ISC Mathematics 2026 Specimen Paper with Answers
All 30 questions of the ISC Mathematics 2026 Specimen Paper, in printed order. Open a question to read it in full and see its answer.
- Q1 In subparts (i) to (xi) choose the correct options and in subparts (xii) to (xv), answer the questions as instructed. If A and B… 15 marks · Short answer
- Q2 If $x = e^{\frac{x}{y}}$, then prove that $\frac{dy}{dx} = \frac{x-y}{x \log x}$ (Understanding) Find the values of $'a'$ for… 2 marks · Derivation
- Q3 Find the point on the curve $y = (x-2)^2$ at which the tangent is parallel to the chord joining the end points $(2,0)$ and… 2 marks · Short answer
- Q4 Show that the general solution of the differential equation: $\frac{dy}{dx} = y \cot 2x$ is… 2 marks · Derivation
- Q5(i) If $\int x^5 \cos(x^6)\,dx = k \sin(x^6) + C$, find the value of ‘$k$’. (Evaluate) 2 marks · Numerical
- Q5(ii) Evaluate: $\int_{0}^{5} \frac{\sqrt{x}}{\sqrt{5-x} + \sqrt{x}}\,dx$ (Evaluate) 2 marks · Numerical
- Q6 A music streaming app uses the function: $f(x) = \tan^{-1}\left(\frac{x}{10}\right)$ to assign a mood score based on the number… 2 marks · Numerical
- Q7 Solve: $\sin^{-1}(x) + \sin^{-1}(1-x) = \cos^{-1} x$. (Application) 4 marks · Short answer
- Q8 If $y = (A + Bx)e^{-2x}$, prove that: $\frac{d^2y}{dx^2} + 4\frac{dy}{dx} + 4y = 0$ (Application) 4 marks · Derivation
- Q9(i) Solve the following differential equation: $x \frac{dy}{dx} = y - x \tan\left(\frac{y}{x}\right)$ (Application) 4 marks · Short answer
- Q9(ii) Solve the following differential equation: $(1 + x^2)\frac{dy}{dx} + 2xy = 4x^2$ (Application) 4 marks · Short answer
- Q10(i) Three friends go to a restaurant to have pizza. They decide who will pay for the pizza by tossing a coin. It is decided that each… 4 marks · Short answer
- Q10(ii) A school offers students the choice of three modes for attending classes: • Mode A: Offline (in-person) – 40% of students • Mode… 4 marks · Short answer
- Q11 To raise money for an orphanage, students of three schools A, B and C organised an exhibition in their residential colony, where… 6 marks · Short answer
- Q12(i) Evaluate: $\int \frac{x^2+x+1}{(x+2)(x^2+1)}\,dx$ (Evaluate) 6 marks · Short answer
- Q12(ii) Evaluate: $\int_{\pi/4}^{3\pi/4} \frac{x}{1+\sin x}\,dx$ (Evaluate) 6 marks · Short answer
- Q13(i) A person has manufactured a water tank in the shape of a closed right circular cylinder. The volume of the cylinder is… 6 marks · Short answer
- Q13(ii) A Dolphin jumps and taken a path given by the equation $h(t) = \frac{1}{2}(-7t^2 + 3t + 2)$, ($t \ge 0$), $h(t)$ is the height of… 6 marks · Short answer
- Q14 In a school, three subject teachers English, Math, and Science sometimes give surprise tests on the same day. Based on past… 6 marks · Short answer
- Q15 In subparts (i) and (ii) choose the correct options and in subparts (iii) to (v), answer the questions as instructed. Consider… 5 marks · Short answer
- Q16(i) A building is to be constructed in the form of a triangular pyramid ABCD as shown in the figure. Let the angular points be… 2 marks · Short answer
- Q16(ii) What are the values of $x$ for which the angle between the vectors $2x^2\hat{i} + 3x\hat{j} + \hat{k}$ and… 2 marks · Short answer
- Q17 Given, B and C lie on the line $\frac{x+3}{5} = \frac{y-1}{2} = \frac{z+4}{3}$ and $BC = 5$ units find the area of $\Delta ABC$… 4 marks · Short answer
- Q18 Find the area bounded by the curve, $y = x(4 - x)$ and the $x$-axis from $x = 0$ to $x = 5$ as shown in the figure given above… 4 marks · Short answer
- Q19 In subparts (i) and (ii) choose the correct options and in subparts (iii) to (v), answer the questions as instructed. Which… 5 marks · Short answer
- Q20(i) A real estate company is going to build a new residential complex. The land they have purchased can hold at the most 500… 2 marks · Short answer
- Q20(ii) The demand function of a monopoly is given by $x = 100 - 4p$. Find the quantity at which the Marginal Revenue will be zero… 2 marks · Short answer
- Q21 A survey of 50 families to study the relationships between expenditure on accommodation in (₹ $x$) and expenditure on food and… 4 marks · Short answer
- Q22(i) A linear programming problem is given by $Z = px + qy$ where $p, q > 0$ subject to the constraints: $x + y \le 60$… 4 marks · Short answer
- Q22(ii) The feasible region for an L.P.P. is shown in the adjoining figure: Based on the given graph, answer the following questions… 4 marks · Short answer