Series RTEQ.P. Code 93/1/0
Roll No.
RTE LEARN · CBSE PATTERN 2026–27
MATHEMATICS
Annual 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 value of sin 30° + cos 60° is
(A) 1/2(B) 1(C) √3(D) 0
2.1
If tan A = 3/4 and A is acute, then sin A is
(A) 3/5(B) 5/3(C) 3/4(D) 4/5
3.1
(1 + tan² A) cos² A is equal to
(A) tan² A(B) 1(C) 0(D) sec² A
4.1
sin (90° − A) is equal to
(A) tan A(B) sec A(C) sin A(D) cos A
5.1
A pole 6 m high casts a shadow 2√3 m long. The angle of elevation of the Sun is
(A) 45°(B) 60°(C) 90°(D) 30°
6.1
A ladder reaches a height of 9 m on a wall and makes 60° with the ground. Its length is
(A) 9√3 m(B) 18 m(C) 3√3 m(D) 6√3 m
7.1
The number of tangents that can be drawn to a circle from a point outside it is
(A) 2(B) 0(C) infinitely many(D) 1
8.1
A point is 13 cm from the centre of a circle of radius 5 cm. The length of the tangent from it is
(A) √194 cm(B) 18 cm(C) 12 cm(D) 8 cm
9.1
PA and PB are tangents to a circle with centre O. If ∠APB = 80°, then ∠AOB is
(A) 160°(B) 80°(C) 100°(D) 90°
10.1
The area of a sector of angle 90° of a circle of radius 14 cm is (π = 22/7)
(A) 154 cm²(B) 308 cm²(C) 616 cm²(D) 77 cm²
11.1
The length of an arc of angle 60° of a circle of radius 21 cm is (π = 22/7)
(A) 44 cm(B) 66 cm(C) 11 cm(D) 22 cm
12.1
The volume of a cube of edge 4 cm is
(A) 48 cm³(B) 64 cm³(C) 96 cm³(D) 16 cm³
13.1
A cone and a cylinder have the same base radius and height. The ratio of their volumes (cone : cylinder) is
(A) 3 : 1(B) 1 : 2(C) 1 : 3(D) 2 : 3
14.1
The curved surface area of a hemisphere of radius r is
(A) 2πr²(B) πr²(C) 3πr²(D) 4πr²
15.1
The median of a data is 22 and its mean is 20.5. Using Mode = 3 Median − 2 Mean, its mode is
(A) 24(B) 21.5(C) 25(D) 26
16.1
The class mark of the class 10–25 is
(A) 17.5(B) 12.5(C) 35(D) 15
17.1
The probability of an impossible event is
(A) 0(B) 1(C) 0.5(D) −1
18.1
A die is thrown once. The probability of getting a prime number is
(A) 1/3(B) 1/2(C) 2/3(D) 1/6
19.1
Assertion (A): The probability of an event cannot be 1.2.Reason (R): The probability of an event always lies between 0 and 1 (both included).
(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): sin θ = 4/3 for some acute angle θ.Reason (R): For any angle θ, sin θ ≤ 1.
(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
Find the total surface area of a solid hemisphere of radius 21 cm. (Use π = 22/7.)
ORThe angle of elevation of the top of a vertical tower from a point on level ground, 20 m away from its foot, is 60°. Find the height of the tower. (Use √3 = 1.73.)
22.2
The radii of two solid cylinders are in the ratio 2 : 3 and their heights are in the ratio 5 : 3. Find the ratio of their volumes.
OREvaluate: 4(sin² 60° + cos² 30°) − 2 tan² 45°.
23.2
Two concentric circles have radii 7 cm and 14 cm. Find the area of the ring between them. (Use π = 22/7.)
24.2
Two tangents PA and PB are drawn from an external point P to a circle with centre O. If ∠APB = 70°, find ∠AOB.
25.2
Find the mean of the following data. Class 0–10 has frequency 5, class 10–20 has frequency 7, class 20–30 has frequency 6 and class 30–40 has frequency 2.
SECTION C
This section comprises 6 short answer (SA) type questions of 3 marks each.
