Goa Board Class 10 Question Paper June 2018 Mathematics English

2018 VI 18 0230 Seat No. :

Time : 2½ Hours MATHEMATICS (E)

Subject Code

S 0 2 1

Total No. of Questions : 8 (Printed Pages : 7) Maximum Marks : 80
INSTRUCTIONS : i) Answer each main question on a fresh page.
ii) All questions are compulsory.
iii) The question paper consists of eight questions, each of 10 marks.
iv) There is no overall choice. However, internal choice has been
provided in three questions of three 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 will be supplied on request.
vii) Use of calculator and mathematical tables is not permitted.
1. A) Select and write the most appropriate alternative from those given below :
If α and β are the zeroes of a quadratic polynomial 2x2 – 5x – 7, then the
1 1
value of + = __________ [1]
α β
5
a)
7
7
b)
5
−5
c)
7
−7
d)
5
B) Use Euclid’s division algorithm to find the HCF of 81 and 135. [2]
C) Assuming that 7 is an irrational number, prove that 5 – 4 7 is also an
irrational number. [3]
D) If two zeroes of the polynomial 3x4 – 10x3 – 17x2 + 30x + 24 are 3 and – 3 ,
then find the other two zeroes. [4]

S-021 -1- P.T.O.

2. A) Select and write the most appropriate alternative from those given below : [1]
A box contains some discs which are numbered from 5 to 15. If one discs is
drawn at random from the box, then the probability of getting a multiples of
3 or 4 is _____________.
6
a)
11
5
b)
11
3
c)
10
7
d)
10
B) A die and a coin are thrown once simultaneously. Find the probability of getting : [2]
i) A prime number and a head
ii) A number greater than 4 and a tail.
C) Find the roots of ANY ONE of the following quadratic equations. [3]
i) 4x2 + 11x – 20 = 0 (By Factorisation method)
ii) 4x2 + 12x – 7 = 0 (By using quadratic formula)
D) A group of students planned a picnic and estimated the expenditure to be
` 5,000. Five more students joined the group so the expenditure was increased
by ` 1,000, but the average expenses per student was decreased by ` 10.
Find the total number of students who went for the picnic. [4]

3. A) Select and write the most appropriate alternative from those given below : [1]
A car takes ‘y’ hours to travel from a city A to city B with a speed of ‘x’ km/hour,
then the distance between the two cities can be written as _________ km.
a) x + y
x
b)
y
c) x.y
y
d)
x
B) The numerator of a fraction is greater than the denominator by 2. If 1 is added
to the numerator the value of the fraction becomes 2.
Represent the above statements by two linear equations in x and y. [2]

S-021 -2-

C) Find the solution of ANY ONE of the following linear equations : [3]
i) 2x + 5y = – 4
3x – 2y = 13 (By Elimination method)
ii) 3x + 2y = 6
4x – 3y = 25 (By Cross-multiplication method)
D) Find the solution of the following pair of linear equations graphically. [4]
x – y = 7 and 3x + 2y = 6
Rewrite and complete the following tables.
x–y=7 3x + 2y = 6
x x
y y
(Plot atleast 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 ‘n’ terms of an A.P. whose first term is 8 and the last term is
62, is 700. Therefore the A.P. consists of ___________ terms.
a) 15
b) 20
c) 25
d) 30
B) The following table shows the weight of 30 students of a class.

Weight (kg) No. of students
35-40 5
40-45 7
45-50 11
50-55 7

Find the median of the above data upto two decimal places. [2]
C) A man started saving money from the first week of January 2017. He saved
` 25 in the first week, ` 40 in the second week, ` 55 in the third week and so
on, till the last week of December 2017. Find the total saving of the
man in the year 2017. [3]

S-021 -3- P.T.O.

D) The distribution given below shows the daily wages of the employees working
in a factory : [4]
Wages (Rs.) No. of employer Class-mark Deviation fidi
C.I. fi xi di = xi – a
300-350 5 – – –
350-400 9 – – –
400-450 16 – – –
450-500 9 – – –
500-550 5 – – –
550-600 6 – – –
∑ fi = 50 ∑ fidi =
Taking the class-mark denoted by ‘a’ of the class interval (400-450) as the
assumed mean, rewrite and complete the table and also find the mean of the
daily wages by the assumed mean method.

