ISC Class 12 Mathematics: The Ultimate Topic-Wise PYQ Breakdown (2017–2025)

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 (2017, 2018, 2019, 2020, 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.

Matrices: If the matrix $\begin{pmatrix}6 & -x^2 \\ 2x-15 & 10\end{pmatrix}$ is symmetric, find the value of $x$. (2017, [3 marks])
Inverse Trigonometry: Prove that $\frac{1}{2}\cos^{-1}\left(\frac{1-x}{1+x}\right) = \tan^{-1}\sqrt{x}$. (2017, [3 marks])
Complex Numbers: If $a+ib = \frac{x+iy}{x-iy}$, prove that $a^2+b^2=1$ and $\frac{b}{a} = \frac{2xy}{x^2-y^2}$. (2017, [3 marks])
Determinants: Using properties of determinants, prove that: $\begin{vmatrix}a & b & b+c \\ c & a & c+a \\ b & c & a+b\end{vmatrix} = (a+b+c)(a-c)^2$. (2017, [5 marks])
Linear Equations: Given that $A = \begin{pmatrix}1 & -1 & 0 \\ 2 & 3 & 4 \\ 0 & 1 & 2\end{pmatrix}$ and $B = \begin{pmatrix}2 & 2 & -4 \\ -4 & 2 & -4 \\ 2 & -1 & 5\end{pmatrix}$, find $AB$. Using this result, solve the system of equations: $x-y=3$, $2x+3y+4z=17$ and $y+2z=7$. (2017, [5 marks])
Inverse Trigonometry: Solve the equation for $x$: $\sin^{-1}x + \sin^{-1}(1-x) = \cos^{-1}x$, where $x \ne 0$. (2017, [5 marks])
Boolean Algebra: If $A, B$ and $C$ are the elements of Boolean algebra, simplify the expression $(A'+B')(A+C') + B'(B+C)$. Draw the simplified circuit. (2017, [5 marks])
Complex Numbers: Prove that locus of $z$ is a circle and find its centre and radius if $\frac{z-i}{z-1}$ is purely imaginary. (2017, [5 marks])
Binary Operations: The binary operation $*: \mathbb{R} \times \mathbb{R} \to \mathbb{R}$ is defined as $a*b = 2a+b$. Find $(2*3)*4$. (2018, [2 marks])
Matrices: If $A = \begin{pmatrix}5 & a \\ b & 0\end{pmatrix}$ and $A$ is a symmetric matrix, show that $a=b$. (2018, [2 marks])
Inverse Trigonometry: Solve: $3\tan^{-1}x + \cot^{-1}x = \pi$. (2018, [2 marks])
Determinants: Without expanding at any stage, find the value of: $\begin{vmatrix}a & b & c \\ a+2x & b+2y & c+2z \\ x & y & z\end{vmatrix}$. (2018, [2 marks])
Functions: If the function $f(x) = \sqrt{2x-3}$ is invertible then find its inverse. Hence prove that $(f \circ f^{-1})(x) = x$. (2018, [4 marks])
Inverse Trigonometry: If $\tan^{-1}a + \tan^{-1}b + \tan^{-1}c = \pi$, prove that $a+b+c = abc$. (2018, [4 marks])
Determinants: Use properties of determinants to solve for $x$: $\begin{vmatrix}x+a & b & c \\ c & x+b & a \\ a & b & x+c\end{vmatrix} = 0$ and $x \ne 0$. (2018, [4 marks])
Linear Equations: Using matrices, solve the following system of equations: $2x-3y+5z=11$, $3x+2y-4z=-5$, $x+y-2z=-3$. (2018, [6 marks])
Matrices: Using elementary transformation, find the inverse of the matrix: $\begin{bmatrix}0 & 1 & 2 \\ 1 & 2 & 3 \\ 3 & 1 & 1\end{bmatrix}$. (2018, [6 marks])
Functions: If $f: \mathbb{R} \to \mathbb{R}, f(x) = x^3$ and $g: \mathbb{R} \to \mathbb{R}, g(x) = 2x^2+1$, then find $f \circ g(x)$ and $g \circ f(x)$. (2019, [2 marks])
