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MATHEMATICS
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A well defined collection of objects or ideas is known as a SET.
Set Theory is a comparatively new concept in Mathematics.
It was developed by Georg Cantor (1845-1918).
Cantor's work between 1874 and 1884 is the origin of Set Theory.
VýS$Æý‡$ÐéÆý‡… l íœ{ºÐ]lÇ l 11 l 2016
2
10th Class Special - Maths
Prepared by:
MýSsêt MýSÑ™èl, çÜ*PÌŒæ AíÜòÜt…sŒæ
Mø§ýl…yéç³NÆŠæ, Ð]l$çßæº*»Œæ¯]lVýSÆŠæ.
PAPER - I
1. REAL NUMBERS
Target - 10 grade points
A. A–B = {3, 4}
4. Write all subsets of A = {1, 2, 3}?
A. Subsets of A = P(A) = [φ, {1}, {2}, {3},
{1, 2}, {2, 3}, {3, 1}, A}
IMPORTANT QUESTIONS
1 Mark
1. ''LCM of two numbers, is a multiple of
HCF of that two numbers''. Is it true or
not? Justify your answer by giving an
example?
A. LCM of 4, 6 is 12 and HCF of 4, 6 is 2 ..
LCM 12 is a multiple of HCF 2
2. Three bells toll at intervals of 9, 12, 15
minutes respectively. If they start tolling
together, after what time will they next toll
together?
A. Three bells will toll together at the time =
LCM of 9, 12, 15 = 180 minutes = 3 hours.
After 3 hours they will toll together.
3
3. Express as decimal form?
8
A. 3 = 3 × 125 = 375 = 0.375
8 8 × 125 1000
4. From the given factor tree find x?
4
x
y
2 Marks
1. Simplify log 12 + 2 log 3 –3log 2 as logN.
Determine the value of N?
2. Explain why 7×11×13+13 and 7×6×5× 4
×3×2×1+5 are composite numbers?
4 Marks
1. Prove that √5 is an irrational?
x+ y 1
 = (log x + log y). Then find
 3  2
x y
the value of y + x ?

2. If l og 
IMPORTANT QUESTIONS
1 Mark
1. Is the following sets are equal sets? or not?
Justify your answer?
A = {x:2<x<3, x∈N}
B = {x: 2<x<3, x∈Q}
A. A = {x:2<x<3, x∈N} = φ
(∵ there is no natural no. between 2 and 3.
B = {x: 2<x<3, x∈Q} = sets of rational
numbers between 2 and 3 are infinite.
∴ A and B are not equal sets.
2. Write the following set in set builder form
{1, 8, 27, 64, ...1000}?
A. {1, 8, 27, 64, ....1000} = {x3:x∈N, x≤10}
3. From the venn Diagram find A–B?
1
2
4
5
7
1 Mark
1. Find the quadratic polynomial 2, 3 are the
sum and product of its zeroes respectively?
A. α + β = 2, αβ =3
Required polynomial
= K{x2–(α + β)x+αβ}, K≠0
=K {x2–2x+3}, K≠0
2. If the length and breadth of a rectangle are
the zeroes of a polynomial x2–7x+12 then
find the area?
A. Let the zeroes are α, β.
Area of rectangle =
= lb = αβ =
c 12
=
= 12sq.units
a 1
3. If the zeroes of polynomial kx2–5x+6 are
multiplicatively inverse, then find K?
2. SETS
3
10x – 5, if two of its zeroes are
5
5
,−
?
3
3
4. PAIR OF LINEAR
EQUATIONS IN
TWO VARIABLES
IMPORTANT QUESTIONS
1 Mark
1. "The cost of 5 tables and 7 chairs is Rs.
6400." Represent this situation in a linear
equation?
A. Let the cost of 1 table is Rs.x, the cost of 1
chair is Rs.y then 5x+7y = 6400.
IMPORTANT QUESTIONS
A. y = 5 × 3 = 15
x = 4 × y = 4 × 15 = 60
B
4 Marks
1. A = {x: x is a prime, x<10}
B = {x: x is an odd number, x<10}
C = {x: x is a factor of 24}
D = {x: x is a multiple of 3, x≤18}
Write the above sets in Roster form and
find A∪B, B∩C, C–D, D–A?
2. If A = {1, 2,3} and B = {3, 4, 5} then show
that A–B, A∩B, B–A are mutually disjoint
sets?
4 Marks
1. Verify that 3, –1, –1/3 are the zeroes of the
cubic polynomial P(x) = 3x3–5x2–11x–3,
then verify the relationship between the
zeroes and coefficients?
2. Obtain all other zeroes of 3x4 + 6x3 – 2x2 –
3. POLYNOMIALS
5
3
A
2 Marks
1. A represents the set of 3 multiples and B
represents the set of 4 multiples. Which
set of multiples does A∩B represents?
2. P is the set of factors of 5, Q is the set of
factors of 25 and R is the set of factors of
125. Is it right to say that P⊂Q, Q⊂R and
R⊂P. Explain?
2. If α and β are the zeroes of the polynomial f(x) = x2–5x+k such that α – β = 1, then
find the value of k?
µ
8
1
then the product
α
1 c 6
of zeroes = α. = =
α a k
6
⇒1= ⇒ k = 6
k
A. Let the zeroes are α,
4. From the figure find the sum of the
zeroes?
–4 –3 –2 –1
1
a
2 b
−3
1
1
A. From the given equations a = 7 , b = 4
2
2
a
b
1
1
Since a ≠ b
2
2
∴ The equations are consistent
2 Marks
1. The larger of two supplementary angles
exceeds the smaller by 18o. Find the
angles?
2. Find out the pair of linear equations
9x+3y+12=0 and 18x+6y+24=0 are intersect at a point, or parallel or coincident?
4 Marks
1. A fraction becomes 4/5 if 1 is added to
both numerator and denominator. If, however 5 is subtracted from both numerator
and denominator, the fraction becomes
1/2. What is the fraction?
2. Draw the graph of the equations 2x–y=5
and 3x+2y = 11. Find the solution of the
equations from the graph?
3. Solve the following pairs of equations by
reducing them to a pair of linear equations
6x+3y = 6xy; 2x+4y=5xy?