5. A) Select and write the most appropriate alternative from those given below : [1]
PA and PB are tangent segments drawn from external point ‘P’ to a circle with
centre ‘O’ at A and B respectively. If ∠ AOB and ∠ APB are in the ratio 3 : 2,
then ∠ APO = ___________°.
a) 72
b) 36
c) 108
d) 90
B) Given : A circle with centre ‘O’ is inscribed in Δ ABC, where AB = AC. The
sides AB, BC and AC touches the circle at points P, Q and R respectively.
Prove that : ‘Q’ is a mid-point of BC.
A

P R
O

B C
Q
[3]
S-021 -4-

C) Draw a circle with centre ‘A’ and radius 3.5 cm, then take a point ‘P’ at a
distance of 8.5 cm from the centre of the circle. Using a pair of compasses
and ruler, construct two tangent segments PX and PY to the circle. Measure
and state the length of tangent segments. [3]
D) Using a pair of compasses and ruler, construct Δ ABC with sides AB = 6.5 cm,
BC = 7.2 cm and ∠ ABC = 60°. Then construct ΔA′BC′ whose sides are
3
of the corresponding sides of Δ ABC. [3]
4
6. A) Select and write the most appropriate alternative from those given below : [1]
If 3 Sin A – 4 Cos A = 0, then the value of Tan A = ______________.
7
a)
4
4
b)
7
4
c)
3
3
d)
4
B) Attempt ANY ONE of the following : [3]
9
i) In Δ ABC, if ∠ ABC = 90° and Tan A = .
C
40

B
A
Find :
a) The length of AC
b) The value of Sec A
c) The value of Sin C.
ii) Evaluate the following expression using known numerical values of
trigonometrical ratios :
2sin260° – 6 cot2 45° + 5 cosec2 30°.
C) Prove the following identity. [2]
1 − sin A
= sec A − tan A
1 + sin A
D) i) If the points A(6, 1), B(8, 2), C(9, 4) and D(x, y) are the vertices of a
parallelogram, taken in order, find the value of x and y. [2]
ii) Find the area of the triangle whose vertices are A(–5, 7), B(4, –5) and C(4, 5). [2]
S-021 -5- P.T.O.

7. A) Select and write the most appropriate alternative from those given below : [1]
In Δ ABC, points P and Q are on sides AB and AC respectively such that
PQ||BC. If AP : PB = 1 : 2 and ar(APQ) = 6 sq.units, then
ar(
PBCQ) = __________ sq. units.
a) 12
b) 18
c) 36
d) 48
B) With reference to the given figure and given condition, write only the proof
with reasons of the following theorem. In Δ ABC, AB2 + BC2 = AC2 and
Δ PQR is constructed such that PQ = AB, QR = BC and ∠ Q = 90°. [3]
Prove that :
Δ ABC is right angled triangle.

A
P

B C
Q R

C) Given : In
PQRS, PQ||SR, diagonals PR and QS intersect at X, line
through R parallel to PS intersect diagonal SQ on producing at Y. (S – Q – Y).
2
⎛ PX ⎞ QX
Prove that : ⎜ ⎟ = . [3]
⎝ RX ⎠ XY
Y

P Q

X

S R

S-021 -6-

D) The shadow of a tower AB, standing on a level ground is found to be 50 m
longer when the sun’s altitude is 30° than when it is 60° find the height of the
tower (take 3 = 1.73 ). A [3]

30° 60°
D B
50 m C
8. A) Select and write the most appropriate alternative from those given below :
i) If the area of a circle is numerically equal to twice the circumference then
its radius is ___________ cm. [1]
a) 16
b) 8
c) 4
d) 2
ii) The total surface area of a right circular cylinder with radius of its base
3 cm and height 2 cm is ________ sq.cm. [1]
a) 15 π
b) 30 π
c) 18 π
d) 36 π
B) A container, opened from the top and made up of a metal sheet, is in the form of
a frustum of a cone of height 21 cm, with radii of its lower and upper ends 6 cm
22
and 10 cm respectively. Find the volume of the container (Take π = ) [2]
7
C) In the given figure, ABCDEF is a regular hexagon of side 10 cm. Taking AB
and DE as radii two sectors are drawn as shown in the figure. Taking
π = 3.14 and 3 = 1.73, find the area of shaded region. [3]
F E

A D

B C
D) A metallic ball of radius 10.5 cm is melted and recast into 126 cones of equal
size. If the height of the cones formed is 3 cm, then find the radius of the each
22
cone formed (Take π = ). [3]
7
____________
S-021 -7-