List of top Questions asked in UP Board XII

Differentiate \(\log(\log x)\), \(x>1\) with respect to \(x\).

Examine the continuity of the function \(f(x)=2x^2-1\) at \(x=3\).

Find the direction cosines of the \(x\), \(y\) and \(z\) axes.

Find the area of the region bounded by the curve \(y=\sin x\), \(x=0\) and \(x=2\pi\).

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

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

Find the order and degree of the differential equation \(\left(\dfrac{d^2y}{dx^2}\right)^3 + \left(\dfrac{dy}{dx}\right)^2 + \sin\left(\dfrac{dy}{dx}\right) + 1 = 0\).

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

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

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

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

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 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 \(\dfrac{4}{27}\pi h^3\tan^2\alpha\).

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

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

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

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 \(\dfrac{x+3}{3}=\dfrac{y-4}{5}=\dfrac{z-8}{6}\).

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

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 \(|\vec a+\vec b+\vec c|\).