List of top Mathematics Questions asked in UP Board XII

If \(A=\begin{bmatrix}1&2&3\\3&-2&1\\4&2&1\end{bmatrix}\), then show that \(A^3-23A-40I=0\).

Evaluate: \(\displaystyle\int_0^\pi \frac{x\,dx}{a^2\cos^2x+b^2\sin^2x}\)

Find the area bounded by the ellipse \(\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1\).

Find the general solution of the differential equation \(x\cos\left(\dfrac yx\right)\dfrac{dy}{dx} = y\cos\left(\dfrac yx\right)+x\).

Prove that the volume of the largest right circular cone that can be inscribed in a sphere of radius \(R\) is \(\dfrac{8}{27}\) of the volume of the sphere.

Solve the system of linear equations by Matrix Method: \(2x-3y+5z=11\), \(3x+2y-4z=-5\), \(x+y-2z=-3\).

Find the shortest distance between the lines \(\vec r=\hat i+\hat j-\lambda(2\hat i+\hat j+\hat k)\) and \(\vec r=2\hat i+\hat j+\hat k+\mu(3\hat i-5\hat j+2\hat k)\).

If \(A=\begin{bmatrix}2&3\\1&-4\end{bmatrix},\ B=\begin{bmatrix}1&-2\\-1&3\end{bmatrix}\), then find \((AB)^{-1}\).

Is the function \(f(x)=\begin{cases}x+5,&x\le1\\x-5,&x>1\end{cases}\) continuous at \(x=1\)?

If the function \(f:\left[0,\dfrac{\pi}{2}\right]\to R\) is given by \(f(x)=\cos x\) and the function \(g:\left[0,\dfrac{\pi}{2}\right]\to R\) is given by \(g(x)=\sin x\), then prove that \(f\) and \(g\) are one-one, but \(f+g\) is not one-one.

Write \(\tan^{-1}\left(\dfrac{\cos x-\sin x}{\cos x+\sin x}\right)\), \(\dfrac{-\pi}{4}<x<\dfrac{3\pi}{4}\) in the simplest form.

A family has two children. If it is known that at least one child is a boy, then find the probability of both children being boys.

Three vectors \(\vec a, \vec b\) and \(\vec c\) satisfy the condition \(\vec a+\vec b+\vec c=0\). If \(|\vec a|=3, |\vec b|=4\) and \(|\vec c|=5\), then find the value of \(\vec a\cdot\vec b+\vec b\cdot\vec c+\vec c\cdot\vec a\).

Solve: \(\displaystyle\int \cos^2 x\, dx\)

Find the angle between the lines \(\vec r = 3\hat i+2\hat j-4\hat k+\lambda(\hat i+2\hat j+2\hat k)\) and \(\vec r = 5\hat i-2\hat j+\mu(3\hat i+2\hat j+6\hat k)\).

Evaluate: \(\displaystyle\int_0^1 \frac{\tan^{-1}x}{1+x^2}\, dx\)

Minimize \(z=3x+2y\) under the following constraints by graphical method: \(x+y\ge8,\ x\ge0,\ 3x+5y\le15,\ y\ge0\).

Prove that on \(R\), the function \(f(x)=e^{2x}\) is increasing.

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.

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

If \(P(A)=\dfrac{7}{13}\), \(P(B)=\dfrac{9}{13}\) and \(P(A\cap B)=\dfrac{4}{13}\), then find \(P(A/B)\).

Prove that \((\vec a+\vec b)\cdot(\vec a+\vec b) = |\vec a|^2+|\vec b|^2\) if and only if \(\vec a\) and \(\vec b\) are perpendicular. It is given that \(\vec a\ne0,\ \vec b\ne0\).

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

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