MATHEMATICS (STANDARD) — Code 041

Sample Question Paper • Class X • Academic Session 2025–26

Maximum Marks: 80 Time: 3 hours

General Instructions

1. This question paper contains 38 questions. All questions are compulsory.
2. The paper is divided into five Sections A, B, C, D and E.
3. Section A: Questions 1–18 are MCQs and Questions 19–20 are Assertion–Reason (1 mark each).
4. Section B: Questions 21–25 are Very Short Answer (VSA) (2 marks each).
5. Section C: Questions 26–31 are Short Answer (SA) (3 marks each).
6. Section D: Questions 32–35 are Long Answer (LA) (5 marks each).
7. Section E: Questions 36–38 are Case Study (4 marks each; subparts 1,1,2).
8. No overall choice. Internal choice is provided as specified in the paper.
9. Draw neat figures where required. Take \(\pi=\dfrac{22}{7}\) wherever required if not stated.
10. Use of calculators is not allowed.

Section A

Section A consists of 20 questions of 1 mark each

1. If \(a=2^{2}\cdot 3^{x},\; b=2^{2}\cdot 3\cdot 5,\; c=2^{2}\cdot 3\cdot 7\) and \(\operatorname{LCM}(a,b,c)=3780\), then \(x\) equals
   1. 1
   2. 2
   3. 3
   4. 0
2. The shortest distance (in units) of the point \((2,3)\) from the \(y\)-axis is
   1. 2
   2. 3
   3. 5
   4. 1
3. If the lines \(3x+2ky=2\) and \(2x+5y+1=0\) are not parallel, then \(k\) has to be
   1. \(\dfrac{15}{4}\)
   2. \(\ne \dfrac{15}{4}\)
   3. any rational number
   4. any rational number having 4 as denominator
4. A quadrilateral \(ABCD\) is drawn to circumscribe a circle. If \(BC=7\,\text{cm},\, CD=4\,\text{cm}\) and \(AD=3\,\text{cm}\), then the length of \(AB\) is
   1. 3 cm
   2. 4 cm
   3. 6 cm
   4. 7 cm
5. If \(\sec\theta+\tan\theta=x\), then \(\sec\theta-\tan\theta\) will be
   1. \(x\)
   2. \(x^2\)
   3. \(\dfrac{2}{x}\)
   4. \(\dfrac{1}{x}\)
6. Which of the following is not a quadratic equation?
   1. \((x+2)^2=2(x+3)\)
   2. \(x^2+3x=(-1)(1-3x)^2\)
   3. \(x^3-x^2+2x+1=(x+1)^3\)
   4. \((x+2)(x+1)=x^2+2x+3\)

7. The picture of Olympic rings is made by five congruent circles of radius \(1\,\text{cm}\) intersecting so that the chord joining the points of intersection has length \(1\,\text{cm}\). Total area of all dotted regions (ring thickness negligible) is  

   

   1. \(4\left(\dfrac{\pi}{12} - \dfrac{\sqrt{3}}{4}\right)\,\text{cm}^2\)
   2. \(\left(\dfrac{\pi}{6} - \dfrac{\sqrt{3}}{4}\right)\,\text{cm}^2\)
   3. \(4\left(\dfrac{\pi}{6} - \dfrac{\sqrt{3}}{4}\right)\,\text{cm}^2\)
   4. \(8\left(\dfrac{\pi}{6} - \dfrac{\sqrt{3}}{4}\right)\,\text{cm}^2\)
   Alternative Question for Visually Impaired

   The area of the circle inscribed in a square of side \(6\,\text{cm}\) is:

