10
c) 2
d) –5
Ram drove 8 miles west, 6 miles north, 3 miles east and 6 miles north.
How for was Ram from his starting point?
a) 17
b) 13
c) 19
d) 21
If 3x + 2y = 8 and 2x + 3y = 12, what is the arithmetic mean of x and y.
a) 2
b) 3
c) 4
d) cannot determine
104 | Cambridge institute/ Mathematics
30. The value of sin 105o is
a)
3
2 2
b)
1 3
2
c)
3 1
2 2
d)
1 3
2
31. The two A.M. between 160 and 172 are
a) 168, 164
b) 164, 168
c) 165, 170
d) 148, 184
32. If a solid metal sphere of radius 1 foot is melted and recast to form
spheres of radios 1 inch, how many of these smaller spheres can be
made?
a) 36
b) 144
c) 1432
d) 1728
33. A printer that can print 1 page in 5 seconds sh town for 3 minutes to
shuts cool off after every hours of operation. How many minutes will
the printer take to print 3600 pages.
a) 300
b) 312
c) 18,00
d) 18,897
34. 750 times 45 equals P. Therefore, 750 times 44 equals.
a) P-45
b) P-750
c) P-1
d) 750P
35. How many degrees has the minute hand moved on a clock from 4:00
p.m. to 4:12 p.m.?
a) 12
b) 36
c) 72
d) 90
36. In an AS, S11 = 77, find t6
a) 17
b) 71
c) 21
d) 7
37. The sum of three numbers in A.P. is 21 and the sum of their squares is
155 then the common difference of the A.P. is
a) 2
b) -2
c) 2
d) 3
38. If 2 < x < 4 and 3 < y < 7, what is the largest integer value of x + y?
a) 9
b)11
c) 12
d) 10
39. Which of the following infinite series has its sum,
a) 2, 4, 8, 16 ………
b) 1, -1, 1, -1,…………
d) 16, 8, 4, 2 ………
d) 2, -4, 8, 16,……….
40. For two matrices A and which relation is false.
a) (A1)1 = A
b) (A + B)1 = A1 + B1
c) (AB)1 = A1B1
d) (A – B)1 = A1 – B1
Cambridge institute/ Mathematics | 105
41. The length of the minute hand of a wall clock is 10 cm. Find the distance
traveled by it in 21 minutes.
a) 22 cm
b) 21 cm
c) 210 cm
d) 1 cm
42. The value of (xa - b)a + b (xb - c)b + c (xc – a)c + a is
a) 1
b) 0
c) x
d) none of the above
43. Cumulative frequency is to be calculate to find
a) Mean
b) Median
c) Mode
d) Range
44. Out of 10 liters of milk, bought at the rate of Rs. 10 per liter, if 4 liters
are lost by leakage. What percentage is gained of lost by selling the
remainder at Rs. 15 per liters.
a) 0%
45. If
b) 15%
c) 11
1
9
%
d) 10%
x4
2
(x – 1)
then
3
6
a) x
5
3
b) x
46. For what values of x is
8
3
c) x
5
4
d) x
5
3
x
= 1?
x
a) x = 0
b) x 0
c) x 0
d) x > 0
47. When a is divided by 7, the remainder is 1. When b is divided by 7, the
remainder is 2. What is the remainder if ab is divided by 7?
a) 3
b)1
c) 2
d) 6
2
48. If 0 x 360 and 4 Sin x + 4 Cos x -1 = 0, which of the following sets
contains all values of x?
a) (60, 120)
b) (120, -120)
c) (60, -60)
d) (120, 240)
49. Which of the following is a quadratic equation with roots of
3
4
and
1
?
2
a) 8x2 + 2x -3 = 0
b) 8x2 + 5x = 3 = 0
c) 8x2 + 5x – 3 = 0
d) 8x2 -2x – 3 = 0
50. Which of the following is the general term of the sequence 11, 9, 7, …..,
when n is the number of the terms
a) n
b) 11 + n
c) 11-2n
d) 11-2 (n-1)
106 | Cambridge institute/ Mathematics
51. The base of an isosceles triangle lies on the x – axis. What is the sum of
the slopes of the three sides?
a) 0
b) 1
c) -1
d) cannot determine
52. If (a, Sin a) and (b, Sin b) are any two points on the graph of y = Sin x,
then the greatest value of Sin a – Sin b is
a) 1
b) 2
c) 0
d) 180o
1
53. If f (x)
then,
Sinx
1
1
a) f (x) =
b) –f (x) = f (-x)
c) f (x) = -f (x) d) f (-x) = f
f ( x)
x
54. If 2Sin2x = Sinx, then a value of x is
a) 120o
b) 60o
c) 45o
d) 0o
55. If 7 more than x is 1 less that twice x, which of the following numbers
falls between 3/x and 4/x?
a) 1/4
b) 5/16
c) 7/16
d) 5/8
56. The average rate of a class of 35 students is 15 years. If the teacher’s age
is also included the average age increases by one year. Find the age of
the teacher in years.
a) 51 yrs
b) 50 yrs
c) 52 yrs
d) none of these
57. The lines 4x + 5y + 6 = 0 and 5x – 4y + 3 = 0 are
a) Parallel
b) Co- incident
c) Intersecting c) none of these
58. A man bought 420 apples for Rs. 280 but 84 were rotten yet he earned
20% how did he sell?
a) Rs. 1.25
b) Rs. 1
c) Rs. 1.50
d) Rs. 0.95
59. If a and b are positive number with a3 = 3 and a5 = 12b2. What is the
ratio of a to b?
a) 2:3
b) 3:2
c) 2:1
d) 1:2
60. Brian gave 20% of his baseball cards to scott and 15% to Adam. If he still
had 520 cards, how many did he have originally?
a) 7426/7
b) 800
c) 700
d) 1320
Cambridge institute/ Mathematics | 107
61. The probability that a boy will get a scholarship is 0.75 and that a girl
will get is 0.72. What is the probability that at least one of them will get
the scholarship?
a) 0.93
b) 1.47
c) .78
d) None of these
62. What should be the value of k so that 9x2 + kx +
a) 6
b) 3
c)
1
3
1
4
is a perfect square?
d)
1
6
63. First bell rings in 1:45 and 2nd bell in 2: 05. What portion of an hour is
the interval between the two bells?
a) 1/3
b) 1/4
c) 2/3
d) 4/5
64. At the rate of 5% per annum, if the interest doubles the amount. Find
the time taken
a) 50 yrs
b) 60 yrs
c) 40 yrs
d) 20 yrs
65. If 8 men can do a work in 12 days. In how many days will the same work
be completed when 4 men are sent out?
a) 12 days
b) 24 days
c) 16 days
d) 20 days
66. Four persons are engaged in the business in the ratio 18:10:8:4. What
percentage does the person occupies who have the least share of the
whole business.
a) 10%
b) 4%
c) 20%
d) 0%
67. ½ the perimeter of a rectangle is 89. The difference between length and
breadth is 4. Find the length and breadth.
a) (6, 2)
b) (4, 0)
c) (8, 0)
d) (5,1)
68. If a rubber ball is dropped from a height of 1 meter and continues to
rebound to a height that is 9/10 of its previous fall, find the total
distance in meter that it travels on falls only.
81
a)
b) 1
c) 9
d) 10
100
69. 10% more than 10% less than x is what percent of 10x?
a) 9%
b) 9.9%
c) 10%
d) 99%
70. The co-ordinate of the foot of the perpendicular from (6, 8) to the line
through (1, 5) and (9, 3) is
108 | Cambridge institute/ Mathematics
71.
72.
73.
74.
75.
76.
77.
a) (-4, -5)
b) (4, 5)
c) (5, 4)
d) (1, 2)
The equation of the line through (4, 1) which is perpendicular to the line
x – 2y – 4 = 0 is
a) x – 2y – 2 = 0
b) 2y – x + 2 = 0
c) 2x + y + 9 = 0 d) 2x + y – 9 = 0
What number should be subtracted from each of the numbers 54, 71, 75
and 99 so that the remainders may be proportional?
a) 3
b) 7
c) 2
d) none of the above
A fraction becomes 6/5, if the numerator is multiplied by 2 and
denominator is reduced by 5. But if the numerator is increased by 8 and
the denominator is doubled, the fraction becomes 2/5. Find the
fraction.
a) 12/25
b) 13/25
c) 10/25
d) 15/25
Points A and B are 90 km apart from each other on a highway. A car
starts from A and another from B at the same time. If they go in the
same direction, they meet in 9 hours and if they go in opposite
directions, they meet in 9/7 hours. Find their speeds.
a) (30, 30)
b) (40, 30) kmph
c) (30, 35)
d) (40, 35)
A sailor goes 8 km downstream in 40 minutes and returns back to the
starting point in 1 hour. Find the speed of the sailor in still water.
a) 2 km/hr
b) 20 km/m
c) 10 km/h d) non of the above
For what values of k the equations
kx - y = 2, 6x – 2y = 3 has no solution.
a) K 3
b) k = 3
c) k = 4
d) k = -3
If x is the length of a median of an equilateral triangle, then its area is
a)
x2
b)
x
3
c)
2
x2 3
3
x2
d)
2
78. The ratio of the length of a rod and its shadow is 1: 3.
The angle of elevation of the sun is
a) 30o
b) 45o
79. If 2A = 3B = 4C then A:B:C is
a) 2:3:4
b) 4:3:2
c) 60o
d) 90o
c) 6:4:3
d) 3:4:2
Cambridge institute/ Mathematics | 109
80. Rs. 1360 have been divided among A, B, C such that A gets (2/3) of what
B gets and B gets (1/4) of what C gets. Then, B’s share is
a) Rs. 120
b) Rs. 160
c) Rs. 240
d) Rs. 320
81. If the sum of five consecutive odd integers is 735 what is the largest of
these integers?
a) 155
b) 151
c) 145
d) 143
82. If – 7 x 7 and 0 y 12, what is the greatest possible value of y – x?
a) 19
b) 7
c) 14
d) 0
83. When 423, 890 is rounded off to the nearest thousand, how many digits
will be changed?
a) 1
b) 2
c) 3
d) 4
84. The population of a town doubled every 10 years from 1960 to 1990.
What was the percent increase in population during this time?
a) 100%
b) 200%
c) 800%
c) 700%
85. If M be the median and m the mode, of the following set of numbers.
10, 70, 20, 70, 90. What is the average of M and m?
a) 50
b) 55
c) 60
d) 62.5
86. If a is increased by 10% and b is decreased by 10%, the resulting
numbers will be equal. What is the ratio of a to b?
