ISC Mathematics 2026 Improvement Paper with Answers
All 54 questions of the ISC Mathematics 2026 Improvement Paper, in printed order. Open a question to read it in full and see its answer.
- Q1(i) If $\begin{bmatrix} 2x & 3 \end{bmatrix} \begin{bmatrix} 1 & 2 \\ -3 & 0 \end{bmatrix} \begin{bmatrix} x \\ 3 \end{bmatrix} = O$… 1 mark · MCQ
- Q1(ii) The equation $\sin^{-1} x - \cos^{-1} x = \cos^{-1}\left(\frac{\sqrt{3}}{2}\right)$ has: 1 mark · MCQ
- Q1(iii) $R$ is not an equivalence relation. If $A = \{1, 2, 3\}$ and the relation $R = \{(1, 1), (2, 2), (3, 3), (2, 3)\}$, then 1 mark · Assertion-reason
- Q1(iv) If $A$ and $B$ are matrices of same order, then $AB' - BA'$ is a: 1 mark · MCQ
- Q1(v) Consider the matrix $A$ of order $3 \times 3$ with $|A| = 5$. If any two columns are interchanged in $A$, then $|A|$ is: 1 mark · MCQ
- Q1(vi) Statement 1: The sum, difference and product of two continuous functions are also continuous. Statement 2: The function $f(x)$ is… 1 mark · MCQ
- Q1(vii) The value of $\int_{-2}^{1} |x + 1| \, dx$ is: 1 mark · MCQ
- Q1(viii) A die is thrown three times. If the first throw is a four, then the probability of getting 15 as a sum is: 1 mark · MCQ
- Q1(ix) Consider the graph given below of the function $y = x^{2/3}$ and answer the question that follows: Statement 1: The function is… 1 mark · MCQ
- Q1(x) The system of linear equations $x + y + z = 6$, $x + \lambda y + 2z = 7$ and $3x + y + z = 12$ has a unique solution if: 1 mark · MCQ
- Q1(xi) If $\tan^{-1} x + \tan^{-1} y = \frac{\pi}{4}$, then $x + y + xy = 1$. Which one of the following is correct? 1 mark · Assertion-reason
- Q1(xii) Evaluate: $\int \frac{10^x \log 10 + 10x^9}{10^x + x^{10}} \, dx$ 1 mark · Short answer
- Q1(xiii) Find the general solution for the following differential equation: $\frac{dy}{dx} = (1 + x^2)(1 + y^2)$ 1 mark · Short answer
- Q1(xiv) Differentiate $y = x^x$ with respect to $x$. 1 mark · Short answer
- Q1(xv) Find the range of the function shown in the graph below: 1 mark · Short answer
- Q2(i) Prove that the function $f: \mathbb{Z} \to \mathbb{Z}$, defined by $f(x) = x - 5$ is a bijective function. 2 marks · Derivation
- Q2(ii) For what values of $x$ is the function $f(x) = \frac{4x+3}{6x-4}$ invertible? Also, find the inverse function. 2 marks · Short answer
- Q3 Using determinants, find the value of $x$ if the points $(x, -2)$, $(5, 2)$ and $(8, 8)$ are collinear. 2 marks · Numerical
- Q4(i) A water treatment plant uses a large cylindrical storage tank to collect treated water before distribution. The tank has a… 2 marks · Numerical
- Q4(ii) Find the equation of the tangent to the curve $x^2 + xy - 3 = 0$ at $(1, 2)$. 2 marks · Short answer
- Q5 Prove that $y = -(8x^2 + 4x + 5)$ is an increasing function in $\left(-\infty, -\frac{1}{4}\right]$. 2 marks · Derivation
- Q6 If $A$ and $B$ are two independent events such that $P(A \cup B) = \frac{3}{5}$ and $P(A) = \frac{1}{2}$, calculate $P(B)$. 2 marks · Numerical
- Q7(i) If $\tan^{-1} 2x + \tan^{-1} 3x = \frac{\pi}{4}$, then: Prove that $6x^2 + 5x - 1 = 0$. Hence, solve the equation… 4 marks · Short answer
- Q7(ii) If $\cos^{-1} x + \cos^{-1} y + \cos^{-1} z = \pi$, then find the value of $x^2 + y^2 + z^2 + 2xyz$. 4 marks · Numerical
- Q8(i) If $A = \begin{bmatrix} 1 & -2 & 0 \\ 2 & -1 & -1 \\ 0 & -2 & 1 \end{bmatrix}$, then find $A^{-1}$. Using $A^{-1}$, solve the… 4 marks · Short answer
- Q8(ii) Using properties of determinants, solve for $x$: $\begin{vmatrix} x+a & b & c \\ a & x+b & c \\ a & b & x+c \end{vmatrix} = 0$ 4 marks · Numerical
- Q9 If $y = x + \tan x$, show that $\cos^2 x \frac{d^2 y}{dx^2} - 2y + 2x = 0$. 4 marks · Derivation
- Q10 A football club purchased footballs manufactured at three different factories in Chennai, Mumbai and Delhi. Given below is the… 4 marks · Numerical
