MATHEMATICS
(Standard)
,
, DELHI
2023 CBSE Solved Paper
Time allowed : 3 hours Maximum Marks : 80
GENERAL INSTRUCTIONS:
Read the following instructions very carefully and strictly follow them:
(i) This question paper contains 38 questions. All questions are compulsory.
(ii) This question paper is divided into five Sections - A, B, C, D and E.
(iii) In Section A, Questions no. 1 to 18 are multiple choice questions (MCQs) and questions number 19 and 20 are Assertion-
Reason based questions of 1 mark each.
(iv) In Section B, Questions no. 21 to 25 are very short answer (VSA) type questions, carrying 2 marks each.
(v) In Section C, Questions no. 26 to 31 are short answer (SA) type questions, carrying 3 marks each.
(vi) In Section D, Questions no. 32 to 35 are long answer (LA) type questions carrying 5 marks each.
(vii) In Section E, Questions no. 36 to 38 are case study based questions carrying 4 marks each. Internal choice is provided in
2 marks questions in each case-study.
(viii) There is no overall choice. However, an internal choice has been provided in 2 questions in Section B, 2 questions in
Section C, 2 questions in Section D and 3 questions in Section E.
22
(ix) Draw neat diagrams wherever required. Take π = wherever required, if not stated.
7
(x) Use of calculators is not allowed.
(c) Mode = 2 Median – 3 Mean
Section-A (d) Mode = 2 Mean – 3 Median
1. The ratio of H.C.F to L.C.M of the least composite number 7. The pair of linear equations 2x = 5y + 6 and 15y = 6x – 18
and the least prime number is: (1 Mark)
represents two lines which are: (1 Mark)
(a) 1 : 2 (b) 2 : 1 (c) 1 : 1 (d) 1 : 3
(a) Intersecting (b) Parallel
2. The roots of the equation x2 + 3x – 10 = 0 are: (1 Mark)
(c) Coincident (d) Either intersecting or parallel
(a) 2, –5 (b) –2, 5 (c) 2, 5 (d) –2, –5
8. If a, b are zeroes of the polynomial x2 – 1, then value of
3. The next term of the A.P. 6, 24, 54 is: (1 Mark)
(a + b) is: (1 Mark)
(a) 60 (b) 96 (c) 72 (d) 216 (a) 2 (b) 1 (c) –1 (d) 0
4. The distance of the point (– 1, 7) from x-axis is: (1 Mark)
9. If a pole 6 m high casts a shadow 2 3 m long on the
(a) –1 (b) 7 (c) 6 (c)50 ground, then sun’s elevation is: (1 Mark)
5. What is the area of a semi-circle of diameter ‘d’? (a) 60° (b) 45° (c) 30° (d) 90°
[OS]* (1 Mark)
10. secq when expressed in terms of cot q, is equal to: (1 Mark)
1 1 2 1 1 2
(a) πd 2 (b) πd (c) πd 2 (d) πd 1 + cot 2 θ
16 4 8 2 (a) (b) 1 + cot 2 θ
cot θ
6. The empirical relation between the mode, median and
mean of a distribution is: (1 Mark) 1 + cot 2 θ 1 − cot 2 θ
(a) Mode = 3 Median – 2 Mean (c) (d)
(b) Mode = 3 Mean – 2 Median cot θ cot θ
*[OS] denotes Out of the Syllabus questions.
, 11. Two dice are thrown together. The probability of getting 17. If a, b are the zeroes of the polynomial p(x) = 4x2 – 3x – 7,
the difference of numbers on their upper faces equals 1 1
to 3 is: (1 Mark) then + is equal to: (1 Mark)
α β
1 2 1 1 7 −7 3 −3
(a) (b) (c) (d) (a) (b) (c) (d)
9 9 6 12 3 3 7 7
12. 6 cm 18. A card is drawn at random from a well-shuffled pack of
A C R
52 cards. The probability that the card drawn is not an
ace is: (1 Mark)
x
5 cm
3 cm
1 9 4 12
(a) (b) (c) (d)
13 13 13 13
Questions number 19 and 20 are Assertion and Reason based
B Q P
questions carrying 1 mark each. Two statements are given, one
In the given figure, DABC ~ DQPR, If AC = 6 cm , BC = 5 cm, labelled as Assertion (A) and the other is labelled as Reason (R).
QR = 3 cm and PR = x; then the value of x is: (1 Mark) Select the correct answer to these questions from the codes (a),
(a) 3.6 cm (b) 2.5 cm (c) 10 cm (d) 3.2 cm (b), (c) and (d) as given below.
