List of top Mathematics Questions asked in UP Board XII

If \(R_1\) and \(R_2\) are equivalence relations in the set A, prove that \(R_1 \cap R_2\) is also an equivalence relation in A.
Find the value of \(\int \frac{1}{x^2 - a^2} dx\).
If two vectors \(\vec{a}\) and \(\vec{b}\) are such that \(|\vec{a}| = 2\), \(|\vec{b}| = 3\) and \(\vec{a} \cdot \vec{b} = 4\), find \(|\vec{a} - \vec{b}|\).
The Cartesian equation of a line is \(\frac{x+3}{2} = \frac{y-5}{4} = \frac{z+6}{2}\). Find its vector equation.
Find the value of \(\frac{dy}{dx}\) if \(y = \tan^{-1}\left(\frac{3x - x^3}{1 - 3x^2}\right)\), where \(-\frac{1}{\sqrt{3}}<x<\frac{1}{\sqrt{3}}\).
A die is thrown once. If the event 'the number obtained on the die is a multiple of 3' is represented by E and 'the number obtained on the die is even' is represented by F, tell whether the events E and F are independent.
Find the principal value of \(\cos^{-1}\left(-\frac{1}{\sqrt{2}}\right)\).
Let \(\vec{a} = \hat{i} + 2\hat{j}\) and \(\vec{b} = 2\hat{i} + \hat{j}\). Is \(|\vec{a}| = |\vec{b}|\)? Are the vectors \(\vec{a}\) and \(\vec{b}\) equal?
Find the value of \(\int x^3 e^{x^4} dx\).
If the orders of the matrices A and B are \(m \times n\) and \(n \times p\) respectively, then the order of AB will be
The direction cosines of the vector \(\hat{i} + \hat{j} - 2\hat{k}\) are
If the matrix A = \(\begin{bmatrix} 0 & 2y & z
x & y & -z
x & -y & z \end{bmatrix}\) satisfies the equation AA' = I, then find the values of x, y and z.
Given any two events A and B are such that \( P(A) = \frac{1}{2}, P(B) = \frac{1}{4} \) and \( P(A \cap B) = \frac{1}{8} \), then find \( P(\text{not A and not B}) \).
If a function \( f: \mathbb{R} \to \{ x \in \mathbb{R} : x \in (-1, 1) \} \) is defined as \( f(x) = \frac{x}{1+|x|}, \ x \in \mathbb{R} \), then prove that f is one-one and onto.
If \( x\sqrt{1+y} + y\sqrt{1+x} = 0\), \(-1 < x < 1\), then prove that \(\frac{dy}{dx} = -\frac{1}{(1+x)^2} \).
If \( A = \begin{bmatrix} 3 & 3 & 1 \\ 3 & 4 & 1 \\ 4 & 3 & 1 \end{bmatrix} \), then verify that \( A \, (\text{adj } A) = |A| I \) and find \( A^{-1} \).
Prove that: \(\int_0^{\pi/4} \log_e(1 + \tan \theta) \, d\theta = \frac{\pi}{8} \log_e 2\).
Solve the integral \(\int \frac{\sec^2 2\theta}{(\cot \theta - \tan \theta)^2} \, d\theta\).
Show that \( y = c_1 e^{ax} \cos(bx) + c_2 e^{ax} \sin(bx) \), where \( c_1, c_2 \) are constants, is a solution of the differential equation \( \frac{d^2y}{dx^2} - 2a\frac{dy}{dx} + (a^2 + b^2)y = 0 \).
If \( A = \begin{bmatrix} 1 & 3 & 3 \\ 1 & 4 & 3 \\ 1 & 3 & 4 \end{bmatrix} \), then verify that \( A \cdot \text{adj}(A) = |A| I \) and find \( A^{-1} \).
If \( x\sqrt{1+y} + y\sqrt{1+x} = 0 \) for \( -1<x<1 \), then prove that \( \frac{dy}{dx} = -\frac{1}{(1+x)^2} \).
Express the matrix \( A = \begin{bmatrix} 3 & 3 & -1 \\ -2 & -2 & 1 \\ -4 & -5 & 2 \end{bmatrix} \) as the sum of a symmetric and a skew-symmetric matrix.
Does the relation defined by \( R = \{(x, y) : y \text{ is divisible by } x\} \) on the set \( A = \{1, 2, 3, 4, 5, 6\} \), an equivalence relation?
For two invertible matrices A and B of order n, prove that \( (AB)^{-1} = B^{-1}A^{-1} \).
Does the function \( f(x) = \begin{cases} x+5 & \text{if } x \le 1 \\ x-5 & \text{if } x > 1 \end{cases} \) continuous at \( x = 1 \)?