Goa Board Class 10 Sample Paper 2023 Maths

Sample Paper
&
Question Paper Pattern
2022
2023

GOA BOARD

GOA BOARD OF SECONDARY AND HIGHER SECONDARY EDUCATION
PORTION AND MARKS DISTRIBUTION FOR STD X (2022-2023)
SUBJECT: MATHEMATICS
FIRST INTERNAL TEST (LEVEL 1 and LEVEL 2)
Ch.no. Name of the chapter Marks
2 Polynomials 5
3 Pair of Linear equations in two variables 7
7 Co-ordinate Geometry 5
10 Circles 3
Total 20
SECOND INTERNAL TEST (LEVEL 1 and LEVEL 2)
Ch.no. Name of the chapter Marks
1 Real numbers 4
15 Probability 4
8 Introduction to Trigonometry 7
12 Areas related to Circles 5
Total 20

THIRD INTERNAL TEST (20marks) (LEVEL 1 and LEVEL 2)
INNOVATIVE TEST on any two /three chapters given below or
Presentation/Project/Assignment based on anyone of the following chapters

Ch.no. Name of the chapter
6 Triangles
7 Co-ordinate Geometry
9 Some Applications of Trigonometry
13 Surface areas and Volumes
14 Statistics

FIRST TERM EXAM (MCQs) (LEVEL 1 and LEVEL 2)
Ch.no. Name of the chapter Marks
1 Real Numbers 5
2 Polynomials 5
3 Pair of Linear equations in two variables 7
15 Probability 3
8 Introduction to Trigonometry 7
7 Co-ordinate Geometry 5
10 Circles 3
12 Areas related to Circles 5
Total 40
SECOND TERM EXAM (Subjective) (LEVEL 1 and LEVEL 2)
Ch.no. Name of the chapter Marks
4 Quadratic Equations 8
5 Arithmetic Progressions 5
14 Statistics 7
6 Triangles 5
9 Some Applications of Trigonometry 3
11 Constructions 6
13 Surface Areas and Volumes 6
Total 40

PORTION FOR STD X MATHEMATICS (LEVEL2)
Name of the Chapter Portion
1)Real Numbers whole topic is included for evaluation
2)Polynomials a) Concept of a Polynomial, degree-types
b) Zero of a Linear Polynomial ,Quadratic
Polynomial-relation between zeroes and
coefficients
c)Finding a Quadratic Polynomial given
sum and product of zeroes/zeroes
d)To find the Quotient and remainder
when a Cubic Polynomial is divided by a
Linear polynomial and to express in the
form:
Dividend =divisor x Quotient +Remainder
3)Pair of Linear equations in two a) General form of a pair of linear
variables equations in two variables
b) Conditions for a pair of Linear
equations in two variables to have-a
unique solution, no solution, infinitely
many solutions -finding the value of the
unknown
c)Find the solution of a pair of linear
equations in two variables by
(I) Elimination method
(II)Substitution method (one
equation should have coefficient of x and
y as one)
(III)Graphical method (one
equation should have coefficient of x and
y as one and the other equation should
have coefficient of any one x or y as one)

4)Quadratic equations a) Concept of a Quadratic equation-
standard form
b) Finding the Roots of a Quadratic
equation by
(I)Factorisation method
(II)Quadratic formula
C)Nature of Roots based on discriminant
5)Arithmetic Progressions a) Concept of an AP-first term, common
difference
b) Direct sums based on nth term, sum of
n terms of an AP
6)Triangles a) Concept of Similarity of Triangles-Tests
for similarity of Triangles
b) Concept of theorem on Areas of
Similar Triangles (Proof not for
evaluation)
c)B.P.T., Pythagoras theorem and
Converse of Pythagoras theorem (Proofs
for evaluation)
d)Simple numerical applications of the
above 4 theorems
7)Co-ordinate Geometry Concept of (I) Distance Formula
(II)Section Formula (III)Area of Triangle
Formula and direct questions based on
them
8)Introduction to Trigonometry a) Concept of Trigonometry
b) Trigonometric ratios and their
relationships, k method
c) Proving Sin2θ+Cos2θ=1 with the figure
d)Expressions involving Trigonometric
ratios of some specific angles:
0⁰, 30⁰,45⁰,60⁰,90⁰
e) Trigonometric ratios of

complementary angles

9)Some applications of Trigonometry a) Heights and Distances: Angle of
Elevation and Angle of Depression
b) Simple problems on heights and
Distances. Problems should have only
one right triangle with either angle of
elevation or Depression.
10)Circle a) Concept of Tangent,
Thm.10.1(proof not for Evaluation)
Thm.10.2(with Proof)
b) Simple numerical applications
11)Constructions a) Construction of Tangents to a Circle
from a point outside the circle
b) Construction of Similar Triangles as per
given scale factor.
Note : Angles can also be drawn using a
protractor
12)Areas related to Circles a) Perimeter and Area of a Circle
b) Areas of Segment, Sector, Quadrant
of a Circle and Semicircle
c)Simple applications to find areas of
shaded region involving only two plane
figures
13)Surface Areas and Volumes Whole topic is included for evaluation
14)Statistics a) Concept of Mean, Median, Mode
b) To find Mean of grouped data by
Direct method
c)To find Mode of grouped data.

15)Probability a) Concept of Theoretical Probability
b) Probability of a Sure event and an
Impossible event, 0≤P(E)≤1, P (not E)
c)Simple problems based on

coins(max2), Dice (only 1), playing cards,
numbered cards, items in a box.

PORTION FOR STD X MATHEMATICS (LEVEL 1)
a) Everything is included from ch. 1 to ch.15.
b) In the topic of Triangles , Rider and numerical applications based
on the theorems will be tested.
c)In the topic of Constructions , a pair of compasses and ruler to be
used to draw specific angles
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GOA BOARD OF SECONDARY AND HIGHER SECONDARY EDUCATION
FIRST INTERNAL TEST
MODEL PAPER ( 2022-2023)
STD : X MAX MARK S: 20
SUBJECT : MATHEMATICS (E) : LEVEL 1 and LEVEL 2
TIME : 1 ℎ𝑟

________________________________________________________________________

Weightage to Content/Subject Units

Sr. No. Units Marks

1 Polynomials 05

2 Pair of Linear Equations in Two Variables 07

3 Coordinate Geometry 05

4 Circles 03
TOTAL 20

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GOA BOARD OF SECONDARY AND HIGHER SECONDARY EDUCATION
FIRST INTERNAL TEST
MODEL PAPER (2022-2023)
STD : X MAX MARKS: 20
SUBJECT : MATHEMATICS (E) : LEVEL 1 TIME : 1 ℎ𝑟
__________________________________________________________________________
Q.1.A) Select and write the correct alternative from those given below. (1)
The value of ‘k’ for which the pair of linear equations 3𝑥 − 4𝑦 = 7 𝑎𝑛𝑑 6𝑥 + 𝑘𝑦 = 5
has no solution is:
a) 8 b) 2 c) – 2 d) – 8
B) Find the solution of ANY ONE of the following pair of linear equations: (3)
i) 2𝑥 − 3𝑦 = 13 𝑎𝑛𝑑 4𝑥 + 5𝑦 = −7 ( 𝐵𝑦 𝐸𝑙𝑖𝑚𝑖𝑛𝑎𝑡𝑖𝑜𝑛 𝑚𝑒𝑡ℎ𝑜𝑑 )
ii) 7𝑥 + 2𝑦 = 11 𝑎𝑛𝑑 5𝑥 − 𝑦 = 20 (𝐵𝑦 𝐶𝑟𝑜𝑠𝑠 𝑚𝑢𝑙𝑡𝑖𝑝𝑙𝑖𝑐𝑎𝑡𝑖𝑜𝑛 𝑚𝑒𝑡ℎ𝑜𝑑)
C) Find all the zeroes of the polynomial 𝟐𝒙𝟑 + 𝒙𝟐 − 𝟔𝒙 − 𝟑 if two of its zeroes (3)
are √3 𝑎𝑛𝑑 − √3 .
D) The area of a rectangle gets reduced by 8 cm², if its length is reduced by 5 cm and (3)
breadth is increased by 3 cm. If we increase the length by 3 cm and the breadth by 2 cm,
the area increases by 74 cm². Find the length and the breadth of the rectangle.

Q.2.A) Select and write the correct alternative from those given below. (1)
The distance between the points A(2, –3) and B(2, 2) is :
(a) 2 units (b) 3 units (c) 4 units (d) 5 units
B) Attempt the following. (2)
i) If m and n are the zeroes of the polynomial 𝑥 2 − 7𝑥 + 12 then find the
1 1
value of 𝑚 + 𝑛
ii) Determine if –2 is a zero of the polynomial 𝑝(𝑥) = 𝑥 3 + 𝑥 2 − 2𝑥

C) In the adjoining figure, ∆ ABC is right angled (3)
at B such that BC = 6 cm and AB = 8 cm.
Find the radius of the circle.

