Assignmen ts --x

ASSIGNMENTS:
LINEAR EQUATIONS
Solve the following system of linear equations by Substitution
method:
Q. 1.
Q. 2.
Solve the following system of linear equations by Elimination
Method(Equating coefficients)
Q. 3.
Q. 4.
Solve the following system of linear equations by Cross Multiplication
method
Q. 5.
Q. 6.
Solve the following pairs of equations by reducing them to a pair of
linear equations.
Q. 7.

Q. 8.
Solve the following pair of linear equations in two variables:
Q. 9.
Q. 10. Solve:
Q. 11. For what value of k will the following system of linear equations
have no solution
,
Q. 12. Solve for x and y:
,
Q. 13. Find the values of p and q for which the following system of
equations has infinite number of solution;
Q. 14. Solve and hence find the value of m
for which

Q. 15. Solve the following equations graphically
Shade the region between the two lines and x-axis.
Q. 16. Solve graphically: . Determine
the vertices of the triangle formed by the lines and x-axis.
Q. 17. The present age of a father is equal to the sum of the ages of his
5 children. 12 years hence the sum of the ages of his children will be
twice the ages of their father. Find the present age of the father.
Q. 18. Two places A and B are 80 km apart from each other on a
highway. A car stats from A and another from B at the same time. If
they move in the same direction, they meet in 8 hours and if they
move in opposite directions the meet in 1hour and 20 minutes. Find
the seed of the cars.
Q. 19. A train covered a certain distance at a uniform speed. If the
train would have 6 km/hr. aster it would have taken 8 hrs. less than
the scheduled time. And if the train were slower by 6 km/hr. it would
have taken 12 hours more than the scheduled time. Find the length of
the journey.
Q. 20. A man travel 600 km partly by train and partly by car . If he
covers 400 km by train and the rest by car , it takes him 6 hours 30
minutes. But if he travels 200 km. by train and the rest by car , he
takes half an hour longer. Find the speed of the train and that of
the car.
(POLYNOMIALS)
Q. 1. If and are the zeroes of the quadratic polynomial :
find .
Q.2. Find the quadratic polynomial, the sum and product of whose
zeros are -3 and 2respectively.

Q. 3. Find the sum and product of zeroes of the
polynomial
Q. 4. Verify that are the zeroes ofthe cubic
polynomial
Q. 5. Divide the polynomial p(x) by the polynomial q(x) and find the
quotient and remainder.
Q.6 Find the value for K for which x4 + 10x3 + 25x2 + 15x + K exactly
divisible by x + 7. (Ans : K= - 91)
Q.7 If two zeros ofthe polynomial f(x) = x4 - 6x3 - 26x2 + 138x – 35 are
2±√ 3.Find theother zeros. (Ans:7, -5)
Q.8 Find the Quadratic polynomial whose sum and product of zeros
are √2 + 1 ,
𝟏
√𝟐 + 𝟏
(Ans. x2 – (2 √2) x + 1)
Q.9 If 𝜶, 𝜷 are the zeros of the polynomial x2 + 8x + 6 form a Quadratic
polynomial
whose zeros are 1 +
𝜶
𝜷
, 1 +
𝜷
𝜶
( Ans: x2- 32/3x+32/3)
Q.10 On dividing the polynomial 4x4 - 5x3 - 39x2 - 46x – 2 by the
polynomial g(x) the quotient is x2 - 3x – 5 and the remainder is -5x +
8.Find the polynomial g(x). (Ans:4 x2+7x+2)
{Hint p(x) = g (x) q (x) + r (x)
let p(x) = 4x4 – 5x3 – 39x2 – 46x – 2
q(x) = x2 – 3x – 5 and r (x) = -5x + 8}
Q.11 Ifthe squared difference ofthe zeros of the quadratic polynomial
x2 + px + 45 is equal to 144 , find the value of p. (Ans: ±18).
Q.12 If, are the zeros of a Quadratic polynomial suchthat 𝜶 + 𝜷 = 24,
𝜶 - 𝜷 = 8. Find a Quadratic polynomial having and as its zeros.
(Ans: k(x2– 24x + 128))
Q.13 Ifthe ratios of the polynomial ax3+3bx2+3cx+dare in AP, Prove
that 2b3-3abc+a2d=0

{Hint
Let p(x) = ax3 + 3bx2 + 3cx + d and 𝜶, 𝜷 , r are their three Zeros
but zero are in AP
let 𝜶 = m – n , 𝜷 = m, r = m + n , Sum = -b/a, product =c/a}
Q.14 Ifone zero of the polynomial 3x2 - 8x +2k+1 is seven times the
other, find the zeros and the value of k (Ans k=
2/3)
Q.15 If 2, ½ are the zeros of px2+5x+r, prove that p= r.
(Coordinate Geometry)
1. Show that the points A(2, -2), B(14, 10), C(11, 13) and D(-1, 1)
are the vertices of a rectangle.
2. Determine the ratio in which the points (6, a) divides the join of
A(-3, -1) and B(-8, 9). Also find the value of a.
3. Find the point on the x-axis which is equidistant from the points
(-2, 5) and (2, -3).
4. The co-ordinates of the mid-point of the line joining the points
(3p, 4) and (-2, 2q) are (5, p). Find the values of p and q.
5. Two vertices of a triangle are (1, 2) and (3, 5). If the centroid of
the triangle is at the origin, find the coordinates of the third
vertex.
6. If ‘a’ is the length of one of the sides of an equilateral triangle
ABC, base BC lies on x-axis and vertex B is at the origin, find the
coordinates of the vertices of the triangle ABC.
7. The coordinates of the mid-point of the line joining the points
(2p+1, 4) and (5, q – 1) are (2p, q). Find the value of p and q.
8. The coordinates of two vertices A and B of a triangle ABC are (1,
4) and (5, 3) respectively. If the coordinates of the centroid of
triangle ABC are (3, 3), find the coordinates of the third vertex C.
9. Find the value of k for which the points with coordinates (3, 2),
(4, K) and (5, 3) are collinear.
10. Find the value of k for which the points with coordinates
(2, 5), (k, 11/2) and (4, 6) are collinear.
Answers
1. (4, -4) 3. 3 : 2, 5 4. (-2, 0)

