CONCEPT BOOSTER : MATHS
1.
Given tan
 
sin

, p, tan

+ sin

= q, show that q 2

p 2
 4 qp  0
.
2.
Given cos

+ sin

=
2 cos 
, show that cot

= 1 +
2
.
3.
Given that cos

+ sin

= k and sec

+ cosec

= 1, show that k 2   2 k   .
4.
Given that tan

+ sec

= k, show that
5.
Given that cos cos


 k
and cos sin


  k k
2
2


1
1

. Show that cos
( k 2 ec


2

 2
.
)
 sec 2 
.
6.
Given that cosec
 
sin

= k and sec
 
cos

= l , show that k 2 l 2 (k 2 + l 2 +3) = 1.
7.
Given that x = r sin A cos C, y = r sin A sin C and z = r cos A. Find r 2 in terms of x, y and z.
Ans. y 2 + x 2 + z 2 .
8.
Eliminate ‘  ’ from the following : x sin 
= y cos

, x sin 3

+ y cos 3

= sin

cos

.
9.
Eliminate

from the following : cosec
 
sin

= k and sec
 
cos

= 1.
Ans. k
2
3  3
2 
 k
2
3  
2
3


 1
10.
Given that a sin

+ b cos
 =± a 2  b 2  c 2
. Show that c = ± (a cos  b sin

).
11.
Given sin

(1 + sin

) = 1, prove that cos 2  ( 1  cos 2  )  1
.
12.
Given that sin 2   sin   1
, find the value of cos 12   3 cos 10   3 cos 8   cos 6   2 cos 4  
13.
Given that sec

+ tan

= p, find the value of sec

.
2 cos sec
2




1
2
1 p
. Ans.2

1 p
.
14.
Given that tan 2
Ans. (2


= 1

k 2 , find the value of tan 3

cosec

+ sec

.
15.
Given that sin

+ sin 2

+ sin 3

= 1, show that cos 4
 
4 cos 2

+ 8 = 4 sec 2

.
Ans. 4sec
k
2
2 ) 3/2 .

= 8

4 cos 2

+ cos 4

16.
Find the value of sin
42  43 '  cos 68  27 '
. Ans. 1.0457.
17.
Find the value of tan 46°22  
cos 62°50
 
sin 72°39

. Ans.
1 .
6378
.
18.
Given sin

= 0.8679 and cos

= 0.7589, find the value of sin (90°
  ) + cos(90°  
). Ans.
1.1480.
19.
In the right angled

ABC,
 A = 38°32 
and CB = 20 cm. Find AB.
Ans. 25.12 cm.
20.
The angle of elevation of the top of a multi-storeyed flat, from a point A on the ground, was
42°45 
. After moving 30 m towards the building to a point B, the angle of elevations was
44°42° 
. Find the height of the building, nearest to metre. Ans. 390
21.
From the top of a building AB, 700 metres high, the angles of depression of the top and bottom of a vertical lamp-post CD were observed to be 31° and 61° respectively. Find the (i) horizontal distance between the building and the lamp-post. (ii) height of the lamp-post CD. Ans. 466.8 m.
22.
A man walks 400 m up a slope of 14° and 200 m up a slope of 18°. Find his height above the starting point, correct to the nearest metre. Ans. 159 m.

23.
A person sitting on top of a tree 20 m high, observers the angle of elevation of the top of a tower as 58°36 
and the depression of the bottom of the tower as 31°24

. Find (i) the distance between the tree and the tower (ii) height of the tower. Ans. 32.76m, 73.66 m(cor. 2 dec. pl)
24.
The angle of elevation of a plane from a point p on the ground was 61°. After 30 seconds, the angle of elevation was 24°. If the plane was flying at a constant height of 1000 m, find the speed of the plane. Ans. 56.39 m/sec.
25.
The angle of elevation of a cloud from the top of a light house was 31° and the angle of depression of its reflection in the sea was 45°. If the height of the light house was 100 m and the distance of cloud along the line of sight was 400 m, find the height of the cloud above the water level. Ans. 274.44m
26.
A rectangular paving stone ABCD, 3m by 1 m rests against a vertical wall as shown in the figure. Find the height of A above the ground, if the inclination of the shorter side is ‘  ’. Ans.
(sin