Inverse Trigonometry: Solve: $\sin(2\tan^{-1}x) = 1$. (2019, [2 marks])
Determinants: Using determinants, find the values of $k$, if the area of triangle with vertices $(-2, 0)$, $(0, 4)$ and $(0, k)$ is $4$ square units. (2019, [2 marks])
Matrices: Show that $(A+A')$ is a symmetric matrix, if $A = \begin{pmatrix}2 & 4 \\ 3 & 5\end{pmatrix}$. (2019, [2 marks])
Functions: If $f: A \to A$ and $A = \mathbb{R} - \{\frac{8}{5}\}$, show that the function $f(x) = \frac{8x+3}{5x-8}$ is one-one onto. Hence, find $f^{-1}$. (2019, [4 marks])
Inverse Trigonometry: Solve for $x$: $\tan^{-1}\left(\frac{x-1}{x-2}\right) + \tan^{-1}\left(\frac{x+1}{x+2}\right) = \frac{\pi}{4}$. (2019, [4 marks])
Inverse Trigonometry: If $\sec^{-1}x = \text{cosec}^{-1}y$, show that $\frac{1}{x^2} + \frac{1}{y^2} = 1$. (2019, [4 marks])
Determinants: Using properties of determinants prove that: $\begin{vmatrix}x & x(x^2+1) & x+1 \\ y & y(y^2+1) & y+1 \\ z & z(z^2+1) & z+1\end{vmatrix} = (x-y)(y-z)(z-x)(x+y+z)$. (2019, [4 marks])
Linear Equations: Solve the following system of linear equations using matrix method: $\frac{1}{x} + \frac{1}{y} + \frac{1}{z} = 9$, $\frac{2}{x} + \frac{5}{y} + \frac{7}{z} = 52$, $\frac{2}{x} + \frac{1}{y} - \frac{1}{z} = 0$. (2019, [6 marks])
Binary Operations: Determine whether the binary operation $*$ on $\mathbb{R}$ defined by $a*b = |a-b|$ is commutative. Also, find the value of $(-3)*2$. (2020, [2 marks])
Inverse Trigonometry: Prove that: $\tan^2(\sec^{-1}2) + \cot^2(\text{cosec}^{-1}3) = 11$. (2020, [2 marks])
Determinants: Without expanding at any stage, find the value of the determinant: $\Delta = \begin{vmatrix}20 & a & b+c \\ 20 & b & a+c \\ 20 & c & a+b\end{vmatrix}$. (2020, [2 marks])
Matrices: If $\begin{pmatrix}2 & 3 \\ 5 & 7\end{pmatrix}\begin{pmatrix}1 & -3 \\ -2 & 4\end{pmatrix} = \begin{pmatrix}-4 & 6 \\ -9 & x\end{pmatrix}$, find $x$. (2020, [2 marks])
Functions: If the function $f: \mathbb{R} \to \mathbb{R}$ be defined as $f(x) = \frac{3x+4}{5x-7}, (x \ne \frac{7}{5})$ and $g: \mathbb{R} \to \mathbb{R}$ be defined as $g(x) = \frac{7x+4}{5x-3}, (x \ne \frac{3}{5})$, show that $(g \circ f)(x) = (f \circ g)(x)$. (2020, [4 marks])
Inverse Trigonometry: If $\cos^{-1}\frac{x}{2} + \cos^{-1}\frac{y}{3} = \theta$, then prove that $9x^2 - 12xy\cos\theta + 4y^2 = 36\sin^2\theta$. (2020, [4 marks])
Inverse Trigonometry: Evaluate: $\cos(2\cos^{-1}x + \sin^{-1}x)$ at $x = \frac{1}{5}$. (2020, [4 marks])
Determinants: Using properties of determinants, show that $\begin{vmatrix}x & p & q \\ p & x & q \\ q & q & x\end{vmatrix} = (x-p)(x^2+px-2q^2)$. (2020, [4 marks])
Linear Equations: Solve the following system of linear equations using matrices: $x-2y = 10$, $2x-y-z = 8$, $-2y+z = 7$. (2020, [6 marks])

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.

