ISC Class 12 Mathematics: The Ultimate Topic-Wise PYQ Breakdown (2023–2025)
Welcome back to Prime Maths! As we accelerate our preparation for the upcoming board exams, one of the most powerful strategies you can employ is analyzing Previous Year Questions (PYQs). It’s not just about solving problems; it’s about understanding the pattern, the weightage, and the exact language the council uses.
To make your revision seamless, I have sifted through the recent ISC Class 12 Mathematics examination papers (2023, 2024, 2025, and the 2025 Improvement Exam) and segregated the essential questions by topic. Grab your notebooks, and let's dive in!
📌 Algebra & Inverse Trigonometric Functions
This section tests your foundational logic and matrix operations. Pay close attention to the properties of determinants and inverse trigonometric identities.
- Relations: Determine if the relation $R$ on $\{1, 2, 3\}$ given by $R=\{(1,1), (2,2), (1,2), (3,3), (2,3)\}$ is reflexive, symmetric, or transitive. (2023, [1 mark])
- Matrices: Find the value of $k$ for which the matrix $\begin{bmatrix}0 & k \\ -6 & 0\end{bmatrix}$ is a skew-symmetric matrix. (2023, [1 mark])
- Functions: If $f(x)=[4-(x-7)^3]^{1/5}$ is a real invertible function, find $f^{-1}(x)$. (2023, [2 marks])
- Determinants: Evaluate the determinant without expanding:
$$\begin{vmatrix} 5 & 5 & 5 \\ a & b & c \\ b+c & c+a & a+b \end{vmatrix}$$ (2023, [2 marks]) - Inverse Trigonometry: Solve for $x$: $5\tan^{-1}x + 3\cot^{-1}x = 2\pi$. (2023, [2 marks])
- Linear Equations: Use the matrix method to solve the system of equations: $\frac{2}{x} + \frac{3}{y} + \frac{10}{z} = 4$, $\frac{4}{x} - \frac{6}{y} + \frac{5}{z} = 1$, and $\frac{6}{x} + \frac{9}{y} - \frac{20}{z} = 2$. (2023, [6 marks])
- Inverse Trigonometry: Evaluate the value of $\csc(\sin^{-1}(\frac{-1}{2})) - \sec(\cos^{-1}(\frac{-1}{2}))$. (2024, [1 mark])
- Matrices: Determine if $AB-BA$ is a symmetric or skew-symmetric matrix given that $A$ and $B$ are symmetric matrices of the same order. (2024, [1 mark])
- Inverse Trigonometry: Solve for $x$: $\sin^{-1}(\frac{x}{2}) + \cos^{-1}x = \frac{\pi}{6}$. (2024, [4 marks])
- Inverse Trigonometry: If $\sin^{-1}x + \sin^{-1}y + \sin^{-1}z = \pi$, show that $x^2-y^2-z^2+2yz\sqrt{1-x^2}=0$. (2024, [4 marks])
- Matrices: Find the value of $A^{16}$ if $A=\begin{bmatrix}0 & a \\ 0 & 0\end{bmatrix}$. (2025, [1 mark])
- Relations: Write the smallest equivalence relation from the set $A$ to $A$, where $A=\{1,2,3\}$. (2025, [1 mark])
- Inverse Trigonometry: Find the value of $\tan^{-1}x - \cot^{-1}x$ if $(\tan^{-1}x)^2 - (\cot^{-1}x)^2 = \frac{5\pi}{8}$. (2025, [2 marks])
- Determinants: Prove the determinant identity $\begin{vmatrix} 1 & 1 & 1 \\ x & y & z \\ x^3 & y^3 & z^3 \end{vmatrix} = 0$ if $x+y+z=0$. (2025, [4 marks])
- Linear Equations: Find the value of $\mu$ if the system of equations $2x+3y-8=0$, $7x-5y+3=0$, and $4x-6y+\mu=0$ is consistent. (2025 IE, [1 mark])
- Matrices: For what value of $a$ is the matrix $A=\begin{bmatrix} 1 & -2 & 3 \\ 1 & 2 & 1 \\ a & 2 & -3 \end{bmatrix}$ not invertible? (2025 IE, [1 mark])
- Determinants: Using properties of determinants, prove that:
$$\begin{vmatrix} 1 & 1 & 1 \\ a & b & c \\ a^3 & b^3 & c^3 \end{vmatrix} = (a-b)(b-c)(c-a)(a+b+c)$$ (2025 IE, [4 marks])
📌 Calculus
Calculus forms the major chunk of the paper. Focus heavily on differential equations and properties of definite integrals.
