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CBSE CLASS XII
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Mathematics
List of top Mathematics Questions asked in CBSE CLASS XII
The differential equation \( \frac{dy}{dx} = F(x, y) \) will not be a homogeneous differential equation, if \( F(x, y) \) is:
CBSE CLASS XII
Mathematics
Differential Equations
For any two vectors \( \vec{a} \) and \( \vec{b} \), which of the following statements is always true?
CBSE CLASS XII
Mathematics
Vectors
The coordinates of the foot of the perpendicular drawn from the point \( (0, 1, 2) \) on the \( x \)-axis are given by:
CBSE CLASS XII
Mathematics
Properties of Lines
\( \int_{0}^{\pi/2} \frac{\sin x - \cos x}{1 + \sin x \cos x} \, dx \) is equal to:
CBSE CLASS XII
Mathematics
Some Properties of Definite Integrals
The function \( f(x) = x^3 - 3x^2 + 12x - 18 \) is:
CBSE CLASS XII
Mathematics
Functions
If \( \begin{vmatrix} -a & b & c
a & -b & c
a & b & -c \end{vmatrix} = kabc \), then the value of \( k \) is:
CBSE CLASS XII
Mathematics
Determinants
Let \( f : \mathbb{R}_+ \to [-5, \infty) \) be defined as \( f(x) = 9x^2 + 6x - 5 \), where \( \mathbb{R}_+ \) is the set of all non-negative real numbers. Then, \( f \) is:
CBSE CLASS XII
Mathematics
Functions
If the sum of all the elements of a \( 3 \times 3 \) scalar matrix is 9, then the product of all its elements is:
CBSE CLASS XII
Mathematics
Matrix
If \( A = \begin{bmatrix} -1 & a & 2 \\ 1 & 2 & x \\ 3 & 1 & 1 \end{bmatrix} \) and \( A^{-1} = \begin{bmatrix} 1 & -1 & 1 \\ -8 & 7 & -5 \\ b & y & 3 \end{bmatrix} \), find the value of \((a + x) - (b + y)\).
CBSE CLASS XII
Mathematics
Matrix
The traffic police has installed Over Speed Violation Detection (OSVD) system at various locations in a city. These cameras can capture a speeding vehicle from a distance of 300 m and even function in the dark.
\vspace{0.3cm} \includegraphics[width=\linewidth]{latex2.png} \vspace{0.3cm} A camera is installed on a pole at the height of 5 m. It detects a car travelling away from the pole at the speed of 20 m/s. At any point, \(x\) m away from the base of the pole, the angle of elevation of the speed camera from the car C is \(\theta\). \vspace{0.3cm} On the basis of the above information, answer the following questions: \begin{enumerate} Express \( \theta \) in terms of the height of the camera installed on the pole and \( x \). Find \( \frac{d\theta}{dx} \). \begin{enumerate} Find the rate of change of angle of elevation with respect to time at an instant when the car is 50 m away from the pole.
OR
If the rate of change of angle of elevation with respect to time of another car at a distance of 50 m from the base of the pole is \( \frac{3}{101} \) rad/s, then find the speed of the car. \end{enumerate} \end{enumerate}
CBSE CLASS XII
Mathematics
Miscellaneous
Evaluate:
\[ \int_{-2}^{2} \frac{x^3 + |x| + 1}{x^2 + 4|x| + 4} \, dx. \]
CBSE CLASS XII
Mathematics
Some Properties of Definite Integrals
Find:
\[ \int \frac{(3 \cos x - 2) \sin x}{5 - \sin^2 x - 4 \cos x} \, dx. \]
CBSE CLASS XII
Mathematics
Integration
Using integration, find the area of the ellipse:
\[ \frac{x^2}{16} + \frac{y^2}{4} = 1, \]
included between the lines \(x = -2\) and \(x = 2\).
CBSE CLASS XII
Mathematics
Integration
Evaluate:
\[ \int_1^3 (|x - 1| + |x - 2| + |x - 3|) \, dx. \]
CBSE CLASS XII
Mathematics
Some Properties of Definite Integrals
Solve the following differential equation:
\[ (\tan^{-1}y - x) \, dy = (1 + y^2) \, dx. \]
CBSE CLASS XII
Mathematics
Differential Equations
Find the equation of a line \( l_2 \) which is the mirror image of the line \( l_1 \) with respect to line \( l : \frac{x}{1} = \frac{y - 1}{2} = \frac{z - 2}{3} \), given that line \( l_1 \) passes through the point \( P(1, 6, 3) \) and is parallel to line \( l \).
CBSE CLASS XII
Mathematics
Circles
If \( y = (\tan x)^x \), then find \( \frac{dy}{dx} \).
CBSE CLASS XII
Mathematics
Differentiation
If \( \sqrt{1 - x^2} + \sqrt{1 - y^2} = a(x - y) \), prove that \( \frac{dy}{dx} = \frac{\sqrt{1 - y^2}}{\sqrt{1 - x^2}} \).
CBSE CLASS XII
Mathematics
Differentiation
A relation \( R \) on set \( A = \{1, 2, 3, 4, 5\} \) is defined as \[ R = \{(x, y) : |x^2 - y^2| <8\}. \] Check whether the relation \( R \) is reflexive, symmetric, and transitive.
CBSE CLASS XII
Mathematics
Relations and functions
A function \( f \) is defined from \( R \to R \) as \( f(x) = ax + b \), such that \( f(1) = 1 \) and \( f(2) = 3 \). Find the function \( f(x) \). Hence, check whether the function \( f(x) \) is one-one and onto.
CBSE CLASS XII
Mathematics
Relations and functions
Find:
\[ \int \frac{x^2}{(x^2 + 4)(x^2 + 9)} \, dx. \]
CBSE CLASS XII
Mathematics
Integration
E and F are two independent events such that \( P(\overline{E}) = 0.6 \) and \( P(E \cup F) = 0.6 \). Find \( P(F) \) and \( P(\overline{E} \cup \overline{F}) \).
CBSE CLASS XII
Mathematics
Conditional Probability
Evaluate:
\[ \int_0^{\frac{\pi^2}{4}} \frac{\sin\sqrt{x}}{\sqrt{x}} \, dx \]
CBSE CLASS XII
Mathematics
Some Properties of Definite Integrals
If \( x^{1/3} + y^{1/3} = 1 \), find \( \frac{dy}{dx} \) at the point \( \left( \frac{1}{8}, \frac{1}{8} \right) \).
CBSE CLASS XII
Mathematics
Differentiation
Find:
\[ \int x \sqrt{1 + 2x} \, dx \]
CBSE CLASS XII
Mathematics
Integration
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