math

SAMPLE Paper 3

Mox Marks

Tume Allowed

3 hrs.]

Generol Instructions: As given in Sample Paper 1

Section A

Determine the value of k for which the indicated value of x is a solution:

kx - 4 0; x =
Find the sum of the following AP: 2, 7, 12, .

upto 10 terms.

Find the ratio in which the joining of points (-3, 10) and (6, 8) is divided by point (- 1, 6).
AFind the area of a quadrant of a circle whose circumference is 22 cm.

Section B
S. Find discriminant of the following quadratic equation and examine the nature of real roots (f they eie
7y 7y+4y+ 5 0.
.Find the sum of the first 17 terms of the AP whose nth term is given by t, = 7-4n.
7 An figure, O is the centre of the circle, radius of the circle is 3 cm and PA is a tangent drawn to the arte

from point P. If OP = x cm and AP 6 cm, then find the value of x.

6 Cm

3 cm

X Cm

2000 tickets of a lottery were sold and there are 8 prizes on these tickets. Your friend has purchased one
lottery ticket. What is the probability that your friend wins a prize?
The radii of two circles are 19 cm and 9 cm respectively. Find the radius of the circle which has circumference

equal to the sum of the circumferences of the two circles.
A0,The diameter of a solid metallic sphere is 16 cm. The sphere is melted and recast into 8 equal solid sphenca

balls. Determine the radius of the balls.

Section C
The sum of an integer and its reciprocal is

find the integer.
12

12 Find the 12th term from the end in the AP 56, 63, 70, .329.
Solve for x: x-
= 3, x 0.
A. Construct a triangle of sides 4 cm, 5 cm and 6 cm and then a triangle similar to it, whose sides are o
corresponding sides of the first triangle.
A box contains 5 red marbles, 8 white marbles and 4 green marbles. One marble is taken out of the box
random. Find the probability that the marble taken out will be () red () white (iü) not green?

A piggy bank contains hundred 50p coins, fifty 1 coins, twenty

2 coins and ten 5 coins. If it is equa
likely that one of the coins will fall out when the piggy bank is turned upside down, find the probability
the coin () will be a 50p coin (i) will not be a ? 5 coin.
24 Tagether wis EAD Mathematics-X

ordi-
hreTIeS o a paralielogram taken in order are (-1 0, (3. 1) and (2, 2) respectively. Find tne o

TeS

the fourth vertex

mg distance fomula show that the points A B and C are collinear: A(2, 3), B(3. 4), C(6, 7)

are of the segment of a circle of radius 12 cm whose corresponding sector has a centrai angie

onng piass is in the shape of a frustum of a cone of height 14 cm. The radii of its two circular ends are

c- an 2 m

ind the capacity of the glass. (Take 7 =

Section D

a aves 60 km at a uniform speed. If the speed had been 5 km/h more, it would have taken 1 hour

less ioe seme joumey. Find the speed of the train.

sitve integers o O g are in AP such that a a + a, = 33 and a, x 0z x 0^ = 1155. Find the
ges

and o

Thres

Avlape Panchayat constructed a ircular tank to serve as a bird bath. A fencing was made in the shape of
TEa ABCD to circumscribe the cirde. Prove that AB + CD = AD + BC.

he alue does the village Panchayat depict through this action?
24h ure PA and P are tangents to circle drawn from an extenal point P. CD is a third tangent touching the
airde

Q F PB = 7 cm and CQ = 2.5 cm, find the length of CP

a

The engths of tangents drawn from an external point (point outside the circle) to a circle are equal.
PRovE t
Acus artist is limbing a 20 m long rope, which is tightly stretched and tied from the top of a vertical pole
tothe ground. Find the height of the pole if the angle made by the rope with the ground level is 30.

Two men standing on either side of a cliff of 80 m high, observes the angles of elevation of the top of the
dif to be 30 and 60 respectively. Find the distance between the two men.
2Find the area of the quadrilateral formed by joining the points: A(- 4, -2), B(- 3,-5), C(3,

2) and

DI2, 3)

29n figure, OACB is quadrant of a circle with centre 0 and radius 8 cm. If OD = 5 cm, find
the area of the quadrant OACB. í
the area of the shaded region. (Take n =

Value Based Question

Sample Papers 25

SNN onsisting of a right circular cone of height 120 cm and radius 60 cm standing on a hem

bott
dNs

m is placed upright in a right circular cylinder full of water uch that
and

it touches
its height
the
is 180 G

Find the wlume of water ieft in the cylinder, if the radius of the cylinder is 60 cm and its height ie 1tom

