Class 10 CBSE Maths Basic Boards Paper 2023

Maximum Marks: 80

Time Allowed: Three hours

(Candidates are allowed additional 15 minutes for only reading the paper.
They must NOT start writing during this time).

This paper is divided into four sections – A, B, C and D.

Answer all questions.

Section A consists of one question having sub-parts of one mark each.

Section B consists of seven questions of two marks each.

Section C consists of nine questions of three marks each, and

Section D consists of three questions of five marks each

. Internal choices have been provided in two questions each in Section B, Section C and Section D.

The intended marks for questions are given in brackets [ ].

All working, including rough work, should be done on the same sheet as and adjacent to the rest of the answer.

Answers to sub parts of the same question must be given in one place only.

A list of useful physical constants is given at the end of this paper.

A simple scientific calculator without a programmable memory may be used for calculations.

Class 10 CBSE Maths Basic Boards Paper 2023

Section-A Section - A consists of 20 questions of 1 mark each.

Question 1

Let E be an event such that P(not E) =1/5, then P(E) is equal to : (a) 1/5 (b) 2/5 (c) 0 (d) 4/5

Question 2

If p (x) = x2 + 5x + 6, then p (-2) 1s: (a) 20 (b) 0 (c) -8 (d) 8

Question 3

The mode of the numbers 2, 3, 3, 4,5, 4,4, 5, 3,4, 2, 6, 7 is: (a) 2 (b) 3 (c) 4 (d) 5

Question 4

How many tangents can be drawn to a circle from a point on it? (a) One (b) Two (c) Infinite (d) Zero

Question 5

A quadratic equation whose one root is 2 and the sum of whose roots is zero, is: (a) x2+4=0 (b) x2-2=0 (c) 4x2-1=0 (d) x2-4=0

Question 6

Which of the following is not a quadratic equation ? (a) 2(x-1)2 =4x2 - 2x + 1 (b) 2x - x2 =x2 + 5 (c) (√2x +√3)2 + x2= 3x2 - 5x (d) (x2 + 2x)2=x4+3+4x3

Question 7

A quadratic polynomial whose sum and product of zeroes are 2 and -1 respectively is : (a) x2+2x+1 (b) x2-2x-1 (c) x2+2x-1 (d) x2 -2x+1

Question 8

(HCF * LCM) for the numbers 30 and 70 is - (a) 2100 (b) 21 (c) 210 (d) 70

Question 9

The length of the arc of a circle of radius 14 cm which subtends an angle of 60° at the centre of the circle is: (a) 44/3 cm (b) 88/3 cm (c) 308/3 cm (d) 616/3 cm

Question 10

If the radius of a semi-circular protractor is 7cm, then its perimeter is: (a) 11 cm (b) 14cm (c) 22cm (d) 36cm

Question 11

The angle of elevation of the top of a 15 m high tower at a point 15√3m away from the base of the tower is : (a) 30° (b) 45° (c) 60° (a) 90°

Question 12

(2/3 sin 0° -4/5 cos0°) is equal to: (a) 2/4 (b) -4/5 (c) 0 (d) -2/15

Question 13

From a well-shuffled deck of 52 cards, a card is drawn at random. What is the probability of getting king of hearts ? (a) 1/52 (b) 1/26 (c) 1/13 (d) 12/13

Question 14

The number (5-3√5+√5) is: (a) an integer (b) rational number (c) an irrational number (d) a whole number

Question 15

If the pair of linear equations x -y = 1, x + ky =5 has a unique solution x = 2, y= 1, then the value of k is : (a) - 2 (b) -3 (c) 3 (d) 4

Question 16

If ▵ABC ~ ▵DEF and ∠A = 47°, ∠E = 83°, then ∠C is equal : (a) 47° (b) 50° (c) 83° (d) 130°

Question 17

The length of the tangent from an external point A to a circle, of radius 3 cm, is 4 cm. The distance of A from the centre of the circle is: (a) 7cm (b) 5cm (c) √7cm (d) 25cm

Question 18

The pair of linear equations x + 2y + 5 = 0 and -3x-6y + 1=0 has: (a) aunique solution (b) exactly two solutions (c) infinitely many solutions (d) no solution

Question 19

Assertion (A) : If one root of the quadratic equation 4x2-10x+(k—4)=0 is reciprocal of the other, then value of k is 8. Reason (R) : Roots of the quadratic equation x2-x + 1 =0 are real.

