Goa Board Class 10 Question Paper Mar 2019 Mathematics English

2019 IV 08 0930 Seat No.

Time : 2½ Hours MATHEMATICS (E)
Subject Code

S 0 2 1
Total No. of Questions : 8 (Printed Pages : 13) Maximum Marks : 80

INSTRUCTIONS : (i) Answer each main question on a fresh page.
(ii) All questions are compulsory.
(iii) The question paper consists of 8 questions, each of
10 marks.
(iv) There is no overall choice. However, internal choice
has been provided in three questions of 3 marks
each.
(v) In questions on constructions, the drawing should
be clear and exactly as per the given measurements.
The construction lines and arcs should also be
maintained.
(vi) Graph paper is provided at the last page of the main
answer booklet.
(vii) Use of calculator and Mathematical tables is not
permitted.
1. (A) Select and write the most appropriate alternative from those given
below : 1
The product of two numbers is 864. If their HCF is 12, then their
LCM is ...............
(a) 12
(b) 72
(c) 852
(d) 876

(B) Attempt the following : 2

(i) Find the sum of the zeroes of the quadratic polynomial

2x2 – 7x – 15.

2
(ii) Find the zeroes of the quadratic polynomial x – 11x.

3 2
(C) On dividing the polynomial 2x – 5x + 8x – 5 by a polynomial g(x),

the quotient and remainder are (2x – 3) and (3x – 2) respectively.

Find g(x). 3

(D) Prove that 7 is an irrational number. 4

2. (A) Select and write the most appropriate alternative from those given

below : 1

If P(E) = 0.07, then the probability of getting an event ‘‘not E’’ is ................

(a) 0.03

(b) 0.93

(c) 1.00

(d) 1.07

(B) A card is drawn from a well shuffled deck of 52 playing cards.

Find the probability of getting : 2

(i) an Ace

(ii) a red face card.

(C) Find the roots of ANY ONE of the following quadratic equations : 3

(i) 7x2 – 17x + 6 = 0 (by factorisation method)

(ii) 3x2 + 10x – 8 = 0 (by quadratic formula method)

1
(D) Two pipes A and B running together can fill a tank in 3 minutes.
3

If pipe B takes 5 minutes more than pipe A to fill the tank

separately, then find the time in which each pipe would fill the tank

separately. 4

3. (A) Select and write the most appropriate alternative from those given

below : 1

Three years hence, the ages of two friends will be x and y years

respectively. Therefore the sum of their ages two years ago

was ...................... years.

(a) x + y + 10

(b) x + y + 5

(c) x + y – 10

(d) x + y – 5

(B) The following is a pair of linear equations in two variables : 2

2x + 3y = 7

(k + 1)x + (2k – 1) y = 4k + 1.

Answer the following questions with reference to the given pair of
equations :

(i) Write down the condition for infinitely many solutions.

(ii) Find the value of k.

(C) Find the solution of ANY ONE of the following pair of linear
equations : 3

(i) 3x – 2y = 5 and

5x + 3y = 21 (By elimination method)

(ii) 2x + 5y = 29

7x – 2y = 4 (By cross-multiplication method)

(D) Find the solution of the following pair of linear equations
graphically : 4

x + y = 8 and

3x – y = 4
Rewrite and complete the following tables :

x y 8 3x y 4

x x

y y

(Plot at least 3 points for each line using a graph paper)

4. (A) Select and write the most appropriate alternative from those given

below : 1

The sum of first 20 odd natural numbers is ...............

(a) 100

(b) 210

(c) 400

(d) 420

(B) The following table shows the ages of 50 people in a locality : 2

Ages in Years Number of People

5—15 10

15—25 12

25—35 15

35—45 13

Find the Median of the above given data.

(C) A factory manufacturing electric bulbs increases the production uni-

formly by a fixed number every month. If in the third month the

production is 600 electric bulbs and in the seventh month the produc-

tion is 800 electric bulbs, then find the total production of electric bulbs

in the year. 3

(D) The following table shows the donation collected by a club from
60 donors : 4

Donation No. of Class Deviation fidi

in ` (C.I.) Donors Mark d = x – a
i i

(f ) (x )
i i

0—20 5 — — —

20—40 12 — — —

40—60 14 — — —

60—80 15 — — —

80—100 8 — — —

100—120 6 — — —

Total f = 60 fd = —
i i i

Taking the class-mark denoted by ‘a’ of the class interval (40–60) as
the assumed mean, rewrite and complete the table. Also find the mean
of the donation by the assumed mean method.