5. 4, 2 6. (-4, -7)
7.
8. 3, 3 9. (3, 2) 10. 5/2
( Probability)
1. (1) A dice is thrown once. Find the probability of getting
(a) A number greater than 3
(b) A number less than 5
2. A bag contain 5 black, 7 red and 3 white balls. A ball is drawn
from the bag at random. Find the probability that the ball drawn
is
(a) Red
(b) Black or white
(c) Not black
3. A bag contains 4 red 5 black and 6 white balls. A ball is drawn
from the bag a random. Find the probability that the ball drawn
is
(a) White
(b) Red
(c) Not black
(d) Red or white
4. A card is drawn at random from a pack of 52 playing cards. Find
the probability that the card drawn is neither a queen nor a jack.
5. Tickets numbered from 1 to 20 are mixedup 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?
6. In a single throw of dice, what is the probability of
(a) An odd number on one dice and 6 on the other
(b) A number greater than 4 on each dice
(c) A total of 11
(d) Getting same number on either dice.
7. A die is thrown twice. Find the probability of getting
(a) doublets
(b) number greater than 5 on one dice.
8. Three coins are tossed simultaneously. Find the probability of
getting
(a) Exactly 2 heads
(b) No heads

9. In a simultaneous toss of four coins, What is the probability of
getting:
(a) Less than 2 heads?
(b) Exactly 3 head
(c) More than 2 heads?
10. Three coins are tossed once. Find the probability of:
(a) 3 heads
(b) exactly 2 heads
(c) at least two heads
Answers
1. 1/2, 2/3, 2. 7/15, 8/15,
2/3
3. 2/5, 4/15,
2/3, 2/3
4. 11/13,
5. 2/5 6. 1/6, 1/9,
1/18, 1/6
7. 1/6, 11/36 8. 3/8, 1/8
9. 5/6, 3/8, 5/16 10. 1/8, 3/8, 1/2
1. If 65% of the populations have black eyes, 25% have brown eyes
and the remaining have blue eyes. What is the probability that a
personselectedat random has (i) Blue eyes (ii) Brown or black
eyes (iii) Blue or black eyes
(iv) neither blue nor browneyes (Ans: No.of black eyes = 65
No. of Browneyes = 25
No. of blue eyes = 10
Total no. ofeyes = 180 1/10, 9/10, 3/4, 13/20)
12.A number x is chosenat random from the numbers -3, -2, -1, 0 1, 2,
3. What is the probability that | x|< 2 (Ans: 3/7)
13. A number x is selectedfrom the numbers 1,2,3 and then a second
number y is randomly selectedfrom the numbers 1,4,9. What is the
probability that the product xy of the two numbers will be less than 9?
(Ans: 5/9)
14. A jar contains 54 marbles eachof which is blue , green or white.
The probability of selecting a blue marble at random from the jar is
1/3 and the probability of selecting a greenmarble at random is 4/9
. How many white marbles does the jar contain? (Ans:12 , Let there be
b blue, g green and w white marbles in the marbles in the jar. Then,
b + g + w = 54 , P (Selecting ablue marble) =b/54 )
15. A bag contains 8 red balls and x blue balls,the odd against
drawing a blue ball are 2: 5. What is the value of x? [ hint Probability

of drawing blue balls =
𝒙
𝟖+𝒙
Probability ofdrawing red balls =
𝟖
𝟖+𝒙
,
𝟖
𝟖+𝒙
:
𝒙
𝟖+𝒙
= 2: 5 ans. Is 20]
( Quadratic Equations )
1. Find the value of k for kx2 + 2x - 1 = 0, so that it has two equal
roots
2. Find the value of k for k x2 - 2√ 5 x + 4 = 0, so that it has two equal
roots.
3. Ifthe roots ofthe equation (b - c) x2 + (c - c) x + (a - b) = 0 are equal,
accordingly prove that 2b = a + c.
4. Find the discriminant of the quadratic equation 3x2– 4 √3x + 4 = 0,
and hence find the nature of its roots.
5. Find the value of k for 2 x2 + k x + 3 = 0, so that it has two equal
roots.
6. Find the value of k for k x (x – 2) + 6 = 0, so that it has two equal
roots.
7. Find the value of k for which the equation x2 + 5kx + 16 = 0 has no
real roots.
8 Find the discriminant of the quadratic equation 2x2– 6x + 3 = 0, and
hence find the nature of its roots.
9. Find the value of k for k2 x2 – 2 (2 k - 1) x + 4 = 0, so that it has two
equal roots.
10. Find the value of k for (k+ 1) x2 – 2 ( k - 1) x + 1= 0, so that it has
two equal roots.
11. Determine the positive value of k for which the equation x2 + k x +
64 + 0 and x2 – 8x + k = 0 will bothhave real roots.(3Marks)
12. Find the discriminant of the quadratic equation 2x2 – 3 x + 5 = 0,

