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
Which of these is a perfect square?
(A) 50(B) 48(C) 81(D) 27
2.1
The cube of 5 is
(A) 15(B) 25(C) 125(D) 625
3.1
√144 is
(A) 14(B) 72(C) 12(D) 11
4.1
The smallest number by which 72 must be multiplied to get a perfect square is
(A) 2(B) 8(C) 3(D) 6
5.1
2⁵ × 2³ is equal to
(A) 2²(B) 2¹⁵(C) 2⁸(D) 4⁸
6.1
The value of 5⁰ is
(A) not defined(B) 0(C) 5(D) 1
7.1
3,40,000 in scientific notation is
(A) 3.4 × 10⁴(B) 0.34 × 10⁶(C) 34 × 10⁴(D) 3.4 × 10⁵
8.1
2⁻³ is equal to
(A) −8(B) −6(C) 1/6(D) 1/8
9.1
The Roman numeral XL stands for
(A) 14(B) 90(C) 60(D) 40
10.1
The base-2 number 101 is equal to which number in base 10?
(A) 101(B) 5(C) 3(D) 6
11.1
The sum of the angles of a quadrilateral is
(A) 180°(B) 270°(C) 360°(D) 540°
12.1
A quadrilateral with all sides equal and all angles 90° is a
(A) square(B) rhombus(C) rectangle(D) kite
13.1
The diagonals of a rhombus
(A) bisect each other at right angles(B) are parallel(C) do not meet(D) are always equal
14.1
Which number is divisible by 9?
(A) 1234(B) 2034(C) 5551(D) 7001
15.1
A number is divisible by 4 if
(A) its last digit is 4(B) the number formed by its last two digits is divisible by 4(C) the sum of its digits is divisible by 4(D) it is even
16.1
Using (a + b)(a − b) = a² − b², 53 × 47 equals
(A) 2491(B) 2500(C) 2409(D) 2591
17.1
The ratio 24 : 36 in its simplest form is
(A) 2 : 3(B) 12 : 18(C) 3 : 2(D) 4 : 6
18.1
If 5 pens cost ₹60, then 8 pens cost
(A) ₹480(B) ₹90(C) ₹96(D) ₹100
19.1
Assertion (A): 1000 is a perfect cube.Reason (R): 1000 = 10 × 10 × 10.
(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): Every rectangle is a square.Reason (R): Every square is a rectangle.
(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
Using Identity 1C, calculate 98 × 102.
ORA 40kg mixture has sand and cement in ratio 3:1. How much cement is in the mixture?
22.2
Using the alternating sum method, check if 481 is divisible by 11.
ORFind the smallest multiple of 9 with no odd digits.
23.2
In a parallelogram, one angle is 30°. Find the adjacent and opposite angles.
24.2
Simplify the ratio 90:60 and check if it is proportional to 60:40.
25.2
Expand (6x + 5)² using Identity 1A.
SECTION C
This section comprises 6 short answer (SA) type questions of 3 marks each.
26.3
Expand (a+b)(a²+2ab+b²) fully, combining like terms.
ORHimachal tea: 200g for ₹200. Meghalaya tea: 1kg for ₹800. Determine which tea is more expensive per kg.
27.3
Prove that if all angles of a quadrilateral are 90°, the opposite sides must be equal.
ORFind the mistake and correct it: (a+2)(b+4) = ab + 8.
28.3
A quadrilateral has diagonals that are perpendicular to each other. Is it necessarily a rhombus? Justify.
29.3
If 48a23b is a multiple of 18, find all possible pairs of values for a and b.
30.3
A quadrilateral has four equal sides and one angle of 90°. Prove it must be a square.
31.3
A car travels 90km in 150 minutes. Find the distance covered in 240 minutes at the same speed.
SECTION D
This section comprises 4 long answer (LA) type questions of 5 marks each.
32.5
Prove that the diagonals of a rectangle are equal in length, using congruence.
33.5
A square of sidelength 65 is split into a 60×60 square, a 5×5 square, and two 60×5 rectangles. Using Identity 1A, prove that 65²=4225, and explain the geometric reasoning.
34.5
Find a number that leaves a remainder of 2 when divided by 3, a remainder of 3 when divided by 4, and a remainder of 4 when divided by 5. What is the smallest such number, and why?
35.5
A mixture of 40 kg contains sand and cement in the ratio 3:1. Cement is added to make the new ratio of sand to cement 5:2. Find how much cement was added, showing all steps.
ORThere are 5 bottles in a container. Every day a new container is brought in with the same number of bottles as the total so far. How many bottles will there be after 40 days? Express in scientific notation.
SECTION E
This section comprises 3 source based / case based units of assessment of 4 marks each with sub-parts.
36.
A kheer recipe for 4 people uses 200 g of rice and 300 ml of milk.(i) How much rice is needed for 10 people?1(ii) Write the ratio of rice (in g) to milk (in ml) in its simplest form.1(iii) With 1.2 litres of milk, how many people can be served, and how much rice is needed?2OR(iii) Asha and Ravi share ₹1,200 in the ratio 3 : 5. How much does each get?2
37.
A square hall is covered with square tiles of side 1 ft. Exactly 1764 tiles are used.(i) Is 1764 a perfect square?1(ii) Find the length of a side of the hall.1(iii) Find √1764 by prime factorisation and find the perimeter of the hall.2OR(iii) A cube-shaped box has a volume of 2744 cm³. Find the length of its edge by prime factorisation.2
38.
A sheet of paper is 0.1 mm thick. Each time it is folded in half, the number of layers doubles.(i) How many layers are there after 5 folds?1(ii) How thick is the folded paper after 5 folds?1(iii) Find the number of layers and the thickness after 10 folds, writing the layers as a power of 2.2OR(iii) The distance from the Earth to the Sun is about 150,000,000 km. Write it in scientific notation. Also write 0.000 007 m in scientific notation.2
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