5. QUADRATIC EQUATIONS
IMPORTANT QUESTIONS
2. Father's age is three times the sum of the
ages of his two children. After 5 years, his
age will be twice the sum of the ages of
two children. Find the age of father?
A. Let the present age of the father be x years
and the sum of the present ages of his two
children be y–years. Then according to the
question
x=3y →(1); x+5=2(y+5+5)
⇒ x–2y–15 = 0 →(2)
Solving (1) and (2) we get x=45, y=15
∴ Father age = 45 years.
3. From the figure find K?
3x
+k
2 3 4
x+
y
y=
=2
0
A. zeroes are –2, 1. Their sum = –2+1= –1
A. The lines are parallel
2 Marks
1. Find the zeroes of x2–2x–8 and verify the
relation between the zeroes and coefficients?
4. Check whether the equations 2x–3y=8 and
7x+4y = –9 are consistent or inconsistent?
a
b
c
1
1
1
from a = b ≠ c
2
2
2
1 1
= ⇒k=3
3 k
7
1 Mark
1. The roots of x2–4x+4=0 are equal or not?
Explain?
A. b2–4ac= (–4)2–4(1)(4) = 0
Since b2–4ac=0, the roots are equal
or
x2–4x+4=0 ⇒ (x–2)2=0, x=2, 2. The roots
are equal.
2. "The base of a parallelogram is 4 cm
longer than its altitude and its area is 48
sq. cm." Express this statement in a quadratic equation form?
A. Let altitude h = x, base = x+4 (given)
From the data (x+4) x=48 (∵ Area of parallelogram = bh)
⇒ x2+4x–48=0
3. The product of Ramu's age (in years) five
years ago with his age (in years) 9 years
later is 15. Find Ramu's present age?
A. Let the Ramu's present age = x years
Given (x–5)(x+9)=15
You
should never
remain awake till
late night a day before
the examination as this
can create headache, the
next morning and
you might end
up in trouble.
Aryabhata the famous Indian mathematician gave formulas for the
sum of squares and cubes of natural numbers. His work was
"Arybhateeyam' (499 A.D.). He gave a formula for finding the sum of
"n terms' of an Arithmetic progression starting with any term.
ax 2+bx+
Y
c=0
⇒ x2+4x–60= 0
x2+10x–6x–60= 0
⇒ (x+10)(x–6)= 0, x = –10 or 6
∴ Ramu's present age = 6 years, (x≠–10)
4. Find the nature of the roots from the given
figure?
X
0
A. From the figure, the quadratic equation
has no real roots. Since the curve does not
cuts the X–axis
2 Marks
1. Find the value of 'k' for the quadratic
equation 2x2+kx+3=0, so it has two equal
roots?
2. Find the dimensions of a rectangle field
whose perimeter is 200 meters and area is
2100 sq.m.?
VýS$Æý‡$ÐéÆý‡… l íœ{ºÐ]lÇ l 11 l 2016
A. Arrangement of coins are in A.P. here a=1,
d=1
Total coins in 10 rows=sn =
s10 =
n
[2a+(n–1)d]
2
10
[2(1)+(10–1)1] = 55
2
2 Marks
1. Determine the A.P. whose 3rd term is 5 and
the 7th term is 9?
2. A man repays a loan of Rs. 3250 by paying Rs. 20 in the first month and then
increases the payment by Rs. 15 every
month. How long will it take him to clear
the loan?
3. Which term of the G.P. 2, 2√2, 4, .. is 128?
6. PROGRESSIONS
IMPORTANT QUESTIONS
1 Mark
1. Check whether 62 is a term of the list of
numbers 4, 7, 10, .....?
A. Given list of numbers 4, 7, 10, ... is an A.P.
For this A.P. we have a = 4, d=3
Let we take the nth term of A.P. is 62 i.e.
a+(n–1)d = 62 ⇒ 4+(n–1)3 = 62
n=
61
3
But n should be a positive integer....So, 62
is not a term of the given list of numbers.
2. In a G.P. 3rd term is 24 and 6th term is 192.
Find the common ratio?
A. In a G.P. a3=ar2=24, a6=ar5=192
a 6 ar 5 192
=
=
⇒ r3 = 8 ⇒ r = 2
a 3 ar 2 24
∴ Common ratio = r = 2
3. How many two-digit numbers are divisible by 8?
A. The list of two-digit numbers divisible by
8 is 16, 24, 32, ...96
In this A.P. a=16, d=8
tn = a+(n–1) d= 96 ⇒ 16+(n–1)8 = 96
n=11
∴ There are 11 two-digit numbers divisible by 8.
4. Find the total coins arranged in the given
manner in 10 rows?
joining the points (–3,10) and (6,–8) is
divided by (–1, 6)?
2. Determine x so that 2 is the slope of the
line through P(2,5) and Q(x,3)?
4 Marks
1. If P(2,–1), Q(3,4), R(–2,3) and S(–3, –2)
be four points in a plane, show that PQRS
is a rhombus but not a square. Find the
area of the rhombus?
2 2 2
, , ,.... have their nth term
... and
31 27 9
7. COORDINATE
GEOMETRY
IMPORTANT QUESTIONS
= 32 = 16 × 2 = 4 2 units
But Ganesh says that AB = 4 units.
∴ I cannot agree with Ganesh statement.
2. Find the centre of the circle, whose vertices of a diameter are (–4, 3), (2, 5)?
A. Centre = midpoint of (–4, 3), (2, 5)
 −4 + 2 3 + 5 
=
,
 = (− 1, 4)
2 
 2
3. Find a relation between x and y such that
the point (x,y) is equidistant from the
points (3,0) and (0,4)?
A. Let P = (x, y), A=(3,0), B=(0,4)
Given PA=PB ⇒ PA2 = PB2
⇒ (x–3)2+(y–0)2 = (x–0)2+(y–4)2
⇒ x2–6x+9+y2 = x2+y2–8y+16
–6x+8y–7=0 ⇒ 6x–8y+7=0
Required conditon 6x–8y+7=0
4. In the given figure AD, BE, CF are medians. Find the coordinates of G?
1
F
B
(3,7)
E
G
D
A. Since AD, BE, GF are medians, G is
called centroid of ∆AB
 x + x + x y + y2 + y3 
G= 1 2 3 , 1

3
3


 2 + 3 + 1 −1 + 7 + 3 
=
,
 = (2,3)
3
3


2 Marks
1. Find the ratio in which the line segment
1
C
P
x
2. Find the points of trisection of the line
segment joining the points (5, –6) and (–7,
5)?
