ISC Mathematics 2025 Improvement Paper with Answers
All 49 questions of the ISC Mathematics 2025 Improvement Paper, in printed order. Open a question to read it in full and see its answer.
- Q1(i) Find the value of $\mu$ if the following system of equations is consistent: $2x + 3y - 8 = 0$, $7x - 5y + 3 = 0$… 1 mark · MCQ
- Q1(ii) If $x = a\cos\theta, y = a\sin\theta$, then $\frac{dy}{dx}$ at $\theta = \frac{\pi}{4}$ will be: 1 mark · MCQ
- Q1(iii) $f(x)$ is a quadratic function. 1 mark · Assertion-reason
- Q1(iv) $\int \frac{\cos(\log x)}{x}\,dx$ is equal to: 1 mark · MCQ
- Q1(v) If $A$ is a skew-symmetric matrix of order $3 \times 3$, then $\det(A) = 0$. 1 mark · Assertion-reason
- Q1(vi) The integrating factor of the differential equation $(1 - x^2)\frac{dy}{dx} + xy = ax$ is: 1 mark · MCQ
- Q1(vii) Statement 1: Every scalar matrix is a diagonal matrix. Statement 2: Every diagonal matrix is an identity matrix. 1 mark · MCQ
- Q1(viii) The derivative of $\tan^{-1}\left(\frac{\cos x - \sin x}{\cos x + \sin x}\right)$ with respect to $x$, where… 1 mark · MCQ
- Q1(ix) If $P(A) = \frac{1}{2}, P(B) = \frac{1}{3}$ and $P(A \cup B) = \frac{2}{3}$, then the value of $P(A|B) + P(B|A)$ is: 1 mark · MCQ
- Q1(x) Consider the following graph of $f(x)$, $f: \mathbb{R} \to \mathbb{R}$: Statement 1: The function is one-one function. Statement… 1 mark · MCQ
- Q1(xi) … 1 mark · MCQ
- Q1(xii) In a class of 50 students, 25 study English, 10 study History and 10 study both the subjects. Find the probability that a… 1 mark · Short answer
- Q1(xiii) If $R$ is a relation from $\{11, 12, 13\}$ to $\{8, 10, 12\}$ defined by $y = x - 3$, find the range of $R^{-1}$. 1 mark · One word
- Q1(xiv) For what value of $\alpha$, is the matrix $A = \begin{bmatrix} 1 & -2 & 3 \\ 1 & 2 & 1 \\ \alpha & 2 & -3 \end{bmatrix}$ not… 1 mark · Numerical
- Q1(xv) Consider the two events $A$ and $B$. The probability that at least one of $A$ or $B$ occurs is $0\cdot 6$. If the probability of… 1 mark · Numerical
- Q2 Evaluate $\int_{-a}^a x f(x)\,dx$ given, $f(x) = x + g(x)$ where $g(x)$ is an even function. 2 marks · Short answer
- Q3(i) If $\int x^3 5^{x^2}\,dx = K 5^{x^2} + C$, then find the value of $K$. 2 marks · Numerical
- Q3(ii) Evaluate: $\int e^x \left(\frac{1+x}{(2+x)^2}\right)\,dx$ 2 marks · Short answer
- Q4 Find the equation of the normal to the curve $y = x^2 - 3x + 1$ at the point $(3, 1)$. 2 marks · Short answer
- Q5(i) Evaluate: $\int_0^{\pi/2} \log\tan x\,dx$ 2 marks · Short answer
- Q5(ii) Solve the equation: $\tan^{-1}\left(\frac{x-1}{x-2}\right) + \tan^{-1}\left(\frac{x+1}{x+2}\right) = \frac{\pi}{4}$ 2 marks · Short answer
- Q6 In a market survey, three commodities A, B and C were studied. Each commodity had three varieties. To find out the index number… 2 marks · Short answer
- Q7 Using the properties of determinants, prove that… 4 marks · Derivation
- Q8(i) If $x = \sin t, y = \sin pt$, prove that $(1 - x^2)\frac{d^2y}{dx^2} - x\frac{dy}{dx} + p^2 y = 0$ 4 marks · Derivation
- Q8(ii) In a particular forest, the population density ($P$) of wild animals (in thousands of animals per sq. km.) at a distance '$r$' km… 4 marks · Short answer
- Q9(i) Evaluate: $\int x \tan^{-1} x\,dx$ 4 marks · Short answer