+ 3 cos

).
CONCEPT BOOSTER : MATHS
1.
The sides of a triangle are 3 cm, 4 cm and 6 cm. A second triangle is similar to the first and also has only one of its sides same as one of the sides of the first triangle. All the three sides of the second triangle are integers. Find the area of the second triangle. Ans.
455
cm 2
2.
AB is a common chord of two circles intersecting at A, B. TAC is a tangent at A to circle ABE where DAE is a line through A, terminated by the circles.
Show that
BD
BE is a constant.
3.
In-circle of

ABC touches its sides at D, E, F as shown in the figure. If AB = 8cm, BC = 7cm and
CA = 6 cm, find area area
 DEF
 ABC
. Ans.
13
64
4.
The difference of the radius of two concentric circles is 6 cm. If a chord of length 18 cm of the larger circle is divided into three equal parts by the smaller circle, find the radius of the two circles. Ans.
3
3
4 cm , 1
1
2
5.
In a triangle with sides 5 cm, 12 cm and 13 cm, a semi-circle is drawn inside, tangential to the longest side. Find the area of the semi-circle. Ans.
72 
25 cm 2
6.
A semicircle is drawn on AB = 4 cm. AC = BC. Semi-circles are drawn on AC, BC. The area enclosed by the three semi-circles area shaded.
Ans. 4cm 2
7.
Two non-intersecting (or touching) circles have their centres d cm. apart. The radii are R, r. Find the length of (i) direction common tangent (ii) transverse common tangent, in term of R, r and d.
Ans. (i) d 2  ( R  r ) 2
, (ii) d 2  ( R  r ) 2
8.
If s is the semi-perimeter and r is the in-radius of a triangle, then show that area of triangle = rs.
9.
ABC is a triangle with AC = 5 cm, CB = 6cm and AB = 7cm. I is the in-centre and MN is a tangent parallel to BC. Find the length of MN. Ans. 2cm
10.
A square is inscribed in the in-circle of an equilateral triangle of side 12 cm. Calculate the area of the square. Ans. 24 cm 2
11.
Find the perimeter of the smallest equilateral triangle in which three circles of radii 2 cm, 3 cm and 4 cm can be inscribed, circles touching, each other.
Ans.
21 ( 3  1 ) cm
12.
The in-circle of

ABC touches the sides at D, E, F as shown in the figure. The in-radius = 4cm.
Lengths of AE, CD and BF are three consecutive integers p cm, (p + 1 cm) and (p + 2)cm

respectively. Find the perimeter of

ABC.
Ans. 42cm
13.
Inscribed circle touches the sides of

ABC at D, E, F, AD intersects the in-circle at K and AK =
KD. The lines KB, KC intersect the in-circle at L, M respectively. Show that FLME is an isosceles trapezium.
14.
Given that

,

are the root of x 2 + bx + c = 0, find the value of (
  b )

2 
(
  b )

2
. Ans. b
2 
2 c c
2
.
15.
If the equation x
2  px
 q

0 and x
2  qx
 p

0 have a common root, show that p =
 q

1.
16.
Solve. 4 x
2 
4 x

1

3
 x . Ans. 
4
 x

2
3
.
17.
Find the minimum value 3 x
2 
2 x

1 . Ans.
18.
Find the maximum value of

5 x
2
2
.
3

7 x

4 . For which value of x the expression has this maximum
7 value ? Ans. .
10
19.

,

are the roots of ax 2 + bx + c = 0 (

+ k), (

+ k) are the roots of px 2 + qx + r = 0, then find k in terms of the co-efficients of the two equations. Ans.
1
2

 b
 a q p

 .
20.
Given that 12

4i is a root of x 2 + bx + c = 0, find the value of 2c + 3b.
Ans. 248.
21.
If x
2 
7 x

8

0 and x
2  x

6

0 , then find the interval in which x lies.
Ans. 1 < x < 3
22.
If 2

3i is a root of ax 2 + bx + c = 0, then find the value of 10a

b

c. Ans. 1
23.
If the expression ax 2

6x + 5 has its minimum value at x =
3
2
, find its minimum value. Ans.
1
2
.
CONCEPT BOOSTER : MATHS
Mensuration
1.
The side of an equilateral triangle is 6 cm. Find the area of the square inscribed in the in-circle of the triangle. Ans. 6 cm 2
2.
AD is perpendicular to BC, AD = 8 cm, DC = 6 cm and BD = 15 cm. The

ABC is rotated through
360° about BC. Calculate the volume and total surface of the solid of revolution. Ans. 448 
cm 3 ,
216

cm 2
3.
A spherical metal ball of radius 7 cm lies on a table top. A second metal ball touches the first at a point 10 cm above the table top. Find the volume of the second ball.
67375
3 cm
3
.
4.
The cross-section of a hollow right cylinder is a ring. The area of the inner circle is the same as the area of the ring. If the radius of the inner circle is 3 2 cm, find the volume of matter in 12 cm long hollow cylinder. Find the edge of a cube with same volume. Ans. 6