13. Find the value of k for x2 – 2(k + 10x + k2 ) = 0, so that it has two
equal roots.
14. If-4 is a root of the quadratic equationx2 + px–4=0and the
quadratic equationx2+ px +k=0 has equal roots,find the value of k.
15. Find the discriminant of the quadratic equation 2x2 – 4x + 3 = 0,
16. The difference of squares of two numbers is 180. The square ofthe
smaller number is 8 times the larger number.Find the two numbers.
17. Ina class test,the sum of Amit’s marks in Mathematics and English
is 30. Had she got 2 marks more in Mathematics and 3 Marks less in
English, the product of their marks would have been 210. Find her
marks in the two subjects.
18. A motor boat whose speed is 18 km/hin still water takes 1 hour
more to go 24 km upstream than to returndownstream to the same
spot. Find the speed of the stream.
19. A pole has to be erectedat a point on the boundary of a circular
park of diameter 13 metres in such a way that the differences of its
distances from two diametrically opposite fixedgates A and B on the
boundary is 7 metres.Is it possible to do so? Ifyes, at what distances
from the two gates shouldthe pole be erected?
20. Two water taps together can fill a tank in 9+ 3/8 hours. The tap of
larger diameter takes 10 hours less than the smaller one to fill the
tank separately.Find the time in which eachtap can separately fill the
tank.
REAL NUMBERS
1. Showthat the square of any positive integer is either of the form 4q
or 4q + 1 for some integer q.
2. Show that cube of any positive integer is ofthe form 4m, 4m + 1 or
4m + 3, for some integer m.
3. Showthat the square of any positive integer cannot be of the form
5q + 2 or 5q + 3 for any integer q.

4. Showthat the square of any positive integer cannot be of the form
6m + 2 or 6m + 5 for any integer m.
5. Showthat the square of any odd integer is of the form 4q + 1, for
some integer q.
6. Ifn is an odd integer,then show that n2 – 1 is divisible by 8.
7. Prove that if x and y are both odd positive integers,then x2 + y2 is
even but not divisible by 4.
8. Use Euclid’s division algorithm to find the HCFof 441, 567, 693.
9. Using Euclid’s divisionalgorithm,find the largest number that
divides 1251, 9377 and 15628 leaving remainders 1, 2 and 3,
respectively.
10. Prove that √ 3+ √ 5 is irrational.
11. Showthat 12n cannot end withthe digit 0 or 5 for any natural
number n.
12. On a morning walk, three persons stepoff together and their steps
measure 40 cm, 42 cm and 45 cm, respectively.What is the minimum
distance each should walk so that each can cover the same distance in
complete steps?
13. Showthat the cube of a positive integer of the form 6q + r, q is an
integer and
r = 0, 1, 2, 3, 4, 5 is also of the form 6m + r.
14. Prove that one and only one out of n, n + 2 and n + 4 is divisible by
3, where n is
any positive integer.
15. Prove that one of any three consecutive positive integers must be
divisible by 3.
16. For any positive integer n, prove that n3 – n is divisible by 6.

17. Showthat one and only one out ofn, n + 4, n + 8, n + 12 and n + 16
is divisible
by 5, where n is any positive integer.
[Hint: Any positive integer can be written in the form 5q, 5q+1, 5q+2,
5q+3,5q+4].
ASSIGNMENT OF STATISTICS
Q. 1. If mode of the following data is 15, find k.
10, 15, 17, 15, , 17, 20.
Q. 2. The following table shows the marks obtained by 100 students of
class X in a school during a particular academic session. Find the
mean, median and mode of the distribution.
Marks
No. of
students
Less than 10 7
Less than 20 21
Less than 30 34
Less than 40 46
Less than 50 66
Less than 60 77
Less than 70 92
Less than 80 100
Q. 3. Using step deviation method, find the mean of the following data:
Weight
(in Kg)
25 -
31
31 -
37
37 -
43
43 -
49
49 -
55
55
-
61

No. of
students
10 6 8 12 5 9
Q. 4. Draw a less than ogive for the following data and estimate the
median from it.
C.I.
10 -
20
20 -
30
30 -
40
40 -
50
50 -
60
60 -
70
70 -
80
80 -
90
90 -
100
Frequency 5 14 19 27 43 29 16 12 5
Q. 5. The annual profit earned by 30 shops of a shopping complex in a
locality give rise to the following distribution:
Profit (in lakhs Rs)
Number of Shops
(frequency)
More than or equal
to 5
30
More than or equal
to 10
28
More than or equal
to 15
16
More than or equal
to 20
14
More than or equal
to 25
10
More than or equal
to 30
7
More than or equal
to 35
3
Draw both ogives for the above data and obtain the median profit.
Q. 6. A survey regarding the heights ( in cm ) of 51 girls of class X of a
school was conducted and the following data was obtained:
Height ( in cm ) Number of girls
Less than 140 4