D) i) Find the ratio in which the point P(2,–5) divides the line segment joining the points (2)
A(–3, 5) and B(4, –9) internally.
ii) Find the area of rectangle ABCD if its vertices A(1, –3), B(13, 9), C(10, 12) (2)
and D(–2, 0) are taken in order.

--------------------------------------------- ( ÷ × + – ) ---------------------------------------------
N.B: In Q.1.D), Word problem or Graphical solution may be tested from the topic ‘Pair of L.E
in two variables’.

GOA BOARD OF SECONDARY AND HIGHER SECONDARY EDUCATION
FIRST INTERNAL TEST
MODEL PAPER ( 2022-2023)
STD : X MAX MARK S: 20
SUBJECT : MATHEMATICS (E) : LEVEL 2 TIME : 1 ℎ𝑟
__________________________________________________________________________

Q.1.A) Select and write the correct alternative from those given below. (1)
If 9𝑥 − 7𝑦 = 15 𝑎𝑛𝑑 7𝑥 − 9𝑦 = 13 , then the value of 𝑥 + 𝑦 is:
(a) –3 (b) – 1 (c) 1 (d) 3
B) Find the solution of ANY ONE of the following pair of linear equations: (3)
i) 3𝑥 + 4𝑦 = 18 𝑎𝑛𝑑 7𝑥 − 3𝑦 = 5 ( 𝐵𝑦 𝐸𝑙𝑖𝑚𝑖𝑛𝑎𝑡𝑖𝑜𝑛 𝑚𝑒𝑡ℎ𝑜𝑑 )
ii) 𝑥 − 𝑦 = 7 𝑎𝑛𝑑 2𝑥 + 7𝑦 = −13 ( 𝐵𝑦 𝑆𝑢𝑏𝑠𝑡𝑖𝑡𝑢𝑡𝑖𝑜𝑛 𝑚𝑒𝑡ℎ𝑜𝑑 )
C) Divide the polynomial ( 𝟐𝒙𝟑 − 𝟓𝒙𝟐 − 𝟑𝒙 + 𝟕 ) 𝒃𝒚 ( 𝟐𝒙 − 𝟑 ) and find the quotient (3)
and remainder. Also, express the dividend in the form:
“ Dividend = divisor × quotient + remainder ”
D) Find the solution of the following pair of linear equations graphically : (3)
𝑥 − 𝑦 = 5 𝑎𝑛𝑑 2𝑥 + 𝑦 = 7
Rewrite and complete the following tables.
( Plot at least 3 points for each line on a graph paper )
𝑥−𝑦 =5 2𝑥 + 𝑦 = 7
x x
y y

Q.2.A) Select and write the correct alternative from those given below. (1)
The distance of the point P(12, –5) from the origin is :
(a) 7 unit (b) 17 units (c) 14 units (d) 13 units

B) Attempt the following. (2)
2
i) If the sum of the zeroes of the polynomial 3𝑥 − 2𝑘𝑥 + 6 is 3, then find the value of k .
ii) Find a quadratic polynomial in variable 𝑥 whose zeroes are √5 𝑎𝑛𝑑 − √5 .

C) Given: Point O is the centre of the Circle . Two tangent (3)
segments PA and PB are drawn from an external
point P to the Circle at A and B respectively.

Prove that : PA = PB

D) i) Find the area of ∆ ABC formed by joining the points A(10, –6), B(2, 5) and C(–1, 3). (2)
ii) Find the coordinates of the point P (x, y) which divides the line segment joining (2)

the points A(5, –2) and B(9, 6) internally in the ratio 3 : 1 .

--------------------------------------------- ( ÷ × + – ) ---------------------------------------------

GOA BOARD OF SECONDARY AND HIGHER SECONDARY EDUCATION
SECOND INTERNAL TEST
MODEL PAPER (2022-2023)
STD : X MAX. MARKS: 20
SUB – MATHEMATICS(E) (LEVEL – 1 AND LEVEL -- 2) TIME : 1 Hr

Weightage to Content/Subject Units

Sr. No Units Marks
1 Real Numbers 04
2 Probability 04
3 Introduction to Trigonometry 07
4 Areas Related To Circles 05
TOTAL 20

*******

GOA BOARD OF SECONDARY AND HIGHER SECONDARY EDUCATION
SECOND INTERNAL TEST
MODEL PAPER (2022-2023)
STD : X MAX. MARKS: 20
SUB – MATHEMATICS (E): LEVEL 1 TIME : 1 Hr

Q 1 A) Select and write the most appropriate alternative from those given below. (1)
The L.C.M. and H.C.F. of two numbers are 120 and 12 respectively. If one of
the numbers is 24, then the other number is ___________
(a) 18 (b) 24 (c) 36 (d) 60

B) Attempt the following
i) A box contains tickets numbered from 21 to 50. If one ticket is picked at random, (2)
then find the probability of getting—
a) a prime number b) a number which is a multiple of both 2 as well as 3.

ii) From a deck of 52 playing cards, red kings are removed and from the remaining (2)
cards, a card is picked at random.
Find the probability of getting—
a) a red card. b) a face card.

C) Find the H.C.F of 741 and 1173 using Euclid’s division algorithm. (3)
Or
C) Prove that √5 is an irrational number. (3)

D) In the adjoining figure, ‘O’ is the centre of (2)
a circle with radius 28 cm. If arc AXB
subtends an angle of 900 at the centre, then

find-
i) the area of quadrant O-AXB
ii) the area of minor segment AXB

Q 2 A) Select and write the most appropriate alternative from those given below. (1)
If tan A = cot (2A – 30)0, where 2A is an acute angle, then the value of
A = _____ .
(a) 200 (b) 300 (c) 400 (d) 900

B) Evaluate the following expression using known numerical values of (3)
trigonometrical ratios.

5 tan2300 -- cosec2 600 + 3 sin2 450
5 sin 300

C) Prove the following Identity. (3)

( Cosec A – Sin A)( Sec A – Cos A) = 1
Tan A + Cot A

D) A circular running track of width 3.5 m is to be constructed around a circular (3)
ground of area 5,544 m2. Find the total cost of construction of the track
at Rs 900/- per m2.
********

GOA BOARD OF SECONDARY AND HIGHER SECONDARY EDUCATION
SECOND INTERNAL TEST
MODEL PAPER (2022-2023)
STD : X MAX. MARKS: 20
SUB – MATHEMATICS (E): LEVEL 2 TIME : 1 Hr

Q 1 A) Select and write the most appropriate alternative from those given below. (1)
If the product of two numbers is 432 and their L.C.M. is 72 , then their H.C.F.
is :
(a) 6 (b) 8 (c) 12 (d) 16

B) Attempt the following
i) A box contains 12 red balls , 15 orange balls and 9 blue balls. If one ball is picked (2)
at random , then find the probability of getting-
a) an orange ball. b) not a blue ball.

ii) From a deck of well shuffled 52 playing cards, a card is picked at random. (2)
Find the probability of getting—
a) a diamond. b) a king .

C) Assuming that √3 is irrational , prove that 5 --√3 is also irrational. (3)

Or

C) Find the H.C.F of 168 and 580 using Euclid’s division algorithm. (3)

D) In the adjoining figure, arc AXB subtends an angle of 600 at (2)
the centre ‘O’ of a circle. If the radius of the
circle is 14 cm, then find—

i) length of arc AXB. O

ii) area of sector O-AXB. 600

22
( take ᴨ = 7 ) A B
X

Q 2 A) Select and write the most appropriate alternative from those given below. (1)
If SinP = Cos 280 , where P is an acute angle , then the value of P = _____ .

(a) 170 (b) 280 (c) 620 (d) 900

B) Attempt the following

i) Evaluate : Sec 530 -- Cosec 370 (1)

ii) In the adjoining figure , ∆ABC is a right angled triangle , A (2)
right angled at B.

Prove that :

Sin2 Ө + Cos2 Ө = 1. Ө
B C

C) In ∆PQR , Q = 900 and Tan R = 8 P (3)
15
Then find—
a) the length of PR
b) the value of sin R
c) the value of cosec P
Q R

D) A circle is inscribed in square ABCD of side 14 cm., as shown (3)
in the figure. Find the area of the shaded region. A D
22
(take ᴨ = 7 )

B C

**********

GOA BOARD OF SECONDARY AND HIGHER SECONDARY EDUCATION

ALTO-BETIM GOA 403521

DESIGN OF THE QUESTION PAPER FIRST TERM EXAM (2022-2023)

Class : X Subject : Mathematics - Level 1 ( English )
Time : 1hr 45min Max. Marks : 40

***********************************************************************************************

The weightage or the distribution of marks over different dimensions of the question paper shall be as follows:

1. Weightage to Learning Outcomes
S.No. Learning Outcomes Marks Percentage of Marks
1. Knowledge 8 20%
2. Understanding 20 50%
3. Application 12 30%
Total 40 100%

2. Weightage to Content / Subject Units

S.No. Units Marks
1. Real Numbers 5
2. Polynomials 5
3. Pair of Linear Equations in Two Variables 7
4. Introduction to Trigonometry 7
5. Co-ordinate Geometry 5
6. Circles 3
7. Areas related to circles 5
8. Probability 3
Total 40

3. Weightage to Difficulty level of questions

S.No. Estimated difficulty level of questions Percentage
1 Easy 20%
2 Average 60%
3 Difficult 20%
Total 100%

4. Number of Questions :

There will be 40 questions of 1 mark each.

