TOPPER SAMPLE PAPER 3 Summative Assessment-II MATHEMATICS CLASS X

TOPPER SAMPLE PAPER 3
Summative Assessment-II
MATHEMATICS
CLASS X
M.M: 80
TIME : 3-3
1
Hrs.
2
GENERAL INSTRUCTIONS :
1.
2.
3.
4.
All questions are compulsory.
The question paper consists of 34 questions divided into four
sections, namely
Section A : 10 questions (1 mark each)
Section B : 8 questions (2 marks each)
Section C : 10 questions (3 marks each)
Section D : 6 questions (4 marks each)
There is no overall choice. However, internal choice has been
provided in 1 question of two marks, 3 questions of three marks
and 2 questions of four marks each.
Use of calculators is not allowed.
SECTION A
Q1.
The quadratic equation 2x2- 5x +1=0 has
(a) Two distinct real roots
(b) two equal real roots
(c) no real roots
(d) more than two real roots
Q2.
The 9th term of an AP is 449 and 449th term is 9. The term which
is equal to zero is
(a) 501th
(b) 502th
(c) 458th
(d) 459th
Q3.
Two circles touch each other externally at C and AB is a common
tangent to the circles. Then,  ACB=
(a) 600
(b) 450
(c) 300
(d) 90
Q4. The length of an arc that subtends an angle of 24o at the centre
of a circle with 5 cm radius is
(a)
Q5.
2
3
cm
1
12
(c)