- Rate of Change: An edge of a variable cube is increasing at the rate of 10 cm/sec. How fast will the volume of the cube increase if the edge is 5 cm long? (2023, [1 mark])
- Differentiation: Find the derivative of $\log x$ with respect to $\frac{1}{x}$. (2023, [1 mark])
- Differential Equations: Solve the differential equation: $\frac{dy}{dx} = \csc y$. (2023, [1 mark])
- Integration: Evaluate $\int \cos^{-1}(\sin x) dx$. (2023, [2 marks])
- Differentiation: If $y=e^{ax}\cos bx$, prove that $\frac{d^2y}{dx^2} - 2a\frac{dy}{dx} + (a^2+b^2)y = 0$. (2023, [4 marks])
- Maxima/Minima: Prove that the semi-vertical angle of the right circular cone of given volume and least curved area is $\cot^{-1}\sqrt{2}$. (2023, [6 marks])
- Differential Equations: Find the order and the degree of the differential equation $1+(\frac{dy}{dx})^2 = \frac{d^2y}{dx^2}$. (2024, [1 mark])
- Tangents & Normals: Find a point on the curve $y=(x-2)^2$ at which the tangent is parallel to the line joining the chord through the points (2, 0) and (4, 4). (2024, [2 marks])
- Integration: Evaluate $\int_0^{2\pi} \frac{1}{1+e^{\sin x}} dx$. (2024, [2 marks])
- Differentiation: If $y=3\cos(\log x) + 4\sin(\log x)$, show that $x^2\frac{d^2y}{dx^2} + x\frac{dy}{dx} + y = 0$. (2024, [4 marks])
- Maxima/Minima: A window is designed in the shape of a rectangle surmounted by an equilateral triangle. If the perimeter is 12 m, find the dimensions that admit maximum sunlight. (2024, [6 marks])
- Rate of Change: A cylindrical popcorn tub of radius 10 cm is filled with popcorn at a rate of $314\text{ cm}^3\text{/min}$. Find the rate at which the level of popcorn is increasing. (2025, [1 mark])
- Tangents & Normals: Find the point on the curve $y=2x^2-6x-4$ at which the tangent is parallel to the x-axis. (2025, [2 marks])
- Differentiation: If $x^y = e^{x-y}$, prove that $\frac{dy}{dx} = \frac{\log x}{(1+\log x)^2}$. (2025, [2 marks])
- Differential Equations: Show that $\tan^{-1}x + \tan^{-1}y = C$ is the general solution of the differential equation $(1+x^2)dy + (1+y^2)dx = 0$. (2025, [2 marks])
- Differentiation: Find the derivative of $\tan^{-1}(\frac{\cos x - \sin x}{\cos x + \sin x})$ with respect to $x$. (2025 IE, [1 mark])
- Integration: Evaluate $\int_{-2}^2 x f(x) dx$ given $f(x)=x+g(x)$ where $g(x)$ is an even function. (2025 IE, [2 marks])
- Tangents & Normals: Find the equation of the normal to the curve $y=x^2-3x+1$ at the point (3, 1). (2025 IE, [2 marks])
- Integration: Evaluate $\int_0^{\pi/2} \log(\tan x) dx$. (2025 IE, [2 marks])
- Differentiation: If $x=\sin t$ and $y=\sin pt$, prove that $(1-x^2)\frac{d^2y}{dx^2} - x\frac{dy}{dx} + p^2y = 0$. (2025 IE, [4 marks])
📌 Vectors and 3D Geometry
Visualize these problems before you put pen to paper. Drawing a quick rough sketch can often save you from sign errors in 3D geometry.
- Vectors: Find the area of the parallelogram whose diagonals are $\hat{i}-3\hat{j}+\hat{k}$ and $\hat{i}+\hat{j}+\hat{k}$. (2023, [1 mark])
- 3D Geometry: Write the equation of the plane passing through the point (2, 4, 6) and making equal intercepts on the coordinate axes. (2023, [1 mark])
- Vectors: If $A(1, 2, -3)$ and $B(-1, -2, 1)$ are end points of $\vec{AB}$, find the unit vector in the direction of $\vec{AB}$. (2023, [2 marks])
- 3D Geometry: Find the equation of the plane passing through the point (1, 1, -1) and perpendicular to the planes $x+2y+3z=7$ and $2x-3y+4z=0$. (2023, [4 marks])
- Vectors: If $\vec{a}=3\hat{i}-2\hat{j}+\hat{k}$ and $\vec{b}=2\hat{i}-4\hat{j}-3\hat{k}$, find the value of $|\vec{a}-2\vec{b}|$. (2024, [1 mark])
- Vectors: Find a vector of magnitude 20 units parallel to the vector $2\hat{i}+5\hat{j}+4\hat{k}$. (2024, [1 mark])
- Vectors: If $\vec{a}\times\vec{b} = \vec{a}\times\vec{c}$ where $\vec{a}, \vec{b}$ and $\vec{c}$ are non-zero vectors, prove that either $\vec{b}=\vec{c}$ or $\vec{a}$ and $(\vec{b}-\vec{c})$ are parallel. (2024, [2 marks])
- 3D Geometry: Find the angle between the two planes $x+y+2z=9$ and $2x-y+z=15$. (2025, [1 mark])
- 3D Geometry: The equation of the path traced by a honeybee is $\vec{r}=(\hat{i}+2\hat{j}+3\hat{k})+\lambda(2\hat{i}+3\hat{j}+4\hat{k})$. Find the equation of the parallel path traced by another honeybee passing through the point (2, 4, 5). (2025, [1 mark])
