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CBSE Sample Question Paper (2024-25) CLASS X (Math Standard 041)

Max. Marks: 80

Time Allowed: 3 hours

General Instructions:

Read the following instructions carefully and follow them:

  1. This question paper contains 38 questions.
  2. This Question Paper is divided into 5 Sections A, B, C, D and E.
  3. In Section A, Questions no. 1-18 are multiple choice questions (MCQs) and questions no. 19 and 20 are Assertion- Reason based questions of 1 mark each.
  4. In Section B, Questions no. 21-25 are very short answer (VSA) type questions, carrying 02 marks each.
  5. In Section C, Questions no. 26-31 are short answer (SA) type questions, carrying 03 marks each.
  6. In Section D, Questions no. 32-35 are long answer (LA) type questions, carrying 05 marks each.
  7. In Section E, Questions no. 36-38 are case study-based questions carrying 4 marks each with sub-parts of 1, 1 and 2 marks each respectively.
  8. All Questions are compulsory. However, an internal choice in 2 Questions of Section B, 2 Questions of Section C and 2 Questions of Section D has been provided. An internal choice has been provided in all the 2 marks questions of Section E.
  9. Draw neat and clean figures wherever required.
  10. Take $\pi = {{22} \over 7}$ wherever required if not stated.
  11. Use of calculators is not allowed.

Section-A

Section A consists of 20 questions of 1 mark each.

  1. The graph of a quadratic polynomial $p(x)$ passes through the points $\left( { - 6,0} \right)$, $\left( {0, - 30} \right)$, $\left( {4, - 20} \right)$ and $\left( {6,0} \right)$. The zeroes of the polynomial are
    1. $ - 6,{\rm{ }}0$
    2. $4,{\rm{ }}6$
    3. $ - 30,{\rm{ }} - 20$
    4. $ - 6,{\rm{ }}0$
  2. The value of k for which the system of equations  $3x - ky = 7$  and  $6x + 10y = 3$  is inconsistent, is
    1. $ - 10$
    2. $ - 5$
    3. $5$
    4. $7$
  3. Which of the following statements is not true?
    1. A number of secants can be drawn at any point on the circle.
    2. Only one tangent can be drawn at any point on a circle.
    3. A chord is a line segment joining two points on the circle.
    4. From a point inside a circle only two tangents can be drawn.
  4. If  ${n^{th}}$  term of an A.P. is $7n - 4$ then the common difference of the A.P. is
    1. $7$
    2. $7n$
    3. $ - 4$
    4. $4$
  5. The radius of the base of a right circular cone and the radius of a sphere are each  $5$ cm in length. If the volume of the cone is equal to the volume of the sphere then the height of the cone is
    1. $5$ cm
    2. $20$ cm
    3. $10$ cm
    4. $4$ cm
  6. If  $\tan \theta = {5 \over 2}$ then ${{4\sin \theta + \cos \theta } \over {4\sin \theta - \cos \theta }}$ is equal to
    1. ${{11} \over 9}$
    2. ${3 \over 2}$
    3. ${9 \over {11}}$
    4. $4$

  7. In the given figure, a tangent has been drawn at a point  ${\rm{P}}$  on the circle centred at  ${\rm{O}}$.

    image of a circle with tangent at the extremity of a chord and central angle subtended by the chord

    If $\angle {\rm{TPQ}} = {110^o}$ then $\angle {\rm{POQ}}$ is equal to

    1. ${110^o}$
    2. ${70^o}$
    3. ${140^o}$
    4. ${55^o}$
  8. A quadratic polynomial having zeroes  $ - \sqrt {{5 \over 2}} $ and $\sqrt {{5 \over 2}} $ is
    1. ${x^2} - 5\sqrt 2 x + 1$
    2. $8{x^2} - 20$
    3. $15{x^2} - 6$
    4. ${x^2} - 2\sqrt 5 x - 1$

