Series RTEQ.P. Code 26/0/1
Roll No.
RTE LEARN · CBSE PATTERN 2026–27
MATHEMATICS
Half-Yearly Exam
Time allowed : 3 HoursMaximum Marks : 80
General Instructions :
Read the following instructions carefully and 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, Question numbers 1 to 18 are multiple choice questions (MCQs) and Question numbers 19 to 20 are Assertion–Reason based questions of 1 mark each.
(iv)In Section B, Question numbers 21 to 25 are very short answer (VSA) type questions carrying 02 marks each.
(v)In Section C, Question numbers 26 to 31 are short answer (SA) type questions carrying 03 marks each.
(vi)In Section D, Question numbers 32 to 35 are long answer (LA) type questions carrying 05 marks each.
(vii)In Section E, Question numbers 36 to 38 are case-based integrated units of assessment carrying 04 marks each, with sub-parts.
(viii)There is no overall choice. However, an internal choice has been provided in some questions. Attempt only one of the alternatives in such questions.
(ix)Draw neat figures wherever required. Take π = 22/7 wherever required, if not stated.
(x)Use of calculators is not allowed.
SECTION A
This section comprises 20 multiple choice questions (MCQs) of 1 mark each.
1.1
The HCF of 2³ × 3² and 2² × 3³ is
(A) 36(B) 72(C) 108(D) 216
2.1
The HCF of two numbers is 12 and their product is 1800. Their LCM is
(A) 1800(B) 120(C) 150(D) 180
3.1
Which of the following is an irrational number?
(A) √16(B) 0.333…(C) √8(D) 22/7
4.1
The zeroes of the polynomial x² − 9 are
(A) −3, 3(B) 9, −9(C) 0, 9(D) 3, 3
5.1
If α and β are the zeroes of x² − 5x + 6, then α + β is
(A) 6(B) 5(C) −6(D) −5
6.1
A quadratic polynomial whose sum of zeroes is 3 and product of zeroes is −10 is
(A) x² − 3x − 10(B) x² + 3x − 10(C) x² − 3x + 10(D) x² + 10x − 3
7.1
The pair of equations 2x + 3y = 5 and 4x + 6y = 10 has
(A) a unique solution(B) no solution(C) infinitely many solutions(D) exactly two solutions
8.1
For what value of k do the equations kx + 2y = 5 and 3x + y = 1 have no solution?
(A) −6(B) 3(C) 2/3(D) 6
9.1
If x = 1, y = 2 is a solution of 3x + ky = 9, then k is
(A) 4(B) 6(C) 2(D) 3
10.1
The discriminant of 2x² − 4x + 3 = 0 is
(A) −4(B) 4(C) 8(D) −8
11.1
The roots of x² − 2x = 0 are
(A) 0, −2(B) 2, −2(C) 1, 2(D) 0, 2
12.1
The positive value of k for which x² + kx + 9 = 0 has equal roots is
(A) 3(B) 9(C) 6(D) 18
13.1
The 10th term of the AP 2, 7, 12, … is
(A) 50(B) 45(C) 47(D) 52
14.1
The common difference of the AP 3, 1, −1, −3, … is
(A) −2(B) 2(C) −4(D) 4
15.1
The sum of the first 20 odd natural numbers is
(A) 200(B) 210(C) 400(D) 441
16.1
In Δ ABC, DE ∥ BC with D on AB and E on AC. If AD = 2 cm, DB = 3 cm and AE = 4 cm, then EC is
(A) 4.5 cm(B) 5 cm(C) 6 cm(D) 8 cm
17.1
In Δ ABC and Δ PQR, ∠A = ∠P and ∠B = ∠Q. The triangles are similar by the criterion
(A) SSS(B) SAS(C) AA(D) RHS
18.1
The distance between the points (2, 3) and (5, 7) is
(A) 5 units(B) 7 units(C) 25 units(D) 4 units
19.1
Assertion (A): 6ⁿ can never end with the digit 0 for any natural number n.Reason (R): A number ends with 0 only if its prime factorisation contains both 2 and 5.
(A) Both A and R are true, and R is the correct explanation of A.(B) Both A and R are true, but R is not the correct explanation of A.(C) A is true, but R is false.(D) A is false, but R is true.
20.1
Assertion (A): The point dividing the join of (−1, 7) and (4, −3) in the ratio 2 : 3 is (1, 3).Reason (R): The mid-point of the join of (x₁, y₁) and (x₂, y₂) is ((x₁ + x₂)/2, (y₁ + y₂)/2).