Goa Board Of Secondary And Higher Secondary Education
Blue Print ( First Terminal Examination 2022-23)
Std X : Mathematics Level 1 (English)
Duration : 1hr 45min

Type of Question : VSA type

S.No. Content/Unit Objectives Total
Knowledge Understanding Application
1. Real Numbers 1(1) 9(1) 33(1) 5(5)
25(1) 17(1)
2. Polynomials 5(1) 13(1) 37(1) 5(5)
21(1)
29(1)
3. Pair of Linear Equations 2(1) 10(1) 26(1) 7(7)
in Two Variables 18(1) 30(1) 38(1)
34(1)
4. Introduction to 3(1) 19(1) 39(1) 7(7)
Trigonometry 11(1) 27(1)
31 (1)
35(1)
5. Co-ordinate Geometry 4(1) 12(1) 5(5)
20(1) 36(1)
28(1)
6. Circles 7(1) 3(3)
15(1)
23(1)
7. Areas related to circles 8(1) 16(1) 5(5)
24(1)
32(1)
40(1)
8. Probability 6(1) 14(1) 3(3)
22(1)

Total 8(8) 20(20) 12(12) 40(40)

Note : Figures outside the bracket indicate the question number and figures inside the bracket indicate the marks.

**********************************************************************************************

GOA BOARD OF SECONDARY AND HIGHER SECONDARY EDUCATION
FIRST TERM EXAMINATION MODEL PAPER (2022-2023)

STD X MARKS : 40
SUBJECT : MATHEMATICS - LEVEL 1 ( ENGLISH ) TIME : 1HR 45MIN
_______________________________________________________________________________________________
Instructions :
1. Each question is provided with four alternatives. Choose the correct alternative.
2. Each question carries one mark . There is no negative marking for incorrect choice.

1. The prime factorisation of 1176 is
(A) 22 x 33 x 7 (B) 23 x 32 x 7 (C) 22 x 3 x 72 (D) 23 x 3 x 72

2. If the lines given by 4x + ky = 12 and x + 2y = 3 coincide, then the value of k is
(A) -8 (B) 4 (C) 8 (D) 12

3. If 1 + tan2 36º= sec2 2A where 2A is an acute angle then the value of A is
(A) 18° (B) 36° (C) 54° (D) 72°

4. The midpoint of the line segment joining the points P (-3, 4) and Q ( 7, -2) is
(A) (-2, -1) (B) (2, -1) (C) (-2, 1) (D) (2, 1)

5. One of the zeroes of the quadratic polynomial 4x2- 25 is
(A) 4/25 (B) 2/5 (C) 25/4 (D) 5/2

6. From a deck of 52 playing cards all the ace cards are removed. If a card is then drawn at random, the
probability that it is a face card is
(A) 1/4 (B) 3/14 (C) 3/13 (D) 2/13

7. The length of the tangent drawn from a point P which is at a distance of 7cm from the centre O of a circle
of radius 3cm is
(A) 2√10 cm (B) √58 cm (C) 10 cm (D) 100cm

8. The length of the arc of a circle of radius r and angle with degree measure 𝜃 is
𝜃 𝜃 𝜃 𝜃
(A) 180 x ∏r (B) 360
x ∏r 2 (C) 180 x 2∏r (D) 180 x ∏r2

9. Which of the following is irrational?
3+2√3
(A) √729 (B) ( √3 - 2) ( √3 + 2) (C) (D) √2 (√2 - √8 )
√3

10. If one equation of a pair of consistent linear equations is 3x -2y +4 = 0 then the second equation can be
(A) 6x - 4y +1 = 0 (B) - 6x + 4y +1 = 0 (C) 6x + 4y +1 = 0 (D) 9x - 6y +1 = 0

11. If △PQR is right angled at P and ∠ Q= 60° then the value of cos R is
(A) 1 (B ) 1/2 (C) 1/√2 (D) √3/2

12. The perimeter of a triangle with vertices A(0,6) ; B(0,0) ; C(-8, 0) is
(A) 10 units (B) 12 units (C) 24units (D) 36 units

13. If the product of the zeroes of the quadratic polynomial 5x2 - 20x - m is 3, then the value of m is

(A) - 15 (B) - 4 (C) 4 (D) 15

14. A box contains cards which are numbered from 5 to 102. If a card is drawn at random from the box, then
the probability that it bears a two digit number which is a multiple of 7 is
(A) 13/98 (B) 1/7 (C) 13/97 (D) 14/97

A
15. In the figure MA and MB are tangents to the circle with
centre O and radius 5cm. If OM = 13cm then perimeter M
O
of □AOBM is
(A) 18 cm (B) 27 cm (C) 34 cm (D) 36 cm
B
16. □ABCD is a rectangle . If AB=5cm and the radius of the semicircle
AM=7cm then area of the shaded portion is (Take π = 22/7)
A M D
2 2
(A) 56 cm (B) 73.5 cm
(C) 74.5 cm2 (D) 76.5 cm2
B C
17. Which of the following rational numbers has a terminating decimal expansion?
7 17 27 37
(A) 12 (B) 45 (C) 72 (D) 168

18. The solution of the pair of linear equations 8x - 3y = -2 and 3 x + y = -5 is
(A) x= -1, y= 2 (B) x= 1, y= - 2 (C) x= 1, y= 2 (D) x= -1, y= -2

19. The value of the trigonometric expression : sin 2 30º + cos 2 45º – 7tan 2 60º is
(A) - 81/4 (B ) – 79/4 (C) 79/4 (D) 81/4

20. The ratio in which the point K(1,3) divides the line segment joining the points M(-1, 7 ) and N(4 , -3)
internally is
(A) 1 : 3 (B) 3 : 1 (C) 2 : 3 (D) 3 : 2

21. The quadratic polynomial whose sum and product of zeroes are -9 and 20 respectively is
(A) x2 - 9x - 20 (B) x2 + 9x - 20 (C) x2 - 9x + 20 (D) 2x2 +18x + 40

22. Two dice are thrown simultaneously. The probability that the sum of the numbers appearing on top of both
the dice is 4 is
(A) 1/12 (B ) 1/9 (C) 1/2 (D) 2/3

23. If PA and PB are tangents from an external point P to the circle with centre O such that
∠APB = 80°then the measure of ∠AOP is
(A) 10° (B) 50° (C) 100° (D) 160°

24. The length of the minute hand of a clock is 21cm. Therefore the area swept by the minute hand in
6 minutes is
(A) 13.2 cm2 (B) 15.4 cm2 (C) 23.1 cm2 (D) 138.6 cm2

25. The product of two numbers is 1440. If their HCF is 4 , then their LCM is
(A) 10 (B) 36 (C) 90 (D) 360

26. If the pair of linear equations (k - 7)x - 2y = -5 and 4y – (k+1)x = 2 have no solution then the value of k is
(A) 0 (B) 1 (C) 15 (D) 16

27. The simplified form of (cosecA- cotA) (1+cosA) is
(A) sinA (B) cosA (C) cosecA (D) secA

28. If the area of the triangle with vertices A(4,0) ; B(-5, 0) ; C (2, k) is 36 square units then the value of
k is
(A) - 8 (B ) - 4 (C) 4 (D) 8
29. The remainder when p(x)= 2x3 - 5x2 - 4x - 7 is divided by g(x)= x2 - 2 is
(A) -17 (B) - 3 (C) 3 (D) 17

30. If 114x +156y = 426 and 156x + 114y = 384 ; then the value of x - y is
(A) -3 (B) - 1 (C) 1 (D) 3

31. The value of the trigonometric expression: sin65ºcos 25º + cos65ºsin25º is
(A) - 1 (B) 0 (C) 1 (D) 2

32. If the perimeter of a semi circular wall piece is 108cm , then the diameter of the wall piece is
( Take ∏=22/7)
(A) 21cm (B ) 28cm (C) 42cm (D) 56cm

33. Toffees from two toffee jars containing 210 and 240 toffees are to be packed in small packets. The
maximum number of packets that can be packed if each packet has equal number of toffees is
(A) 9 (B) 18 (C) 30 (D) 36

34. Five years hence Tom will be x years old and Jerry will be y years old. Therefore the sum of their present
ages in years is
(A) x + y – 5 (B) x + y + 5 (C) x + y + 10 (D) x + y - 10

35. If 17sinA = 8 , then tanA =
(A) 8/15 (B ) 15/8 (C) 15/17 (D) 17/15

36. The point which divides the line segment joining the points (7, -6) and (3,4) in the ratio 1:2 internally lies in
the
(A) I quadrant (B) II quadrant (C) III quadrant (D) IV quadrant

