Math PYQ Hitlist | 10 Year Previous Year Questions

01 Relations & Functions

~ 4 Marks Volume

CBSE 2023, 2019 3 Marker (Prove Equivalence)
Let $A = \{x \in \mathbb{Z} : 0 \le x \le 12\}$. Show that the relation $R$ in $A$ given by $R = \{(a, b) : |a - b| \text{ is a multiple of } 4\}$ is an equivalence relation. Find the set of all elements related to 1.
CBSE 2020, 2018 3 Marker (Bijective Function)
Show that the function $f: \mathbb{R} \to \{x \in \mathbb{R} : -1 < x < 1\}$ defined by $f(x)=\frac{x}{1+|x|}, x \in \mathbb{R}$ is one-one and onto function.

02 Inverse Trigonometric Functions

~ 4 Marks Volume

CBSE 2023, 2022 1/2 Marker (Principal Value)
Find the principal value of $\cos^{-1}\left(\cos \frac{7\pi}{6}\right)$ and $\tan^{-1}\left(\tan \frac{3\pi}{4}\right)$.
CBSE 2019, 2017 3 Marker (Simplification)
Write in the simplest form: $\tan^{-1}\left( \frac{\sqrt{1+x^2} - 1}{x} \right), x \neq 0$.

Hint: Put $x = \tan\theta$.

03 Matrices

~ 5 Marks Volume

CBSE 2024, 2020 3 Marker (Polynomial Matrix Eq)
If $A = \begin{pmatrix} 3 & 1 \\ -1 & 2 \end{pmatrix}$, show that $A^2 - 5A + 7I = 0$. Hence find $A^{-1}$.
CBSE 2018 3 Marker (Symmetric & Skew)
Express the matrix $A = \begin{pmatrix} 2 & -2 & -4 \\ -1 & 3 & 4 \\ 1 & -2 & -3 \end{pmatrix}$ as the sum of a symmetric and a skew-symmetric matrix.

04 Determinants

~ 5 Marks Volume

EVERY YEAR (2015-2024) GUARANTEED 5 Marker (Matrix Method)
Solve the system of linear equations using matrix method:
$2x + 3y + 3z = 5$
$x - 2y + z = -4$
$3x - y - 2z = 3$
CBSE 2023 MCQ 1 Marker (Adjoint Property)
If A is a square matrix of order 3 such that $|A| = -4$, then find the value of $|\text{adj A}|$. (Ans: $(-4)^{3-1} = 16$).

05 Continuity & Differentiability

~ 8 Marks Volume

CBSE 2022, 2019 3 Marker (Find unknown constants)
Find the values of $a$ and $b$ such that the function defined by:
$f(x) = \begin{cases} 5, & x \le 2 \\ ax + b, & 2 < x < 10 \\ 21, & x \ge 10 \end{cases}$
is a continuous function.
CBSE 2020, 2017 3 Marker (Logarithmic Diff)
Differentiate w.r.t $x$: $y = (\sin x)^x + x^{\sin x}$.
CBSE 2024, 2018 4 Marker (Double Derivative)
If $y = ae^{mx} + be^{nx}$, show that $\frac{d^2y}{dx^2} - (m+n)\frac{dy}{dx} + mny = 0$.

06 Application of Derivatives

~ 9 Marks Volume

CBSE 2023, 2019, 2016 GUARANTEED 5 Marker (Word Prob)
Show that the right circular cylinder of given surface and maximum volume is such that its height is equal to the diameter of the base.
CBSE 2024 Case Study 4 Marker (Increasing/Decreasing)
Find the intervals in which the function $f(x) = 2x^3 - 3x^2 - 36x + 7$ is (a) strictly increasing, (b) strictly decreasing.

07 Integrals

~ 9 Marks Volume

EVERY YEAR (King's Rule) 3/4 Marker (Definite Integral)
Evaluate: $\int_{0}^{\pi/2} \frac{\sqrt{\sin x}}{\sqrt{\sin x} + \sqrt{\cos x}} \, dx$.
(Standard Answer is always $\frac{\pi}{4}$)
CBSE 2022, 2018 3 Marker (Modulus Integral)
Evaluate: $\int_{-1}^{2} |x^3 - x| \, dx$.
Hint: Find roots (0, 1, -1) and break limits.
CBSE 2020 3 Marker (By Parts - Special Form)
Integrate: $\int e^x \left(\frac{1-x}{1+x^2}\right)^2 \, dx$ type questions reducible to $e^x[f(x)+f'(x)]$.

08 Application of Integrals

~ 6 Marks Volume

CBSE 2023, 2019 4 Marker (Area Bounded)
Find the area of the region bounded by the parabola $y^2 = 4ax$ and its latus rectum.
CBSE 2020, 2017 4 Marker (Area between 2 curves)
Find the area of the region $\{(x, y) : x^2 \le y \le |x|\}$.

09 Differential Equations

~ 7 Marks Volume

CBSE 2024, 2021 4 Marker (Linear Differential Eq)
Find the general solution of the differential equation: $\frac{dy}{dx} + y \cot x = 2x + x^2 \cot x, \quad (x \neq 0)$.
CBSE 2019, 2018 4 Marker (Homogeneous DE)
Solve the differential equation: $x \frac{dy}{dx} = y - x \tan\left(\frac{y}{x}\right)$.

10 Vector Algebra

~ 7 Marks Volume

CBSE 2022, 2020 2/3 Marker (Dot Product)
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 to $\vec{a}, \vec{b}$ and $\vec{c}$.
CBSE 2023 MCQ 1 Marker (Cross Product)
Find the area of a parallelogram whose adjacent sides are determined by the vectors $\vec{a} = \hat{i} - \hat{j} + 3\hat{k}$ and $\vec{b} = 2\hat{i} - 7\hat{j} + \hat{k}$.

11 Three Dimensional Geometry

~ 7 Marks Volume

EVERY YEAR (2015-2024) GUARANTEED 4 Marker (Shortest Dist)
Find the shortest distance between the lines:
$\vec{r} = (\hat{i} + 2\hat{j} + \hat{k}) + \lambda(\hat{i} - \hat{j} + \hat{k})$
$\vec{r} = (2\hat{i} - \hat{j} - \hat{k}) + \mu(2\hat{i} + \hat{j} + 2\hat{k})$
CBSE 2019, 2016 4 Marker (Foot of Perpendicular)
Find the coordinates of the foot of perpendicular drawn from the point $P(0,2,3)$ to the line $\frac{x+3}{5} = \frac{y-1}{2} = \frac{z+4}{3}$. Also find the length of perpendicular.

12 Linear Programming

~ 5 Marks Volume

13 Probability

~ 8 Marks Volume

ALMOST EVERY YEAR 5 Marker / Case Study (Bayes' Theorem)
An insurance company insured 2000 scooter drivers, 4000 car drivers and 6000 truck drivers. The probability of an accident is 0.01, 0.03 and 0.15 respectively. One of the insured persons meets with an accident. What is the probability that he is a scooter driver?
CBSE 2020, 2016 3 Marker (Probability Distribution)
Two cards are drawn simultaneously (without replacement) from a well-shuffled pack of 52 cards. Find the mean and variance of the number of red cards.
CBSE 2022 2 Marker (Independent Events)
If A and B are two independent events such that $P(A) = 0.3$ and $P(A \cup B) = 0.5$, then find $P(B)$.

CBSE 10-Year PYQ Hitlist 🎯