List of top Questions asked in UP Board XII

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?
Does the function \( f(x) = \begin{cases} x+5 & \text{if } x \le 1 \\ x-5 & \text{if } x > 1 \end{cases} \) continuous at \( x = 1 \)?
If \( y = e^{\tan^{-1}x} \), prove that \( (1 + x^2)\frac{d^2y}{dx^2} + (2x - 1)\frac{dy}{dx} = 0 \).
If \( A = \begin{bmatrix} 0 & -\tan{\frac{\alpha}{2}} \\[4pt] \tan{\frac{\alpha}{2}} & 0 \end{bmatrix} \), then prove that \[ (I + A) = (I - A)\begin{bmatrix} \cos{\alpha} & -\sin{\alpha} \\[4pt] \sin{\alpha} & \cos{\alpha} \end{bmatrix}. \]
If \( \begin{bmatrix} x+y & 2 \\ 5+z & xy \end{bmatrix} = \begin{bmatrix} 6 & 2 \\ 5 & 8 \end{bmatrix} \), then find the values of \(x, y, z\).
A relation \(R = \{(a, b) : a = b - 1,\; b \geq 3\}\) is defined on set \(\mathbb{N}\), then
If \(y = 600 e^{-7x} + 500 e^{7x}\), then show that \(\frac{d^2y}{dx^2} = 49y\).
Find the area bounded by the ellipse \(\frac{x^2}{16} + \frac{y^2}{25} = 1\).
There are 10 white and 5 black balls in a bag. Two balls are drawn one by one. First ball is not placed back before the second is taken out. Assume that the taking out of each ball from the bag is equally likely. What is the probability that both balls taken out are white?
Prove that \(|\vec{a} \cdot \vec{b}| \le |\vec{a}| |\vec{b}|\) is always true for any two vectors \(\vec{a}\) and \(\vec{b}\).
Solve: \(\int \frac{3x - 5}{x^3 - x^2 - x + 1} dx\)
Solve the system of equations
x + y + z = 2,
2x + y - 3z = 0,
x - y + z - 4 = 0 by matrix method.
Prove that the radius of the right circular cylinder of maximum curved surface inscribed in a cone is half of the radius of the cone.
There are two children in a family. It is known that there is at least one child is a boy. Then find the probability that both children are boys.
Find the interval in which the given function \( f(x) = x^2 - 4x + 6 \) is (i) Increasing (ii) Decreasing.
Find the differential coefficient of \(y = x^{\cos x} + (\sin x)^x\) with respect to x.
Find the shortest distance between the lines \(\vec{r} = (2\hat{i} + \hat{j} – \hat{k}) + \mu(3\hat{i} – 5\hat{j} + 2\hat{k})\) and \(\vec{r} = (\hat{i} + \hat{j}) + \lambda(2\hat{i} - \hat{j} + \hat{k})\).
Find the minimum value of Z = 3x + 7y by the graphical method under the following constraints:
\(x + y \le 8, 3x + 5y \ge 0, x \ge 0, y \ge 0\)
Show that (3\(\hat{i}\) - 4\(\hat{j}\) - 4\(\hat{k}\)), (2\(\hat{i}\) - \(\hat{j}\) + \(\hat{k}\)) and (\(\hat{i}\) - 3\(\hat{j}\) - 5\(\hat{k}\)) are the position vectors of vertices of a right angle triangle.
If \( A = \{1, 2\} \) and \( B = \{3, 4, 5\} \), then find all number of relations from A to B.
If \( y = A \cos \theta + B \sin \theta \), then prove that \( \frac{d^2y}{d\theta^2} = -y \).
Find the Cartesian equation of a line which passes through point (3, -2, -5) and is parallel to the vector \( (3\hat{i} + 2\hat{j} - 2\hat{k}) \).
Solve the differential equation \( \frac{dy}{dx} = e^x \cos x \).
If \( \vec{a}, \vec{b} \) and \( \vec{c} \) are vectors and \( \vec{a} + \vec{b} + \vec{c} = \vec{0} \), then find the value of \( (\vec{a} \cdot \vec{b} + \vec{b} \cdot \vec{c} + \vec{c} \cdot \vec{a}) \).