37. If the zeroes of the quadratic polynomial x2 + ( m+1 )x + n are 2 and -3 , then
(A) m = -7, n = - 1 (B) m = 5 , n = - 1 (C) m = - 2 , n = - 6 (D) m = 0, n = - 6

38. If the digit in the unit’s place of a two digit number is 3x and the digit in the ten’s place is y then the two
digit number formed after interchanging the digits is
(A) 3xy (B) 3x + y (C) 3x + 10y (D) 30x +y

39. If sinA – cosA = 0 then the value of sin4A - sin2A is
(A) -3/ 4 (B ) - 1/4 (C) 1/4 (D) 3/4

40. If the radius of a wheel is 0.63m, then the distance covered by the wheel in 400 revolutions is
(Take π = 22/7)
(A)142.56m (B) 1584m (C) 14256m (D) 15840m

***********************************************************************************

ANSWER KEY

GOA BOARD OF SECONDARY AND HIGHER SECONDARY EDUCATION
FIRST TERM EXAMINATION MODEL PAPER (2022-2023)

STD X SUBJECT : MATHEMATICS - LEVEL 1 ( ENGLISH )

Q No Answer
1. D 23 x 3 x 7 2
2. C 8
3. A 18°
4. D (2, 1)
5. D 5/2
6. A 1/4
7. A 2√10 cm
8. A 𝜃
180
x ∏r
9. C 3 + 2√3
√3

10. C 6x + 4y +1 = 0
11. D √3/2
12. C 24units
13. A -15
14. A 13/98
15. C 34cm
16. B 73.5cm2
17. C 27/72
18. D x= -1, y= -2
19. A - 81/4
20. C 2:3
21. D 2x2 +18x + 40
22. A 1/12
23. B 50
24. D 138.6 cm2
25. D 360
26. C 15
27. A sinA
28. A -8
29. A -17
30. B -1
31. C 1
32. C 42
33. C 30
34. D x + y - 10
35. A 8/15
36. D IV quadrant
37. D m = 0, n = - 6
38. D 30x +y

39. B -1/4
40. B 1584m

GOA BOARD OF SECONDARY AND HIGHER SECONDARY EDUCATION
FIRST TERM EXAMINATION MODEL PAPER (2022 – 2023)
DESIGN OF THE QUESTION PAPER

STD : X MARKS : 40
SUBJECT: MATHEMATICS – LEVEL 2 (ENGLISH) TIME: 1 HOUR 45 MINUTES
______________________________________________________________________________________________

The weightage or the distribution of marks over different dimensions of the question paper shall be as follows:

1. Weightage to Learning Outcomes:
Sr. No. Learning Outcomes Marks Percentage of Marks
1. Knowledge 08 20 %
2. Understanding 24 60 %
3. Application 08 20 %
Total 40 100 %

2. Weightage to Content/Subject Units
Sr. No. Units Marks
1. Real Numbers 05
2. Polynomials 05
3. Pair of Linear Equations in Two Variables 07
4. Introduction to Trigonometry 07
5. Coordinate Geometry 05
6. Circles 03
7. Areas Related to Circles 05
8. Probability 03
Total 40

3. Weightage to Difficulty level of questions:
Sr. No. Estimated difficulty level of questions Percentage
1. Easy 20 %
2. Average 60 %
3. Difficult 20 %
Total 100 %

4. Number of Questions:
There will be 40 questions of 1 mark each

GOA BOARD OF SECONDARY AND HIGHER SECONDARY EDUCATION
FIRST TERM EXAMINATION MODEL PAPER (2022 – 2023)
BLUE PRINT

STD : X MARKS : 40
SUBJECT: MATHEMATICS – LEVEL 2 (ENGLISH) TIME: 1 HOUR 45 MINUTEs

Type of Questions : VSA type

S. No. Content/Unit Objectives Total
Knowledge Understanding Application
1. Real Numbers 2(1) 5(5)
15(1)
24(1)
29(1)
38(1)

2. Polynomials 1(1) 6(1) 5(5)
12(1) 17(1)
34(1)

3. Pair of Linear Equations in Two 4(1) 7(7)
Variables 8(1)
11(1)
20(1)
27(1)
32(1)
36(1)

4. Introduction to Trigonometry 3(1) 9(1) 7(7)
14(1) 22(1)
18(1) 31(1)
39(1)

5. Coordinate Geometry 7(1) 37(1) 5(5)
16(1)
25(1)
33(1)

6. Circles 28(1) 5(1) 19(1) 3(3)

7. Areas Related to Circles 26(1) 13(1) 5(5)
35(1) 23(1)
40(1)

8. Probability 10(1) 3(3)
21(1)
30(1)

Total 8(8) 24(24) 8(8) 40(40)

Note: Figures outside the bracket indicate the question number and figures inside the bracket indicate the
marks.

GOA BOARD OF SECONDARY AND HIGHER SECONDARY EDUCATION

FIRST TERM EXAMINATION MODEL PAPER (2022-2023)

STD: X MARKS: 40
SUBJECT: MATHEMATICS-LEVEL 2 (ENGLISH) TIME: 1 HOUR 45 MINUTES
---------------------------------------------------------------------------------------------------------------------------------------------------------
Instructions:
1. Each question is provided with four alternatives. Choose the correct alternative.
2. Each question carries one mark. There is no negative marking for incorrect choice.
____________________________________________________________________________________________
1. The product of zeroes of the quadratic polynomial 2𝓍² - 5𝓍 + 7 is
−5 −7 5 7
(A) 2 (B) 2 (C) 2 (D) 2

2. The prime factorisation of 504 is
(A) 2² × 3³ × 7 (B) 2³ × 3² × 7 (C) 2³ × 3 × 7² (D) 2² × 3² × 7²

3. If tan (A + 40)° = cot 32°, where A is an acute angle, then the value of 𝐴 is
(A) 18° (B) 50° (C) 58° (D) 72°

4. The solution of the pair of linear equations 4𝓍 + y = 7 and 𝓍 – y = 3 is
(A) 𝓍 = - 2, y = - 1 (B) 𝓍 = - 2, y = 1 (C) 𝓍 = 2, y = - 1 (D) 𝓍 = 2, y = 1

5. From an external point Q, the length of the tangent to a circle is 15 cm. If the radius of the circle is 8 cm,
then the distance of Q from the centre is
(A) √23 cm (B) 7 cm (C) 17 cm (D) √161 cm

6. The quadratic polynomial having the sum and product of its zeroes as -2 and -5 respectively is
(A) y² +2y -5 (B) y² - 2y - 5 (C) y² - 2y + 5 (D) y² + 2y + 5

7 The distance between the points P(1, 4) and Q(4, 0) is
(A) √41 units (B) 3 units (C) 5 units (D) 25 units

8 The equation which has (2, -3) as one of its solution is
(A) 𝓍 – 2y = 7 (B) 3𝓍 + 2y = 12 (C) 5𝓍 + y = 9 (D) 𝓍 + 2y = - 4

9 A In figure, ∆ABC is right angled at B. If AB = 10 cm, then the length of BC is
20√3 10 10
(A) cm (B) cm (C) 10√3 cm (D) cm
3 3 √3
30°
B C
10 A die is thrown once. The probability of getting an odd prime number is
1 1 1 2
(A) 6 (B) 3 (C) 2 (D) 3

11 The value of ‘k’ for which the pair of linear equations 2𝓍 + 3y = 4 and 4𝓍 - ky = 8 has infinitely many
solutions is
−8 8
(A) -6 (B) 3 (C) 3 (D) 6

12 The degree of the polynomial (2p² - 5) (3 – 4p³) is
(A) 2 (B) 3 (C) 5 (D) 6

13. In the given figure, a circle is inscribed in a trapezium of height 14 cm. If the

lengths of the parallel sides of the trapezium are 10 cm and 20 cm, then the
area of the shaded region is (Take 𝜋 = 22/7)
(A) 56 cm² (B) 188 cm² (C) 266 cm² (D) 364 cm²
14. If cos A = 3, then sec²A is
5
4 9 5 25
(A) (B) (C) (D)
5 25 3 9

15 The HCF of 84 and 108 is
(A) 4 (B) 12 (C) 756 (D) 9072

16 The area of ∆ABC with vertices A(-4, 2), B(2, 2) and C(0, -4) is
(A) 6 sq. units (B) 12 sq. units (C) 18 sq. units (D) 20 sq. units

17 The polynomial which when divided by (𝓍 + 1) gives (𝓍² - 2) as the quotient and 5 as the remainder is
(A) 𝓍³ + 𝓍² - 2𝓍 + 3 (B) 𝓍³ + 2𝓍² - 3𝓍 + 5 (C) 𝓍³ - 𝓍² - 𝓍 + 2 (D) 𝓍³ - 𝓍² - 3𝓍 + 6