5. (A) Select and write the most appropriate alternative from those given
below : 1

TP and TQ are tangents drawn from an external point T to a circle
with centre O, at P and Q respectively. If POQ = 130°, then the measure
of PTO = .................

(a) 25°

(b) 50°

(c) 65°

(d) 130°

(B) Given : Point O is the centre of a circle. QS and QT are two tangent

segments drawn from an external point Q to the circle at S and T

respectively. Prove that : 3

QS = QT

S

O
Q

T

(Write only the proof with reasons)

(C) Draw a circle with centre O and radius 3.5 cm. Take a point P at a

distance of 8 cm from the centre of the circle. Using a pair of

compasses and ruler construct two tangents PA and PB to the circle.

Measure and state the length of the tangent segments. 3

(D) Using a pair of compasses and ruler construct PQR with sides

PQ = 7 cm, QR = 8.5 cm and PR = 6.5 cm. Then construct P QR
3
whose sides are of the corresponding sides of PQR. 3
5

6. (A) Select and write the most appropriate alternative from those given

below : 1

If cosec 3A = sec (A – 22) where 3A is an acute angle, then the measure

A = .....................

(a) 11°

(b) 17°

(c) 28°

(d) 58°

(B) Attempt ANY ONE of the following : 3

17
(i) In DRY, R = 90° and cosec Y = :
8

D

R Y

Find :

(a) the length of YR

(b) the value of cot D

(c) the value of cos Y

(ii) Evaluate the following expression using known numerical values

of trigonometric ratios :

1
3 tan2 30° – 2 sec2 45° + cos 60°.
3
(C) Prove the following identity : 2

tan sec 1
sec tan .
tan sec 1

(D) Attempt the following :

(i) Find the area of triangle ABC whose vertices are A(4, 3),

B(12, 5) and C(4, 6). 2

(ii) Find the value of k, if the point P(3, 4) is equidistant from the

points A(5, k) and B(k, 7). 2

7. (A) Select and write the most appropriate alternative from those given

below : 1

In ABC, points D and E are on the sides of BC and AC respectively

such that B–D–C, A–E–C and DE AB. If CE = 4 cm, AE = 5 cm

and BD = 4.5 cm, then BC = ................. cm

(a) 3.6

(b) 5.6

(c) 8.1

(d) 9

(B) With reference to the given figure and the given conditions, write only
the proof with reasons of the following theorem : 3
D
P

E F Q R

Given : In DEF, DE2 + EF2 = DF2, PQR is constructed such that
PQ = DE, QR = EF and Q = 90°.
Prove that : DEF is a right-angled triangle.
(C) Given : ABC is a right-angled triangle, right angled at C. Line segment
CD is drawn perpendicular to side AB of ABC. 3
Prove that :
1 1 1
2
= 2
+
CD BC AC2
C

A B
D

(Write only the proof with reasons).

(D) Two pillars AB and CD are 50 m apart and the height of pillar CD

is double the height of pillar AB as shown in the figure. From a point

P on the line joining the feet of the pillars, an observer observes the

top A of the pillar AB and top C of the pillar CD at angles of elevation

30° and 60° respectively. Find the height of the pillar AB and pillar

CD. (Take 3 1.7 ) 3

C

A

30° 60°
B D
P
50 m

8. (A) Select and write the most appropriate alternative from those given

below : 2

(i) The length of an arc of a circle of radius 15 cm and subtending

an angle 36° at the centre of the circle is ............... cm.

(a) 3

(b) 5

(c) 15

(d) 30

(ii) The diameter of a circle whose circumference is 14 cm is

................. cm.

(a) 7

(b) 14

(c) 21

(d) 28

(B) A toy is in the form of a hemisphere surmounted by a conical top of

the same base radius as shown in the figure. If the radius of the base

of conical top is 5 cm and height of the toy is 17 cm,

5 cm

find : 2

(i) The slant height of the cone.

(ii) The curved surface area of the hemisphere.

(Do not substitute for )

(C) In the adjoining figure a piece of cardboard is in the shape of trapezium

ABCD where AB DC and BCD = 90º. A quadrant with centre C

and radius 3.5 cm is drawn. If AB = BC and DE = 2 cm, then
22
find the area of the shaded region. 3
7

A B

3.5 cm

C
D E

2
(D) The surface area of a solid metallic sphere is 616 cm . The sphere is

melted and recast into smaller cones, each of diameter 3.5 cm and height
22
14 cm respectively. Find the number of such cones formed.
7

3