   1. \(36\pi\)
   2. \(18\pi\)
   3. \(12\pi\)
   4. \(9\pi\) \(\text{cm}^2\).
8. A pair of dice is tossed. The probability of not getting the sum eight is
   1. \(\dfrac{5}{36}\)
   2. \(\dfrac{31}{36}\)
   3. \(\dfrac{5}{18}\)
   4. \(\dfrac{5}{9}\)
9. If \(2\sin(5x)=\sqrt{3},\; 0^\circ\le x\le90^\circ\), then \(x=\,?\)
   1. \(10^\circ\)
   2. \(12^\circ\)
   3. \(20^\circ\)
   4. \(50^\circ\)
10. The sum of two numbers is 1215 and their HCF is 81. The possible pairs of such numbers are
    1. 2
    2. 3
    3. 4
    4. 5
11. If the area of the base of a right circular cone is \(51\,\text{cm}^2\) and its volume is \(85\,\text{cm}^3\), then the height of the cone equals
    1. \(\dfrac{5}{6}\,\text{cm}\)
    2. \(\dfrac{5}{3}\,\text{cm}\)
    3. \(\dfrac{5}{2}\,\text{cm}\)
    4. \(5\,\text{cm}\)
12. If the zeroes of \(a x^2+bx+c\;(a,c\ne0)\) are equal, then
    1. \(c\) and \(b\) have opposite signs
    2. \(c\) and \(a\) have opposite signs
    3. \(c\) and \(b\) have same signs
    4. \(c\) and \(a\) have same signs
13. The area (in \(\text{cm}^2\)) of a sector of radius \(21\,\text{cm}\) cut off by an arc of length \(22\,\text{cm}\) is
    1. 441
    2. 321
    3. 231
    4. 221
14. If \(\triangle ABC\sim\triangle DEF,\; AB=6\,\text{cm},\, DE=9\,\text{cm},\, EF=6\,\text{cm},\, FD=12\,\text{cm}\), then the perimeter of \(\triangle ABC\) is
    1. 28 cm
    2. 28.5 cm
    3. 18 cm
    4. 23 cm
15. If the probability that a randomly chosen letter from “Mathematics” is a vowel is \(\dfrac{2}{2x+1}\), then \(x=\,?\)
    1. \(\dfrac{4}{11}\)
    2. \(\dfrac{9}{4}\)
    3. \(\dfrac{11}{4}\)
    4. \(\dfrac{4}{9}\)
16. The points \(A(9,0),\, B(9,-6),\, C(-9,0),\, D(-9,6)\) are vertices of a
    1. Square
    2. Rectangle
    3. Parallelogram
    4. Trapezium
17. The median of a set of 9 distinct observations is 20.5. If each observation is increased by 2, then the new median
    1. increases by 2
    2. decreases by 2
    3. becomes twice the original
    4. remains the same
18. The length of a tangent drawn to a circle of radius \(9\,\text{cm}\) from a point at a distance of \(41\,\text{cm}\) from the centre is
    1. 40 cm
    2. 9 cm
    3. 41 cm
    4. 50 cm

DIRECTIONS: In the question number 19 and 20, a statement of Assertion (A) is followed by a statement of Reason (R). Choose the correct option:

1. Both assertion (A) and reason (R) are true and reason (R) is the correct explanation of assertion (A)
2. Both assertion (A) and reason (R) are true and reason (R) is not the correct explanation of assertion (A)
3. Assertion (A) is true but reason (R) is false.
4. Assertion (A) is false but reason (R) is true.

1. Assertion (A): The number \(5^n\) cannot end with the digit 0, where \(n\in\mathbb{N}\).
   Reason (R): A number ends with 0 if its prime factorization contains both 2 and 5. 
    
2. Assertion (A): If \(\cos A+\cos^2A=1\), then \(\sin^2A+\sin^2 2A=1\).
   Reason (R): \(\sin^2A+\cos^2A=1\). 
    

SECTION- B

Section B consists of 5 questions of 2 marks each

21. (A) The A.P. \(8,10,12,\dots\) has 60 terms. Find the sum of the last 10 terms.
     

    OR

    
    (B) Find the middle term of A.P. \(6,13,20,\dots,230\).

22. If \(\sin(A + B) = 1\) and \(\cos(A - B) = \dfrac{\sqrt{3}}{2}\), \(0^\circ < A,B < 90^\circ\), find the measure of angles \(A\) and \(B\).
23. If \(AP\) and \(DQ\) are medians of \(\triangle ABC\) and \(\triangle DEF\) respectively, where \(\triangle ABC\sim\triangle DEF\), prove that \(\dfrac{AB}{DE}=\dfrac{AP}{DQ}\).

