Chapter
Page
1.
Real Numbers
5
2.
Polynomials
11
3.
Pair of Linear Equations in two Variables
17
4.
Similar Triangles
26
5.
Trigonometry
39
6.
Statistics
49
Sample Paper
[Class-X – Maths]
60
4
1.
Euclid’s division lemma :
For given positive integers ‘a’ and ‘b’ there exist unique whole numbers ‘q’
and ‘r’ satisfying the relation a = bq + r, 0 r < b.
2.
Euclid’s division algorithms :
HCF of any two positive integers a and b. With a > b is obtained as follows:
Step 1 : Apply Euclid’s division lemma to a and b to find q and r such that
a = bq + r , 0 r < b.
Step 2 : If r = 0, HCF (a, b) = b
Step 3 : if r 0, apply Euclid’s lemma to b and r.
3.
The Fundamental Theorem of Arithmetic :
Every composite number can be expressed (factorized) as a product of
primes and this factorization is unique, apart from the order in which the
prime factors occur.
4.
5.
6.
p
Let x  , q 0 to be a rational number, such that the prime
q
factorization of ‘q’ is of the form 2m5n, where m, n are non-negative integers.
Then x has a decimal expansion which is terminating.
p
Let x  , q 0 be a rational number, such that the prime factorization
q
of q is not of the form 2m5n, where m, n are non-negative integers. Then
x has a decimal expansion which is non-terminating repeating.
is irrational, where p is a prime. A number is called irrational if it cannot
p
be written in the form q where p and q are integers and q 0.
5
[Class-X – Maths]
1.
2.
5 × 11 × 13 + 7 is a
(a)
prime number
(b)
composite number
(c)
odd number
(d)
none
Which of these numbers always ends with the digit 6.
(a)
4n
(b)
2n
(c)
6n
(d)
8n
where n is a natural number.
3.
4.
5.
6.
7.
For a, b (a b) positive rational numbers
____
(a)
Rational number
(c)

a

a
b

a
b

(b)
irrational number
(d)
0
is a
2
b

If p is a positive rational number which is not a perfect square then p is
(a)
integer
(b)
rational number
(c)
irrational number
(d)
none of the above.
All decimal numbers are–
(a)
rational numbers
(b)
irrational numbers
(c)
real numbers
(d)
integers
In Euclid Division Lemma, when a = bq + r, where a, b are positive
integers which one is correct.
(a)
0 < r b
(b)
0 r < b
(c)
0<r<b
(d)
0 r b
Which of the following numbers is irrational number
(a)
3.131131113...
(b)
4.46363636...
(c)
2.35
(d)
b and c both
[Class-X – Maths]
6
21
8.
The decimal expansion of the rational number
7 2

5
4
will terminate
after ___ decimal places.
9.
10.
11.
12.
13.
14.
(a)
8
(b)
4
(c)
5
(d)
never
HCF is always
(a)
multiple of L.C.M.
(b)
Factor of L.C.M.
(c)
divisible by L.C.M.
(d)
a and c both
The product of two consecutive natural numbers is always.
(a)
an even number
(b)
an odd number
(c)
a prime number
(d)
none of these
Which of the following is an irrational number between 0 and 1
(a)
0.11011011...
(b)
0.90990999...
(c)
1.010110111...
(d)
0.3030303...
pn = (a × 5)n. For pn to end with the digit zero a = __ for natural no. n
(a)
any natural number
(b)
even number
(c)
odd number
(d)
none.
A terminating decimal when expressed in fractional form always has
denominator in the form of —
(a)
2m3n, m, n 0
(b)
3m5n, m, n 0
(c)
5n 7m, m, n 0
(d)
2m5n, m, n 0
33
– 55is
44
(a)
An irrational number
(b)
A whole number
(c)
A natural number
(d)
A rational number
7
[Class-X – Maths]
15.
16.
If LCM (x, y) = 150, xy = 1800, then HCF (x, y) =
(a)
120
(b)
90
(c)
12
(d)
0
Which of the following rational numbers have terminating decimal
expansion?
(a)
91
2100
(c)
5 2 2 3 73
(b)
64
455
(d)
29
73
343
18 
17.
Solve
50. What type of number is it, rational or irrational.
18.
Find the H.C.F. of the smallest composite number and the smallest prime
number.
19.
If a = 4q + r then what are the conditions for a and q. What are the values
that r can take?
20.
What is the smallest number by which 5 3 be multiplied to make it
a rational number? Also find the number so obtained.
21.
What is the digit at unit’s place of 9n?
22.
Find one rational and one irrational no. between
23.
State Euclid’s Division Lemma and hence find HCF of 16 and 28.
24.
State fundamental theorem of Arithmetic and hence find the unique
factorization of 120.
25.
Prove that
26.
Prove that 5 
1
5
2
is irrational number..
2
.3 is irrational number.
7
[Class-X – Maths]
8
3 and
5.
27.
Prove that
2
28.
.7 is not rational
number.
Find HCF and LCM of 56 and 112 by prime factorisation method.
29.
Why 17 + 11 × 13 × 17 × 19 is a composite number? Explain.
30.
Check whether 5 × 6 × 2 × 3 + 3 is a composite number.
31.
Check whether 14n can end with the digit zero for any natural number, n.
32.
If the HCF of 210 and 55 is expressible in the form 210 × 5 + 55y then
find y.
33.
Find HCF of 56, 96 and 324 by Euclid’s algorithm.
34.
Show that the square of any positive integer is either of the form 3m or
3m + 1 for some integer m.
35.
Show that any positive odd integer is of the form 6q + 1, 6q + 5 where q
is some integer.
36.
Two milk containers contains 398 l and 436 l of milk. The milk is to be
transferred to another container with the help of a drum. While transferring
to another container 7l and 11l of milk is left in both the containers
respectively. What will be the maximum capacity of the drum.
37.
Show that
.7 is not a rational
number.
38.
39.
A sweet seller has 420 Kaju burfis and 130 Badam burfis. He wants to
stack them in such a way that each stack has the same number and they
take up the least area of the tray.
i.
What is the number of burfis that can be placed in each stack for
this purpose?
ii.
Which value of the sweet seller is reflected in the question?
An Army contingent of 616 members is to march behind an army band of
32 members in a parade, the two groups are to march in the same number
9
[Class-X – Maths]
of columns. What is the maximum number of columns in which they can
march ? What values are reflected by the members?
40.
In the morning Ankita and Salma walk around a rectangular park. Ankita
takes 18 minutes to cover one round of the park while Salma takes 12
minutes for the same. Suppose they both start at the same point and at
the same time and go in the same direction, find
i.
After how many minutes will they meet again at the starting point?
ii.
What values of Ankita and Salma are reflected in the question?
1.
b
2. c
3.
a
4. c
5.
c
6. b
7.
a
8. b
9.
b
10. a
11.
b
12. b
13.
d
14. d
15.
c
16. c
17.
30, rational
18. 2
19.
r, q whole no. 0 r < 4
20.
21.
1
23. 4
24.
2×2×2×3×5
28. HCF = 56, LCM = 112
30.
Yes
31. No
32.
HCF (210, 55) = 5, 5 = 210 × 5 + 55y y = – 19
33.
4
34. a = 3q + r
35.
a = 6q + r
36. 17 litre
38.
10, Economic value
39.
8, Work in groups, Discipline etc.
40.
36 minutes, Health Awareness etc.
[Class-X – Maths]
10

5

3,2
1.
Polynomials of degrees 1, 2 and 3 are called linear, quadratic and cubic
polynomials respectively.
2.
A quadratic polynomial in x with real coefficient is of the form ax 2 + bx + c,
where a, b, c are real numbers with a 0.
3.
The zeroes of a polynomial p(x) are precisely the x–coordinates of the
points where the graph of y = p(x) intersects the x-axis i.e. x = a is a zero
of polynomial p(x) if p(a) = 0.
4.
A polynomial can have at most the same number of zeros as the degree
of polynomial.
5.
For quadratic polynomial ax2 + bx + c (a 0)
Sum of zeros 
Product of zeros 
6.
b
a
c
.
a
The division algorithm states that given any polynomial p(x) and polynomial
g(x), there are polynomials q(x) and r(x) such that :
p(x) = g(x).q (x) + r(x), g(x) 0
where r(x) = 0 or degree of r(x) < degree of g(x).
1.
A real no. is a zero of the polynomial f(x) if
(a)
(b)
(c)
(d)
11
none
[Class-X – Maths]
2.
3.
4.
5.
6.
7.
8.
9.
The zeroes of a polynomial f(x) are the coordinates of the points where the
graph of y = f(x) intersects
(a)
x-axis
(b)
y-axis
(c)
origin
(d)
(x, y)
If is a zero of f(x) then ____ is one of the factors of f(x)
(a)
(x – 
(b)
(x –
(c)
(x + 
(d)
(2x – 
If (y – a) is factor of f(y) then ___ is a zero of f(y)
(a)
y
(b)
a
(c)
2a
(d)
2y
If 1 is one of the zeroes of the polynomial x2 – x + k, the value of k is:
(a)
0
(b)
–2
(c)
1
(d)
3
Cubic polynomial x = f(y) cuts y-axis at atmost
(a)
one point
(b)
two points
(c)
three points
(d)
four points
Polynomial x2 + 1 has ___ zeroes
(a)
only one real
(b)
no real
(c)
only two real
(d)
one real and the
other non-real.
If 1 is one of the zeroes of the polynomial ax2 – bx + 1 then.
(a)
a–b=0
(b)
a–b–1=0
(c)
a+1=b
(d)
a + b +1 = 0
If degree of polynomial f(x) is ‘n’ then maximum number of zeroes of f(x)
would be –
(a)
n
(b)
2n
(c)
n+1
(d)
n–1
[Class-X – Maths]
12
10.
11.
12.
13.
If 2 is a zero of both the polynomials, 3x2 + ax – 14 and 2x – b then
a – 2b = ___
(a)
–2
(b)
7
(c)
–8
(d)
–7
If zeroes of the polynomial ax2 + bx + c are reciprocal of each other then
(a)
a=c
(b)
a=b
(c)
b=c
(d)
a=–c
The zeroes of the polynomial h(x) = (x – 5) (x2 – x–6) are
(a)
–2, 3, 5
(b)
–2, –3, –5
(c)
2, –3, –5
(d)
2, 3, 5
If are the zeroes of the polynomial x2 + x + 1, then + =
(a)
–1
(b)
0
(c)
1
(d)
2
14.
If and are the zeroes of the polynomial 2x2 – 7x + 3. Find the sum of
the reciprocal of its zeroes.
15.
If are the zeroes of the polynomial p(x) = x2 – a (x + 1) – b such that
16.
17.
If are the zeroes of the polynomial x2 – (k + 6) x + 2 (2k – 1). Find
1
k if .
2
If (x + p) is a factor of the polynomial 2x2 + 2px + 5x + 10 find p.
18.
Find a quadratic polynomial whose zeroes are 5 3 2 and 5 3 2 .
19.
If



1
and – 2 are respectively product and sum of the zeroes of a quadratic
5
polynomial. Find the polynomial.
13
[Class-X – Maths]

20.
If one of the zero of the polynomial
g(x) = (k2 + 4) x2 + 13x + 4k is recoprocal of the other, find k.
21.
If be the zeroes of the quadratic polynomial 2 – 3x – x2 then find the
value of + (1 + 
22.
Form a quadratic polynomial, one of whose zero is
of zeroes is 4.
23.
If sum of the zeroes of kx2 + 3k + 2x is equal to their product. Find k.
24.
If one zero of 4x2 – 9 – 8kx is negative of the other, find k.
25.
Find the quadratic polynomial whose zeroes are 2 and –3. Verify the
relation between the coefficients and the zeroes of the polynomial.
26.
If one zero of the polynomial (a2 + a) x2 + 13x + 6a is reciprocal of the
other, find value (s) of a.
27.
–5 is one of the zeroes of 2x 2 + px – 15. Quadratic polynomial
p(x2 + x) + k has both the zeroes equal to each other. Then find k.
28.
What should be subtracted from the polynomial 2x3 + 5x2 – 14x + 10 so
that the resultant polynomial is a multiple of (2x – 3).
29.
If f(x) = 2x4 – 5x3 + x2 + 3x – 2 is divided by g(x), the quotient is
q(x) = 2x2 – 5x + 3 and r(x) = – 2x + 1 find g(x).
30.
If (x – 2) is one of the factors of x3 – 3x2 – 4x + 12, find the other zeroes.
31.
If and are the zeroes of the polynomial x2 – 5x + k such that – 
= 1, find the value of k.
32.
If are zeroes of quadratic polynomial 2x2 + 5x + k, find the value of
2 
5 and the sum
21
2 – =.
4
33.
Obtain all zeroes of x4 – x3 –7x2 + x + 6 if 3 and 1 are zeroes.
34.
If the two zeroes of the polynomial 2x4 – 2x3 – px2 + qx – 6 are –1 and
2, find p and q.
[Class-X – Maths]
14
35.
If
2 
3
and 2 
3
 are two zeroes of x
4–
4x3 – 8x2 + 36x – 9
find the other two zeroes.
36.
What must be subtracted from 8x4 + 14x3 – 2x2 + 7x – 8 so that the
resulting polynomial is exactly divisible by 4x2 + 3x – 2.
37.
When we add p(x) to 4x4 + 2x3 – 2x2 + x – 1 the resulting polynomial is
divisible by x2 + 2x – 3 find p(x).
38.
Find a and f if (x4 + x3 + 8x2 + ax + f) is a multiple of (x2 + 1).
39.
If the polynomial 6x4 + 8x3 + 17x2 + 21x + 7 is divided by 3x2 + 1 + 4x
then r(x) = (ax + b) find a and b.
40.
If Honesty and Dishonesty are the zeroes of the polynomial ax2 + bx + c,
a 0 and are the reciprocal to each other then
41.
42.
i.
Find a relation between a and c
ii.
What values from the question should be adopted by the people
in their life?
An NGO distributed K number of Books on moral-education to the students.
If K is the zero of polynomial x2 – 100x – 20000 then
i.
How many books were distributed by the NGO?
ii.
Write any three moral values which you should adopt in your life
by reading such books.
If 3 is one of the zero of the polynomial x3 – 12x2 + 47x – 60 and the
remaining two zeroes are the number of plants, planted by the two students
then
i.
Find the total number of plants, planted by both the students.
ii.
What value is reflected in this question?
1.
b
2. a
3.
a
4. b
15
[Class-X – Maths]
5.
a
6. c
7.
b
8. c
9.
a
10. d
11.
a
12. a
13.
a
14.
15.
1
1 17

3
16. k = 7
17.
p=2
18. x2 – 10x + 7
19.
x
21.
k=–5
22. x2 – 4x – 1
23.
2
 3
24. 0
25.
x2 + x – 6
26. 5
27.
p 7, k 
29.
g(x) = x2 – 1
30. –2, 3
31.
k=6
32. k = 2
33.
–2, –1
34. 1, –3
35.
±3
36. 14x – 10
37.
61x – 65
38. r(x) = 0
2
2x 
1
5
20. 2
7
4
28. 7
39.
a 1x f 7 0 

a 1 and f 7

r (x) = x + 2 = ax + b a = 1 and b = 2
40.
a = c, Honesty
41.
(i) 200
(ii) Honesty, Discipline etc.
42.
(i) 9
(ii) Love towards nature, plantation etc.
[Class-X – Maths]
16
1.
The most general form of a pair of linear equations is :
a1x + b1y + c1 = 0
a2x + b2y + c2 = 0
Where a1, a2, b1, b2, c1, c2 are real numbers and a12 + b12 0, a22 + b22 0.
2.
3.
The graph of a pair of linear equations in two variables is represented by
two lines;
(i)
If the lines intersect at a point, the pair of equations is consistent.
The point of intersection gives the unique solution of the equation.
(ii)
If the lines coincide, then there are infinitely many solutions. The
pair of equations is consistent. Each point on the line will be a
solution.
(iii)
If the lines are parallel, the pair of the linear equations has no
solution. The pair of linear equations is inconsistent.
If a pair of linear equations is given by a1x + b1y + c1 = 0 and a2x + b2y
+ c2 = 0
(i)
(ii)
a1b
1 the pair of linear equations is consistent. (Unique
a2b2
solution).
a1bc
1 1 the pair of linear equations is inconsistent
a2b2c2
(No solution).
17
[Class-X – Maths]
(iii)
a1bc
1 1 the pair of linear equations is dependent and
a2b2c2
consistent (infinitely many solutions).
1.
2.
3.
4.
5.
6.
Every linear equation in two variables has ___ solution(s).
(a)
no
(b)
one
(c)
two
(d)
infinitely many
a1bc
1 1 is the condition for
a2b2c2
(a)
intersecting lines
(b)
parallel lines
(c)
coincident lines
(d)
none
For a pair of linear equations in two variables to be consistent and dependent
the pair must have
(a)
no solution
(b)
unique solution
(c)
infinitely many solutions
(d)
none of these
Graph of every linear equation in two variables represent a ___
(a)
point
(b)
straight line
(c)
curve
(d)
triangle
Each point on the graph of pair of two lines is a common solution of the
lines in case of ___
(a)
Infinitely many solutions
(b)
only one solution
(c)
no solution
(d)
none of these
If the system of equations 6x – 2y = 3 and Kx – y = 2 has unique solution
then K is
(a)
K=3
(b)
K=4
(c)
K 3
(d)
K 4
[Class-X – Maths]
18
7.
8.
9.
10.
11.
12.
13.
One of the common solution of ax + by = c and y-axis is _____
(a)
c
0, b 

