List of top Questions asked in UP Board XII

Find the general solution of the differential equation \(y - x \frac{dy}{dx} = x + y \frac{dy}{dx}\).
Find the shortest distance between the lines whose vector equations are \(\vec{r}=(\hat{i}+2\hat{j}+3\hat{k})+\lambda(\hat{i}-3\hat{j}+2\hat{k})\) and \(\vec{r}=(4\hat{i}+5\hat{j}+6\hat{k})+\mu(2\hat{i}+3\hat{j}+\hat{k})\).
Find the general solution of the differential equation \(x\frac{dy}{dx} + 2y = x^2\) (\(x \neq 0\)).
A die was thrown twice and the sum of the numbers which appeared was found to be 6. Find the conditional probability that the number 4 appears at least once.
Maximize \(Z = x + 2y\) by graphical method under the constraints \(x+y \le 1, -x+y \le 0, x \ge 0, y \ge 0\).
Find the area of the parallelogram whose diagonals are \( \vec{a} = 3\hat{i} + \hat{j} - 2\hat{k} \) and \( \vec{b} = \hat{i} - 3\hat{j} + 4\hat{k} \).
If \( \theta \) be the angle between two unit vectors \( \hat{a} \) and \( \hat{b} \), prove that \( \sin\frac{\theta}{2} = \frac{1}{2} |\hat{a} - \hat{b}| \).
Sand is falling from a pipe at the rate of 12 cm\(^3\)/second. The falling sand forms such a cone on the ground that its height is always one-sixth of the radius of its base. At which rate is the height of the cone formed by sand increasing while its height is 4 cm?
Evaluate : \(\int \frac{dx}{\sqrt{x^2 - a^2}}\).
Let a relation R be defined in the set \(\mathbb{N} \times \mathbb{N}\) as follows: \((a, b) R (c, d)\) if and only if \(a + d = b + c\). Prove that R is an equivalence relation.
Find the projection of the vector \(\vec{a} = 2\hat{i} + 3\hat{j} + 2\hat{k}\) on the vector \(\vec{b} = \hat{i} + 2\hat{j} + \hat{k}\).
In a leap year, selected at random, find the probability that there are 53 Tuesdays.
Find the principal value of \( \text{cosec}^{-1}(-\sqrt{2}) \).
Test whether the function defined by \( f(x) = x^2 - \sin(x) + 5 \) is continuous at \( x = \pi \).
Evaluate: \( \int \text{cosec}\,x(\text{cosec}\,x + \cot x) \,dx \).
Find the direction-cosines of the y-axis.
Differentiate \(x^{\sin x}\) with respect to \(x\), while \(x>0\).
Find the area of the region bounded by the ellipse \(\frac{x^2}{9^2} + \frac{y^2}{4^2} = 1\).
Let [x] represents the greatest integer which is less than or equal to x. Then the function \(f: \mathbb{R} \to \mathbb{R}\) defined by \(f(x) = [x]\) will be
The order of the differential equation \( \left( \frac{d^3y}{dx^3} \right)^2 + \log x \left( \frac{d^2y}{dx^2} \right)^3 + 5y = \cos x \) will be
If the coordinates of the points P and Q are respectively (2, 3, 0) and (–1, –2, –4), the vector \( \vec{PQ} \) will be
Minimize Z = 200x + 500y by graphical method subject to the following constraints:
x + 2y \(\ge\) 10, 3x + 4y \(\le\) 24, x \(\ge\) 0, y \(\ge\) 0.
For two vectors \(\vec{a}\) and \(\vec{b}\) prove that \(|\vec{a} + \vec{b}| \le |\vec{a}| + |\vec{b}|\).
Solve: \(ydx - (x + 2y^2)dy = 0\).
Prove that the semi-vertical angle of a right circular cone of given surface and maximum volume is \(\sin^{-1}(1/3)\).