ISC Mathematics 2026 Question Paper with Answers
All 54 questions of the ISC Mathematics 2026 Question Paper, in printed order. Open a question to read it in full and see its answer.
- Q1(i) If $A$ is a square matrix of order $3$ and its determinant is $|A| = -3$, then the value of $|-4A|$ is: 1 mark · MCQ
- Q1(ii) Consider the function $f$ given by $f(x) = \log x$, $x > 0$, then the function $f$ is: 1 mark · MCQ
- Q1(iii) If events $A$ and $B$ are mutually exclusive, such that $P(A) = \frac{1}{5}$ and $P(B) = \frac{2}{3}$, then the value of… 1 mark · MCQ
- Q1(iv) $f(x) = \begin{cases} 1 + x, & x \le 2 \\ 5 - x, & x > 2 \end{cases}$ at $x = 2$ is not differentiable. 1 mark · Assertion-reason
- Q1(v) The value of $\int_{0}^{3/2} |x| \, dx$ is: 1 mark · MCQ
- Q1(vi) Statement 1: If $0 < x < \frac{\pi}{2}$ then the value of $\tan^{-1}(\cot x) = \frac{\pi}{2} - x$ Statement 2… 1 mark · MCQ
- Q1(vii) How many possible matrices can be formed of order $3 \times 3$ if each entry is either $0$ or $1$? 1 mark · MCQ
- Q1(viii) Observe the graph given below and answer the question that follows. Statement 1: $f(x)$ increases in $(-\infty, -1)$ and… 1 mark · MCQ
- Q1(ix) If set $A$ contains four elements and set $B$ contains five elements, then the number of one-one and onto mapping from $A \to B$… 1 mark · MCQ
- Q1(x) The solution of $\frac{dy}{dx} - y = 1$, $y(0) = 1$ is given by: 1 mark · MCQ
- Q1(xi) The system of three linear equations in three unknown variables can be written in the matrix form as $AX = B$. It has a unique… 1 mark · Assertion-reason
- Q1(xii) If $x = e^{y + e^{y + e^{y + \dots \infty}}}$, $x > 0$ then find $\frac{dy}{dx}$. 1 mark · Short answer
- Q1(xiii) Solve for $x$: $\begin{vmatrix} 1 & -2 & 5 \\ 2 & x & -1 \\ 0 & 4 & 2x \end{vmatrix} = 86$. 1 mark · Short answer
- Q1(xiv) Find the principal value of $\sec^{-1}(-\sqrt{2})$. 1 mark · Short answer
- Q1(xv) A relation $R$ on the set $A = \{a, b, c\}$ is defined by $R = \{(a, b), (b, a)\}$. Is the relation $R$ symmetric? Justify. 1 mark · Short answer
- Q2 Using properties of determinant, show that… 2 marks · Derivation
- Q3(i) Let $f(x) = 4 - (x - 7)^3$ be an invertible function, then find $f^{-1}(x)$. 2 marks · Short answer
- Q3(ii) Find the range of the function $f(x) = \frac{1}{3 - 2\sin x}$. 2 marks · Short answer
- Q4 Evaluate: $\int \frac{e^x}{1 + e^{2x}} \, dx$ 2 marks · Short answer
- Q5 A die marked 1, 2, 3 in red and 4, 5, 6 in green is thrown. Let A be the event 'Number appearing is odd' and B be the event… 2 marks · Derivation
- Q6(i) The surface of a spherical balloon is increasing at the rate of $4\text{ cm}^2/\text{sec}$. Find the rate of change of volume… 2 marks · Short answer
- Q6(ii) Find the equation of the normal at $(1, 2)$ to the curve $x^2 = 4y$. 2 marks · Short answer
- Q7 Vinayak runs a bakery shop. He sells three items: Sandwiches (₹$x$ per unit), Fruit juices (₹$y$ per unit) and Cookies (₹$z$ per… 4 marks · Short answer
- Q8(i) If $\tan^{-1}\left(\frac{\sqrt{1+x^2} - \sqrt{1-x^2}}{\sqrt{1+x^2} + \sqrt{1-x^2}}\right) = \alpha$, prove that… 4 marks · Derivation
- Q8(ii) Solve for $x$: $2\tan^{-1}\left(\frac{1}{3}\right) + \sec^{-1}\left(\frac{5\sqrt{2}}{7}\right) = \tan^{-1} x$ 4 marks · Short answer
- Q9 If $y = x^3 \log\left(\frac{1}{x}\right)$, then prove that $x\frac{d^2 y}{dx^2} - 2\frac{dy}{dx} + 3x^2 = 0$. 4 marks · Derivation
- Q10(i) A sports store owner conducts a game 'weekend-surprise' every Friday for his customers. He fills two bags with cricket balls of… 4 marks · Short answer
- Q10(ii) Children of a society practise building human pyramids for 16 days to participate in the pyramid building competition during the… 4 marks · Case based