3
cm
(d)
5
3
cm
(b) 20 m
(c) 30 m
(d) 40 m
(b)
1
2
(c)
1
4
(d)
1
6
The value of x, for which the points (x,-1), (2,1) and (4,5) lie on
a line is
(a) 0
Q8.
cm
The probability that a randomly chosen number from one to
twelve is a divisor of twelve is
(a)
Q7.
3
2
An observer 1.5 m tall is 28.5 m away from a tower. The angle
of elevation of the top of the tower from his eyes is 45 0. The
height of the tower is
(a) 10 m
Q6.
(b)
(b) 1
(c) 2
(d) 3
 2 20 
The ratio in which the point R 
divides the join of
,
7 
 7
P(-2,-2) and Q(2,-4) is
(a) 3:4
(b) 4:3
(c) 2:1
(d) 1:2
Q9.
A pendulum swings through an angle of 300 and describes an arc
8.8cm in length. The length of the pendulum is
(a) 8.8 cm
(b) 15.8 cm
(c) 16.8 cm
(d) 17 cm
Q10. Three cubes whose edges measure 3cm, 4cm and 5 cm
respectively are melted to recast a single cube. The surface area
of the new cube is
(a) 216 cm2
(b) 200 cm2
(c) 215 cm2
(d) 220 cm2
SECTION B
Q11. If the centroid of the triangle formed by points P(a, b),Q(b, c)
and R(c, a) is at the origin, then find the value of a + b + c.
OR
Find the area of the quadrilateral ABCD whose vertices are
A(1,1), B(7,-3), C(12,2) and D(7,21) respectively.
Q12. An A.P. consists of 60 terms. If the first and the last term be 7
and 125 respectively, find the 32nd term.
Q13. A box contains 90 discs which are numbered from 1 to 90. If one
disc is drawn at random from the box, find the probability that it
bears
(i) A two digit number (ii) a perfect square number
Q14. Find two consecutive positive integers, sum of whose squares is
25.
Q15. Prove that the tangents at the extremities of any chord make
equal angles with the chord.
Q16. In given figure, find the area of the circle not included in the
rectangle whose sides are 8cm and 6cm respectively.
Q17. The perimeter of a sector of a circle of radius 5.2 cm is 16.4 cm,
find the area of the sector.
Q18. A circle touches all the four sides of a quadrilateral ABCD. Prove
that
AB + CD = BC + DA
SECTION C
Q19. If a  b  c, prove that the points (a,a2), (b,b2) and (c,c2) can
never be collinear.
Q20. A body falls 8 metres in the first second of its motion, 24 metres
in the second, 40 metres in the third second and so on. How
long will it take to fall 2048 metres?
Q21. Draw a pair of tangents to a circle of any convenient radius,
which are inclined to the line joining the centre of the circle and
the point at which they intersect at an angle of 45°. Also write
the steps of construction.
Q22. From a balloon vertically above a straight road, the angles of
depression of two cars at an instant are found to be 450 and 600.
If the cars are 100m apart, find the height of the balloon.
OR
The angles of elevation of the top of a tower from two points at a
distance a and b metres from the base and in the same straight
line with it are complementary. Prove that the height of the
tower is ab metres.
Q23. A bag contains 12 balls out of which x are white.
(i) If one ball is drawn at random, what is the probability that it
will be a white ball?
(ii) If 6 more white balls are put in the bag, then the probability
of drawing a white ball will be double than that in (i). Find x.
OR
Two customers are visiting a particular shop in the same week
(Monday to Saturday). Each is equally likely to visit the shop on
any day of the week. What is the probability that both will visit
the shop on
(i) the same day?
(ii) two different days?
(iii) consecutive days?
Q24. In an equilateral triangle of side 24 cm, a circle is inscribed
touching its sides. Find the area of the remaining portion of the
triangle. (take 3 =1.732)
OR
A round table cover has six equal designs as shown in the figure.
If the radius of the cover is 28 cm, find the cost of making the
design at the rate of Rs. 0.35 per cm2.
Q25. A canal is 300cm wide and 120cm deep. The water in the canal
is flowing with a speed of 20km/h. How much area will it irrigate
in 20 minutes if 8cm of standing water is desired?
Q26. From a solid cylinder whose height is 2.4 cm and diameter
1.4cm, a conical cavity of the same height and same diameter is
hollowed out. Find the total surface area of the remaining solid
to the nearest cm2.(use =3.1416)
Q27. Solve for x, using the quadratic formula:
abx2 _ (a2+b2)x + ab=0
Q28. If D, E and F are the mid points of sides BC, CA and AB
respectively of a ∆ABC, whose vertices are A(-4,1), B(6,7) and
C(2,-9), then prove that,
Ar. ∆DEF =
1
(Ar ∆ABC).
4
SECTION D
Q29. Find the value of the middle most term (s) of the A.P. :
-11, -7, -3, ...., 49
OR
The sum of the first three terms of an AP is 33. If the product of
the first and the third term exceeds the second term by 29, find
the AP.
Q30. The angle of elevation of the top of a tower from a point on the
same level as the foot of the tower is  . On advancing p metres
towards the foot of the tower, the angle of elevation becomes β.
Show that the height h of the tower is given by
p tan  tan 
 tan   tan  
Q31. A passenger train takes 2 hour less for a journey of 300 km, if
its speed is increased by 5km/hr from its usual speed. Find its
usual speed.
OR
In a class test, the sum of the marks obtained by a student P in
Mathematics and Science is 28. Had he got 3 more marks in
Mathematics and 4 marks less in Science, the product of marks
obtained in the two subjects would have been 180. Find the
marks obtained in the two subjects separately.
Q32. Prove that the lengths of tangents drawn from an external point
to a circle are equal.
Q33. In given figure, a circle is inscribed in a quadrilateral ABCD in
which  B=900. If AD=23 cm, AB=29 cm, and DS=5 cm, find
the radius r of the circle.
Q34. A solid consisting of a right circular cone, standing on a
hemisphere, is placed upright, in a right circular cylinder, full of
water, and touches the bottom. Find the volume of water left in
the cylinder, having been given that the radius of the cylinder is
3 cm and its height is 6 cm, the radius of the hemisphere is 2 cm
and the height of the cone is 4 cm.
SOLUTIONS
SECTION A
Ans1. Option (c)
2x2-
5x +1=0
D= b2-4ac
=
 5
2
-4(2)(1)
= 5-8 = -3
Since D < 0, therefore the given quadratic equation has no real
roots.
1 mark
Ans2. Option (c)
a+8d = 449 and,
a+448d = 9
on solving we get, 440d = -440 i.e. d = -1
therefore, a-8 = 449 i.e. a = 457
let its nth term be zero.
an = a+(n-1)d
0 = 457+(n-1)(-1)
457 = n-1 or n = 458
1 mark
Ans3. Option (d)
Lengths of tangents drawn from an external point to a circle are
equal.
PA=PC
(from P)
Therefore,  PAC=  PCA = x (say)
Also, PC=PB
(from P)
 PBC=  PCB=y
In ∆ABC,
 ABC+  ACB+  BAC=180
y + (x + y) + x = 180
2(x + y) = 180
x+ y = 90
 ACB = 900.
1 mark
Ans 4.
Option (a)
The length of an arc that subtends an angle of 24o at the centre
of a circle with a 5 cm radius is
24
2
2
x2.5 
x2.5 
360
30
3
The length of the arc is
Ans5.
Option (c)
2
3
cm.
1 mark
Let AC be the tower of height h metres and ED be the observer
of height 1.5 m at a distance of DC=28.5 m from the tower AC.
In right ∆AED,
Tan45 =
1=
AB
EB
AB
28.5
(as EB = DC = 28.5 m )
AB = 28.5 m
Height of the tower = h = AB + BC = AB+DE
= 28.5 + 1.5 = 30m
Ans6. Option (b)
Total number of outcomes = 12
1 mark
Numbers which are divisors of 12 are 1,2,3,4,6,12.
Number of favourable outcomes = 6
P(divisor of 12) =
6
1

