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CBSE CLASS X β MATHEMATICS (STANDARD) SAMPLE QUESTION PAPER
SET β 9 | ACADEMIC SESSION 2026β2027
General Instructions:
- This question paper contains 38 questions divided into 5 Sections: A, B, C, D and E.
- All Questions are compulsory.
- Section A comprises 20 Multiple Choice Questions (MCQs) of 1 mark each (Q1 to Q18 are MCQs and Q19 & Q20 are AssertionβReason based).
- Section B comprises 5 Very Short Answer (VSA) questions of 2 marks each (Q21 to Q25).
- Section C comprises 6 Short Answer (SA) questions of 3 marks each (Q26 to Q31).
- Section D comprises 4 Long Answer (LA) questions of 5 marks each (Q32 to Q35).
- Section E comprises 3 Case-Based Integrated Units of Assessment of 4 marks each (Q36 to Q38) with sub-parts of values 1, 1 and 2 marks respectively. Internal choice is provided in the 2-mark question.
- Use of calculators is strictly prohibited. Use $\pi = 22/7$ wherever required unless stated otherwise.
SECTION A β MULTIPLE CHOICE QUESTIONS (1 Mark Each)
Q1.
The HCF of the smallest prime number and the smallest composite number is:
[1]
Q2.
If one zero of the polynomial $f(x) = (k^2 + 4)x^2 + 13x + 4k$ is reciprocal of the other, then the value of $k$ is:
[1]
Q3.
If the pair of equations $3x – y + 8 = 0$ and $6x – ky = -16$ represents coincident lines, then the value of $k$ is:
[1]
Q4.
The quadratic equation $2x^2 – \sqrt{5}x + 1 = 0$ has:
[1]
Q5.
In an AP, if $d = -4$, $n = 7$, and $a_n = 4$, then the first term $a$ is:
[1]
Q6.
The point on the y-axis which is equidistant from the points $A(6, 5)$ and $B(-4, 3)$ is:
[1]
Q7.
If $\triangle ABC \sim \triangle PQR$ such that $\text{Perimeter}(\triangle ABC) = 36\text{ cm}$, $\text{Perimeter}(\triangle PQR) = 24\text{ cm}$, and $PQ = 10\text{ cm}$, then the length of $AB$ is:
[1]
Q8.
If $\tan\theta = \frac{a}{b}$, then the value of $\frac{a\sin\theta – b\cos\theta}{a\sin\theta + b\cos\theta}$ is:
[1]
Q9.
$(1 + \tan^2\theta)(1 – \sin\theta)(1 + \sin\theta)$ is equal to:
[1]
Q10.
If the ratio of the height of a vertical pole to the length of its shadow on horizontal ground is $\sqrt{3} : 1$, then the angle of elevation of the sun is:
[1]
Q11.
If $PA$ and $PB$ are tangents to a circle with centre $O$ from an external point $P$ such that $\angle APB = 70^\circ$, then $\angle AOB$ is:
[1]
Q12.
If the area of a circle is $154\text{ cm}^2$, then its perimeter (circumference) is:
[1]
Q13.
If a sphere of radius $r$ is melted and recast into a cone of base radius $r$, then the height of the cone is:
[1]
Q14.
If the mean and mode of a frequency distribution are 28 and 16 respectively, then the median is:
[1]
Q15.
A bag contains cards numbered from 1 to 25. A card is drawn at random. The probability that the number on the card is divisible by both 2 and 3 is:
[1]
Q16.
The distance between the points $(0, 5)$ and $(-5, 0)$ is:
[1]
Q17.
Which of the following cannot be the empirical probability of an event?
[1]
Q18.
The decimal expansion of $\frac{23}{2^3 \times 5^2}$ will terminate after:
[1]
Q19.
Assertion (A): The HCF of two numbers is 8 and their product is 384, then their LCM is 48.
Reason (R): For any two positive integers $a$ and $b$, $\text{HCF}(a, b) \times \text{LCM}(a, b) = a \times b$. [1]
(a) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of (A).
(b) Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of (A).
(c) Assertion (A) is true but Reason (R) is false.
(d) Assertion (A) is false but Reason (R) is true.