(b)
b
0, c 

(c)
c
b , 0

(d)
c
0, b 

For x = 2 in 2x – 8y = 12 the value of y will be
(a)
–1
(b)
1
(c)
0
(d)
2
The pair of linear equations is said to be inconsistent if they have
(a)
only one solution
(b)
no solution
(c)
infinitely many solutions.
(d)
both a and c
On representing x = a and y = b graphically we get ____
(a)
parallel lines
(b)
coincident lines
(c)
intersecting lines at (a, b)
(d)
intersecting lines at (b, a)
A motor cyclist is moving along the line x – y = 2 and another motor cyclist
is moving along the line x + y = 2. Tell whether the y
(a)
move parallel
(b)
move coincidently
(c)
collide somewhere
(d)
none
For 2x + 3y = 4, y can be written in terms of x as—
(a)
y
4 2x
3
(b)
y
4 3x
2
(c)
x
4 3y
2
(d)
y
4 2x
3
For what value of p, the pair of linear equations 2x + py = 8 and x + y =
6 has a unique solution x = 10, y = –4.
(a)
–3
(b)
3
(c)
2
(d)
6
19
[Class-X – Maths]
14.
15.
16.
The point of intersection of the lines x – 2y = 6 and y-axis is
(a)
(–3, 0)
(b)
(0, 6)
(c)
(6, 0)
(d)
(0, –3)
Graphically x – 2 = 0 represents a line
(a)
parallel to x-axis at a distance 2 units from x-axis.
(b)
parallel to y-axis at a distance 2 units from it.
(c)
parallel to x-axis at a distance 2 units from y-axis.
(d)
parallel to y-axis at a distance 2 units from x-axis.
If ax + by = c and lx + my = n has unique solution then the relation
between the coefficients will be ____
(a)
am lb
(b)
am = lb
(c)
ab = lm
(d)
ab lm
17.
Form a pair of linear equations for : If twice the son’s age is added to
father’s age, the sum is 70. If twice the father’s age is added to the son’s
age the sum is 95.
18.
Amar gives9000 to some athletes of a school as scholarship everyy
month. Had there been 20 more athletes each would have got 160 less..
Form a pair of linear equations for this.
19.
Write a pair of linear equations which has the unique solution x = –1 and
y = 4. How many such pairs are there?
20.
What is the value of a for which (3, a) lies on 2x – 3y = 5.
21.
Solve :
x + 2y – 8 = 0
22.
2x + 4y = 16
Dinesh is walking along the line joining (1, 4) and (0, 6), Naresh is walking
along the line joining (3, 4,) and (1,0). Represent on graph and find the
point where both of them cross each other.
[Class-X – Maths]
20
23.
Solve the pair of linear equations
x – y = 2 and x + y = 2. Also find p if p = 2x + 3
24.
Check graphically whether the pair of linear equations 3x + 5y = 15,
x – y = 5 is consistent. Also check whether the pair is dependent.
25.
For what value of p the pair of linear equations
(p + 2) x – (2 p + 1)y = 3 (2p – 1)
2x – 3y = 7
has unique solution.
26.
Find the value of K so that the pair of linear equations :
(3 K + 1) x + 3y – 2 = 0
(K2 + 1) x + (k–2)y – 5 = 0 is inconsistent.
27.
Given the linear equation x + 3y = 4, write another linear equation in two
variables such that the geometrical representation of the pair so formed is
(i) intersecting lines (ii) parallel lines (iii) coincident lines.
28.
Solve x – y = 4, x + y = 10 and hence find the value of p when
y = 3 x –p
29.
Determine the values of a and b for which the given system of linear
equations has infinitely many solutions:
2x + 3y = 7
a(x + y) – b(x – y) = 3a + b – 2
30.
The difference of two numbers is 5 and the difference of their reciprocals
1
is
. Find the numbers
10
31.
Solve for x and y :
x 1y 1x 1y 1
3
28;9 3
21
2
[Class-X – Maths]
32.
Solve for x and y
x+y=a+b
ax – by = a2 –b2
33.
Solve for x and y
139x 56 y 641
56x 139 y 724
34.
Solve for x and y
51
2
x yx y
155
2
x yx y
35.
Solve the following system of equations graphically
x + 2y = 5, 2x – 3y = – 4.
Write the coordinates of the point where perpendicular from common
1 
solution meets x-axis. Does the point – , 1lie on any of the lines?
2 
36.
Draw the graph of the following pair of linear equations
x + 3y = 6
and
2x – 3y = 12
Find the ratio of the areas of the two triangles formed by first line, x = 0,
y = 0 & second line x = 0, y = 0.
37.
Solve for x and y
1121

22 2x 3 y 7 3x 2y 
74
2 for 2x + 3y 0 and 3x – 2y 0
2x 3 y 3 x 2 y 
[Class-X – Maths]
22
38.
Solve for p and q
39.
p qp q
2,6, p 0, q 0.
pqpq
On selling a T.V. at 5% gain and a fridge at 10% gain, a shopkeeper gains
Rs. 2000. But if he sells the T.V. at 10% gain & the fridge at 5% loss, he
gains Rs. 1500 on the transaction. Find the actual price of the T.V. and the
fridge.
40.
2
x

3
y
2,
4
x

9
y
1 ; x 0, y 0
41.
If from twice the greater of two numbers, 20 is subtracted, the result is the
other number. If from twice the smaller number, 5 is subtracted, the result
is the greater number. Find the numbers.
42.
In a deer park the number of heads and the number of legs of deer and
visitors were counted and it was found that there were 39 heads and 132
legs. Find the number of deers and visitors in the park, using graphical
method.
43.
A two digit number is obtained by either multiplying the sum of the digits
by 8 and adding 1; or by multiplying the difference of the digits by 13 and
adding 2. Find the number. How many such numbers are there.
1
In an examination one mark is awarded for every correct answer and
4
mark is deducted for every wrong answer. A student answered 120 questions
and got 90 marks. How many questions did he answer correctly?
44.
45.
A boatman rows his boat 32 km upstream and 36 km down stream in 7
hours. He can row 40 km upstream and 48 km downstream in 9 hours.
Find the speed of the stream and that of the boat in still water.
46.
In a function if 10 guests are sent from room A to B, the number of guests
in room A and B are same. If 20 guests are sent from B to A, the number
of guests in A is double the number of guests in B. Find number of guests
in both the rooms in the beginning.
47.
In a function Madhu wished to give Rs. 21 to each person present and
found that she fell short of Rs. 4 so she distributed Rs. 20 to each and
23
[Class-X – Maths]
found that 1 were left over. How much money did she gave and howw
many persons were there.
48.
A mobile company charges a fixed amount as monthly rental which includes
100 minutes free per month and charges a fixed amount there after for
every additional minute. Abhishek paid 433 for 370 minutes and Ashish
paid 398 for 300 minutes. Find the bill amount under the same plan, if
Usha use for 400 minutes.
49.
Ravi used 2 plastic bags and I paper bag in a day which cost him 35
while Mahesh used 3 plastic bags and 4 paper bags in a day which cost
him 65.
50.
i.
Find the cost of each bag.
ii.
Which bag has to be used and what value is reflected by using it?
Two schools want to award their selected students on the values of
Discipline and Punctuality. First school wants to award these values to its
3 and 2 students respectively, while second wants to award for the same
values to its 4 and 1 students respectively. If the amount of award for each
value given by both the school are same and total amount spent by each
school is 600 and 550 respectively then.
i.
Find the award money for each value.
ii.
Which more values of the students may be chosen for the award?
1.
d
2. c
3.
c
4. b
5.
a
6. c
7.
a
8. a
9.
b
10. c
11.
c
12. d
[Class-X – Maths]
24
13.
b
14. d
15.
b
16. a
17.
Father’s age = x years, Son’s age = y years
x + 2y = 70, 2x + y = 95
18.
No. of athletes = x, No. of athletes increased = y
19.
Infinite
20.
21.
Infinite
22. (2, 2)
23.
(2, 0) P = 7
24. No
25.
p 4
26.
k 1, k 
28.
(7, 3), 18
29. a = 5, b = 1
30.
–5, –10 or 10, 5
31. (7, 13)
32.
x = 1, y = b
33. (2, –1)
34.
(3, 2)
35. (1, 2), (1, 0), yes
36.
(6, 0), 1 : 2
37. (2, 1)
1
3
19
2
40.
1 1
2, 4

(4, 9)
42.
27, 12
43. 41 or 14(2)
44.
96
45. 2 km/hr, 10km/hr.
46.
100, 80
47. Rs. 101, 5
38.
39.
100000
41. 15, 10
49.
1
Rs. 298, Rs. Rs. 448
2
(i) 15, 5,(ii) Environment friendly
50.
(i) 100, 150
48.
20000,
(ii) Honesty, regularity etc.
25
[Class-X – Maths]
1.
Similar Triangles : Two triangles are said to be similar if their corresponding
angles are equal and their corresponding sides are proportional.
2.
Criteria for Similarity :
in and 
(i)
AAA Similarity : ~ when = = and
= 
(ii)
SAS Similarity :
ABC ~ DEF when
ABC ~ DEF ,
(iii)
3.
ABBC
and B E
DEEF
ABACBC
.
DEDFEF
The proof of the following theorems can be asked in the examination :
(i)
Basic Proportionality Theorem : If a line is drawn parallel to one
side of a triangle to intersect the other sides in distinct points, the
other two sides are divided in the same ratio.
(ii)
The ratio of the areas of two similar triangles is equal to the square
of the ratio of their corresponding sides.
(iii)
Pythagoras Theorem : In a right triangle, the square of the
hypotenuse is equal to the sum of the squares of the other two
sides.
[Class-X – Maths]
26
(iv)
1.
2.
Converse of Pythagoras Theorem : In a triangle, if the square of
one side is equal to the sum of the squares of the other two sides
then the angle opposite to the first side is a right angle.
~ If DE = 2 AB and BC = 3cm then EF is equal to _______.
(a)
1.5 cm
(b)
3 cm
(c)
6 cm
(d)
9 cm
In AB || EW if AD = 4 cm, DE = 12cm and DW = 24 cm then the
value of DB = ____
(a)
4 cm
(b)
8 cm
(c)
12 cm
(d)
16 cm
3.
A
D
Q
Q
c
f
b
e
O
O
B
C
E
F
a
d
In the figure the value of cd = ________
4.
(a)
ae
(b)
af
(c)
bf
(d)
be
If the corresponding medians of two similar triangles are in the ratio 5 : 7,
then the ratio of their sides is :
(a)
25 : 49
(b)
7:5
(c)
1:1
(d)
5:7
27
[Class-X – Maths]
5.
6.
AD is the bisector of If BD = 4cm. DC = 3cm and AB = 6 cm. AC is
equal to
(a)
4.2 cm
(b)
4.5 cm
(c)
4.8 cm
(d)
5 cm
In the figure, is similar to ______
16 cm
B
A
53°
53°
24
cm
36 cm
C
D
7.
(a)

(b)

(c)

(d)

~ 
(a)
(c)
8.
(b)
2 MB
2
MB
If ~ and
will be PR.
(d)
2 MD
2
MD
ar ABC 9
, AB = 18 cm, BC = 15 cm what
ar PQR 4
(a)
10 cm
(b)
9 cm
(c)
4 cm
(d)
18 cm
[Class-X – Maths]
28
9.
10.
11.
In D and E are points on side AB and AC respectively such that
DE || BC and AD : DB = 3 : 1. If EA = 3.3 cm then AC =
(a)
1.1 cm
(b)
4.4 cm
(c)
4 cm
(d)
5.5 cm
ABC and BDE are two equilateral triangles such that D is the midpoint of
BC. Ratio of the areas of triangles ABC and BDE is—
(a)
2:1
(b)
1:2
(c)
4:1
(d)
1:4
In DE || BC. In the figure, the value of x is ______
A
x
x+3
D
E
x+1
x+5
B
(a)
1
(b)
–1
(c)
3
(d)
–3
ar (PRQ )BC1
It is given that ~ withthen ar (BCA ) is equal toQR3
12.
13.
C
(a)
9
(b)
3
(c)
1
3
(d)
1
9
The altitude of an equilateral triangle, having the length of its side 12cm is
(a)
12 cm
(b)
29
6 2 cm
[Class-X – Maths]
(c)
14.
15.
16.
6 cm
(d)
6 3 cm
A right angled triangle has its area numerically equal to its perimeter. The
length of each side is an even number & the hypotenuse is 10 cm. What
is the perimeter.
(a)
26 cm
(b)
24 cm
(c)
30 cm
(d)
16 cm
The perimeters of two similar triangles ABC and PQR are respectively 36
cm and 24 cm. If PQ = 10cm then AB = ........
(a)
10 cm
(b)
20 cm
(c)
25 cm
(d)
15 cm
In figure ~ If BC = 8 cm, PQ = 4cm AC = 6.5 cm, AP = 2.8
cm, find AB and AQ.
B
P
A
Q
C
11
CD . Prove that CA2 = AB2 + BC 2
32
17.
In 
18.
If is an equilateral triangle such that AD BC then prove that
AD2 = 3 DC2
19.
An isosecles triangle ABC is similar to triangle PQR. AC = AB = 4 cm,
RQ = 10 cm and BC = 6 cm. What is the length of PR? What type of
triangle is ?
[Class-X – Maths]

30
20.
In the adjoining figure, find AE.
A
E
8 cm
4 cm
6 cm
B
21.
3 cm
C
D
ar PQR 1
QR . Find.
In DE || QR and DE 
4ar PDE 
P
D
E
Q
22.
In triangles ABC and PQR if = and
the value of
23.
R
PR
?
AC
ABC is a right angled at B, AD and CE are two medians drawn from A
and C respectively. If AC = 5 cm and AD 
24.
ABBC1
then what is
PQQR2
35
cm. Find CE.
2
In the adjoining figure DE || BC. What is the value of DE.
31
[Class-X – Maths]
A
10
cm
D
E
2c
m
B
C
3 cm
25.
Legs (sides other then the hypotenuse) of a right triangle are of lengths
16 cm and 8 cm. Find the length of the side of the largest square that can
be inscribed in the triangle.
26.
In the following figure, DE || AC and
BEBC
. Prove that DC || AP
ECCP
A
D
B
27.
E
C
P
Two similar triangles ABC and PBC are made on opposite sides of the
same base BC. Prove that AB = BP.
[Class-X – Maths]
32
28.
I a quadriatnl eral
ABCD, = 90°, AD2 = AB2 + BC2 + CD2. Prove that
= 90°.
D
C
B
A
2.
29.
Find
ar (trapezium BCED).
A
D
E
3 cm
B
C
9 cm
30.
Amit is standing at a point on the ground 8m away from a house. A mobile
network tower is fixed on the roof of the house. If the top and bottom of
the tower are 17m and 10m away from the point. Find the heights of the
tower and house.
31.
In a right angled triangle PRQ, PR is the hypotenuse and the other two
sides are of length 6cm and 8cm. Q is a point outside the triangle such
that PQ = 24cm RQ = 26cm. What is the measure of 
32.
PQRS is a trapezium. SQ is a diagonal. E and F are two points on parallel
sides PQ and RS respectively. Line joining E and F intersects SQ at G.
Prove that SG × QE = QG × SF.
33.
Two poles of height a metres and b metres are apart. Prove that the height
of the point of intersection of the lines joining the top of each pole to the
ab
foot of the opposite pole ismetres..
a b
33
[Class-X – Maths]
D
B
O
bm
am
h
x
C
L
y
A
34.
Diagonals of a trapezium PQRS intersect each other at the point O, PQ||RS
and PQ = 3 RS. Find the ratio of the areas of triangles POQ & ROS.
35.
In a rhombus, prove that four times the square of any sides is equal to the
sum of squares of its diagonals.
36.
ABCD is a trapezium with AE || DC. If ABD is similar to Prove that
AD = BC.
37.
In a triangle, if the square of one side is equal to the sum of the squares
of the other two sides, then prove that the angle opposite to the first side
is a right angle.
38.
If BL and CM are medians of a triangle ABC right angled at A. Prove that
4 (BL2 + CM2) = 5 BC2.
39.
ABCD is a rectangle in which length is double of its breadth. Two equilateral
triangles are drawn one each on length and breadth of rectangle. Find the
ratio of their areas.
40.
In figure DE || BC and AD : DB = 5 : 4. Find
ar DEF
ar CFB
A
E
D
F
C
B
[Class-X – Maths]
34
41.
Four friends went to a restaurant. They ordered four triangular shaped
pizzas with different topings. One of them told to others they should divide
their pizzas in four equal parts so that all can taste all the four pizzas but
the three fail to do so. Fourth friend took a straw and he make a point at
mid of each side and join all mid points and cut it by a knife into four equal
parts.
Prove that
i.
ii.
42.
ar (DEF )1

ar (ABC )4 if D, E, F are the midpot of sides AB, BC & CA
of a triangular pizza.
Which value is described in this equation?
A poster competition on “Save Earth” is organised by Directorate of
Education. Medals in the shape of similar triangles were distributed to first
and second position holder, first prize in form of and second in the
form of 
12
A
B
SAVE
EARTH
15
9
C
i.
Find the value of x and y.
P
8
SAVE
EARTH
Q
ii.
Y
x
R
Which mathematical concept was used in this question?
35
[Class-X – Maths]
iii.
What is the ratios of areas of triangle ABC and PQR.
iv.
What moral values of a child was shown in this poster competition.
43.
From a village A to city B road passes through a mountain C. It was
constructed so that AC CB, AC = 2x km and CB = 2 (x + 7) km. It was
proposed that a 26 km long highway will be constructed which will connect
city B to village A directly. If a person goes to city B from village A through
this highway by bus then what distance was reduced in his journey. Write
the values of this question.
44.
Ashok has a quadrilateral shaped field of rice. He divide it into two by one
diagonal and distribute it to one son and one daughter. Each get the rice
in proportion to the area which they got. Both don’t know how much area
they got. Ashok using second diagonal distribute the rice in such ratio.
i.
ar (ABC )AO

ar (DBC ) DO
Prove the above relation
C
A
O
B
ii.
45.
D
Which value of Ashok was shown in this questions?
Two advertising balloons showing “Pulse Polio” and “Save Girl Child" were
tied by two wires ML and NR such that they makes an angle 40° with the
ground. A person standing at the point K. Length of wire ML is ‘a’ metre,
MN is ‘b’ metre and NK is ‘c’ metre.
i.
Find the length of wire NR when both the balloons are on a same
straight line.
ii.
What moral values are shown in this question.
[Class-X – Maths]
36
save
girl
child
L
Pulse
polio
R
x
40°
M
40°
b
N
K
c
1.
c
2.
b
3.
a
4.
d
5.
b
6.
d
7.
a
8.
a
9.
b
10.
c
11.
c
12.
a
13.
d
14.
b
15.
d
16.
AB = 5.6 cm, AQ = 3.25 cm
18.
~ 
19.
20.
22.
21.
20
cm , isosceles triangle
3
16 : 1
1
1
2
23.
25 – 6 5 cm
4
24.
2.5 cm
25.
16
cm
3
29.
240 cm2
30.
9 m, 6 m
37
[Class-X – Maths]
31.
90°
34.
9:1
39.
4:1
40.
25
81
41.
(ii) Cooperation, healthy competition, intelligency, quick decision
42.
ar (ABC )9
(ii) similar triangle

(i) x = 10, y = 6, ar (PQR )4
(iii) Awareness about pollution, Saving of water, Energy conservation
43.
(i) 8 Km (ii) Saving of time, Economic saving, Save energy.
44.
(i) Intelligency, Self decision, Responsibility.
45.
(i) x 
ac
b c
(ii) Save girl child, health awareness.
[Class-X – Maths]
38
1.
Trigonometric Ratios : In = 90°, for angle ‘A’
sin A 
Perpendicular
Hypotenuse
C
Base
cos A 
Hypotenuse
Hy
p
Perpendicular
tan A 
Base
nu
ote
se
Perpendicular