10
11
9
9
a)
b)
c)
d)
10
11
9
9
87. What is the maximum number of points of intersection between square
and a circle?
a) less than 4
b) 4
c) 6
d) 8
88. A 15- gallon mixture of 20% alcohol has 5 gallons of water sated to it.
The stringing of the mixture, as a percent, is approximately.
1
2
a) 12
b) 15
c) 20
d) 16
2
3
89. Two ships leave from the same port at 11:30 A.M. If one sails due east
at 20 miles per hour and the other due south at 15 miles per hour, how
many miles apart are the ships at 2:30 P.M.?
a) 75
b) 25
c) 50
d) 80
110 | Cambridge institute/ Mathematics
90. If a = b, x < y then
a) a + x > y + b
b) a + x < y + b
c) a + x = y
d) a + x = y + b
91. What single discount is equivalent to two successive discounts of 10%
and 15%?
a) 25%
b) 24%
c) 24.5%
d) 23.5%
92. In a right angled triangle x < y < z which is true>
a) x + y = z
b) x2 – y2 = z
c) x2 = y2 + z2 d) x2 + y2 = z2
93. If each of the dimensions of a rectangle is increased 100%, the area if
increased as 100%
a) 100%
b) 200%
c) 300%
d) 400%
94. A women wants to earn Rs. 76 a year. How much money must be invest
at the rate of 2% to earn her desired amount.
a) Rs. 38
b) Rs. 3800
c) Rs. 380
d) Rs. 760
o
95. Find the circular measure of 60
2
3
a) /3
b) /4
c)
d)
5
4
96. A circle is inscribed in a given square and another circle is circumscribed
about the same square. What is the ratio of the area of inscribed to the
area of the circumscribed circle?
a) 1:4
b) 1:2
c) 2:3
d) 3:4
97. Three circles are tangent externally to each other and have radii of 2
inches, 3 inches and 4 inches, respectively. How many inches are in the
perimeter of the triangle formed by joining the centers of the three
circles?
a) 9
b) 12
c) 15
d) 18
98. The cost of a bicycle including sales tax is Rs. 1760. If the sales tax is paid
at the rate of 10%, find the list price of the cycle.
a) Rs. 1820
b) Rs. 1600
c) Rs. 1760
d) Rs. 800
99. The tax that is added on the value of good while they are transferred
from one party to another is called:
a) Excise duty
b) Sales Tax
c) VAT
d) none
100.The formula for finding the annual single depreciation on the original
costs is:
Cambridge institute/ Mathematics | 111
V S
V S
V
A
P
b)
c)
d)
P
100
Sn
n
n
101.If the radius of the right circular cylinder is r, and the height is 1/3 r then
the curved surface area is:
a) 1/3r2
b) r3
c) r2
d) none
a)
102.A semi circle of radius 14 3 is bent into a conical cup find the volume of
the cup.
a) 3423 cm2
b) 4323 cm3
c) 2343 cm3
d) none
103.The sum of roots of the quadratic equation x2- 3 x-6=0 is:
a) - 3
b) 6
c) –6
d) 3
104.Find the area of a square that can be inscribed in a circle of radius 5 cm.
a) 25 cm2
b) 12.5 cm2
c) 16cm2
d) 50cm2
105.The roots of the equation 2x2 – 6x = 0 are
a) 1,3
b) 0.3
c) 3,3
d) 3,1/3
2
4
106.The square root of 1+2x +x = 0 are
a) ± (1+x2)
b) ± (1+x4)
c) ± (1+2x2)
d) ± (1+x2)
107.The L.C.M. and H.C.F of the two numbers are 840 and 14 respectively
and if one of the numbers is 42 then the other number is
a) 84
b) 280
c) 868
d) 42
108.A man bought an article for Rs. 1 and sold it for Rs. 1.20. What is the
gain percent?
a) 12%
b) 20%
c) 1.2%
d) 10%
109.The simplified form of (27)4/3 is
a) 9
b) 999
c) 88
d) 81
3
2
110.If a = 1/8, the value of a is
a) 1/24
b) 1/6
c) 1/3
d) ¼
111.The value of tan 70º is
a) 2 3
b)
1
2 3
c)
3 1
3 1
d)
3 1
2 3
112.What is the common ratio of the geometric progression 1, 0.1, 0.01,
0.001?
a) 10
b) 1/100
c) 1/10
d) 1
112 | Cambridge institute/ Mathematics
113.In a triangle ABC, a =3cm, b=4cm and c + 5cm then the area of the
triangle is
a) 12cm2
b) 6 cm2
c) 10cm2
d) 15cm2
114.If x-2y+3=0, then the y-intercept of the line as
a)
1
2
b)
1
3
c)
3
2
d)
3
2
115.Which term of the progression 2,4,6,8, 8 is 98?
a) 48
b) 46
c) 50
d) 49
116.If 3x +2y = 11 and 2x +3y = 17, what is the arithmetic mean of x and y?
a) 2.5
b) 2.8
c)5.6
d) 1.4
2
117.If x = 9 is a solution of the equation x -a=0, which of the following is a
solution of x4 – a = 0?
a) –18
b) –3
c) 0
d) none
118.If a-b = 10, a2 – b2 = 20, what is the value of b?
a) –6
b) –4
c) 4
d) 6
119.The circumference of the second circle is 2 feet longer than the
circumference of the first circle. How many feet longer is the radius of
the second than that of the first?
a)
1
2
b)
1
c) 2
d)
120.What is the area of a rectangle whose length is twice its width and
whose perimeter is equal to that of a square?
a) 2/3
b) 8/9
c) 18/5
d) 12/7
121.A jar contains 10 red marbles and 30 green ones. How many red
marbles should be added to the jar so that 60% of the marbles will be
red?
a) 45
b) 35
c) 50
d) 70
122.George drove 8 miles west, 6 miles north, 3 miles east and 6 more miles
north. How far was Tony from his starting point?
a) 13
b) 17
c) 19
d) 21
123.At a speed of 48 miles per hour, how many minutes will be required to
drive 32 miles?
a) 40
b) 45
c) 50
d) 2400
Cambridge institute/ Mathematics | 113
124.If the average of 2, 7 and x is 12. What is the value of x?
a) 9
b) 21
c) 12
d) 27
125.If the sum of five consecutive odd integers is 735. What is the largest of
these integers?
a) 150
b) 155
c) 145
d) 151
126.What is the largest prime factor of 1001?
a) 11
b) 7
c) 13
d) 101
127.25% of 220 equals 5.5% of W. What is the value of W?
a) 100
b) 101
c) 55
d) 1000
128.If 2 x 4 and 3 y 7, hat is the largest integer value of x +y?
a) 9
b) 11
c) 12
d) 10
129.If the sum of three consecutive integers is less than 75, what is the
greatest possible value of the smallest of the three integers?
a) 22
b) 23
c) 24
d) 25
130.The average of 10 numbers is –10. If the sum of six of them is 100, what
is the average of the other four?
a) –100
b) 0
c) 050
d) 100
131.How many positive integers less than 100 have remainder 3 when
divided by 7?
a) 12
b) 9
c) 10.
d) 13
132.What is the smallest number that is divisible by both 34 and 35?
a) 1
b) 34
c) 35
d) none
x
y
100
133.If 3 x 3 = 3 , what is the arithmetic mean of x and y?
a) 50
b) 100
c) 25
d) 200
134.What is the circumference of a circle whose area is 10?
a) 5
b) 20
c) 20
d) 20
100
50
135.If 50 = k (100 ), what is the value of the k?
a) 250
b) 2550
c) 5050
d) none
2x-4
x
136.For what value of x is 8 = 16 ?
a) 2
b) 3
c) 8
d) 6
137.In a departmental store 100 pounds of cake was divided into packages,
each of which weighed 4/7 pounds. How many packages were there?
114 | Cambridge institute/ Mathematics
a) 175
b) 157
c) 751
d) 715
138.What fraction of a week is 98 hours?
a) 7/24
b) 24/98
c) ½
d) 7/12
139.5/8 of 24 is equal to 15/7 of what number?
a) 7
b) 8
c) 15
d) 7/225
140.The area of a right triangle is 12 sq. inches. The ratio of its legs is 2:3.
