List of top Mathematics Questions asked in UP Board XII

If the ordered pairs \((2x-3,-5)\) and \((x,y-1)\) are equal, then find the value of \(x\) and \(y\).

Find the area of the parallelogram whose adjacent sides are given by \(\vec a = 3\hat i+\hat j+4\hat k\) and \(\vec b = \hat i-\hat j+\hat k\).

If \(A^2=A\), then simplify \((I+A)^2 - 7A\), where \(A\) is a square matrix.

If the radius of a circle is \(r = 6\) cm, then the rate of change of Area with respect to radius is:

Evaluate: \(\displaystyle\int_{-\pi/2}^{\pi/2} (x^3 + x\cos x)\, dx\)

The value of \(\displaystyle\int_0^{2/3} \frac{dx}{4+9x^2}\) is:

If \(A\) is any matrix and \(K\) is any constant, then \((KA)'\) is:

The value of \(\tan^{-1}(\sqrt{3}) - \cot^{-1}(-\sqrt{3})\) is:

The function \(f: R \to R\) defined by \(f(x) = 3x + 5,\ \forall x \in R\) is:

Find the value of \(\displaystyle\int\left(\sqrt{\cot x}+\sqrt{\tan x}\right)dx\).

Find the area of the circle \((x-2)^2+y^2=4\).

Solve by matrix method the system of equations: \(x-y+2z=7,\ 3x+4y-5z=-5,\ 2x-y+3z=12\).

Prove that \(\displaystyle\int_0^{\pi/2}\log\sin x\,dx=-\dfrac{\pi}{2}\log2\).

Prove that the matrix \(A=\begin{bmatrix}2&-1\\3&4\end{bmatrix}\) satisfies the equation \(A^2-6A+11I=O\), where \(I\) is the identity matrix of order \(2\times2\) and \(O\) is the zero matrix of order \(2\times2\). With the help of it, find \(A^{-1}\).

Prove that the height of the cylinder of maximum volume inscribed in a right circular cone of semi-vertical angle \(\alpha\) and height \(h\) is one-third of the height of the cone, and the maximum volume of the cylinder is \(\dfrac{4}{27}\pi h^3\tan^2\alpha\).

\(x^2\,dy+\dfrac12(xy+y^2)\,dx=0\); find the particular solution, given that \(y=1\) when \(x=1\).

If the position vectors of the vertices \(A,B,C\) of a \(\triangle ABC\) are respectively \(\vec a,\vec b\) and \(\vec c\), then prove that the area of \(\triangle ABC\) will be \(\dfrac12\left|\vec a\times \vec b+\vec b\times \vec c+\vec c\times \vec a\right|\).

Maximize \(Z=105x+90y\) under the constraints \(2x+y\le80,\ x+y\le50,\ x\ge0,\ y\ge0\) by graphical method.

Let three vectors \(\vec a,\vec b\) and \(\vec c\) be such that \(|\vec a|=3,|\vec b|=4,|\vec c|=5\) and each of them is perpendicular to the sum of the other two vectors, then find the value of \(|\vec a+\vec b+\vec c|\).

Solve the differential equation \(y\,dx+(x-y^2)\,dy=0\).

Find the Cartesian equation of the line which passes through the point \((-1,4,-5)\) and is parallel to \(\dfrac{x+3}{3}=\dfrac{y-4}{5}=\dfrac{z-8}{6}\).

It is given that the numbers obtained on throwing two dice are different. Find the probability that the sum of both numbers is 4.

The length \(x\) of a rectangle is decreasing at the rate of 3 cm/min and the breadth \(y\) is increasing at the rate of 2 cm/min. When \(x=5\) cm and \(y=3\) cm, find the rate of change of the area of the rectangle.

If \(x=a\sin^3t,\ y=b\cos^3t\), then find \(\dfrac{dy}{dx}\) at the point \(t=\dfrac{\pi}{2}\).

Show that \(\sin^{-1}\dfrac{12}{13}+\cos^{-1}\dfrac{4}{5}+\tan^{-1}\dfrac{63}{16}=\pi\).