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 SUBJECT: MATHEMATICS STANDARD (041) 
 SAMPLE QUESTION PAPER 
 CLASS – X (2026 – 27) 
Maximum Marks: 80  Time Allowed: 3 hours
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, Question number 1 to 18 are Multiple Choice Questions (MCQs) and Question
number 19 and 20 are Assertion - Reason based questions of 1 mark each.
(iv) In Section B, Question number 21 to 25 are Very Short Answer (VSA)-type questions, carrying
2 marks each.
(v) In Section C, Question number 26 to 31 are Short Answer (SA)-type questions carrying 3 marks
each.
(vi) In Section D, Question number 32 to 35 are Long Answer (LA)-type questions carrying 5 marks
each.
(vii) In Section E, Question number 36 to 38 are case study-based questions carrying 4 marks each.
(viii) There is no overall choice. However, an internal choice has been provided in 2 questions in
Section B, 2 questions in Section C and 2 questions in Section D. An internal choice has been
provided in all the sub parts having 2 marks of questions in section E.
(ix) Draw neat and clean figures wherever required. Take π = 
22
7  wherever required, if not stated. 
(x) Use of calculators is not allowed.
 SECTION – A  (20 x 1 = 20) 
This section comprises of 18 multiple choice questions and two questions of assertion and 
reasoning type of 1 mark each. 
respectively, is: 
(A) 13 (B) 26 (C) 39 (D) 65
1 
2. If f(x) = px2 + qx + r, p ≠ 0 and p + r = q, then one of the zeroes of f(x) is: 
 (A)  
q
p              (B) 
r
p                 (C) −
r
p              (D) −
q
p
1 
3. Tarun correctly solved a pair of linear equations in two variables and found their 
only point of intersection as (5, – 1). One of the lines was x – y = 6. 
Which of the following could have been the other line? 
I: 3x – 3y = 18 
II: 2x – 3y = 13 
III: 2x – 3y = 16 
(A) I only (B) II only (C) I and II (D) II and III
1 
4. If (1 – p) is a root of the quadratic equation x2 + px + 1 – p = 0, then its roots are: 
(A) 0, 1 (B) – 1, 1 (C) 0, – 1 (D) – 1, 2
1 
*There is no change in the Question Paper Design and Assessment Pattern for Academic Session 2026-27
Q. No. Questions Marks 
1. The greatest number which divides both 134 and 188, leaving remainders 4 and 6
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5. The middle term of the A.P.: 10, 7, 4, …, – 62 is: 
(A) – 26 (B) – 29 (C) – 32 (D) – 35
1 
1 
7. The point which lies on the perpendicular bisector of the line segment joining the 
points A (– 3, – 4) and B (3, 4) is:   
(A) (0, 0) (B) (0, 3) (C) (3, 0) (D) (0, 4)
1 
8. If tan (A + B) = √3 and tan (A – B) = 
1
√3, 0° < A + B < 90 °, A > B, then the value 
of A and B respectively are: 
(A) 60°, 30° (B) 60°, 45° (C) 45°, 15° (D) 60°,15°
  1 
9. T-shirts marked with nu mbers 4 to 99 are placed in a box. Gunika is fond of
numbers. She randomly takes out a T -shirt from this box. The probability that she
gets a T-shirt marked with a number that is either a perfect square or a perfect cube
is:
 (A) 
1
12  (B) 
1
32  (C) 
11
96  (D) 
5
48
1 
10. The diameter of a car wheel is 21 cm. The number of complete revolutions it will 
make in moving 66 km is: 
(A) 104 (B) 105 (C) 106 (D) 107
1 
11. Two cubes each of volume 64 cm3 are joined end to end to form a solid. The surface 
area of the resultant cuboid is: 
(A) 192 cm2 (B) 160 cm2 (C) 96 cm2 (D) 80 cm2
1 
12. The mean age of a combined group of men and women is 35 years. If the mean 
ages of the men and women are 38 years and 30 years respectively, then the 
percentage of women in the group is: 
(A) 15 (B) 25.5 (C) 35 (D) 37.5
1 
13. Two dice are rolled simultaneously. The probability of getting number less than 4 
on each die is: 
 (A) 
1
4  (B) 
1
9  (C) 
1
36  (D) 
1
6
1 
(A) 2√17 units (B) 2√15 units
(C) √101 units (D) 10 units
1 
15. The arc of a circle is of length 6𝜋 cm and the sector it bound has an area of 24𝜋 cm2. 
The radius of the circle is: 
(A) 4 cm (B) 8 cm (C) 16 cm (D) 18 cm
1 
1 
17. A flying kite is tied to a point on the ground with the help of a string. The string 
makes an angle θ with the ground level such that tan θ = 
12
5 . If the length of the
string is 52 m, then the height of the kite above the ground is: 
(A) 40 m (B) 45.5 m (C) 48 m (D) 50 m
1 
*There is no change in the Question Paper Design and Assessment Pattern for Academic Session 2026-27
6. The perimeters of two similar triangles are 56  cm and 70 cm respectively. If one
side of the first triangle is 14 cm, then the corresponding side of the second triangle
(in cm) is:  
(A) 5 (B) 7.5 (C) 10 (D) 17.5
14.  PM is a median of  Δ PQR with vertices P (5,  – 6),  Q (6,  4) and R (0,  0).  The
length of PM is:
(A) 23 (B) 26 (C) 25 (D) 30
16. The mean and median of the data are 35.5 and 32 respectively. The value of
mode for this data is: 
                                                                                                    Page 3 of 6 
 
