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CBSE CLASS X – MATHEMATICS (STANDARD) SAMPLE QUESTION PAPER
SET – 8 | ACADEMIC SESSION 2026–2027
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 Assertion–Reason 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 – MULTIPLE CHOICE QUESTIONS (1 Mark Each)
Q1.
The largest number which divides $70$ and $125$, leaving remainders $5$ and $8$ respectively, is:
[1]
Q2.
If $\alpha$ and $\beta$ are the zeroes of the polynomial $f(x) = x^2 – p(x + 1) – c$, then $(\alpha + 1)(\beta + 1)$ is equal to:
[1]
Q3.
The value of $k$ for which the pair of linear equations $3x + y = 1$ and $(2k – 1)x + (k – 1)y = 2k + 1$ has no solution is:
[1]
Q4.
If the quadratic equation $ax^2 + bx + c = 0$ has equal roots, then the value of $c$ is:
[1]
Q5.
The sum of the first 16 terms of the AP: $10, 6, 2, \dots$ is:
[1]
Q6.
The perimeter of a triangle with vertices $(0, 0)$, $(1, 0)$, and $(0, 1)$ is:
[1]
Q7.
If $\triangle ABC \sim \triangle EDF$ and $\triangle ABC$ is not similar to $\triangle DEF$, then which of the following is not true?
[1]
Q8.
If $\sin\theta – \cos\theta = 0$, then the value of $\sin^4\theta + \cos^4\theta$ is:
[1]
Q9.
$(\sec A + \tan A)(1 – \sin A)$ is equal to:
[1]
Q10.
If the angle of elevation of a tower from a distance of $100\text{ m}$ from its foot is $60^\circ$, then the height of the tower is:
[1]
Q11.
If two tangents inclined at an angle of $90^\circ$ are drawn to a circle of radius $5\text{ cm}$, then the length of each tangent is:
[1]
Q12.
If the perimeter of a semi-circular protractor is $36\text{ cm}$, then its diameter is:
[1]
Q13.
If the radius of the base of a cylinder is doubled and the height is halved, then its curved surface area will be:
[1]
Q14.
For the following distribution, the class mark of the median class is:
$$\begin{array}{|c|c|c|c|c|c|} \hline \text{Marks} & 0-10 & 10-20 & 20-30 & 30-40 & 40-50 \\ \hline \text{Frequency} & 5 & 8 & 12 & 15 & 10 \\ \hline \end{array}$$ [1]
$$\begin{array}{|c|c|c|c|c|c|} \hline \text{Marks} & 0-10 & 10-20 & 20-30 & 30-40 & 40-50 \\ \hline \text{Frequency} & 5 & 8 & 12 & 15 & 10 \\ \hline \end{array}$$ [1]
Q15.
A card is drawn from a well-shuffled deck of 52 playing cards. The probability that the card drawn is neither an ace nor a king is:
[1]
Q16.
The point which lies on the perpendicular bisector of the line segment joining the points $A(-2, -5)$ and $B(2, 5)$ is:
[1]
Q17.
A box contains 5 red, 4 blue, and 3 green pens. If a pen is picked at random, the probability that it is blue is:
[1]
Q18.
If $p$ and $q$ are co-prime numbers, then $p^2$ and $q^2$ are:
[1]
Q19.
Assertion (A): If the zeroes of the quadratic polynomial $ax^2 + bx + c$ are both positive, then $a, b,$ and $c$ all have the same sign.
Reason (R): If $\alpha, \beta > 0$, then $\alpha + \beta = -\frac{b}{a} > 0$ and $\alpha\beta = \frac{c}{a} > 0$. [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): If $\alpha, \beta > 0$, then $\alpha + \beta = -\frac{b}{a} > 0$ and $\alpha\beta = \frac{c}{a} > 0$. [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): The value of $\sin\theta$ increases as $\theta$ increases from $0^\circ$ to $90^\circ$.
Reason (R): In a right-angled triangle, as the acute angle $\theta$ increases, the length of the opposite side increases relative to the hypotenuse. [1]
Reason (R): In a right-angled triangle, as the acute angle $\theta$ increases, the length of the opposite side increases relative to the hypotenuse. [1]
SECTION B – VERY SHORT ANSWER QUESTIONS (2 Marks Each)
Q21.