1/3 cm
5.
ABCD is a face of a cuboid, X, Y are the mid-points of AD, BC respectively. The perimeter of
AXYB is 6 2
Ans. 48 cm 2 cm. Find the total surface of the cube.
6.
A cylinder with radius 4 cm and height 9 cm is full of water. A sphere of radius ‘r’ cm is completely immersed in the water and then taken out of the cylinder. If the volume of the left over water in the cylinder is 108

cm 3 , find r. Ans. 3 cm
7.
An igloo was in the shape of an inverted hemispherical shell. The maximum internal height was 1.5 m. Find the circumference of the outer circle at the base, given that the volume of ice in the igloo =
37 
12 m
3
. Ans. 4
 m

8.
Two right cylindrical logs are placed touching each other, floor and a vertical wall of a shed. The
9.
radius of the cross-section of the larger long is 15 cm. Both the logs are 4 cm long. Find the curved surface of the smaller log. Ans. 2056

cm 2
Given that a 
2
( 3 3 
1
3 3  1 )
 1
, show that 4a 
1
3 3  1 .
10.
Without extracting the roots, find which is greater : 10  7 or 11  8 ?
11.
If
12.
x x
 4
 4


Show that x x
10
5
 3
 3


1
7
1
9  4
, find x.
3
 a  7 b , find the value of a

b.
CONCEPT BOOSTER PHYSICS : MOTION
1.
A body dropped from top of a tower fall through 40 m during the last two seconds of its fall. What is the height of tower (g = 10 m/s 2 )? Ans. 45m
2.
A body covers a distance of 20 m in the 7 th second and 24 m in the 9 th second. How much shall it cover in 15 th sec. Ans. 36 m.
3.
A particle experiences constant acceleration for 6 seconds after starting from rest. If it travels a distance s
1
in the first two seconds and a distance s
2
in the next 2 seconds and a distance s
3
in the last
2 seconds, the calculate the ratio of s
1
: s
2
: s
3
. Can a body be said to be at rest as well as in motion at the same time?
Ans. 1 : 3 : 5
4.
A person is running with a uniform speed of 5m/s to catch a bus at rest. When the person is 12m behind the bus, the bus starts and moves with uniform acceleration of 1m/s 2 . When does the person catch the bus?
Ans. 6s or 4s.
5.
A body starts from rest and moves with a uniform acceleration. Find the ratio of the distance covered in n sec. Ans.
2 n  n 2
1
6.
A car accelerates from rest at a constant rate

for some time after which it decelerates at a constant rate

to come to rest. If the total time elapsed is t, find the maximum velocity acquired by the car.
Ans.





 



.
7.
In a car race, car A takes a time of t sec less than car B at the finish and passes the finishing point with a velocity v m/s more than the car B. Assuming that the cars start from rest and travel with constant acceleration a
A
and a
B
respectively, show that v  t a
A a
B
.
8.
If a body falls freely from a height h on a sandy surface & it buries into sand upto a depth of x, then prove that the retardation produced by sand is given by given by a  g
 
 h x x 
 .
9.
Prove that for a body starting from rest and moving with uniform acceleration, the ratio of distances covered in t
1
sec., t
2
sec, t
3
sec, etc. are in the ratio t 2
1
: t 2
2
: t 2
3
etc.
A body moving with a velocity v is stopped by application of brakes after covering s. If the same body moves with a velocity nV. then prove that it stops after covering a distance n 2 s by the application of same brake force.
1.
Simplify :

 x a x b

 a  b

 x b x c

 b  c

 x c x a
CONCEPT BOOSTER : MATHS

 c  a
. Ans. 1

2.
Simplify :

 a  2 b 4 a 3 b  8




 a 3 b  4 a  2 b 4


 4
. Ans. a 15 b

20 .
3.
Given that x y  y x
, show that


 x y
4.
Given 3 a = 4 b = 12 c , show that c 


 y x
 ab x x y
1
( a  b )

 0
0
.
5.
Find the value of :
( a x  y  a x  z  1 )  1  ( a y  z  a y  x  1 )  1  ( a z  x  a z  y  1 )  1
6.
Given that ( l n ) m
7.
Ans. 1.
If a = x m+n y
= l
, b = x l nm , show that (n) n+ l
, y m , c = x l +m y n m