A
Base
B
Base
cot A 
Perpendicular
sec A 
cosec A 
2.
Hypotenuse
Base
Hypotenuse
Perpendicular
Reciprocal Relations :
sin 
cos 
1
,
cosec 
1
sec 
cosec 
1
sin 
sec 
1
cos 
,
39
[Class-X – Maths]
tan 
3.
cot 
,
1
tan 
Quotient Relations :
tan 
4.
1
cot 
sin 
cos 
cos 
sin 
cot 
,
Indentities :
sin2 + cos2 = 1 sin2 = 1 – cos2 and cos2 = 1 – sin2 
1 + tan2 = sec2 tan2 = sec2 – 1 and sec2 – tan2 = 1
1 + cot2 = cosec2 cot2 = cosec2 – 1 and cosec2 – cot2 = 1
5.
Trigonometric Ratios of Some Specific Angles

0°
30°
45°
60°
90°
0
1
2
1
sin A
3
2
1
1
3
2
1
cos A
1
2
0
tan A
0
2
2
1
1
3
3
Not defined
2
cosec A
Not defined
2
2
3
1
2
sec A
1
cot A
Not defined
2
3
2
Not defined
1
[Class-X – Maths]
1
3
40
3
0
6.
Trigonometric Ratios of Complementary Angles
sin (90° – = cos 
cos (90° – = sin 
tan (90° – = cot 
cot (90° – = tan 
sec (90° – = cosec 
cosec (90° – = sec 
Note : In the following questions 0° 90°
1.
2.
3.
4.
If x = a sin and y = a cos then the value of x2 + y2 is _______
(a)
a
(b)
a2
(c)
1
(d)
1
a
The value of cosec 70° – sec 20° is _____
(a)
0
(b)
1
(c)
70°
(d)
20°
If 3 sec – 5 = 0 then cot = _____
(a)
5
3
(b)
4
5
(c)
3
4
(d)
3
5
(b)
1
(d)
22
If = 45° then sec cot – cosec tan is
(a)
(c)
0
2
41
[Class-X – Maths]
5.
6.
7.
If sin (90 – cos = 1 and is an acute angle then = ____
(a)
90°
(b)
60°
(c)
30°
(d)
0°
The value of (1 + cos (1 – cos cosec2= _____
(a)
0
(b)
1
(c)
cos2 
(d)
sin2 
is a right-angled isosceles triangle then cos T + cos R + cos Y is
_____
(a)
(c)
8.
2
2
1
(b)
22
(d)
1
(b)
x 1
2x
x
2
2
1
x
2
(c)
10.
2
If sec + tan = x, then sec =
(a)
9.
1
x 1
2x
(d)
x
2
1
x

The value of cot sin cos is _______
22
(a)
cot cos2 
(b)
cot2
(c)
cos2 
(d)
tan2 
If sin – cos = 0, 0 90° then the value of is _____
(a)
cos 
(b)
45°
(c)
90°
(d)
sin 
[Class-X – Maths]
42
sin 
11.
can be written as
2
1 sin 
(a)
cot

(b)
sin 
(d)
tan 
1 sin 
is equal to
1 sin 
(a) sec2+ tan2
(b)
sec – tan 
sec2– tan2
(d)
sec + tan 
(c)
12.
cos 
(c)
13.
sin 
In an isosceles right-angled = 90°. The value of 2 sin A cos A
is _____
(a)
1
(b)
1
2
1
(c)
(d)
2
2
14.
15.
If
sin 20sin 70

2
2
2 cos 69cos 21 
2

2
sec 60
 then K is ______
K
(a)
1
(b)
2
(c)
3
(d)
4
1
If tan 
7
, then
cosec sec 
2
2
2
2
cosec sec 

(a)
3
4
(b)
5
7
(c)
3
7
(d)
1
12
43
[Class-X – Maths]
16.
17.
3
In = 90° and sin R 
, write the value of cos P.
5
If A and B are acute angles and sin A = cos B then write the value of
A + B.
18.
If 4 cot = 3 then write the value of tan + cot 
19.
Write the value of cot2 30° + sec2 45°.
20.
Given that 16 cot A = 12, find the value of
21.
If = 30° then write the value of sin + cos2 
22.
If 1 tan 
2
sin A cos A

sin A cos A
2
then what is the value of 
3
23.
Find the value of if
24.
If and are complementary angles then what is the value of
3 tan 23 0.
cosec sec – cot tan 
25.
If tan (3x – 15°) = 1 then what is the value of x.

26.
27.
Simplify :
tan2 60° + 4 cos2 45° + 3 (sec2 30° + cos2 90°)
28.
Evaluate
cos 58
2
sin 32
29.
cos 38 cosec 52 
3
tan 15 tan 60 tan 75 
If sin + sin2 = 1 then find the value of cos2 + cos4 
–
30.
– 36° are acute angles then find the
value of 
[Class-X – Maths]
44
31.
Prove that
cosec4
32.
33.
If sin (3x + 2y) = 1 and cos 3x 2y 
3
then find the value of x and y.
2
, where 0 (3x + 2y) 
If sin (A + B) = sin A cos B + cos A sin B then find the value of
(a)
sin 75°
(b)
cos 15°
34.
Prove that
35.
Prove that
36.
– cosec2 = cot2 + cot4 
cos Acos A
cos A, A 45.
1 tan A 1 cot A
sec 1

sec 1
Find the value of
sec 1
2cosec 
sec 1
sin2 5° + sin2 10° + sin2 15° + .... + sin2 85°
37.
Prove that
tan sec 1cos 
.
tan sec 1 1 sin 
38.
If 2 sin 3x 15 
sin
2
2x
3 then find the value of
10 tan
2
x
5.
39.
Find the value of sin 60° geometrically.
40.
1
Let p = tan + sec then find the value of p p .
41.
Find the value of
2
tan cot 90 sec cosec 90 sin 35 sin 55 
tan 10tan 20tan 30 tan 70 tan 80 
45
2
[Class-X – Maths]
43.
cos cos 
m andn show that (m2 + n2) cos2 = n2.
cos sin 
Prove that cos 1° cos 2° cos 3°.........cos 180° = 0.
44.
sin cos sin cos 2 sec 
Prove that sin cos sin cos .
45.
tan 1
If A, B, C are the interior angles of a triangle ABC, show that
42.
If
2
2
AAB C B C 
sin coscos sin1.

222 2 
46.
In the given right triangle, if base and perpendicular are represented by
‘Hardwork’ and ‘Success’ and are in the ratio 1 : 1 then find What
mathematical concept has been used? What value is depicted from the
problem?
A
B
47.
C
If x = sin2y = cos2
Where x represents Honesty and y represents Hardwork.
48.
a.
What you will get when honesty is added with hardwork?
b.
Which mathematical concept is used?
c.
What value is depicted here?
If punctuality and regularity are two measurable quantities are numberically
equal to A and B respectively such that
[Class-X – Maths]
46
1
Sin(A – B
2)
1
Cos (A B ) 
2
where 0° A + B 90°
then find A and B.
Which one more value other than punctuality & regularity, would you like
to adopt in your life?
1.
b
2.
a
3.
c
4.
a
5.
d
6.
b
7.
a
8.
b
9.
a
10.
b
11.
d
12.
d
13.
a
14.
d
15.
a
16.
cos P 
17.
90°
18.
25
12
19.
5
20.
7
21.
5
4
22.
30°
23.
30°
24.
1
25.
x = 20.
26.
10°
27.
9
28.
1
29.
1
30.
42°
47
3
5
[Class-X – Maths]
32.
33.
x = 20, y = 15
3 1
,
22
3 1
22
36.
17
2
38.
13
12
40.
2 sec 
41.
23
46.
45°, Trigonometry, Hardwork and Success
47.
(a) 1, (b) Trigonometry, (c) Honesty and Hardwork
48.
A = 45°, B = 15°
Honesty, Hardwork, Responsibility, Cooperation
[Class-X – Maths]
48
1.
The mean for grouped data can be found by :
fixi .
fi
(i)
The direct method X 
(ii)
The assumed mean method
X a 
fidi ,
fi
where di = xi –a.
(iii)
The step deviation method
2.
fiui
fi
xi a
.
h
The mode for the grouped data can be found by using the formula :
X a 
h, where u i 
f1 f 0
mode l h
2f 1 f 0 f 2 
l = lower limit of the modal class.
f1 = frequency of the modal class.
f0 = frequency of the preceding class of the modal class.
f2 = frequency of the succeeding class of the modal class.
h = size of the class interval.
Modal class - class interval with highest frequency.
49
[Class-X – Maths]
3.
The median for the grouped data can be found by using the formula :
n 2 Cf median l h