3. Find the area of the triangle formed by
joining the mid-points of the sides of the
triangle whose vertices are (0,–1) (2, 1),
(0, 3). Find the ratio of this area to the area
of the given triangle?
z
B
8. SIMILAR TRIANGLES
IMPORTANT QUESTIONS
1 Mark
1. "Basic Proportionality theorem is applicable only for right angled triangles". Can
you agree or not? Discuss?
A. No, I cannot agree, because Basic
Proporationality theorem is applicable for
any triangle.
2. In ∆ABC, AD is the
bisector of ∠A. If BD = 4 cm, DC = 3 cm,
AB = 6 cm, determine AC?
A
B
C
D
A. We know that ∆ABD ∼ ∆ADC
(∵SAS Similarity)
2. ABC is a right triangle at C. Let BC = a,
CA = b, AB = c and let 'P' be the length of
perpendicular from C on AB prove that
i) pc = ab
2
BC
81
9
9
=
= ⇒ BC = × 26 = 18c.m.
26
169 13
13
4. From the figure find AD?
A
D
4
C
B
12
1
9. TANGENTS AND
SECANTS TO A CIRCLE
IMPORTANT QUESTIONS
1 Mark
1. If PA and PB are two tangents from a point
P to a circle with centre O and are inclined
to each other at an angle of 80o, then find
POA ?
A
40
40
O
P
B
Area ∆ABC  BC 
81  BC 
=
=
 ⇒

Area ∆DEF  EF 
169  26 
E
1
3. State and prove Basic Proportionality theorem?
4. State and Prove Pythagoras theorem?
5. Construct an Isosceles triangle whose base
is 9 cm and altitude is 5 cm. Then, draw
another triangle whose sides are 1½ times
the corresponding sides of the Isosceles
triangle?
BD AB
4
6
∴
=
⇒ =
DC AC
3 AC
⇒ AC = 4.5cm
9
D
Q
1
A.
y
ii) p 2 = a 2 + b2
PAPER - II
2
C
(1,3)
1
A
3. ∆ABC, ∆DEF are similar triangles and
their areas are respectively 81cm2 and 169
cm2. If EF =26 cm, then Find BC?
A (2,–1)
2 Marks
1. Can you show that sum of the squares of
the sides of a rhombus is equal to the sum
of the squares of its diagonals?
2. BL and CM are medians of a triangle
ABC, right angled at A. Prove that
4(BL2+CM2) = 5BC2?
Prove that x + y = z
1 Mark
1. Ganesh says that the distance between two
points A(4,2), B(8,6) is 4 units Do you
agree with Ganesh or not? Why?
A. Distance between
AB = (8 − 4) 2 + (6 − 2) 2 = 16 + 16
A. From the given figure DE = BC = 12
AE = AB – BE = 9–4=5
From Pythagoras theorem
AD2 = AE2+ED2 = 52+122=169
AD = 169 =13
4 Marks
1. AB, CD, PQ are perpendicular to BD.
AB = x, CD=y and PQ = z
4 Marks
1. Determine 32 into four parts which are in
A.P. such that the product of extremes is to
the product of means is 7:15?
2. If the Geometric progressions 162, 54, 18,
equal. Find the value of n?
4 Marks
1. A shop keeper buy a number of books for
Rs. 1200. If he had bought 10 more books
for the same amount, each book would
have cost Rs. 20 less. How many books
did he buy?
2. A train travels 360 km at a uniform speed.
If the speed had been 5km/h more it would
have taken 1 hour less for the same journey. Find the speed of the train?
3
10th Class Special - Maths
A. OPA =
1
1
APB = (80 o ) = 40 o
2
2
The
syllabus has
been understood
effectively and the basics
of the topic are clear in
your mind, you will be able
to solve any given
problem with minimal efforts on
your part.
Carl Friedrich Gauss (1777-1855) the great German mathematician,
proposed a formula to find the Sum of first "n' terms in Arithmetic
Progression. He contributed significantly to many fields like number
theory, algebra, geophysics, optics etc.
VýS$Æý‡$ÐéÆý‡… l íœ{ºÐ]lÇ l 11 l 2016
4
10th Class Special - Maths
OAP = 90o (∵ OA ⊥ AP)
POA = 180o − (90o + 40o ) = 50o
2. Two Parallel lines touch the circle at
points A and B. If area of the circle is 25π
sq.cm then find the length of AB?
A.
quadrilateral ABCD at points PQRS, then
show that AB+CD=BC+DA?
3. Draw a pair of tangents to a circle of
radius 5 cm which are inclined to each
other at an angle 60o?
A
10. MENSURATION
IMPORTANT QUESTIONS
•
1 Mark
1. How many balls, each of radius 1 cm, can
be made from a solid sphere of lead of
radius 2 cm?
B
Area of circle πr2 = 25π sq.cm
r = 5cm.
AB = 2r = 10 cm
3. Two concentric circles of radii 5 cm and 3
cm are drawn. Find the length of the chord
of the larger circle which touches the
smaller circle?
Volume of solid sphere of lead
A. No. of balls =
Volume of small ball
4
π(2) 3
=3
=8
4
π(1) 3
3

4 Marks
1. A sphere, a cylinder and a cone are of the
same radius and same height. Find the
ratio of their curved surface areas?
2. A tent is in the form of a cylinder of diameter 4.2m and height 4m surmounted by a
cone of equal base and height 2.8 m. Find
the capacity of the tent and the cost of
canvas for making the tent at Rs. 100 per
Sq.m.?
3. A 20 m deep well with diameter 7m. is
dug and the earth from digging is evenly
spread out to form a platform 22m by 14
m, find the height of the platform?
0 r
r2 1
P
B
A
22 
of radius 3.5 cm  π =  ?
7 

2. A right circular cylinder has base radius 14
cm and height 21 cm, then find volume of
the right circular cylinder?
3. If the volume of a cube is 1728 cm3, then
find the height of the cube?
