ISC Mathematics 2025 Practice Paper with Answers
All 134 questions of the ISC Mathematics 2025 Practice Paper, in printed order. Open a question to read it in full and see its answer.
- Q1 If $h(x) = 4^x$ and $h^{-1}(x) = 2$, then value of $x$ is: 1 mark · MCQ
- Q2 If $a + \frac{\pi}{2} < 2\tan^{-1} x + 3\cot^{-1} x < b$, then $a$ and $b$ are respectively: 1 mark · MCQ
- Q3 Which one of the following is true? 1 mark · MCQ
- Q4 If a matrix $A = [a_{ij}]_{2 \times 2}$, where $a_{ij} = \begin{cases} 1, & i \neq j \\ 0, & i = j \end{cases}$, then $A^{-1}$ is: 1 mark · MCQ
- Q5 If the value of 3rd order determinant is $5$, then the value of determinant formed by replacing its element by its co-factor is: 1 mark · MCQ
- Q6 If $A = \begin{bmatrix} 0 & 5 & -y \\ -5 & 0 & x \\ y & -x & 0 \end{bmatrix}$, then the value of… 1 mark · MCQ
- Q7 If $D = \begin{vmatrix} p & p & p \\ p & p+x & p \\ p & p & p+y \end{vmatrix}$ for $p \neq 0, x \neq 0, y \neq 0$ then $D$ is… 1 mark · MCQ
- Q8 If $\operatorname{adj}(A) = \begin{bmatrix} 2 & 3 & 5 \\ x & 5 & 1 \\ 3 & 3 & 4 \end{bmatrix}$ and $|A| = 4$, then the value of… 1 mark · MCQ
- Q9 Which of the following could be a sketch of the function $y = \frac{d}{dx}(x \log x)$? 1 mark · MCQ
- Q10 If $\int \left(\frac{2-x}{(x-1)^2}\right) e^x \, dx = e^x f(x) + c$, then $f(x)$ will be: 1 mark · MCQ
- Q11 An integrating factor of the differential equation: $x \frac{dy}{dx} + y \cdot P = x \cdot e^x \cdot x^{-1/2} \log x$, where $P$… 1 mark · MCQ
- Q12 Identify the correct answer of $\int_0^\pi \sin^{2024} x \cos^{2023} x \, dx$: 1 mark · MCQ
- Q13 Three fair dice are thrown. What is the probability of getting a total of 15 given that they exhibit three different numbers that… 1 mark · MCQ
- Q14 Rohit and Vishal, two below-average students in a class, are attempting a Mathematics problem during revision classes. Their… 1 mark · MCQ
- Q15 The value of… 1 mark · MCQ
- Q16 Four students are playing a game. In a box, there are four strips of paper with four expressions written on each one. The student… 1 mark · MCQ
- Q17 If $L$ is the foot of the perpendicular drawn from a point $P(a, b, c)$ on the $XY$ plane, what is the co-ordinate of $L$? 1 mark · MCQ
- Q18 Find the equation of the plane passing through the point $(2, -1, 3)$ and containing the line… 1 mark · MCQ
- Q19 Rohit joins a career counselling institute as a counsellor. The manager says, “Within a year I want a breakeven point. For that… 1 mark · MCQ
- Q20 The fixed cost of a new product is ₹ 200 and the variable cost per unit is ₹ $x$. If the demand function $p(x) = 30$, then what… 1 mark · MCQ
- Q21 In statistical modelling, regression analysis is a set of statistical processes for estimating the relationships between a… 1 mark · MCQ
- Q22 The function represented by the given graph is not differentiable at which of the following points: 1 mark · MCQ
- Q23 In an examination, a candidate takes three tests namely $\alpha, \beta, \gamma$ in succession and the probability of failing the… 1 mark · MCQ
- Q24 In the picture given above, take a look at the double-arrowed lines drawn on the overpass. This is an example of skew lines in… 1 mark · MCQ
- Q25 The graph below shows the two lines of regressions for a certain set of $x$ and $y$ values. What is the mean of $x$ and mean of… 1 mark · MCQ
- Q26 A point $x = c$ is called the critical point of a function if: 1 mark · MCQ
- Q27 Let $f(x)$ be a function such that $f'(x) = g(x)$ and $f''(x) = -f(x)$. Let $h(x) = \{f(x)\}^2 + \{g(x)\}^2$. Then, consider the… 1 mark · MCQ
- Q28 Statement I: For any two real numbers $a$ and $b$, we define $a\text{ R }b$ if $\sec^2 a - \tan^2 b = 1$. Then, R is transitive… 1 mark · MCQ