f
l = lower limit of the median class.
n = number of observations.
Cf = cumulative frequency of class interval preceding the
median class.
f = frequency of median class.
h = class size.
4.
Empirical Formula : Mode = 3 median - 2 mean.
5.
Cumulative frequency curve or an Ogive :
(i)
Ogive is the graphical representation of the cumulative frequency
distribution.
(ii)
Less than type Ogive :
(iii)
(iv)
•
Construct a cumulative frequency table.
•
Mark the upper class limit on the x = axis.
More than type Ogive :
•
Construct a frequency table.
•
Mark the lower class limit on the x-axis.
To obtain the median of frequency distribution from the graph :
•
Locate point of intersection of less than type Ogive and
more than type Ogive :
Draw a perpendicular from this point on x-axis.
•
[Class-X – Maths]
The point at which it cuts the x-axis gives us the median.
50
1.
2.
3.
4.
5.
6.
7.
Mean of first 10 natural numbers is
(a)
5
(b)
6
(c)
5.5
(d)
6.5
If mean of 4, 6, 8, 10, x, 14, 16 is 10 then the value of ‘x’ is
(a)
11
(b)
12
(c)
13
(d)
9
The mean of x, x + 1, x + 2, x + 3, x + 4, x + 5 and x + 6 is
(a)
x
(b)
x+3
(c)
x+4
(d)
3
The median of 2, 3, 2, 5, 6, 9, 10, 12, 16, 18 and 20 is
(a)
9
(b)
20
(c)
10
(d)
9.5
The median of 2, 3, 6, 0, 1, 4, 8, 2, 5 is
(a)
1
(b)
3
(c)
4
(d)
2
Mode of 1, 0, 2, 2, 3, 1, 4, 5, 1, 0 is
(a)
5
(b)
0
(c)
1
(d)
2
If the mode of 2, 3, 5, 4, 2, 6, 3, 5, 5, 2 and x is 2 then the value of ‘x’ is
(a)
2
(b)
3
(c)
4
(d)
5
51
[Class-X – Maths]
8.
The modal class of the following distribution is
Class Interval
Frequency
9.
10.
11.
12.
13.
10–15
15–20
20–25
25–30
30–35
4
7
12
8
2
(a)
30–35
(b)
20–25
(c)
25–30
(d)
15–20
A teacher ask the students to find the average marks obtained by the
class students in Maths, the student will find
(a)
mean
(b)
median
(c)
mode
(d)
sum
The empirical relationship between the three measures of central tendency is
(a)
3 mean = mode + 2 median
(b)
3 median = mode + 2 mean
(c)
3 mode = mean + 2 median
(d)
median = 3 mode – 2 mean
Class mark of the class 19.5 – 29.5 is
(a)
10
(b)
49
(c)
24.5
(d)
25
Measure of central tendency is represented by the abscissa of the point
when the point of intersection of ‘less than ogive’ and ‘more than ogive’ is
(a)
mean
(b)
median
(c)
mode
(d)
None of these
The median class of the following distribution is
Class Interval :
Frequency :
[Class-X – Maths]
0–10
10–20
20–30
30–40
40–50
50–60
60–70
4
4
8
10
12
8
4
52
14.
15.
16.
17.
18.
19.
20.
(a)
20–30
(b)
40–50
(c)
30–40
(d)
50–60
The mean of 20 numbers is 17, if 3 is added to each number, then the new
mean is
(a)
20
(b)
21
(c)
22
(d)
24
The mean of 5 numbers is 18. If one number is excluded then their mean
is 16, then the excluded number is
(a)
23
(b)
24
(c)
25
(d)
26
The mean of first 5 prime numbers is
(a)
5.5
(b)
5.6
(c)
5.7
(d)
5
The sum of deviations of the values 3, 4, 6, 8, 14 from their mean is
(a)
0
(b)
1
(c)
2
(d)
3
If median = 15 and mean = 16, then mode is
(a)
10
(b)
11
(c)
12
(d)
13
The mean of 11 observations is 50. If the mean of first six observations
is 49 and that of last six observations is 52, then the sixth observation is
(a)
56
(b)
55
(c)
54
(d)
53
The mean of the following distribution is 2.6, then the value of ‘x’ is
Variable
1
2
3
4
5
Frequency
4
5
x
1
2
53
[Class-X – Maths]
(a)
24
(b)
3
(c)
8
(d)
13
21.
The mean of 40 observations was 160. It was detected on rechecking that
the value of 165 was wrongly copied as 125 for computing the mean. Find
the correct mean.
22.
Find ‘x’ if the median of the observations in ascending order 24, 25, 26,
x + 2, x + 3, 30, 31, 34 is 27.5.
23.
Find the mean of the following data.
S.No.
x:
10
12
14
16
18
20
f:
3
5
6
4
4
3
24.
Find the value of ‘p’, if mean of the following distribution is 7.5
Variable :
3
5
7
9
11
13
Frequency :
6
8
15
p
8
4
25.
From the cumulative frequency table, write the frequency of the class
20–30.
Marks
Number of Students
Less than 10
1
Less than 20
14
Less then 30
36
Less than 40
59
Less than 50
60
[Class-X – Maths]
54
26.
Following is a cumulative frequency curve for the marks obtained by
40 students. Find the median marks obtained by the student.
27.
The following ‘more than ogive’ shows the weight of 40 students of a class.
What is the lower limit of the median class.
55
[Class-X – Maths]
28.
The mean of the following frequency distribution is 62.8. Find the values
of x and y.
Class Interval :
0–20
20–40
40–60
60–80
80–100
100–120
Total
5
x
10
y
7
8
50
Frequency :
29.
The following frequency distribution gives the daily wages of a worker of
a factory. Find mean daily wage of a worker.
Daily Wage (in
30.
0
More than 250
12
More than 200
21
More than 150
44
More than 100
53
More than 50
59
More than 0
60
The median distance of the following data is 46. Find x & y.
No. of students
10-20
20-30
30-40
40-50
50-60
60-70
70-80
Total
12
20
x
65
y
25
18
230
Find the mean, median and mode of the following :
Class Interval :
Frequency :
32.
Number of Workers
More than 300
Distance (m)
31.
)
0–10
10–20
20–30
30–40
40–50
50–60
60–70
6
8
10
15
5
4
2
The following frequency distribution shows the marks obtained by 100
students in a school. Find the mode.
[Class-X – Maths]
56
Marks
33.
Number of Students
Less than 10
10
Less than 20
15
Less than 30
30
Less than 40
50
Less than 50
72
Less than 60
85
Less than 70
90
Less than 80
95
Less than 90
100
Draw ‘less than’ and ‘more than’ ogives for the following distribution
Marks :
20–30
30–40
40–50
50–60
60–70
70–80
80–90
8
12
24
6
10
15
25
No. of Students :
Also find median from graph.
34.
The mode of the following distribution is 65. Find the values of x and y, if
sum of the frequencies is 50.
Class Interval :
Frequency :
35.
0–20
20–40
40–60
60–80
80–100
100–120
120–140
6
8
x
12
6
y
3
Following is the table people violating traffic rules under different age
groups:
Age (In yrs)
No. of persons
violating traffic rules
18-20
20-22
22-24
24-26
26-28
28-30
63
50
35
27
16
9
Find mean. Why should we obey traffic rules. What values are depicted
here?
36.
The following table shows the ages of patients in a private hospital after
Parliamentary Elections held in November for free treatment for those who
cast their votes.
57
[Class-X – Maths]
Age
(In yrs)
18-28
28-38
38-48
48-58
58-68
No. of
Patients
6
11
21
23
14
68 onwards
Total
5
80
Find the mode of the above data. What values are depicted here?
37.
Following table shows distance covered by 50 students of a school in a
shot-put competition.
Distance (M)
No. of St.
0-20
20-40
40-60
60-80
80-100
6
11
17
12
4
Construct cumulative frequency distribution table. Find median. What values
are depicted here?
38.
The number of persons living in an old age home of a city are as follows:
Age (In yrs)
No. of old Persons
50-55
55-60
60-65
65-70
70-75
75-80
10
12
17
13
16
22
Find the mean and median. What steps should be taken to improve the
condition of old people in our society. What values are shown in this
question?
39.
During the last two months the quantity of PNG used in a locality of 30
houses was recorded as follows:
Quantity (M3)
38-40
40-42
42-44
44-46
46-48
48-50
50-52
No. of families
3
2
4
5
14
4
3
Find mean, median & mode of the above data.
40.
The haemoglobin level of 35 students of a class is as follows:
Hb Level
Less then
8
No. of Students
3
Less then
10
7
Construct "less than type" of give.
What value is depicted in the question?
[Class-X – Maths]
58
Less then
12
Less then
14
13
23
Less then
16
35
1.
c
2.
b
3.
b
4.
a
5.
b
6.
c
7.
a
8.
b
9.
a
10.
b
11.
c
12.
b
13.
c
14.
a
15.
d
16.
b
17.
a
18.
d
19.
a
20.
c
21.
161
22.
x = 25
23.
14.8
24.
p=3
25.
22
26.
40
27.
147.5
28.
x = 8, y = 12
30.
x = 34, y = 46
29.
182.50
31.
Mean = 30,
Median = 30.67, mode = 33.33
32.
41.81
33.
60
34.
x = 10, y = 5
35.
(a) 22.1 years
(b) discipline
(c) knowledge &
obeying traffic rules
36.
mode – 49.81 Social responsibility
37.
m edi = 49.an41m , (A w areness f physiorcal ftiness), (I portmance of sports),
(H ealhy com petton)tii
38.
(a) (m ean) = 66.88,
(m edian) = 67.3
(b) (generaton gap) (devel ng m oraliopi
social values), (motivation)
(c) (moral values)
39.
45.8, 46.5, 47.9
40.
(Awarness for fitness), (Moral values).
59
[Class-X – Maths]
Time allowed : 3 hours
Maximum Marks : 90
General Instructions
1.
All questions are compulsory.
2.
The question paper consists of 31 questions divided into four sections A,
B, C and D. Section A comprises of 4 questions of 1 mark each. Section
B comprises of 6 questions of 2 marks each. Section C comprises of 10
questions of 3 marks each and Section D comprises of 11 questions of 4
marks each.
3.
Question numbers 1 to 4 in Section A are multiple choice questions where
you are to select one correct option out of the given four.
4.
Use of calculator is not permitted.
1.
In the given figure XY QR and
PX 1
then :
XQ 2
P
Y
X
Q
[Class-X – Maths]
R
60
1
(a)
(c)
XY = QR
(b)
XY=
QR
3
1
(XY)2 = (QR)2
(d)
XY=
QR
2
2.
If sin (60° + A) = 1, then the value of cosec A is:
(a)
1
(b)
0
(c)
1
2
(d)
2
(1– sin = k, then the value of k is:
3.
4.
(a)
0
(b)
1
(c)
–1
(d)
2
If the mode of observation 2, 3, 5, 4, 2, 6, 3, 5, 5, 2 and x is 2, then the
value of x is :
(a)
2
(b)
3
(c)
4
(d)
5
5.
Whether 6n can end with the digit 0 for any natural number. Give reason
for your answer.
6.
If the product of the zeroes of the polynomial ax2 – 6x – 6 is 4, find the
value of a.
7.
8.
ABC 9
,PQR 4 AB = 18 cm, BC = 15 cm, find the
ar
If ~ 
ar
value of PR.
If 16 cot A = 12 then find the value of
sin A cos A
.
sin A – cos A
9.
Find HCF of 56, 96 and 324 by Euclid's algorithm.
61
[Class-X – Maths]
10.
Find x, if the median of the observations in ascending order
24, 25, 26, x + 2, x + 3, 30, 31, 34 is 27.5.
11.
If the areas of two similar triangles are equal, then prove that the triangles
are congruent.
12.
Find the other zeroes of the polynomial for x4 – 20x3 + 23x2 + 5x – 6, if
two of its zeroes are 2 and 3.
13.
Use Euclid’s division lemma to show that the square of any positive integer
is either of the form 3m or 3m + 1 for some integer m.
14.
If sin (3x + 2y) = 1 and cos 3x 2 y 
3
find the value of x and y.
2
when 0 3x + 2y 90° then
15.
Divide 5x3 – 13x2 + 21x – 14 by 3 – 2x + x2 and verify the division lemma.
16.
Solve for x and y, when x ±y
17.
304
10
x yx y
4055
13
x yx y
The mean of the following frequency distribution is 62.8 and the sum of all
the frequencies is 50. Find the values of x and y.
Class interval
Frequency
0–20
20–40
40–60
60–80
80–100
100–120
5
x
10
y
7
8
18.
The diagonals of a trapezium PQRS intersect each other at the point O,
PQ||RS and PQ = 3 RS, find the ratio of the areas of and 
19.
If A, B and C are interior angles of a then show that
AB C
sin cos
2 2
[Class-X – Maths]
62
20.
Find the median weight of the following data :
Weight in (kg.)
38–40
40–42
42–44
44–46
46–48
48–50
50–52
No. of Students
3
2
4
5
14
4
3
21.
Prove that 5 is an irrational number and hence prove 2 –
irrational number.
22.
Draw the graph of the following pair of linear equations x + 3y = 6 and
2x – 3y = 12. Find the ratio of the areas of the two triangles formed by first
line, x = 0, y = 0 and second line x = 0, y = 0.
23.
Prove that :
5 is an
tan cot 
1 sec . cosec .
1 cot 1 tan 
24.
Find the value of K so that the pair of linear equations:
3K 1x 3 y 2 0
K 2 1x K 2y 5 0
is inconsistent.
25.
Change the distribution to a more than type distribution and draw its ogive
Class Interval
50-55
55-60
60-65
65-70
70–75
75–80
2
8
12
24
38
16
Frequency
26.
On dividing 4x3 – 8x2 + 8x + 1 by a polynomial d(x), the quotient and
remainder are (2x2 – 3x + 2) and (x + 3) respectively, find d(x).
27.
State and prove converse of pythagoras theorem.
28.
Prove the following identity :
cos A1 sin A
2 sec A.
1 sin Acos A
63
[Class-X – Maths]
29.
In an isosceles with AB =AC, BD is the perpendicular drawn from
the vertex B to the side AC. Prove that BD2 – CD2 = 2CD.AD.
30.
Evaluate :
sec .cosec 90 – – tan . cot 90 – sin2 55 sin2 35 
tan10.tan20.tan60 .tan70 .tan80 
31.
In a city, the number of old aged citizens lived in an old age home is as
given below
Age (in years)
No. of People
50–55
10
55–60
12
60–65
17
65–70
13
70–75
16
75–80
22
(a)
Find mean of the above data?
(b)
Why the old people are neglected in the society and what are the
various steps to be taken to improve the status of old people in the
society?
(c)
What value is depicted in this question?
1.
b
2.
d
3.
c
4.
a
6.
a = –3/2
7.
10 cm
8.
7
9.
4
1
10.
x = 25
[Class-X – Maths]
12.
64
±2
13.
x = 20, y = 15
14.
A = 60°, B = 30°
15.
Q = 5x – 3, R = –5
16.
x = 5, y = 3
17.
x = 8, y = 12
18.
9:1
20.
46. 5 Kg.
22.
1:2
24.
K = –1, K 
26.
d (x) = 2x – 1
19
2
2
30.
31.
3
(a) 66.8.
(b) Generation gap, lack of social and moral values.
(c) Impart social and moral values in the society.
65
[Class-X – Maths]
S.No. Chapter
Page
1.
Quadratic Equations
68
2.
Arithmetic Progression
77
3.
Co-ordinate Geometry
84
4.
Some Applications of Trigonometry
92
5.
Circle
100
6.
Constructions
116
7.
Mensuration
120
8.
Probability
133
Sample Papers
141
67
[Class-X – Maths]
[Class-X – Maths]
68
CHAPTER 1
QUADRATIC EQUATIONS
1.
The equation ax2 + bx + c = 0, a 0 is the standard form of a quadratic
equation, where a, b and c are real numbers.
2.
A real number is said to be a root of the quadratic equation ax2 + bx
+ c = 0, a 
2
polynomial ax2
+ bx + c and the roots of the quadratic equation ax2 + bx + c = 0 are the
same.
3.
If we can factorise ax2 + bx + c = 0, a 0 into product of two linear factors,
then the roots of the quadratic equation can be found by equating each
factors to zero.
4.
The roots of a quadratic equation ax2 + bx + c = 0, a 0 are given by
b 
5.
6.
7.
2
b 4ac
, provided that b2 – 4ac 0.
2a
A quadratic equation ax2 + bx + c = 0, a 0, has ___
(a)
Two distinct and real roots, if b2 – 4ac > 0.
(b)
Two equal and real roots, if b2 – 4ac = 0.
(c)
Two roots are not real, if b2 – 4ac < 0.
A quadratic equation can also be solved by the method of completing the
square.
(i)
a2 + 2ab + b2 = (a + b)2
(ii)
a2 – 2ab + b2 = (a – b)2
Discriminant of the quadratic equation ax2 + bx + c = 0, a 0 is given by
D = b2 – 4ac.
69
[Class-X – Maths]
1.
2.
3.
4.
5.
The roots of ax2 + bx + c = 0, a 0 are real if b2 – 4ac is
(a)
0
(b)
>0
(c)
<0
(d)
None of these
Maximum number of solutions of a quadratic equation are :
(a)
0
(b)
1
(c)
2
(d)
3
If the equation x2 – (2 + m) x + (– m2 – 4m – 4) = 0 has coincident roots,
then
(a)
m = 0, m = 1
(b)
m = 2, m = 2
(c)
m = – 2, m = – 2
(d)
m = 6, m = 1
If one root of x2 – ax – 4 = is negative of the other then the value of a is:
(a)
0
(b)
1
(c)
2
(d)
4
(b)
x2 + 1 = (x + 3)2
Which is a quadratic equation?
(a)
x
1
2
x
6.
7.
1
(c)
x (x + 2)
(d)
x
(a)
x2 + 5x + 6 = 0
(b)
x2 + 5x – 6 = 0
(c)
x2 – 5x – 6 = 0
(d)
x2 – 5x + 6 = 0
.
x
If the roots of a quadratic equation are 2 and 3, then the equation is:
Roots of the equations x2 – 3x + 2 = 0 are
(a)
1, –2
(b)
–1, 2
(c)
–1, –2
(d)
1, 2
[Class-X – Maths]
70
8.
9.
10.
11.
12.
13.
14.
If the roots of a quadratic equation are equal, then discriminant is
(a)
1
(b)
0
(c)
greater than 0
(d)
less than zero.
1
If one root of 2x2 + kx + 1 = 0 is –
(a)
3
, then the value of ‘k’ is
2
(b)
–3
(c)
5
(d)
–5
The sum of the roots of the quadratic 5x2 – 6x + 1 = 0 is
(a)
6
5
(b)
1
5
(c)
5
 6
(d)
1
 5
The product of the roots of the quadratic equation 2x2 + 5x – 7 = 0 is
(a)
5
2
(b)
7
 2
(c)
5
 2
(d)
7
2
If the roots of the quadratic 2x2 + kx + 2 = 0 are equal then the value of
‘k’ is
(a)
4
(b)
–4
(c)
±4
(d)
± 16
If the roots of x2 + px + 12 = 0 are in the ratio 1 : 3 then p = _____
(a)
±2
(b)
8
(c)
4
(d)
8
If the sum and product of roots of a quadratic equation are 
respectively, then the equation is
71
75
and
22
[Class-X – Maths]
15.
(a)
2x2 + 7x + 5 = 0
(b)
2x2 – 7x + 5 = 0
(c)
2x2 – 7x – 5 = 0
(d)
2x2 + 7x – 5 = 0
If one root of ax2 + bx + c = 0, a 0 is the reciprocal of the other then
c = .....
(a)
b
(b)
a
(c)
c
(d)
0
16.
If one root of the equation x2 + 7x + k = 0 is –2, then find the value of k
and the other root.
17.
If the roots of 4x2 + 3px + 9 = 0 are real and distinct, find the value of p?
18.
For what value of m the sum of the roots of mx2 + 6x + 4m = 0 be equal
to the product of the roots ?
19.
The product of two consecutive odd integers is 63. Represent this in form
of a quadratic equation.
20.
Find the roots of the equation : x 
21.
Find the value of c for which one roots of 4x2 – 2x + (c – 4) = 0 is
reciprocal of the other.
22.
Divide 51 in to two parts such that their product is 378.
23.
Find ‘k’ so that (k – 12) x2 + 2 (k – 12) x + 2 = 0 has equal roots.
(k 12).
24.
If (–5) is a root of the equation 2x2 + px – 15 = 0 and the equation
p(x2 + x) + k = 0 has equal roots, find values of p and k.
25.
Find the value of K, so that the difference of the roots of x2 – 5x + 3
(k – 1) is 11.
26.
The difference of two numbers is 5 and the difference of their reciprocals
1
. Find the numbers.is
10
[Class-X – Maths]
11
4 , x 0.
x4
72
27.
If the roots of the equation (b – c)x2 + (c – a) x + (a – b) = 0 are equal,
then prove that 2b = a + c.
28.
Find the nature of the roots of the following quadratic equations. If roots
are real, find them.
5x2 – 3x + 2 = 0.
(a)
29.
(b)
2x2 – 9x + 9 = 0.
Sum of two numbers is 15, if sum of their reciprocals is
numbers.
3
. Find the
10
Solve the following quadratic equations:30.
2x 3 x 3 
225 5
x 3 2x 3 
31.
1 1 11
, a, b, x 0 and x – a b 
a bxabx
32.
4 3x
33.
ab x2 + (b2 – ac) x – bc = 0.
34.
35.
36.
2
x 3, x 
3
2
5x 2 3 0.
x 1 x 310
, x 2, x 4.
x 2x 43
1111
, x 4, x 7.
x 4x 730
Shikha and Mona each wish to grow 100 m2 rectangular vegetable garden.
They used 30m wire each for fencing three sides of the garden letting
compound wall of their houses act as the fourth side. Find the possible
sides of their gardens.
37.
A shop keeper buys a number of pens for 80. If he had bought 4 more
pens for the same amount, each pen would have cost him 1 less. Howw
many pens did he buy?
38.
A two digit number is such that the product of the digits is 35, when 18 is
added to the number, the digits inter change their places. Find the number.
39.
Three consecutive positive integers are such that the sum of the square
of the first and the product of the other two is 46, find the integers.
73
[Class-X – Maths]
40.
A motor boat whose speed is 9 km/h in still water goes 12 km down
stream and comes back in a total time 3 hours. Find the speed of the
stream.
41.
A train travels 360 km at uniform speed. If the speed had been 5 km/hr
more it would have taken 1 hour less for the same journey. Find the speed
of the train.
42.
The hypotenuse of right angled triangle is 6cm more than twice the shortest
side. If the third side is 2 cm less than the hypotenuse, find the sides of
the triangle.
43.
By a reduction of 2 per kg in the price of tomato. Anita can purchase
2 kg tomato more for 224. Find the original price of tomato per kg.
44.
Two trains leave a Railway Station at the same time. The first train travels
due west and the second train due North. Speed of first train is 5 km/h
faster than the second train.If after two hours they are 50 km apart, find
the average speed of the faster train.
45.
A girl is twice as old as her brother. Four years hence, the product of their
ages will be 160. Find their present ages.
46.
Two years ago a man’s age was three times the square of his son’s age.
Three years hence his age will be four times his son’s age. Find their
present ages.
47.
In a cricket match against Sri Lanka, Sehwag took one wicket less than
twice the number of wickets taken by Unmukt. If the product of the number
of wickets taken by these two is 15, find the number of wickets taken by
each.
48.
8
Two pipes running together can fill aminutes.
cistern inIf 2one pipe
11
takes 1 minute more than the other to fill the cistern, find the time in which
each pipe would fill the cistern alone.
49.
6500 were divided equally among a certain number of students. Had
there been 15 more students, each would have got 30 less. Find the
original number of students. What are the values reflected in this question?
[Class-X – Maths]
74
50.
A takes 10 days less than the time taken by B to finish a piece of work.
If both A and B together can finish the work in 12 days, find the time taken
by B to finish the work alone. What are the moral values reflected in this
question which are to be adopted in our life?
1.
b
2. c
3.
c
4. a
5.
a
6. d
7.
d
8. b
9.
a
10. a
11.
b
12. c
13.
b
14. a
15.
b
16. k = 10, second root = – 5
17.
p < –4 or p > 4
18.
3
– 2
19.
x2 + 2x – 63 = 0
20.
1
4, 4
21.
c=8
22. 9, 42
23.
k = 14
24.
7
7, 4
26.
10, 5
27.
Hint : For equal roots D = 0.
29.
5, 10
30.
32.
34.
6, 1
32
,
43
5
5, 2
25. k = –7
28. (a) Not real roots.
(b) Roots are real, 3,
31. –a, –b
33.
3
.
2
cb
,
ba
35. 1, 2
75
[Class-X – Maths]
36.
5 × 20, 10 × 10
37. 16
38.
57
39. 4, 5, 6
40.
3 km/hr.
41. 40 km/hr.
42.
26 cm, 24 cm, 10 cm
43. Rs. 16
44.
20 km.
45. 12, 6
46.
29 yrs., 5 yrs.
47. 3, 5
48.
5, 6 min
49. 50, Equality
50.
30 days, Unity
[Class-X – Maths]
76
CHAPTER 2
ARITHMETIC PROGRESSION
1.
Sequence : A set of numbers arranged in some definite order and formed
according to some rules is called a sequence.
2.
Progression : The sequence that follows a certain pattern is called
progression.
3.
Arithmetic Progression : A sequence in which the difference obtained by
subtracting any term from its preceding term is constant throughout, is
called an arithmetic sequence or arithmetic progression (A.P.).
The general form of an A.P. is a, a + d, a + 2d, ..... (a : first term
d : common difference).
4.
General Term : If ‘a’ is the first term and ‘d’ is common difference in an
A.P., then nth term (general term) is given by an = a + (n – 1) d .
5.
Sum of n Terms of an A.P. : If ‘a’ is the first term and ‘d’ is the common
difference of an A.P., then sum of first n terms is given by
Sn 
n
Sn 
n
2a n 1d 
2
If ‘l’ is the last term of a finite A.P., then the sum is given by
6.
(i)
a l .
2
If an is given, then common difference d = an – an–1.
(ii)
If sn is given, then nth term is given by an = sn – sn–1.
(iii)
If a, b, c are in A.P., then 2b = a + c.
(iv)
If a sequence has n terms, its rth term from the end = (n – r + 1)th
term from the beginning.
77
[Class-X – Maths]
1.
2.
3.
Three numbers in A.P. have sum 24. The middle term is—
(a)
6
(b)
8
(c)
3
(d)
2
If nth term of on A.P. is 2n + 7, then 7th term of the A.P. is
(a)
15
(b)
21
(c)
28
(d)
25
5 23n
n , then sum of its 10 terms
22
If the sum of n terms of an A.P. is
is
4.
5.
6.
7.
(a)
250
(b)
230
(c)
225
(d)
265
If nth term of the A.P. 4, 7, 10, ________ is 82, then the value of n is
(a)
29
(b)
27
(c)
30
(d)
26
If a, b and c are in A.P. then
a c
2
(a)
a
b c
2
(b)
b
(c)
c
a b
2
(d)
b=a+c
12th term of the A.P. x – 7, x – 2, x + 3 is
(a)
x + 62
(b)
x – 48
(c)
x + 48
(d)
x – 62
Common difference of A.P. 8
(a)
1
8
[Class-X – Maths]
123
, 8 , 8 , ________ is
888
1
(b)
18
78
(c)
8.
9.
10.
11.
12.
13.
14.
1
8 8
(d)
1
nth term of the A.P. –5, –2, 1, ________ is
(a)
3n + 5
(b)
8 – 3n
(c)
8n – 5
(d)
3n – 8
If nth term of an A.P. is 5 – 3n, then common difference of the A.P. is
(a)
2
(b)
–3
(c)
–2
(d)
3
If 5, 2k – 3, 9 are in A.P., then the value of ‘k’ is
(a)
4
(b)
5
(c)
6
(d)
–5
Sum of first 10 natural numbers is
(a)
50
(b)
55
(c)
60
(d)
65
9th term from the end of the A.P. 7, 11, 15, _______ 147 is
(a)
135
(b)
125
(c)
115
(d)
110
If the sum of n terms of an A.P. is n2, then its nth term is
(a)
2n – 1
(b)
2n + 1
(c)
n2 – 1
(d)
2n – 3
The sum of 3 numbers in A.P. is 30. If the greatest number is 13, then its
common difference is
(a)
4
(b)
3
(c)
2
(d)
5
79
[Class-X – Maths]
15.
16.
17.
The sum of 6th and 7th terms of an A.P. is 39 and common difference is
3, then the first term of the A.P. is
Is
(a)
2
(b)
–3
(c)
4
(d)
3
.32, ______ an A.P.? If yes, then
find its next two terms.
Find an A.P. whose 2nd term is 10 and the 6th term exceeds the 4th term
by 12.
2,
8,
18,
18.
Which term of the A.P. 41, 38, 35... is the first negative term? Find the term
also.
19.
The first and the last term of an AP are 4 and 81 respectively. If common
difference is 7. Find the number of terms.
20.
Find the number of terms in an A.P. whose first term and 6th term are 3
and 23 respectively and sum of all terms is 406.
21.
How many two digits numbers between 6 and 102 are divisible by 6.
22.
If sn the sum of first n terms of an A.P. is given by sn = 3n2 – 4n, then find
its nth term and common difference.
23.
The sum of 4th and 8th terms of an A.P. is 24 and sum of 6th and 10th terms
is 44. Find A.P.
24.
Find the sum of odd positive integers between 1 and 199.
25.
How many terms of the A.P. 22, 20, 18, _____ should be taken so that
their sum is zero?
26.
4k + 8, 2k2 + 3k + 6, 3k2 + 4k + 4 are the angles of a triangle. These form
an A.P. Find value of k.
27.
If 11 times of 11th term is equal to 17 times of 17th term of an A.P. find its
28th term.
28.
Find an A.P. of 8 terms, whose first term is
117
.and last term is
26
[Class-X – Maths]
80
29.
The fourth term of an A.P. is equal to 3 times the first term and the seventh
term exceeds twice the third term by 1. Find the first term and common
difference of the A.P.
30.
Find the middle term of the A.P. 20, 16, 12, ......, –176.
31.
If 2nd, 31st and last terms of on A.P. are
Find the number of terms in the A.P.
31 113
,and respectively..
4 22
32.
Find the number of terms of the A.P. 57, 54, 51, ______ so that their sum
is 570.
33.
The sum of three numbers in A.P. is 24 and their product is 440. Find the
numbers.
34.
Find the sum of the first 40 terms of an A.P. whose nth term is 3 – 2n.
35.
In an A.P., the first term is 2, the last term is 29 and the sum of the terms
is 155. Find common difference ‘d’.
36.
The sum of 5th and 9th terms of an A.P. is 8 and their product is 15. Find
the sum of first 28 terms of the A.P.
37.
Find the sum of all the three digits numbers each of which leaves the
remainder 3 when divided by 5.
38.
The sum of first six terms of an A.P. is 42. The ratio of the 10th term to the
30th term is 1 : 3. Find first term and 11th term of the A.P.
39.
Kriti, starts a game and scores 200 points in the first attempt and she
increases the points by 40 in each attempt. How many points will she
score is the 30th attempt?
40.
The eight term of on A.P. is half the second term and the eleventh term
exceeds one-third of its fourth term by 1. Find a15.
41.
The sum of first 8 terms of an A.P. is 140 and sum of first 24 terms is 996.
Find the A.P.
42.
The digits of a three digits positive number are in A.P. and the sum of digits
is 15. On subtracting 594 from the number the digits are interchanged.
Find the number.
81
[Class-X – Maths]
43.
A picnic group for Shimla consists of students whose ages are in A.P., the
common difference being 3 months. If the youngest student Neeraj is just
12 years old and the sum of ages of all students is 375 years. Find the
number of students in the group.
44.
The sum of first 20 terms of an A.P. is one third of the sum of next 20
terms. If first term is 1, then find the sum of first 30 terms.
45.
The sum of first 16 terms of an A.P. is 528 and sum of next 16 terms is
1552. Find the first term and common difference of the A.P.
46.
Nidhi saves Rs. 2 on day 1, Rs. 4 on day 2, Rs. 6 on day 3 and so on.
How much money she save in month of Feb. 2011? What values of Nidhi
are depicted in the question?
47.
An old lady Krishna Devi deposited Rs. 120000 in a bank at 8% interest
p.a. She uses the annual interest to give five scholarships to the students
of a school for their overall performances each year. The amount of each
scholarship is Rs. 300 less than the preceding scholarship. Find the amount
of each scholarship. What values of lady are depicted here?
48.
Ram asks the labour to dig a well upto a depth of 10 metre. Labour
charges are Rs. 150 for first metre and Rs. 50 for each subsequent metre.
As labour was uneducated, he claims Rs. 550 for the whole works. What
should be the actual amount to be paid to the labour? What value of Ram
is depicted in the question if he pays Rs. 600 to the labour?
49.
200 logs are stacked such that 20 logs are in the bottom row, 19 in the
next now, 18 in the row next to it and so on. In how many rows are the
200 logs placed? What value is depicted in the pattern of logs?
50.
Puru and Ashu live in two different villages 165 km apart. They want to
meet each other but there is no fast means of transport. Puru travels 15km
the first day, 14 km the second day, 13 km the third day and so on. Ashu
travels 10 km the first day, 12 km the second dry, 14 km the third day and
so on. After how many days will they meet. What value of Puru and Ashu
is depicted here?
[Class-X – Maths]
82
1.
b
2. b
3.
d
4. b
5.
b
6. c
7.
a
8. d
9.
b
10. b
11.
b
12. c
13.
a
14. b
15.
d
16.
17.
4, 10, 16, ...............
18. 15th term, –1
19.
12
20. 14
21.
15
22. 6n – 7, 6
23.
–13, –8, –3, 2 ...............
24. 9800
25.
23
26. 0, 2
27.
0
28.
29.
3, 2
30. –76, –80
31.
59
32. 19 or 20
33.
5, 8, 11
34. –1520
35.
3
36. 217, 7 d 
Yes,
50,
72
1, 5 7 9 11 13 15 17
, , , , ,,,
26666666
1
2
37.
99090
38. First term = 2, 11th term = 22
39.
1360
40. 3
41.
7, 10, 13, 16, ...............
42. 852
43.
25
44. 900
45.
3, 4
46. Rs. 812, Saving, Economy
47.
Rs. 2520, Rs. 2220, Rs. 1920, Rs. 1620, Rs. 1320 Love for children,
charity, small savings
48.
Rs. 600, Honesty, Sincerity
49.
16, Space saving, creativity, Reasoning, Balancing.
50.
6 days, Affection for each other.
83
[Class-X – Maths]
CHAPTER 3
CO-ORDINATE GEOMETRY
1.
The length of a line segment joining A and B is the distance between two
x x
points A (x1, y1) and B (x2, y2) is
2.
2
2
1
y 2 y1 
The distance of a point (x, y) from the origin is
2
.
x 2 y 2 . The distance
of P from x-axis is y units and from y-axis is x-units.
3.
The co-ordinates of the points p(x, y) which divides the line segment
joining the points A(x1, y1) and B(x2, y2) in the ratio m1 : m2 are
m 1x 2 m 2 x 1 m 1y 2 m 2 y 1 
,
m m
m1 m 212
m1
we can take ratio as k : 1, k 
.
m2
4.
5.
The mid-points of the line segment joining the points P(x1, y1) and
Q(x2, y2) is
x1 x 2 y1 y2 
,