 
18. 2 cards of diamonds and 4 cards of spades are missing from a pack of 52 cards. 
A card is drawn at random from this pack. The probability of getting a card of 
heart is: 
         (A) 
13
52      (B) 
13
46                  (C) 
11
52        (D) 
11
46   
1 
 Assertion-Reason Based Questions 
Directions: Questions number 19 and 20 ar e Assertion and Reason based 
questions carrying 1 mark each. Two statements are given, one labelled 
Assertion (A) and the other labelled Reason (R). 
Select the correct answer from the options (A), (B), (C) and (D) as given below:   
(A) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct 
explanation of Assertion (A). 
(B) Both Assertion (A) and Reason (R) are true but Reason (R) is not the 
correct explanation of Assertion (A). 
(C) Assertion (A) is true but Reason (R) is false. 
(D) Assertion (A) is false but Reason (R) is true. 
 
 
 
 
 
 
 
 
 
 
19. Assertion (A): If three vertices of a parallelogram taken in order are (–1, –6), (2, –
5) and (7, 2), then its fourth vertex is (4,1). 
Reason (R):    Diagonals of a parallelogram bisect each other. 
1 
20. Assertion (A): If zeroes of the polynomial (2k – 1) x2 + 4x – 3 are reciprocal of each 
other, then k = –1. 
Reason (R) :    If a = c , then zeroes of the polynomial ax2 + bx + c , a ≠ 0 are 
reciprocal of each other. 
1 
                                                        SECTION – B                                  (5 x 2 = 10) 
This section comprises of 5 Very Short Answer (VSA) type questions of 2 marks each. 
21(A). A hall has a length of 9.75 m, breadth of 6.75 m and height of 5.25 m. What is the 
length of the longest unmarked ruler that can exactly measure the  dimensions of 
the hall? 
2 
 
 OR 
21(B). There are three bells placed at different swings in a park, which toll at inter vals of 
5, 6 and 8 minutes, respectively. They all toll together when the park opens for the 
visitors at 10:00 a.m.. How many more times do they all toll together till the park is 
closed at 7:00 p.m.?   
22(A). ABCD is a trapezium in which AB is parallel to DC and its diagonals intersect each 
other at the point O. Show that 
AO
BO = 
CO
DO . 
2 
 
 OR 
22(B). In the given figure, if ∠PQR = ∠QSP, PQ = 6 cm and PS = 3 cm, then find the length 
of PR. 
 