- Q9(ii) Evaluate: $\int \frac{\cos x}{\sin^2 x + 4\sin x + 5}\,dx$ 4 marks · Short answer
- Q10 In a school, there are 30 teachers in the examination committee. Out of these, 20 never commit any error in their work. Two… 4 marks · Short answer
- Q11(i) Evaluate: $\int \frac{1}{\operatorname{cosec} x + \cot x}\,dx, x \neq 0$ 6 marks · Short answer
- Q11(ii) Evaluate: $\int_0^{\pi/2} \frac{\sqrt{\tan x}}{1 + \sqrt{\tan x}}\,dx$ 6 marks · Short answer
- Q12(i) Solve the differential equation: $x(x - 1)\frac{dy}{dx} - y = x^2(x - 1)^2$. Find the particular solution satisfying the… 6 marks · Short answer
- Q12(ii) Find the particular solution of the differential equation: $(x + y)\,dy + (x - y)\,dx = 0$, given that $y = 1$ when $x = 1$. 6 marks · Short answer
- Q13 Naman has to attend his friend's marriage. He can reach the wedding venue by train, bus or a two-wheeler. The probability of… 6 marks · Short answer
- Q14 Consider the function $y = \sin^{-1}(2x\sqrt{1 - x^2}), x \in \left[-\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}}\right]$ Find the… 6 marks · Short answer
- Q15(i) Consider the figure, shown below: Statement 1: The position vector of $P$ is $\frac{2\vec{c} + 3\vec{a}}{5}$ Statement 2: The… 1 mark · MCQ
- Q15(ii) There are two solar panels, A and B placed parallel to each other on the terrace of a building. The equation of the surface of… 4 marks · Short answer
- Q16(i) Find the shortest distance between the lines… 2 marks · Short answer
- Q16(ii) Find the equation of the straight line passing through $(1, 2, 1)$ and parallel to the line joining the points $(1, 4, 6)$ and… 2 marks · Short answer
- Q17 Consider the two vectors $\vec{a} = 2\hat{i} - 3\hat{j} + 4\hat{k}$ and $\vec{b} = 5\hat{i} + q\hat{j} - \hat{k}$. Calculate… 4 marks · Short answer
- Q18(i) Using integration, find the area of the region bounded by the line $x - y + 2 = 0$, the curve $x = \sqrt{y}$ and $y$-axis. 4 marks · Short answer
- Q18(ii) Find the area enclosed by the parabola $y^2 = x$ and the line $y = 2x$. 4 marks · Short answer
- Q19(i) Which one of the following statements is correct? 1 mark · MCQ
- Q19(ii) Observe the graph shown below of Cost, Revenue versus Quantity. Statement 1: A denotes Profit area, B denotes Break-even point… 1 mark · MCQ
- Q19(iii) Consider the following table: $x$ 3 $a$ $y$ 2 $b$ Complete the table if $\bar{x} = 4, \bar{y} = 2$. If $b_{yx} = \frac{1}{2}$ and… 3 marks · Short answer
- Q20(i) Consider the data given below: $x$ $y$ Mean 10 20 Standard deviation 2 3 and $r = 0\cdot 66$. Find the regression coefficients. 2 marks · Short answer
- Q20(ii) Out of the two regression lines given below, find the line of regression of $x$ on $y$. $x + 2y - 5 = 0$ and $2x + 3y - 8 = 0$. 2 marks · Short answer
- Q21(i) The cost function $C(x)$ of producing '$x$' quantities of a product is given by $C(x) = 500x^2 + 2500x + 5000$. If each unit of… 4 marks · Short answer
- Q21(ii) The demand function for a product is given by $p = 1500 - 9x - 2x^2$ Find the revenue function. Write the marginal revenue… 4 marks · Short answer
- Q22 A cooperative society of farmers has 50 hectares of land to grow two crops, X and Y. The profits from crops, X and Y per hectare… 4 marks · Short answer