- 3D Geometry: Find the equation of the plane passing through the points (2, 2, -1), (3, 4, 2) and (7, 0, 6). (2025, [2 marks])
- Vectors: Consider position vectors $\vec{OA}=2\hat{i}-2\hat{j}+\hat{k}$, $\vec{OB}=\hat{i}+2\hat{j}-2\hat{k}$ and $\vec{OC}=2\hat{i}-\hat{j}+4\hat{k}$. Find the area of the triangle ABC whose sides are $\vec{AB}$ and $\vec{BC}$. (2025, [2 marks])
- 3D Geometry: Find the shortest distance between the lines $\vec{r}=(\hat{i}+2\hat{j}+3\hat{k})+\lambda(2\hat{i}+3\hat{j}+4\hat{k})$ and $\vec{r}=(2\hat{i}+4\hat{j}+5\hat{k})+\mu(4\hat{i}+6\hat{j}+8\hat{k})$. (2025 IE, [2 marks])
- Vectors: Consider vectors $\vec{a}=2\hat{i}-3\hat{j}+4\hat{k}$ and $\vec{b}=5\hat{i}+q\hat{j}-\hat{k}$. Calculate $q$ if both vectors are perpendicular to each other. (2025 IE, [1 mark])
📌 Probability and Linear Programming (LPP)
These questions are incredibly scoring if you read the word problems carefully. Define your variables clearly in LPP, and always double-check your sample space in probability.
- Probability: A bag contains 19 tickets, numbered from 1 to 19. Two tickets are drawn randomly in succession with replacement. Find the probability that both the tickets drawn are even numbers. (2023, [1 mark])
- Probability: The probability of event A occurring is $\frac{1}{3}$ and event B occurring is $\frac{1}{2}$. If A and B are independent, find the probability of neither A nor B occurring. (2023, [2 marks])
- Probability Distribution: A box contains 30 fruits, out of which 10 are rotten. Two fruits are selected at random one by one without replacement. Find the probability distribution and mean of the number of unspoiled fruits. (2023, [6 marks])
- LPP: Solve the Linear Programming Problem graphically: Maximise $z=5x+2y$ subject to $x-2y\le 2$, $3x+2y\le 12$, $-3x+2y\le 3$, $x\ge 0, y\ge 0$. (2023, [6 marks])
- Probability: Evaluate $P(A\cup B)$ if $2P(A)=P(B)=\frac{5}{13}$ and $P(A|B)=\frac{2}{5}$. (2024, [2 marks])
- LPP Formulation: Aman has ₹1500 to purchase rice and wheat. Each sack of rice and wheat costs ₹180 and ₹120 respectively. He can store a maximum of 10 bags and will earn a profit of ₹11 per bag of rice and ₹9 per bag of wheat. Formulate a Linear Programming Problem to maximise profit. (2024, [4 marks])
- Probability: Three critics review a book. Odds in favour of the book are 5:2, 4:3 and 3:4 respectively. Find the probability that all critics are in favour of the book. (2025, [1 mark])
- Probability: Pia, Sia and Dia displayed 15, 5 and 10 of their paintings respectively in an art exhibition. A person bought three paintings. Find the probability that he bought one painting from each of them. (2025, [2 marks])
- LPP Formulation: Two different types of books have to be stacked. Type 1 weighs 1 kg with a thickness of 6 cm. Type 2 weighs 1.5 kg with a thickness of 4 cm. The 96 cm long shelf holds a maximum of 21 kg. Formulate an LPP to maximize the number of books. (2025, [4 marks])
- Probability: In a class of 50 students, 25 study English, 10 study History and 10 study both. Find the probability that a randomly selected student studies either English or History. (2025 IE, [1 mark])
- Probability Distribution: In a school committee of 30 teachers, 20 never commit errors. Two teachers are selected at random. Let $X$ be the number of selected teachers who never make an error. Find the probability distribution and its mean. (2025 IE, [4 marks])
- LPP: A cooperative society has 50 hectares to grow crops X and Y (profits ₹10500 and ₹9000 per hectare). A liquid herbicide constraint limits usage to 20 litres and 10 litres per hectare respectively, with a maximum total of 800 litres. Formulate the LPP to maximize profit and solve graphically. (2025 IE, [4 marks])
A Final Word of Advice:
Don't just read these questions—solve them. Write out the steps completely as if you were sitting in the exam hall. If you get stuck on any of these, let me know in the comments below, and we can tackle the solutions together in an upcoming post.
Keep practicing, and stay focused!