13. The distance of the point (–6, 8) from origin is: (1 Mark) (a) Both Assertion (A) and Reason (R) are true and Reason
(a) 6 (b) –6 (c) 8 (d) 10 (R) is the correct explanation of the Assertion (A).
(b) Both Assertion (A) and Reason (R) are true, but Reason
14. In the given figure, PQ is a tangent to the circle with
(R) is not the correct explanation of the Assertion (A).
centre O. If ∠OPQ = x, ∠POQ = y, then x + y is: (1 Mark)
(c) Assertion (A) is true, but Reason (R) is false.
Q
(d) Assertion (A) is false, but Reason (R) is true.
19. Assertion (A): The probability that a leap year has 53
x y
P O 2
Sundays is .
7
Reason (R): The probability that a non-leap year has 53
5
(a) 45° (b) 90° (c) 60° (d) 180° Sundays is . (1 Mark)
7
15. In the given figure, TA is a tangent to the circle with centre O 20. Assertion (A): a, b, c are in A.P. if and only if 2b = a + c.
such that OT = 4 cm, ∠OTA = 30°, then length of TA is:
(1 Mark) Reason (R): The sum of first n odd natural numbers is n2.
(1 Mark)
Section-B
O
21. Two numbers are in the ratio 2 : 3 and their L.C.M is 180.
m
4c What is the H.C.F. of these numbers? (2 Marks)
30°
T A 22. If one zero of the polynomial p(x) = 6x2 + 37x – (k – 2) is
(a) 2 3 cm (b) 2 cm (c)
2 2 cm (d) 3 cm reciprocal of the other, then find the value of k.(2 Marks)
16. In DABC, PQ||BC. If PB = 6 cm, AP = 4 cm , AQ = 8 cm, 23. (a) Find the sum and product of the roots of the quadratic
find the length of AC. (1 Mark) equation 2x2 – 9x + 4 = 0. (2 Marks)
A OR
(b) Find the discriminant of the quadratic equation
4x2 – 5 = 0 and hence comment on the nature of roots
P Q of the equation.
24. If a fair coin is tossed twice, find the probability of getting
‘atmost one head’. (2 Marks)
B C 2 2 2
25. (a) Evaluate 5cos 60° + 4sec 30° − tan 45° (2 Marks)
(a) 12 cm (b) 20 cm (c) 6 cm (d) 14 cm sin 2 30° + cos 2 30°
164 MATHS
(Standard)
,
, DELHI
2023 CBSE Solved Paper
Time allowed : 3 hours Maximum Marks : 80
GENERAL INSTRUCTIONS:
Read the following instructions very carefully and strictly follow them:
(i) This question paper contains 38 questions. All questions are compulsory.
(ii) This question paper is divided into five Sections - A, B, C, D and E.
(iii) In Section A, Questions no. 1 to 18 are multiple choice questions (MCQs) and questions number 19 and 20 are Assertion-
Reason based questions of 1 mark each.
(iv) In Section B, Questions no. 21 to 25 are very short answer (VSA) type questions, carrying 2 marks each.
(v) In Section C, Questions no. 26 to 31 are short answer (SA) type questions, carrying 3 marks each.
(vi) In Section D, Questions no. 32 to 35 are long answer (LA) type questions carrying 5 marks each.
(vii) In Section E, Questions no. 36 to 38 are case study based questions carrying 4 marks each. Internal choice is provided in
2 marks questions in each case-study.
(viii) There is no overall choice. However, an internal choice has been provided in 2 questions in Section B, 2 questions in
Section C, 2 questions in Section D and 3 questions in Section E.
22
(ix) Draw neat diagrams wherever required. Take π = wherever required, if not stated.
7
(x) Use of calculators is not allowed.
