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CBSE CLASS XII
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Mathematics
List of top Mathematics Questions asked in CBSE CLASS XII
Show that a function \( f : \mathbb{R} \to \mathbb{R} \) defined by \( f(x) = \frac{2x
{1 + x^2} \) is neither one-one nor onto. Further, find set \( A \) so that the given function \( f : \mathbb{R} \to A \) becomes an onto function.}
CBSE CLASS XII
Mathematics
Functions
Find the general solution of the differential equation:
\[ y \, dx = (x + 2y^2) \, dy. \]
CBSE CLASS XII
Mathematics
Differential Equations
The position vectors of vertices of \( \Delta ABC \) are \( A(2\hat{i} - \hat{j} + \hat{k}) \), \( B(\hat{i} - 3\hat{j} - 5\hat{k}) \), and \( C(3\hat{i} - 4\hat{j} - 4\hat{k}) \). Find all the angles of \( \Delta ABC \).
CBSE CLASS XII
Mathematics
Circles
A relation \( R \) is defined on \( \mathbb{N} \times \mathbb{N} \) (where \( \mathbb{N} \) is the set of natural numbers) as:
\[ (a, b) \, R \, (c, d) \iff a - c = b - d. \]
Show that \( R \) is an equivalence relation.
CBSE CLASS XII
Mathematics
Functions
The vertices of triangle \( \triangle ABC \) are \( A(1, 1, 0), B(1, 2, 1) \), and \( C(-2, 2, -1) \). Find the equations of the medians through A and B. Use the equations so obtained to find the coordinates of the centroid.
CBSE CLASS XII
Mathematics
Circles
Find the probability that the airplane will not crash.
CBSE CLASS XII
Mathematics
Conditional Probability
(a) If \( y = (\log x)^2 \), prove that \( x^2 y'' + x y' = 2 \).
CBSE CLASS XII
Mathematics
Differentiation
Find the particular solution of the differential equation given by:
\[ 2xy + y^2 - 2x^2 \frac{dy}{dx} = 0; \quad y = 2, \, \text{when } x = 1. \]
CBSE CLASS XII
Mathematics
Differential Equations
Find:
\[ \int \frac{1}{x \left[(\log x)^2 - 3 \log x - 4\right]} \, dx. \]
CBSE CLASS XII
Mathematics
Integration
Find:
\[ \int x^2 \log(x^2 - 1) \, dx. \]
CBSE CLASS XII
Mathematics
Integration
Evaluate:
\[ \int_{-2}^{2} \sqrt{\frac{2 - x}{2 + x}} \, dx. \]
CBSE CLASS XII
Mathematics
Integration
Find:
\[ \int \frac{e^{4x} - 1}{e^{4x} + 1} \, dx. \]
CBSE CLASS XII
Mathematics
Integration
A pair of dice is thrown simultaneously. If \( X \) denotes the absolute difference of the numbers appearing on top of the dice, then find the probability distribution of \( X \).
CBSE CLASS XII
Mathematics
Conditional Probability
If \( y = \csc(\cot^{-1} x) \), then prove that \( \sqrt{1 + x^2} \frac{dy}{dx} - x = 0 \).
CBSE CLASS XII
Mathematics
Differentiation
(a) Find the value of
\( \sin^{-1}\left( -\frac{1}{2} \right) + \cos^{-1}\left( -\frac{\sqrt{3}}{2} \right) + \cot^{-1}\left( \tan \frac{4\pi}{3} \right) \).
CBSE CLASS XII
Mathematics
Functions
Determine whether the function \( f(x) = x^2 - 6x + 3 \) is increasing or decreasing in \( [4, 6] \).
CBSE CLASS XII
Mathematics
Differentiation
Assertion (A):
For any symmetric matrix \( A \), \( B'A B \) is a skew-symmetric matrix.
Reason (R):
A square matrix \( P \) is skew-symmetric if \( P' = -P \).
CBSE CLASS XII
Mathematics
Matrix
Let \( E \) be an event of a sample space \( S \) of an experiment, then \( P(S | E) \) is:
CBSE CLASS XII
Mathematics
Conditional Probability
Assertion (A):
For two non-zero vectors \( \vec{a} \) and \( \vec{b} \), \( \vec{a} \cdot \vec{b} = \vec{b} \cdot \vec{a} \).
Reason (R):
For two non-zero vectors \( \vec{a} \) and \( \vec{b} \), \( \vec{a} \times \vec{b} = -\vec{b} \times \vec{a} \).
CBSE CLASS XII
Mathematics
Vector Algebra
(a) Find the value of
\( \sin^{-1}\left( -\frac{1}{2} \right) + \cos^{-1}\left( -\frac{\sqrt{3}}{2} \right) + \cot^{-1}\left( \tan \frac{4\pi}{3} \right) \).
CBSE CLASS XII
Mathematics
Differentiation
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
The common region determined by all the constraints of a linear programming problem is called:
CBSE CLASS XII
Mathematics
Linear Programming Problem and its Mathematical Formulation
The function \( f(x) = x^3 - 3x^2 + 12x - 18 \) is:
CBSE CLASS XII
Mathematics
Functions
For any two vectors \( \vec{a} \) and \( \vec{b} \), which of the following statements is always true?
CBSE CLASS XII
Mathematics
Vectors
The anti-derivative of \( \sqrt{1 + \sin 2x}, \, x \in \left[ 0, \frac{\pi}{4} \right] \) is:
CBSE CLASS XII
Mathematics
Integration
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