Tate

length
n e

capsule is in the shape of a cylinder with two hemispheres stuck to each of its ends. The

7

the entie capsule is 14 mm and diameter of the capsule is 5 mm. Find ts surtace area. (Take T 2

5 mm

14 mm
Solutions

x =

-
4 is a solution
kr 4 = 0
4)

k * (-4) - 4 = 0
= -4k = -12

16 4k -4 = 0

k 3

2ere

a 2 d 7

-
2 =
5, n = 10

S,- [20

(n -11d

Sy 2x 2 (10 15

= 5445]

= 5x 49 245

L pont Ci-L

divides the joining of A(- 3, 10) and B(6, -8) in the ratio k 1.

Icoorcinate of C = 6*1x-3

-i = 6k -3

Crmeence of e orde = 22 cm
22 cm

Ae o e qudr=

S. Given quadratic equation
7y4y + 5 0
a 7, b 4 and c 5

D b'- 4ac
Here

-(4)-

4 x 7 x5

-16 140
= -
124
D--124
Dis negative

t quation has no real roots.

t, 7- 4n
7-4x 1 3
t 7-4 x 2 = -1

d

= -1 - 3 = 4

So, sum of 17 terms,

S

12 x 3 (17 -1)x -4]

16-64) = (-58) -=0

17 x -29 = -493
7. In right aOAP

OP? = OA + AP2
(3)+(6
= 9 + 36

x 45

X = 3/5 cm

Number of lottery tickets

20000

Total number of prizes = 8

8

Probability to win a prize
= 2000
2000

1
250
9. Circurmference of 1st circle
= 2n x 19 387 cm

Circumference of 2nd circle

2n x 9 18n cm
Let the radius of required circle
= x cm
A.T.Q.
2TX= 38n + 18r

2nx 56n

56T 28 cm
2n
10.

16

Radius of metallic sphere
8 cm

Volume of the spherear'n (8)' cm'

Sample Papers

Let the radius of shperical balls = x cm

Volume of spherical balls = nx) cm
3

Volume of 8 spherical balls

8 xrx cm

-

cm

A.T.Q

3
-8x

= 64

x

= 4 cm

11. Let the integer = x

A.T.Q
12

12x+12 = 145x
12x

145x +12 = 0
12x 144x x + 12 =0

12x(x

12) -1(x -12) = 0

x 12) (12x

1) =0

X = 12 or x =

Rejecting x =

x is an integer

x = 12

12. 12th term from the end of AP 56, 63, 70,.. 322, 329 is 12th term of the AP_329, 322,. 56th

a 329, d = -7

Here,
a2 329

11x -7 = 329 77= 252

x- -3

13
13

-1 3x
-3x 1 = 0

D b-4ac = (-3) -4 x 1x-1 = 13
-bD --3)13

3/133
X
2a

2x1

2

Tagether wth EAD Mathematics-x

28

14

4 cm-

CAB' is the required triangle.
15. Total number of marbles = 5 8+ 4 17
( Number of red marbles = 5

Required probability =

(Number of white marbles = 8

Probability of getting white marbles
17
() Number of green marbles = 4

Probability of getting green marbles =

Probability of not getting a green marble = 1 -

16. Total number of coins

10050

20+ 10 = 1800
( Number of 50p coins

100

Probability of getting a S0p coin =
80
() No. of 2 coins = 20

20

Required probability80

18
180
17. Let the coordinates of fourth vertex D (« y)
Dx. y)
Mid point of AC:

Mid point of BD

)

A(-1, 0)

8(3. 1)
Diagonals of a parallelogram bisect each other
Coordinates of mid point of BD coordinates of mid point of AC
(3

)-
3

and

1
2
3+X 1 and 1 y = 2

Sample Papers

29

m

* = -2 and y

= l

Coordinates of fourth vertex are (-2, 1).

18. AB 3-2)+(4-3)

=
2 unit

BC 6-3)+(7-4)

=
/18 3V2 unit

AC =V(6-2)+(7-3)

= 32 = 4/2 unit

AB BC= 2 + 3V2
42 AC

Now

A, B and C are collinear.

