List of top Mathematics Questions asked in UP Board XII

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?
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.
Evaluate : \(\int \frac{dx}{\sqrt{x^2 - a^2}}\).
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 \( x = \pi \).
Find the direction-cosines of the y-axis.
Differentiate \(x^{\sin x}\) with respect to \(x\), while \(x>0\).
Evaluate: \( \int \text{cosec}\,x(\text{cosec}\,x + \cot x) \,dx \).
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
If the coordinates of the points P and Q are respectively (2, 3, 0) and (–1, –2, –4), the vector \( \vec{PQ} \) will be
Solve: \((\tan^{-1}y - x)dy = (1+y^2)dx\).
Find the area of the region bounded by the ellipse \(\frac{x^2}{9^2} + \frac{y^2}{4^2} = 1\).
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
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 \(\sin^{-1}(1/3)\).
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.
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.
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.
For the differential equation \(xy\frac{dy}{dx} = (x+2)(y+2)\), find the curve passing through the point \((1, -1)\).
If \(y = x^{x^{x^{\dots \text{to infinity}}}}\), prove that \(x \frac{dy}{dx} = \frac{y^2}{1 - y \log x}\).
Evaluate: \(\int \sqrt{3 - 2x - x^2} dx\).