Reason (R): For any two positive integers $a$ and $b$, $\text{HCF}(a, b) \times \text{LCM}(a, b) = a \times b$. [1]
(a) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of (A).
(b) Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of (A).
(c) Assertion (A) is true but Reason (R) is false.
(d) Assertion (A) is false but Reason (R) is true.
Q20.
Assertion (A): If the coordinates of the mid-point of the line segment joining $A(2, p)$ and $B(q, 4)$ is $(3, 5)$, then $p = 6$ and $q = 4$.
Reason (R): The midpoint coordinates are given by $\left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)$. [1]
Reason (R): The midpoint coordinates are given by $\left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)$. [1]
SECTION B β VERY SHORT ANSWER QUESTIONS (2 Marks Each)
Q21.
Prove that $\sqrt{3}$ is an irrational number.
[2]
Q22.
In $\triangle ABC$, $D$ and $E$ are points on sides $AB$ and $AC$ respectively such that $DE \parallel BC$. If $\frac{AD}{DB} = \frac{3}{4}$ and $AC = 14\text{ cm}$, find the length of $AE$.
[2]
OR
A vertical pole of length $7.5\text{ m}$ casts a shadow $5\text{ m}$ long on the ground and at the same time a tower casts a shadow $24\text{ m}$ long. Find the height of the tower.
Q23.
Find the coordinates of the point of trisection of the line segment joining $A(1, -2)$ and $B(-3, 4)$ that is closer to $A$.
[2]
Q24.
If $\sin(A – B) = \frac{1}{2}$ and $\cos(A + B) = \frac{1}{2}$, where $0^\circ < A + B \le 90^\circ$ and $A > B$, find the values of $A$ and $B$.
[2]
OR
Prove that:
$$\frac{1 + \cos\theta}{\sin\theta} + \frac{\sin\theta}{1 + \cos\theta} = 2\csc\theta$$
Q25.
A quadrilateral $ABCD$ is drawn to circumscribe a circle. Prove that $AB + CD = AD + BC$.
[2]
SECTION C β SHORT ANSWER QUESTIONS (3 Marks Each)
Q26.
Find the zeroes of the quadratic polynomial $f(x) = 4\sqrt{3}x^2 + 5x – 2\sqrt{3}$ and verify the relationship between the zeroes and its coefficients.
[3]
Q27.
Solve the following pair of linear equations for $x$ and $y$:
$$2x + 3y = 7$$
$$(k – 1)x + (k + 2)y = 3k$$
Find the value of $k$ for which the system has infinitely many solutions.
[3]
OR
The perimeter of a rectangular garden is $36\text{ m}$. If the length is $4\text{ m}$ more than its width, find the dimensions of the garden.
Q28.
Find the sum of all multiples of 7 lying between 100 and 500.
[3]
Q29.
Prove the trigonometric identity:
$$\frac{\tan\theta + \sec\theta – 1}{\tan\theta – \sec\theta + 1} = \frac{1 + \sin\theta}{\cos\theta}$$
[3]
Q30.
Prove that the tangents drawn at the ends of a diameter of a circle are parallel.
[3]
OR
In two concentric circles, a chord of length $16\text{ cm}$ of the larger circle touches the smaller circle of radius $6\text{ cm}$. Find the radius of the larger circle.
Q31.
Two dice are rolled simultaneously. Find the probability that:
(a) The product of numbers obtained is 12.
(b) The sum of numbers is at most 5.
(c) The same number appears on both dice (a doublet). [3]
(a) The product of numbers obtained is 12.
(b) The sum of numbers is at most 5.
(c) The same number appears on both dice (a doublet). [3]
SECTION D β LONG ANSWER QUESTIONS (5 Marks Each)
Q32.
A motorboat whose speed in still water is $18\text{ km/h}$ takes 1 hour more to go $24\text{ km}$ upstream than to return downstream to the same spot. Find the speed of the stream.
[5]
OR
Two water taps together can fill a tank in $9\frac{3}{8}\text{ hours}$. The tap of larger diameter takes 10 hours less than the smaller one to fill the tank separately. Find the time in which each tap can separately fill the tank.
Q33.
State and prove Basic Proportionality Theorem (Thales Theorem).
[5]
Q34.