18 The simplified form of √1 − 𝑐𝑜𝑠²𝐴 is
(A) sin²A (B) cos²A (C) cos A (D) sin A

19 From an external point T, TP and TQ are two tangent segments to a circle at P and Q respectively. If O is
the centre of the circle and ∠POQ = 130°, then ∠PTO is
(A) 25° (B) 50° (C) 65° (D) 130°

20 The pair of linear equations 4𝓍 – 9y = 10 and 3𝓍 – 6y = 5 represents
(A) intersecting lines (B) parallel lines (C) coincident lines (D) skew lines

21 The probability of getting a red king from a well shuffled deck of 52 playing cards is
1 1 3 3
(A) 52 (B) 26 (C) 26 (D) 13

22 If ∆ ABC is right angled at C, then the value of cosec (A + B) is
(A) 0 (B) 1 (C) 2 (D) not defined

23 Two concentric circles have radii 20 cm and 15 cm. Therefore, the area of the region between the outer
and inner circles is
(A) 175𝜋 cm² (B)625𝜋 cm² (C) 35𝜋 cm² (D) 5𝜋 cm²

24 The rational number having a terminating decimal expansion is
8 5 23 19
(A) 15 (B) 7 (C) 20 (D) 12

25 If the point P(𝓍, y) is equidistant from A(5, 1) and B(-1, 5), then
(A)3𝓍 = 2y (B) 𝓍 = 5y (C) 2𝓍 = 3y (D) 2𝓍 = y

26 The area covered by the minute hand of a clock of length 6 cm in 20 minutes is
(A) 2𝜋 (B) 6𝜋 (C) 10𝜋 (D) 12𝜋

27 If (-1, 3) is the solution of the equation 3𝓍 – ky = 9, then the value of k is
(A) -2 (B) -4 (C) 2 (D) 4

28 The length of the longest chord of a circle of radius 5.4 cm is
(A) 1.8 cm (B) 2.7 cm (C) 5.4 cm (D) 10.8 cm

29 If the product of two numbers is 756 and their HCF is 6, then their LCM is
(A) 14 (B) 21 (C) 108 (D) 126

30 If a ball is drawn at random from a bag containing 7 red, 5 blue and 8 yellow balls, then the probability
of getting a ball which is not yellow is
2 3 3 13
(A) 5 (B) 5 (C) 4 (D) 20

31 The value of the trigonometric expression sin²20° + sin²70° - cosec²45° is
(A) -1 (B) 0 (C) ½ (D) 1

32 If 5𝓍 + 7y = 15 and 7𝓍 + 5y = 21, then the value of 𝓍 + y is
(A) -2 (B) -3 (C) 2 (D) 3

33 The y-coordinate of the point which divides the line segment joining the points P(6, 4) and Q(2, -8) in
the ratio 1 : 3 internally is
(A) -5 (B) 1 (C) 3 (D) 5

34 On dividing the polynomial (𝓍³ + 3𝓍² - 4𝓍 – 12) by (𝓍 – 2), the quotient is
(A) 𝓍² + 𝓍 – 6 (B) 𝓍² + 5𝓍 – 6 (C) 𝓍² + 5 𝓍 + 6 (D) 𝓍² + 𝓍 + 6

35 In a circle of radius 7 cm, if an arc subtends an angle of 90° at the centre, then the length of the arc
22
is (Take 𝜋 = 7 )
(A) 11 cm (B) 22 cm (C) 38.5 cm (D) 77 cm

36 The pair of linear equations k𝓍 + 6y = 7 and 2𝓍 – 3y = 8 has a unique solution for all the values of ‘k’
other than
(A) k = -4 (B) k = -9 (C) k = 4 (D) k = 9

37 If A(1, 1), B(-1, 2), C(2, 5) and D(a, 4) are the vertices of a parallelogram ABCD, then the value of ‘a’ is
(A) -4 (B) -3 (C) 3 D) 4

38 Which of the following real numbers is irrational?
(A) √49 (B) 2 + √6 (C) √3 × 4√3 (D) (5 - √7)(5 + √7)

39 If tan 3𝜃 = sin45° cos 45° + sin 30°, then the value of 𝜃 is
(A) 15° (B) 20° (C) 30° (D) 45°

40 The area of a circle that can be inscribed in a square of side 8 cm is
(A) 4𝜋 cm² (B) 8𝜋 cm² (C) 16𝜋 cm² (D) 64𝜋 cm²

8 cm

GOA BOARD OF SECONDARY AND HIGHER SECONDARY EDUCATION
FIRST TERM EXAMINATION MODEL PAPER (2022 – 2023)
ANSWER KEY

STD: X MARKS: 40
SUBJECT: MATHEMATICS – LEVEL 2 (ENGLISH) TIME : 1 HOUR 45 MINUTES
______________________________________________________________________________________________

Q. No Answer Q. No Answer
7 1
1 D 21 B
2 26
2 B 2³ x 3² x 7 22 B 1
3 A 18° 23 A 175 𝜋 cm²
23
4 C x = 2, y = - 1 24 C
20
5 C 17 cm 25 A 3x = 2y
6 A y² + 2y – 5 26 D 12𝜋 cm²
7 C 5 units 27 B -4
8 D x + 2y = -4 28 D 10.8 cm
9 C 10√3 cm 29 D 126
1
10 B 30 B 3/5
3
11 A -6 31 A -1
12 C 5 32 D 3
13 A 56 cm² 33 B 1
14 D 25 34 C x² + 5x + 6
9
15 B 12 35 A 11 cm
16 C 18 sq. units 36 A K = -4
17 A x³ + x² - 2x + 3 37 D 4
18 D sin A 38 B 2 + √6
19 A 25° 39 A 15°
20 A Intersecting 40 C 16𝜋 cm²

***********************************************************************************************

GOA BOARD OF SECONDARY AND HIGHER SECONDARY EDUCATION
DESIGN OF THE SSCE QUESTION PAPER ( FOR ACADEMIC YEAR 2022 – 2023 )
SECOND TERM EXAMINATION
CLASS : X
MATHEMATICS ( E ) LEVEL – 1
TIME : 1 HOUR 45 MINUTES MARKS : 40

1. Weightage of Learning Outcomes

Sr. Learning Outcomes Marks Percentage of
No. Marks
1 Knowledge 8 20%
2 Understanding 16/13* 40%
3 Application 10 25%
4 Skill 6/9* 15%
Total 40 100%

2. Weightage to Content /Subject Units
Sr. No. Units Marks
1 Quadratic Equations 8
2 Arithmetic Progression 5
3 Statistics 7
4 Triangles 5
5 Some Applications of Trigonometry 3
6 Constructions 6
7 Surface Areas and Volumes 6
Total 40

3. Weightage to Forms of Questions
Sr. Forms of Questions Marks Number Total
No. for each of Marks
question questions
1 Long Answer Type (LA) 04 02 08
2 Short Answer Type (SA-II) 03 07 21
3 Short Answer Type (SA-I) 02 04 08
4 Very Short Answer Type (VSA) 01 03 03

Total 40

4. Weightage to Difficulty Level of Questions
Sr. No. Estimated difficulty Percentage
level of questions
1 Easy 20%
2 Average 60%
3 Difficult 20%
Total 100%

5. Number of Main Questions and Scheme of options:

There will be 04 main questions of 10 marks each. There is no overall
choice. However internal choice will be provided in one question of 3
marks

GOA BOARD OF SECONDARY AND HIGHER SECONDARY EDUCATION
BLUE PRINT ( SECOND TERMINAL EXAMINATION )
STD X : MATHEMATICS (E) – LEVEL 1
Duration : 1hour 45 Minutes Maximum Marks : 40
Sr. Objectives Understanding Application Skill Total
No knowledge
Forms of VSA SA-I SA-II LA VSA SA-I SA-II LA VS SA-I SA-II LA VSA SA-I SA-II LA
questions A
Content /Marks 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4
1 Statistics Q2A(1) Q1B(2) Q2D(4) 3(7)
2 Arithmetic Q2B(2) Q2C(3) 2(5)
Progression
3 Quadratic Q1A(1) Q1C(3) Q1D(4) 3(8)
Equations
4 Constructions Q3B(3) 2(6)
Q3C(3)
5 Some Applications Q3D(3) 1(3)
of Trigonometry
6 Surface Areas and Q4A(2) Q3A(1) Q4D(3) 3(6)
Volumes
7 Triangles Q4B(2) Q4C(3) 2(5)
Total 2 (2) 3(6) 1(1) 1(2) 3(9) 1(4) 2(6) 1(4) 2(6) 16(40)

GOA BOARD OF SECONDARY AND HIGHER SECONDARY EDUCATION
SECOND TERM EXAMINATION
MODEL PAPER ( 2022-2023)

Class : X Max Marks : 40
Subject :Mathematics(E) Level – 1 Time :1 hour 45 minutes

INSTRUCTIONS : I)Answer each main question on a fresh page.