Less than 145 11
Less than 150 29
Less than 155 40
Less than 160 46
Less than 165 51
Find the median height.
Q. 7. Following is the age distribution of a group of students. Draw the
cumulative & frequency polygon, cumulative & frequency curve (less
than type) and hence obtain the median value.
Age Frequency
5 – 6 40
6 – 7 56
7 – 8 60
8 – 9 66
9 – 10 84
10 – 11 96
11 – 12 92
12 – 13 80
13 – 14 64
14 – 15 44
15 – 16 22
16 – 17 22
Q. 8. Find the median marks from the following data:-
Marks
00
and
abo
ve
10
and
abo
ve
20
and
abo
ve
30
and
abo
ve
40
and
abo
ve
50
and
abo
ve
60
and
abo
ve
70
and
abo
ve
80
and
abo
ve
90
and
abo
ve
100
and
abo
ve
Numb
er of
Studen
80 77 72 65 55 43 28 16 10 08 00

ts
Q. 9. Find the mean, median and mode of the following data:
Classes
4 −
8
8 −
12
12
−
16
16
−
20
20
−
24
24
−
28
28
−
32
32
−
36
Total
No. of
students
2 12 15 25 18 12 13 3 100
Q. 10. The following frequency distribution gives the monthly
consumption of electricity 68 consumer of a locality. Find the median,
mean and mode of the data and compare them.
Monthly
Consumption of
electricity
No. of Consumers
65 – 85 4
85 – 105 5
105 – 125 13
125 – 145 20
145 – 165 14
165 – 185 8
185 – 205 4
ASSIGNMENT( Surface Area & Volume)
1. The total surface area of a closed right circular cylinder is
65/2cm2 and the circumference of its base is 88cm. Find the volume
of the cylinder.
2. The volume of a vessel in the form of a right circular cylinder
is and its height is 7cm. Find the radius of its base.
3. The height of a cylinder is 15cm and its curved surface area is
660 sq.cm. Find its radius.
4. A cylindrical tank has a capacity of 6160 cu.cm. Find its depth if
the diameter of its base is 28m. Also, Find the area of the inside
curved surface of the tank.

5. The volume of a right circular cylinder is 1100 cu.cm and the
radius of its base is 5cm. Find its curved surface area.
6. If the radius of the base of a right circular cylinder is halved,
keeping the height same, find the ratio of the volume of the reduced
cylinder to that of the original cylinder.
7. 50 circular plates, each of radius 7cm and thickness 0.5cm are
placed one above the other to form a solid right circular cylinder.
Find the total surface area and volume of the cylinder so formed.
8. A well of diameter 3m is dug 14m deep. The earth taken out of it
has been spread evenly all around it to a width of 4m to form an
embankment. Find the height of the embankment formed.
9. The base radii of tow right circular cones of the same height are
in the ratio 3 : 5. Find the ratio of their volume.
10.If h, c and v respectively are the height, the curved surface and
volume of a cone, prove that
11.The circumference of the base of a 16 m high solid cone is 3m.
Find the volume of the cone.
12.A right circular cone of height 4 cm has a curved surface area
47.1 cm2. Find its volume .
13.How many metres of cloth 5m wide will be required to make a
conical tent, the radius of whose base is 7m and whose height is
24m?
14.A right triangle with sides 3cm and 4cm is revolved around its
hypotenuse. Find the volume of the double cone thus generated.
15.The radius of a sphere is 7cm. If the radius be increased by 50%,
find by how much percent its volume is increased.
16.The largest sphere is carved out of a cube of side 7cm. Find the
volume of the sphere.
17.If the surface area of sphere is 616cm2, find its volume.
18.he volume of two spheres are in the ratio 64 : 27. Find their radii
if the sum of their radii is 21cm.
19.The circumference ofthe edge of a hemispherical bowl is 132cm.
Find the capacity of the bowl.
20.The internal and external diameters of a hollow hemispherical
vessel are 24cm and 25cm respectively. If the cost of painting
1cm2of the surface area is Rs. 5.25, find the total cost of painting the
vessel all over .
21.A bucket of height 8cm and made up of copper sheet is in the
form of frustum of a right circular cone with radii of its lower and
upper ends as 3cm and 9cm respectively. Calculate:
(i) The height of the cone of which the bucket is a part

(ii) The volume of water which can be filled in the bucket.
(iii) The area of copper sheet required to make the bucket.
22.A bucket is in the form of a frustum of a cone and holds 28.490
litres of water. The radii of the top and bottom are 28cm and 21cm
respectively. Find the height of the bucket
23.The radii of the faces of a frustum of a cone are 3cm and 4cm and
its height is 5 cm. Find its volume.
24.A cone of radius 10cm is divided into two parts by drawing a
plane through the mid-point of its axis, parallel to its base compare
the volume of two parts.
25.A hollow cone is cut by a plane parallel to the base and the upper
portion is removed. If the curved surface of the remainder is 8/9 of
the curved surface of the whole cone, find the ratio of the line
segments into which the attitude of the cone is divided by the plane.
26.Marble of diameter 1.4cm are dropped into a cylindrical beaker
of a diameter 7cm, containing some water. Find the number of
marbles that should be dropped into the beaker so that the water
level rises by 5.6cm.
27.Sphere of diameter 5cm is dropped into a cylindrical vessel
partly filled with water. The diameter of the base of the vessel is
10cm. If the sphere is completely submerged, by how much will the
level of water rise?
28.A cone is 8.4cm high and the radius of its base is 2.1cm. It is
melted and recast into a sphere. Find the radius of the sphere.
29.A cone and a hemisphere have equal bases and equal volumes.
Find the ratio of their heights.
30.A conical vessel whose internal radius is 5cm and height 24cm is
full of water. The water is emptied into a cylindrical vessel with
internal radius 10cm. Find the height to which the water rises in the
cylindrical vessel.
31.A spherical cannon ball, 28cm in diameter is melted and cast
into a right circular conical mould, the base of which is 35cm in
diameter. Find the height of the cone.
32.The radii of the internal and external surface of a metallic shell
are 3cm and 5cm respectively. It is melted and recast into a solid
right circular cylinder of height cm. Find the diameter of the
base of the cylinder.
33.A solid metallic sphere of diameter 21cm is melted and recasted
into a number of smaller cones, each of diameter 7cm and height
3cm. Find the number of cones so formed.