12 2
1 mark
Ans7. Option (b)
Area of a triangle = 0
[x x1 + 2x5 + 4x(-1)] – [2x-1 + 4x1 + xx5] = 0
(x+10-4) - (-2+4+5x) = 0
(x+6)-(5x+2)=0
-4x+4=0
x=1
Ans8. Option (a)
 2 20 
Let R 
divides PQ in the ratio K:1
,
7 
 7
 2K  2 4K  2   2 20 
 K 1 , K 1    7 , 7 

 

2K  2
2
4K  2
20


and
K 1
7
K 1
7
1 mark
14K-14 = -2K-2
16K = 12
K=
12 3

16 4
Thus R divides PQ in the ratio 3:4.
1 mark
Ans9. Option (c)
Length of arc (l) = 2r

360
8.8  360  7 = 30  2  22  r
r=
8.8  6  7
cm = 16.8 cm
22
1 mark
Ans10. Option (a)
Let x be the edge of the new cube. Then,
Volume of the new cube = sum of the volumes of three cubes
x3 = 33 + 43 + 53
x3 = 216 i.e. x= 6cm
Surface area of the new cube = 6x2 = 6  (6)2 cm2 = 216 cm2
1 mark
SECTION B
Ans11.Let P(x1,y1)= P(a, b), Q(x2,y2) = Q(b, c)and R(x3,y3) = R(c, a)
be the vertices of ∆PQR.
We know that the coordinates of centroid of a triangle is given
 x  x2  x3   y1  y2  y3  
by  1
,
  i.e.
3
3
 


 a  b  c b  c  a
,


3
3


1 mark
Also, given that centroid is at origin i.e. its coordinates are (0,0).
ab  c
So, 
  0  a  b  c  0.
3


1 mark
OR
Area of quadrilateral ABCD
= area of triangle ABC + area of
Area of triangle ABC =
triangle ACD
1
[1(-3-2) +7(2-1)+12(1+3)]
2
1
(-5+7+48) = 25 sq. units
1 mark
2
1
Area of triangle ACD =
[1(2-21) + 12(21-1)+7(1-2)]
2
=
1
(-19+240-7) = 107 sq. units
2
Area of quadrilateral ABCD = 25 + 107
1
mark
2
=
1
mark
2
= 132 sq. Units
1
mark
2
Ans12.Here, n = 60, a = 7 and t60 = 125
7+59d = 125
d=2
1
mark
2
Therefore, 32nd term (t32) = a + 31d = 7 + 31 x 2 = 69
1 mark
Ans13.Total number of outcomes = 90
(i) Two digit numbers are 10,11,12,13,.............90
Number of favourable outcomes = 81
81
9