II) All questions are compulsory

III)The question paper consists of 4 questions each of 10 marks

IV) There is no overall choice however internal choice has been
provided in one question of 3 marks

V) In questions of construction the drawing should be clear
and exactly as per the given measurements. The
construction lines and arcs should also be maintained

VI) Use of calculator and mathematical tables is not permitted.

Q1A) Write the equation x + 1 = 2 in the form of ax2 + bx +c = 0 (1)
x

B) The distribution below shows the weight of 30 employees in a (2)
factory.Calculate the median weight of the employees.
(Write your answer correct upto one decimal place)

Weight in Kg No. of employees
40 – 45 6
45 – 50 7
50 – 55 8
55 – 60 9

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

i) 5x2 + 2x – 39 = 0 ( By Factorization method )
ii) 3x2 + 6x + 1 = 0 (By completing the square method )

D) A number consists of 2 digits . The digit in ten’s place is greater (4)
than the digit in units place by 4. The sum of the squares of the
digits is 15 less than the original number. Find the original
number .

Q2A) Write the modal class of the following frequency distribution table (1)
Class Interval 0 – 20 20 - 40 40 - 60 60 - 80
Frequency 6 10 18 4

B) For the AP: 3, 15,27 , 39,------------- (2)
Find :
i) The 21st term.
ii) Which term of the AP is 363 ?

C) A contract on construction job specifies a penalty for delay of completion
beyond a certain date as follows: Rs 200 for the first day and then Rs 50 (3)
more on every succeeding day . Find the total amount the contractor has
to pay as penalty for delaying the job by 30 days.

D) The following table shows the marks scored by 60 students in a test
Marks No. of Class Marks Deviation fidi (4)
Scored students xi d i = xi - a
(C.I) (fi)
0 – 20 10 _________ ________ _________
20 – 40 12 _________ ________ ________
40 -60 8 _________ ________ ________
60 – 80 14 _________ ________ ________
80 – 100 9 _________ ________ ________
100 - 120 7 _________ ________ ________
Total Σfi = 60 Σfi .di=_____

Taking the class mark denoted by ‘a’of class interval 60 – 80 as the
assumed mean , rewrite and complete the table and find the mean of the
marks obtained by the assumed mean method.

Q3A) The inner radii of the 2 circular ends of a toy bucket are 10cm and 6cm (1)
respectively. If the height of the bucket is 15cm then find the capacity of
the bucket (Take 𝜋 = 22/7)

B) Draw a line segment AB = 7.5cm. Taking A as the centre and radius 3cm (3)
draw a circle and taking B as centre draw another circle of radius 2.5 cm
.Using a pair of compasses and ruler construct tangents to each circle
from the centre of the other circle.

C) Using a pair of compasses and ruler , Construct ΔABC with BC = 6cm , (3)
AB = 5.5cm and Լ ABC = 60o . Then construct ΔA’BC’ whose sides are 5 of
3
the corresponding sides of ΔABC.

D) A tree ‘AB’ casts a shadow when the sun is 30o above the horizon . But (3)
when it rises 45o above the horizon , the length of the shadow reduces by
20 metres. Find the height of the tree correct upto one place of decimal .
( Take √3 = 1.73)
A

45o 30o

B C
D 20m

Q4A) A wooden article was made by scooping out a hemisphere from one end (2)
of a cylinder . If the height of the cylinder is 10 cm and its base is of
diameter 7cm then find :

i)The Curved surface area of the hemispherical part
(Do not substitute for 𝜋)

ii)TheCurved surface area of the cylindrical part
(Do not substitute for 𝜋)

B) In ΔPQR , Լ PQR = 90o and Line segment MN ‖ QR as shown in the figure (2)
.If PM = 9cm , MQ = 2cm and ar(ΔPMN) = 81sqcm then Find ar(ΔPQR)

P

M N

Q R

C) In ΔABC , P and Q are points on the sides AB and AC respectively such (3)
that PQ ‖ BC.
Prove that the median AD drawn from A to BC bisects PQ

A

P E Q

D C
B

D) A container shaped like a right circular cylinder having diameter 12cm (3)
and height 15cm is full of ice cream . This ice cream is to be filled into
cones of height 12cm and diameter 6cm, having a hemispherical shape on
the top . Find the number of such cones which can be filled with the

ice cream.

SSCE Second Term Examination
Mathematics ( E ) – Level 1 ( 2022 – 2023 )
Model paper : Answers and Marking Scheme

Note : Any Alternative method unless otherwise specified should be
considered for full marks.

Q1A) x+1=2
x
2
x +1=2
x
2
x + 1 = 2x ½
x2 – 2x + 1 = 0 ½

B) cf
6 ½
13
21
30

n = 30
n = 30 = 15
2 2
Median class = 50 – 55
l = 50
cf = 13
f=8
h=5

Median = l + n – cf
2 xh
f

= 50 + 15 – 13 x 5
8 ½+½
= 50 + 2 x 5
8
= 50 + 10
8
= 50 + 1.25
= 51.25
= 51.3 kg ½

C)i) 5x2 + 2x – 39 = 0
5x2 + 15x – 13x – 39 = 0 ½
5x ( x + 3 ) – 13( x + 3 ) = 0 ½
( x + 3 ) ( 5x – 13 ) = 0 ½
Either x + 3 = 0 or 5x – 13 = 0 ½
x = -3 or x = 13 ½
5
∴roots are -3 and 13 ½
5

ii) 3x2 + 6x + 1 = 0
3x2 + 6x = -1 ½
Dividing by 3 on both the sides
x2 + 2x = -1
3
Last term = 1 x 2 2 =1 ½
2

Adding 1 on both the sides of the equation

x2 + 2x + 1 = -1 + 1 ½
3
2
(x + 1) = -1 + 3
3
2
(x + 1) = 2
3
(x+ 1) =± √2 ½
√3
Either x + 1 = √2 or x + 1 = - √2
√3 √3 ½
x = √2 - 1 or x = - √2 - 1
√3 √3
∴roots are √2 - 1 or x = - √2 - 1 ½
√3 √3

D) Let the digit in the units place be x ½
∴ the digit in the tens place is x + 4
Original number = 10 ( x + 4 ) + x
= 10x + 40 + x
= 11x + 40 ½
2 2
x + ( x + 4 ) = 11x + 40 – 15 ½
x2 +x2 + 8x + 16 = 11x + 25
2x2 + 8x – 11x + 16 – 25 = 0
2x2 – 3x – 9 = 0 ½
2x2 – 6x + 3x – 9 = 0
2x( x – 3 ) + 3 ( x – 3 ) = 0

( x – 3 ) ( 2x + 3 ) = 0 ½
Either x – 3 = 0 or 2x – 3 = 0

x = 3 or x = - 3 ½
2
x = - 3/2 is discarded ½
∴x=3
∴ original number = 11 x 3 + 40
= 33 + 40
= 73 ½

Q2A) 40 – 60 1

B)i) an = a + (n – 1 ) d
a21 = 3 + ( 21 – 1 ) 12 ½
= 3 + 20 x 12
= 3 + 240
= 243 ½

ii) an = 363
a=3
d = 15 – 3 = 12
an = a + (n – 1 ) d
363 = 3 + ( n – 1 ) 12 ½
363 – 3 = (n – 1 ) 12
360 = ( n – 1 ) 12
30
360 = n – 1
12
30 = n – 1
30 + 1 = n
n = 31 ½

C) AP : 200 , 250 , 300 , 350 ,------------ ½
a = 200
d = 250 – 200 = 50
n = 30
Sn = n 2a+ ( n – 1 ) d
2
15
= 30 2 x 200 + ( 30 – 1 ) 50 ½+½
2

= 15 400 + (29 x 50) ½

= 15 ( 400 + 1450) ½
= 15 x 1850
= 27750

∴ the contractor has to pay Rs 27,750 as penalty for ½
delaying the job by 30 days.