  9. Consider the frequency distribution of 45 observations.

    Class0 - 1010 - 2020 - 3030 - 4040 - 50
    Frequency5915106

    Ther upper limit of the median class is

    1. $20$
    2. $10$
    3. $30$
    4. $40$

  10. ${\rm{O}}$  is the point of intersection of two chords ${\rm{AB}}$  and  ${\rm{CD}}$  of a circle.

    image of a circle with chords AB and CD intersecting at O

    If $\angle {\rm{BOC}} = {80^{\rm{o}}}$ and ${\rm{OA}} = {\rm{OD}}$ then $\Delta {\rm{ODA}}$ and $\Delta {\rm{OBC}}$ are

    1. equilateral and similar
    2. isosceles and similar
    3. isosceles but not similar
    4. not similar
  11. The roots of the quadratic equation ${x^2} + x - 1 = 0$ are
    1. irrational and distinct
    2. not real
    3. rational and distinct
    4. real and equal
  12. If  $\theta = {30^{\rm{o}}}$,  then the value of  $3\tan \theta $  is
    1. $1$
    2. ${1 \over {\sqrt 3 }}$
    3. ${3 \over {\sqrt 3 }}$
    4. not defined
  13. The volume of a solid hemisphere is  ${{396} \over 7}$ cm3. The total surface area of the solid hemisphere (in sq.cm) is
    1. ${{396} \over 7}$
    2. ${{594} \over 7}$
    3. ${{549} \over 7}$
    4. ${{604} \over 7}$
  14. In a bag containing $24$ balls, $4$ are blue, $11$ are green and the rest are white. One ball is drawn at random. The probability that the drawn ball is white in colour is
    1. ${1 \over 6}$
    2. ${3 \over 8}$
    3. ${{11} \over {24}}$
    4. $5\over 8$
  15. The point on the $x$- axis nearest to the point $\left( { - 4, - 5} \right)$ is
    1. $\left( { 0, 0} \right)$
    2. $\left( { - 4, 0} \right)$
    3. $\left( { - 5, 0} \right)$
    4. $\left( {\sqrt {41} ,0} \right)$
  16. Which of the following gives the middle most observation of the data?
    1. Median
    2. Mean
    3. Range
    4. Mode
  17. A point on the x-axis divides the line segment joining the points  ${\rm{A}}\left( {2, - 3} \right)$  and  ${\rm{B}}\left( {5,6} \right)$  in the ratio  $1:2$.  The point is
    1. $\left( {4,0} \right)$
    2. $\left( {{7 \over 2},{3 \over 2}} \right)$
    3. $\left( {3,0} \right)$
    4. $\left( {0,3} \right)$
  18. A card is drawn from a well-shuffled deck of playing cards. The probability of getting red face card is
    1. ${3 \over {13}}$
    2. ${1 \over 2}$
    3. ${3 \over {52}}$
    4. ${3 \over {26}}$

Question No.19 and 20 consist of two statements - Assertion (A) and Reason (R). Answer these questions by selecting the appropriate option given below:

  1. Both A and R are true, and R is the correct explanation of A.
  2. Both A and R are true, and R is not the correct explanation of A.
  3. A is true but R is false.
  4. A is false but R is true

 

  1. Assertion (A): HCF of any two consecutive even natural numbers is always  $2$.

    Reason (R): Even natural numbers are divisible by  $2$.

  2. Assertion (A): If the radius of sector of a circle is reduced to its half and angle is doubled then the perimeter of the sector remains the same.

    Reason (R): The length of the arc subtending angle  $\theta$  at the centre of a circle of radius  $r$  is  ${{\pi r\theta } \over {{{180}^{\rm{o}}}}}$.

 

Section-B

Question No. 21 to 25 are of two marks each 

  1.  

    1. Find the H.C.F and L.C.M of  $480$  and  $720$  using the Prime factorisation method

    OR

    1. The H.C.F of  $85$  and  $238$  is expressible in the form  $85m - 238$.  Find the value of  $m$.

  2.  

    1. Two dice are rolled together bearing numbers  $4, 6, 7, 9, 11, 12$. Find the probability that the product of numbers obtained is an odd number.