A solid toy is in the form of a hemisphere surmounted by a right circular cone of the same base radius. The height of the cone is $2\text{ cm}$ and the diameter of the base is $4\text{ cm}$. Determine the volume of the toy. If a right circular cylinder circumscribes the toy, find the difference between the volumes of the cylinder and the toy. (Take $\pi = 3.14$).
[5]
OR
A tent is in the shape of a cylinder surmounted by a conical top. If the height and diameter of the cylindrical part are $2.1\text{ m}$ and $4\text{ m}$ respectively, and the slant height of the top is $2.8\text{ m}$, find the area of the canvas used for making the tent. Also, find the cost of the canvas at the rate of βΉ500 per $\text{m}^2$. (Note that the base of the tent will not be covered with canvas).
Q35.
The median of the following frequency distribution is 525. Find the values of $x$ and $y$, if the total frequency is 100:
$$\begin{array}{|c|c||c|c|} \hline \text{Class Interval} & \text{Frequency} & \text{Class Interval} & \text{Frequency} \\ \hline 0-100 & 2 & 500-600 & 20 \\ 100-200 & 5 & 600-700 & y \\ 200-300 & x & 700-800 & 9 \\ 300-400 & 12 & 800-900 & 7 \\ 400-500 & 17 & 900-1000 & 4 \\ \hline \end{array}$$ [5]
$$\begin{array}{|c|c||c|c|} \hline \text{Class Interval} & \text{Frequency} & \text{Class Interval} & \text{Frequency} \\ \hline 0-100 & 2 & 500-600 & 20 \\ 100-200 & 5 & 600-700 & y \\ 200-300 & x & 700-800 & 9 \\ 300-400 & 12 & 800-900 & 7 \\ 400-500 & 17 & 900-1000 & 4 \\ \hline \end{array}$$ [5]
SECTION E β CASE-BASED INTEGRATED UNITS (4 Marks Each)
Q36. Case Study 1 (Arithmetic Progression):
[4]
(b) Find the annual fixed increase in production ($d$). [1]
(c) Find the total production of fans in the first 10 years. [2]
A factory produces 600 electric fans in the 3rd year and 700 fans in the 7th year. Assuming that the production increases uniformly by a fixed number of units every year:
(a) Find the production of fans in the 1st year ($a$). [1]
(b) Find the annual fixed increase in production ($d$). [1]
(c) Find the total production of fans in the first 10 years. [2]
OR
In which year will the annual production reach 1000 fans? [2]
Q37. Case Study 2 (Coordinate Geometry):
[4]
(b) Write the coordinates of the red flag posted by Preet. [1]
(c) Find the straight-line distance between both flags. [2]
On a sports field, lines are drawn with chalk at $1\text{ m}$ intervals. 100 flower pots are placed at distances of $1\text{ m}$ along boundary line $AD$. Niharika runs $1/4\text{th}$ the distance $AD$ on the 2nd line and posts a green flag. Preet runs $1/5\text{th}$ the distance $AD$ on the 8th line and posts a red flag.
(a) Write the coordinates of the green flag posted by Niharika. [1]
(b) Write the coordinates of the red flag posted by Preet. [1]
(c) Find the straight-line distance between both flags. [2]
OR
If Rashmi posts a blue flag exactly halfway between the line segment joining the two flags, find the line number and distance for her flag. [2]
Q38. Case Study 3 (Some Applications of Trigonometry):
[4]
(b) Find the height of the temple up to the roof. (Use $\sqrt{3} \approx 1.732$). [1]
(c) Calculate the height of the flagstaff on top of the temple roof. [2]
A vertical flagstaff stands on top of a flat-roofed temple. From a point on the ground $100\text{ m}$ away from the base of the temple, the angles of elevation of the bottom and the top of the flagstaff are observed to be $30^\circ$ and $45^\circ$ respectively.
(a) Draw a neat labelled mathematical diagram representing this situation. [1]
(b) Find the height of the temple up to the roof. (Use $\sqrt{3} \approx 1.732$). [1]
(c) Calculate the height of the flagstaff on top of the temple roof. [2]
OR
Find the direct line-of-sight distance from the point of observation to the top of the flagstaff. [2]