ISC Mathematics 2025 Question Paper with Answers
All 51 questions of the ISC Mathematics 2025 Question Paper, in printed order. Open a question to read it in full and see its answer.
- Q1(i) If $A = \begin{bmatrix} 0 & a \\ 0 & 0 \end{bmatrix}$, then $A^{16}$ is: 1 mark · MCQ
- Q1(ii) Which of the following is a homogenous differential equation? 1 mark · MCQ
- Q1(iii) Consider the graph of the function $f(x)$ shown below: Statement 1: The function $f(x)$ is increasing in $(\frac{1}{2}, 2)$… 1 mark · MCQ
- Q1(iv) $\int_0^1 \frac{x^4 - 1}{x^2 + 1}\,dx$ is equal to: 1 mark · MCQ
- Q1(v) Consider the two events $A$ and $B$ such that $n(A) = n(B)$ and $P(A/B) = P(B/A)$. 1 mark · Assertion-reason
- Q1(vi) The existence of unique solution of the system of equations $x + y = \lambda$ and $5x + ky = 2$ depends on: 1 mark · MCQ
- Q1(vii) A cylindrical popcorn tub of radius $10\text{ cm}$ is being filled with popcorns at the rate of $314\text{ cm}^3$ per minute. The… 1 mark · MCQ
- Q1(viii) If $f(x) = \begin{cases} x+2 & x < 0 \\ -x^2-2 & 0 \le x < 1 \\ x & x \ge 1 \end{cases}$ then the number of point(s) of… 1 mark · MCQ
- Q1(ix) If Set $A$ has $m$ elements, Set $B$ has $n$ elements and $n < m$, then the number of one-one function(s) from $A \to B$ is zero. 1 mark · Assertion-reason
- Q1(x) Let $X$ be a discrete random variable. The probability distribution of $X$ is given below: $X$ $30$ $10$ $-10$ $P(X)$… 1 mark · MCQ
- Q1(xi) Statement 1: If '$A$' is an invertible matrix, then $(A^2)^{-1} = (A^{-1})^2$ Statement 2: If '$A$' is an invertible matrix, then… 1 mark · MCQ
- Q1(xii) Write the smallest equivalence relation from the set $A$ to $A$, where $A = \{1, 2, 3\}$. 1 mark · One word
- Q1(xiii) For what value of $x$, is $A = \begin{bmatrix} 0 & 1 & -2 \\ -1 & 0 & 3 \\ x & -3 & 0 \end{bmatrix}$ a skew symmetric matrix? 1 mark · One word
- Q1(xiv) Three critics review a book. Odds in favour of the book are $5:2$, $4:3$ and $3:4$ respectively for the three critics. Find the… 1 mark · Numerical
- Q1(xv) Evaluate: $\int \frac{5}{\sqrt{2x+7}}\,dx$ 1 mark · Short answer
- Q2 Find the point on the curve $y = 2x^2 - 6x - 4$ at which the tangent is parallel to the $x$-axis. 2 marks · Short answer
- Q3 Find the value of $\tan^{-1} x - \cot^{-1} x$, if $(\tan^{-1} x)^2 - (\cot^{-1} x)^2 = \frac{5\pi^2}{8}$ 2 marks · Short answer
- Q4(i) If $x^y = e^{x-y}$, prove that $\frac{dy}{dx} = \frac{\log x}{(1+\log x)^2}$ 2 marks · Derivation
- Q4(ii) If $f(x) = \log(1 + x) + \frac{1}{1+x}$, show that $f(x)$ attains its minimum value at $x = 0$. 2 marks · Derivation
- Q5 Three shopkeepers Gaurav, Rizwan and Jacob use carry bags made of polythene, handmade paper and newspaper. The number of… 2 marks · Short answer
- Q6(i) Differentiate $\sin^{-1}\left(\frac{2^{x+1}\cdot 3^x}{1 + (36)^x}\right)$ with respect to $x$. 2 marks · Short answer
- Q6(ii) Show that $\tan^{-1} x + \tan^{-1} y = C$ is the general solution of the differential equation $(1 + x^2)\,dy + (1 + y^2)\,dx = 0$ 2 marks · Derivation
- Q7 If $x + y + z = 0$ then show that $\begin{vmatrix} 1 & 1 & 1 \\ x & y & z \\ x^3 & y^3 & z^3 \end{vmatrix} = 0$, using properties… 4 marks · Derivation
- Q8(i) Evaluate: $\int \frac{\cos x}{3\cos x - 5}\,dx$ 4 marks · Short answer
- Q8(ii) Evaluate: $\int (\log x)^2\,dx$ 4 marks · Short answer
- Q9(i) If $x = \tan\left(\frac{1}{a}\log y\right)$ then show that $(1 + x^2)\frac{d^2y}{dx^2} + (2x - a)\frac{dy}{dx} = 0$ 4 marks · Derivation