 
 
 
 
For visually Impaired Candidates: 
In a right-angled triangle ABC, right -angled at B, a perpendicular BD is drawn  to 
the hypotenuse AC. Prove that the triangle BCD is similar to the triangle ACB. 
*There is no change in the Question Paper Design and Assessment Pattern for Academic Session 2026-27                     
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23. 
If tanθ = 
1
√5, then find the value of 
cosec2θ −  sec2θ 
cosec2θ + sec2θ . 
2 
24. Find the area of a sector of  a circle with diameter 28 cm, If  the length of the 
corresponding arc is 22 cm. 
2 
25. Two dice are thrown simultaneously. Find the probability that the product of the 
numbers appearing on them is a prime number.  
2 
 SECTION – C  (6 x 3 = 18) 
This section comprises of 6 Short Answer (SA) type questions of 3 marks each. 
26. Given that √3 is irrational number, prove that (2 + 3 √3) is an irrational number. 3 
27. If α and β are the zeroes of the quadratic polynomial 2 𝑥2 − 8𝑥 + 5, then find the 
value of     (𝛼 + 
1
𝛽) × (𝛽 + 
1
𝛼 ). 
3 
28(A). Ridhi drew a polygon with n sides. The smallest exterior angle is 8 ° and each  
subsequent exterior angl e is 4 ° more than the previous exterior angle. Find  the
number of sides of the polygon that Ridhi had drawn. 
3 
OR 
28(B). Find the sum of integers between 1 and 400 that are multiples of 4 as well as of 5. 
29. Prove that the parallelogram circumscribing a circle is a rhombus. 3 
30. In what ratio does the x –axis divide the line segment jo ining the points ( – 4, – 6) 
and (–1, 7)? Also, find the coordinates of the point of division.  
3 
31(A). During a math class, Ms. Isha wrote the expression given below on the board and 
asked the students to simplify it. 
cos θ
1 − sin θ +  1 − sin θ
cos θ
Jyoti solved it in her note book as follows: 
cos θ
1−sin θ + 
1−sin θ
cos θ
 = 
cos2θ+(1−sinθ)2
(1−sin θ)×cosθ  …. (step 1) 
 = 
cos2θ+cos2θ
(1−sin θ)×cosθ  …. (step 2) 
 = 
2cos2θ
(1−sin θ)×cosθ  …. (step 3) 
 = 
2cosθ
(1−sin θ)  …. (step 4) 
Identify the step in which Jyoti has made error(s), if any. Rectify the same and find 
the correct answer. 
3 
OR 
31(B). 
If 
1
sin 𝑥 − cos 𝑥 =
cosec 𝑥
√2 , then prove that (
1
sin 𝑥 + cos 𝑥)
2
=
sec2𝑥
2
*There is no change in the Question Paper Design and Assessment Pattern for Academic Session 2026-27
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 SECTION – D  (4 x 5= 20) 
This section comprises of 4 Long Answer (LA) type questions of 5 marks each. 
32. In a rectangular park measuring 50 m × 40 m, the gram panchayat plans to 
construct a rectangu lar swimming pool in the middle , surrounded by a uniform -
width grass strip throughout the park. Find the dimensions of the swimming pool, if 
the area of the grassy strip is 1184 m². 
5 
33. Prove that a line drawn parallel to one side of a triangle to inter sect the other two 
sides in distinct points, divides the other two sides in the same ratio.  
Using the above theorem solve the following: 
PQRS is a trapezium with PQ || SR. X and Y are points on non-parallel sides PS 
and QR respectively such that XY || PQ. Show that 
PX
XS =
QY
YR  . 
5 
34(A). A tent is in the shape of a cylinder surmounted by a conical top. If the height  and 
radius of the cylindrical part are 3 m and 14 m respectively, and the  total height of 
the tent is 13.5 m, then find the area of th e canvas required for  making the tent,  
keeping a provision of 26 m2 of canvas for stitching and wastage. Also, find the cost 
of the canvas to be purchased at the rate of ₹ 250 per m2.
5 
OR 
34(B). A solid wooden toy is in the form of a hemisphere surmo unted by a cone of  the
same radius. The radius of hemisphere is 3.5 cm and the total wood used in making 
the toy is 166 
5
6 cm3. Find the height of the conical part. Also, find the cost of painting 
the hemispherical part of the toy at the rate of ₹ 15 per cm2. 
5 
OR 
35(B). The angles of depression of two ships from the top of a lighthouse and on the same 
side of it are found to be 45 0 and 300. If the ships are 200m apart  and one ship is 
exactly behind the other then find the height of lighthouse.  
 SECTION – E  (3 x 4 = 12) 
This section comprises of 3 case-study-based questions of 4 marks each with sub parts. 
 Each case study questions has three sub parts (i), (ii), (iii) of marks 1, 1, 2 respectively.  
36. A student entrepreneur started a company that manufactures sanitizers in two sizes 
– small and large. The cost of a small bottle of sanitizer is ₹10 and that of a large
bottle is ₹15. In June, the company sold 1000 bottles and recorded a total sale of
₹12,750. Seeing the increased demand, the company decided to increase the price
of both sanitizer bo ttles by ₹2 each. In the next month, the company sold 2500
bottles and recorded a total sale of ₹34,250.
Based on the above information, answer the following questions:
(i) Form a linear equation in two variables representing the sale for June.
(ii) Form a linear equation in two variables representing the sale for July.
(iii) (A) How many sanitizer bottles of each type were actually sold in June?
OR 
(iii) (B) How many sanitizer bottles of each type were sold in July?
1 
1 
2 
2 
*There is no change in the Question Paper Design and Assessment Pattern for Academic Session 2026-27
35(A). Manish is standing on level ground and observes a kite flying 200 m away from  
him at an angle of elevation of 300. Mahesh, standing on the roof of a 50 m high  
building on the opposite side of the kite, observes the same kite at an angle of  
elevation of 450. Find the distance between the kite and Mahesh. 
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37. 
 