24. (A) A horse, a cow and a goat are tied by ropes of length \(14\,\text{m}\) at corners \(A,B,C\) respectively of a grassy triangular field \(ABC\) with sides \(35\,\text{m}, 40\,\text{m}, 50\,\text{m}\). Find the total grazed area.
     

    OR

    
    (B) Find the area of the major segment (in terms of \(\pi\)) of a circle of radius \(5\,\text{cm}\) formed by a chord subtending \(90^\circ\) at the centre.

25. A \(\triangle ABC\) is drawn to circumscribe a circle of radius \(4\,\text{cm}\) such that \(BD=10\,\text{cm}\) and \(DC=8\,\text{cm}\). If \(\operatorname{ar}(\triangle ABC)=90\,\text{cm}^2\), find \(AB\) and \(AC\).

    

    Alternative Question for Visually Impaired

    A circle is inscribed in a right triangle \(ABC\) right-angled at \(B\). If \(BC=7\,\text{cm}\) and \(AB=24\,\text{cm}\), find the inradius.

SECTION- C

Section C consists of 6 questions of 3 marks each.

26. In the figure, \(XY\) and \(X'Y'\) are two parallel tangents to a circle with centre \(O\) and another tangent \(AB\) with point of contact \(C\) intersects \(XY\) at \(A\) and \(X'Y'\) at \(B\). Prove that \(\angle AOB=90^\circ\).

    

    Alternative Question for Visually Impaired

    Two tangents \(PA\) and \(PB\) are drawn from an external point \(P\) to a circle with centre \(O\). Prove that \(\angle APB=2\angle OAB\).

27. In a workshop, the numbers of teachers of English, Hindi and Science are 36, 60 and 84 respectively. Find the minimum number of rooms required, if in each room the same number of teachers are seated and all are of the same subject.
28. Find the zeroes of \(2x^2-(1+2\sqrt{2})x+\sqrt{2}\) and verify the relation between zeroes and coefficients.

29. If \(\sin\theta+\cos\theta=\sqrt{3}\), prove that \(\tan\theta+\cot\theta=1\).
     

    OR

    Prove that \(\dfrac{\cos A-\sin A+1}{\cos A+\sin A-1}=\csc A+\cot A\).

30. On a particular day, Vidhi and Unnati couldn’t decide on who would get to drive the car. They had one coin each and flipped their coin exactly three times. The following was agreed upon:
    Rule I: If Vidhi gets two heads in a row, she would drive the car; 
    Rule II: If Unnati gets a head immediately followed by a tail, she would drive the car. 
    Who has greater probability to drive the car that day? Justify your answer.

31. (A) The monthly incomes of Aryan and Babban are in the ratio \(3:4\) and their expenditures are in the ratio \(5:7\). If each saves ₹\(15{,}000\) per month , find their monthly incomes.
     

    OR

    
    (B) Solve graphically: \(2x+y=6\), \(2x-y-2=0\). Find the area of the triangle formed by the two lines and the \(x\)-axis.
     

    Alternative Question for Visually Impaired

    Five years hence, fathers age will be three times the age of son. Five years ago, father was seven times as old as his son. Find their present ages.

SECTION- D

Section D consists of 4 questions of 5 marks each

32. A train travels \(63\,\text{km}\) at average speed \(v\) and then \(72\,\text{km}\) at \(v+6\,\text{km/h}\). If the total time is 3 hours, find \(v\). [5]
33. Prove: If a line is drawn parallel to one side of a triangle to intersect the other two sides at distinct points, the other two sides are divided in the same ratio (Basic Proportionality Theorem). Hence, in \(\triangle PQR\), a line \(\ell\) intersects \(PQ\) and \(PR\) at \(L\) and \(M\) respectively with \(LM\parallel QR\). If \(PL=5.7\,\text{cm},\, PQ=15.2\,\text{cm},\, MR=5.5\,\text{cm}\), find \(PM\) (in cm).

34. (A) From a solid right circular cone (height \(6\,\text{cm}\), base radius \(12\,\text{cm}\)), a right circular cylindrical cavity (height \(3\,\text{cm}\), radius \(4\,\text{cm}\)) is hollowed out such that the bases form concentric circles. Find the surface area of the remaining solid (in terms of \(\pi\)).
     