Show that $3\sqrt{2}$ is an irrational number, given that $\sqrt{2}$ is irrational.
[2]
Q22.
In the given figure, $\angle D = \angle E$ and $\frac{AD}{DB} = \frac{AE}{EC}$. Prove that $\triangle BAC$ is an isosceles triangle.
[2]
OR
A vertical stick $20\text{ cm}$ long casts a shadow $15\text{ cm}$ long on the ground. At the same time, a tower casts a shadow $45\text{ m}$ long on the ground. Find the height of the tower.
Q23.
Find the ratio in which the point $P(m, 6)$ divides the line segment joining the points $A(-4, 3)$ and $B(2, 8)$. Hence, find the value of $m$.
[2]
Q24.
Evaluate:
$$\frac{2\cos^2 60^\circ + 3\sin^2 45^\circ – 4\tan^2 30^\circ}{2\sin 30^\circ\cos 30^\circ\tan 60^\circ}$$
[2]
OR
If $\sin(A + 2B) = \frac{\sqrt{3}}{2}$ and $\cos(A + 4B) = 0$, where $A$ and $B$ are acute angles, find the values of $A$ and $B$.
Q25.
In the given figure, $PA$ and $PB$ are tangents to the circle with centre $O$ such that $\angle APB = 50^\circ$. Write the measure of $\angle OAB$.
[2]
SECTION C – SHORT ANSWER QUESTIONS (3 Marks Each)
Q26.
If $\alpha$ and $\beta$ are the zeroes of the quadratic polynomial $p(x) = 3x^2 – 4x + 1$, find a quadratic polynomial whose zeroes are $\frac{\alpha^2}{\beta}$ and $\frac{\beta^2}{\alpha}$.
[3]
Q27.
Solve the following system of linear equations by the elimination method:
$$21x + 47y = 110$$
$$47x + 21y = 162$$
[3]
OR
A 2-digit number is 4 times the sum of its digits and twice the product of its digits. Find the number.
Q28.
The ratio of the sums of $m$ and $n$ terms of an AP is $m^2 : n^2$. Show that the ratio of the $m^{\text{th}}$ and $n^{\text{th}}$ terms is $(2m – 1) : (2n – 1)$.
[3]
Q29.
Prove the trigonometric identity:
$$\frac{\cos A – \sin A + 1}{\cos A + \sin A – 1} = \csc A + \cot A$$
[3]
Q30.
In the given figure, $PQ$ is a chord of length $8\text{ cm}$ of a circle of radius $5\text{ cm}$. The tangents at $P$ and $Q$ intersect at a point $T$. Find the length of $TP$.
[3]
OR
Prove that the intercept of a tangent between two parallel tangents to a circle subtends a right angle at the centre of the circle.
Q31.
Two different dice are rolled together. Find the probability of getting:
(a) A doublet of even numbers
(b) A sum of at least 10
(c) A product of numbers equal to 12 [3]
(a) A doublet of even numbers
(b) A sum of at least 10
(c) A product of numbers equal to 12 [3]
SECTION D – LONG ANSWER QUESTIONS (5 Marks Each)
Q32.
A motor boat whose speed is $24\text{ km/h}$ in still water takes 1 hour more to go $32\text{ km}$ upstream than to return downstream to the same spot. Find the speed of the stream.
[5]
OR
Solve for $x$:
$$\frac{1}{a + b + x} = \frac{1}{a} + \frac{1}{b} + \frac{1}{x}, \quad a \neq 0, b \neq 0, x \neq -(a + b), x \neq 0$$
Q33.
State and prove Basic Proportionality Theorem.
Using this theorem, prove that in $\triangle ABC$, if a line $DE \parallel BC$ intersects $AB$ at $D$ and $AC$ at $E$, then: $$\frac{AD}{AB} = \frac{AE}{AC}$$ [5]
Using this theorem, prove that in $\triangle ABC$, if a line $DE \parallel BC$ intersects $AB$ at $D$ and $AC$ at $E$, then: $$\frac{AD}{AB} = \frac{AE}{AC}$$ [5]
Q34.