 
 
 
 
 
 
 
 
 
(
i
) 
In a technolog y park in Hyderabad, a company installed a circular meditation 
garden for its employees as shown in the figure.  
 
 
 
 
 
 
 
 
Arjun, a designer, stood at a point P outside the garden and drew two tangents PA 
and PB to the circular boundary. The radius of the garden is 9 m, and the distance 
of point P from the centre O is 18 m. For preparing safety guidelines, Arjun needed 
the lengths of the tangents. During a demonstration to interns, he asked them to 
compute the angle formed between the two tangents using the properties of circle. 
Based on the above information, answer the following questions: 
(i)   What is the measure of ∠OAP? 
(ii)   Calculate the length of the tangent PA. 
(iii)    (A) Calculate the measure of ∠AOB. 
OR 
(iii)     (B) In the above design, if PA and PB were inclined at 60°, then what 
would be their length? 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
1 
1 
2 
 
 
2 
38. 
 
 
 
 
 
 
 
A school conducted a weekly test for Class X students, before the commencement 
of the pre-board examination and recorded their scores (out of 50). To analyse 
performance patterns, the academic coordinator grouped the marks into intervals. 
The grouped frequency distribution is as below: 
 
Marks Obtained 0 – 10 10 – 20 20 – 30 30 – 40 40 – 50 
Number of students 3 6 12 15 14 
 
The academic coordinator computed the central te ndencies to judge overall 
learning level, the most common performance range and to understand consistency 
across the batch for planning the remedial sessions. 
Based on the above information, answer the following questions: 
(i) Identify the class with the most common performance range. 
(ii) Find the class interval containing the median. 
(iii)    (A) What is the average performance of the students? 
OR 
(iii)        (B) Find the mode of the data. 
 
 
 
 
 
 
 
 
 
 
 
 
 
1 
1 
2 
 
2 
 
 
*There is no change in the Question Paper Design and Assessment Pattern for Academic Session 2026-27