34.A solid metallic cylinder of radius 14cm and height 21cm is
melted and recast into 72 equal small spheres. Find the radius of
one such sphere.
35.The diameter of a sphere is 42cm. It is melted and drawn into a
cylindrical wire of 28cm diameter. Find the length of the wire.
36.The largest sphere is carved out of a cube of side 7cm. Find the
volume of the sphere.
37.A solid sphere of radius 6cm is melted into a hollow cylinder of
uniform thickness. If the external radius of the base of the cylinder
is 5cm and its height is 32cm. find the uniform thickness of the
cylinder.
38.A spherical shell of lead, whose external diameter is 18cm, is
meltedand recast into a right circular cylinder,whose height is 8cm
and diameter 12cm. Determine the internal diameter of the shell.
39.The diameters of the internal and external surfaces of a hollow
spherical shell are 6cm and 10cm respectively. If it is melted and
recast into a solid cylinder of diameter 14cm, find the height of the
cylinder.
40.A hemispherical bowl of internal radius 9cm. is full of water.
This water is to be filled in cylindrical bottles of diameter 3cm and
height 4cm. Find the number of bottles needed to fill the whole
water of the bowl.
41.The internal and external radii of a hollow sphere are 3cm and
5cm respectively. The sphere is melted to form a solid cylinder of
height cm. Find the diameter and curved surface area of the
cylinder.
ASSIGNMENT( COMBINED SOLID FIGURES)
1. A solid is in the form of a right circular cone mounted on a
hemisphere. The radius of the hemisphere is 2.1cm and the
height of the cone is 4cm. The solid is placed in a cylindrical tub,
full of water, in such a way that the whole solid is submerged in
water. If the radius of the cylinder is 5cm and its height is 9.8cm,
find the volume of the water left in the cylindrical tub.
2. A solid wooden toy is in the shape of a right circular cone
mounted on a hemisphere. If the radius of the hemisphere is

4.2cm and the total height of the toy is 10.2cm, find the volume of
the wooden toy.
3. A toy is in the form of a cone mounted on a hemisphere of radius
7cm. The total height of the toy is 14.5cm. Find the volume and
the total surface area of the toy..
4. A toy is in the form of a cone mounted on a hemisphere of radius
3.5cm. The total height of the toy is 15.5cm. Find the total surface
area and the volume of the toy.
5. A building is in the form of a hemispherical vaulted done and
contains of air. If the internal diameter of the building is
equal to its total height above the floor, find the height of the
building.
6. A circus tent is in the shape of a cylinder surmounted by a cone.
The diameter of the cylindrical part is 24m and its height is 11m.
If the vertex of the tent is 16m above the ground, find the area of
canvas required to make the tent.
7. An iron pillar has some part in the form of a right circular
cylinder and remaining in the form of a right circular cone. The
radius of the base of each of cone and cylinder is 8cm. The
cylindrical part is 240cm high and the conical part is 36cm light.
Find the weight of the pillar if one cu.cm of iron weight 7.8
grams.
8. A circus tent is made of canvas and is in the form of a right
circular cylinder and a right circular cone above it. The diameter
and height of the cylindrical part of the tent are 176m and 5m
respectively. The total height of the tent is 21m. Find the total
cost of the tent if the canvas used costs Rs. 12 per square meter.
9. A circus tent has cylindrical shape surmounted by a conical roof.
The radius of the cylindrical base is 20m. The height of
cylindrical and conical portions are 4.2m and 2.1m respectively.
Find the volume of tent.
10.A petrol tank is a cylinder of base diameter 21cm and 18cm
length fitted with conical ends each of axis length 9cm.
determine the capacity of the tank.

Answers(Surface Area & Volume)
1. 36960cm 2. 8cm 3. 7cm
4. 10m, 880m2 5. 440sqm 6. 1 : 4
7. 3850cm 8. 1.125m 9. 9 : 25
10. 462cu m 12. 37.68cm 13. 110m
14. 3017cm3 15. 237.5% 16. 179.67cm3
17. 1437.33cm 18. 9cm, 12cm 19. 19404cm3
20. Rs. 10101.18 21. 12cm,
,
22. 15cm
23. 193.8cm3 24. 1 : 7 25. 1 : 2
26. 150 27. 5/6cm 28. 2.1cm
29. 2 : 1 30. 2cm 31. 35.84cm
32. 7cm 33. 126 34. 3.5cm
35. 63 cm 36. 179.67 cm 37. 1 cm
38.
39. 8/3 cm 40. 54
41. 14 cm, 117.3cm3
Answers(Combined solid figures)
1. 732.12cu cm 2. 266.11cm 3. 231cm3, 203.94cm2
4. 214.5cm2,243.8 cm3 5. 4m 6. 1320m2
7. 395.37kg 8. Rs. 178200 9. 6160m
10. 8316cm3
ASSIGNMENT(TROGOMETRY)
1. Simplify:
2. Evaluate:
3. Evaluate:

4. Evaluate:
Prove the following indentities: Answers
1. 1 2. -1 3. 11/6 4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.

15.
16.
17.
18.
19.
20.
21.
22.
23.
24.
25.
26.
27.