- Q9(ii) The graph of $f(x) = -x^3 + 27x - 2$ is given below: Find the slope of the above graph. Find the co-ordinates of turning points… 4 marks · Short answer
- Q10 Pia, Sia and Dia displayed their paintings in an art exhibition. The three artists displayed $15$, $5$ and $10$ of their… 4 marks · Short answer
- Q11(i) Prove: $\int_0^{\frac{3\pi}{4}} \frac{x\,dx}{1 + \sin x} = (\sqrt{2} - 1)\pi$ 6 marks · Derivation
- Q11(ii) Evaluate: $\int \frac{dx}{(x-1)^2(x^2 + 1)^2}$ 6 marks · Short answer
- Q12(i) Solve the differential equation: $(x + 5y^2)\frac{dy}{dx} = y$ when $x = 2$ and $y = 1$ 6 marks · Short answer
- Q12(ii) Find the particular solution of the differential equation: $(x^2 - 2y^2)\,dx + 2xy\,dy = 0$, when $x = 1$ and $y = 1$ 6 marks · Short answer
- Q13 Observe the two graphs, Graph 1 and Graph 2 given below and answer the questions that follow. Which one of the graphs represents… 6 marks · Short answer
- Q14 An international conference takes place in a metropolitan city. International leaders, scientists and industrialists participate… 6 marks · Short answer
- Q15(i) $(\vec{a} + \vec{b}) \cdot (\vec{b} - \vec{a}) = 2 (a^2 + b^2)$ 1 mark · Assertion-reason
- Q15(ii) The angle between the two planes $x + y + 2z = 9$ and $2x - y + z = 15$ is: 1 mark · MCQ
- Q15(iii) Show that points $P(-2, 3, 5)$, $Q(1, 2, 3)$ and $R(7, 0, -1)$ are collinear. 1 mark · Short answer
- Q15(iv) Two honeybees are flying parallel to each other in the garden to collect the nectar. The path traced by the bees is given in the… 2 marks · Short answer
- Q16(i) Find the equation of the plane passing through the points $(2, 2, -1)$, $(3, 4, 2)$ and $(7, 0, 6)$. 2 marks · Short answer
- Q16(ii) Find the equation of the plane passing through the points $(2, 3, 1)$, $(4, -5, 3)$ and parallel to $x$-axis. 2 marks · Short answer
- Q17 Consider the position vectors of $A$, $B$ and $C$ as $\vec{OA} = 2\hat{i} - 2\hat{j} + \hat{k}$… 4 marks · Short answer
- Q18(i) The equation $y = 4 - x^2$ represents a parabola. Make a rough sketch of the graph of the given function. Determine the area… 4 marks · Short answer
- Q18(ii) A farmer has a field bounded by three lines $x + 2y = 2$, $y - x = 1$, $2x + y = 7$. Using integration, find the area of the… 4 marks · Short answer
- Q19(i) The total revenue received from the sale of '$x$' units of a product is $R(x) = 3x^2 + 36x + 5$. Then, the actual revenue for… 1 mark · MCQ
- Q19(ii) Read the following statements and choose the correct option. (I) The correlation coefficient and the regression coefficients are… 1 mark · MCQ
- Q19(iii) Consider the following data: $x$ $y$ $x - \bar{x}$ $y - \bar{y}$ $(x - \bar{x})^2$ $(y - \bar{y})^2$ $(x - \bar{x})(y - \bar{y})$… 3 marks · Short answer
- Q20(i) Find the regression line of best fit from the following data: $\sum x = 24$, $\sum y = 44$, $\sum xy = 306$, $\sum x^2 = 164$… 2 marks · Short answer
- Q20(ii) Two lines of regression are given as $4x + 3y + 7 = 0$ and $3x + 4y + 8 = 0$. Identify the line of regression of $x$ on $y$. 2 marks · Short answer
- Q21(i) A utensil manufacturer produces '$x$' dinner sets per week and sells each set at ₹ $p$, where $x = \frac{600 - p}{8}$. The cost… 4 marks · Short answer
- Q21(ii) The Average Cost of producing '$x$' units of commodity is given by: $AC = \frac{x}{200} + 30 + \frac{5000}{x}$ Find the Cost… 4 marks · Short answer
- Q22 Two different types of books have to be stacked in the shelf of a library. The first type of book weighs 1 kg and has a thickness… 4 marks · Short answer