11. TRIGONOMETRY
= 2 r12 − r22 (r1 > r2 )
= 2 (5) 2 − 3 2 = 8cm
A. If a, b are multiplicative inverse to each
other then ab=1
A. Length of chord
sin θ
A
4
F
3
E
•
B
D
C
5
A. Perimeter of ∆ABC
= 2 (AF+BD+CE)
= 2 (4+3+5) = 24 cm
(∵AF=AE=4, BD = BF
= 3, CE=CD=5)
1
Volume of cone = πr 2 h
3
=
22 

 use π =

7 

D
2. "A cylinder and cone have bases of equal
radii and are of equal heights. Then their
volumes are in the ration 3:1". Explain?
A. A cylinder and cone have equal radii r, and
equal height h.
Volume of cylinder = π r2h
Ratio of their volumes
2 Marks
1. Prove that the lengths of two tangents
drawn from an external point to a circle
are equal?
2. Find the area of the shaded region in figure, if ABCD is a square of side 7 cm and
APD and BPC are semicircles
3
πr 2 h
= = 3 :1
1 2
1
πr h
3
here
sin θ
1 − cos θ 1 + cos θ 1 − cos 2 θ sin 2 θ
.
=
=
=1
sin θ
sin θ
sin 2 θ
sin 2 θ
1 − cos θ
is multiplicative inverse to
sin θ
1 + cos θ
sin θ
∴
A. If θ=45o sinθ = sin 45o =
C
 A + B
tan 
 = cot
2 ?
 2 
4 Marks
1. Prove that
1 + cos θ
= cos ecθ + cot θ ?
1 − cos θ
2. In a right angle triangle ∆ABC, right angle
at B, if tan A = √3 then find the value of
i) sinA cosC + cosA sinC
ii) cos A cosC – sinA sinC
3. If secθ+ tan θ=k. then prove that
sin θ =
k2 −1
?
k2 +1
3. Two cubes each of volume 8 cm3 are
joined end to end, then find the surface
area of the resulting cuboid?
A. Volume of a cube = a3 = 8 ⇒ a = 2 cm
side of cube a = 2 cm
length of resulting cuboid = l = 4 cm
breadth b = 2 cm
height h = 2 cm
Surface area of the cuboid = 2 (lb+bh+hl)
= 2(4×2+2×2+2×4)=40 cm2
4. From the figure find the slant height of circular cone?
1
2
3. In a ∆ABC, sin A =
3
then find the area of
5
A. In ∆ABC, sin A =
3
5
From the figure
C
5
A. From the figure r=5 cm
h=12 cm
Slant height
L = r 2 + h 2 = 52 + 122 = 13c.m.
2 Marks
1. Find the total surface area of a hemisphere
h
θ
∴ θ= 45o
∴ The sun rays make 45o angle with the
ground at that time.
2. The angle of elevation of the top of a
tower from a point on the ground which is
30 m away from the foot of the tower is
30o. Find the height of the tower?
A
3
h
B
= AC2 –
C
h
h
= 1 = tan 45 o
h
5
A
BC2
C
30o
B
30m
AB = 16 = 4
Area of ∆ABC = ×base×height
B
A
tan θ =
1
2
12
IMPORTANT QUESTIONS
1 Mark
1. The height of the pole and the length of its
shadow are equal. Then, what is the angle
of elevation of the sun rays with the
ground at that time?
A. From the given data
the triangle?
AB2
L
12. APPLICATIONS OF
TRIGONOMETRY
B
∴ sinθ = cosθ, if θ = 45o
P
4 Marks
1. A chord of a circle of radius 12 cm subtends an angle of 120o at the centre. Find
the area of the corresponding minor segment of the circle (use π = 3.14 and √3 =
1.732)?
2. If a circle touches all the four sides of a
1
1
, cos (A+B) = ,
2
2
0o<A+B ≤ 90o A>B, then find A and B?
3. If A, B and C are interior angles of a triangle ABC, then show that
1
2
AB2 =52 –32 =16
7 cm
2. If sin (A–B) =
2. For what value of 'θ', the trigonometric
ratio sine, cosine are equal?
cosθ = cos 45o =
C
A
2 Marks
1. Is it right to say that sin (A+B) = sinA+
SinB justify your answer?
IMPORTANT QUESTIONS
1 Mark
1. "The mutiplicative inverse of
1 − cos θ 1 + cos θ ''
is
explain?
4. From the figure find the perimeter of
∆ABC?
A. Sin 75o + cos55o
= sin (90o–15o) + cos (90o–35o)
= cos 15o + sin 35o
1
= × 4 × 3 = 6sq.units
2
4. Express tanθ in terms of secθ?
A. From the trigonometric identity
1+tan2θ= sec2θ
tan2θ = sec2θ–1
tan θ = sec2 θ − 1
5. Express sin 75o+cos55o in terms of
trigonometric ratios of angles between 0o
and 45o?
Making brief
notes of the topics
always helps and you
should prepare brief notes
of whatever your teachers
share with you. These
help you prepare
effectively as and
when the exams
approach.
In a right angled triangle the square of the hypotenuse is equal to the
sum of the squares of the other two sides. Pythagoras (570 BC-495
BC), the great Greek mathematician announced it. More than 50
proofs are available for this theorem.
h
1
h
tan 30 =
⇒
=
30
3 30
o
30 30 × 3 30 3
h=
=
=
= 10 3m
3
3
3× 3
2 Marks
1. A circus artist is climbing a 20 m long
rope, which is tightly stretched and tied
from the top of a vertical pole in the
ground. Find the height of the pole if the
angle made by the rope with the ground
level is 30o?
2. The angle of elevation of a ladder leaning
against a wall is 60o and the foot of the
ladder is 9.5m away from the wall. Find
the length of the ladder?
4 Marks
1. A tree breaks due to storm and the broken
part bends so that the top of the tree touches the ground by making 300 angle with
the ground. The distance between the foot
of the tree and the top of the tree on the
ground is 6m. Find the height of the tree
before falling down?
2. Two pillars of equal height and on either
side of a road, which is 100 m wide. The
angles of elevation of the top of the pillars
are 60o and 30o at a point on the road
between the pillars. Find the position of
the point between the pillars and the
height of each pillar?