Question 20

Assertion (A) : A tangent to a circle is perpendicular to the radius through the point of contact. Reason (R) : The lengths of tangents drawn from an external point to a circle are equal.

Section-B Section - B comprises of Very Short Answer (VSA) questions of 2 marks each.

Question 21

(A) Find the discriminant of the quadratic equation 3x2 — 2x + 1/3 = 0 and hence find the nature of its roots. OR (B) Find the roots of the quadratic equation x2 -x-2=0.

Question 22

In the adjoining figure, A, B and C are points on OP, OQ and OR respectively such that AB||PQ and : AC||PR. Show that BC||QR.

Question 23

If sin α=1/2,then find the value of (3 cos α -4 cos3α)

Question 24

(A) Find the coordinates | o “ a which divides the join of A(-1, 7) and B (4, -3) in the ratio 2:3 OR (B) If the points A(2,3),B(-5,6),C(6,7) and D (p,4) are the vertices of a parallelogram ABCD,find the value of p.

Question 25

PA and PB are tangents drawn to the circle with center O as shown in the figure. Prove that ∠APB=2 ∠OAB

Section-C This section comprises short answer (SA) type questions of 3 marks each.

Question 26

Find the area of the sector of a circle of radius 7 cm and of central angle 90°.Also,find the area of corresponding major sector

Question 27

If α, β of the quadratic polynomial x2- 5x+6,form another quadratic polynomial whose zeroes are 1/α,1/β

Question 28

A die is rolled once. Find the probability of getting: (i) an even prime number. (ii) anumber greater than 4, (111) an odd number.

Question 29

Prove that 1+tan2A/1+cot2A=sec2A -1

Question 30

(A) Prove that the lengths of tangents drawn from an external point to a circle are equal. OR (B) Two concentric circles with centre O are of radii 3 cm and 5 cm. Find the length of chord AB of the larger circle which touches the smaller circle at P

Question 31

(A) If we add 1 to the numerator and subtract 1 from the denominator, a fraction reduces to 1. It becomes 1/2 if we only add 1 to the denominator, What is the fraction ? OR (B) For which value of 'k' will the following pair of linear equations have no solution ? 3x + y=1 (2k—1)x+(k-1)y=2k+1

Question 32

(A) Find the sum of first 51 terms of an A.P. whose second and third terms are 14 and 18, respectively, OR (B) The first term of an A.P. is 5,the last term is 45 and the sum is 400.Find the number of terms and the common difference

Question 33

The distribution below gives the weight of 30 students of a class.Find the median weight of the students:

Question 34

Question 35

Section-E This section comprises 3 case study based questions of 4 marks each.

Question 36

Question 37

Question 38

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Section-A

Section - A consists of 20 questions of 1 mark each.

Let E be an event such that P(not E) =1/5, then P(E) is equal to :

(a) 1/5
(b) 2/5
(c) 0
(d) 4/5

If p (x) = x² + 5x + 6, then p (-2) 1s:

(a) 20
(b) 0
(c) -8
(d) 8

The mode of the numbers 2, 3, 3, 4,5, 4,4, 5, 3,4, 2, 6, 7 is:

(a) 2
(b) 3
(c) 4
(d) 5

How many tangents can be drawn to a circle from a point on it?

(a) One
(b) Two
(c) Infinite
(d) Zero

A quadratic equation whose one root is 2 and the sum of whose roots is zero, is:

(a) x²+4=0
(b) x²-2=0
(c) 4x²-1=0
(d) x²-4=0

Which of the following is not a quadratic equation ?

