List of top Mathematics Questions asked in UP Board XII

Find the projection of the vector \(\vec{a} = 2\hat{i} + 3\hat{j} + 2\hat{k}\) on the vector \(\vec{b} = \hat{i} + 2\hat{j} + \hat{k}\).
Evaluate : \(\int \frac{dx}{\sqrt{x^2 - a^2}}\).
Find the principal value of \( \text{cosec}^{-1}(-\sqrt{2}) \).
Find the direction-cosines of the y-axis.
Test whether the function defined by \( f(x) = x^2 - \sin(x) + 5 \) is continuous at \( x = \pi \).
Evaluate: \( \int \text{cosec}\,x(\text{cosec}\,x + \cot x) \,dx \).
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
If the coordinates of the points P and Q are respectively (2, 3, 0) and (–1, –2, –4), the vector \( \vec{PQ} \) 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
Solve: \(ydx - (x + 2y^2)dy = 0\).
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 \(|\vec{a} + \vec{b}| \le |\vec{a}| + |\vec{b}|\).
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 \((\frac{\pi}{2}, \pi)\).
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.
If \(y = x^{x^{x^{\dots \text{to infinity}}}}\), prove that \(x \frac{dy}{dx} = \frac{y^2}{1 - y \log x}\).
For the differential equation \(xy\frac{dy}{dx} = (x+2)(y+2)\), find the curve passing through the point \((1, -1)\).
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.
The Cartesian equation of a line is \(\frac{x+3}{2} = \frac{y-5}{4} = \frac{z+6}{2}\). Find its vector equation.
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 \(|\vec{a} - \vec{b}|\).
Find the value of \(\frac{dy}{dx}\) if \(y = \tan^{-1}\left(\frac{3x - x^3}{1 - 3x^2}\right)\), where \(-\frac{1}{\sqrt{3}}<x<\frac{1}{\sqrt{3}}\).
Find the value of \(\int \frac{1}{x^2 - a^2} dx\).
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.