3. An aeroplane at an altitude of 200 meters
observes the angles of depression of opposite points on the two banks of a river to be
45o and 60o. Find the width of the river?
4. As observed from the top of a light house,
100m above sea level, an angle of depression of a ship, sailing directly towards it,
changes from 30o to 45o. Determine the
distance travelled by the ship during the
period of observation?
13. PROBABILITY
IMPORTANT QUESTIONS
Rs.1''
The no. of favourable out comes to the
event E =25
∴ P(E) =
25 1
=
100 4
3. Express the probability of the complement
event of event E, interms of the probability of event E?
A. Let complement event of event E = E
4. From the figure find the probability of getting a ball of prime number?
13 12
1
2
15
11 10
3
4
9
5
8
6
7
A. From the figure number of all possible out
comes = 15
The numbers of outcomes favourable to
the event getting a ball of prime numbers
= 6. ({2, 3, 5, 7, 11, 13})
∴ Probability =
6 2
=
15 5
2 Marks
1. Harpreet tosses two different coins simulataneously. What is the probability that he
gets atleast one head?
2. One card is drawn from a well -shuffled
deck of 52 cards. Find the probability of
getting a face card?
3. A bag contains two black, three white, 5
yellow balls. One ball is drawn at random
from the bag. Find the probability that the
ball is drawn is not yellow?
4 Marks
1. Two dice are rolled simulataneously and
counts are added. Complete the table
given below?
Event sum
on 2 dice
2
Probability
1
36
3
4
5
1 Mark
1. Are the probabilities of getting a head
when the coin is tossed, and getting an
even number, when a die is thrown once,
equal or not? explain?
A. coin
E = the event "getting a head"
Probability P(E)
2. A box contains 100 discs which are numbered from 1 to 100. If one disc is drawn
at random from the box, find the probability that it bears
i) Two-digit numbers divisible by 5
ii) Factor of 100
iii) A perfect square number.
Number of outcomes favourable to E 1
=
=
Number of all possible outcomes
2
14. STATISTICS
Die
E = getting an even number = [2, 4, 6]
P(E) =
3 1
=
6 2
∴ Probability in both situations are equal.
2. A kiddy bank contains twenty five Rs.1
coins, forty three Rs.2 coins, thirty two
Rs.5 coins. If it is equally likely that one
of the coins will fall out when the kiddy
bank is turned upside down. What is the
probability that the coin will be Rs.1?
A. The no. of possible out comes = 25 + 32 +
43 = 100
Let E denote the event ''the coin will be
⇒ 25×7= p+150 ⇒ p=175−150= 25.
3. The heights of the students in a class are as
follows (in feets)
5.2, 5.6, 5.3, 4.9, 5.1, 5.2, 5.5, 5.3, 4.8, 5.7,
5.3, 4.8.
Find the mode of the data?
A. A mode is that value among the observations which occurs most frequently.
∴ P(E) = 1 − P(E)
14
5
10th Class Special - Maths
IMPORTANT QUESTIONS
1 Mark
1. "Mean takes into consideration extreme
values of the data only" will you agree
with the statement? or not? why?
A. Not agree, because mean takes into
account all the observations, and lies
between the extremes.
2. Mean of observations 10, 15, 20, p, 30, 35,
40 is 25. Then find the value of P?
A. Mean == Sum of observations
No.of observations
10 + 15 + 20 + p + 30 + 35 + 40
⇒ 25 =
7
∴ Mode of the given data = 5.3
4. Find the mode from the given ogive curve
if n=60?
60
50
Less than
cumulative frequency
A. From the given data
VýS$Æý‡$ÐéÆý‡… l íœ{ºÐ]lÇ l 11 l 2016
40
n/2
30
20
les
st
n
ha
og
iv
e
10
X
5
10 15
20
25 30
Upper limits
n 60
A. n = 60, = = 30
2 2
n
Locate value on the Y-axis.
2
From this point, draw a line parallel to the
6
7
8
9
5
36
10 11 12
1
36
X-axis cutting the curve at a point. From
this point, draw a perpendicular to theXaxis. Foot of this perpendicular determines
the median of the data
∴ Median = 10
2 Marks
1. Write the formula of median for a grouped
data. Explain the symbols in words?
2. Convert the given below distribution to a
'more than' type cumulative frequency distribution?
Class Interval
50-55
55-60
60-65
65-70
70-75
75-80
Frequency
2
8
12
24
38
16
4 Marks
1. A student noted the number of cars passing through a spot on a road for 100 periods each of 3 minutes, and summarised
this in the table given below.
Find the mode of the data?
No. of Cars
0-10
10-20
20-30
30-40
40-50
50-60
60-70
70-80
Frequency
7
14
13
12
20
11
15
8
2. The following distribution show the daily
pocket allowance of children of a localilty.
The mean pocket allowance is Rs. 18.
Find the missing frequency f?
Daily Pocket
Allowance
(in Rs.)
11-13
13-15
15-17
17-19
19-21
21-23
23-25
No. of
Children
7
6
9
13
f
5
4
3. During the medical check-up of 35 students of a class, their weights were recorded as follows?
Weight
(in kg)
less than 38
less than 40
less than 42
less than 44
less than 46
less than 48
less than 50
less than 52
No.
of students
0
3
5
9
14
28
32
35
Draw a less than type ogive curve for the
given data. Hence obtain the median
weight from the graph.
Do
not leave the
things to the end as
this can create unnecessary pressure on you
and can affect your
examination
results eventually.
Probability theory had its origin in the 16th century when an Italian physician
and mathematician J.Cardan wrote the first book on the subject, The Book
on Games of Chance. James Bernoulli, A.DeMoivre, and Pierre Simon
Laplace are among those who made significant contributions to this field.
VýS$Æý‡$ÐéÆý‡… l íœ{ºÐ]lÇ l 11 l 2016
6
10th Class Special - Maths
x
PAPER - I
ANSWERS
1. REAL NUMBERS
1) 2; 2) 4; 3) 2; 4) 1; 5) 4; 6) 1; 7) 1
1. 7×11×13 + 13 is
( )
1) a prime number
2) a composite number
3) an odd number
4) divisible by 5
2. The LCM of two numbers is 1200. Which
of the following cannot be their HCF
( )
1) 600
2) 500
3) 400
4) 200.