26.3
A solid metallic cylinder of radius 6 cm and height 15 cm is melted and recast into small solid spheres of radius 3 cm each. How many spheres are formed?
ORIf 15 cot A = 8 and A is an acute angle, find sin A and sec A.
27.3
The marks of 50 students are grouped as follows: 0–10 has 5 students, 10–20 has 7, 20–30 has 20, 30–40 has 10 and 40–50 has 8. Find the median marks.
ORTwo dice are thrown together. Find the probability that (i) the sum of the numbers is 8, (ii) the product of the numbers is 12, (iii) the sum of the numbers is a prime number.
28.3
The marks of 50 students are grouped as follows: 10–20 has 7 students, 20–30 has 12, 30–40 has 18, 40–50 has 9 and 50–60 has 4. Find the mode of the data.
29.3
A quadrilateral ABCD is drawn so that all four of its sides touch a circle. If AB = 6 cm, BC = 7 cm and CD = 5 cm, find AD.
30.3
A chord of a circle of radius 14 cm subtends a right angle at the centre. Find the area of the corresponding minor segment. (Use π = 22/7.)
31.3
Prove that sin A/(1 + cos A) + (1 + cos A)/sin A = 2 cosec A.
SECTION D
This section comprises 4 long answer (LA) type questions of 5 marks each.
32.5
The angle of elevation of the top of a vertical tower from a point A on level ground is 30°. A man walks 60 m straight towards the foot of the tower and reaches a point B, from where the angle of elevation of the top is 60°. Find the height of the tower and the original distance of A from the foot of the tower. (Use √3 = 1.73.)
33.5
A toy is in the shape of a cone of base radius 3 cm and height 4 cm mounted on a hemisphere of the same radius 3 cm. Taking π = 3.14, find (i) the slant height of the cone, (ii) the total volume of the toy, (iii) the total surface area of the toy.
34.5
Prove that the lengths of the two tangents drawn from an external point to a circle are equal. Hence, if PA and PB are tangents from a point P to a circle with centre O and radius 6 cm, and OP = 10 cm, find the length PA and the perimeter of quadrilateral OAPB.
ORFrom the top of a 60 m high building, the angles of depression of the top and the bottom of a tower are 30° and 60° respectively. Find the height of the tower and its horizontal distance from the building. (Use √3 = 1.732)
35.5
The marks of 40 students are grouped as follows: 0–20 has 5 students, 20–40 has 9, 40–60 has 12, 60–80 has 8, 80–100 has 4 and 100–120 has 2. Find (i) the mean marks and (ii) the median marks.
ORThe mean of the following distribution is 18. Class 11–13 has frequency 7, 13–15 has 6, 15–17 has 9, 17–19 has 13, 19–21 has f, 21–23 has 5 and 23–25 has 4. (i) Find the missing frequency f. (ii) Write the modal class.
SECTION E
This section comprises 3 source based / case based units of assessment of 4 marks each with sub-parts.
36.
A lighthouse is 75 m high. From its top, a guard sees two ships on the same side of the lighthouse, in line with its foot. The angles of depression of the ships are 45° and 30°. (Use √3 = 1.732)(i) Find the distance of the nearer ship (angle of depression 45°) from the foot of the lighthouse.1(ii) Find the distance of the farther ship from the foot of the lighthouse.1(iii) Find the distance between the two ships.2OR(iii) A boat is seen at an angle of depression of 60°. Find its distance from the foot of the lighthouse.2
37.
A wooden toy is a cone mounted on a hemisphere of the same radius 3.5 cm. The total height of the toy is 15.5 cm. (Use π = 22/7)(i) Find the height of the cone.1(ii) Find the slant height of the cone.1(iii) Find the total surface area of the toy.2OR(iii) Find the volume of the toy.2
38.
The marks of 40 students in a test are: 0–10: 5 students, 10–20: 8, 20–30: 15, 30–40: 7, 40–50: 5.(i) Write the modal class.1(ii) Write the median class.1(iii) Find the mode of the data.2OR(iii) Find the mean of the data.2
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