(A) Both A and R are true, and R is the correct explanation of A.(B) Both A and R are true, but R is not the correct explanation of A.(C) A is true, but R is false.(D) A is false, but R is true.
SECTION B
This section comprises 5 very short answer (VSA) type questions of 2 marks each.
21.2
Solve: x + y = 17 and x − y = 5.
ORFind the point on the x-axis which is equidistant from (2, −5) and (−2, 9).
22.2
How many terms are there in the AP 7, 13, 19, …, 205?
ORIn triangles ABC and DEF, AB = 6 cm, BC = 8 cm, CA = 10 cm, DE = 9 cm, EF = 12 cm and FD = 15 cm. Are the triangles similar? Give a reason.
23.2
Find the zeroes of the polynomial 6x² − 7x − 3 and verify that their sum equals −b/a.
24.2
Form a quadratic polynomial whose sum and product of zeroes are 5/2 and −3 respectively.
25.2
Find the coordinates of the point which divides the line segment joining (1, −3) and (−4, 7) internally in the ratio 2 : 3.
SECTION C
This section comprises 6 short answer (SA) type questions of 3 marks each.
26.3
Given that √5 is irrational, prove that 3 + 2√5 is irrational.
ORIf α and β are the zeroes of x² − 5x + k such that α − β = 1, find the value of k.
27.3
The sum of the digits of a two-digit number is 11. If 27 is added to the number, its digits get reversed. Find the number.
ORTwo bells ring at intervals of 18 minutes and 24 minutes. They ring together at 8:00 a.m. At what time will they next ring together?
28.3
Solve by the elimination method: 2x + 3y = 13 and 5x − 2y = 4.
29.3
In what ratio does the x-axis divide the line segment joining (3, −6) and (5, 3)? Also find the point of division.
30.3
In triangle ABC, D is on AB and E is on AC with AD = 4 cm, DB = 6 cm, AE = 5 cm and EC = 7.5 cm. Is DE ∥ BC? Give reasons.
31.3
Show that the points A(1, 7), B(4, 2), C(−1, −1) and D(−4, 4) are the vertices of a square.
SECTION D
This section comprises 4 long answer (LA) type questions of 5 marks each.
32.5
State and prove the Basic Proportionality Theorem (Thales theorem).
33.5
The 5th term of an AP is 17 and its 9th term is 29. Find (i) the AP, (ii) its 20th term, (iii) the sum of its first 20 terms, and (iv) which term of the AP is 74.
34.5
A train travels 300 km at a uniform speed. If the speed had been 10 km/h more, it would have taken 1 hour less for the same journey. Find the speed of the train.
ORA boat goes 30 km upstream and 24 km downstream in 7 hours. It goes 18 km upstream and 48 km downstream in 7 hours. Find the speed of the boat in still water and the speed of the stream.
35.5
If α and β are the zeroes of the polynomial 2x² − 7x + 3, find the value of (i) α² + β², (ii) 1/α + 1/β, (iii) α³ + β³. Hence form a quadratic polynomial whose zeroes are 1/α and 1/β.
ORProve that √3 is irrational. Hence show that 2 + 5√3 is irrational.
SECTION E
This section comprises 3 source based / case based units of assessment of 4 marks each with sub-parts.
36.
A rectangular park is to be made whose breadth is 3 m less than its length. Its area must be 4 m² more than the area of a triangular park already made, whose base equals the breadth of the rectangular park and whose height is 12 m.(i) If the length of the rectangular park is x m, write its breadth.1(ii) Write the area of the triangular park in terms of x.1(iii) Form a quadratic equation and find the length and breadth of the rectangular park.2OR(iii) If the length is 7 m, find the area of each park and check the given condition.2
37.
Rahul saves ₹50 in the first week and increases his saving by ₹20 every week, so his weekly savings form an AP.(i) How much does he save in the 10th week?1(ii) In which week does he save ₹450?1(iii) Find his total savings in the first 15 weeks.2OR(iii) In how many weeks will his total savings be ₹1,650?2
38.
In a classroom, four friends sit at the points A(3, 4), B(6, 7), C(9, 4) and D(6, 1).(i) Find the distance AB.1(ii) Find the mid-point of AC.1(iii) Show that ABCD is a square.2OR(iii) Find the point on the x-axis that is equidistant from A and C.2
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