    OR

    1. How many positive three-digit integers have the hundredth digit  $8$  and unit's digit  $5$ ? Find the probability of selecting one such number out of all three-digit numbers.
  3. Evaluate:  ${{2{{\sin }^2}{{60}^{\rm{o}}} - {{\tan }^2}{{30}^{\rm{o}}}} \over {{{\sec }^2}{{45}^{\rm{o}}}}}$
  4. Find the point(s) on the $x$-axis which is at a distance of  $\sqrt {41} $ units from the point $\left( {8, - 5} \right)$.
  5. Show that the points  ${\rm{A}}\left( { - 5,6} \right)$, ${\rm{B}}\left( {3,0} \right)$ and ${\rm{C}}\left( {9,8} \right)$  are the vertices of an isosceles triangle.

 

Section-C

Question No. 26 to 31 are of three marks each 

  1.  

    1. In $\Delta {\rm{ABC}}$,  ${\rm{D}}$,  ${\rm{E}}$  and  ${\rm{F}}$  are the mid points of  ${\rm{BC}}$ ,  ${\rm{CA}}$  and  ${\rm{AB}}$  respectively. Prove that $\Delta {\rm{FBD}} \sim \Delta {\rm{DEF}}$ and $\Delta {\rm{DEF}} \sim \Delta {\rm{ABC}}$.

      a scalene triangle ABC with mid points D, E, F joined

    OR

    1. In $\Delta {\rm{ABC}}$,  ${\rm{P}}$  and  ${\rm{Q}}$  are points on  ${\rm{AB}}$  and  ${\rm{AC}}$  respectively such that   ${\rm{PQ}}$  is parallel to  ${\rm{BC}}$ .Prove that the median  ${\rm{AD}}$  drawn from  ${\rm{A}}$  on  ${\rm{BC}}$  bisects  ${\rm{PQ}}$. 

      a scalene triangle ABC with median AD intersecting PQ at R

  2. The sum of two numbers is $18$ and the sum of their reciprocals is ${9 \over {40}}$. Find the numbers.
  3. If $\alpha $ and $\beta $ are zeroes of a polynomial $6{x^2} - 5x + 1 = 0$ then form a quadratic polynomial whose zeroes are ${\alpha ^2}$ and ${\beta ^2}$.
  4. If $\cos \theta + \sin \theta = 1$, then prove that $\cos \theta - \sin \theta = \pm 1$.

  5.  

    1. The minute hand of a wall clock is 18 cm long. Find the area of the face of the clock described by the minute hand in 35 minutes.

    OR

    1. AB is a chord of a circle centred at O such that $\angle {\rm{AOB}} = {60^{\rm{o}}}$. If OA = 14 cm then find the area of the minor segment. (take $\sqrt 3 = 1.73$) 

      a circle with chord AB subtending 60 degrees at the centre

  6. Prove that $\sqrt 3 $ is an irrational number.

 

Section-D

Question No. 32 to 35 are of five marks each 

  1.  

    1. Solve the following system of linear equations graphically: $x + 2y = 3$, $2x - 3y + 8 = 0$

    OR

    1. Places  ${\rm{A}}$  and  ${\rm{B}}$  are $180$ km apart on a highway. One car starts from  ${\rm{A}}$  and another from  ${\rm{B}}$  at the same time. If the car travels in the same direction at different speeds, they meet in $9$ hours. If they travel towards each other with the same speeds as before, they meet in one hour. What are the speeds of the two cars?

  2. Prove that the lengths of tangents drawn from an external point to a circle are equal.
    Using the above result, find the length BC of $\Delta {\rm{ABC}}$. Given that, a circle is inscribed in $\Delta {\rm{ABC}}$ touching the sides AB, BC and CA at R, P and Q respectively and $AB = 10$ cm, $AQ = 7$ cm ,$CQ = 5$ cm. 

    a circle inscribed in a triangle

  3. A boy whose eye level is $1.35$ m from the ground, spots a balloon moving with the wind in a horizontal line at some height from the ground. The angle of elevation of the balloon from the eyes of the boy at an instant is  ${60^{\rm{o}}}$.  After $12$ seconds, the angle of elevation reduces to  ${30^{\rm{o}}}$. If the speed of the wind is $3$ m/s then find the height of the balloon from the ground. (use $\sqrt 3 = 1.73$)