28.
29.
30.
31.
32.
33.
34.
35. If determine
36. If Show taht
37. If Show
that
38. If Show that
39. If fi nd the value of
40. If find the value of

41. If show that
42. prove
that
43. If prove that
44. If and
prove that
45. If prove that
ASSIGNMENT (Heights & Distances)
1. From the top of tower 60m high, the angles of depression of the
top and bottom of a building whose base is in the same straight
line with the base of the tower are observed to be 300 and
600 respectively. Find the height of the building.
2. An aeroplane flying horizontally at a height of .1.5 km above the
ground is observed at a certain point on earth to subtend and
angle of 600. After 15 seconds, its angle of elevation at the same
point is observed to be 300. Calculate the speed of the aeroplane
in km/hr.
3. A vertical tower is surmounted by a flagstaff of height h metres.
At a point on the ground, the angle of elevation of the bottom
and top of the flag staff are and respectively. Prove that
the height of the tower is
4. If 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 lake

is , prove that the distance of the cloud from the point of
observation is
5. A tower in a city is 750m high and a multi-storeyed hotel at the
city centre is 50m high. The angle of elevation of the top of the
tower at the top of the hotel is 300. A building, h metres high, is
situated on the straight road connecting the tower with the city
centre at a distance the city centre at a distance of a 1.0 km from
the tower. Find the value of h if the top of the hotel, the top of the
building and the top of the tower are in a straight line. Also find
the distance of the tower from the city centre.
6. In the adjoining figure, ABCD is a trapezium in which AB || CD.
Line segments RS and LM are drawn parallel to AB such that AJ =
JK = KP. If AB = O.5m and AP = BQ = 1.8m, find the length of AP,
BD, RS and LM.
7. The angle of elevationof a jet plane from a point A on the ground
is 600. After a flight of 15 seconds, the angle of elevation changes
to 300. If the jet plane is flying at a constant height of
find the speed of the jet plane.
8. Determine the height of a mountain if the elevation if the
elevation of its top at an unknown distance from the base is
450 and at a distance 10km further off from the mountain, along
the same line, the angle of elevation is 300 (USE tan300 =
0.5774).
9. The angle of elevation of the top of a rock from the top and foot
of a 100m high tower are respectively 300 and 450. Find the
height of the rock.
10. The angle of elevation of the top Q of a vertical tower PQ
from a point X on the ground is 600. At a point Y, 40m vertically
above X, the angle of elevation is 450. Find the height of the
tower PQ and the distance XQ.
11. An aeroplane, when 3000m high, pass vertically above
another aeroplane at an instance when the angles of elevation of

the two aeroplanes from the same point on the ground are
600 and 450 respectively. Find the vertical distance between the
two aeroplans.
12. A pole 5m high is fixed on the top of a tower. The angle of
elevation of the top of the pole observed from a point A on the
ground is 600 and the angle of depression of the point ‘A’ from
the top of the tower is 450. Find the height of the tower.
13. A tower is 50m high. Its shadow is x m shorter when the
sun’s altitude is 450 than when it is 300. Find x.
14. From the top of a tower, the angle of depression of two
objects on the same side of the tower are found to
be If the distance between the object is ‘p’
metres, show that the height ‘h’ of the tower is given
by .
Also, ditermine the height of the tower
if
15. A 7m long flagstaff is fixed on the top of a tower on the
horizontal plane. From a point on the ground, the angles of
elevation of the top and bottom of the flagstaff are 600 and
450 respectively. Find the height of the tower correct to one
place of decimal.
16. A round balloon of radius r subtends an angle at the eye
of the observer while the angle of elevation of its centre is ,
Prove that the height of the centre of the balloon
is
17. The angle of elevation of the top of a tower as observed
from a point in a horizontal line through the foot of the tower is
300. When the observer moves towards the tower a distance of
100m, he finds the angle of elevation of the top to be 600. Find
the height of the tower and the distance ofthe first positionfrom
the tower.
18. The angle of elevation of the top of a tower from a point A
on the ground is 300. On moving a distance of 20m of towards
the foot of the tower to a point B, the angle of elevationincreases
to 600. Find the height of the tower and distance of the tower
from the point A.

19. A vertical tower stands on a horizontal plane and is
surmounted by a flagstaff of height 30m. At a point on the plane
the angle of elevation of the bottom of the flagstaff is 450 and of
the top of the flagstaff is 600. Determine the height of the tower
and the horizontal distance.
20. From the top of a tower 50m high, the angle of depression
of the top and bottom of a pole are observed to be 450 and
600respectively. Find the height of the pole if the pole and tower
stand on the same line.
21. The angles of elevation of the top of a tower, as seen from
two points A and B situated in the same line and at distances p
and q respectively from the foot of the tower, are
complementary. Prove that the height of the tower is .
22. From a window (60 metres high above the ground) of a
house in a street the angles of elevation and depression of the
top and the foot of another house on opposite side of street are
600 and 450 respectively. Show that the height of the opposite
house is metres.
23. If 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 the height of the cloud is
24. From an aeroplane vertically above a straight horizontal
plane, the angles of depression of two consecutive kilometer
stones on the opposite sides of the aeroplane are found to be
and . Show that the height of the aeroplane is .
25. A man standing on the deck of a ship, which is 10m above
water level, observes the angle of elevation of the top of a hill as
600 and angle of depression of the base of the hill is 300. Find the
distance of the hill from the ship and height of the hill.
26. As observed from the top of a light-house, 100m high
above sea level, the angle of depression of a ship, sailing directly
towards it, changes from 300 to 600. Determine the distance
traveled by the ship during the period of
observation .
27. The angle of elevation of a cloud from a point 200m above
the lake is 300 and the angle of depressionofthe reflectionof the
cloud in the lake is 600. Find the height of the cloud.