Limits: Using L'Hospital's Rule, evaluate: $\lim_{x \to \pi/2}(x\tan x - \frac{\pi}{4}\sec x)$. (2017, [3 marks])
Integration: Evaluate: $\int \frac{1}{x^2}\sin^2(\frac{1}{x}) dx$. (2017, [3 marks])
Definite Integrals: Evaluate: $\int_0^{\pi/4} \log(1+\tan\theta) d\theta$. (2017, [3 marks])
Differential Equations: Solve: $\frac{dy}{dx} = 1-xy+y-x$. (2017, [3 marks])
Mean Value Theorems: Verify Lagrange's mean value theorem for the function $f(x) = x(1-\log x)$ and find the value of '$c$' in the interval $[1, 2]$. (2017, [5 marks])
Differentiation: If $y = \cos(\sin x)$, show that: $\frac{d^2y}{dx^2} + \tan x\frac{dy}{dx} + y\cos^2x = 0$. (2017, [5 marks])
Maxima & Minima: Show that the surface area of a closed cuboid with square base and given volume is minimum when it is a cube. (2017, [5 marks])
Integration: Evaluate: $\int \frac{\sin 2x}{(1+\sin x)(2+\sin x)} dx$. (2017, [5 marks])
Area Under Curves: Draw a rough sketch of the curve $y^2 = 4x$ and find the area of the region enclosed by the curve and the line $y=x$. (2017, [5 marks])
Differential Equations: Solve: $(x^2-yx^2)dy + (y^2+xy^2)dx = 0$. (2017, [5 marks])
Continuity: Find the value of constant '$k$' so that the function $f(x) = \frac{x^2-2x-3}{x+1}$ for $x \ne -1$, and $k$ for $x = -1$ is continuous at $x = -1$. (2018, [2 marks])
Approximations: Find the approximate change in the volume '$V$' of a cube of side $x$ metres caused by decreasing the side by $1\%$. (2018, [2 marks])
Integration: Evaluate: $\int \frac{x^3+5x^2+4x+1}{x^2} dx$. (2018, [2 marks])
Differential Equations: Find the differential equation of the family of concentric circles $x^2+y^2=a^2$. (2018, [2 marks])
Differentiability: Show that the function $f(x) = x^2$ for $x \le 1$ and $\frac{1}{x}$ for $x > 1$ is continuous at $x = 1$ but not differentiable. (2018, [4 marks])
Mean Value Theorems: Verify Rolle's theorem for the function $f(x) = e^{-x}\sin x$ on $[0, \pi]$. (2018, [4 marks])
Differentiation: If $x = \tan(\frac{1}{a}\log y)$, prove that $(1+x^2)\frac{d^2y}{dx^2} + (2x-a)\frac{dy}{dx} = 0$. (2018, [4 marks])
Integration: Evaluate: $\int \tan^{-1}\sqrt{x} dx$. (2018, [4 marks])
Tangents & Normals: Find the points on the curve $y=4x^3-3x+5$ at which the equation of the tangent is parallel to the x-axis. (2018, [4 marks])
Rate of Change: Water is dripping out from a conical funnel of semi-vertical angle $\pi/4$ at the uniform rate of $2 \text{ cm}^2/\text{sec}$ in the surface, through a tiny hole at the vertex of the bottom. When the slant height of the water level is $4\text{ cm}$, find the rate of decrease of the slant height of the water. (2018, [4 marks])
Differential Equations: Solve: $\sin x\frac{dy}{dx} - y = \sin x \cdot \tan\frac{x}{2}$. (2018, [6 marks])
Differential Equations: The population of a town grows at the rate of $10\%$ per year. Using differential equation, find how long will it take for the population to grow 4 times. (2018, [6 marks])
Maxima & Minima: A cone is inscribed in a sphere of radius $12\text{ cm}$. If the volume of the cone is maximum, find its height. (2018, [6 marks])
Integration: Evaluate: $\int \frac{x-1}{\sqrt{x^2-x}} dx$. (2018, [6 marks])
Definite Integrals: Evaluate: $\int_0^{\pi/2} \frac{\cos^2x}{1+\sin x\cos x} dx$. (2018, [6 marks])
Area Under Curves: Draw a rough sketch of the curve and find the area of the region bounded by curve $y^2 = 8x$ and the line $x = 2$. (2018, [6 marks])
Area Under Curves: Sketch the graph of $y = |x+4|$. Using integration, find the area of the region bounded by the curve $y = |x+4|$ and $x = -6$ and $x = 0$. (2018, [6 marks])
Continuity: $f(x) = \frac{x^2-9}{x-3}$ is not defined at $x = 3$. What value should be assigned to $f(3)$ for continuity of $f(x)$ at $x=3$? (2019, [2 marks])