The area22
of the triangle formed by the points (x1, y1), (x2, y2) and (x3, y3)

is the numeric
value of the expressions
1
x y y 3 x 2 y 3 y 1x 3 y 1 y 2 .21 2
6.
If three points are collinear then we can not draw a triangle, so the area
will be zero i.e.
|x1(y2 – y3) + x2 (y3 – y1) + x3(y1 – y2)| = 0
[Class-X – Maths]
84
1.
2.
3.
4.
5.
6.
7.
P is a point on x axis at a distance of 3 unit from y axis to its left. The
coordinates of P are
(a)
(3, 0)
(b)
(0, 3)
(c)
(–3, 0)
(d)
(0, –3)
The distance of point P (3, –2) from y-axis is
(a)
3 units
(b)
(c)
–2 units
(d)
2 units
13 units
The coordinates of two points are (6, 0) and (0, –8). The coordinates of
the mid point are
(a)
(3, 4)
(b)
(3, –4)
(c)
(0, 0)
(d)
(–4, 3)
If the distance between (4, 0) and (0, x) is 5 units, the value of x will be
(a)
2
(b)
3
(c)
4
(d)
5
(a)
(a, 0)
xy
7 intersects y-axis are
ab
(b)
(0, b)
(c)
(0, 7b)
(d)
The coordinates of the point where line
(2a, 0)
The area of triangle OAB, the coordinates of the points A (4, 0)
B (0, –7) and O origin, is
(a)
11 sq. units
(b)
18 sq. units
(c)
28 sq. units
(d)
14 sq. units
11 2
The distance between the points P , 5 and Q , 5 is
33
(a)
6 units
(b)
4 units
(c)
3 units
(d)
2 units
85
[Class-X – Maths]
8.
9.
xy
The line
1 intersects both the axis at P and Q, the coordinates
24
of the mid point of PQ are
(a)
(1, 2)
(b)
(2, 0)
(c)
(0, 4)
(d)
(2, 1)
A line is drawn through a point P (3, 2) parallel to x – axis. The distance
of the line from x-axis is–
(a)
2 units
(c)
10.
11.
12.
13.
14.
13 units
(b)
3 units
(d)
none.
The distance between the line 2x + 4 = 0 and x – 5 = 0 is
(a)
9 units
(b)
1 unit
(c)
5 units
(d)
7 units
The distance between the points (5 cos 35°, 0) and (0, 5 cos 55°) is
(a)
10 units
(b)
5 units
(c)
1 unit
(d)
2 units
The points (–4, 0), (4, 0) and (0, 3) are the vertices of a :
(a)
right triangle
(b)
Isosceles triangle
(c)
equilateral triangle
(d)
Scalene triangle
The perimeter of triangle formed by the points (0, 0), (2, 0) and (0, 2) is
(a)
4 units
(b)
6 units
(c)
6 2 units
(d)
4 2 2 units
AOBC is a rectangle whose three vertices are A (0, 3), O (0, 0), B (5, 0).
The length of its diagonal is :
(a)
5 units
(c)
[Class-X – Maths]
34 units
86
(b)
3 units
(d)
4 units
15.
If the centroid of the triangle formed by (9, a), (b, –4) and (7, 8) is (6, 8)
then (a, b) is
(a)
(4, 5)
(b)
(5, 4)
(c)
(5, 2)
(d)
(3, 2)
16.
Find the value of a so that the point (3, a) lies on the line represented by
2x – 3y = 5.
17.
The co-ordinates of vertex P of are (–4, 2) and the point S (2, 5)
is the mid point of QR. What will be the co-ordinates of centroid.
18.
What is the value of a if the points (3, 5) and (7, 1) are equidistant from
the point (a, 0)?
19.
20.
b a
Prove that the points 0, a , , and (b, 0) are collinear.
.
AB is diameter of a circle
2 2 with centre at origin. What are the coordinates
of B if coordinates of A are (3, –4)?
21.
If C is a point lying on the line segment AB joining A (1, 1) & B (2, –3)
such that 3AC = BC. Find the coordinates of C.
22.
For what value of p, are the points (–3, 9), (2, p) and (4, –5) collinear?
23.
If the coordinates of the vertices of a triangle are A (6, 7), B (4, –5),
C (x, 2x) and its area is 3 square units. Find x.
24.
Find the coordinates of point P if P and Q trisect the line segment joining
the points A (1, –2) and B (–3, 4).
25.
The
–1, 7) and it passes through the point
(–3, –1). If diameter is 20 units, find 
26.
Find the abscissa of a point where ordinate is 4 and which is at a distance
5 units from (5, 0).
27.
Find a point on y-axis which is equidistant from the points (–2, 5) and
(2, –3).
87
[Class-X – Maths]
28.
The mid point of the line segment joining the points (5, 7) and (3, 9) is also
the mid point of the line segment joining the points (8, 6) and (a, b). Find
a and b.
29.
Find the coordinates of the point which divides the line segment joining the
points (1, 3) and (2, 7) in the ratio 3 : 4.
30.
If the points (x, y), (–5, –2) and (3, –5) are collinear, prove that 3x + 8y
+ 31 = 0
31.
The point K (1, 2) lies on the perpendicular bisector of the line segment
joining the points E (6, 8) and F (2, 4). Find the distance of the point K
from the line segment EF.
32.
The area of triangle ABC is 5 square units. Two of its vertices are A (2,
1) and B (3, –2). The third vertex C lies on x – y – 3 = 0, find coordinates
of C.
33.
Find the distance between the points A (a, b) and B (b, a) if a – b = 4.
34.
Three vertices of a parallelogram taken in order are (–3, 1), (1, 1) and
(3, 3). Find the coordinates of fourth vertex.
35.
Triangle ABC is an isosceles triangle with AB = AC and vertex A lies on
y-axis. If the coordinates of B and C are (–5, –2) and (3, 2) respectively
then find the coordinates of vertex A.
36.
If A (3, 0), B (4, 5), C (–1, 4) and D (–2, –1) are four points in a plane,
show that ABCD is a rhombus but not a square.
37.
Find the coordinates of a point which is
38.
3
of the way (3, 1) to (–2, 5).
4
The area of a triangle with vertices (6, –3), (3, K) and (–7, 7) is 15 sq. unit.
Find the value of K.
39.
Determine the ratio in which the line 3x + y = 9, divides the line segment
joining the points A (1, 3) and B (2, 7).
40.
A point P on the x-axis divides the line segment joining the points (4, 5)
and (1, –3) in certain ratio. Find the coordinates of point P.
41.
In right angled = 90° and AB 34 unit. The coordinates of
points B, C are (4, 2) and (–
then find the value of y.
[Class-X – Maths]
88
42.
If the point P (3, 4) is equidistant from the points A (a + b, b – a) and B
(a – b, a + b) then prove that 3b – 4a = 0.
43.
The vertices of quadrilateral ABCD are A (–5, 7), B (–4, –5), C (–1, –6)
and D (4, 5). Find the area of quadrilateral ABCD.
44.
The line segment joining the points A (2, 1) and B (5, –8) is trisected at
the points P and Q such that P is nearer to A. If P also lies on line given
by 2x – y + k = 0, find the value of k.
45.
The line segment joining the points (3, –4) and (1, 2) is trisected at the
46.
5
point P and Q. If the coordinates of P and Q are (p, –2) and , q 
3
respectively, find the values of p and q.
A teacher is asked three students to stand to form a triangle at the points
P (–1, 3), Q (1,–1) and R (5, 1). Suddenly a fourth boy came and shows
his interest in participating the activity. She asked him to stand at point mid
way between Q & R. What is his distance from P. What values of the
teacher appears when she agreed the fourth boy to participate?
47.
To solve a riddle a girl is asked to join the three points A (7, 5), B (2, 3)
and C (6, –7) with a sketchpen. After joining these points a triangle is
obtained by her. What type of triangle is it? What values are depicted in
the question?
48.
The coordinates of the houses of Mona and Nishi are (7, 3) & (4, –3)
respectively. The coordinates of their school are (2, 2). If they both start
for school at the same time in the morning and reaches at the same time,
who walks fast? What values are depicted from the question?
49.
Prove that the points A (a, o), B (o, b) & C (1, 1) are collinear if
50.
For social awareness regarding bad effects of pollution a school started a
campaign for which 1000 students were asked to make triangular badges
with coordinates (5, 6), (1, 5), (2, 1) If cost of 1cm2 of paper is Rs. 2 find
the total expenditure incurred in this campaign. What values are depicted
in the question?
89
11
1
ab
[Class-X – Maths]
1.
c
2. a
3.
b
4. b
5.
c
6. d
7.
c
8. a
9.
a
10. d
11.
b
12. b
13.
d
14. c
15.
c
16. a 
17.
(0, 4)
18. a = 2
20.
(–3, 4)
22.
p=–1
24.
1
3 , 0

25. = 2, –4
26.
2, 8
27. (0, 1)
28.
a = 0, b = 10
31.
5 units
10 33 
,29. 
77 

32. (5, 2)
33.
4 2 units
34. (–1, 3)
35.
(0, –2)
3
37. 4 , 4 

38.
K
21
13
[Class-X – Maths]
1
3
5 
21. , 0 
3 
23. x = 8
39. 3 : 4
90
43.
17
8 , 0

72 Square Units
45.
p
40.
41. y = –1, 5
44. K = – 8
7
46.
,q 0
3
5 Units, interest in Mathematics, Friendship, Cooperation
47.
(a) Right Angled Triangle
(b) Sports, Activeness, Critical thinking.
48.
(a) Mona, (b) Time bound, Reality
50.
Rs. 11500
91
[Class-X – Maths]
CHAPTER 4
SOME APPLICATIONS OF
TRIGONOMETRY
1.
Line of Sight : The line of sight is the line drawn from the eyes of an
observer to a point in the object viewed by the observer.
2.
Angle of Elevation : The angle of elevation is the angle formed by the line
of sight with the horizontal, when it is above the horizontal level i.e. the
case when we raise our head to look at the object.
3.
Angle of Depression : The angle of depression is the angle formed by the
line of sight with the horizontal when it is below the horizontal i.e. case
when we lower our head to look at the object.
1.
The length of the shadow of a man is equal to the height of man. The
angle of elevation is
2.
3.
(a)
90°
(b)
60°
(c)
45°
(d)
30°
The length of the shadow of a pole 30m high at some instant is 10 3 m.
The angle of elevation of the sun is
(a)
30°
(b)
60°
(c)
45°
(d)
90°
Find the angle of depression of a boat from the bridge at a horizontal
distance of 25m from the bridge, if the height of the bridge is 25m.
[Class-X – Maths]
92
4.
5.
(a)
45°
(b)
60°
(c)
30°
(d)
15°
The tops of two poles of height 10m and 18m are connected with wire. If
wire makes an angle of 30° with horizontal, then length of wire is
(a)
10m
(b)
18m
(c)
12m
(d)
16m
From a point 20m away from the foot of the tower, the angle of elevation
of the top of the tower is 30°. The height of the tower is
20
6.
(a)
20 3 m
(b)
(c)
40 3 m
(d)
m
3
40
m
3
1
The ratio of the length of a tree and its shadow is 1 :
. The angle of
3
elevation of the sun is
7.
(a)
30°
(b)
45°
(c)
60°
(d)
90°
A kite is flying at a height of 50 3 m above the level ground, attached
to string inclined at 60° to the horizontal, the length of string is
8.
(a)
100 m
(b)
50 m
(c)
150 m
(d)
75 m
In given Fig. 1 the perimeter of rectangle ABCD is
D
C
10 m
30°
A
B
Fig. 1
93
[Class-X – Maths]
9.
10.
11.
(a)
40 m
(b)
20

3 1 m

(c)
60 m
(d)
10

3 1 m

A tree is broken at a height of 10 m above the ground. The broken part
touches the ground and makes an angle of 30° with the horizontal. The
height of the tree is
(a)
30 m
(b)
20 m
(c)
10 m
(d)
15 m
The angle of elevation of the top of a 15 m high tower at a point 15 m away
from the base of the tower is
(a)
30°
(b)
45°
(c)
60°
(d)
75°
In given Fig. 2, D is mid point of BC, = 1 and = 2 then
tan 1 : tan 2 is equal to
A
1
C
2
D
B
Fig. 2
(a)
2:1
(b)
1:2
(c)
1:1
(d)
1:3
[Class-X – Maths]
94
12.
8
In given Fig. 3, tan 
(a)
16 m
(c)
32 m
if PQ = 16 m, then the length of PR is
15
(b)
34 m
(d)
30 m
P

R
Q
Fig. 3
13.
14.
15.
The height of a tower is 50 m. When angle of elevation changes from 45°
to 30°, the shadow of tower becomes x metres more, the value of x is
(a)
50 m
(b)
(c)
50 3 m
(d)
50
50


3 1 m
m
3
The angle of elevations of a building from two points on the ground 9m and
16m away from the foot of the building are complementary, the height of
the building is
(a)
18 m
(b)
16 m
(c)
10 m
(d)
12 m
In the given Fig. 4, If ABCD is a rectangle then AL + AM =
70
(a)
70 3m
(b)
(c)
210 3
(d)
95
m
3
70 m
[Class-X – Maths]
L
D
C
60°
45 m
M
60°
A
B
60 m
Fig. 4
16.
From a point on the ground the angle of elevations of the bottom and top
of a water tank kept on the top of the 30m high building are 45° and 60°
respectively. Find the height of the water tank.
17.
The shadow of a tower standing on the level ground is found to be 60m
shorter when the sun’s altitude changes from 30° to 60°, find the height
of tower.
18.
The angle of elevation of a cloud from a point h metres above a lake is
and the angle of depression of its reflection in the lake is prove that
2h sec 
the distance of the cloud from the point of observation is tan tan .
19.
The angle of elevation of a bird from a point on the ground is 60°, after
50 seconds flight the angle of elevation changes to 30°. If the bird is flying
at the height of 500 3 m. Find the speed of the bird.
20.
The angle of elevation of a jet fighter plane from a point A on the ground
is 60°. After a flight of 15 seconds, the angle of elevation changes to 30°.
If the jet is flying at a speed of 720 km/h. Find the constant height at which

the jet is flying. Take
3 1.732 .