(c) Mode = 2 Median – 3 Mean
Section-A (d) Mode = 2 Mean – 3 Median
1. The ratio of H.C.F to L.C.M of the least composite number 7. The pair of linear equations 2x = 5y + 6 and 15y = 6x – 18
and the least prime number is: (1 Mark)
represents two lines which are: (1 Mark)
(a) 1 : 2 (b) 2 : 1 (c) 1 : 1 (d) 1 : 3
(a) Intersecting (b) Parallel
2. The roots of the equation x2 + 3x – 10 = 0 are: (1 Mark)
(c) Coincident (d) Either intersecting or parallel
(a) 2, –5 (b) –2, 5 (c) 2, 5 (d) –2, –5
8. If a, b are zeroes of the polynomial x2 – 1, then value of
3. The next term of the A.P. 6, 24, 54 is: (1 Mark)
(a + b) is: (1 Mark)
(a) 60 (b) 96 (c) 72 (d) 216 (a) 2 (b) 1 (c) –1 (d) 0
4. The distance of the point (– 1, 7) from x-axis is: (1 Mark)
9. If a pole 6 m high casts a shadow 2 3 m long on the
(a) –1 (b) 7 (c) 6 (c)50 ground, then sun’s elevation is: (1 Mark)
5. What is the area of a semi-circle of diameter ‘d’? (a) 60° (b) 45° (c) 30° (d) 90°
[OS]* (1 Mark)
10. secq when expressed in terms of cot q, is equal to: (1 Mark)
1 1 2 1 1 2
(a) πd 2 (b) πd (c) πd 2 (d) πd 1 + cot 2 θ
16 4 8 2 (a) (b) 1 + cot 2 θ
cot θ
6. The empirical relation between the mode, median and
mean of a distribution is: (1 Mark) 1 + cot 2 θ 1 − cot 2 θ
(a) Mode = 3 Median – 2 Mean (c) (d)
(b) Mode = 3 Mean – 2 Median cot θ cot θ
*[OS] denotes Out of the Syllabus questions.
, 11. Two dice are thrown together. The probability of getting 17. If a, b are the zeroes of the polynomial p(x) = 4x2 – 3x – 7,
the difference of numbers on their upper faces equals 1 1
to 3 is: (1 Mark) then + is equal to: (1 Mark)
α β
1 2 1 1 7 −7 3 −3
(a) (b) (c) (d) (a) (b) (c) (d)
9 9 6 12 3 3 7 7
12. 6 cm 18. A card is drawn at random from a well-shuffled pack of
A C R
52 cards. The probability that the card drawn is not an
ace is: (1 Mark)
x
5 cm
3 cm
1 9 4 12
(a) (b) (c) (d)
13 13 13 13
Questions number 19 and 20 are Assertion and Reason based
B Q P
questions carrying 1 mark each. Two statements are given, one
In the given figure, DABC ~ DQPR, If AC = 6 cm , BC = 5 cm, labelled as Assertion (A) and the other is labelled as Reason (R).
QR = 3 cm and PR = x; then the value of x is: (1 Mark) Select the correct answer to these questions from the codes (a),
(a) 3.6 cm (b) 2.5 cm (c) 10 cm (d) 3.2 cm (b), (c) and (d) as given below.
13. The distance of the point (–6, 8) from origin is: (1 Mark) (a) Both Assertion (A) and Reason (R) are true and Reason
(a) 6 (b) –6 (c) 8 (d) 10 (R) is the correct explanation of the Assertion (A).
(b) Both Assertion (A) and Reason (R) are true, but Reason
14. In the given figure, PQ is a tangent to the circle with
(R) is not the correct explanation of the Assertion (A).
centre O. If ∠OPQ = x, ∠POQ = y, then x + y is: (1 Mark)
(c) Assertion (A) is true, but Reason (R) is false.
Q
(d) Assertion (A) is false, but Reason (R) is true.
19. Assertion (A): The probability that a leap year has 53
x y
P O 2
Sundays is .
7
Reason (R): The probability that a non-leap year has 53
5
(a) 45° (b) 90° (c) 60° (d) 180° Sundays is . (1 Mark)
7
15. In the given figure, TA is a tangent to the circle with centre O 20. Assertion (A): a, b, c are in A.P. if and only if 2b = a + c.
such that OT = 4 cm, ∠OTA = 30°, then length of TA is:
(1 Mark) Reason (R): The sum of first n odd natural numbers is n2.
(1 Mark)
Section-B
O
21. Two numbers are in the ratio 2 : 3 and their L.C.M is 180.
m
4c What is the H.C.F. of these numbers? (2 Marks)
30°
T A 22. If one zero of the polynomial p(x) = 6x2 + 37x – (k – 2) is
(a) 2 3 cm (b) 2 cm (c)
2 2 cm (d) 3 cm reciprocal of the other, then find the value of k.(2 Marks)
16. In DABC, PQ||BC. If PB = 6 cm, AP = 4 cm , AQ = 8 cm, 23. (a) Find the sum and product of the roots of the quadratic
find the length of AC. (1 Mark) equation 2x2 – 9x + 4 = 0. (2 Marks)
A OR
(b) Find the discriminant of the quadratic equation
4x2 – 5 = 0 and hence comment on the nature of roots
P Q of the equation.
24. If a fair coin is tossed twice, find the probability of getting
‘atmost one head’. (2 Marks)
B C 2 2 2
25. (a) Evaluate 5cos 60° + 4sec 30° − tan 45° (2 Marks)
(a) 12 cm (b) 20 cm (c) 6 cm (d) 14 cm sin 2 30° + cos 2 30°
164 MATHS