- Q11 A manufacturing company produces airtight cylindrical containers for storing sensitive chemicals. For safety, each container is… 6 marks · Case based
- Q12(i) Evaluate: $\int \frac{dx}{x^3 + 4x^2 + 4x + 1}$ 6 marks · Short answer
- Q12(ii) Using properties of definite integral, prove that: $\int_0^\pi \frac{x}{1 + \sin x} \, dx = \pi$ 6 marks · Derivation
- Q13(i) Find the particular solution of the following differential equation: $x^2 dy - (2xy + y^2) dx = 0$ given that $y = 1$ when… 6 marks · Short answer
- Q13(ii) Solve the differential equation $\frac{dy}{dx} + y \tan x = \sin 2x$, when $x = 0$ and $y = 1$. 6 marks · Short answer
- Q14 A school conducted an inter house quiz competition. The number of students who participated in the competition was 15. The scores… 6 marks · Case based
- Q15(i) For any three vectors $\vec{a}$, $\vec{b}$ and $\vec{c}$, Statement 1: $\vec{a} \cdot (\vec{b} \times \vec{c})$ is a scalar… 1 mark · MCQ
- Q15(ii) A unit vector in the direction of the sum of the vectors $2\hat{i} + 3\hat{j} - \hat{k}$ and $4\hat{i} - 3\hat{j} + 2\hat{k}$ is: 1 mark · MCQ
- Q15(iii) For vector $\vec{a}$, $\vec{a} \cdot \hat{i} = \vec{a} \cdot \hat{j} = \vec{a} \cdot \hat{k}$ and $|\vec{a}| = 300$. Then… 1 mark · MCQ
- Q15(iv) Find the equation of the plane parallel to ZOX plane with Y intercept 3. 1 mark · Short answer
- Q15(v) Find the angle between the line $\frac{x-1}{2} = \frac{y-2}{1} = \frac{z-3}{-3}$ and the plane $2x + y - 3z + 4 = 0$. 1 mark · Short answer
- Q16(i) A school’s media team identifies three points A, B and C on a large bulletin board to position three charts. The position vectors… 2 marks · Numerical
- Q16(ii) Three forces act simultaneously on a space satellite in such a way that the satellite remains in equilibrium. The forces are… 2 marks · Numerical
- Q17(i) Find the equation of the plane passing through the points $(2, -3, 1)$ and $(-1, 1, -7)$ and perpendicular to the plane… 4 marks · Short answer
- Q17(ii) Find the foot of perpendicular from the point $P(2, 3, -8)$ to the line $\frac{x+1}{2} = \frac{y-3}{-2} = \frac{z+4}{1}$. Also… 4 marks · Short answer
- Q18 Find the area of the region bounded by the curves $y^2 = x$ and $y^2 = 4 - 3x$. 4 marks · Short answer
- Q19(i) Statement 1: Profit is maximum when marginal revenue equals marginal cost. Statement 2: At maximum profit, the slope of the… 1 mark · MCQ
- Q19(ii) If $\bar{x} = 28$, $\bar{y} = 32$, $b_{yx} = 1.8$ and $b_{xy} = 0.5$, then the regression equation of $y$ on $x$ is: 1 mark · MCQ
- Q19(iii) The total cost function and the total revenue function of a commodity are given by $C(x) = x + 40$ and $R(x) = ax$ respectively… 1 mark · Numerical
- Q19(iv) A company produces a product whose cost function is $C(x) = x^3 - 27x + 2000$. Find the number of products when the marginal cost… 1 mark · Numerical
- Q19(v) Justify which scatter diagram given above represents a regression equation with positive slope. 1 mark · Short answer
- Q20(i) The cost of manufacturing $x$ units of a commodity is $5x^2 + 30x + 5$. Find the range of output where Average Cost is increasing. 2 marks · Short answer
- Q20(ii) The Revenue function for $x$ units of a commodity is $R(x) = x^3 + 3x^2 - 7x + 16$. Calculate the Marginal Revenue at $x = 5$. 2 marks · Numerical
- Q21 A school runs two training modules for sports competition. The data for a sample of 6 batches is given below. $X$: Average number… 4 marks · Short answer
- Q22(i) An ice-cream factory makes ice creams of two flavours: chocolate and mango. - Each chocolate ice cream needs 5 minutes of mixing… 4 marks · Short answer
- Q22(ii) The feasible region for a Linear Programming Problem is shown in the following graph. Based on the given graph, answer the… 4 marks · Short answer