Increasing & Decreasing Functions: Prove that the function $f(x) = x^3-6x^2+12x+5$ is increasing on $\mathbb{R}$. (2019, [2 marks])
Integration: Evaluate: $\int \frac{\sec^2x}{\text{cosec}^2x} dx$. (2019, [2 marks])
Limits: Using L'Hospital's Rule, evaluate: $\lim_{x \to 0}\frac{8^x-4^x}{4x}$. (2019, [2 marks])
Differentiability: Show that the function $f(x) = |x-4|, x \in \mathbb{R}$ is continuous, but not differentiable at $x=4$. (2019, [4 marks])
Mean Value Theorems: Verify the Lagrange's mean value theorem for the function $f(x) = x+\frac{1}{x}$ in the interval $[1, 3]$. (2019, [4 marks])
Differentiation: If $y = e^{\sin^{-1}x}$ and $z = e^{-\cos^{-1}x}$, prove that $\frac{dy}{dz} = e^{\pi/2}$. (2019, [4 marks])
Rate of Change: A $13\text{ m}$ long ladder is leaning against a wall, touching the wall at a certain height from the ground level. The bottom of the ladder is pulled away from the wall, along the ground, at the rate of $2 \text{ m/s}$. How fast is the height on the wall decreasing when the foot of the ladder is $5\text{ m}$ away from the wall? (2019, [4 marks])
Integration: Evaluate: $\int \frac{x(1+x^2)}{1+x^4} dx$. (2019, [4 marks])
Definite Integrals: Evaluate: $\int_{-6}^3 |x+3| dx$. (2019, [4 marks])
Differential Equations: Solve the differential equation: $\frac{dy}{dx} = \frac{x+y+2}{2(x+y)-1}$. (2019, [4 marks])
Maxima & Minima: The volume of a closed rectangular metal box with a square base is $4096 \text{ cm}^3$. The cost of polishing the outer surface of the box is ₹4 per square cm. Find the dimensions of the box for the minimum cost of polishing it. (2019, [6 marks])
Maxima & Minima: Find the point on the straight line $2x+3y=6$ which is closest to the origin. (2019, [6 marks])
Definite Integrals: Evaluate: $\int_0^{\pi} \frac{x\tan x}{\sec x+\tan x} dx$. (2019, [6 marks])
Area Under Curves: Draw a rough sketch and find the area bounded by the curve $x^2 = y$ and $x+y=2$. (2019, [6 marks])
Differentiation: Find $\frac{dy}{dx}$ if $x^3+y^3=3axy$. (2020, [2 marks])
Rate of Change: The edge of a variable cube is increasing at the rate of $10 \text{ cm/sec}$. How fast is the volume of the cube increasing when the edge is $5\text{ cm}$ long? (2020, [2 marks])
Definite Integrals: Evaluate: $\int_4^5 |x-5| dx$. (2020, [2 marks])
Differential Equations: Form a differential equation of the family of the curves $y^2 = 4ax$. (2020, [2 marks])
Mean Value Theorems: Verify Rolle's theorem for the function, $f(x) = -1+\cos x$ in the interval $[0, 2\pi]$. (2020, [4 marks])
Differentiation: If $y = e^{m\sin^{-1}x}$, prove that $(1-x^2)\frac{d^2y}{dx^2} - x\frac{dy}{dx} = m^2y$. (2020, [4 marks])
Tangents & Normals: The equation of tangent at $(2, 3)$ on the curve $y^2 = px^3+q$ is $y = 4x-7$. Find the values of '$p$' and '$q$'. (2020, [4 marks])
Limits: Using L'Hospital's rule, evaluate: $\lim_{x \to 0}\frac{xe^x-\log(1+x)}{x^2}$. (2020, [4 marks])
Integration: Evaluate: $\int \frac{dx}{\sqrt{5x-4x^2}}$. (2020, [4 marks])
Integration: Evaluate: $\int \sin^3x\cos^4x dx$. (2020, [4 marks])
Differential Equations: Solve the differential equation $(1+x^2)\frac{dy}{dx} = 4x^2-2xy$. (2020, [4 marks])
Maxima & Minima: Show that the radius of a closed right circular cylinder of given surface area and maximum volume is equal to half of its height. (2020, [6 marks])
Maxima & Minima: Prove that the area of right-angled triangle of given hypotenuse is maximum when the triangle is isosceles. (2020, [6 marks])
Integration: Evaluate: $\int \tan^{-1}\sqrt{\frac{1-x}{1+x}} dx$. (2020, [6 marks])
Integration: Evaluate: $\int \frac{2x+7}{x^2-x-2} dx$. (2020, [6 marks])