1

(m) n

1 = m n.
find the value of a m
 n b n
 l c l
 m
. Ans. 1
8.
Given x =
1
2
( a  a  1 )
find a in term of x. Ans. x  x 2  1
9.
Given a x = b y = c z = d u and ab = cde, find the value of
1 x

1 y

1
 z
1 u
. 0
1 1 1
10.
Given a 3
 b 3
 c 3
11.
Given that a x (bc)

1

0 , show that (a + b + c) 3
= a, b y (ca)

1 = b and c z
27abc = 0. Ans. 27abc.
(ab)

1 = c. Show that xyz

xy

yz

zx = 0.
3
12.
Evaluate ( 52

6 43 ) 2 . Ans. 70

43

414 .
13.
Given that (81) x = 3 3

1
( 343 ) y
, find the value of 38x + 5y. Ans.
3
2
.
14.
Given a x = b y = c z and b 2 = ac, show that y x

2 z x
 z

0 .
15.
Given x
 3
2

1
3
2
, find the value of x
3 
3 x . Ans.
5
2
.
CONCEPT BOOSTER : MATHS
1.
A  7  6 and B  6  5 , then
(a) A > B (b) A= B (c) A < B (d) A

b (e) none
2.
A can complete a work in 3 days, B in 4 days and C in 5 days. If they complete the same work together and get Rs. 14,100 as remuneration, then the share of C, in rupees, will be :
(a) 2800 (b) 3000 (c) 3200 (d) 3600 (e) None
3.
The perimeter of a square whose area is equal to that of a circle with perimeter 2
 x, is :
(a) 2
 x (b)  x (c) 4
 x (d) 4
 x (e) none
4.
What is the area of an equilateral triangle inscribed in a circle of unit radius?
(a) 3 3 sq.units
(b)
3 3
2
sq.units
(c)
3 3
4
sq.units
(d)
3 3
16
sq.units
(e) none
5.
A rectangular tank is 80 × 40 cm 3 . Water flows into it through a pipe 40 cm 2 are the opening at the speed of 10 km per hour. The rise in the level of water in the tank in
1
2 hours is :
(a)
3
2 cm (b)
4
3 cm (c)
5
8 cm (d) 6 cm (e) none
6.
The ratio of the areas of the incircle and circmcircle of a square is
(a) 1 : 2 (b) 2 : 1 (c) 1 : 4 (d) 4 : 1 (e) none

7.
If a
1 / 3  b
1 / 3
(a) 3abc
+ c
1 / 3
= 0, then (a + b + c) 3 is equal to :
(b) 9abc (c) 27 abc (d) 0
8.
A  7  6 and B  6  5 , then
(d) A

b
9.
(a) A > B
Solve x pq
 x qr
 x pr
(b) A= B (c) A < B
 p  q  r
(a) 0 (b) 1 (c) pqr (d)
1 pqr
10.
Solve
(a)

2 x
.
6 y
2,

1)
 24 and
11.
The expressions a
 bc
2
2 x
.
3 y
and

(b) (2, 1) (c) (

( a

48 : b )( a
 c )
2, 3)
are :
(d) (3, 2)
(a) always equal
(c) equal when a
 b
 c

1
(b) never equal
(d) equal when a
 b
 c

0
(e) none
(e) none
(e) none
(e) none
(c) A
 
1 , B

3 (d) A

3 , B
 
1
(e) none
12.
When simplified the product ( 1

(a)
1 n
(b)
2 n
1
3
)( 1
(c)

1
4
)( 1

2 ( n

1 ) n
1
5
)....( 1

1 / n )
(d)
becomes : n ( n
2

1 )
(e) None
13.
The fraction
5 x

11
2 x
2  x

6
was obtained by adding the two fractions and B must be :
(a) A

5 x , B
 
11 (b) A
 
11 , B

5 x x
A

2 and
A
2 x

3
. The values of A
(e) None
14.
If
( 2 x

3 ), ( x

1 )
are two factors of
( 2 x
2  x )
2 
4 ( 2 x
2  x )

3
, then the remaining two factors are
(a)
(c)
( 2 x

1 )
( 2 x

1 )
and
and
( x

1 )
( x

1 )
(b) ( 2 x

1 )
(d) ( 2 x

1 )
and ( x

1 )
and ( x

1 ) (e) None
If 0 .
04

0 .
4
 a

0 .
4

0 .
04

(a) 0.016 b , then a b
is :
(b) 0.16 (c) 1 (d) 16 (e) None