21.
From a window 20m high above the ground in a street, the angle of
elevation and depression of the top and the foot of another house opposite
side of the street are 60° and 45° respectively. Find the height of opposite
house.
22.
An aeroplane flying at a height of 1800m observes angles of depressions
of two points on the opposite bank of the river to be 60° and 45°, find the
width of the river.
[Class-X – Maths]
96
23.
The angle of elevation of the top of the tower from two points A and B
which are 15m apart, on the same side of the tower on the level ground
are 30° and 60° respectively. Find the height of the tower and distance of
point B from the base of the tower. Take

3 1.732

24.
The angle of elevation of the top of a 10m high building from a point P on
the ground is 30°. A flag is hoisted at the top of the building and the angle
of elevation of the top of the flag staff from P is 45°. Find the length of the
flag staff and the distance of the building from point P.
25.
The angle of elevation of a bird from a point 12 metres above a lake is 30°
and the angle of depression of its reflection in the lake is 60°. Find the
distance of the bird from the point of observation.
26.
The shadow of a vertical tower on level ground increases by 10 m when
sun’s attitude changes from 45° to 30°. Find the height of the tower, upto
one place of decimal

3 1.73 .
27.
A man on a cliff observes a boat at an angle of depression of 30°, which
is approaching the shore to point ‘A’ immediately beneath the observer with
a uniform speed, 12 minutes later, the angle of depression of the boat is
found to be 60°. Find the time taken by the boat to reach the shore.
28.
A man standing on the deck of a ship, 18m above the water level observes
that the angle of elevation and depression of the top and the bottom of a
cliff are 60° and 30° respectively. Find the distance of the cliff from the ship
and height of the cliff.
29.
A person standing on the bank of a river observes that the angle of
elevation of the top of a tree standing on the opposite bank is 60°. When
he moves 40m away from the bank he finds the angle of elevation to be
30°. Find the height of the tree and the width of the river.
30.
An aeroplane, when 300 m high, passes vertically above another plane at
an instant when the angle of elevation of two aeroplanes from the same
point on the ground are 60° and 45° respectively. Find the vertical distance
between the two planes.
31.
The angle of depression of the top and bottom of a 10m tall building from
the top of a tower are 30° and 45° respectively. Find the height of the tower
and distance between building and tower.
32.
A boy standing on a horizontal plane, finds a bird flying at a distance of
97
[Class-X – Maths]
100m from him at an elevation of 30°. A girl, standing on the roof of 20m
high building, finds the angle of elevation of the same bird to be 45°. Both
the boy and girl are on the opposite sides of the bird. Find the distance
of bird from the girl.
33.
As observed from the top of a light house, 100 m high above sea level,
the angle of depression of a ship, sailing directly towards it, changes from
30° to 60°. Determien the distance travelled by the ship during the period
of observation. Use
34.
3 1.732
.
The angle of elevation of a building from two points P and Q on the level
ground on the same side of the building are 54° and 36° respectively. If the
distance of the points P and Q from the base of the building are 10m and
20m respectively, find the height of the building. Take
35.

2 1.414
A pole of height 5m is fixed on the top of the tower. The angle of elevation
of the top of the pole as observed from a point A on the ground is 60° and
the angle of depression of the point A from the top of the tower is 45°. Find

the height of tower. Take
3 1.732

36.
Anand is watching a circus artist climbing a 20m long rope which is tightly
stretched and tied from the top of vertical pole to the ground. Find the
height of the pole if the angle made by the rope with the ground level is
30. What value is experienced by Anand?
37.
A fire in a building 'B' is reported on telephone in two fire stations P an
Q, 20 km apart from each other on a straight road. P observes that the
fire is at an, angle of 60° to the road, and Q observes, that it is at an angle
of 45° to the road. Which station should send its team and how much
distance will this team has to travel? What value is depicted from the
problem?
38.
A 1.2m tall girl spots a balloon on the eve of Independence Day, moving
with the wind in a horizontal live at a height of 88.2 m from the ground.
The angle of elevation of the balloon from the of the girl at an instant is
60°. After some time, the angle, of elevation reduces to 30°. Find the
distance travelled by the balloon. What value is depicted here?
[Class-X – Maths]
98

1.
c
2. b
3.
a
4. d
5.
b
6. c
7.
a
8. b
9.
a
10. b
11.
a
12. b
13.
b
14. d
15.
a
16.
17.
30 3 m
19. 20 m/sec.
20.
2598 m
21.
22.
600 3 
23.
Height = 12.97 m, distance = 7.5 m
24.
Length of flag staff 10
25.
24 3 m
26. 13.6 m
27.
18 minutes
28. 18 3 m, 72 m
29.
Height = 34.64 m, Width of the river = 20 m.
30.
1000 3 
31.
Height 5 3 
32.
30 m
33. 115.47 m
34.
14.14 m
35. 6.83 m
36.
10m, Fun, Entertainment
37.
Station P, 14.64, Reasoning ability. Thinking Skill.
38.
58 3m , Fun & game, Entertainment


30
20



3 1 m

3 1 m

3m


2 1 m, Distance of the building 10 3 m.

3m


3m

distance 5 3 
99

3m
[Class-X – Maths]
CHAPTER 5
CIRCLE
1.
Tangent to a Circle : It is a line that intersects the circle at only one point.
2.
There is only one tangent at a point of the circle.
1.
In the given fig. 1 PQ is tangent then + is equal to
Q
P
o
Fig. 1
2.
(a)
120°
(b)
90°
(c)
80°
(d)
100°
If PQ is a tangent to a circle of radius 5cm and PQ = 12 cm, Q is point
of contact, then OP is
(a)
13 cm
(b)
(c)
7 cm
(d)
[Class-X – Maths]
100
17 cm
119 cm
3.
In the given fig. 2 PQ and PR are tangents to the circle, = 70°, then
is equal to
Q
70°
P
o
R
Fig. 2
4.
(a)
35°
(b)
70°
(c)
40°
(d)
50°
In the given fig. 3 PQ is a tangent to the circle, PQ = 8 cm, OQ = 6 cm
then the length of PS is
Q
o
P
S
Fig. 3
5.
(a)
10 cm
(b)
2 cm
(c)
3 cm
(d)
4 cm
In the given fig. 4 PQ is tangent to outer circle and PR is tangent to inner
circle. If PQ = 4 cm, OQ = 3 cm and OR = 2 cm then the length of PR is
101
[Class-X – Maths]
Q
P
o
R
Fig. 4
6.
(a)
5 cm
(b)
(c)
4 cm
(d)
21 cm
3 cm
In the given fig. 5 P, Q and R are the points of contact. If AB = 4 cm,
BP = 2 cm then the perimeter of is
A
B
P
c
Q
R
o
Fig. 5
(a)
12 cm
(b)
8 cm
(c)
10 cm
(d)
9 cm
[Class-X – Maths]
102
7.
In the given fig. 6 the perimeter of is
A
3cm
R
Q
2 cm
C
B
P
5 cm
Fig. 6
8.
9.
(a)
10 cm
(b)
15 cm
(c)
20 cm
(d)
25 cm
The distance between two tangents parallel to each other to a circle is 12
cm. The radius of circle is
(a)
13 cm
(b)
6 cm
(c)
10 cm
(d)
8 cm
In the given fig. 7 a circle touches all sides of a quadrilateral. If AB = 6 cm,
BC = 5 cm and AD = 8 cm. Then the length of side CD is
D
C
B
A
Fig. 7
(a)
6 cm
(b)
8 cm
(c)
5 cm
(d)
7 cm
103
[Class-X – Maths]
10.
11.
In a circle of radius 17 cm, two parallel chords are drawn on opposite
sides of diameter. The distance between two chords is 23 cm and length
of one chord is 16 cm, then the length of the other chord is
(a)
34 cm
(b)
17 cm
(c)
15 cm
(d)
30 cm
In the given fig. 8 P is point of contact then is equal to
B
o
40°
P
A
Fig. 8
12.
(a)
50°
(b)
40°
(c)
35°
(d)
45°
In the given fig. 9 PQ and PR are tangents to the circle with centre O, if
= 45° then is equal to
Q
o
45°
R
Fig. 9
(a)
90°
(b)
110°
(c)
135°
(d)
145°
[Class-X – Maths]
104
P
13.
In the given fig. 10 O is centre of the circle, PA and PB are tangents to
the circle, then is equal to
A
o
Q
40°
P
B
Fig. 10
14.
(a)
70°
(b)
80°
(c)
60°
(d)
75°
In the given fig. 11 is circumscribed touching the circle at P, Q and
R. If AP = 4 cm, BP = 6 cm, AC = 12 cm, then value of BC is
A
4
P
R
6
B
Q
C
Fig. 11
15.
(a)
6 cm
(b)
14 cm
(c)
10 cm
(d)
18 cm
In the given fig. 12 is subscribing a circle and P is mid point of side
BC. If AR = 4 cm, AC = 9 cm, then value of BC is equal to
(a)
10 cm
(b)
11 cm
(c)
8 cm
(d)
9 cm
105
[Class-X – Maths]
A
R
B
Q
P
C
Fig. 12
16.
If d1, d2 (d2 > d1) be the diameters of two concentric circles and c be the
length of chord of a circle which is tangent to the other circle. Prove that
d22 = c2 + d12.
17.
The length of tangent to a circle of radius 2.5 cm from an external point
P is 6 cm. Find the distance of P from the nearest point of the circle.
18.
TP and TQ are the tangents from the external point T of a circle with
centre O. If = 30°, then find the measure of 
19.
In the given fig. 13 AP = 4 cm, BQ = 6 cm and AC = 9 cm. Find the semi
perimeter of 
A
m
4c
m
9c
C
R
P
Q
Fig. 13
[Class-X – Maths]
106
6 cm
B
20.
An incircle is drawn touching the sides of a right angled triangle whose
sides are a, b, c where c is the hypotenuse. Prove that the radius r of the
ab –c
circle is r
2
21.
Prove that the tangent at any point of a circle is perpendicular to the radius
through the point of contact.
22.
Prove that in two concentric circles the chord of the larger circle which
touches the smaller circle is bisected at the point of contact.
23.
In the given Fig. (14) AC is diameter of the circle with centre O and A is
point of contact, then find x.
C
x
o
B
40°
P
A
Q
Fig. 14
40.
In the given fig. (15) PA and PB are tangents from point P. Prove that
KN = AK + BN.
A
K
o
C
P
N
B
Fig. 15
25.
In the given fig. (16) PQ is chord of length 6 cm of the circle of radius 6
cm. TP and TQ are tangents. Find 
107
[Class-X – Maths]
P
o
6 cm
T
Q
Fig. 16
26.
A circle is inscribed in a having sides AB = 12 cm, BC = 8 cm and
AC = 10 cm find AD, BE, CF.
C
F
E
A
B
D
Fig. 17
27.
On the side AB as diameter of a right angled triangle ABC a circle is
drawn intersecting the hypotenuse AC in P. Prove that PB = PC.
28.
Two tangents PA and PB are drawn to a circle with centre O from an
external point P. Prove that = 2 
A
O
P
B
Fig. 18
[Class-X – Maths]
108
29.
If an isosceles triangle ABC in which AB = AC = 6 cm is inscribed in a
circle of radius 9 cm, find the area of the triangle.
30.
In the given fig. (19) AB = AC, D is the mid point of AC, BD is the diameter
of the circle, then prove that AE = 1/4 AC.
A
E
B
D
C
Fig. 19
31.
In the given fig. 20 OP is equal to the diameter of the circle with centre
O. Prove that is an equilateral triangle.
A
P
o
B
Fig. 20
32.
In the given fig. (21) AB = 13 cm, BC = 7 cm. AD = 15 cm. Find PC.
A
R
B
S
C
o
4 cm
P
Q
D
109
[Class-X – Maths]
33.
In the given fig. 22 from an external point P, a tangent PT and a line
segment PAB is drawn to a circle with centre O. ON is perpendicular on
the chord AB.
PA. PB = PN2 – AN2
Prove that : (i)
(ii)
PN2 – AN2 = OP2 – OT2
(iii)
PA. PB = PT2
T
A
O
B
A
N
Fig. 22
34.
AB is a diameter and AC is a chord of a circle with centre ) such that
= 30°. The tangent at c intersect extended AB at a point D. Prove
that BC = BD
35.
In the given fig. (23) PA and PB are tangents to the circle with centre O.
Prove that OP is perpendicular bisector of AB.
A
o
P
B
Fig. 23
[Class-X – Maths]
110
36.
In the given fig. (24) find the radius of the circle.
A
cm
23
R
29 cm
B
5 cm
S
o
r
Q
C
P
D
Fig. 24
37.
In the given fig. (25) if radius of circle r = 3 cm. Find the perimeter of

A
C
3 5 c mo
3 5 c m
B
Fig. 25
38.
A circle touches the side BC of a at P and AB and AC produced
at Q and R respectively. Prove that AQ is half the perimeter of 
39.
In the given fig. (26) XP and XQ are tangents from X to the circle with
centre O. R is a point on the circle. Prove that
XA + AR = XB + BR.
111
[Class-X – Maths]
P
A
O
R
X
B
Q
Fig. 26
40.
In the given fig. (27) PQ is tangent and PB is diameter. Find the value of
x and y.
P
y
A
o
x
y
Q
35°
B
Fig. 27
41.
42.
Distance between villages A and B is 7 km, B and C is 5 km and C and
A is 8 km. Village Pradhan wants to dig a well for three villages A, B and
C such that which is equidistant from all the villages.
i.
What should be the position of a well?
ii.
Which moral value is shown in this question of village Pradhan?
People of a circular villages wants to construct a road near-their village.
Road cannot go inside the village so they wants that the road should be
at minimum distance from the village.
i.
Which road will be at minimum distance from the centre of the
village?
ii.
Which moral value of people of village reflected here?
[Class-X – Maths]
112
43.
In the given figure 28, four roads touch to a circular village of radius 1700
m (Kanpur). Savita got a contract for constructing road AB and CD while
Vijay got a construct for constructing road AD and BC.
Prove that :
i.
AB + CD = AD + BC
ii.
Which moral value was shown in this questions.
A
B
KHANPUR
D
44.
C
Two roads starting from point P touch a circular path at A and B as shown
in the Figure. Sarita walks 10 km from P to A and Ramesh goes from P
to B at the same time.
i.
If Sarita wins in this walking race them find the distance covered
by Ramesh.
ii.
What value is described here.
A
10 km
P
B
Fig. 29
113
[Class-X – Maths]
45.
When Rahim was returning to home he saw a circular pit and informed to
municipal corporation for this circular pit immediately. Municipal corporation
employee fencing this it as shown in figure 30.
i.
Find the total length of fencing.
ii.
What mathematical concept was used to solve this problem?
iii.
Which value of Rahim was show here?
5 ft
D
C
3 ft
4 ft
A
6 ft
Fig. 30
1.
b
2. a
3.
c
4. d
5.
b
6. a
7.
c
8. b
9.
d
10. d
11.
a
12. c
13.
a
14. b
15.
a
17. 4 cm
18.
60°
19. 15 cm
23.
40°
25. 120°
26.
AD = 7 cm, BE = 5 cm, CF = 3 cm
[Class-X – Maths]
114
B
29.
2
32. 5 cm
36.
11 cm.
37. 32 cm
40.
x = 35°, y = 55°
41.
(i) A, B, C, will be on circumference of the circle and well lie on the centre
of the circle. (ii) Social, honesty, equality, love towards humanity.
42.
(i) tangent on circle (ii) Economic value
43.
(ii) Gender equality
44.
(i) 10 km (ii) Gender equality, healthy competition
45.
(i) 36 feet (ii) tangent are equal from the external point
(iii) Moral and social responsibility, logical reasoning.
115
[Class-X – Maths]
CHAPTER 6
CONSTRUCTIONS
1.
Construction should be neat and clean and as per scale given in question.
2.
Steps of construction should be provided only to those questions where it
is mentioned.
1.
A pair of tangents cannot be constructed to a circle at an angle of
2.
3.
4.
(a)
185°
(b)
90°
(b)
120°
(d)
60°
To divide a line segment AB in the Ratio 2 : 3, first a ray AX is drawn so
that is an acute angle and then at equal distances. Points are
marked on the ray AX such that the minimum number of these points is
(a)
3
(b)
6
(c)
5
(d)
2
To divide a line segment AB in the ratio 3:8, a ray AX is drawn first such
that is an acute angle and then points A1, A2, A3,... are located at
equal distances on the ray AX and the points B is joined to
(a)
A12
(b)
A11
(c)
A3
(d)
A8
By geometrical constructions, which of the following is possible to divide
a line segment in the given ratio?
[Class-X – Maths]
116
(a)
2 3 : 2 – 3 
(c)
5.
2:
(b)
6 :2
(d)
1
2

3 2) :