ZBAC

60° and AB = AC [radius of a semi circle

ABAC is an equilaterals triangle

19. Consider the figure,

Area of AABC

x (12) = 36v3 cm

xn 24r cm
360
60

Area of sector BAC =

Area of segment Area of sector BAC Area of triangle BACc

(247 36/3) cm2

= 12(2m 3/3) cm

20. Radii of the frustum of a cone: R = 4 cm, R

= 2 cm and h = 14 cmn

Volume of the frustum =(Rí+R$+R,R,)
3 xx

(4+ 2 4 x 2) cm
3

16 + 4 8) = 410.67 cm*

21. Let speed of the train = x km/hr

Distance 360 km

Time taken = hr
X
If speed of the train becomes (x + 5) km/hr
Distance= 360 km

Time taken = 360 hr
X+5
360360 =1
X+5
A.TQ
X

360(x+ 5)- 360x
= 1

x(x +5)

1800 = x* + 5x

Sx 1800 0
x+ 45)
-40) = 0
X=45 or x = 40

. Speed of train = 40 km/hr
22. Let a1 a -d, 02 = a and a3 = a +d
Rejecting x*

a1+a2+ ag 33
a-d+ a +a +d= 33
30 Tgether uteh EAD Mathematics-x

3a 33

a = 11

Also,
(a d)x (o)x (a d) = 1155
(11 d)x (11) x (11 d) = 1155
1155
using ())
(11 d) (11 d)
11
121 - d= 105
16

dt4
When a = 11 and d = 4
a11 - 4 = 7, a, = a = 11 and a, 11 4 15
When a = 11 and d = 4
=11 + 4 = 15, o, = 11, o, = 11 - 4 7

23. Here,
Similarly,

AP AS (tangents from an external point are equal)
BP BQ
CR CQ

.iv)

DR = DS
Adding, () + () + (iüð) + (iv), we get

AP+ BP+ CR DR = AS BQCQ + DS
(AP+BP) + (CR +DR) = (AS + DS) (BQ + CO)

AB CD = AD+ BC

Hence proved.
Value reflected Care towards nature, love for creatures.
PB PA (tangents from an external point are equal)

24. Here,

PB 7 cm

PA 7 cm
PA CP+ AC

AC CQ (tangents from an external point are equal)
PA CP CQ
7 CP 2.5

Now

Also,

CP 7 -2.5 = 4.5 cm

25. Given: A circle C(o, ). P is a point outside the circle and PA and PB are tangents to a circle.

To Prove: PA = PB

Construction: Draw OA, OB and OP
Proof: Consider triangle OAP and OBP.

O
2OAP = 2OBP = 90°

[Radius is perpendicular to the tangent at the point of contact]

OA 0B (radii)

OP is common

from (0. (), (id)

AOAP AOBP

AP BP (cpct) [Hence proved).

20 20 m-

26. Let AB be pole and AC be rope
AC = 20 m and ZACB = 30°

30 CC

Sample Papers 31

In rnght ABC

AB sin 30
AC

AB

AB 10 m
20

Let AB be the clift and two men are standing at C and D.

Now

AB 80 m

ACB = 30, ZADB

= 60°

In nght AABC

AB tan 30
BC

80 m
80 BC = 80/3 m
3
30°

BC
In nght ABD
= V3

A5 = tan 60°
BD

80
m

BD = /3

NoW

CD BC + BD
80/380

= 320m

320/3 m

28 Consider the figure

Area of AABC = M2

- y) + *zV

- yi) + X0

- y»
D(2. 3)
H-S- -2)+ (3)F2- (-2)]+ 3-2 - (-5)1

4 x -3 3
x 0 3

x 31
A(4,-2)

= 112 0 9 sq. units

Area AACD

= A-2

-
3) 3 13 -2)]

21-2 -2)11
120150=

sq. units

Area of quadrilateral ABCD

= area of AABC + area of AACD
3 56 sq. units

29

Area of quadrant OACB =
8x8 -2 cm
82 7petken wd

EAD Mathematics- X

Area Aa

Ara shated reiNn

A

140

Height of the coun al part 1

cm

Volume of conial rt

- 14400NN Cm

Raius ot hemisphee o cn

Vwdume of hemispheie

0)- 1440NNN m

wlume of solid

1440DA 1440NN

88UNNM cm
Volume of cylinier-

180
-0480004 cm

Volume of water left 648AXM

8ANNDA

- 3oXXX)

C

Radius of hemispere -mm

Cutved surtace area of hemisphere r-

Suttace area of twe hemisphere

a

23N mn

length of cylnderical part (14

) 9mm

Radius of cylindrv al part

Cuved surface arva of cylndical part 2h 2n

man

45 m

total surface area of the capsule - 25a

45n 70s mm

0
20 mn

Nampe Papwrs 83