A solid toy is in the form of a hemisphere surmounted by a right circular cone. The height of the cone is $4\text{ cm}$ and the diameter of the base is $8\text{ cm}$. Determine the volume of the toy. If a cube of side $8\text{ cm}$ circumscribes the toy, find the difference between the volumes of the cube and the toy. (Take $\pi = 3.14$).
[5]
OR
A solid metallic sphere of radius $10.5\text{ cm}$ is melted and recast into a number of smaller cones, each of radius $3.5\text{ cm}$ and height $3\text{ cm}$. Find the total number of cones so formed.
Q35.
The median of the following data is $50$. Find the values of $p$ and $q$, if the sum of all the frequencies is $90$:
$$\begin{array}{|c|c||c|c|} \hline \text{Marks} & \text{Frequency} & \text{Marks} & \text{Frequency} \\ \hline 20-30 & p & 50-60 & 20 \\ 30-40 & 15 & 60-70 & q \\ 40-50 & 25 & 70-80 & 8 \\ \hline \end{array}$$ [5]
$$\begin{array}{|c|c||c|c|} \hline \text{Marks} & \text{Frequency} & \text{Marks} & \text{Frequency} \\ \hline 20-30 & p & 50-60 & 20 \\ 30-40 & 15 & 60-70 & q \\ 40-50 & 25 & 70-80 & 8 \\ \hline \end{array}$$ [5]
SECTION E – CASE-BASED INTEGRATED UNITS (4 Marks Each)
Q36. Case Study 1 (Arithmetic Progression):
[4]
(a) Find the number of solar panel installations completed in the 12th week. [1]
(b) In which week will the team install 75 solar panels? [1]
(c) Find the total cumulative number of installations completed in the first 20 weeks. [2]
A construction company installs solar panels on rooftops. The installation team starts with 15 installations in the first week and increases the weekly installations by 4 units every successive week to meet clean energy targets.
Based on the above information, answer the following questions:
(a) Find the number of solar panel installations completed in the 12th week. [1]
(b) In which week will the team install 75 solar panels? [1]
(c) Find the total cumulative number of installations completed in the first 20 weeks. [2]
OR
If the company receives a contract for a total of 1000 installations, find the number of weeks required to complete the target. [2]
Q37. Case Study 2 (Coordinate Geometry):
[4]
(b) Find the distance between Bhuvan ($B$) and Chahat ($C$). [1]
(c) Check whether the three friends sit in a straight line (collinear). [2]
Three friends, Ananya, Bhuvan, and Chahat, sit at positions $A(3, 1)$, $B(6, 4)$, and $C(8, 6)$ on a park lawn mapped onto a Cartesian plane.
(a) Find the distance between Ananya ($A$) and Bhuvan ($B$). [1]
(b) Find the distance between Bhuvan ($B$) and Chahat ($C$). [1]
(c) Check whether the three friends sit in a straight line (collinear). [2]
OR
Find the coordinates of a water booth $D$ such that $B$ is the midpoint of the line segment joining $A$ and $D$. [2]
Q38. Case Study 3 (Some Applications of Trigonometry):
[4]
(b) Find the height of the balloon above the ground at that instant. (Take $\sqrt{3} \approx 1.732$). [1]
(c) Find the angle of elevation of the balloon as observed from observer $Q$ at the same instant. [2]
A hot air balloon rises vertically from a launching pad on the ground. Two observers, $P$ and $Q$, stand on the ground on the same side of the launching pad at distances of $40\text{ m}$ and $100\text{ m}$ respectively from the pad. At a certain instant, the angle of elevation of the balloon from observer $P$ is $60^\circ$.
(a) Draw a neat labelled mathematical diagram representing this situation. [1]
(b) Find the height of the balloon above the ground at that instant. (Take $\sqrt{3} \approx 1.732$). [1]
(c) Find the angle of elevation of the balloon as observed from observer $Q$ at the same instant. [2]
OR
Find the direct line-of-sight distance from observer $P$ to the balloon. [2]