P(two digit number) =
90 10
(ii) Perfect square numbers from 1 to 90 are
1,4,9,16,25,36,49,64,81
9
1

P(perfect square number) =
90 10
1 mark
1 mark
Ans14. let the required numbers be x and x+1
Given x2 + (x+1)2 = 25
1 mark
2x2 +2x-24 =0  x2 + x – 12 = 0
(x+4) (x-3) = 0
X = -4 or x = 3
Reject x = -4
1
mark
2
Therefore the given consecutive positive integers are
X = 3 and x + 1 = 3 + 1 = 4
1
mark
2
Ans15.
Let AB be a chord of a circle with centre O, and let AP and BP be
the tangents at A and B respectively. Suppose the tangents meet
at P. join OP. suppose OP meets A at C.
1
mark
2
We have to prove that  PAC=  PBC
In triangles PCA and PCB, we have
PA = PB
(tangents from an external point are equal)
 APC =  BPC (PA and PB are equally inclined to OP)
And, PC = PC (common)
∆PAC
∆PBC
  PAC =  PBC
Ans16.
( by SAS )
( by cpct )
1 mark
1
mark
2
Diameter of the circle = diagonal BD of the rectangle ABCD
In right triangle BCD,
BD2 = BC2+BD2 = 62+82 = 100
BD2 = 100 i.e.BD = 10 cm
Radius of the circle =
10
= 5cm
2
Area of the circle = r2 = 3.14(5)2 = 78.50 cm2
Area of rectangle ABCD = 6  8 = 48 cm2
Area of the circle not included in the rectangle
= area of the circle – area of rectangle ABCD
1
mark
2
1
mark
2
1
mark
2
= 78.50 – 48 = 30.50 cm2
1
mark
2
Ans17.
Let OAB be the given sector. Then,
Perimeter of sector OAB = 16.4 cm
1
mark
2
OA+OB+ arc AB = 16.4 cm
5.2+5.2+arc AB =16.4
Arc AB = 6 cm i.e. l =6cm
Area of sector =
Ans18.
1
1
 lr =
 6  5.2 = 15.6 cm2
2
2
1
mark
2
1 mark
Since tangents drawn from an external point to a circle are equal
in length.
AP = AS (From A) ………..(i)
BP = BQ
(From B) ………(ii)
CR = CQ (From C) ………(iii)
DR = DS (From D) ………(iv)
1 mark
Adding (i) ,(ii), (iii) and (iv), we get,
(AP+BP)+(CR+DR) = (AS+DS)+(BQ+CQ)
AB+CD = AD+BC.
1 mark
SECTION C
Ans19. If the area of the triangle formed by joining the given points is
zero, then only the points are collinear.
Area of triangle =
1
x1  y2  y3   x2  y3  y1   x3  y1  y2 
2
Here, x1 = a, y1 = a2; x2 = b, y2 = b2; x3 = c, y3 = c2
Substituting the values in the formula for area of a triangle, we
get
Area of triangle =
1
a b2  c2  b c2  a2  c a2  b2 
2