- Q29 Statement I: $f: \mathbb{R} \to \mathbb{R}$ given by $f(x) = \frac{1}{x} - 2$, is neither injective nor surjective. Statement II… 1 mark · MCQ
- Q30 In the third-order matrix, $a_{ij}$ denotes the element of the $i^{\text{th}}$ row and $j^{\text{th}}$ column… 1 mark · MCQ
- Q31 $y = x^5 + x^3, x \in \mathbb{R}$ is a function. Rina, Abha and Saurabh have given their opinions about the function in the… 1 mark · MCQ
- Q32 Statement I: $f(x) = \begin{cases} x^2 \sin\left(\frac{1}{x}\right), & \text{if } x \neq 0 \\ 0, & \text{if } x = 0 \end{cases}$… 1 mark · MCQ
- Q33 Statement I: $\int_{-2}^2 \frac{x^2}{1 + 2^x} \, dx = \frac{8}{3}$. Statement II… 1 mark · MCQ
- Q34 For any given events A: Statement I: Event A and null event $\emptyset$ are always independent. Statement II: Event A and sure… 1 mark · MCQ
- Q35 Which of the following statement(s) DOES NOT/DO NOT hold true related to regression analysis: Statement I: $|r|$ is the geometric… 1 mark · MCQ
- Q36 The relation $f: \{m, n, p, q\} \to \{11, 12, 13, 14\}$ defined by $f = \{(m, 11), (n, 12), (p, 13)\}$ is a bijective function… 1 mark · Assertion-reason
- Q37 Let $A = \begin{bmatrix} d_1 & 0 & 0 \\ 0 & d_2 & 0 \\ 0 & 0 & d_3 \end{bmatrix}$, then… 1 mark · Assertion-reason
- Q38 If $E_1$ and $E_2$ are two mutually exclusive events associated with a random experiment and $E$ is an event such that… 1 mark · Assertion-reason
- Q39 $\vec{PQ} \times (\vec{RS} + \vec{ST}) \neq \vec{0}$. The vectors $\vec{PQ}, \vec{QR}, \vec{RS}, \vec{ST}, \vec{TU}$ and… 1 mark · Assertion-reason
- Q40 A company uses a demand function $p = \frac{a}{x+b} - c$, where $a, b, c \in \mathbb{R}$ and $x = \text{number of units}$. The… 1 mark · Assertion-reason
- Q41 The curve in the graph below is not a one-one function: Assertion (A): The curve in the graph below is not a one-one function… 1 mark · Assertion-reason
- Q42 Let $f(x)$ be a polynomial function of degree 7 such that $\frac{d}{dx}(f(x)) = (x-2)^3(x+1)^2(7x-2)$ has a local minimum at… 1 mark · Assertion-reason
- Q43 If $y = \sin^{-1}(x\sqrt{x})$, then $\frac{dy}{dx} = \frac{3\sqrt{x}}{2\sqrt{1-x^3}}$. Assertion (A): If… 1 mark · Assertion-reason
- Q44 Degree of the differential equation $a\left(\frac{dy}{dx}\right)^2 + b\frac{dy}{dx} = c$ cannot be determined. Assertion (A)… 1 mark · Assertion-reason
- Q45 Maximum value of the function is 0. Shown below the graph of $f(x) = -2|x-3|$. Assertion (A): Maximum value of the function is 0… 1 mark · Assertion-reason
- Q46 If the domain of the function $f(x) = \sqrt{\cos^{-1}(3x) + \frac{\pi}{4}}$ is $[a, b]$, then find the value of $a+b$. 1 mark · Short answer
- Q47 Each triangular face of the pyramid of Peace of Kazakhstan is made up of 25 smaller triangles as shown in figure below: Using the… 1 mark · Short answer
- Q48 The figure given below shows the graph of a function $y = f(x)$. What is the derivative of the function? 1 mark · Short answer
- Q49 Find the angle between the normal to the planes: $\vec{r} \cdot (\hat{i} - \hat{j} + \hat{k}) = 3$ and… 1 mark · Short answer
- Q50 The Average Revenue (AR) and Marginal Revenue (MR) of a company on selling $x$ units (on x-axis) is shown in the following graph… 1 mark · Short answer
- Q51 $f(x) = x^2 + 2x - 3$ for $x \ge -1$. Given that the minimum value of $x^2 + 2x - 3$ occurs when $x = -1$, explain why $f(x)$ has… 1 mark · Short answer
- Q52 Explain why $x + 4y = 12$, $x, y \in \mathbb{N}$ is NOT a symmetric relation. 1 mark · Short answer
- Q53 If $\cos^{-1}\frac{x}{a} - \cos^{-1}\frac{y}{b} = \frac{\pi}{3}$ and… 1 mark · Short answer
- Q54 The matrix $A = \begin{bmatrix} a & 6 & b \\ 2c & x+1 & 8 \\ -1 & -8 & y-3 \end{bmatrix}$ is a skew symmetric matrix, then find… 1 mark · Short answer
- Q55 The displacement, $x\text{ m}$ of a particle from a fixed point at time '$t$' seconds is given by… 1 mark · Short answer
