List of top Questions asked in UP Board XII

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 \(\frac{x^2}{9^2} + \frac{y^2}{4^2} = 1\).
Solve: \((\tan^{-1}y - x)dy = (1+y^2)dx\).
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: \(ydx - (x + 2y^2)dy = 0\).
If \(A = \begin{bmatrix} 2 & -3 & 5
3 & 2 & -4
1 & 1 & -2 \end{bmatrix}\), find \(A^{-1}\). Using \(A^{-1}\) solve the following system of equations:
2x - 3y + 5z = 11
3x + 2y - 4z = -5
x + y - 2z = -3.
Prove that the semi-vertical angle of a right circular cone of given surface and maximum volume is \(\sin^{-1}(1/3)\).
Show that the function \(f(x) = \log \sin x\) is increasing in the interval \((0, \frac{\pi}{2})\) and decreasing in \((\frac{\pi}{2}, \pi)\).
If \(y = x^{x^{x^{\dots \text{to infinity}}}}\), prove that \(x \frac{dy}{dx} = \frac{y^2}{1 - y \log x}\).
Evaluate: \(\int \sqrt{3 - 2x - x^2} dx\).
For the differential equation \(xy\frac{dy}{dx} = (x+2)(y+2)\), find the curve passing through the point \((1, -1)\).
A person has taken the contract of a construction work. The probability of strike is 0.65. The probabilities of the construction work being completed on time in the circumstances of no strike and strike are respectively 0.80 and 0.32. Find the probability of the construction work being completed on time.
Find the value of \(\frac{dy}{dx}\) if \(y = \tan^{-1}\left(\frac{3x - x^3}{1 - 3x^2}\right)\), where \(-\frac{1}{\sqrt{3}}<x<\frac{1}{\sqrt{3}}\).