Area Under Curves: Draw a rough sketch of the curves $y^2 = x$ and $y^2 = 4-3x$ and find the area enclosed between them. (2020, [6 marks])
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: If $\vec{a}, \vec{b}, \vec{c}$ are three mutually perpendicular vectors of equal magnitude, prove that $(\vec{a}+\vec{b}+\vec{c})$ is equally inclined with vectors $\vec{a}, \vec{b}$ and $\vec{c}$. (2017, [5 marks])
Vectors: Find the value of $\lambda$ for which the four points with position vectors $6\hat{i}-7\hat{j}$, $16\hat{i}-19\hat{j}-4\hat{k}$, $\lambda\hat{j}-6\hat{k}$ and $2\hat{i}-5\hat{j}+10\hat{k}$ are coplanar. (2017, [5 marks])
3D Geometry: Show that the lines $\frac{x-1}{2}=\frac{y+1}{-3}=\frac{z+10}{8}$ and $\frac{x-4}{1}=\frac{y+3}{-4}=\frac{z+1}{7}$ intersect. Find the coordinates of their point of intersection. (2017, [5 marks])
3D Geometry: Find the equation of the plane passing through the point $(1,-2,1)$ and perpendicular to the line joining the points $A(3,2,1)$ and $B(1,4,2)$. (2017, [5 marks])
Vectors: Find $\lambda$ if the scalar projection of $\vec{a} = \lambda\hat{i}+\hat{j}+4\hat{k}$ on $\vec{b} = 2\hat{i}+6\hat{j}+3\hat{k}$ is $4\text{ units}$. (2018, [2 marks])
3D Geometry: The Cartesian equation of a line is: $2x-3 = 3y+1 = 5-6z$. Find the vector equation of a line passing through $(7,-5,0)$ and parallel to the given line. (2018, [2 marks])
3D Geometry: Find the equation of the plane through the intersection of the planes $\vec{r} \cdot (\hat{i}+3\hat{j}-\hat{k}) = 9$ and $\vec{r} \cdot (2\hat{i}-\hat{j}+\hat{k}) = 3$ and passing through the origin. (2018, [2 marks])
Vectors: If $A, B, C$ are three non-collinear points with position vectors $\vec{a}, \vec{b}, \vec{c}$ respectively, then show that the length of the perpendicular from $C$ on $AB$ is $\frac{|(\vec{a}\times\vec{b})+(\vec{b}\times\vec{c})+(\vec{c}\times\vec{a})|}{|\vec{b}-\vec{a}|}$. (2018, [4 marks])
Vectors: Show that the four points $A, B, C$ and $D$ with position vectors $4\hat{i}+5\hat{j}+\hat{k}, -\hat{j}-\hat{k}, 3\hat{i}+9\hat{j}+4\hat{k}$ and $4(-\hat{i}+\hat{j}+\hat{k})$ respectively, are coplanar. (2018, [4 marks])
3D Geometry: Find the image of a point having position vector: $3\hat{i}-2\hat{j}+\hat{k}$ in the plane $\vec{r} \cdot (3\hat{i}-\hat{j}+4\hat{k}) = 2$. (2018, [6 marks])
Vectors: If $\vec{a}$ and $\vec{b}$ are perpendicular vectors, $|\vec{a}+\vec{b}| = 13$ and $|\vec{a}| = 5$, find the value of $|\vec{b}|$. (2019, [2 marks])
3D Geometry: Find the length of the perpendicular from origin to the plane $\vec{r} \cdot (3\hat{i}-4\hat{j}-12\hat{k}) + 39 = 0$. (2019, [2 marks])
3D Geometry: Find the angle between the two lines $2x = 3y = -z$ and $6x = -y = -4z$. (2019, [2 marks])
Vectors: If $\vec{a} = \hat{i}-2\hat{j}+3\hat{k}, \vec{b} = 2\hat{i}+3\hat{j}-5\hat{k}$, prove that $\vec{a}$ and $\vec{a}\times\vec{b}$ are perpendicular. (2019, [4 marks])
Vectors: If $\vec{a}$ and $\vec{b}$ are non-collinear vectors, find the value of $x$ such that the vectors $\vec{r} = (x-2)\vec{a}+\vec{b}$ and $\vec{s} = (3+2x)\vec{a}-2\vec{b}$ are collinear. (2019, [4 marks])
3D Geometry: Find the equation of the plane passing through the intersection of the planes $2x+2y-3z-7 = 0$ and $2x+5y+3z-9 = 0$ such that the intercepts made by the resulting plane on the x-axis and the z-axis are equal. (2019, [4 marks])
3D Geometry: Find the equation of the lines passing through the point $(2, 1, 3)$ and perpendicular to the lines $\frac{x-1}{1} = \frac{y-2}{2} = \frac{z-3}{3}$ and $\frac{x}{-3} = \frac{y}{2} = \frac{z}{5}$. (2019, [4 marks])
Vectors: Write a vector of magnitude of $18\text{ units}$ in the direction of the vector $\hat{i}-2\hat{j}-2\hat{k}$. (2020, [2 marks])