Find the hypotenuse of the triangle.
a) 13
b) 26
c) 3 13
d) 52
141.A gallon of water is equal to 231 cubic inches. How many gallons of
water are needed to fill a tank of dimension 11" high, 14" long and 9"
wide?
a) 6
b) 8
c) 9
d) 14
142.A rectangular block of metal weight 3 kg. What will be the weight of the
block of the same metal if the edges are twice as large?
a) 3/8
b) ¾
c) 3/2
d) 3
143.If the length of a rectangle is increased by 30% and the altitude is
decreased by 20%, then its area is increase by
a) 4%
b) 5%
c) 10%
d) 25%
144.The average temperatures for five days were 82º, 86º, 91º, 79º, and
91º. What is the median of these temperatures?
a) 82
b) 86
c) 84
d) none
145.The water level of a swimming pool of size 75' x 42' is to be increased by
4'. How many cubic feet of water to be added to accomplish this?
a) 1050
b) 1450
c) 1500
d) none
Cambridge institute/ Mathematics | 115
Try yourself-II
b
c
If 2a =2 2 = 25 2 , In which of the following are a, b and c arranged in
descending order of value?
a.a, b, c
b.c, b, a
c. b, c, a
d. None
2. If x is increased by 25% then by what percent is x2 increased?
a.56.25%
b. 156.25
c. 6.25
d. None
3. A man’s annual income is Rs 60000. The first 20000 is tax free and he has to pay
5% on the next 20000 and 10% on the remainder. How much does he pay as
income tax per year?
a.2500
b. 2800
c. 3000
d. None
4. A dealer is selling an article at a discount 5% on the market price.
What is the
cost price, if the market price is 12% above the cost price?
a.125
b. 120
c. 100
d. None
5. A single discount equivalent to the successive discount 10% and 5% will be
a. 14.5%
b. 10%
c. 3.5%
d. None
3
6. At what rate percent per annum will the principle be 8 of the amount in 8 years?
a. 13.2%
b. 15%
c. 20.83%
d. None
7. If A : B = 6 : 7 and B : C = 8 : 7, then A : C will be
a. 48 : 49
b. 7 : 9
c. 30 : 7
d. None
8. What number must be added to each of the numbers 6, 15, 20, 43 to make them
proportional?
a. 4
b. –1
c. 3
d. None
9. What sum will amount to Rs 2782.50 in 2 years at compound interest, if the rates
are 5% and 6% for the successive years?
a. 1500
b. 2500
c. 1800
d. None
10. The present value of a computer is Rs 350000. If its value depreciates by 8% in
first year and by 10% in the second year, then its value after 2 years will be
a. Rs 289800
b. Rs 200000
c. Rs 250000
d. None
x+2
1
11. If f(x – 1) = f (- 3) where f(x) = 2x – 1 , x 2 x belongs to R then the value of x will
be
a. –1
b. 5
c. –2
d. None
1
12. If f (x) = x + x , then f(b2) will be equal to
a. {f(b)}2 – 2
b. {f(b)}+ 2
c. {f(b)}2
d. None
1.
116 | Cambridge institute/ Mathematics
13. If U = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}, A = {0, 1, 2, 3, 4, 5} & {2, 4, 6}, then (A – B’)’ will be
a. {0, 1, 3, 5, 6, 7, 8, 9} b. {1, 4, 5}
c. {0, 1, 3, 5, 6, 7, 8,} d. None
14. If A and B are two sets such that (A) = 17, (A B) = 38 and (A B) = 2, then (B –
A) will be
a. 10
b. 5
c. 15
d. None
15. The sun’s rays are inclined to the ground at an angle of 600. The length of the
shadow of a pole of 15m high will be
a. 7.52m
b. 8.6m
c. 9.32m
d. None
2
Cos – 3Cos + 2
16. If
= 1, then the value of will be
Sin2
a. 500
b. 300
c. 600
d. None
17. If 2Cos2 + Sin = 0, then the value of will be
a. 450
b. 300
c. 900
d. None
18. The points (3,3), (9,0) and (12,21) are the vertices of
a. lying angled triangle
b. an equilateral triangle
c. a right angled triangle
d. None
19. The value of x, such that AB = BC where A, B and C are the points (6,-1), (1,3) and
(x,8) respectively will be
a. 5 or -3
b. 2 or -3
c. 5 or 1
d. None
20. The equation of the diagonals of a rectangle sides are x = -1, x = 2, y = 6 are
a. 8x – 3y + 2 = 0; 8x + 3y - 10=0
b. x – 2y + 1 = 0; y = 5
c. x – 2y = 0; 2x – 3y = 7
d. None
21. The equation of the line passing through the point (1,4) and intersecting the line
x – 2y -11 = 0 on the Y – axis will be
a. x + y + 3 = 0
b. 19x – 2y – 11 = 0 c. y = 3x + 2
d. None
22. The lines y = 2, y - 3 x = 5 and y + 3 x = 4 from a triangle which is
a. an equilateral triangle
b. a right angled triangle
c. an isosceles triangle
d. None
23. If the area of a quadrilateral, whose angular points are (1,2), (-5,6), (7,-4) and (K,
-2) is zero. Then the value of K will be
a. -3
b. 3
c. 5
d. None
24. From a cube of edge 14cm, a cone of maximum size is carried out. The volume of
the cone will be
2
a. 702cm2
b. 618 cm3
c. 718 3cm3
d. None
Cambridge institute/ Mathematics | 117
25. If the equation axy + bx + cy + d = 0 represents a pair of straight lines then bc
equal to
a
1
a. b
b. d
c. ad
d. None
26. The condition ax2 + by2 + 2bxy + 2gx + 2fy + c = 0 represents pair of lines is
a. abc + 2fgh – af2 – bg2 – ch2 = 0
b. abc - 2fgh + af2 – bg2 + ch2 = 0
c. abc + 2fgh – af2 – bg2 + ch2 = 0
d. None
27. The value of ‘a’ for which lines represented ax2 + 5xy + 2y2 = 0 are mutually
perpendicular if
25
a. 8
b. – 2
c. 2
d. None
28. If the pair of opposite angles of a quadrilateral are supplementary, then the
quadrilateral is
a. a parallelogram
b. cyclic
c. square
d. None
29. In a cyclic quadrilateral ABCD, if AB = DC then B equals to
a. A
b. D
c. D
d. None
30. Modulus of a vector a will be equal to
a. a . a
b. a .a
c. a
d. None
31. If a , b and c three mutually perpendicular vectors such that each one is of
magnitude unity, then a + b + c is equal to
a. 3
b. 1
32. The matrix A satisfying the equation:
1
1
1 4
a.
0 1
b.
0 1
c. 3
d. None
1 3
1 1
xA=
equals to
0 1
0 1
1 0
c.
0 1
d. None
10 1
4 5
33. If
+ X = 0 5 then the value of X equals to
3 6
6 6
1 0
a.
3 1
34. The value of
b.
2 5
3
16
0 0
c.
0 0
d. None
c. 3
d. None
will be
33 2
a. – 1
b. 2/3
118 | Cambridge institute/ Mathematics
35. In a 100 tosses of a coin, 56 heads were observed, what is the empirical
probability of getting a tail in the next toss?
a. 0.72
b. 0.44
c. 0.12
d. None
36. Two dice are tossed. What probability of getting a sum of 6 or 5 on one of the
dice?
a. 7/18
b. 1/7
c. 3/5
d. None
37. If the sum of two sides of a right angled triangle is 17 cm and the hypotenuse is
13 cm. Then the length of sides are
a. 5 cm and 2 cm
b. 12 cm and 5 cm c. 10 cm and 3 cm d. None
38. One of the factors of 2x2 + 5x – 3 = 0 is
a. x + 3
b. x + 5
c. x – 3
d. None
3
2
39. One of the factors of 2x + 7x – 4x – 14 = 0 is
a. x – 5
b. 2x + 7
c. x – 1
d. None
40. The roots of the equation ax2 + bx + c = 0 will be rational number if b2 – 4ac is
a. 0
b. perfect square
c. 2
d. None
41. A checker is placed on a rectangular table 3 inches from one side of the table and
4 inches from the adjacent side. How far, in inches, is the checker from the nearest
corner of the table?
a. 3
b. 5
c. 5
d. none
42. David’s income was increased by 10% and later decreased by 10%, what is the
total change percent in David’s income?
a. 11%
b. 1%
c. 11.5%
d. none
1
43. If the numbers 8 and 12 are increased by 25% and 33 /3 % respectively. What
will be the average increment?
a. 30%
b. 15%
c. 10%
d. none
44. A dealer ordinarily makes a profit of 16%. If his cost goes down by 20% and he
decreases his price by 10%, what percent does he gain?