- Q11 A van is carrying a large amount of money in cash to deposit it in two ATM machines on a hill station. The location of these… 6 marks · Case based
- Q12(i) Evaluate: $\int \frac{\sin x \, dx}{\cos x(1 - \sin x)}$ 6 marks · Short answer
- Q12(ii) Using properties of definite integral, calculate the value of: $\int_{0}^{\pi/2} \frac{\sin^2 x}{1 + \sin x \cos x} \, dx$ 6 marks · Short answer
- Q13 A gardener wants to plant saplings on a day when rain is not predicted. According to the forecast by the weather department, -… 6 marks · Case based
- Q14(i) Solve the following differential equation: $x^2 dy + (xy + y^2) dx = 0$ 6 marks · Short answer
- Q14(ii) Find the particular solution for the following differential equation: $\sqrt{1 - y^2} \, dx = (\sin^{-1} y - x) dy$, given that… 6 marks · Short answer
- Q15(i) A scalar is multiplied by a unit vector, then the resultant is: Statement 1: A vector with the magnitude of the scalar. Statement… 1 mark · MCQ
- Q15(ii) The projection of $\hat{\imath} + 2\hat{\jmath} - 3\hat{k}$ on $2\hat{\imath} - \hat{\jmath} + \hat{k}$ is: 1 mark · MCQ
- Q15(iii) The direction cosines of the line passing through the points $P(2, 3, 5)$ and $Q(-1, 2, 4)$ are: 1 mark · MCQ
- Q15(iv) Find the area of a parallelogram whose adjacent sides are given by the vectors… 1 mark · Short answer
- Q15(v) Find the equation of the plane with intercepts $3, -4$ and $2$ on $x, y$ and $z$ axes respectively. 1 mark · Short answer
- Q16(i) Three drone cameras A, B and C are recording a hockey match. Their positions with respect to a control tower O are given by the… 2 marks · Derivation
- Q16(ii) If $\vec{a}$ and $\vec{b}$ are mutually perpendicular vectors, $|\vec{a} + \vec{b}| = 13$ and $|\vec{a}| = 5$, then find the… 2 marks · Short answer
- Q17(i) The paths traced by two hot air balloons are: $\frac{x - 1}{2} = \frac{y - b}{3} = \frac{z - 3}{4}$ and… 4 marks · Short answer
- Q17(ii) A school is preparing the stage for its annual day function. They want to place a hanging mic and a hanging light on the stage. -… 4 marks · Case based
- Q18 Find the area of the region bounded by $y = \sqrt{4 - x^2}$ and $x$ axis using integration. 4 marks · Short answer
- Q19(i) Statement 1: Two regression coefficients cannot have the same sign. Statement 2: Both the regression coefficients can be… 1 mark · MCQ
- Q19(ii) Which one of the following statements is true about Marginal Revenue? 1 mark · MCQ
- Q19(iii) The cost of manufacturing $x$ units of a commodity is $27 + 12x + 3x^2$. Find the output for which Average Cost is decreasing. 1 mark · Short answer
- Q19(iv) For two variables $x$ and $y$, if $\sigma_x = 5$, $r = \frac{-1}{2}$, $b_{yx} = \frac{-2}{7}$ then find the value of $\sigma_y$. 1 mark · Short answer
- Q19(v) The total cost function for production and marketing of a product is given by $C(x) = \frac{3x^2}{4} - 7x + 3$, where $x$ is the… 1 mark · Short answer
- Q20(i) A company produces a commodity with ₹36,000 as a fixed cost. The variable cost is estimated to be 25% of the total revenue… 2 marks · Case based
- Q20(ii) A school is organising an art and craft exhibition. The management has decided to donate the profit earned from the sale of… 2 marks · Short answer
- Q21(i) Consider the following data of a bivariate distribution: - The mean of the variables $x$ and $y$ are 25 and 30 respectively. -… 4 marks · Case based
- Q21(ii) If the regression lines of a bivariate distribution are $4x - 5y + 33 = 0$ and $20x - 9y - 107 = 0$, then Calculate the… 4 marks · Case based
- Q22 Raunak is a small-scale entrepreneur who sells sewing machines in a rural market. He wants to expand his business but has two… 4 marks · Short answer