1 mark
1
mark
2
1 mark
It is given that a
b
c
Area of the triangle can never be 0
Ans20. The AP is 8, 24, 40, ….., and the sum is 2048.
We have to determine n.
1
mark
2
1
mark
2
Here a = 8, d = 16
Since, Sn = 2048, we have
2048 =
n
[2x8+(n-1)16]
2
1 mark
2048 = n[8+8n-8]
2048 = 8n2
n2 = 256 i.e. n = 16
negative)
(as n being number of terms, it can’t be
1 mark
Hence, the body will take 16 seconds to fall 2048 metres.
1
mark
2
Ans21. The steps of construction are as follows:
(i) Draw a circle of any convenient radius with O as centre.
(ii) Take a point A on the circumference of the circle and join OA.
Draw a perpendicular to OA at point A.
(iii) Draw a radius OB, making an angle of 90 with OA.
(iv) Draw a perpendicular to OB at point B. Let both the
perpendiculars intersect at point P.
(v) Join OP.
PA and PB are the required tangents, which make an angle of
45 with OP.
1 mark
2 marks
Ans22.
1
mark
2
Let the height of the balloon at P be h metres. Let A and B be the
two cars. Thus AB=100m.
From right ∆PAQ, AQ = PQ = h
From right ∆PBQ,
(as tan 45 =1)
1 mark
PQ
= tan60 =
BQ
3
or
h

h  100
3 or h =
3 (h-100)
1 mark
Therefore, h =
100 3
3 1
= 50(3+ 3 )
i.e. the height of the balloon is 50(3+ 3 ) m.
1
mark
2
OR
1
mark
2
Let AB be the tower of height h metres and C and D be two
points at a distance of a and b respectively from the base of the
tower.
AC = a
m
AD = b
m
(a>b)
In right ∆CAB,
tanx =
AB
h
 tanx =
………………….(i)
AC
a
1 mark
In right ∆DAB,
Tan(90-x) =
AB
AD

cotx =
h
……………(ii)
b
1
mark
2
From (i) and (ii),
tanx.cotx =
1=
h2
ab
h2
ab
 h2 = ab
or h =
Thus, the height of the tower is
ab metres.
1 mark
ab metres.
Ans23. There are 12 balls in the bag. Out of these 12 balls, one ball
can be chosen in 12 ways.
Therefore, total number of outcomes = 12
There are x white balls out of which one can be chosen in x
ways.
Therefore, favourable number of outcomes = x
Hence, p1=P(Getting a white ball) =
x
12
1 mark
If 6 more white balls are put in the bag, then
Total number of balls in the bag = 12+6 = 18
Number of white balls in the bags = x + 6
Therefore p2=P(Getting a white ball) =
x6
18
It is given that
p2 = 2p1
x6
x
2
18
12
x6
x

18
6
6(x+6) = 18x
6x+36 = 18x
12x = 36
1
mark
2
1
mark
2
1
mark
2
x=
36
=3
12
1
mark
2
OR
Total number of equally likely outcomes for visiting shop in the
same week = 6x6=36
(i) Same days are (M,M),(T,T),(W,W),(Th,Th),(F,F),(SAT,SAT)
Number of favourable outcomes = 6
6 1

P(same day) =
1 mark
36 6
(ii) Number of different days = 36-6 = 30
Number of favourable outcomes = 30
30
5
P(two different days) =
1 mark

36
6
(iii) Favourable outcomes for (I customer,II customer)
= (M,T),(T,W),(W,Th),(Th,F),(F,SAT) and for
(II customer,I customer)
= (M,T),(T,W),(W,Th),(Th,F),(F,SAT)
Number of favourable outcomes = 10
P(consecutive days) =
Ans24.
10
5

36 18
1 mark
Let ABC be an equilateral triangle of side 24cm, and let AD be an
altitude from A on BC. Since the triangle is equilateral, so AD
bisects BC.
Therefore BD=CD=12 cm
1
mark
2
The centre O of the inscribed circle will coincide with the centroid
of ∆ABC
Therefore OD=
AD
3
(as centroid divides each median in the ratio 2:1)
1
mark
2
In right ∆ABD, we have
AB2 = AD2+BD2
[Using Pythagoras Theorem]
242 = AD2+122
AD = 12
cm
Therefore OD =
AD
1