a. 28.2%
b. 30.50%
c. 15%
d. none
45. The principal value which amounts to Rs 1200 at 8% p. a. S.I. after 9 years will be
29
11
a. 697 43
b. 697 43
c. 697
d. none
46. If (3x +1): (5x +3) is the triplicate ratio of 3:4, then the value of x will be
a. 17
b. 17/57
c. 57
d. none
47. If x : y = 2 : 3 and y : z = 4 : 7 then x : y: z will be
a. – 1 : 5 : 7
b. 8 : 12 : 21
c. 3 : 5 :1
d. none
Cambridge institute/ Mathematics | 119
48. Two quantities are in the ratio 7 : 4. If the greater quantity is 24.5, then the smaller
quantity will be
a. 14
b. 15
c. 10
d. none
49. If a, b, c, d and e are in continued proportion, then a : e will be equal to
a. a4 : b4
b. a3 : d
c. a : d
d. none
50. In a mixture of 35 litres, the ratio of milk to water is 4 :1. Another 7 litres of water
is added to the mixture. Then the ratio of milk to water in the resulting mixture
will be
a. 2 : 1
b. 3 : 5
c. 10 :13
d. none
51. The difference between C.I and S.I on sum of Rs 4800 for 2 years at 5% per annum
will be
a. Rs 10
b. Rs 30
c. Rs 12
d. none
52. In what time will a sum of Rs 1562.50 produce Rs 195.10 at 4% per annum
compound interest?
a. Two years
b. Three years
c. Ten years
d. none
53. The compound interest on Rs 50,000 for 2 years at 10% per year, compounded
half yearly will be
a. Rs 23205
b. Rs 25000
c. Rs 10000
d. none
54. At what rate percent per annum compound interest will be Rs 2304 amount to Rs
2500 in 2 years
a. 25 /6 %
b. 30%
c. 17%
d. none
55. In a group of 50 students, 25 play hockey 30 play football and 8 play neither
game.The number of students who play both games will be
a. 10
b. 15
c. 13
d. none
56. The Value of tan / 3 . Sin / 3 + Sin / 4 . Cos / 3 + Cos / 2. Sin / 3 will be
a. 3 / 2
b. -1 / 2
c. 3 / 2
d. none
0
0
57. The value of Cosec 35 – Sec 55 will be
a. 0
b. -1
c. 10
d. none
58. The incenter of a triangle, the equation whose sides are 3x + 4 y = 0; 5x – 12y = 0
and y – 15 = 0 will be
a. ( -1, 2)
b. (8, 1)
c. (1, 8)
d. none
59. The orthocenter of the triangle formed by the lines whose equation are X – y + 1
= 0, x – 2y + 4 = 0 and 9x – 3y + 1 = 0 will be
a. (-1, 4)
b. (4 – 1)
c. (0, 5)
d. none
60. If the lines 2x + 3ay -1 = 0 and 3x +4y +1 = 0 are mutually perpendicular, then the
value of a is
a. -½
b. 3
c. 5
d. none
120 | Cambridge institute/ Mathematics
61. A rectangle with one side 4 cm, is inscribed in a circle of radius 2.5 cm. The area
of the rectangle will be
a. 2 cm2
b. 7 cm2
c. 12 cm2
d. none
62. Two right circular cones X and Y are made, X having three times the radius of Y
and Y having half the volume of X. Then the ratio of heights of X and Y will be
a. 1:9
b. 9:1
c. 2:9
d. none
63. A sector of a circle or radius 35 cm has an angle of 1440. It is folded so that the
two bounding radii are joined together to form a cone. Then the total surface area
of the cone will be
a. 2156 cm2
b. 2000 cm2
c. 1800 cm2
d. none
64. The largest sphere is carved out of a cube of wood of side 21 cm. Then the volume
of the remaining wood will be
a. 4410 cm3
b. 4010 cm3
c. 4900 cm3
d. none
65. A circular hall has a hemispherical roof. The greatest height is equal to the inner
diameter. If the capacity of the hall is 48510 m3, then the area of the floor will be
a. 1218 m2
b. 1386 m2
c. 1300 m2
d. none
66. The Range of the relation R = { (x, y): x + 2y 6 and x, y N }
a. { 1 , 2 }
b. { 0 , 2 }
c, { 1 , 5 }
d. none
67. The equation x2+k1y2+k2xy = 0 represent a pair of perpendicular lines if
a. k1= -1
b. k1 = 2k2
c. 2 k1 = 2k2
d. none
68. If x2 – 10xy + 12y2 + 5x – 16y – 3 = 0, represents a pair of st. lines, then the value
of is
a. 4
b. 3
c. 2
d. none
69. Any four vertices of a regular pentagon line on a
a. circle
b. square
c. parallelogram d. none
70. If two circles touch, the point of contact lies on a
a. St. line
b. quadrilateral
c. square
d. none
71. The domain of the Relation R where R = { (x) : y = x + 8/ x ; x, y N and x 9} will
be
a. {x, 2, 3}
b. {1, 2, 4. 8}
c. {1, 0, 4, 8}
d. none
72. A sum of money is divided between Mary and David in the ratio 5:8. If Mary’s
Share is Rs 225, then the total amount of money will be
a. 300
b. 400
c. 585
d. none
1 2
3 1
73. If A =
1 2
3 1
a.
then the value of A2 – 2A –5I equals to
b. 5
c. 0
d. none
Cambridge institute/ Mathematics | 121
74. A dealer gains the selling price of 10 radio sets by selling 60 sets. His percentage
profit will be
a. 10%
b. 15%
c. 20%
d. none
75. A father is 2 times as old as his son. 16 years ago the age of the father was three
times the age of the son what is the present age of the father?
a. 64 yrs
b. 10 yrs
c. 80 yrs
d. none
76. A box contains 17 good oranges and 3 bad oranges. If 3 oranges were drawn one
after the other. Then, what is the probability that all the three oranges are good.
a. ½
b. 20 / 27
c. 34 / 57
d. none
2
77. One of the factors 2x + 5x – 3 = 0 is
a. x + 3
b. x + 5
c. x – 3
d. none
78. Which of the following sets is a null set?
a. {x: x=0}
b. {x: x2-2=0, x is rational x is real}
c. {x: x2+4x+0}
d. The set of circles passing through three co-linear points.
1
1
79. The function f: X→Y, x={x: x є R, 2 ≤ x ≤ 2) defined by ƒ(x) = x is
a. one to one ‘onto’
b. one to one ‘onto’
c. many to one ‘onto’
d. many to one ‘onto’
80. One can buy a dozen oranges in Re 1 and sale by gaining 25% profit. How many
oranges would he sale in Re 1?
a. 8
b. 9
c. 10
d. none of the above
81. A number of sphere of radius 1cm are dropped into water contained in a
cylindrical vessel of diameter 6cm. If the spheres are completely immersed and
rise in water level by 4cm, the number of sphere immersed are
a. 1
b. 3
c. 9
d. 27
82. The probability of drawing a diamond or a queen form a pack of 52 cards is
3
4
17
1
a. 13
b. 13
c. 52
d. 4
83. A car traveling at 75km/hr takes 30 minutes for a journey. How long will the car
take to travel the same distance if it is at 25 km/hrs.
a. 75 minutes
b. 30 minutes
c. 90 minutes
d. 120 minutes
84. If a sum of money will be as half much again as it is in 10 yrs the rate of interest
is
a. 5%
b. 10%
c. 15%
d. neither one
1
10a
85. If a+ a = 9, the value of a2 + a + 1 is
a. 0
b. 1
c. 9
d. 10
122 | Cambridge institute/ Mathematics
86. If 2160 = 2a 3b 5c the solution set of a, b, c, is
a. {4,3,0}
b. {1,0,3}
c. {4,3,1}
d. { 2,3,4}
87. If the length of shadow of tree 243 m is 9 3 , the altitude of the sun is
a. 00
b. 450
c. 600
d. 900
88. The non-isometric transformation is
a. translation
b. reflection
c. rotation
d. enlargement
89. The relation between A.M. and G.M. in series is,
a. A.M. <G.M.
b. A.M.>G.M.
c. A.M. =G.M.
d. neither of one
90. Any three given points can be shown collinear by
a. equating the slopes
b. showing area of triangle zero
c. making the equation of st. line joining any two points passing through the
remaining point.
d. all above
91. In an equilateral triangle, which of the following are coincide?
a. circumcentre & incentre
b. incentre & orthocentre
c. circumcentre & orthocentre
d. all of threes coincide
92. The distance between two parallel lines x+3y =6 & 2x +6y =20
a. 4 units
b. 6 units
c. 20 units
d. 24 units
93. How many times the hour hand is faster than the minute hand of a clock
a. 6
b. 12
c. 24
d. 60
94. The diagonals are at right angles in
a. rectangle
b. rhombus
c. parallelogram d. trapezium
j
95. The matrix of order 2×2 for aij = (i) is
1 2
a.
1 4
1 1
b.
2 4
1 1
c.
4 2
1 4
d.
2 1
1-Tan2A
96. Cos2X =1+Tan2A , the relation between X and A is
a. X=2A
b. A=2X
c. X=A
d. relation can not be determined
97. Average height of 25 student of a class 5.5 feet. If average height of 15 students in
the class is 5.7 feet, the average height of remaining student is
a. 5.5 feet
b. 5.2 feet
c. 5.7 feet d. insufficient information
Cambridge institute/ Mathematics | 123
x y
98.If 2x +3y =1, what is 2 +3 in terms of y?
y
a. 5
1-3y
3-5y
3y+4
b. 2
c. 12
d. 15
99. Three equal circles each of radius 2cm touch each other. The area of triangle is
a. 0.12cm2 (app)
b. 0.65cm2 (app)
c. 0.314cm2 (app) d. 4 3 cm2
A
B
C
100. What is the circumference of a circle whose area is 10?
a. 5
b. 10
e.