28. The angle of depression of the top and bottom of a tower,
as seen from the top of a 100m high cliff, are 300 and
600 respectively. Find the height of the tower.
Answers
1. 40m 2. 415.68 km/h
5. 172.6m, 1212.4m 6. BD = AC = 2.0785m, Rs.
1.1928m, LM = 1.8856m.
7. 720km/h 8. 13.66m
9. 236.5m 10. 94.64m, 109.3m
11. 1268m 12. 6.83m
13. 3660cm 14. 43.25m
15. 9.589m 17. 86.6m, 150m
18.17.3m, 30m 19. 40.98m, 40.98m
20. 21.13m 25. 17.3m, 40m
26. 115.46m 27. 400m,
28. 66.67m
ASSIGNMENT(AREA RELATED TO CIRCLE)
1. Find the areaof a ∆OAB with < AOB = 120° & OA = OB = 18 cm.
Fig. 1
2. Find the areaof sector ofangle 120° and radius 18 cm.

Fig. 2
3. Find the areaof the segment corresponding to sector AOB of
angle 120° and radius 18 cm.
Fig. 3
4. A chord of a circle ofradius 10 cm subtends a right angle at the
centre. Find the following.
(i) Area of minor sector (ii) Area of minor segment
(iii) Area of major segment (iv) Area of minor segment
(Use π=3.14)
5. It is proposedto add two circular ends, to a square lawn whose
side measures 58cm, the centre ofeach circle being the point of
intersectionofthe diagonals of the square. Find the areaof the
whole lawn.
6. In a circle of radius 21 cm, an arc subtends an angle of 60º at the
centre. Find:
(i) length of the arc (ii)areaofsector formed by the arc

(iii)areaofsegment formedby the corresponding chordof the arc.
7. The length of an arc subtending an angle of72 at the center is 44
cm. Find the areaof the circle.
8. A park is in the form of a rectangle 120 m by 100 m. At the
centre of the park, there is a circular lawn. The areaof the park
excluding the lawn is 11384 sq. m. Find the radius of the circular
lawn.
9. A chord 10 cm long is drawn in a circle of radius 50 cm. Find the
area ofthe minor and the major segment made by PQ.
10. An athletic track,14 m wide, consists oftwo straight
sections 120 m long joining semicircular ends whose inner
radius is 35 m. Calculate the area of the track.
SIMILAR TRIANGLES
1. ABC is a right-angled triangle,right-angledat A. A circle is inscribed
in it. The
lengths of the two sides containing the right angle are 6cm and 8 cm.
Find the
radius of the in circle.
(Ans: r=2)
2. ABC is a triangle. PQis the line segment intersecting AB in P and AC
in Q such
that PQparallel to BC and divides triangle ABC into two parts equal in
area. Find
BP: AB.
3. Ina right triangle ABC, right angled at C, P and Q are points of the
sides CA and

CB respectively,whichdivide these sides in the ratio 2: 1.
Prove that 9AQ2= 9AC2 +4BC2
9BP2= 9BC2 + 4AC2
9 (AQ2+BP2) = 13AB2
4.In an equilateral triangle ABC, the side BC is trisectedat D.
Prove that 9AD2 = 7AB2
5. Prove that three times the sum of the squares ofthe sides ofa
triangle is equal to four times the sum of the squares of the medians of
the triangle.[ Hint AB2
+ AC2
= 2{AD2
+
𝑩𝑪 𝟐
𝟒
}]
6. ABC is an isosceles triangle is whichAB=AC=10cm. BC=12. PQRS is a
rectangle inside the isosceles triangle.GivenPQ=SR=y cm, PS=QR=2x.
Prove that x = 6 -
𝟑𝒚
𝟒
. [hint AL = 8 cm , ΔBPQ∼ΔBAL]
7. If ABC is an obtuse angled triangle, obtuse angled at B and if AD CB
Prove that AC2 =AB2 + BC2 +2BCxBD
8. If ABC is an acute angled triangle,acyte angledat B and if AD CB
Prove that AC2 =AB2 + BC2 -2BCxBD.
9. Prove that in any triangle the sum of the squares of any two sides is
equal to twice the square of half of the third side together with twice
the square of the median,which bisects the third side.
[hintTo prove AB2 + AC2 = 2AD2 + 2 (1/2BC)2, Draw AE ┴BC ]
10.If A be the areaof a right triangle and b one of the sides
containing the right angle, prove that the length of the altitude on
the hypotenuse is
𝟐𝑨𝒃
√ 𝒃 𝟒+𝟒𝑨 𝟐
.
11. ABC is a right triangle right-angled at C and AC= √3BC. Prove
that <ABC=60o. [HINTTanB =
𝑨𝑪
𝑩𝑪
]
12. ABCD is a rectangle.ADE and ABF are two triangles suchthat <E=
<F as shown in the figure. Prove that AD x AF=AE x AB.