3. The rational number between √2 and √3 is
( )
3. POLYNOMIALS
6
1)
5
3
2)
4
3
3)
2
4
4)
5
4. The smallest number by which √27 should
be multiplied so as to get a rational number is
( )
1) √27
2) 3√3
3) √3
4) 3
( )
5. log20162016 =
1) 0
2) 1
3) –1
4) 2016
6. The exponential form of log264=6 is ( )
2) 25 = 64
1) 43 = 64
6
4) 23 = 6
3) 2 = 64
7. The sum of the exponents of the prime
factors in the prime factorisation of 196, is
( )
1) 1
b) 2
c) 4
d) 6
1)
−7
7
3
2) 3
3) 7
4) 7
3. If the product of two zeroes of the polynomial f(x) = 2x3 +6x2 – 4x +9 is 3, then its
third zero is =
( )
1)
−3
3
2
1. Roster form of {x:x∈N, x2+x–20=0}is
( )
1) {–5, 4} 2) {4}
3) {5}
4) {4, 5}
2. From the figure, cardinal number of set A
( )
µ
B
2
1 A
6
7
8
10
9
1) 2
2) 3
3) 4
4) 5
3. If n (A) = 7, n(B)= 8, n (A∩B)= 5 then
n (A∪B) =
( )
1) 12
2) 10
3) 9
4) 6
4. If A, B are disjoint sets then
( )
1) A∩B = φ
2) A∪B = φ
3) A–B = φ
4) B–A = φ
5. From the figure which statement is true
( )
B
A
a
d
h b g
4) x + 3
10
2.
4. What value of 'K', the pair of equations
3x+4y+2=0 and 9x+12y+k=0 represent
coincident lines
( )
1) 5
2) 6
3) –5
4) –6
5. Solution for the equations √3x+√5y=0 and
√7x+√11y=0 is
( )
1) x=3, y=5
2) x=7, y=11
3) x=1, y=1
4) x=0, y=0
6. A pair of linear equations in two variables
are 2x–y=4 and 4x–2y=6. This pair of
equations are
( )
1) Consistent equations
2) Dependent equations
3) Inconsistent equations 4) Cannot say
−9
9
2) 2
3) 2
4)
2
4. The following is the graph of the polynomial. Find the zeroes of the polynomial
from the given graph
( )
ANSWERS
3.
4.
5.
6.
1) 2b = a+c
2) b=a+c
3) b = ac
4) b = ac
If the sum of first k terms of an A.P. is
3k2–k and its common difference is 6 then
the first term is
( )
1) 1
2) 2
3) 3
4) 4
Find the sum of first 15 multiples of 8( )
1) 960 2) 1000 3) 940
4) 1060
In a G.P. 3rd term is 24 and 6th term is 192,
( )
then 10th term is
1) 1024 2) 2048 3) 3072 4) 4024
In a garden there are 32 rose flowers in
first row and 29 flowers in 2nd row, 26
flowers in 3rd row, then how many rose
( )
flowers are there in the 6th row
1) 14
2) 15
3) 16
4) 17
The common difference of an Arithmetic
progression, whose 3rd term is 5 and 7th
term is 9, is
( )
1) 1
2) 2
3) 3
4) 4
1) 2; 2) 1; 3) 2; 4) 2; 5) 4; 6) 3
ANSWERS
5. QUADRATIC EQUATIONS
1) 1; 2) 2; 3) 1; 4) 3; 5) 4; 6) 1
7. COORDINATE
GEOMETRY
–4 –3 –2 –1 0
1 2 3 4
1. The distance between (x1,y1) and (x2, y2)
is
( )
x
1) –2, 3 2) 1, 3
3) –2, 1 4) 3, 0
5. Which of the following is a polynomial
with only one zero
( )
1) P(x) = 2x2–3x+4 2) P(x) = x2–2x+1
4) P(x) = 5
3) P(x) = 2x2–3
6. Observe the given rectangle figure then its
area in polynomial function is
( )
10-x
2. SETS
5
)
−3
3
x + 10
3
y
1) 2; 2) 2; 3) 3; 4) 3; 5) 2; 6) 3; 7) 3
4
is equal to
(
ANSWERS
3
1 1
+
α β
p(x)= 4x2+3x+7, then
2) 3 + 10
3)
1. The degree of a constant polynomial is
( )
1) 0
2) 1
3) 2
4) 3
2. If α, β are the zeroes of the polynomial
x
1) 3 − 10
x+3
1) A(x) = x2 + 7x+30
2) A(x)= –x2+7x+30
3) A(x) = x2–7x+30 4) A(x)= –x2–7x+30
7. What is the coefficient of the first term of
the quotient when 3x3+x2+2x+5 is divided
( )
by 1+2x+x2
1) 1
2) 2
3) 3
4) 5
1) 0
2)
2
1) x + x =
3) x − =
1) 1; 2) 4; 3) 2; 4) 3; 5) 2 ; 6) 2; 7) 3
5
2
5
2
4) None
1
x
2) x + =
5
2
5
2
6. From the figure the roots of the quadratic
equation are
( )
2
1
0
–3–2–1
( x2 + x1 ) 2 + (y 2 + y1 ) 2
2)
( x2 − x1 ) 2 + (y 2 + y1 ) 2
3)
( x2 + x1 ) 2 + (y 2 − y1 ) 2
4)
( x2 − x1 ) 2 + (y 2 − y1 ) 2
2. The triangle with vertices (–2, 1), (2,–2)
and (5, 2) is
( )
1) Scalene
2) Equilateral
3) Isosceles
4) Right angled isosceles
3. The co-ordinates of the centroid of the triangle whose vertices are (8,–5), (–4,7) and
(11, 13) are
( )
1) (2,2) 2) (3,3) 3) (4,4) 4) (5,5)
4. 'Heron's formula to find the area of a triangle is
( )
1)
2)
4) None
4. PAIR OF LINEAR
EQUATIONS IN
TWO VARIABLES
c
3) 5
4. If one root of the equation 4x2–2x+(λ–4)
=0 be the reciprocal of the other, then λ =
( )
1) 8
2) 7
3) 6
4) 5
5. The sum of a number and its reciprocal is
5/2 Represent this situation as
( )
1
x
ANSWERS
µ
1. If the product of two consecutive natural
numbers is 72, then the natural numbers
are
( )
1) 6, 7
2) 7, 8
3) 8, 9
4) 9, 10
( )
2. The roots of x2–2x–(r2–1)=0 are
1) 1–r, –r–1
2) 1–r, 1+r
3) 1, r
4) 1–r, r
3. If α, β are the roots of the quadratic equa( )
tion √2x2+7x+5√2=0, then αβ =
1)
(s − a)(s − b)(s − c)
s(s + a)(s + b)(s + c)
3) s(s − a)(s − b)(s − c)