20
c. 10
d. 2 10
101.If 2x = 32, what is x2?
a. 24
b. 25
c. 25
d. 30
a
b
100
102.If 3 × 3 = 3 , what is the average (arithmetic mean) of a and b?
a. 55
b. 50
c. 45
d. 60
103. If a + b = 5, a – b = 1, the value of a/b is
a. 3/2
b. 6
c. 2/3
d. 1 e. 4
104. What is the difference in degree measurement of the angle made by minute hand
and hour hand of the clock at 12:35 and 12:36 O’clock?
a. 5.60
b. 5.50
c. 50 d. 6.20
e. 4.50
x+2
x
x+6
105. If 17 = 16 , what is the value of 19 ?
a. 1/2
b. 1
c. 3/2
106.If 10a + 10b = 35, what is the arithmetic mean of a & b?
a. 1.75
b. 3.5
c. 7
c
7
107.If (7a).(7b. = 7d , expression d in terms of a, b & c is
c
a. ab
b. c - a - b
c. a + b - c
d. 2
e.3
d. 10 e.51
d. c – ab
108.If -7 x 7 & 0 y 12, what is the greatest possible value of y – x?
a. -19
b. 5
c. 7
d. 17 e.19
124 | Cambridge institute/ Mathematics
c
e.a + b
109. Which of the following is equal to (78 79 )10 ?
a. 727
b. 882
c. 7170
d. 49170 e.49720
3x – 4z
110. If x:y:z = 3:4:5, then the value of2x – y + 4z is
a. -1/2
b. -1
c. -2
d. -1/2.5
111. Ramesh bought a $60 sweater on sale at 5% off. How much did he pay including
5% sales tax?
a. $54.15
b. $57.0 c. $59.85 d. $57.75
e. $60.0
112. What is the greatest value of Sin θ+ Cos θ?
a. 1
b. - 2
c. 2
d. 1/ 2 e. 3 /2
113. The area of the triangle ABC will zero if
a. AB + BC = AC
b. AB. BC = 2AC c. AB2 + BC2 = AC2
d. AB/AC = BC
114. The point of intersection of perpendicular bisectors of the triangle is called
a. centroid
b. in centre
c. circum centre d. ex centre
115. The internal bisectors of the angle of a triangle intersect at the point named
a. Centroid
b. Hypocentre
c. circum centre d. Ortho centre
e.in centre
116. The two vectors a & b will be orthogonal if
a. a = b
b. a b = a b
c. a . b = 0
d. b and c both
117. Two vectors p & q will be oppositely directed if
a. p . q = p . q
b.
p-q
= -1
c. p - q = 0
d. p + q = 0
p.q
118. If Sin2x – Cos2x = 1 then Sinx – Cosx = ………..
a. 0
b. -1
c. 1
119. Which of the following is rational number?
d. - 2
a. x
b.
120. Which of the following is not correct?
a. 0 Sin 1
b. -1 Cos A 1
d. Sin 600
c. ½
c. -1.5 Sin 1.5 d. - tan
121. If the Co-ordinates of A and B are (2, 3) & (3, 4) then the unit vector along AB is
a.
1
2
1
2
2
2
b.
c. i + j
1
d.
1
Cambridge institute/ Mathematics | 125
122. If the sides of a triangle are 6, 8 &10 cm respectively. The area of the triangle will
be
1
1
1
a. 2 6 8 cm2
b. 2 6 10 cm2 c. 2 8 10 cm2
d. none of the above
123. If the interest of Re.1 for 5 months is 10 paisa. The rate of interest is
a. 30%
b. 36%
c. 32%
d. 24%
3
3
124. If a + b = 3, ab =2 the value of a – b = ……………….
a. -7
b. 7
c. 9
d. 12
125. The equation ax2 + by2 + 2gx + 2fy + c = 0 represents a circle if
a. a = b
b. a b
c. b a
d. a = b = 0
126. The determinant of a matrix is a
a. non-negative quantity
b. a negative quantity
c. integer value
d. any value
127. If the curved surface area of a cylinder is numerically equal to the volume of the
cylinder then the area of the base of cylinder is
a. 4
b. 2
c. 27
d. 3
128. If ax = by = cz and abc = 1 then x, y, z are in
a. A.P
b. G.P
c. H.P
d. None of the above
129. A real number which when added to its square gives the cube of that number.
That number is
a. 0
b. 1/2
c. -1
d. 2
130. The line 2x + 3y = 5 meets x-axis at
a. (0, 2/5)
b. (-5/2, 0)
c. (2/5, 0)
d. (5/2, 0)
131. Which of the following is true?
a. second quartile = median
b. mean = standard deviation
c. first quartile = third quartile
d. none of the above
132. The cross sectional area of cylindrical rod is
a. r2
b. 4r2
c. 2rh
d. 2r (r + h)
133.If A = {1,2}, B = {4,5} than AXB is
a. {(1,2}, (1,4), (1,5), (1,1)}
b. {1,4}, (1,5), (2,4), (2,5)}
c. {(1,2), (2,4), (2,2), (2,5)
d. {(1,1), (2,2), (4,4), (5,5)}
134.If AXB = {1,2), (2,5), (2,3) then A is
a. {1,3}
b. {3,5}
c. {1,2}
d. {2,3}
135.If f is a rule from the set A to the set B, then
a. A is called the domain of f
b. B is called the co-domain of A
c. A is called the pre-image of B
d. B is called the range of f
126 | Cambridge institute/ Mathematics
136.The function f: AB is onto if
a. f (A) is equal to B
b. f (A) is a proper subset of B
c. A is equal to B
d. A is a subset of B
137.The function f: R R defined by f(x) = x2 is
a. one-to-one
b. into
c. surjective
d. bijective
138.A universal set is
a. The superset of every set under consideration.
b. The subset of every set under consideration.
c. The set of all real numbers.
d. None of the above
139.Which is a null set?
a. {x:x = x}
b. {x:x x}
c. x: x = x2}
d. {x:x x2)
140.Which is true?
a. AB = {x:x A and x A and x B}
b. AB = {x:xA and xB}
c. AB = {x:xA and xA)
d. AB = {x:xA and xB}
141.Set A and B have 3 and 6 elements respectively. What can be the maximum
number of elements in A B?
a. 3
b. 6
c. 9
d. 18
142.If A and B are sets with n(A) = 8, n(B) = 5, n(AB) = 3,then minimum n (AB) is
a. 13
b. 11
c. 10
d. 8
143.If A = [1,2], B = [2,3] then A-B equals to
a. [1,2]
b. [2,1]
c. [1,3]
d. [2,3]
144.If a,b,c, are in A.P. as well as in G.P. then
a. b2>ac
b. b2<ac
c. b2=ac
d. none
x y
y 2x
145. If
z x 3 2
then y =
=
x 8 1
a. 2
b. 3
c. 4
d. 5
146. P is a matrix of order 23 & Q is of order 32 then PQ is of order,
a. 23
b. 22
c. 32
d. 33
147.The inverse of a diagonal matrix is
a. diagonal matrix
b. scalar matrix
c. unit matrix
d. null matrix
st
148.The 1 term of G.P is 16 and the sum up to infinity 32, then the common ratio is
1
1
a. 2
b. 2
c. 3
d. 1
Cambridge institute/ Mathematics | 127
149.The fourth term of a G.P. is 2, then the product of 1st 7 terms is
a. 25
b. 26
c. 27
d. 210
150.In 1970 the populations of town A and town B were the same. From 1970 to 1980.
however, the population of town A increased by 60% while the population of
town B decreased by 60% while the population of town B decreased by 60%. In
1980, the population of town was what percent of the population of town A?
a. 25%
b. 36%
c. 40%
d. 60% e. 120%
151.At Hary's Discount Hardware everything is sold for 20% less than the price
marked. If Harry buys tools kits for $80, what price should he mark them if he
wants to make a 20% profit on his cost?
a. $96
b. $100
c. $112
d. $120
e. $125
152.What is 10% of 20% of 30%?
a. 0.006%
b. 0.6%
c. 6%
d. 60%
e. 6000%
153.On a test consisting of 80 questions, Marie answered 75% of the first 60 questions
correctly. What percent of the other 20 questions did she need to answer
correctly for her grade on the entire exam to be 80%.
a. 85%
b. 87.5%
c. 90%
d. 95%
e. 100%
154.Brain gave 20% of his baseball cards to Scott and 15% to Adam. If he still had 520
cards, how many did he have originally?
a. 900
b. 750
c. 800
d. 450
155.After Michael gave 110 baseball cards to Sally and 75 to Heidi, he still had 315
left. What percent of his cards did Michael give away?
a. 47%
b. 37%
c. 57%
d. 67%
156.In January, the value of a stock increased by 25%; and in February, it decreased
by 20%. How did the value of the stock at the end of February compare with its
value at the beginning of January?
a. It was less
b. It was the same
c. It was 5% greater.
d. It was more than 5% greater
e.It depends on the value of the stock.