D) Marks Number Class Deviation fidi
scored of Marks di = x i – a
(C.I ) students xi a = 70
(fi)
0 – 20 10 10 -60 -600
20 – 40 12 30 1/2 -40 1/2 -480 1/2

40 – 60 8 50 -20 -160
60 – 80 14 70 0 0 3
80 – 100 9 90 1/2 20 1/2 180 1/2

100 – 7 110 40 280
120
Total Σfi = 60 Σfidi = -780

Mean = a + Σfidi
Σfi
13 ½
= 70 – 780
60
= 70 – 13 ½
= 57

Q3A) r1 = 10 cm
r2 = 6cm
h = 15cm
V( Frustum of the cone ) = 1 𝜋h ( r12 + r22 +r1r2)
3
5
= 1 x 22 x 15 ( 102 + 62 + 10 x 6 ) ½
3 7
28
= 22 x 5 x 196
7

= 3080 cm3 ½

B) To draw line segment AB = 7.5cm ½
To draw a circle with centre A and radius 3cm ½
To draw a circle with centre B and radius 2.5cm ½
To bisect line segment AB ½
To mark points on the two circles ½
To draw tangent segments to each circle from the centre of
both the circles . ½

C) To construct ΔABC with the given data 1
To draw a ray making an acute angle with side BC ½
To locate 5 points on the ray using a pair of compasses ½
To join B3C and construct B5C’ ‖ B3C ½
To draw C’A’ ‖ CA ½
ΔABC is the required triangle

D) Let the height of the tree by ‘x’ metres
and BD = y metres
In ΔABD

Tan 45o = x
Y
1=x
y
x=y ½

In ΔABC
Tan 30o = x
Y + 20 ½

1= x
√3 x + 20 ½

x + 20 = x√3
20 = x√3 - x
20 = x ( √3 - 1 )
20 =x
√3 - 1
½
20 x √3 + 1 = x
√3 - 1 √3 + 1
20(√3 + 1 ) =x
(√3 - 1 ) (√3 + 1 )
20 x ( 1.73 + 1 ) = x
3–1
10 ( 2.73 ) = x
27.3 = x ½

∴The height of the tree is 27.3 metres ½

Q4A)i) C.S.A (hemispherical part ) = 2𝜋r2
= 2 x 𝜋 x ( 3.5 )2 ½
= 24.5 𝜋 sq. cm ½

ii) C.S.A (cylindrical part ) = 2𝜋rh

=2 x 𝜋 𝑥 3.5 𝑥 10 ½
= 70𝜋sq.cm ½

B) ԼPMN = ԼPQR Corresponding angles ½
ԼPNM = ԼPRQ

ΔPMN ~ ΔPQR By AA similarity test ½

ar (ΔPMN ) = PM 2
ar ( ΔPQR ) PQ

2
81 = 9
ar ( ΔPQR ) 11 ½

ar ( ΔPQR ) = 81 x 11 x 11
9x 9
= 121 sq .cm ½

C) In ΔAPE and ΔABD ½
ԼPAE = ԼBAD --------- common angle
ԼAPE = ԼABD --------- corresponding angles
ΔAPE ~ ΔABD ------- By AA similarity test
AE = PE ------------- corresponding sides of similar
AD BD triangles are proportional ½

Similarly ΔAEQ ~ ΔADC ------------ By AA similarity test

AE = EQ ------------- corresponding sides of similar
AD DC triangles are proportional ½

∴ PE = EQ ½
BD DC

PE = BD
EQ DC
=1 Since BD = DC ½

∴PE = EQ ½
∴ Median AD drawn from A to BC bisects PQ

D) Let the number of cones to be filled with icecream be ‘n’
Rcylinder = 6cm rcone = 3cm ½
Hcylinder = 15cm hcone = 12cm

V( Cylinder) = n x V( each cone) + V ( each hemisphere)

𝜋R2H =n x 1 𝜋𝑟2h + 2 𝜋r3 ½
3 3

𝜋R2H = n x 1 𝜋𝑟2 ( h + 2r )
3

6 x 6 x 15 = n x 1 x 3 x 3 ( 12 + 2 x 3 ) ½+½
3

6 x 6 x 15 = n x 3 x 18

2 1 5
n = 6 x 6 x 15
3 x 18 ½
3
1

n = 10

∴ the number of cones to be filled with icecream are 10 ½

GOA BOARD OF SECONDARY AND HIGHER SECONDARY EDUCATION

DESIGN OF THE SSCE QUESTION PAPER ( FOR ACADEMIC YEAR 2022 – 2023 )
SECOND TERMINAL EXAMINATION
MATHEMATICS ( E ) - LEVEL 2
TIME : 1HR 45 MINUTES MAX MARKS : 40

1. Weightage to Learning Outcomes

Sr. No Learning Outcomes Marks Percentage of Marks
1 Knowledge 08 20%
2 Understanding 18 45%
3 Application 08 20%
4 Skill 06 15%
Total 40 100%

2. Weightage to Content / Subject Units
Sr. No Units Marks
1 Quadratic Equations 8
2 Arithmetic Progressions 5
3 Statistics 7
4 Triangles 5
5 Some Applications of Trigonometry 3
6 Constructions 6
7 Surface Areas and Volumes 6
Total 40

3. Weightage to Forms Of Questions
Sr. No Forms of Questions Marks for Number of Total Marks
each questions
question
1 Long Answer Type (LA) 04 02 08
2 Short Answer Type (SA-II) 03 07 21
3 Short Answer Type (SA-I) 02 04 08
4 Very Short Answer Type (VSA) 01 03 03
Total 16 40

4. Weightage to Difficulty level of questions :

Sr. No Estimated difficulty level of questions Percentage
1 Easy 20%
2 Average 60%
3 Difficult 20%
Total 100%

5. Number of Main Questions :
There will be 04 main questions of 10 marks each.

GOA BOARD OF SECONDARY AND HIGHER SECONDARY EDUCATION
SECOND TERM EXAMINATION
MODEL PAPER ( 2022 – 2023 )
STD : X MAX MARKS : 40
SUBJECT : MATHEMATICS (E) - LEVEL 2 TIME : 1hr 45 minutes

INSTRUCTIONS : i) Answer each main question on a fresh page .
ii) All questions are compulsory.
iii) The question paper consists of four questions , each of 10 marks
iv) There is no overall choice.
v) In questions on constructions , the drawing should be clear
and exactly as per the given measurements. The construction
lines and arcs should be maintained.
vi) Use of calculators and mathematical tables is not permitted.

Q1A) Find the class size of the class intervals 25 – 55 , 55 – 85 , 85 – 115,… (1)

B) Attempt the following : (2)
(i) Write the first four terms of an AP having first term as 19 and
common difference as − 3 .
(ii) State with reason if the given list of numbers is an AP or not.
1 , 3 , 9 , 27 ,…

C) Answer the following questions with reference to the given
Arithmetic Progression : 11 , 15 , 19 , 23 , … (3)
(i) Find the 20𝑡ℎ term of the AP.
(ii) Find the sum of the first 12 terms of the AP.
(iii) Find which term of the AP is 91.

D) The following table shows the donation given by 50 students
towards a Charitable trust.
Donation in Rs No of students Class marks 𝑓𝑖 𝑥𝑖
C.I 𝑓𝑖 𝑥𝑖

0 - 20 5
20 - 40 8
40 - 60 10
60 - 80 12
80 - 100 7

100 - 120 8
∑ 𝑓𝑖 = 50 ∑ 𝑓𝑖 𝑥𝑖 =
Rewrite and complete the table and find the mean donation by using the
Direct Method . (4)

Q2A) Attempt the following : (2)
(i) If the sum and product of the roots of the quadratic equation
are 5 and – 6 respectively , then write the quadratic equation in 𝓍 .
(ii) If one root of the quadratic equation 2𝑥 2 + 𝑚 𝓍 − 15 = 0 is – 5,
then find the value of m .

B) Find the mode of the following frequency distribution table : (2)

Class Interval Frequency
20 - 30 5
30 - 40 12
40 - 50 20
50 - 60 8

C) Find the roots of the Quadratic Equation 5𝑥 2 − 14𝑥 + 8 = 0
by using the “ Factorisation Method .“ (3)

D) Find the roots of the Quadratic Equation 3𝑥 2 − 4𝑥 − 7 = 0
by using the “ Quadratic Formula Method.“ (3)

3A) Find the total surface area of a hemisphere of radius 7cm. (1)
( Do not substitute the value of 𝜋 )

B) Draw a line segment AB of length 7.5cm . Taking A as centre and
radius 3cm , draw a circle. Using a pair of compasses and ruler ,
Construct tangents BP and BQ to the circle .
Measure and state the length of the tangent segments. (3)

C) Construct ∆ PQR with sides PQ = 6.5cm , QR = 7cm and ∠ PQR = 60° .
Using a pair of compasses and ruler , construct ∆ P’QR’ similar to
3
∆ PQR whose sides are of the corresponding sides of ∆ PQR . (3)
4

D) From the top ‘A’ of a tower ‘ AB ‘ a man finds that the angle of
depression of a car at point ‘C’ on the ground to be 30° . If the car
is at a distance of 30m from the foot of the tower ,
then find the height of the tower. A (3)
( Take √3 = 1.73 ) 30 °

B 30° C
30m

4A) Two identical solid cubes of side 2cm are joined end to end .
Find the volume of the resulting cuboid. (1)

B) D and E are points on the sides AB and AC respectively of ∆𝐴𝐵𝐶 ,
such that ∠ ADE = ∠ ABC . If AD = 1.5cm , AB = 6cm , AE = 3cm and
DE = 3.5cm , then
find i) EC
ii) BC (2)
A

D 3.5cm
E

B C

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

E F R
Q

Given : In ∆ DEF , 𝐷𝐸 2 + 𝐸𝐹 2 = 𝐷𝐹 2
∆ PQR is constructed such that PQ = DE , QR = EF and ∠ Q = 900

Prove that : ∆ DEF is a right angled triangle. (3)

D) Attempt the following :
(i ) In the figure given below an open steel bucket is in the shape
of a frustum of a right circular cone of height 15cm . If the radii of its
lower and upper ends are 12cm and 20cm respectively , then
find

a) The slant height of the bucket

b) The curved surface area of the bucket (2)
( Do not substitute the value of 𝜋 )

20cm
m

15cm
m

12cm
m

ii) A solid metallic cylinder of base diameter 6cm and height 32cm is melted
to form 8 solid spheres of the same size.
Find the radius of each sphere. (2)

xxxxxxxxxxxxxxx The End xxxxxxxxxxxxxxxxxx

SSC SECOND TERM EXAMINATION ( MODEL PAPER) Marks
MATHEMATICS (E) - Level 2 ( S – 2024 )
MODEL ANSWERS AND MARKING SCHEME

NOTE : Any alternative method unless otherwise specified
should be considered for full credit.