- Q56 Evaluate: $\int_{-\pi/2}^{\pi/2} \sin|x| \, dx$. 1 mark · Short answer
- Q57 Rahul, a class XII student, has a probability of getting a grade A in the examination of three subjects namely Mathematics… 1 mark · Short answer
- Q58 If $A$ and $B$ are two events such that $P(\bar{A}) = 0.3$, $P(B) = 0.4$, $P(A \cap \bar{B}) = 0.5$, then find the value of… 1 mark · Short answer
- Q59 There are 10 cookies in a box. Six have chocolate centres and four have jam-filled centres. Shweta randomly chooses a cookie from… 1 mark · Short answer
- Q60 A biased dice is tossed and the respective probabilities for various faces to turn up are: Face 1 2 3 4 5 6 Probability 0.10 0.24… 1 mark · Short answer
- Q61 If $|\vec{a}| = 1$, $|\vec{b}| = 2$, $|\vec{c}| = 3$ and $\vec{a} + \vec{b} + \vec{c} = 0$, then verify… 1 mark · Short answer
- Q62 If $\vec{p} \times \vec{q} = \vec{0}$ and $\vec{p} \cdot \vec{q} = 0$, then what conclusion can we draw? 1 mark · Short answer
- Q63 A line makes angles $\alpha, \beta$ and $\gamma$ with the x, y and z axes respectively. Given that… 1 mark · Short answer
- Q64 A variable plane at unit distance from the origin cuts the co-ordinate axes at A, B and C. The centroid of the triangle formed by… 1 mark · Short answer
- Q65 The angle between the line $\frac{x+1}{1} = \frac{y-1}{2} = \frac{z-2}{2}$ and the plane $2x - y + \sqrt{k}z + 4 = 0$ is… 1 mark · Short answer
- Q66 The cost function of a product is given by $C(x) = \frac{x^3}{3} - 45x^2 - 900x$ where $x$ is the number of units produced. What… 1 mark · Short answer
- Q67 A company paid ₹ 8475 towards the rent of the building and ₹ 7625 interest on the loan. The cost of producing one unit of a… 1 mark · Short answer
- Q68 The total cost of producing and marketing $x$ units of bulbs by a wholesaler is given by $C(x) = \frac{1}{3}x^2 + e^{2x} + 3e$… 1 mark · Short answer
- Q69 Nikhil is investigating the growth patterns of a certain species of slug and measures their thickness, $t$, and length, $l$. The… 1 mark · Short answer
- Q70 If the covariance of $x$ and $y$ is $58.08$, the variance of $x$ is $121$ and the standard deviation of $y$ is $8$ then, find the… 1 mark · Short answer
- Q71 The values of two functions $f$ and $g$ for certain values of $x$ are given in the following table: $x$ -2 0 3 $f(x)$ -12 -4 8… 2 marks · Short answer
- Q72 “A function $f$ is called a self-inverse function if $f^{-1}(x) = f(x)$ for all values of $x$ in the domain.” Let… 2 marks · Short answer
- Q73 Rahul sits on a speed boat in an island to do water sports, which is moving along a curve $y = \frac{1}{\cos x \sin x}$ in water… 2 marks · Short answer
- Q74 The function defined by $f(x) = \sin^{-1}\left(\frac{1-x^2}{1+x^2}\right)$ is differentiable at $x = 0$. Is this statement true… 2 marks · Short answer
- Q75 Seema enjoys a roller coaster ride in Ferrari world by first going downwards and then upwards to the maximum height. The relation… 2 marks · Short answer
- Q76 A child is solving an equation $x^2 + y^2 = 1$ and finds that this is the solution of the following differential equation… 2 marks · Short answer
- Q77 Given $2f(x) = \log|x|^a - bx^2 + x$, if $f(x)$ has extreme values at $x = -1$ and $x = 2$, find the value of $a$ and $b$. 2 marks · Short answer
- Q78 The solution of $x\,dy - y\,dx = 0$ represents a family of straight lines passing through the origin. Justify. 2 marks · Short answer
- Q79 Find the function ‘f’ which satisfies the equation $\frac{df}{dx} = 2f$, given that $f(0) = e^3$. 2 marks · Short answer
- Q80 Evaluate: $\int 2^{2^{2^x}} \cdot 2^{2^x} \cdot 2^x \, dx$. 2 marks · Short answer
- Q81 Evaluate: $\int \{f(ax+b)\}^n \cdot f'(ax+b) \, dx, n \neq -1$. 2 marks · Short answer