   12 3  cm  4 3 cm
3
3

Area of the incircle = (OD)2
1
mark
2


1
mark
2
2
 22
 22

=   4 3  cm2    48 cm2  150.85 cm2
7

7

Area of the triangle ABC =
=
3
(Side)2
4
3
(24)2 = 249.4 cm2
4
1
mark
2
Therefore, area of the remaining portion of the triangle
= (249.4-150.85)cm2
1
mark
2
= 98.55 cm2
OR
Area of one design =
= 
2
60    28
– area of ∆OAB
360
2
28
6
3
28
4
2
1 mark
= 282(0.5238-0.433)
=282x0.0908
= 71.1872
=71.19
cm2 (approx.)
1 mark
Total cost = 6  71.19x 0.35
= Rs149.50
1 mark
Ans25.Volume of water flows in the canal in one hour = width of the
canal x depth of the canal x speed of the canal water
= 3x1.2x20x1000m3 = 72000m3.
In 20 minutes the volume of water =
1 mark
72000  20
m3
60
= 24000 m3.
1 mark
Area irrigated in 20 minutes, if 8cm, i.e. 0.08 m standing water
24000
is required =
= 300000 m2 = 30 hectares.
1 mark
0.08
Ans26.
Radius of the cylinder = 0.7cm
Height of the cylinder = 2.4 cm
Surface area of the cylinder = 2rh+r2
= [2  0.7  2.4+(0.7)2]
= 3.85  cm2
1 mark
Slant height of the cone is given by
l2 = (0.7)2+(2.4)2 = 6.25
 l = 2.5 cm
1
mark
2
surface area of interior conical portion = rl = x0.7  2.5
= 1.75
1
mark
2
Total surface area = 3.85 +1.75 = 5.60 cm2
= 5.6  3.1416 = 17.59 cm2.
1 mark
Ans27.Here, abx2-(a2+b2)x + ab = 0
X=
=
=


2
  a2  b2   4abab


2ab
(a2 +b2 ) ±


 a4  b4   2a2b2


2ab
(a2 +b2 ) ±

(a2 +b2 ) ± a2  b2

1
mark
2
1
mark
2
2ab
Therefore, x =
1 mark
a b
or
b
a
1 mark
Ans28. Let A = (x1,y1) = (-4,1), B = (x2,y2) = (6,7) and
C = (x3,y3)= (2,-9)
1
[x1 (y2 - y3 )+x2 ( y3- y1) +x3 (y1 - y2)]
2
Area of ∆ABC =
Area of ∆ABC =
1
[ -4(7+9) + 6(-9-1) + 2(1-7)]
2
Area of ∆ABC =
1
[ -64-60-12]  = 68 sq. Units.
2
1 mark
Coordinates of D, E and F are
  6  2  7  9  
  4  2  1  9  
  4  6  1  7  
D 
,
,
,
, E
 and F 

2 
2
2 
2
2 
 2


i.e. D(4,-1), E(-1,-4) and F(1,4)
Area of ∆DEF =
1
mark
2
1
[ 4(-4-4) -1(4+1) + 1(-1+4)]
2
1
[ -32-5+3]  = 17 sq. Units.
2
1
1
(Area of ∆ABC) =
(68) = 17 = Area of ∆DEF
4
4
Area of ∆DEF = 
1 mark
1
mark
2
SECTION D
Ans29. Here, a =-11, d =-7-(-11) =4,an = 49
We have an = a+(n-1)d
1
mark
2
So, 49 = -11+(n-1)  4
i.e., 60 = (n-1)  4
i.e., n = 16
1 mark
As n is an even number, there will be two middle terms which
are
16
 16 
th and [
  1] th
2
 2 
i.e., 8th term and the 9th term.
1 mark
a8 = a+7d = -11+7  4 =17
1
mark
2
a9 = a+8d = -11+8  4 = 21
1
mark
2
So, the values of the two middle most terms are 17 and 21,
1
respectively.
mark
2
OR
Let the three terms in AP be d, a, a + d
so, a-d+a+a+d=33
or a = 11
1 mark
Also, (a-d)(a+ d) = a+29
i.e.,
a2-d2 = a+29
i.e.,
121-d2 = 11+29
i.e.,
d2 = 81
1 mark
i.e.,
d=±9
1 mark
So there will be two APs and they are:2, 11, 20,...
And 20, 11, 2, ...
1 mark
Ans30.
1 mark
Let CQ be the tower of height h metres and A and B be the
points of observation. AB = p metres and let BC = x metres.
In right ∆ACQ,
CQ
= tan 
AC
………………(i)