4) None
5. From the figure, if area of ∆ABC=5sq.
units, then the area of given parallelogram
is ____ sq.units
( )
1) 5
2) 10
3) 2.5
4) 15
123
D
e f
1) A∩B = {d, g}
2) A–B = {a, b, h}
3) B–A = {c, e, f}
4) All the above
6. If A⊂B then A∩B=
( )
1) A
2) B
3) φ
4) A∪B
7. If the union of two sets is one of the set
itself, then the relation between the two
sets is
( )
1) one set is a subset of other set
2) disjoint sets
3) equal number of elements in both thesets
4) empty
1. If the line y = px–2 passes through the
point (3, 2), then the value of P is
( )
1)
3
4
2)
4
3
3) 3
4) 4
2. If the pair of lines 2x+y+5=0 and 4x+2y
+10=0 represent ____ lines
( )
1) Coincident lines
2) Lines through origin
3) parallel lines
4) Intersecting lines
3. The age of a son is one-third of the age of
his father. If the present age of father is x
years, then the age of the son after 10
years is
( )
C
1) –2, 1 2) –1, 2 3) 0, 1
4) 0, 2
7. The roots of the quadratic equation
x2 − 8 1
= are
x 2 + 20 2
1) ±2
2) ±3
(
3) ±4
)
4) ±6
ANSWERS
B
6. If a straight line passing through the points
P (x1,y1), Q(x2, y2) is making an angle 'θ'
with positive X-axis, then the slope of the
straight line is
( )
y 2 + y1
1) x + x
2
1
1) 3; 2) 2; 3) 3; 4) 1; 5) 2; 6) 1; 7) 4
2) θ
y 2 − y1
3) x − x 4) sin θ
2
1
ANSWERS
6. PROGRESSIONS
1. If a, b, c are in A.P., then
A
(
)
1) 4; 2) 4; 3) 4; 4) 3; 5) 2; 6) 3
Hipparchus, a Greek mathematician established the relationships
between the sides and angles of a triangle. The first trigonometric
table was apparently compiled by Hipparchus, who is now consequently known as "the father of trigonometry'.
PAPER - II
1. In triangles ABC and DEF, ∠A = ∠E =
40o, AB:ED = AC:EF and ∠F = 65o, then
∠B =
( )
2) 65o
3) 75o
4) 85o
1) 35o
2. Sides of two similar triangles are in the
ratio 4:9. Areas of these triangles are in the
ratio
( )
1) 2:3
2) 4:9
3) 81:16 4) 16:81
3. In an equilateral triangle ABC, if AD⊥BC,
then
( )
1) 2 AB2 = 3 AD2 2) 4 AB2 = 3 AD2
3) 3 AB2 = 4 AD2 4) 3 AB2 = 2 AD2
4. If ∆ABC is an isoscles triangle and D is a
point on BC such that AD ⊥BC, then ( )
1) AB2– AD2 = BD.DC
2) AB2–AD2=BD2–DC2
3) AB2+AD2=BD.DC
4) AB2+AD2=BD2–DC2
5. In the figure ∆ACB ∼ ∆APQ. If AB = 6
cm, BC = 8 cm and PQ = 4 cm then AQ =
____ cm
( )
B
A
P
Q
C
1) 2cm 2) 3 cm 3) 4 cm 4) 5 cm
6. A Vertical stick 20m long casts a shadow
10m long on the ground. At the same time,
a tower casts a shadow 50 m long on the
ground. The height of the tower is ( )
1) 100m 2) 120 m 3) 25 m 4) 200 m
7. In the figure, AD bisects ∠A. AB = 6cm
BD = 8 cm, DC = 6 cm. Then the value of
AC is
( )
A
D
2) 4.5 cm 3) 5 cm
C
4) 5.5 cm
2
3
2
3) 102 sq cm
3
4
3
2
4) 101 sq cm
3
1) 104 sq cm
5. The number of pair of tangent can be
drawn to a circle, which are parallel to
each other are
( )
1) 0
2) 2
3) 4
4) Infinite
6. How many tangent lines can be drawn to a
circle from a point outside the circle ( )
1) 1 2) 2
3) 3
4) 4
1. The surface areas of two spheres are in the
ratio 1:4. then, the ratio of their volumes is
( )
1) 1 :4
2) 1 : 8
3) 1 :16 4) 1 : 64
2. Diagonal of a cuboid is
( )
1) lbh
2) 2 (lb+ bh+hl)
2
2
2
3) l + b + h
4) none
3. The ratio of the volume of a cube to that of
the sphere which will exactly fit inside the
cube is
( )
1) 6: π
2) 4: π
3) 2: π 4) 3: π
4. How many lead balls of radius 2cm can be
made from a ball of radius 4 cm
( )
1) 1
2) 2
3) 4
4) 8
5. If a sphere, a cylinder and cone are of the
same radius and same height, then the
ratio of their curved surface is
( )
1) 4 : 4: 5
2) 2 : √3 : 5
3) 4 : 4 : √5
4) None
6. Total surface area of a solid hemisphere
( )
whose radius is 7cm is ____ cm2
1) 327 π 2) 144 π 3) 147 π 4) 189 π
7. The area of the square that can be
inscribed in a circle of radius 8cm is ____
( )
(in cm2)
1) 256
2) 128
3) 64 2 4) 64
11. TRIGONOMETRY
B
D
1) 8 cm 2) 28 cm 3) 14 cm 4) 56 cm
4. If the length of the minute hand of a clock
3)
4
5
4)
5
4
1
1)
2
2) 2
(
3
3)
2
)
4) 1
1) 1; 2) 2; 3) 3; 4) 2; 5) 2; 6) 4; 7) 2
12. APPLICATIONS OF
TRIGONOMETRY
2 tan 30 0
1. The value of
=
1 + tan 2 300
(
)
1) sin 60°
2) cos 60°
3) tan 60°
4) sin 30°
2. If A+B = 90°, sin A = 3/4, then secB is
( )
1)
3
4
2)
4
3
3)
1
4
4)
1
3
3. The value of 1+tan 5o. cot 85o is equal to
( )
2) cos2 5o
1) sin2 5o
3) sec2 5o
4) cosec2 5o
4. (sec A + tan A) (1–sinA) is equal to ( )
1) sec A
2) cos A
3) cosecA
4) sin A
24
5. If sin A =
then cot A =
25
25
7
24
1)
2)
3)
24
24
7
(
4)
25
7
)
1. The height of the tower is 100cm. When
the angle of elevation of sun is 30o, then
shadow of the tower is
( )
1) 100 √3 m
2) 100m
100
m
4)
3
2. If the height and length of the shadow of a
man are the same, then the angle of elevation of the sun is
( )
2) 60o
3) 45o 4) 15o
1) 30o
3. The tops of two poles of height 20m and
14 m are connected by a wire. If the wire
makes an angle of 30o with horizontal,
then the length of the wire is
( )
1) 6m
2) 8m
3) 10m 4) 12m
4. A ladder 'x' meters long is laid against a
wall making an angle 'θ' with the ground.