  4. Find the mean and median of the following data:

    Class

    Frequency

    85-90

    15

    90-95

    22

    95-100

    20

    100-105

    18

    105-110

    20

    110-115

    25

    OR

    The monthly expenditure on milk in  $200$  families of a Housing Society is given below

    Monthly Expenditure

    (in ₹)

    Number of Families

    1000-1500

    24

    1500-2000

    40

    2000-2500

    33

    2500-3000

    $x$

    3000-3500

    30

    3500-4000

    22

    4000-4500

    16

    4500-5000

    7

    Find the value of $x$ and also find the mean expenditure

 

Section-D

Question No. 36 to 38 are case-based/data-based questions each carrying four marks.

  1. Ms. Sheela visited a store near her house and found that the glass jars are arranged one above the other in a specific pattern. On the top layer, there are $3$  jars. In the next layer, there are $6$  jars. In the 3rd layer from the top there are $9$  jars and so on till the 8th layer. On the basis of the above situation answer the following questions.
    1. Write an AP whose terms represent the number of jars in different layers starting from the top. Also, find the common differencee. (1 Mark)
    2. Is it possible to arrange $34$  jars in a layer if this pattern is continued? Justify your answer. (1 Mark)
    3.  

      1. If there are '$n$' number of rows in a layer then find the expression for finding the total number of jars in terms of $n$. Hence find ${{\rm{S}}_8}$. (2 Marks)

        OR

      2. The shopkeeper added 3 jars in each layer. How many jars are there in the 5th layer from the top? (2 Marks)

  2. a triangle shaped cabinet with shelves

    two triangles ABC and DEF with PQ parallel to EF

    Triangle is a very popular shape used in interior design. The picture given above shows a cabinet designed by a famous interior designer. Here the largest triangle is represented by $\Delta {\rm{ABC}}$ and the smallest one with a shelf is represented by $\Delta {\rm{DEF}}$. $PQ$ is parallel to $EF$.

    1. Show that $\Delta {\rm{DPQ}} \sim \Delta {\rm{DEF}}$. (1 Mark)
    2. If ${\rm{DP}} = 50{\rm{ cm}}$ and ${\rm{PE}} = 70{\rm{ cm}}$,find ${{{\rm{PQ}}} \over {{\rm{EF}}}}$.
    3.  

      1. If $2{\rm{AB}} = 2{\rm{DE}}$ and $\Delta {\rm{DPQ}} \sim \Delta {\rm{DEF}}$, then show that ${{{\rm{perimeter of }}\Delta {\rm{ABC}}} \over {{\rm{perimeter of }}\Delta {\rm{DEF }}}}$ is constant. (2 Marks)

        OR

      2. If AM and DN are medians of triangles ABC and DEF respectively then prove that $\Delta {\rm{ABM}} \sim \Delta {\rm{DEN}}$. (2 Marks)
  3. Metallic silos are used by farmers for storing grains. Farmer Girdhar has decided to build a new metallic silo to store his harvested grains. It is in the shape of a cylinder mounted by a cone. Dimensions of the conical part of a silo is as follows: Radius of base = 1.5 m Height = 2 m Dimensions of the cylindrical part of a silo is as follows: Radius = 1.5 m Height = 7 m On the basis of the above information answer the following questions.
    1. Calculate the slant height of the conical part of one silo. (1 Mark)
    2. Find the curved surface area of the conical part of one silo. (1 Mark)
    3.  

      1. Find the cost of the metal sheet used to make the curved cylindrical part of 1 silo at the rate of Rs 2000 per m2. (2 Marks)

        OR

      2. Find the total capacity of one silo to store grains. (2 Marks)

Click HERE to download the PDF version of the Mathematics (Standard) Sample Paper for Class 10 for 2025 directly from the CBSE website (https://cbseacademic.nic.in).

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