157.Charlie bought a $60 radio on sale at 5% off. How much did he pay, including 5%
sales tax?
a. $54.14
b. $57.00
c. $57.75
d. $59.85
e. $60.00
158.If a is a positive number, 400% of a is what percent of 400a?
a. 0.01
b. 0.1
c. 1
d. 10
e.100
128 | Cambridge institute/ Mathematics
159.At Harry's Discount Hardware everything is sold for 20% less than the price
marked. If Harry buys tool kits for $80, what price should he mark them if he
wants to make a 20% profit on his cost?
a. $96
b. $100
c. $112
d. $120
e.$125
4
7
160.If 7 of the 350 sophomores at Adams High School are girls, and 8 of the girls play
on a team, how many sophomore girls do not play on a team?
a. 150
b. 200
c. 250
d. 300
e.350
161.Brain gave 20% of his baseball cards to Scott and 15% to Adam. If he still had 520
cards, how many did he have originally?
a. 600
b. 700
c. 800
d. 900
e.1000
162.One Monday, a store owner received a shipment of books. On Tuesday, she sold
2
half of them; on Wednesday, after two more sold, she had exactly 5 of the books
left. How many books were in the shipment?
a. 10
b. 20
c. 30
d. 40
e.50
163.A competition offers a total of $250,000 in prize money to be shared by the top
three contestants. If the money is to be divided among them in the ratio of 1:3:6
what is the value of the largest prize?
a. $25,000
b. $75,000
c. $100,000
d. $ 125,000
e.$ 150,000
164.If the sum of four consecutive odd integers is S, then, in terms of s, what is the
greatest of these integers?
s – 12
s–6
s+6
s + 12
a. 4
b. 4
c. 4
d. 4
s + 16
e. 4
165.From 1994 to 1995 the number of boys in the school chess club decreased by
20%, and the number of girls in the club increased by 20%. The ratio of girls to
boys in the club in 1995 was how many times the ratio of girls to boys in the club
in 1994?
2
4
5
3
a. 3
b. 5
c. 1
d. 4
e.2
Cambridge institute/ Mathematics | 129
166.From 1980 to 1990, Michael's weight increased by 25%. If his weight was W
kilogram in 1990.What was it in 1980 ?
a. 1.75W
b. 1.25 W
c. 1.20W
d. 0.80 W
e.0.75
167.The average of 10 numbers is -10. If the sum of six of them is 100, what is the
average of the other four?
a. -100
b. -50
c. 0
d. 50 e. 100
168.What is the largest prime factor of 255?
a. 5
b. 15
c. 17
d. 51 e.255
169.If 120% of a is equal to 80% of b, which of the following is equal to a+b?
a. 1.5a
b. 2a
c. 2.5a
d. 3a e.5a
170.A region inside a semicircle of radius r is shaded. What is it's the area?
1
1
1
2
a. 4 2
b. 3 2
c. 2 2
d. 3 2
3
e.4 2
171.If 5(3x-7)=20, what is 3x-8?
a. 1
b. 2
c. 2.5
d. 3 e.5
172.A French class has 12 boys and 18 girls. What fraction of the class are boys?
2
3
2
3
3
a. 5
b. 5
c. 3
d. 4 e.2
173.Father is 5 times the son. In 5 years time he will be thrice as much as his son. Find
the son age
a. 6
b. 5
c. 8
d. 10 e.None
174.When an article is sold for Rs. 2,000 there is a loss of 10%. To earn a gain of 10%
the article must be sold for
a. Rs. 2444.44
b. Rs. 2420
c. Rs. 300
d. Rs. 60
e.None
175.The simple interest on Rs. 600 at 6% for six month is
a. Rs. 218
b. Rs. 21.80
c. Rs. 35
d. Rs.18
e.None
176.In sum of Rs. 5000 became Rs. 6000 in 3 years at a certain rate of compound
interest, what will the sum be in 6 years
a. Rs. 7200
b. Rs.1100
c. Rs. 7000
d. Rs. 6600
e.None
130 | Cambridge institute/ Mathematics
177.If the sum of five consecutive odd integers is 735. What is the largest of these
integers?
a. 150
b. 155
c. 145
d. 151
e.141
178.What is the area of a rectangle whose length is twice its width and whose
perimeter is equal to that of a square whose area is 1?
a. 1
b. 6
c. 2/3
d. 4/3
e.8/9
179.Ram and Shyam start moving in opposite direction when they are 9 km. far from
each other, the distance traveled by Ram exceeds twice the distance traveled by
Shyam by 3. Find the distance traveled by Shyam
a. 7km
b. 2km
c. 6km
d. 4km
e.None
180. A boy is 3 years older then his sister. Two years ago the sum of their ages was
19. How old is the boy now?
a. 13 years
b. 12 years
c. 11years
d. 10 years
e.None
181.The ratio of two nos. is 10 and their difference is 18.what is the value of the
smaller no.?
a. 2
b. 5
c. 10
d. 21 e.27
182.In a class of 200 students, 40% are girls.25% of the boys and 10% of the girls
signed up for a tour to Washington DC. What % of the class signed up for the tour?
a. 19%
b. 23%
c. 25%
d. 27%
e.35%
183.The age of B is half the sum of the ages of A and C. If B is 2 years younger than A
and C is 32 years old, then the age of B must be
a. 28
b. 30
c. 32
d. 34 e.36
184.Salary was first increased by 10% and then decreased by 10%. What is the total
percentage change in salary?
a. 2.2%
b. 1.5%
c. 1%
d. 3% e.3.3%
185.If the side of the square increases by 40%, then the area of the square increases
bya. 50%
b. 80%
c. 96%
d. 160% e.None
n
186.What is the value of the largest integer’s n such that 112/2 is an integer?
a. 1
b. 2
c. 3
d. 4 e.5
187.The average of four members is 20.if one of the nos. is removed, the average of
the remaining nos. is 15. What no. was removed?
a. 10
b. 15
c. 30
d. 35 e.45
188.Suppose x is divisible by 8 but not by 3. Then which of the following can’t be an
integer?
a. x/2
b. x/4
c. x/6
d. x/8 e.x
Cambridge institute/ Mathematics | 131
189.If x is an even positive integer then other consecutive even integer is
a. (x+2)
b. (x+1)
c. x2
d. 2x e.None
190.After paying an income tax of 5%, a man has Rs. 7600 left. What is his income?
a. Rs.800
b. Rs.8000
c. Rs.4000
d. Rs.16000
e.None
191.In a two digit number, the unit’s digit is twice the ten’s digit. If the digits are
reversed, the new number is 27 more than the original number. Find the number.
a. 63
b. 18
c. 36
d. 72 e.None
192.If length of sides of cuboids is reduced to half, its surface area becomes
a. ¼
b. 1/3
c. ½
d. double
e.None
193.If the radius of a sphere is doubled, its volume becomes …….. the original volume.
a. 16 times
b. 4 times
c. 8 times
d. double
e.None
194.The L.C.M. and H.C.F. of the two numbers are 840 and 14 respectively and if one
of the numbers is 42 then the other number is
a. 84
b. 280
c. 868
d. 42
e.None
195.If 10 cows eat as much as 6 oxen, how many oxen will eat as much as 15 cows?
a. 10
b. 6
c. 9
d. 15
196.If the price of petrol oil increases by 25% by how much percent should a
consumer reduce his consumption of petrol to maintain his previous
expenditure?
a. 0%
b. 20%
c. 25%
d. 1%
197.Shyam sold two radios for Rs. 960 each. If he gained 20% from one radio and lost
20% from the other, then loss percent on his total outlay is
a. 0%
b. 1%
c. 3%
d. 4%
198.In what time will a sum of money double itself at 3% per annum simple interest?
a. 10 yrs
b. 33.33yrs
c. 40 yrs
d. none of these
199.The cost of using 2 bulbs of 45 watt each for 4 hrs. daily for a month of 30 days at
an average rate of Rs. 4.50 per unit is
a. Rs. 40.10
b. 42.25
c. 45.50
d. 48.60
200.The price of an article is Rs. 6,000.00 after 20% VAT then the price of an article
before the VAT is
a. Rs. 1000
b. Rs. 1100
c. 1200
d. 1500
132 | Cambridge institute/ Mathematics
201.A bus moving with a velocity of 60 km/hr covers 420 km in 7 hours. How long
does it take to cover 960 km with a velocity of 40 km/hr ?
a. 24 hrs
b. 16 hrs
c. 12 hrs
d. 6 hrs
202.If 40 litres of milk and water mixture have ratio 3 : 1 how much milk should be
added in the mixture such that the ratio becomes 4 : 1.
a. 8 ltr
b. 10 ltr
c. 12 ltr
d. 15 ltr
203.The ratios of salt and water in the salt water solution taken in 3 beakers are 1 : 2,
3 : 5 and 4:9. If all three solutions are poured into a single vessel, then the ratio of
salt and water in the vessel is
a. 3 :5
b. 1 : 2
c. 4 :9
d. 3 : 5
204.The length of a rectangle is increased by 60%. By what percent would the width
have to be decreased to maintain the same area?
a. 37.5%
b. 60%
c. 75%
d. 120%
205.If P is the length of the median of an equilateral triangle, then area is
a. p2
b. p23
c. p23
d. p2
206.The radius of the wheel of a vehicle is 70cm. The wheel makes 10 revolutions in
5 seconds, then the speed of the wheel is
a. 32.72 km/hr
b. 36.25 km/hr
c. 31.68 km/hr d. 29.46 km/hr
207.An ant moves 4.4 cm per second. How long will it take to go round a circular dish
of radius 21cm?
a. 30 sec
b. 45 sec
c. 1 minute
d. 3/2 minute
208.If a is increased by 10% and b is decreased by 10% and b is decreased by 10%,
the resulting numbers will be equal. What is the ratio of a to b?
