ISC Mathematics 2025 Specimen Paper with Answers
All 32 questions of the ISC Mathematics 2025 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. A matrix… 15 marks · Short answer
- Q2(i) If $x^y = y^x$, then find $\frac{dy}{dx}$. [Understanding] 2 marks · Short answer
- Q2(ii) Find the interval in which the function $f(x) = x^2 e^{-x}$ is strictly increasing or decreasing. [Understanding] 2 marks · Short answer
- Q3 Evaluate: $\int_0^{\sqrt{2}} [x^2] \\, dx$ [Evaluate] 2 marks · Short answer
- Q4 Find the equation to the tangent at $(0,0)$ on the curve $y = 4x^2 - 2x^3$. [Analysis] 2 marks · Short answer
- Q5(i) Evaluate: $\int \frac{2x^3 - 1}{x^4 + x} \\, dx$ [Evaluate] 2 marks · Short answer
- Q5(ii) Evaluate: $\int e^x \csc x (1 - \cot x) \\, dx$ [Evaluate] 2 marks · Short answer
- Q6 Find the value of: $\tan^{-1} \left(\frac{x}{y}\right) + \tan^{-1} \left(\frac{y-x}{y+x}\right)$ [Application] 2 marks · Short answer
- Q7 Solve: $\sin^{-1}(x) + \sin^{-1}(1-x) = \cos^{-1} x$. [Application] 4 marks · Short answer
- Q8 Evaluate: $\int \sqrt{\sec\frac{x}{2} - 1} \\, dx$ [Application] 4 marks · Short answer
- Q9(i) A kite is being pulled down by a string that goes through a ring on the ground 8 meters away from the person pulling it. If the… 4 marks · Short answer
- Q9(ii) If $y = (x + \sqrt{a^2 + x^2})^m$, prove that $(a^2 + x^2)\frac{d^2y}{dx^2} + x\frac{dy}{dx} - m^2y = 0$. [Analysis] 4 marks · Derivation
- 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) Students of under graduation submitted a case study on “Understanding the Probability of Left-Handedness in Children Based on… 4 marks · Case based
- 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) Solve the differential equation… 6 marks · Short answer
- Q12(ii) Evaluate: $\int_0^\pi (\sin^4 x + \cos^4 x) \\, dx$ Hence evaluate… 6 marks · Short answer
- Q13(i) A cone of maximum volume is inscribed in a given sphere. Then prove that ratio of the height of the cone to the diameter of the… 6 marks · Derivation
- Q13(ii) A given quantity of metal is to be cast into a solid half circular cylinder with a rectangular base and semi-circular ends. If… 6 marks · Derivation
- Q14 Kiran plays a game of throwing a fair die 3 times but to quit as and when she gets a six. Kiran gets +1 point for a six and −1… 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) Find a vector of magnitude 9 units and perpendicular to the vectors $\vec{a} = 4\hat{i} - \hat{j} + \hat{k}$ and… 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(i) Show that the line whose vector equation is $\vec{r} = (2\hat{i} - 2\hat{j} + 3\hat{k}) + \lambda(\hat{i} - \hat{j} + 4\hat{k})$… 4 marks · Derivation
- Q17(ii) Find the equation of the plane containing the line $\frac{x}{-2} = \frac{y-1}{3} = \frac{1-z}{1}$ and the point $(-1, 0, 2)$… 4 marks · Short answer
- Q18 Sketch the region enclosed bounded by the curve, $y = x|x|$ and the ordinates $x = -1$ and $x = 1$. Evaluate… 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) The cost function of a commodity is $C(x) = 200 + 20x - \frac{1}{2} x^2$ (in rupees). Find the range where AC falls. [Application] 2 marks · Short answer
- Q20(ii) The demand function of a monopoly is given by $x = 100 - 4p$. Find the quantity at which the MR will be zero. [Application] 2 marks · Short answer
- Q21(i) A survey of 50 families to study the relationships between expenditure on accommodation in (₹ x) and expenditure on food and… 4 marks · Short answer
- Q21(ii) The random variables have regression lines $3x + 2y - 26 = 0$ and $6x + y - 31 = 0$. Calculate Mean value of $x$ and $y$… 4 marks · Short answer
- Q22 A linear programming problem is given by $Z = px + qy$ where $p, q > 0$ subject to the constraints… 4 marks · Short answer