    OR

    
    (B) An empty cone (radius \(3\,\text{cm}\), height \(12\,\text{cm}\)) is filled with ice-cream such that the lower part equal to one-sixth of the cone’s volume is unfilled, but a hemisphere is formed on the top. Find the volume of ice-cream.

35. (A) If the mode of the following distribution is \(55\), find \(x\). Hence, find the mean.

    Class Interval	Frequency
    0–15	10
    15–30	7
    30–45	\(x\)
    45–60	15
    60–75	10
    75–90	12

     

    OR

    
    (B) Heights (in cm) of \(51\) girls were recorded as a “less than” cumulative frequency:

     

    Heights (cm)	Number of girls
    less than 140	04
    less than 145	11
    less than 150	29
    less than 155	40
    less than 160	46
    less than 165	51

    Find the median height. If the mode is \(148.05\), find the mean using the empirical formula.

SECTION- E

Section E consists of 3 case study-based questions of 4 marks each.

36. In a class, the teacher asks every student to write an example of A.P. Two boys Aryan and Roshan writes the progression as \(-5,-2,1,4,\dots\) and \(187,184,181,\dots\) respectively. Now the teacher asks his various students the following questions on progression.
    Help the students to find answers for the following:
    1. Find the sum of the common difference of two progressions.
    2. Find the 34th term of Roshan’s progression.

    3. (A) Find the sum of first 10 terms of Aryan’s progression.
        

       OR

       (B) Which term of the two progressions are equal?

37. A group of class X students goes to picnic during winter holidays. The position of three friends Aman, Kirti and Chahat are shown by the points \(P,Q\) and \(R\).

    

    1. Find the distance between \(P\) and \(R\).
    2. Is \(Q\) the midpoint of \(PR\)? Justify by finding the midpoint of \(PR\).

    3. (A) Find the point on the \(x\)-axis equidistant from \(P\) and \(Q\).
        

       OR

       
       (B) Let \(S\) be a point which divides \(PQ\) in ratio \(2:3\). Find the coordintates of \(S\).

    Alternative Question for Visually Impaired

    A group of class X students goes to picnic during winter holidays. Aman, Kirti and Chahat are three friends. The position of three friends Aman, Kirti and Chahat are shown by the points P, Q and R. The coordinates of \(P(2,5),\; Q(4,4),\; R(8,3)\) are given.

    1. Find the distance between \(P\) and \(R\).
    2. Is \(Q\) the midpoint of \(PR\)? Justify by finding the midpoint of \(PR\).

    3. (A) Find the point on the \(x\)-axis equidistant from \(P\) and \(Q\).
        

       OR

       
       (B) Let \(S\) be a point which divides \(PQ\) in ratio \(2:3\). Find the coordintates of \(S\).

38. India gate (formerly known as All India war memorial) is located near Karthavya path. (formerly Rajpath) at New Delhi. It stands as a memorial to 74187 soldiers of Indian Army, who gave their life in the first world war. This \(42\) \(m\) tall structure was designed by Sir Edwin Lutyens in the style of Roman triumphal arches. A student Shreya of height \(1\) \(m\) visited India Gate as a part of her study tour.

    

    1. What is the angle of elevation from Shreya’s eye to the top of India Gate, if she is standing at a distance of \(41\) \(m\) away from the India Gate?
    2. If Shreya observes the angle of elevation from her eye to the top of India Gate to be \(60^\circ\), then how far is the she standing from the base of the India Gate?

    3. (A) If the angle of elevation from Shreya’s eye changes from 45° to 30°, when she moves some distance back from the original position. Find the distance she moves back.
        

       OR

       
       (B) If Shreya moves to a point which is at a distance of \(\dfrac{41}{\sqrt{3}}\) \(m\) from the India Gate, then find the angle of elevation made by her eye to the top of India Gate.

Click HERE to download the PDF version of the Mathematics (Standard) Sample Paper for Class 10 for 2026 directly from the CBSE website (https://cbseacademic.nic.in).

Click HERE for the Science Sample Paper.