9
10
9
11
a. 10
b. 9
c. 11
d. 9
Cambridge institute/ Mathematics | 133
USEFUL FORMULA OF MATHEMATICS
Trigonometry measurements of angles
1.
right angle = 90 degree = 900 =100 grades = 100g
2. 10 =60 minutes = 60'
3. 1' = 60 seconds = 60"
4. 1g= 100 minutes = 100
5. 1800= 200g = radians= 2 right angles
6. The circular measure of an angle subtended at the centre of the circle by an arc of
length l is equal to the ratio of the length / to the radius of the circle i.e. =
7. Each interior angle of a regular polygon of n sides is equal to
2n –4
n rt.angles
TRIGONOMETRICAL RATIOS (CIRCULAR FUNCTION)
1. Sin2x + Cos2x = 1
2. Sec2x = 1 + tan2x
3. Cosec2x = 1 + Cot2x
4. |sinx| and |Cosx| 1[ i.e Sin2x 1 and Cos2x 1]
5. |Secx| 1and |Cosecx|1 [Sec2x and Cosec2x 1]
6. Sin(A +B +C) = Cos A Cos B CosC { Tan A + Tan B + Tan C-TanA.TanB.TanC}
7. Cos(A+B+C)=CosA.CosB.CosC{1–TanA.TanB–TanB.TanC–TanC.TanA)
PROPERTIES OF TRIANGLE
1.
2
2
SinA = bc s(s – a)(s – b)(s – c) = bc
2.
2
2
SinB = bc s(s – a)(s – b)(s – c) = ac
134 | Cambridge institute/ Mathematics
1
3.
2
2
SinC = bc s(s – a)(s – b)(s – c) = ab
4.
1
1
1
= 2 ab SinC = 2 bc SinA = 2 ca SinB
5.
abc
= 4R
6.
= s(s – a)(s – b)(s – c) [HEROE’S FORMULAE]
Relation between sides & angle of a [sine formulae In any triangle ABC]
7.
a
b
c
abc
SinA = SinB = SinC = 2R = 2
8.
i. CosA =
b2 + c2 – a2
[Cosine formula]
2bc
ii. Cos B=
a 2 + b2 – c 2
2ac
iii. Cos C=
9.
a 2 + b2 – c 2
2ab
i. a = b Cos C + c Cos B [projection formulae]
ii. b = c CosA + a Cos C
iii. c = a Cos B + b Cos A
A
10 i. Sin 2 =
(s – b)(s – c)
B
ii. Sin 2 =
bc
(s – c)(s – a)
ac
C
iii. sin2 =
(s – a)(s – b)
A
iv. Cos 2 =
ab
B
v. Cos 2 =
s(s – b)
ac
C
vi. Cos2 =
s(s – a)
bc
s(s – c)
ab
A
11. i. Tan 2 =
(s – b)(s – c) (s – b)(s – c)
=
= s(s – a)
s(s – a)
B
ii. Tan 2 =
(s – a)(s – c) (s – c)(s – a)
= s(s – b)
s(s – b) =
C
iii. Tan 2 =
(s – a)(s – b) (s – a)(s – b)
=
= s(s – c)
s(s – c)
Cambridge institute/ Mathematics | 135
IF A + B + C =
1. Sin2A+Sin2B+Sin2C=4SinA.SinB.Sinc
A
B
C
2. Sin A + Sin B +Sin C = 4Cos 2 .Cos2 . Cos2
A
B
C
3. CosA + Cos B + Cos C = 1 + 4Sin 2 .Sin2 .Sin2
4. Sin2A+Sin2B+Sin2c = 2+2CosA.CosB.CosC
5. Cos2A+Cos2B+Cos2C = 1-2CosA.CosB.Cosc
6.
A
B
C
A
B
C
Sin2 2 +Sin22 +Sin22 = 1 + 2Sin2 .Sin2 .Sin2
A
B
C
A
B
C
7. Cos2 2 +Cos22 +Cos22 = 1 + 2Cos 2 .Cos2 .Cos2
8. Tan A + Tan B +Tan C =Tan A Tan B Tan C
A
B
B
C
C
A
9. Tan2 ,Tan2 +Tan2 , Tan2 + Tan2 ,Tan 2 = 1
CIRCUM RADIUS IN RADIUS FORMULAE & EX-RADIUS
a
b
c
abc
1. R= 2SinA = 2SinB = 2SinC =
)
4
A
B
C
2. r = s(s – a) Tan 2 = s(s – b)Tan2 = s(s – c) Tan2
A
B
C
3. r = 4R Sin 2 .Sin2 .Sin2
4. r1 = Radius of ex-circle opposite to
A
B
C
A
<A= s–a = 4R Sin 2 .Cos2 .Cos2 = sTan 2
5. r2 = Radius of ex-circle opposite to
A
B
C
B
<B = s–b = 4R Cos2 .Sin2 .Sin2 = sTan2
136 | Cambridge institute/ Mathematics
T-RATIOS OF COMPOUND ANGLES
Reduction of Formulae
1.
Sin 2 + = Cos
9.
2.
Cos 2 ± = +sin
10. Cos2 ± = Sin
3.
Sin( ± ) = + Sin
11.
4.
3
Sin 2 ± = –Cos
3
12. Cos 2 ± = ± Sin
5.
3
Tan 2 ± = Cot
13. Sin(–) = –Sin
6.
Cos (–) = Cos
14. Tan(–) = Tan
7.
Sin (2 – ) = –Sin
15. Cos (2 – ) = Cos
8.
Tan (2 – )= -Tan
Tan 2 ± = + Cot
Tan ( ± ) = + Tan
Cambridge institute/ Mathematics | 137
T-Ratio for sum and difference
1.
Sin (A + B) = Sin A. Cos B + Cos A. Sin B
2.
Sin (A - B) = SinA. Cos B –Cos A.Sin B
3.
Cos (A ± B) = Cos A . Cos B + Sin A. Sin B
4.
Sin (A + B). .Sin (A - B) = Sin2A - Sin2B = Cos2B - Cos2A.
5.
Cos (A + B) .Cos (A - B) = Cos2A - Sin2B = Cos2B -Sin2A
6.
TanA ± TanB
Tan(A±B)=1± TanA.TanB [here A n + /2][B n + /2,A±Bk + /2]
7.
1 + Tan
Tan 4 + =
1 – Tan
8.
1 – Tan
Tan 4 – =
1 + Tan
9.
Cot (A B) =
CotA CotB+ 1
CotB±cotA
10. Sin (A + B + C) = Sin A. Cos B .Cos C + Cos A. Sin B. Cos C + Cos A. Cos B. Sin C SinA.SinB.SinC
11. Cos (A + B + C) = Cos A. Cos B. Cos C - CosA. Sin B.Sin C- Sin A. Sin B. Cos C.SinA.CosB.Sin C
Tana + TanB + TanC – TanA.TanB.TanC
12. Tan (A + B + C) = 1 – TanA.TanB –TanBTanC –TanC.TanA
Sum and difference into product
a)
C +D
C–D
Sin C + Sin D= 2Sin 2 .Cos 2
b)
C +D
C–D
Sin C – Sin D = 2 Cos 2 .Cos 2
c)
C +D
C–D
Cos C + Cos D = 2Cos 2 .Cos 2
d)
C +D
C–D
Cos C –Cos D = 2Sin 2 .Sin 2
138 | Cambridge institute/ Mathematics
e)
Sin(A±B)
TanATanB=CosA.CosB
[here A n + /2;Bm]
f)
Sin(B±A)
CotA±CotB=SinA.SinB
[here A n,/Bm + /2]
g)
CosA±SinA= 2sin 4 A = 2cos 4 A
h)
1
Tan A Cot A = (SinA.CosB)
Product into sum or difference
i)
2SinA.CosB = Sin(A+B) + Sin(A-B)
j)
2CosA . SinB = Sin (A + B) – Sin ( A – B)
k)
2 SinA . SinB = Cos (A-B) – Cos (A +B)
l)
2CosA . CosB = Cos (A+B) +Cos (A-B)
m)
Sin (A+B) . Sin (A-B) = Sin2A- Sin2B = Cos2B- Cos2A
n)
Cos (A+B). Cos(A-B) = Cos2A-Sin2B =Cos2B-Sin2A
T-RATIO OF MULITPLE ANGLES
2Tan
1 + Tan2
1.
Sin2 =2 Sin.Cos=
2.
Cos2= Cos2 – Sin2 = 1-Sin2 = 2Cos2 -1=
3.
Tan 2=
4.
From (2) we get
1 – Tan2
1 + Tan2
2Tan
[ (2n + 1)4 ]
1 + Tan2
a)
1+ Cos2
Cos2 =
or 1 + Cos= 2Cos2/2
2
b)
Sin2 =
1 Cos2
or 1 - Cos= 2Sin2/2
2
Cambridge institute/ Mathematics | 139
5.