3D Geometry: Find the angle between the two lines: $\frac{x+1}{2} = \frac{y-2}{5} = \frac{z+3}{4}$ and $\frac{x-1}{5} = \frac{y+2}{2} = \frac{z-1}{-5}$. (2020, [2 marks])
3D Geometry: Find the equation of the plane passing through the point $(2,-3, 1)$ and perpendicular to the line joining the points $(4, 5, 0)$ and $(1,-2, 4)$. (2020, [2 marks])
Vectors: Prove that $\vec{a} \cdot [(\vec{b}+\vec{c})\times(\vec{a}+3\vec{b}+4\vec{c})] = [\vec{a} \vec{b} \vec{c}]$. (2020, [4 marks])
Vectors: Using vectors, find the area of the triangle whose vertices are: $A(3,-1, 2), B(1,-1,-3)$ and $C(4,-3, 1)$. (2020, [4 marks])
3D Geometry: Find the image of the point $(3,-2, 1)$ in the plane $3x-y+4z=2$. (2020, [4 marks])
3D Geometry: Determine the equation of the line passing through the point $(-1, 3,-2)$ and perpendicular to the lines: $\frac{x}{1} = \frac{y}{2} = \frac{z}{3}$ and $\frac{x+2}{-3} = \frac{y-1}{2} = \frac{z+1}{5}$. (2020, [4 marks])

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 problem is given to three students whose chances of solving it are $1/4, 1/5$ and $1/3$ respectively. Find the probability that the problem is solved. (2017, [3 marks])
Probability: In a class of 60 students, 30 opted for Mathematics, 32 opted for Biology and 24 opted for both Mathematics and Biology. If one of these students is selected at random, find the probability that: (i) The student opted for Mathematics or Biology. (ii) The student has opted neither Mathematics nor Biology. (iii) The student has opted Mathematics but not Biology. (2017, [5 marks])
Probability: Bag A contains 1 white, 2 blue and 3 red balls. Bag B contains 3 white, 3 blue and 2 red balls. Bag C contains 2 white, 3 blue and 4 red balls. One bag is selected at random and then two balls are drawn from the selected bag. Find the probability that the balls drawn are white and red. (2017, [5 marks])
Probability (Bayes' Theorem): A fair die is rolled. If face 1 turns up, a ball is drawn from Bag A. If face 2 or 3 turns up, a ball is drawn from Bag B. If face 4 or 5 or 6 turns up, a ball is drawn from Bag C. Bag A contains 3 red and 2 white balls, Bag B contains 3 red and 4 white balls and Bag C contains 4 red and 5 white balls. The die is rolled, a Bag is picked up and a ball is drawn. If the drawn ball is red, what is the probability that it is drawn from Bag B? (2017, [5 marks])
Probability Distribution: An urn contains 25 balls of which 10 balls are red and the remaining green. A ball is drawn at random from the urn, the colour is noted and the ball is replaced. If 6 balls are drawn in this way, find the probability that: (i) All the balls are red. (ii) Not more than 2 balls are green. (iii) Number of red balls and green balls are equal. (2017, [5 marks])
LPP Formulation: A farmer has a supply of chemical fertilizer of type A which contains $10\%$ nitrogen and $6\%$ phosphoric acid and of type B which contains $5\%$ nitrogen and $10\%$ phosphoric acid. After soil test, it is found that at least $7\text{ kg}$ of nitrogen and same quantity of phosphoric acid is required for a good crop. The fertilizer of type A costs $₹5.00\text{ per kg}$ and the type B costs $₹8.00\text{ per kg}$. Using Linear programming, find how many kilograms of each type of the fertilizer should be bought to meet the requirement and for the cost to be minimum. Find the feasible region in the graph. (2017, [5 marks])
Probability: If $A$ and $B$ are events such that $P(A) = 1/2, P(B) = 1/3$ and $P(A \cap B) = 1/4$ then find $P(A/B)$ and $P(B/A)$. (2018, [2 marks])
Probability: In a race, the probabilities of $A$ and $B$ winning the race are $1/3$ and $1/6$ respectively. Find the probability of neither of them winning the race. (2018, [2 marks])
