Series RTEQ.P. Code 23/0/1
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 point (−3, 5) lies in the
(A) second quadrant(B) third quadrant(C) fourth quadrant(D) first quadrant
2.1
Every point on the x-axis has
(A) x = y(B) x + y = 0(C) y = 0(D) x = 0
3.1
The distance of the point (6, 8) from the origin is
(A) 48 units(B) 10 units(C) √14 units(D) 14 units
4.1
The distance between the points (1, 2) and (4, 6) is
(A) 25 units(B) √7 units(C) 5 units(D) 7 units
5.1
The degree of the polynomial 3x − 7 is
(A) 0(B) 1(C) 2(D) 3
6.1
The zero of the polynomial 2x − 6 is
(A) 1/3(B) −3(C) 6(D) 3
7.1
A taxi charges ₹50 as a fixed fare and ₹12 per km. The fare for x km is
(A) 12x − 50(B) 12x + 50(C) 50x + 12(D) 62x
8.1
For y = 4x + 1, when x increases by 1, y increases by
(A) x(B) 1(C) 4(D) 5
9.1
Which of the following is irrational?
(A) 3/7(B) √7(C) 0.75(D) √25
10.1
The decimal expansion of 1/8 is
(A) terminating(B) non-terminating and repeating(C) non-terminating and non-repeating(D) none of these
11.1
A rational number between 1/3 and 1/2 is
(A) 5/12(B) 2/3(C) 3/5(D) 1/4
12.1
0.3333… (0.3 recurring) written in the form p/q is
(A) 10/3(B) 3/10(C) 33/100(D) 1/3
13.1
The value of (√5 + √2)(√5 − √2) is
(A) √3(B) 10(C) 3(D) 7
14.1
(a + b)² is equal to
(A) 2a + 2b(B) a² + b²(C) a² + 2ab + b²(D) a² − 2ab + b²
15.1
Using an identity, 101 × 99 equals
(A) 10001(B) 9909(C) 9899(D) 9999
16.1
x² − 16 factorises as
(A) (x + 4)(x − 4)(B) (x + 8)(x − 2)(C) (x − 16)(x + 1)(D) (x − 4)²
17.1
If a + b + c = 0, then a³ + b³ + c³ equals
(A) abc(B) 3abc(C) a + b + c(D) 0
18.1
Using an identity, 103² equals
(A) 10906(B) 10069(C) 1069(D) 10609
19.1
Assertion (A): √2 is an irrational number.Reason (R): The square root of every prime number is irrational.
(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): (x + 3)² = x² + 9.Reason (R): (a + b)² = a² + 2ab + b².
(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
Express 1.272727… (27 repeating) as a fraction in lowest terms.
ORFind the midpoint of (−6, 4) and (10, −8).
22.2
Write two rational numbers and one irrational number between 1 and 2.
ORFactor 2x² + 7x + 6.
23.2
The vertices of a triangle are (0, 0), (4, 0) and (0, 3). Show that it is right-angled and find its hypotenuse.
24.2
A tile pattern has 3n + 2 tiles at stage n. Find the number of tiles at stage 20, and the stage that has 80 tiles.
25.2
Using an identity, find 103 × 97.
SECTION C
This section comprises 6 short answer (SA) type questions of 3 marks each.
26.3
Find the point on the x-axis that is equidistant from (1, 5) and (7, 3).
ORP (−2, 3), Q (4, 3) and R (4, −1) are three corners of a rectangle. Find the fourth corner S, the area, and the length of a diagonal.
27.3
Expand (p − 2q)³ and use it to find 98³.
ORFind five rational numbers between 3/5 and 4/5.
28.3
Show that A (1, 1), B (4, 5) and C (7, 9) lie on a straight line.
29.3
If a + b + c = 9 and ab + bc + ca = 26, find (a) a² + b² + c², (b) a³ + b³ + c³ − 3abc.
30.3
Simplify (x² − 5x + 6)/(x² − 9), assuming the denominator is not zero.
31.3
The relation F = (9/5)C + 32 converts Celsius to Fahrenheit. (a) Convert 25 °C to °F. (b) Find the temperature that has the same value in both scales.
SECTION D
This section comprises 4 long answer (LA) type questions of 5 marks each.
32.5
A rectangle has area x² + 3x − 40. (a) Factor this to find its sides. (b) Find the sides when x = 12. (c) Find its perimeter when x = 12. (d) Using a² − b², show that (x + 8)² − (x − 5)² = 13(2x + 3). (e) Check (d) at x = 12.
OR(a) Prove that √5 is irrational. (b) Hence show that 2 + √5 is irrational.
33.5
(a) Using Brahmagupta's rules, evaluate (−7) × (−8) + (−12) ÷ 4. (b) Classify each as rational or irrational: 0.333…, √16, √5, −3/4, π. (c) Express 0.1333… (only the 3 repeating) as a fraction. (d) Write 11/16 as a decimal. (e) Find a rational number between √2 and √3.
34.5
A phone plan charges ₹350 for 10 GB and ₹650 for 25 GB, following y = ax + b. (a) Find a and b. (b) Find the bill for 40 GB. (c) For what data use is the bill ₹550? (d) What do a and b mean? (e) Another plan charges y = 30x + 50. Which is cheaper for 15 GB, and at what usage do the plans cost the same?
35.5
A (−3, 4), B (3, 4), C (3, −4), D (−3, −4). (a) Name the quadrant of each point. (b) Find AB and BC. (c) Find the diagonal AC. (d) Show that all four points lie on a circle centred at the origin, and give its radius. (e) Find the area of ABCD.
ORp(x) = ax + b and q(x) = cx + d satisfy: p(0) = 5, p(x) − q(x) cuts the x-axis at (2, 0), and p(x) + q(x) = 8x + 2. (a) Find b and d. (b) Find a and c. (c) Write p(x) and q(x). (d) Where does p(x) cut the x-axis? (e) How far apart are the y-intercepts of p and q?
SECTION E
This section comprises 3 source based / case based units of assessment of 4 marks each with sub-parts.
36.
On a town map drawn on a grid (1 unit = 1 km), the school is at S(2, 3), the park at P(2, 6) and the library at L(6, 6).(i) Find the distance between the school and the park.1(ii) Find the distance between the park and the library.1(iii) Find the straight-line distance from the school to the library, and show that triangle SPL is right-angled.2OR(iii) A shop Q is on the x-axis directly below the school. Write its coordinates and find its distance from the library.2
37.
A plant is 5 cm tall and grows 2 cm every week.(i) What is its height after 3 weeks?1(ii) Write a linear polynomial for its height h (in cm) after t weeks.1(iii) After how many weeks will the plant be 25 cm tall?2OR(iii) A second plant is 11 cm tall and grows 1.5 cm every week. After how many weeks will both plants be equally tall, and what will that height be?2
38.
A square garden has side (x + 3) m. In one corner there is a square pond of side x m.(i) Write the area of the garden as an expanded expression.1(ii) Write the area of the garden not covered by the pond.1(iii) If x = 7, find the area not covered by the pond using your expression, and verify it directly.2OR(iii) Factorise 6x + 9. Also factorise x² + 7x + 12, the area of another rectangular plot.2
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