1 A 30 1

B (i) a = 19 , d = − 3
∴ The first four terms of the AP are
a = 19 ½
a + d = 19 + (−3) = 19 – 3 = 16
a + 2d = 19 + 2(−3) = 19 – 6 = 13 ½
a + 3d = 19 + 3(−3) = 19 – 9 = 10 1

(ii) In the given list of numbers : 1 , 3 , 9 , 27 ,…
𝑎2 − 𝑎1 = 3 – 1 = 2
𝑎3 − 𝑎 2 = 9 – 3 = 6 ½
𝑎4 − 𝑎3 = 27 – 9 = 18
Since the given list of numbers does not have a ½
common difference it is not an AP 1

C (i) a = 11 , d = 15 – 11 = 4
𝑎20 = a + 19d
= 11 + 19(4) ½
= 11 + 76
= 87 ½
1
(ii) a = 11 , d = 4 , n = 12 , 𝑆𝑛 = ?
𝑆𝑛 = n 2a + (n - 1) d
2
= 12 2(11) + ( 12 – 1 )4 ½
2
= 6 ( 22 + 44)
= 6 ( 66 ) ½
= 396 1

(iii) a = 11 , d = 4 , 𝑎𝑛 = 91 , n = ?
𝑎𝑛 = a + (n – 1) d
91 = 11 + ( n – 1 ) 4 ½
91 – 11 = ( n – 1 ) 4
80/4 = (n–1)

20 + 1 = n
21 = n
𝑠𝑡
The 21 term of the AP is 91 ½
1

D Donation in Rs No of students Class marks Marks
C.I 𝑓𝑖 𝑥𝑖 𝑓𝑖 𝑥𝑖
0 – 20 5 10 ½ 50 ½
20 – 40 8 30 240
40 – 60 10 50 ½ 500 ½
60 – 80 12 70 840
80 – 100 7 90 ½ 630 ½
100 – 120 8 110 880
Σ𝑓𝑖 = 50 Σ𝑓𝑖 𝑥𝑖 = 3140
Note: Any 2 correct values in column 𝑥𝑖 and 𝑓𝑖 𝑥𝑖 should be given
Mean of donation = Σ𝑓𝑖 𝑥𝑖 ½ mark
Σ𝑓𝑖
= 3140 ½
50
= 62.8 ½
4
2 A (i) The quadratic equation in 𝓍 is
𝑥 2 – ( sum of the roots )𝓍 + product of the roots = 0
𝑥 2 – 5𝓍 + (– 6 ) = 0 ½
2
𝑥 – 5𝓍 – 6 =0 ½
ALTERNATIVE METHOD 1
Sum of the roots = –b = 5
a ½
Product of the roots = c = –6
a
∴ a= 1,b=–5 , c=–6
The quadratic equation in 𝓍 is 𝑥 2 – 5𝓍 – 6 = 0 ½
1
2
(ii) 2𝑥 + m𝓍 – 15 = 0
2 (– 5)2 + m (– 5) – 15 = 0 ½
50 – 5m – 15 = 0
35 – 5m =0
– 5𝑚 = – 35
m = 7 ½
1
B l = 40 , 𝑓1 = 20 , 𝑓0 = 12 , 𝑓2 = 8 (½* mk if all values are correct)
𝑓1 − 𝑓0
Mode = l + 2𝑓 − 𝑓 − 𝑓 xh
1 0 2

= 40 + 20 − 12 x 10 ½ + ½*
2(20)−12 − 8
= 40 + 8 x 10 ½
20

= 40 + 4
= 44 ½
2

C 5𝑥 2 – 14𝓍 + 8 = 0 Marks
5𝑥 2 – 10𝓍 – 4𝓍 + 8 = 0 ½
5𝓍 ( 𝓍 – 2 ) – 4 ( 𝓍 – 2 ) = 0 ½
( 5𝓍– 4 ) ( 𝓍 – 2 ) =0 ½
Either 5𝓍– 4 = 0 OR 𝓍 – 2 = 0 ½
𝓍 =4 𝓍 = 2 ½
5
∴ The roots are 4 and 2 ½
5 3

D 3𝑥 2 – 4𝓍 − 7 = 0
a=3 , b=–4,c= –7
D = 𝑏 2 − 4ac
= − 42 − 4 (3) (– 7)
= 16 + 84
= 100 ½
𝓍 = – b + √𝑏 2 − 4ac
2a
𝓍 = – (−4) + √100 ½
2(3)

𝓍 = 4 + 10 ½
6

𝐸𝑖𝑡ℎ𝑒𝑟 𝑥 = 4 + 10 OR 𝓍 = 4 − 10 ½
6 6

𝑥 = 14 𝓍 = −6
6 6

𝓍= 7 𝓍 = −1 ½
3

∴ The roots are 7 and − 1 ½
3 3

3 A Total surface area of hemisphere = 3Π𝑟 2
= 3Π x 7 x 7 ½

= 147 Π𝑐𝑚2 ½
1

Marks

B To draw line segment AB = 7.5cm ½
To draw a circle with centre A and radius 3cm ½
To bisect line segment AB ½
To mark the points P and Q on the circle ½
To draw the tangent segments from the point B ½
To measure and write the length of the tangent segments
BP = BQ = 6.9 + 0.1cm ½
3

Marks

C To construct ∆ PQR with the given data 1
To draw a ray making an acute angle with side QR ½
To locate four points on the ray using a pair of compasses ½
To join 𝑄3 𝑅 ′ ∥ 𝑄4 𝑅 and 𝑅 ′ 𝑃′ ∥ RP ½+ ½
∆ P’QR’ is the required triangle 3

D In right ∆ ABC
tan 30° = AB
BC
1 = AB ½+ ½
√3 30
30 = AB ½
√3
30√3 = AB
3
10√3 = AB ½
10 x 1.73 = AB ½
17.3m = AB ½

The height of the tower is 17.3 m
3

4 A l = 4cm , b = 2cm , h = 2cm Marks
Volume of the cuboid = l x b x h
=4x 2 x 2 ½
= 16𝑐𝑚3 ½
B 1
∆ ADE ~ ∆ ABC By AA Similarity Criterion
AD = AE ½
AB AC
1.5 = 3
6 AC

AC = 3 x 6
1.5

AC = 12cm
EC = AC – AE = 12 – 3 = 9cm ½

AD = DE
AB BC

1.5 = 3.5 ½
6 BC
BC = 3.5 x 6
1.5

BC = 14cm ½
2

C In ∆ PQR
1) 𝑃𝑄 2 + 𝑄𝑅 2 = 𝑃𝑅 2 By Pythagoras theorem ½
2 2 2
2) 𝐷𝐸 + 𝐸𝐹 = 𝑃𝑅 Since PQ = DE , QR = EF ½
2 2 2
3) 𝐷𝐸 + 𝐸𝐹 = 𝐷𝐹 Given
2 2
4) 𝑃𝑅 = 𝐷𝐹 Statements (2) & (3)
5) PR = DF Statement (4) ½
6) ∆ PQR ≅ ∆ DEF SSS Congruence rule ½
7) ∠ Q = ∠ E C.P.C.T ½
°
8) But ∠ Q = 90 Given
°
9) ∴ ∠ E = 90 From ( 7 ) & ( 8 )
10) ∴ ∆ DEF is a right angled triangle ½
3

D (i)(a) h = 15cm , Upper radius (R) = 20cm , Lower radius (r) = 12cm
Slant height of the bucket = √ℎ2 + ( 𝑅 − 𝑟)2
= √152 + ( 20 − 12)2 ½
= √152 + 82
= √225 + 64
= √289
= 17cm ½

(b) C.S.A of the bucket = Π(R+r)l
= Π ( 20+ 12 ) 17 ½
= Π x 32 x 17
= 544 Π𝑐𝑚2 ½
2
(ii) Cylinder : r = d = 6 = 3cm
2 2
h = 32 cm

Sphere : radius (R) = ?

Volume of 8 spheres = Volume of cylinder
8 x 4 Π 𝑅3 = Π𝑟 2 ℎ
3

8 x 4 x 𝑅3 = 3 x 3 x 32 ½ +½
3

𝑅3 = 3 x 3 x 32 x 3
8x4

𝑅3 = 27 ½
R = 3cm ½
The radius of each sphere is 3cm
2