(a) 2(x-1)² =4x² - 2x + 1
(b) 2x - x² =x² + 5
(c) (√2x +√3)² + x²= 3x² - 5x
(d) (x² + 2x)²=x⁴+3+4x³

A quadratic polynomial whose sum and product of zeroes are 2 and -1 respectively is :

(a) x²+2x+1
(b) x²-2x-1
(c) x²+2x-1
(d) x² -2x+1

(HCF * LCM) for the numbers 30 and 70 is -

(a) 2100
(b) 21
(c) 210
(d) 70

The length of the arc of a circle of radius 14 cm which subtends an angle of 60° at the centre of the circle is:

(a) 44/3 cm
(b) 88/3 cm
(c) 308/3 cm
(d) 616/3 cm

If the radius of a semi-circular protractor is 7cm, then its perimeter is:

(a) 11 cm
(b) 14cm
(c) 22cm
(d) 36cm

The angle of elevation of the top of a 15 m high tower at a point 15√3m away from the base of the tower is :

(a) 30°
(b) 45°
(c) 60°
(a) 90°

(2/3 sin 0° -4/5 cos0°) is equal to:

(a) 2/4
(b) -4/5
(c) 0
(d) -2/15

From a well-shuffled deck of 52 cards, a card is drawn at random. What is the probability of getting king of hearts ?

(a) 1/52
(b) 1/26
(c) 1/13
(d) 12/13

The number (5-3√5+√5) is:

(a) an integer
(b) rational number
(c) an irrational number
(d) a whole number

If the pair of linear equations x -y = 1, x + ky =5 has a unique solution x = 2, y= 1, then the value of k is :

(a) - 2
(b) -3
(c) 3
(d) 4

If ▵ABC ~ ▵DEF and ∠A = 47°, ∠E = 83°, then ∠C is equal :

(a) 47°
(b) 50°
(c) 83°
(d) 130°

The length of the tangent from an external point A to a circle, of radius 3 cm, is 4 cm. The distance of A from the centre of the circle is:

(a) 7cm
(b) 5cm
(c) √7cm
(d) 25cm

The pair of linear equations x + 2y + 5 = 0 and -3x-6y + 1=0 has:

(a) aunique solution
(b) exactly two solutions
(c) infinitely many solutions
(d) no solution

Assertion (A) : If one root of the quadratic equation 4x²-10x+(k—4)=0 is reciprocal of the other, then value of k is 8.

Reason (R) : Roots of the quadratic equation x²-x + 1 =0 are real.

Assertion (A) : A tangent to a circle is perpendicular to the radius through the point of contact.

Reason (R) : The lengths of tangents drawn from an external point to a circle are equal.

Section-B

Section - B comprises of Very Short Answer (VSA) questions of 2 marks each.

(A) Find the discriminant of the quadratic equation 3x² — 2x + 1/3 = 0 and hence find the nature of its roots.

OR

(B) Find the roots of the quadratic equation x² -x-2=0.

If sin α=1/2,then find the value of (3 cos α -4 cos³α)

(A) Find the coordinates | o “ a which divides the join of A(-1, 7) and B (4, -3) in the ratio 2:3

OR

(B) If the points A(2,3),B(-5,6),C(6,7) and D (p,4) are the vertices of a parallelogram ABCD,find the value of p.

Section-C

This section comprises short answer (SA) type questions of 3 marks each.

If α, β of the quadratic polynomial x²- 5x+6,form another quadratic polynomial whose zeroes are 1/α,1/β

Prove that 1+tan²A/1+cot²A=sec²A -1

(A) Prove that the lengths of tangents drawn from an external point to a circle are equal.

OR

(B) Two concentric circles with centre O are of radii 3 cm and 5 cm. Find the length of chord AB of the larger circle which touches the smaller circle at P

(A) If we add 1 to the numerator and subtract 1 from the denominator, a fraction reduces to 1. It becomes 1/2 if we only add 1 to the denominator, What is the fraction ?

OR

(B) For which value of 'k' will the following pair of linear equations have no solution ?

3x + y=1

(2k—1)x+(k-1)y=2k+1

(A) Find the sum of first 51 terms of an A.P. whose second and third terms are 14 and 18, respectively,

OR

(B) The first term of an A.P. is 5,the last term is 45 and the sum is 400.Find the number of terms and the common difference

Section-E

This section comprises 3 case study based questions of 4 marks each.