Probability: $A$ speaks truth in $60\%$ of the cases, while $B$ in $40\%$ of the cases. In what percent of cases are they likely to contradict each other in stating the same fact? (2018, [4 marks])
Probability Distribution: From a lot of 6 items containing 2 defective items, a sample of 4 items are drawn at random. Let the random variable $X$ denote the number of defective items in the sample. If the sample is drawn without replacement, find: (a) The probability distribution of X (b) Mean of X (c) Variance of X. (2018, [6 marks])
LPP Formulation: A manufacturing company makes two types of teaching aids $A$ and $B$ of Mathematics for Class X... The company makes a profit of ₹80 on each piece of type $A$ and ₹120 on each piece of type $B$. How many pieces of type $A$ and type $B$ should be manufactured per week to get a maximum profit? Formulate this as Linear Programming Problem and solve it. Identify the feasible region from the rough sketch. (2018, [6 marks])
Probability: Two balls are drawn from an urn containing 3 white, 5 red and 2 black balls, one by one without replacement. What is the probability that at least one ball is red? (2019, [2 marks])
Probability: If events $A$ and $B$ are independent, such that $P(A) = 3/5, P(B) = 2/3$, find $P(A \cup B)$. (2019, [2 marks])
Probability: Bag A contains 4 white balls and 3 black balls, while Bag B contains 3 white balls and 5 black balls. Two balls are drawn from Bag A and placed in Bag B. Then, what is the probability of drawing a white ball from Bag B? (2019, [4 marks])
Probability (Bayes' Theorem): Given three identical Boxes A, B and C... A person chooses a Box at random and takes out a coin. If the coin drawn is of silver, find the probability that it has been drawn from the Box which has the remaining two coins also of silver. (2019, [6 marks])
Probability Distribution: Determine the binomial distribution where mean is 9 and standard deviation is 3/2. Also, find the probability of obtaining at most one success. (2019, [6 marks])
LPP Formulation: A carpenter has 90, 80 and 50 running feet respectively of teak wood, plywood and rosewood... Formulate the above as a Linear Programming Problem and solve it, indicating clearly the feasible region in the graph. (2019, [6 marks])
Probability: A bag contains 5 white, 7 red and 4 black balls. If four balls are drawn one by one with replacement, what is the probability that none is white? (2020, [2 marks])
Probability: Let $A$ and $B$ be two events such that $P(A) = 1/2, P(B) = p$ and $P(A \cup B) = 3/5$, find '$p$' if $A$ and $B$ are independent events. (2020, [2 marks])
Probability: Three persons A, B and C shoot to hit a target. Their probabilities of hitting the target are 5/6, 4/5 and 3/4 respectively. Find the probability that: (i) Exactly two persons hit the target. (ii) At least one person hits the target. (2020, [4 marks])
Probability Distribution: The probability that a bulb produced in a factory will fuse after 150 days of use is 0.05. Find the probability that out of 5 such bulbs: (i) None will fuse... (ii) Not more than one will fuse... (iii) More than one will fuse... (iv) At least one will fuse after 150 days of use. (2020, [6 marks])
LPP Formulation: A company uses three machines to manufacture two types of shirts, half sleeves and full sleeves. None of the machines can be in operation for more than 40 hours per week... How many of each type of shirts should be made per week to maximize the company's profit? (2020, [6 marks])
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!

Prime Maths

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Saturday, May 16, 2026