[Consider ADE and ABF]
13. ABCD is a parallelogram inthe given figure, AB is divided at P and
CD and Q so that AP:PB=3:2 and CQ:QD=4:1. If PQ meets AC at R, prove
that AR=(3/7)AC. [HINT ΔAPR ∼ΔCQR (AA)]
14. Prove that the area of a rhombus on the hypotenuse of a right-
angled triangle, with one of the angles as 60o, is equal to the sum of
the areas of rhombuses with one of their angles as 60o drawn on the
other two sides. [Hint: Area of Rhombus ofside a & one angle of 60o =
√3/2 a2 ]
15. An aeroplane leaves an airport and flies due north at a speedof
1000 km/h. At the same time, another plane leaves the same airport
and flies due west at a speedof 1200 km/h.Howfar apart will be the
two planes after 1½ hours.
[Ans:300√ (61)Km ON = 1500km (dist = s x t) OW = 1800 km]
N
W O
Arithmetic Progression
Q1.Find A.P. whose first term is a and the commondifference d are
given below:
(a) a = 8, d = 4 (b) a = –90, d = 20 (c) a = 2, d = –1/2(d) a = p, d = –3q
Q2.The nth term of a sequence is 2n + 1. Is the sequence , so formedis
A. P.? If so,find its 12th term.
Q3.Find the value of k for which the following terms are in A.P.
(a) 2k + 1, k2 + k + 1, 3k2 – 3k + 3 (b) k + 2, 4k – 6, 3k – 2 (c) 8k + 4, 6k –
2, 2k + 7
Q4.Which term of A.P. 3, 8, 13, 18,….. is 248?
Q5. How many terms are in the A.P. 6, 3, 0, –3,………..,– 36?
Q6. For what value of n, the nth term of the following A.P.s are equal?
23, 25, 27, 29,……… and –17, –10, –3, 4,………..
Q7. The first term of an A.P. is –3and tenth term is 24. Find the 20th
term.
Q8. The seventhterm of an A.P. is 32 and its 13th term is 62. Find the
A.P.
Q9.The 6th term of an A.P. is 5 times the first term and the eleventh
term exceeds twice the 5th term by 3.
Find the 8th term.

Q10.The 7th term of an A.P. is –4 and its 13th term is –16 .Find the A.P.
Q11.The fifth term of an A.P. is thrice the second term and twelfth
term exceeds twice the 6th term by 1.Find the 16th term.
Q12. Find the 15th term from the end of the A. P. 3, 5, 7, 9,……….,201.
Q13.How many numbers of two digits are divisible by 6 ?
Q14.Find the number of integers between50 and 500 which are
divisible by 7.
Q15. Iffive times the 5th term of an A.P. is equal to 8 times the 8th term,
showthat its 13th term is zero.
Q16.The sum of 4th and 8th terms ofan A.P. is 24 and the sum of the 6th
and 10th terms is 34.Find the first term and the commondifference of
the A.P.
Q17.Which term ofthe A.P. 3, 11, 19,……. is 195 ?
Q18.Find sum ofthe following series:
(a) 72 + 70 + 68 +……. + 40 (b) 5 + 5.5 + 6 +…….to 20 terms
(c) –11 –5+ 1 + … to 10 terms (d) –25 –21–17 ……..to 24 terms
Q19.Find the sum ofthe: (a) first 50 even numbers (b) first 50 odd
numbers
Q20. Find the sum of all the natural numbers:
(a) between 100 and 1000 which are multiple of 5 (b) between50 and
500 which are divisible by 7
(c)between50 and 500 which are divisible by 3 and 5.
Q21.How many terms of the sequence 18, 16, 14,…. should be taken so
that their sum is zero?
Q22.How many terms of an A.P. 1, 4, 7, …. are needed to give the sum
2380?
Q23.If Sn = 3n2 + n, find the A.P.
Q24.Find the nth term of an A.P., sum of whose n terms is 2n2 + 3n.
Q25.The sum of first 9 terms of an A.P. is 171 and that of first 24 terms
is 996. Find the first term and the
commondifference.
Q26.Find the sum offirst 25 terms ofan A.P. whose nth term is given
by tn = 2 – 3n.
Q27.How many terms of A.P. –6, –11/2, –5,….are needed to give the
sum –25 ? Explaindouble answer.
Q28.In an A.P., if the 5th and 12th terms are 30 and 65 respectively,
what is the sum of first 20 terms?
Q29.A man saves Rs 32,000 during first year,Rs 36,000 in the next
year and Rs 40,000 in the third year.
If he continues his savings in this sequence, in how many years will he
saves Rs 2,00,000 ?
Q30.Find the middle term of A.P. 1, 4, 7, ………….,97.

Q31.The sum of three numbers in A.P. is 36 and the sum of their
squares is 450. Find the numbers.
Q32.Find the first negative term of the A.P. 2000, 1990, 1980,
1970,…….
Q33. 8. A club consists ofmembers whose ages are in A.P. the common
difference being 4 months.If the youngest member of the club is 8
year oldand the sum of ages of all the members is 168 years,find the
total number of members inthe club.
Answers
Ans1.(a) 8, 12, 16, 20,… (b) –90, -70, -50, -30,….. (c) 2, 3/2, 1, 1/2, 0,….
(d) p, p – 3q, p – 6q, p – 9q,… Ans2.yes, 25 Ans3.(a) 2 (b) 3 (c) 7.5 Ans4.
50 Ans5. 15 Ans6. 9 Ans7. 54 Ans8. 2, 7, 12, 17,…….Ans9.33 Ans10. 8,
6, 4, 2, ….Ans11. 31 Ans12. 173 Ans13. 15 Ans14. 64 Ans16. –1/2, 5/2
Ans17. 25 Ans18.(a) 952 (b) 195 (c) 160 (d) 504 Ans19.(a) 2550 (b)
2500 Ans20.(a) 98450 (b) 17696 (c) 8325 Ans21. 19 Ans22. 40 Ans23.
4, 10, 16,…… Ans24. 4n + 1 Ans25. 7, 3 Ans26. –925Ans27. 20 or 5
Ans28. 1150 Ans29. 5 years Ans30. 49 Ans31. 9, 12, 15 Ans32. –10
Ans33. 16

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