If we want to directly find the distance
between the foot of the ladder and the foot
of the wall, which trigonometrical ratio
should be considered?
1) sin θ 2) cos θ 3) tan θ 4) cot θ
5. Two persons A and B observe the top of a
pole at an angle of elevation α and β
respectively. If α > β, then
( )
1) A is nearer to the pole than B
2) B is nearer to the pole than A
3) A, B are at the same distance from the
pole
4) Can't compare their distance
6. The angle of elevation of the Sun is 45o.
Then the length of the shadow of a 12 m
high tree is
( )
1) 12m 2) 6m
3) 8m
4) 6√3m
7. A pole 6m high casts a shadow 2√3 m long
on the ground, then Sun's elevation is ( )
3) 30o
4) 90o
1) 45o 2) 60o
ANSWERS
1) 1; 2) 3; 3) 4; 4) 2; 5) 1; 6) 1; 7) 2
13. PROBABILITY
1. Two coins are tossed simultaneously. The
probability of getting a head on only one
of the two coins is
( )
1) 1
2)
1
2
3)
1
4
4)
3
4
1
2
2)
1
6
3)
1
3
4)
3
4
3. A letter is chosen at random from the
English alphabet. The probabilities that
the letter chosen is a vowel is
( )
1)
ANSWERS
3) 100 (√3–1)m
2. The probability of getting a prime number
in a single throw of a die is
( )
1)
10. MENSURATION
9. TANGENTS AND
SECANTS TO A CIRCLE
A
3
5
1) 2; 2) 4; 3) 2; 4) 3; 5) 4; 6) 2
1) 2; 2) 3; 3) 1; 4) 4; 5) 3; 6) 2; 7) 2
E
2)
ANSWERS
1) 3; 2) 4; 3) 3; 4) 1; 5) 2; 6) 1; 7) 2
F
3
4
7. sin 45° + cos 45° =
ANSWERS
C
6. Which of the following values is not a
possible value of sin θ
( )
1)
2) 103 sq cm
ANSWERS
1. A circle may have___parallel tangents( )
1) 3
2) 2
3) 1
4) 4
2. If two concentric circles are radius 5 cm
and 3 cm are drawn, then find the length
of the chord of the larger circle which
touches the smaller circle
( )
1) 7 cm 2) 2 cm 3) 6 cm 4) 8 cm
3. The semi perimeter of ∆ABC = 28 cm
then AF+BD+CE is
( )
7
10th Class Special - Maths
is 14 cm, then find the area swept by the
minute hand in 10 minutes
( )
8. SIMILAR TRIANGLES
B
1) 4cm
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1
26
2)
3
26
3)
5
26
4)
1
52
4. A bag contains 4 red, 5 black and 3 yellow
balls. A ball is taken out of the bag at random. The probability that the ball taken
out is not of red colour is
( )
1)
2
3
2)
1
3
3)
1
4
1)
7
12
2)
12
144
3)
3
12
1
4) 2
5. 12 defective pens are accidentally mixed
with 132 good ones. It is not possible to
just look at a pen and tell whether or not it
is defective. One pen is taken out at random from this lot. The probability that the
pen takenout is a good one is
( )
4)
11
12
6. One card is drawn from a well shuffled
deck of 52 card. The probability of getting
a face card is
( )
1)
40
52
2)
12
52
3)
13
52
4)
4
52
7. If the probability of an event is 0.65, then
the probability of not happening of that
event is
( )
1) 0.35 2) 0.035 3) 1.25 4) 3
ANSWERS
1) 2; 2) 1; 3) 3; 4) 1; 5) 4; 6) 2; 7) 1
14. STATISTICS
1. The most frequently used measure of central tendency is
( )
1) Mean 2) Mode 3) Median 4) None
2. Arun scored 36 marks in English, 44
marks in Hindi, 75 marks in Maths and x
marks in Science. If he has scored an average of 50 marks, find the value of x ( )
1) 45
2) 40
3) 50
4) 48
3. Which of the following cannot be determined graphically
( )
1) Mean 2) Median 3) Mode 4) None
4. The mean of first n odd natural numbers is
( )
1)
n +1
2
2)
n
2
3) n
4) n2
5. The wickets taken over by a bowler in 10
cricket matches are as follows 2, 6, 4, 5, 0,
2, 1, 3, 2, 3, then the mode of this data is
( )
1) 0
2) 1
3) 2
4) 3
6. Cumulative frequency curves are called as
____ curves
( )
1) Median 2) Scale 3) Ogive 4) None
7. Data having two modes is called ____ data
( )
1) Unimodal
2) Bimodal
3) Trimodal
4) None
8. The width of the class interval 40-50 is ( )
1) 40
2) 50
3) 45
4) 10
ANSWERS
1) 1; 2) 1; 3) 1; 4) 3; 5) 3; 6) 3; 7) 2; 8) 4