Tan2 =
Cos2
1 + Cos2
T RATIO OF 3 IN TERMS OF
a)
Sin3= 3Sin-4Sin3
b)
Cos 3= 4Cos3 - 3Cos
c)
Tan 3 =
3Tan – Tan3
[here n + /6]
1 – 3Tan2
T-RATIO OS SUB MULTIPLE ANGLES
1. Sin = 2Sin2 .Cos2 =
2Tan 2
1 + Tan2 2
2. Cos = cos2
cos2 2 – sin2 2 = 1 – 2sin2 2 = 2cos2 2 – 1 =
2Tan 2
3. Tan=
1 + Tan2 2
4. 2Cos2 = 1+Cos
5. 2Sin22 1-Cos
1 – Cos
6. Tan22 =
1 + Cos
7. 2Sin2 = ± 1 + Sin ± 1 – Sin
8. 2Cos2 = ± 1 + Sin + 1 – Sin
T-EQUATIONS & FORMULAE
1. If Sin = 0, Then = n
140 | Cambridge institute/ Mathematics
1 – Tan2 2
1 + Tan2 2
2. If Tan = 0, then = n
1
3. If Cos = 0, then = n + 2
1
4. If Cot =0, then = n + 2
n
5. If Sin =±1,then = (4n ± 1) 2
6. If Cos = 1, then = 2n
7. If Cos = -1, Then = (2n+1)
n
8. If Sin2 = 1, Then = n+2
9. If Cos2 = 1, Then = n
10. If Sin= Sin, then = n+(-1)n
11. If Cos= Cos, then = 2n
12. If Tan= Tan, then= n+
13. If Sin2= Sin2, then= n ±
14. If Cos2= Cos2, then= n ±
15. If Tan2= Tan2, then= n ±
QUADRATIC EQUATION
8. Standard form of quadratic equation is ax2+bx+c = 0 [here a, b, c are numbers a 0,
and x is variable ]a, b, c, are the co-efficient of quadratic equation, a and b are the
coefficient of x2 and x respectively and c is the constant term.
9. If and are the roots of ax2+bx+c = 0; then
=
–b + b2 – 4ac
–b b2 – 4ac
and
=
2a
2a
10. A quadratic equation has two roots.
Let f(x) = ax2 +bx +c =0 be satisfied by and then
f() = f ()=0 and are called the roots of the equation and zeroes
to
the
Cambridge institute/ Mathematics | 141
quadratic expression f(x) = ax2+bx+c = 0
b
c
11. + =a ; = a
12. Equation whose roots are and is x2 – ( + )x + , = 0
13. a2+b+c=0 and a 2 + b +c = 0 when and are roots of the equation ax2+bx+c =
0
14. Complex imaginary and complex irrational roots occur in conjugate pairs.
a) if + i is one root of ax2+bx+c = 0, then – i will be other roots in which ,
R and i = –1 and – i is called complex conjugate of + i
b) If +
is one root of ax2 bx + c = 0, [here a ,b ,c are rational , a 0 then –
will be the other root in which is national
15. Natural of roots
a) If b2 - 4ac < 0, then the roots are imaginary and unequal
b) If b2 - 4ac = 0, then the roots real and equal
c) If b2 - 4ac > 0, then the roots are real and unequal
d) If b2 - 4ac, is a perfect square of a rational number then the roots are national
and unequal otherwise conjugate irrational. N.B: b2 – 4ac are called discriminate
of the quadratic equation and is denoted by D.
16. An equation of nth degree aoxn+a1xn-1+a2xn-2+.......an = 0
has n roots Let the roots be 1, 2, 3..........n then
a1
a1
a) 1 =1 + 2 + 3..........n = – a = (–1) a
0
a2
b) 1,2 = (–1)2 a
0
a3
c) 1,2, 3 = (–1)3 a
0
an
d) 1,2............ n = (–1)n a
0
142 | Cambridge institute/ Mathematics
0
Circle
1. Standard equation of circle
i.
Center = (0,0) and radius = a. equation is x2+y2=a2
ii.
Center = (h, k) and radius = a. equation , is
(x-h)2+(y-k)2= a2
2. General Equation: The equation x2 + y2+ 2gx + 2fy + c = 0 always representation a
circle for all values of g,f and c . Center = (-g, -f) and radius = g2 + f2 –c
N.B. i.
ii.
If g2+f2>c, radius is real
If g2+f2=c, circle becomes a point circle
iii. If g2+f2 <c, radius is imaginary
3. ax2+2hxy+by2+2gx+2fy+c=0 will represents a circle when a =b and h=0
4. Equation of circle
(x - x1) (x - x2) + (y - y1) (y - y2) = 0; (x1, y1)A
5. i.
ii.
B (x2, y2)
If s x2+y2-a2 then s1 x12+y12-a2 and T xx1+yy1-a2
If s
x2+y2+2gx+2fy+c=0 then s1 x21+y12+2gx1+2fy1+c and T
xx1+yy1+g(x+x1)+f(y+y1)+c
6. The eqn of the tangent as (x1, y1) on the circle s = 0 is T = 0
7.
i.
The st line y= mx+c is tangent to the circle x2+y2=a2
when c = ± a 1 + m2
ii. The eqn of any tangent to the circle is y= mx± a 1 + m2
8. The length of the tangent drawn from the point (x1,y1) to the circle s= 0 is S1
Algebra Arithmetic Progression (A.P.)
1. A sequence of the form a, a+d, a+2d, ........... a + (n-1) d is called an A.P. [here a = 1st
term, d= common difference
2. The nth term tn = a+(n-1)d
3. d=2nd term –1st term =3rd term –2nd term and so on.
Cambridge institute/ Mathematics | 143
n
n
4. Sum of an A.P. Sn = 2 [2a+(n-1)d]= 2 (a + 1) [here l = last term of nth term)
5. MEAN:
a+b
(i) Single Arithmetic mean (AM)betn a & b A = 2
a+b
a) Sum of Arithmetic mean (A.M.) betn a&b A1+A2+A3+......................An = 2 n
a+c
b) a,b,c are in A.P, then b = 2 A.M. betn a and c
c) If a,b.c. are in A.P, then A.M. of a and d= A.M. of b and c i.e. a+d=b+c
6. Convenience when sum is given three numbers in A.P. should be taken as a, a – d, a,
a+d, four number in A.P. as a – 3d, a-d, a+d, a+3d; five numbers in A.P. as a –2d, a-d,
a+d, a+2d
Geometric Progression (G.P.)
1. A sequence of the form a, ar, ar2 .................. arn-1 is called a G.P. [here, a= 1st term r =
common ratio]
2nd term 3rdterm
2. r = 1nd term = 2ndterm and so on
3. The nth tn = arn-1
a(1 – rn)
4. Sum of nth terms in G.P. Sn = (1 – r)
a(1 – rn)
[here r<1or (r – 1) [here r > 1]
5.
a
6. Sum of infinity of a G.P. ( s ) =1 – r when |r|<1
7. MEAN: (i) Single geometric mean (G.M.) betn a& b G = ab [here a & b >0]
i.
Product of n G.Ms between a & b = ( ab)n
144 | Cambridge institute/ Mathematics
a
8. Convenience when product id given three numbers in G.P. should be assumed as r ,a,
a a
a a
ar four numbers in G.P. as r3 , r ar, ar3 five numbers as r2 , r ,a, ar, ar2........
Harmonic progression (H. P.)
1 1
1
1
1. Sequence of the forma ,a + d ,a + 2d ... a + (n – 1)d is called an H.P. whereas a, a+d,
a+2d, ……….a+ (n-1) d is called the corresponding A.P. of the H.P. Therefore 1st term
1
1
of the H.P. t1 =a and nth tn =a + (n – 1)d
2. MEAN:
i.
Single harmonic mean : (H..M.) betn a & b
2ab
H = a + b [When a - b ]
2ab
a a–b
ii. If a,b,c are H.P. then b =a + c or c = a – b
2ab 2ab
iii. If a,b,c,d are in H.P. then H.M. of a & d = H.M. of a&d = H.M. b & C i.e. a + b = a + c
ad a + b
bc = a + b
n
3. H.M. n non –zero number a1,a2,a3----------------- an H= 1 1 1
1
+
+
+...
a1 a2 a3
ana1
1 1
a+b
4. Sum of reciprocals of nth H.Ms between a & b = 2 × n
Note at a Glance
1. If a, b> 0 , then their A.M.> G.M. >H.M. [A>G>H]
2. A, G, H form a G.P. i.e. G2= AH
3. i.
ab, (a+d)br, (a+2d) br2 ....... [a+(n-1)d]brn-1 is called arithmetic- geometric
sequence Every term of which constitutes of two factor, the first being a term (Say
rth term ) of an A.P. 0
while the 2nd , rth term of a G.P.
Cambridge institute/ Mathematics | 145
ii.
Sum to n terms of such a series,
ab dbr(1 – rn–1) [a + (n – 1)d]drn
Sn =1 – r + (1 – r)2
–
1–r
ab
abr
[if|r|<1 then S =1 – r + (1 – r)2
lim rn = 0
lim nrn = 0
x
a)
n
n i
x
1 2 3 ........n
i 1
b)
n
n
3
i 3 13 2 3 3 .....n 3
i 1
n ( n 1)
2
n 2 ( n 1) 2
4
n
2
[n(n + 1)(n + 2)]
c) 1.2+2.3+3.4....+n terms =
2
d)
2n =2 n =n(n+1)= sum of 1st n even natural no.
e)
( 2n 1) ( 2i 1) 2 i n n
f)
n
n
i 1
i 1
n
2 2n 2 2
n
2
2 3 ... 2 n
i
g)
2
sum of the 1st odd natural numbers
2( 2 n 1)
2 1
=1+1+1+... to n terms = n
h) 2= 2+2+2+..... to n terms = 2n
146 | Cambridge institute/ Mathematics