3 –2

In the construction of triangle similar and larger to a given triangle as per
given scale factor m : n. Then, it is possible only when
(a)
m>n
(b)
m=n
(c)
m<n
(d)
independent of Scale Factor
1.
a
2. c
3.
b
4. b
5.
a
1.
Draw a line segment AB = 7 cm. Take a point P on AB such that
AP : PB = 3 : 4.
2.
Draw a line segment PQ = 10 cm. Take a point A on PQ such that
PA2
. Measure the length of PA and AQ.
PQ5
3.
Construct a in which BC = 6.5 cm, AB = 4.5 cm and = 60°.
Construct another triangle similar to such that each side of new
4
triangle isof the corresponding sides of .
5
4.
Draw a triangle XYZ such that XY = 5 cm, YZ = 7 cm and = 75°.
3
Now construct a ~ with its sidestimes of the corresponding
2
sides of 
5.
Construct an isoscales triangle whose base is 8 cm and altitude 5 cm and
3
then construct another triangle whose sides aretimes the corresponding
4
sides of the given triangle.
117
[Class-X – Maths]
6.
Draw an isosceles with AB = AC and base BC = 7 cm and vertical
1
angle is 120°. Construct ~ with its sides 1 times of the
3
corresponding sides of 
7.
Draw in which = 90°, PQ = 6 cm, QR = 8 cm. Construct 
~ with its sides equal to 2/3rd of corresponding sides of 
8.
Construct a right angled triangle in which base is 2 times of the
perpendicular. Now construct a triangle similar to it with base 1.5 times
of the original triangle.
9.
Draw an equilateral triangle PQR with side 5cm. Now construct 
PQ1
.such that
PQ´2
10.
Draw a circle of radius 4 cm with centre O. Take a point P outside the
circle such that OP = 6cm. Draw tangents PA and PB to the circle. Measure
the lengths of PA and PB.
11.
Draw a line segment AB = 8 cm. Taking AB as diameter a circle is drawn
12.
Draw a circle of radius OP = 3 cm. Draw = 45° such that OQ = 5
cm. Now draw two tangents from Q to given circle.
13.
Draw a circle with centre O and radius 3.5 cm. Draw two tangents PA and
PB from an external point P such that = 45°. What is the value of
+ 
14.
Draw a circle of radius 4 cm. Now draw a set of tangents from an external
point P such that the angle between the two tangents is half of the central
angle made by joining the points of contact to the centre.
15.
Draw a line segment AB = 9 cm. Taking A and B as centres draw two
circles of radius 5 cm and 3 cm respectively. Now draw tangents to each
circle from the centre of the other.
16.
Draw a circle of radius 3.5 cm with centre O. Take point P such that
OP = 6 cm. OP cuts the circle at T. Draw two tangents PQ and PR. Join
Q to R. Through T draw AB parallel to QR such that A and B are point on
PQ and PR.
[Class-X – Maths]
118
17.
Draw a circle of diameter 7 cm. Draw a pair of tangents to the circle, which
are inclined to each other at an angle of 60°.
18.
Draw a circle with centre O and radius 3.5 cm. Take a horizontal diameter.
Extend it to both sides to point P and Q such that OP = OQ = 7 cm. Draw
tangents PA and QB one above the diameter and the other below the
diameter.
19.
A farmer wants to divide a 7 feet long sugarcane between his daughter
and son equally. Using geometrical construction taking sugarcane as a line
segment (7 an long) divide it into two parts.
20.
i.
Find the length of each part.
ii.
Which moral value was shown in the question?
Traffic police organize a slogan competition on a triangular sheet for aware
the people about traffic rules. It was decided that the winning slogan will
be displayed on road. Using a triangular sheet of in which
BC = 7cm, = 45° and = 105°. Ramesh win the competition.
i.
Construct a similar triangle whose corresponding sides are
times the sides of triangular sheet used by Ramesh.
ii.
What values traffic police wants to inculcate among people?
19.
(i) 3.5 feet (ii) Gender equality
20.
(ii) Obey rules, Social responsibility, Moral value, defence.
119
[Class-X – Maths]
4
3
CHAPTER 7
MENSURATION
AREA OF CIRCLE, SURFACE AREAS
AND VOLUMES
1.
2.
3.
Area of circle = 2 where ‘r’ is the radius of the circle.
r 2
.Area of Semi circle 
2
Area enclosed by two concentric circles
r
= (R2 – r2)
= (R + r) (R – r ); R > r
R
where ‘R’ and ‘r’ are radii of two concentric circles.
4.

l
circumference of the circle 
360
1
=× 2r
360°
5.
r
180

l r
180 an angle then area of
If the arc subtends

r 2 .the corresponding sector is
360
[Class-X – Maths]
120

l
r
r
7.
Angle described by minute hand in 60 minutes = 360°. Angle described by
360
minute hand in 1 minute 6 .
60 
8.
units.
9.
Total
units.
10.
Volume of cylinder of radius r and height h = 2h cubic units.
11.
Curved surface area of cone of radius r, height h and slant height l = 
r 2 h2 .
square units where l 
12.
Total surface area of cone = (l + r) sq. units.
13.
Volume of cone 
12
r h cubic units.
3
2 sq.
14.
units.
2 sq.
15.
16.
Total
17.
Volume of sphere of radius r units 
18.
Volume of hemisphere of radius r units 
units.
2 sq.
units.
43
r cubic units..
3
23
.r cubic units.
3
19.
Curved surface area of frustum = + R) sq. units, where l slant height
of frustum and radii of circular ends are r and R.
20.
Total surface area of frustum = (r + R) + 2 + R2) sq. units.
21.
Volume of Frustum 
1
h r 2 R2 rR cubic units..
3
where l 
h
2
2
R r 
121
[Class-X – Maths]
1.
2.
If the circumference and area of a circle are numerically equal then the
radius of the circle is equal to
(a)
r=1
(b)
r=7
(c)
r=2
(d)
r=0
The radius of a circle is 7 cm. What is the perimeter of the semi circle?
(a)
36 cm
(b)
(c)
3.
(d)
The radius of two circles are 13 cm and 6 cm respectively. What is the
radius of the circle which has circumference equal to the sum of the
circumference of two circles?
(a)
(c)
4.
5.
14 cm
25 cm
(b)
19 cm
(d)
32 cm
The circumference of two circles are in the ratio 4 : 5 what is the ratio of
the areas of these circles.
(a)
4:5
(b)
16 : 25
(c)
64 : 125
(d)
8 : 10
The area of a sector with radius 3 cm and angle of sector 40° is
(a)
2
(b)
(c)
2
(d)
cm2
2
6.
2.
(a)
Then the radius of circle is :
2 cm
(c)
7.
(b)
4 cm
(d)
8 cm
The area of an equilateral triangle is m2, its one side is
(a)
4m
(b)
(c)
33
m
4
(d)
[Class-X – Maths]
122
2m
8.
If ‘C’ and ‘A’ respectively be the circumference and area of the circle of
radius ‘r’, then
(a)
C
1
(b)
Ar
A
2
(c)
9.
10.
11.
12.
13.
1
Cr
2
C = 2 Ar
(d)
A = 2 Cr
The volume of a cuboid is 440 cm3. The area of its base is 66 cm2. What
is its height?
(a)
40
cm
3
(b)
20
cm
3
(c)
440 cm
(d)
66 cm
Three cubes each of side ‘a’ are joined from end to end to form a cuboid.
The volume of the new cuboid is _______ cubic units
(a)
a2
(b)
3a3
(c)
a3
(d)
6a3
Ratio of the volumes of the cylinder and cone having same height and
same radius of the bases, is
(a)
1:1
(b)
1:2
(c)
3:1
(d)
1:3
The radii of the circular ends of a frustum of height 16 cm are 8 cm and
20 cm. The slant height of the frustum is
(a)
28 cm
(b)
10 cm
(c)
44 cm
(d)
20 cm
The radius and height of a solid cylinder is r cm & h cm respectively. It is
divided into equal parts through height. The total surface area of the two
parts is—
(a)
(b)
(c)
(d)
123
[Class-X – Maths]
14.
15.
The ratio of the volumes of two spheres is 8: 27. It r & R are the radii of
the two spheres respectively, then r : (R–r) is
(a)
9:4
(b)
3:2
(c)
3:1
(d)
2:1
During conversion of solid from one shape to another the volume of the
new solid shape will —
(a)
increase
(b)
decrease
(c)
doubled
(d)
remains the same
16.
What is the perimeter of a sector of angle 45° of a circle with radius 7 cm.
17.
The perimeter of semicircular math lab protractor is 108 cm. Find the
diameter of the protractor?
18.
A bicycle wheel makes 5000 revolutions in moving 10 km. Write the
perimeter of wheel.
19.
What is the ratio of the areas of a circle and an equilateral triangle whose
diameter and a side of triangle are equal.
20.
A wire is in the form of a semicircle of radius 7 cm. It is bent into a square.
Find the area of the square.
21.
The cost of fencing a circular field at the rate of Rs. 10 per meter is Rs.
440. What is the radius of the circular field?
22.
Find the perimeter of the shaded region.
4 cm
A
B
D
C
6 cm
[Class-X – Maths]
124
23.
Find the circumference of the circle with centre ‘O’.
P
o
24 cm
7 cm
R
Q
24.
What is the area of the largest triangle that can be inscribed in a semicircle
of radius r cm.
25.
A piece of wire 20 cm long is bent into an arc of a circle subtending an
angle of 60° at the centre then what is the radius of the circle?
26.
The minute hand of a clock is cm long. What is the area described
by the minute hand between 8.00 a.m to 8.05 a.m.?
27.
Find the area of shaded portion.
20 cm
20 cm
20 cm
20 cm
28.
Find the perimeter of the shaded portion.
C
A
14
29.
B
14
D
14
ABCD is a square kite of side 4 cm. What is the area of the shaded
portion.
125
[Class-X – Maths]
A
D
4 cm
4 cm
B
C
4 cm
30.
How many cubes of side 4 cm can be cut from a cuboid measuring
(16 × 12 × 8).
31.
A cylinder, a cone and a hemisphere are of equal base and have the same
height. What is the ratio of their volumes?
32.
The sum of the radius of the base and the height of a solid cylinder is 15
cm. If total surface area is 660 cm2. Write the radius of the base of
cylinder.
33.
Find the height of largest right circular cone that can be cut out of a cube
whose volume is 729 cm3.
34.
Three cubes of the same metal, whose edges are 6, 8, 10 cm are melted
to form a single cube. Find the diagonal of the single cube.
35.
V ol e of riumght circul cylnder i 448
ariscm3, height of cylinder is 7cm.
Find the radius.
36.
If lateral surface area of a cube is 64 cm2. What is its edge?
37.
The area of a rhombus is 24 cm2 and one of its diagonal is 8 cm. What
is other diagonal of the rhombus?
38.
What is the length of the largest rod that can be put in a box of inner
dimensions 30cm, 24 cm and 18 cm?
2.
39.
If radius is 4cm, then find
its height.
40.
A largest sphere is carved out of a cube of side 7 cm. Find the radius.
[Class-X – Maths]
126
41.
If the semi vertical angle of a cone of height 3 cm is 60°. Find its volume.
42.
Find the edge of cube if volume of the cube is equal to the volume of
cuboid of dimensions (8 × 4 × 2) cm.
43.
Find the volume of cone of height 2h and radius r.
44.
A sphere of maximum volume is cut out from a solid hemisphere of radius
6 cm. What is the volume of the sphere cut out?
45.
In the adjoining figure ABC is a right angled triangle, right angled at A.
Semi circles are drawn on AB, AC as diameters. Find the area of shaded
portion.
A
4 cm
3 cm
C
B
46.
In figure, is equilateral triangle. The radius of the circle is 4 cm. Find
the area of shaded portion.
A
4 cm
o
4
4 cm
cm
B
C
127
[Class-X – Maths]
47.
Four cows are tied with a rope of 7 cm at four corners of a quadrilateral
field of unequal sides. Find the total area grazed.
48.
In the figure, find the area of the shaded portion.
5 cm
12 cm
49.
Find the shaded area in the given figure.
28 cm
28 cm
50.
Find the shaded area, in the figure.
14 cm
14 cm
[Class-X – Maths]
128
51.
Three cows are tied to the centre of a circular field of radius 14 m with a
7 m long rope. Each one is given equal sector by putting small height
walls. Find how much area each cow is unable to graze from the field
given to her. What values of farmer is seen from his act.
52.
The wheels of the car are of diameter 80 cm each. How many complete
revolutions does each wheel make in 10 minutes when the car is travelling
at a speed of 66 km/hr?
53.
Find the perimeter of the figure in which a semicircle is drawn on BC as
diameter, = 90°.
A
5 cm
12 cm
B
54.
C
Find the area of shaded region in the figure.
14 cm
9 cm
9 cm
14 cm
55.
The volume and surface area of a sphere are numerically equal. Find the
radius of the sphere.
56.
Water flows in a tank 150 m long and 100 m broad, through a rectagular
pipe whose cross-section is 2 dm × 1.5 dm and the speed of water is 15
km/hr. In what time will the water be 3 m deep?
57.
A cylindrical vessel of 36 cm height and 18 cm radius of the base is filled
with sand. The sand is emptied on the ground and a conical heap of sand
is formed. The height of conical heap is 27 cm. Find the radius of base of
sand.
129
[Class-X – Maths]
58.
The radii of circular ends of bucket are 5.5 cm and 15.5 cm and its height
is 24 cm. Find the total surface area of bucket.
59.
Water flows out through a circular pipe whose internal diameter is 2 cm
at the rate of 6m/sec. into a cylindrical tank. If radius of base of the tank
is 60 cm. How much will the level of the water rise in half an hour?
60.
A toy is in the form of a cone mounted on a cone frustum. If the radius
of the top and bottom are 14 cm and 7 cm. and the height of cone and
toy are 5.5 cm and 10.5 cm respectively. Find the volume of toy (adj. fig.)
5.5 cm
14 cm
10.5 cm
7 cm
61.
A solid consists of a right circular cylinder with a right circular cone at the
top. The height of cone is ‘h’ cm. The total volume of the solid is 3 times
the volume of the cone. Find the height of the cylinder.
62.
From a solid cylinder of height 12 cm & diameter 10 cm, a conical cavity
of same height & same diameter is hollowed out. Find the total surface
area of the remaining solid?
63.
There is a triangular field in front of a classroom. The two sides which are
perpendicular measures 6 m and 8 m. The teacher told students to construct
a circular flower bed which touches the three sides of the field. Find the
radius of circular flower bed. What values are developed in the students
through this activity.
64.
Naresh built a bath tub for birds in his garden which is hollow cylinder with
a hemispherical cavity on the upper edge. The height of the cylinder is
[Class-X – Maths]
130
1.45 m & the diameter is 60 cm. Find the maximum capacity of water
available in the tub in litres. What values of Naresh do you observe.
(Tot = 3.14.al
65.
Dinesh installed underground cylindrical tank for rain water which is 10 m
deep and 3 m in radius. Rain water reaches this tanks from roof through
a cylindrical pipe. The length & breadth of the roof are 10 m & 3.14 m
respectively. If on a particular day the rain falls up to a height of 10 cm on
the roof. What is the height of water in the cylindrical tank. (Assuming there
is no water left in the pipe). What values of Dinesh do you observe in the
question? (Take = 3.14)
1.
c
2. c
3.
b
4. b
5.
b
6. b
7.
d
8. b
9.
b
10. b
11.
c
12. d
13.
c
14. d
15.
d
16. 19.5 cm
17.
42 cm
18. 2m
19.
:
20. 81 cm2
3
22. (16 + cm
21.
14m
23.
25 
24. r2
25.
60 / cm
26. cm2
27.
86 cm2
28. 462 cm2
29.
(16–
31.
3: 1: 2
2
30. 24
32. 7 cm
131
[Class-X – Maths]
33.
9 cm
34. 12 3
35.
8 cm
36. 4 cm
37.
6 cm
38.
39.
2 cm
40. 3.5 cm
41.
27 cm3
42. 4 cm
2
44. 113.14 cm3
r 2h unit3
43.
30 2cm
3
45.
6 cm2
46. (16 – 12
47.
154 cm2
48.
49.
154 cm2
50. 21 cm2
51.
154 m2
52. 4375
53.
37
55.
3 units
56. 100 hour
57.
36 cm
58. 1811.07 cm2
59.
3m
60. 2926 cm3
3
23
) cm
1019
cm 2
14
54. 49 cm2
cm
7
2
61.
62. 660 cm2
h
3
63.
(a) 2m (b) working in groups, cooperation, coordination, love for nature
64.
(a) 56. 52 L,
(b) care for animals
65.
(a) 9 times,
(b) water conversation
[Class-X – Maths]
132
CHAPTER 8
PROBABILITY
1.
The Theoretical probability of an event E written as (E) is
P E 
Number of outcomes favourable to E
Number of all possible outcomes of the experiment.
2.
The sum of the probability of all the elementary events of an experiment
is 1.
3.
The probability of a sure event is 1 and probability of an impossible event
is 0.
4.
If E is an event, in general, it is true that P(E) + P (E ) = 1.
5.
From the definition of the probability, the numerator is always less than or
equal to the denominator therefore 0 P(E) 1.
1.
2.
3.
If E is an event then P(E) + P E = ........ ?
(a)
0
(b)
1
(c)
2
(d)
–1
The probability of an event that is certain to happen is :
(a)
0
(b)
2
(c)
1
(d)
–1
Which of the following can not be the probability of an event :
(a)
2
3
(b)
–3
2
(c)
15%
(d)
0.7
133
[Class-X – Maths]
4.
5.
6.
7.
8.
9.
If P(E) is .65 what is P (Not E)?
(a)
.35
(b)
.25
(c)
1
(d)
0
If P(E) is 38% of an event what is the probability of failure of this event?
(a)
12%
(b)
62%
(c)
1
(d)
0
A bag contains 9 Red and 7 blue marbles. A marble is taken out randomly,
what is the P (red marble)?
(a)
7
16
(b)
9
16
(c)
18
16
(d)
14
16
In a Survey it is found that every fifth person possess a vehicle what is the
probability of a person not possessing the vehicle?
(a)
1
5
(b)
4
5
(c)
3
5
(d)
1
Anand and Sumit are friends what is the probability that they both have
birthday on 11th Nov. (ignoring leap year).
(a)
1
12
(b)
1
7
(c)
1
365
(d)
1
366
The number of face cards in a well shuffled pack of cards are :
(a)
12
(b)
16
(c)
4
(d)
52
[Class-X – Maths]
134
10.
11.
12.
13.
14.
15.
A die is thrown once. What is the probability of getting an even prime
number?
(a)
3
6
(b)
1
6
(c)
1
2
(d)
1
3
The probability of an impossible event is :
(a)
0
(b)
1
(c)
–1
(d)

