CBSE Class 10 Maths Sample Paper 2026: Last-Minute Solved Model Paper for Board Exams
CBSE Class 10 Maths Sample Paper 2026: Last-Minute Solved Model Paper for Board Exams
The countdown to the CBSE Class 10 Mathematics exam 2026 has begun! To score a perfect 80/80, you need more than just formulas; you need to understand the application of concepts. This model paper is designed to simulate the actual board environment, covering Real Numbers, Algebra, Trigonometry, and Statistics.
Whether you are looking for important MCQs for Class 10 Maths or practice questions for the Case Study section, this guide has it all. Let's get into the "Exam Mode" and solve this paper!
Section A: Multiple Choice Questions (1 Mark Each)
Q1. The HCF of the smallest prime number and the smallest composite number is:
(a) 1
(b) 2
(c) 4
(d) 8
Q2. If the zeroes of the quadratic polynomial $x^2 + (a + 1)x + b$ are 2 and -3, then:
(a) $a = -7, b = -1$
(b) $a = 5, b = -1$
(c) $a = 2, b = -6$
(d) $a = 0, b = -6$
Q3. The value of $k$ for which the pair of linear equations $kx - y = 2$ and $6x - 2y = 3$ has a unique solution is:
(a) $k \neq 3$
(b) $k = 3$
(c) $k = 0$
(d) $k \neq 0$
Q4. If the discriminant of $3x^2 - 2x + \frac{1}{3} = 0$ is $D$, then the roots are:
(a) Real and distinct
(b) Real and equal
(c) No real roots
(d) Rational and distinct
Q5. The distance of the point $P(2, 3)$ from the y-axis is:
(a) 2 units
(b) 3 units
(c) 5 units
(d) $\sqrt{13}$ units
Q6. If $\triangle ABC \sim \triangle PQR$ with $\frac{BC}{QR} = \frac{1}{3}$, then $\frac{ar(\triangle PQR)}{ar(\triangle ABC)}$ is:
(a) 9
(b) 3
(c) $1/3$
(d) $1/9$
Q7. If $\cos \theta = \frac{a}{b}$, then $\tan \theta$ is equal to:
(a) $\frac{b}{\sqrt{b^2 - a^2}}$
(b) $\frac{\sqrt{b^2 - a^2}}{b}$
(c) $\frac{\sqrt{b^2 - a^2}}{a}$
(d) $\frac{a}{\sqrt{b^2 - a^2}}$
Q8. The shadow of a tower is $\sqrt{3}$ times its height. The angle of elevation of the sun is:
(a) $30^\circ$
(b) $45^\circ$
(c) $60^\circ$
(d) $90^\circ$
Q9. A tangent $PQ$ at a point $P$ of a circle of radius $5 \text{ cm}$ meets a line through the centre $O$ at a point $Q$ so that $OQ = 12 \text{ cm}$. The length $PQ$ is:
(a) $12 \text{ cm}$
(b) $13 \text{ cm}$
(c) $8.5 \text{ cm}$
(d) $\sqrt{119} \text{ cm}$
Q10. The area of a quadrant of a circle whose circumference is $22 \text{ cm}$ is:
(a) $3.5 \text{ cm}^2$
(b) $9.625 \text{ cm}^2$
(c) $77 \text{ cm}^2$
(d) $154 \text{ cm}^2$
(Questions 11-20 continue with similar MCQ patterns covering Probability, Statistics, and Coordinate Geometry)
Section B: Short Answer Questions (2 Marks Each)
Q21. Prove that $\sqrt{3}$ is an irrational number.
Q22. Find the coordinates of the point which divides the join of $A(-1, 7)$ and $B(4, -3)$ in the ratio $2:3$.
Q23. In the given figure, $PA$ and $PB$ are tangents to the circle from an external point $P$. If $\angle APB = 60^\circ$, find $\angle OAB$.
Section C: Short Answer Questions (3 Marks Each)
Q24. Solve for $x$ and $y$:
Q25. Prove the identity:
Q26. The sum of the first $n$ terms of an AP is $3n^2 + n$. Find its $25^{th}$ term.
Section D: Long Answer Questions (5 Marks Each)
Q27. A motorboat whose speed is $18 \text{ km/h}$ in still water takes 1 hour more to go $24 \text{ km}$ upstream than to return downstream to the same spot. Find the speed of the stream.
Q28. As observed from the top of a $75 \text{ m}$ high lighthouse from the sea-level, the angles of depression of two ships are $30^\circ$ and $45^\circ$. If one ship is exactly behind the other on the same side of the lighthouse, find the distance between the two ships.
Q29. 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.
Section E: Case Study Based Questions (4 Marks Each)
Q30. Case Study: Coordinate Geometry in Sports
Students of a school are standing in rows and columns in their playground for a PT drill. $A, B, C$ and $D$ are the positions of four students as shown in the figure.
(i) Find the distance between $A$ and $B$. (1M)
(ii) Check if
(iii) Find the coordinates of the mid-point of
✅ Answer Key & Solutions
Section A Highlights:
(b) Smallest prime (2), smallest composite (4).
$\text{HCF}(2, 4) = 2$. (d)
$a=0, b=-6$. (a) $k/6 \neq -1/-2 \Rightarrow k \neq 3$.
(b) $D = (-2)^2 - 4(3)(1/3) = 4 - 4 = 0$.
(a) Distance from y-axis is the x-coordinate
$|2| = 2$. (c)
$\text{Use } \sin^2 \theta + \cos^2 \theta = 1 \text{ to find } \sin, \text{ then } \tan = \sin/\cos$. (a)
$\tan \theta = h / h\sqrt{3} = 1/\sqrt{3} \Rightarrow \theta = 30^\circ$.
Section B/C/D Hints:
Q25 (Trig): Take LCM $\rightarrow (\cos^2 A + (1+\sin A)^2) / (\cos A(1+\sin A))$. Expand and use $\sin^2 A + \cos^2 A = 1$.
Q27 (Stream): $\frac{24}{18-x} - \frac{24}{18+x} = 1$. Quadratic: $x^2 + 48x - 324 = 0 \Rightarrow x = 6 \text{ km/h}$.
Q29 (Surface Area): $\text{TSA} = \text{CSA of Cylinder} + \text{CSA of Cone} = 2\pi r h + \pi r l$.
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