- Q82 Evaluate: $\int \frac{dx}{x^{1/2} - x^{1/3}}$. 2 marks · Short answer
- Q83 Derive a condition for which given, $\vec{b} = \vec{c}$, we can say $\vec{a} \times \vec{b} = \vec{c} \times \vec{a}$? 2 marks · Short answer
- Q84 If the angle between $\vec{a} = -3\hat{i} + x\hat{j} + \hat{k}$ and $\vec{b} = x\hat{i} + 2x\hat{j} + \hat{k}$ is acute and the… 2 marks · Short answer
- Q85 To launch a new product in his company, Mr. Rajesh spends ₹ 1 lakh on the infrastructure and the variable cost of the product is… 2 marks · Short answer
- Q86 Anuj and Ashish launched a new fountain pen in their pen-factory which is consisting of ₹ 6400/- as overheads, ₹ 35/- per pen as… 2 marks · Short answer
- Q87 If… 2 marks · Short answer
- Q88 The adjacent figure is the graph of function $y = f(x)$. Give answers to the following questions. Name the type of discontinuity… 2 marks · Short answer
- Q89 If $|\vec{a} - \vec{b}| = |\vec{a} + \vec{b}|$ in the given figure, what inference can you draw? 2 marks · Short answer
- Q90 In Z Square Mall in Kanpur, there is a space to keep 300 cars and the entry fees per car is ₹ 20. It is estimated that if the… 2 marks · Short answer
- Q91 Consider the functions $f(x) = -(x-h)^2 + 2k$ and $g(x) = e^{x-2} + k$, where $h, k \in \mathbb{R}$. Find $f'(x)$. The graphs of… 2 marks · Short answer
- Q92 Evaluate: $\int 2^x [f'(x) + f(x)\log 2] \, dx$. 2 marks · Short answer
- Q93 Let $f(x) = \frac{\log 5x}{kx}$, where $x > 0, k \in \mathbb{R}^+$. Show that $f'(x) = \frac{1-\log 5x}{kx^2}$. The graph of $f$… 4 marks · Long answer
- Q94 A linear programming problem (LPP) is given as: Maximize $Z = x + 2y$ subject to the constraints… 4 marks · Long answer
- Q95 There are two curves given in the first quadrant as: $x^2 + y^2 = \pi^2, y = \sin x$. What are the points of intersection of both… 4 marks · Long answer
- Q96 Let $\vec{\alpha} = 3\hat{i} + \hat{j}$ and $\vec{\beta} = 2\hat{i} - \hat{j} + 3\hat{k}$. If… 4 marks · Long answer
- Q97 Given $f(x) = 2\log(x-2) - x^2 + 4x + 1$ and $f'(x) = \frac{-k(x-p)(x-q)}{(x-k)}$. Find $k + p + q$. The Statement “$f(x)$ is… 4 marks · Long answer
- Q98 Case Study: The length of the perimeter of a slice of a pizza in the form of a sector of circle is 20 cm. $r$ be the radius of… 4 marks · Long answer
- Q99 In a classroom, a teacher explains the properties of a particular curve by saying that this particular curve has beautiful ups… 4 marks · Long answer
- Q100 Answer the following questions: If a real-valued function is given by: $f(x) = \sqrt{25 - x^2}$ is an onto function, then find… 4 marks · Long answer
- Q101 Find the value of: $\cot \left[\sum_{n=1}^{25} \cot^{-1} \left(1 + \sum_{k=1}^n 2k\right)\right]$. 4 marks · Short answer
- Q102 If $f: [1, \infty) \to [2, \infty)$ is given by $f(x) = x + \frac{1}{x}$ then, find $\frac{d}{dx}f^{-1}(x)$. 4 marks · Short answer
- Q103 Find the acute angle between the curves $y = |x^2 - 1|$ and $y = |x^2 - 3|$ at their point of intersection when $x > 0$. 4 marks · Short answer
- Q104 $y = \ln(x+1) - \ln x$ is a curve. The tangent to the curve at the point $P(1, \ln 2)$ meets x-axis at A and y-axis at B. The… 4 marks · Long answer
- Q105 Solve the following differential equation: $\cos^2 x \frac{dy}{dx} + y = \tan x$, given that $y(0) = 1$. Hence, find… 4 marks · Short answer
- Q106 Evaluate: $\int \frac{dx}{\sqrt[4]{(x-1)^3 (x+2)^5}}$. 4 marks · Short answer
- Q107 For any function $f(x)$, we have… 4 marks · Short answer
- Q108 A pot contains 5 red and 2 green balls. At random a ball is drawn from this pot. If a drawn ball is green, then put a red ball in… 4 marks · Short answer
- Q109 Rahul and Divya were playing the snakes and ladders board game. Each one had their own dice to play the game. Rahul was using a… 4 marks · Long answer