From the letters of the word “Mobile”, a letter is selected. The probability
that the letter is a vowel, is
(a)
1
3
(b)
3
7
(c)
1
6
(d)
1
2
The probability of selecting a face card from a deck of 52 cards is:(a)
2
26
(b)
3
13
(c)
4
13
(d)
8
13
When a die is thrown, the probability of getting a number less than 7 is
(a)
1
6
(b)
1
3
(c)
1
(d)
0
If E is an event such that x < P(E) < 1, then the value of x is
(a)
0
(b)
1
(c)
0.5
(d)
2
135
[Class-X – Maths]
16.
A bank A.T.M. has notes of denomination 100, 500 and 1000 in equal
numbers. What is the probability of getting a note of Rs. 1000.
17.
What is the probability of getting a number greater than 6 in a single throw
of a die.
18.
A selection committee interviewed 50 people for the post of sales manager.
Out of which 35 are males and 15 are females. What is the probability of
a female candidate being selected.
19.
A bag contains cards numbering from 5 to 25. One card is drawn from the
bag. Find the probability that the card has numbers from 10 to 15.
20.
In 1000 lottery tickets there are 5 prize winning tickets. Find the probability
of winning a prize if a person buys one ticket.
21.
It is known that in a box of 600 screws, 42 screws are defective. One
screw is taken out at random from this box. Find the probability that it is
not defective.
22.
Two dice are rolled once. What is the probability of getting a doublet?
23.
Two dice are rolled simultaneously. Find the probability that the sum is
more than and equal to 10.
24.
From the well shuffled pack of 52 cards. Two Black kings and Two Red
Aces are removed. What is the probability of getting a face card.
25.
In a leap year what is the probability of 53 Sundays.
26.
A box contains cards numbered from 2 to 101. One card is drawn at
random. What is the probability of getting a number which is a perfect
square.
27.
Tickets numbered from 1 to 20 are mixed up together and then a ticket is
drawn at random. What is the probability that the ticket has a number
which is a multiple of 3 or 7?
28.
A bag contains 5 red balls and ‘n’ green balls. If the P(green ball) = 3 ×
P (red ball) then what is the value of n.
29.
From the data (1, 4, 9, 16, 25, 29). If 29 is removed what is the probability
of getting a prime number.
[Class-X – Maths]
136
30.
A card is drawn from an ordinary pack of playing cards and a person bets
that it is a spade or an ace. What are the odds against his winning the bet.
31.
An arrow pointer is spined which is placed on a fixed circular number plate
numbered from 1 to 12 at equal distance. The pointer is equally likely to
rest at any number. What is the probability that it will rest at
32.
(a)
number 10
(b)
an odd number
(c)
a number multiple of 3
(d)
an even number
In a family having three children, there may be no girl, one girl, two girls
or three girls. Find the probability of :
(a)
33.
exactly one girl
(b)
less than 2 girls
(iii)
3 heads.
A coin is tossed thrice then find the probability of
(i)
2 heads
(ii)
2 tails
34.
The king, queen and jack of clubs are removed from a deck of 52 playing
cards and the remaining cards are shuffled. A card is drawn from the
remaining cards. Find the probability of getting a card of (i) heart;
(ii) queen; (iii) Clubs.
35.
A box contains 5 Red balls, 8 white balls and 4 Green balls. One ball is
taken out of the box at random. What is the probability that ball is (i) red;
(ii) white; (iii) Not green.
36.
12 defective pens are mixed with 120 good ones. One pen is taken out at
random from this lot. Determine the probability that the pen taken out is
not defective.
37.
A number x is selected from the numbers 1, 2, 3 and then a second
number y is randomly selected from the numbers 1, 4, 9. What is the
probability that the product of two numbers will be less than 9?
38.
A box contains 90 discs which are numbered from 1 to 90. If one disc is
drawn at random from the box, find the probability that it bears (i) a two
digit number (ii) a perfect square number (ii) a number divisible by 5.
39.
A game consists of tossing a one rupee coin 3 times and noting its outcome
each time. Anand wins if all the tosses give the same result i.e., three
137
[Class-X – Maths]
heads or three tails and loses otherwise. Calculate the probability that
Anand will lose the game.
40.
A die is thrown twice. What is the probability of getting : (i) The Sum of 7;
(ii) The sum of greater than 10; (iii) 5 will not come up either time.
41.
A card is drawn at randown from a well shuffled deck of playing cards.
Find the probability that the card drawn is
(i)
(iii)
42.
a card of spade or an ace
(ii) a red king
either a king or a queen
(iv) neither a king nor a queen
A jar contains 24 balls, some are green and other are blue. If a ball is
2
drawn at random from the jar, the probability that it is green is . Find
3
the number of blue balls in the jar.
43.
In a lottery, there are 10 prizes and 25 blanks. Find the probability of
getting a prize? Also verify that P(E) + P E = 1
44.
The probability of getting a bad egg in a lot of 400 is 0.035. Calculate the
number of bad eggs in the lot. Also calculate the probability of getting a
good egg from the lot.
45.
A bag contains 12 balls out of which x are white and rest are red balls.
46.
47.
i.
If one ball is drawn at random, what is the probability that it will be
a Red ball?
ii.
If 4 more white balls and 2 more Red balls are put in the bag then
what is the probability of drawing a white ball?
In a class discussion, Himanshu says that probability of an event cannot
be 1.3. He shows:(a)
Truth value
(b)
Economical Value
(c)
Leadership
(d)
Environment Value
P(E) + P E = 1
The Statement depicts ____________ value.
[Class-X – Maths]
138
48.
E be an event associated with a random experiment and 0 < P(E) < x. Find
the maximum value of x, which value is depicted by this statement?
49.
Ramesh got Rs. 24000 as Bonus. He donated Rs. 5000 to temple. He gave
Rs. 12000 to his wife, Rs. 2000 to his servant and gave rest of the amount
to his daughter. Calculate the probability of
i.
wife's share
ii.
Servant's Share
iii.
daughter's share.
Which value's are depicted by Ramesh from the way the amount is
distributed?
50.
240 students reside in a hostel. Out of which 50% go for the yoga classes
early in the morning, 25% go for the Gym club and 15% of them go for
the morning walk. Rest of the students have joined the laughing club. What
is the probability of students who have joined laughing club? Which value
is depicted by the students?
1.
b
2. c
3.
b
4. a
5.
b
6. b
7.
b
8. c
9.
a
10. b
11.
a
12. d
13.
b
14. c
15.
a
16.
1
3
17.
0
18.
3
10
19.
2
7
20.
1
200
139
[Class-X – Maths]
21.
93
100
22.
1
6
23.
1
6
24.
5
24
25.
2
7
26.
9
100
27.
2
5
28. 15
29.
0
30.
1
1
1
1
31.
(i) 12
(ii) 2
(iii) 3
(iv) 2
3
3
1
33.
(i) 8
(ii) 8
(iii) 8
5
8
13
35.
(i) 17
(ii) 17
(iii) 17
37.
5
9
39.
3
4
41.
4
(i) 13
42.
8.
43. 2/7
44.
14, 0.965.
45. (i)
46.
(a) Truth Value
47. Righteous Value
48.
x=1, Righteous value.
49.
1
(i) 2
1
(ii) 26
1
(ii) 12
2
(iii) 13
(iii)
32. (a)
1
4
13
49
3
10
34. (i)
(ii) 49
(iii) 49
9
10
1
1
38. (i)
(ii) 10
(iii) 5
40. (i)
1
6
36.
10
11
1
(ii) 12
12 x
12
5
. Social Value, Religious Value
24
Physical Fitness
10
[Class-X – Maths]
1
(b) 2
25
(iii) 36
11
(iv) 13
1
50.
9
13
140
(ii)
x 4
18
SAMPLE QUESTION PAPER - I
Time allowed : 3 hours
Maximum Marks : 90
General Instructions
1.
All questions are compulsory.
2.
The question paper consists of 31 questions divided into four sections A,
B, C and D. Section A comprises of 4 questions of 1 mark each. Section
B comprises of 6 questions of 2 marks each. Section C comprises of 10
questions of 3 marks each and Section D comprises of 11 questions of 4
marks each.
3.
Question numbers 1 to 4 in Section A are multiple choice questions where
you are to select one correct option out of the given four.
4.
There is no overall choice.
5.
Use of calculator is not permitted.
1.
If one root of 2x2 + kx + 1 = 0 is 
1
, then the value of K is
2
2.
(a)
3
(b)
–3
(c)
5
(d)
–5
In figure, PQ and PR are tangents to the circle with O, such that
= 50°, is equal to :
Q
O
P
R
141
[Class-X – Maths]
3.
4.
(a)
25°
(b)
30°
(c)
40°
(d)
50°
From the letter of the word ‘‘Education’’ a letter is selected, the probability
that the letter is a vowel is :
(a)
7
9
(b)
2
9
(c)
4
9
(d)
5
9
The mid-point of segment AB is the point (0, 4). If the co-ordinates of B
are (–2, 3), then the co-ordinates of A are :
(a)
(2, 5)
(b)
(–2, –5)
(c)
(2, 9)
(d)
(–2, 11)
5.
Find the value of m so that the quadratic equation x2 – 2x (1 + 3m) +
7(3 + 2m) = 0 has equal roots.
6.
How many two digit numbers are there in between 6 and 102 which are
divisible by 6 ?
7.
A circle is inscribed in a having sides PQ = 10 cm, QR = 8 cm and
PR = 12 cm. Find PS and QT.
R
U
P
T
S
Q
8.
Find the area of the shaded region in the given figure.
[Class-X – Maths]
142
A
7cm
7cm
7cm
7cm
B
9.
C
Radius of a wheel of motor cycle is 35 cm. How many turn it should take
in one minute if its speed is 66 km/hour.
10.
22 
If the perimeter of a protractor is 72 cm, calculate its area. Take 
7 
11.
S olve the equaton 2
x2i – 5x + 3 = 0 by the method of completing the
square.
12.
Find the sum of first 15 terms of an A.P. whose nth term is 9 – 5n.
13.
Draw a triangle ABC with side BC = 7 cm, = 45°, = 105°, Then,
4
construct a triangle whose sides aretimes the corresponding sides of
3

14.
Find a point on y-axis which is equidistant from the points (6, 5) and (–4, 3).
15.
In what ratio does the point (½, 6) divides the line segment joining the
points (3, 5) and (–7, 9).
16.
Find the area of the shaded region in the given figure, if PQ = 24 cm,
PR = 7 cm and O is the centre of the circle.
Q
O
R
17.
P
Water in a canal, 6 m wide and 1.5 m deep, is flowing with a speed of
10km/h. How much area in hectares will it irrigate in 30 minute, if 8 cm of
standing water is needed? (1 hectare = 10000 m2)
143
[Class-X – Maths]
18.
The total surface area of a solid right circular cylinder is 231 cm2, its
2
curved surface is rd of the total surface area. Find the radius of the base
3
and height.
19.
Entry fee in a game is 5. A coin is tossed three times in this game. If
Shweta got one or two tails then entry fee is refunded. If she gets three
tails, she receive twice of entry fee amount. Otherwise, she loses entry
fee. Find the probability of following :
(i)
She loses entry fee.
(ii)
She receied twice the amount of entry fee.
(iii)
She received entry fee.
20.
Draw a circle of radius 3 cm. from a point 5 cm away from the centre draw
a pair of tangents on the circle. Also measure the length of tangents.
21.
Usha surveyed a class on World Food Day and observed that the number
of students who like Junk food is 4 more than the number of students who
like home made food. The sum of the squares of the number of two types
of students is 400. Find the number of students who like Junk food and
who like home made food?
22.
If the sum of three consecutive terms of an increasing A.P. is 51 and the
product of first and third terms is 273, find the third term.
23.
An old lady Krishna Devi deposited Rs. 12000 in a bank at 8% simple
interest p.a. She uses the annual interest to give five scholarships to the
students of a school for their overall performances each year. The amount
of each scholarship is 300 less than the preceding scholarship. Answer
the following
24.
(i)
Find amount of each scholarship?
(ii)
What values of the lady are reflected?
Prove that the length of tangents drawn from an external point to a circle
are equal.
[Class-X – Maths]
144
25.
Prove that the intercept of a tangent between a pair of parallel tangents
to a circle subtend a right angle at the centre of the circle. (Figure is given).
A
P
O
B
Q
26.
There is a small island in the middle of a 100 meter wide river and a tall
tree stands on the island. P and Q are points directly opposite to each
other on two banks and in line with the tree. If the angles of elevation of
the top of the tree from P and Q are respectively 30° and 45°, find the
height of the tree.
27.
A bag contains some cards which are numbered between 31 and 96 are
placed. If one card is drawn from the bag, find the probability that the
number on card is :
(i)
a multiple of 3
(ii)
(iii)
a number not more than 40.
(iv)
an odd card no.
a perfect square
28.
The slant height of a frustum of a cone is 4 cm and the perimeter of its
circular ends are 18 cm and 6 cm. Find the curved surface area of the
frustum.
29.
Find the value of K, if the points A (2, 3), B (4, K) and C (6, –3) are
collinear.
30.
A cone of radius 10 cm is divided into two parts by drawing a plane
through the mid-point of its axis, parallel to its base, compare the volume
of the two parts.
31.
As observed from the top of a 75 cm high light house from the sea level,
the angles of depressions of two ships are 30° and 45°. If one ship is
exactly behind the other on the same side of the light house. Find the
distance between two ships.
145
[Class-X – Maths]
1.
a
2. a
3.
d
4. a
5.
–10
,2
6
7.
PS = 4cm, QT = cm
9.
500
3
11.
,1
6. 15
8. 24.5 cm2
10. 308 cm2
12. –465
2
14.
27 
0, 2 
15. 1 : 3
16.
161.53 cm2
17. 56.25 hectare
18.
3.5 cm, 7 cm
i 1 ii 1 iii 3
19.
8
8
20.
4 cm
21. 12, 16
22.
21
23.
(i) Rs. 2520, Rs. 2220, Rs. 1920, Rs. 1620, Rs. 1320
(ii) Love for children kindness
5191
30.
36.6 m
27. (i)
, (ii), (iii), (iv)
1216642
28.
48 cm2
29. K = 0
30.
1:7
31.
[Class-X – Maths]
146
75 3 1m
4
SAMPLE QUESTION PAPER - II
Time allowed : 3 hours
Maximum Marks : 90
General Instructions
1.
All questions are compulsory.
2.
The question paper consists of 31 questions divided into four sections A,
B, C and D. Section A comprises of 4 questions of 1 mark each. Section
B comprises of 6 questions of 2 marks each. Section C comprises of 10
questions of 3 marks each and Section D comprises of 11 questions of 4
marks each.
3.
Question numbers 1 to 4 in Section A are multiple choice questions where
you are to select one correct option out of the given four.
4.
There is no overall choice.
5.
Use of calculator is not permitted.
1.
The angle of elevation of the top of a tower of height 10 3 m from a point
on the ground, which is 30 m away from the foot of the tower is:
2.
3.
(a)
45°
(b)
60°
(c)
30°
(d)
90°
The probability of getting 53 Fridays in a leap year is:
(a)
1
7
(b)
2
7
(c)
4
7
(d)
5
7
If radius of a circle is doubled, its area becomes:
(a)
2 times
(b)
4 times
(c)
8 times
(d)
16 times
147
[Class-X – Maths]
4.
Ratio of the radii of two spheres is 4:5. Find the ratio of the total surface
areas of the spheres.
(a)
16:25
(b)
25:16
(c)
4:5
(d)
5:4
5.
If K + 1, 3k and 4k + 2 be any three consecutive terms of an A.P., find the
value of K.
6.
Prove that the tangents drawn at the ends of a diameter of a circle are
parallel.
7.

22 
If the perimeter of a protractor is 72m, calculate its area. 
8.
7
Find the coordinates of point A if AB is the diameter of the circle having
centre (2, –3) and B (1, 4)
9.
Points (5, –2), (6, 4) and (7, –2) represents the vertices of which type of
triangle.
22 
10.
If the perimeter of a protractor is 72 cm then calculate its area. 
7 
11.
A die is thrown once. Find the probability of getting
(i)
(iii)
an even prime number
(ii)
a multiple of 3
a number greater than 4
12.
The hypotenuse of a right angled triangle is 13 m long. If the base of the
triangle is 7 m more than the other side, find the sides of the triangle.
13.
Find five numbers in A.P. whose sum is 12½ and the ratio of first to the
last term is 2:3.
14.
Draw a triangle ABC in which CA = 6 cm, AB = 5 cm and = 45°
3
Then, construct a triangle similar to the triangle ABC, whose sides are
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of the corresponding sides of the triangle ABC.
15.
From a point P on the ground, the angle of elevation of the top of a 10m
high building is 30°. A flag staff is hoisted at the top of the building and
the angle of elevation of the top of flag staff from P is 45°. Find the length
of the flagstaff.
16.
The line segment joining the points A (3, 2) and B (5, 1) is divided at the
point P in the ratio 1:2 and P lies on the line 3x – 18y + k = 0. Find the
value of k.
cm
14
17.
D
m
7c
Find the area of the shaded region in the given figure:18.
A circus tent is cylindrical to a height of 3 m and conical above it. If its
radius is 105 m and slant height of the cone is 53 m, calculate the total
area of the canvas required.
19.
Solve for x : 12abx 2 9a 2 8b 2 x 6ab 0
20.
If third and nineth terms of an AP are 4 and –8 respectively then which
term of this AP is zero.
21.
On the independence Day, in a group of children, each child gives one gift
to the other. If total numbers of gifts are 90, then find the number of
children in the group. What do we learn from this type of group activities?
22.
A motor boat, whose speed is 15 km/h in still water, goes 30 km downstream
and comes back in a total of 4 hours 30 minutes. Determine the speed of
stream.
23.
124
; x –1, –2, –4
x 1 x 2x 4
149
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24.
Prove that the lengths of tangents drawn from an external points to a circle
are equal.
25.
Two tangents TP and TQ are drawn to a circle with centre O from an
external point To. Prove that 
26.
From the top of a building 60 m high, the angles of depression of the top
and bottom of a lamp post are observed to be 30° and 60° respectively.
Find the height of the lamp post.
27.
The king, queen and jack of clubs are removed from a pack of 52 playing
cards and then the remaining pack is well shuffled. One card is selected
from the remaining cards. Find the probability of getting.
(i)
a heart
(ii)
a king
(ii)
a club
(iv)
a jack
28.
Find the area of the triangle formed by joining the mid-points of the sides
of a triangle ABC whose vertices are A (0, –1), B (2, 1) and C (0, 3). Find
the ratio of this area to the area of the given triangle ABC.
29.
A decorative block is made of two solids - a cube and a hemisphere. The
base of the block is the cube with edge of 7 cm and the hemisphere
attached on the top has a diameter of 4.9 cm. If the block is to be painted,
find the total area to be painted.
30.
A Turkish cap is shaped like a frustum of cone. If its radius on the open
side is 10 cm, radius at the upper base is 4 cm and its slant height is 15
cm, find the area of the material used for making it.
31.
A hemispherical tank full of water is emptied by a pipe at the rate of 3
litres per second. How much time will it take to empty half of the tank, if
22 
use 
it is 3 m in diameter? 
7 
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150
4
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