h = (p+ x)tan  or x =
h
-p
tan 
1 mark
In right ∆BCQ,
CQ
= tanβ
 h = xtanβ
BC
…………………………(ii)
or x =
h
tan 
1 mark
From (i) and (ii),
h
h
-p=
tan 
tan 
 1
1 
P=h 


 tan  tan  
 tan   tan  
P =h 

 tan  tan  
or h =
p tan  t an 
 tan   tan  
1 mark
Hence proved.
Ans31.Let usual speed = x km / hr and
increased speed = (x+5) km/hr
According to question,
300 300

2
x
x 5
(Because Distance/Speed = time)
1 mark
300 
x  5  x

  2
  x  5  x 
1
mark
2
1500 = 2x(x + 5)
x2+5x = 750
x2+5x-750 = 0
1 mark
(x+30) (x-25) = 0
X = -30 or x = 25
1 mark
But x being speed can’t be negative.
1
mark
2
Therefore original speed = x = 25 km/hr
OR
Let the marks obtained by P in Mathematics be x.
Therefore marks obtained by P in Science = 28 – x
New marks in Mathematics = x + 3
New marks in science = 28-x-4 = 24-x
ATQ,
(x+3) (24-x) = 180
1 mark
24x+72-x2-3x = 180
-x2+21x = 180-72
x2-21x+108 = 0
1 mark
x2-12x-9x+108 = 0
x(x-12)-9(x-12) = 0
(x-12)(x-9) = 0
x = 12,9
1 mark
Therefore marks in Mathematics = 12
Marks in Science = 28-12 = 16
and marks in Mathematics = 9
Marks in Science = 28-9 = 19
1 mark
Ans32. Given: A circle with centre O; PA and PB are two tangents to
the circle drawn from an external point P.
To prove: PA = PB
Construction: Join OA, OB, and OP.
2 marks
It is known that a tangent at any point of a circle is
perpendicular to the radius through the point of contact.
 OA  PA and OB  PB
... (1)
In OPA and OPB:
OAP = OBP
(Using (1))
OA = OB
(Radii of the same circle)
OP = PO
(Common side)
Therefore, OPA  OPB
(RHS congruency criterion)
PA = PB (Corresponding parts of congruent triangles are
equal)
Thus, it is proved that the lengths of the two tangents drawn
from an external point to a circle are equal.
2 marks
Ans33. Since tangents drawn from an exterior point to the circle are
equal.
Therefore, DR = DS = 5 cm
Now
AR = AD-DR = 23-5 = 18 cm
But
AR = AQ
Therefore AQ = 18 cm
Also
BQ = AB-AQ = 29-18 = 11 cm
But
BP = BQ
Therefore
BP = 11 cm
1
mark
2
1
mark
2
1
mark
2
1
mark
2
Also
1
mark
2
 OQB =  OPB = 900
[Because OQ and OP are perpendiculars to AB and BC
respectively.]
In quadrilateral BPOQ,
1
mark
2
 QOP+  OPB+  OQB+  B = 3600
 QOP = 3600-(900+900+900)= 900
1
mark
2
Hence, OQBP is a square.
BQ = OQ = OP = BP = 11 cm
1
mark
2
Hence, radius of circle is 11 cm.
Ans34. Volume of water in cylinder when full = (3)2  6
= 54 cm3
Volume of solid =
1 mark
2
1
(2)3 +  (2)2  4
3
3
=
32
cm3
3
1 mark
Volume of water in the cylinder when solid is immersed in it
= 54 - 32/3
1 mark
cm3
1 mark
=136.19