- Q110 Imagine you are at a point A, a café you visit often. Your friend is at the point B, a bookstore a few blocks away on a straight… 4 marks · Short answer
- Q111 Two friends are planning a road trip. One friend stays in City A represented by the position vector… 4 marks · Long answer
- Q112 In the beautiful town of Darjeeling in the Himalayan foothills, the city planning committee wants to construct two major roads to… 4 marks · Long answer
- Q113 The plane $px + y + pz = r$ intersects the co-ordinate axes at A, B and C. Find the co-ordinates of the points A, B and C. Find… 4 marks · Long answer
- Q114 From any point $P(2, 1, 2)$ perpendiculars PM and PN are drawn to ZX and XY planes. If O is the origin, find the equation of the… 4 marks · Long answer
- Q115 Draw a rough sketch and find the area enclosed by the curve $y = -x^2$ and the line $x + y + 2 = 0$. 4 marks · Long answer
- Q116 Find the area of the region bounded by the x-axis, part of the curve $y = 1 + \frac{8}{x^2}$, and the ordinates $x = 2$ and… 4 marks · Short answer
- Q117 Find the area enclosed between the co-ordinate axis and the curve $y^2 = 4a(x + \lambda)$ in the second quadrant. 4 marks · Short answer
- Q118 A study was conducted to investigate the relationship between the number of hours a student studies per week (X) and their scores… 4 marks · Long answer
- Q119 The corner points of the feasible region determined by the system of linear constraints are as shown below: Answer the following… 4 marks · Long answer
- Q120 A movie cinema is considering significantly reducing the price of their popcorn as they believe their customers spend more on… 4 marks · Long answer
- Q121 A part of the graph of the function $f(x) = 2x^3 - 3x^2 - 12x + 8, x \in \mathbb{R}$ is shown below: Answer the following… 4 marks · Long answer
- Q122 In a Kabaddi league, two matches are being played between Jaipur and Delhi. It is assumed that the outcomes of two games are… 4 marks · Long answer
- Q123 Two drones are being used for soil analysis over an area of farmland. Drone A has been programmed to fly on the path given by… 4 marks · Short answer
- Q124 A manufacturing company produces two types of cell phones, Android and iOS. The company has resources to make at the most 300… 4 marks · Long answer
- Q125 Rajesh wants to purchase some fruits from fruit market. 4 kilograms (kgs) apples, 3 kgs grapes and 2 kgs oranges cost him ₹ 600… 6 marks · Long answer
- Q126 In a raffle draw, 1000 raffle tickets are sold for ₹ 1 each. Each has an equal chance of winning. First prize is ₹ 300, second… 6 marks · Long answer
- Q127 Ram and Shyam play a game with a coin. Ram stakes ₹ 1.00 and throws the coin 4 times. If he throws 4 heads, he gets his stake and… 6 marks · Long answer
- Q128 The fuel cost per hour for running a ship is proportional to the square of the speed generated in knots. The fuel cost is ₹ 75/h… 6 marks · Long answer
- Q129 Evaluate: $\int \frac{\tan\frac{x}{4}}{1 - \sin\frac{x}{4}} \, dx$. 6 marks · Short answer
- Q130 Evaluate the following integral: $\int \left(\frac{\cot x + \cot^3 x}{1 + \cot^3 x}\right) \, dx$. 6 marks · Short answer
- Q131 Let $f(x) = \begin{cases} a + \sin^{-1}(x+b), & x \ge 1 \\ x, & x < 1 \end{cases}$, and $f'(1)$ exists. The statement “$f(x)$ is… 6 marks · Long answer
- Q132 Let $f(x) = \cos x + \sqrt{3}\sin x, 0 \le x \le 2\pi$. The following diagram shows the graph of $f$. The $y$-intercept is at… 6 marks · Long answer
- Q133 Consider the following differential equation and answer the questions… 6 marks · Long answer
- Q134 Sonia watches a painting which has its bottom edge 2 meters (m) above eye level and its top edge is 3 m above eye level as shown… 6 marks · Long answer