List of top Mathematics Questions asked in UP Board XII

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 \(\frac{x^2}{16} + \frac{y^2}{25} = 1\).
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.
Solve: \(\int \frac{3x - 5}{x^3 - x^2 - x + 1} dx\)
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 the system of equations
x + y + z = 2,
2x + y - 3z = 0,
x - y + z - 4 = 0 by matrix method.
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?
Find the minimum value of Z = 3x + 7y by the graphical method under the following constraints:
\(x + y \le 8, 3x + 5y \ge 0, x \ge 0, y \ge 0\)
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 interval in which the given function \( f(x) = x^2 - 4x + 6 \) is (i) Increasing (ii) Decreasing.
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.
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.
Solve the differential equation \( \frac{dy}{dx} = e^x \cos x \).
Find the value of \( \int_{-\pi/2}^{\pi/2} \sin^2 x \, dx \).
If \( \vec{a}, \vec{b} \) and \( \vec{c} \) are vectors and \( \vec{a} + \vec{b} + \vec{c} = \vec{0} \), then find the value of \( (\vec{a} \cdot \vec{b} + \vec{b} \cdot \vec{c} + \vec{c} \cdot \vec{a}) \).
If \( y = A \cos \theta + B \sin \theta \), then prove that \( \frac{d^2y}{d\theta^2} = -y \).
If \( R_1 \) and \( R_2 \) be two equivalence relations on a set A, then prove that \( (R_1 \cap R_2) \) also be an equivalence relation on A.
Show that \( f(x) = |x| \) is continuous for all values of x.
The value of \( \int \cos^2 x \, dx \) will be:
Solve the differential equation \( \frac{dy}{dx} = \frac{x^2 - 1}{y^2 + 1} \).
If \( y = A + Be^x \), then prove that \( \frac{d^2y}{dx^2} - \frac{dy}{dx} = 0 \), where A and B are constants.
A function \( f: \mathbb{R} \to \mathbb{R} \) is defined by \( f(x) = 3x \) for all \( x \in \mathbb{R} \). Then function \( f \) will be:
If A is a square matrix and \( A^2 = A \), then \( (A + I)^3 - 7A \) will be: