List of top Mathematics Questions asked in UP Board XII

Solve the differential equation \((x-y)\dfrac{dy}{dx}=x+2y\).

If \(A=\begin{bmatrix}1&2&3\\2&4&5\\3&5&6\end{bmatrix}\), then find \(A^{-1}\).

Prove that vectors \(2\hat i-\hat j+\hat k\), \(\hat i-3\hat j-5\hat k\) and \(3\hat i-4\hat j-4\hat k\) are vertices of a right-angled triangle.

Evaluate \(\displaystyle\int\dfrac{x^2+1}{x^2-5x+6}\,dx\).

If \(y=Ae^{mx}+Be^{nx}\), then show that \(\dfrac{d^2y}{dx^2}-(m+n)\dfrac{dy}{dx}+mny=0\).

If \(\cos y=x\cos(a+y)\) and \(\cos a\ne\pm1\), then prove that \(\dfrac{dy}{dx}=\dfrac{\cos^2(a+y)}{\sin a}\).

Maximize \(Z=10x+3y\) by the graphical method under the following constraints: \(x\ge0;\ y\ge0;\ 5x+3y\le15;\ 2x+5y\le10\).

Prove that \(\tan^{-1}\dfrac15+\tan^{-1}\dfrac17+\tan^{-1}\dfrac13+\tan^{-1}\dfrac18=\dfrac\pi4\).

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

Solve the differential equation: \((1+x^2)\,dy+2xy\,dx=\cot x\,dx\).

Find the value of the determinant \(\begin{vmatrix}a&b&c\\a^2&b^2&c^2\\a^3&b^3&c^3\end{vmatrix}\).

Find the projection of vector \(\hat i+3\hat j+7\hat k\) on vector \(7\hat i-\hat j+8\hat k\).

If \(P(A)=\dfrac13\), \(P(B)=\dfrac12\) and \(P(A\cup B)=\dfrac23\), then show that \(A\) and \(B\) are independent events.

Find the general solution of the differential equation \(\dfrac{dy}{dx}=\dfrac{1+y^2}{1+x^2}\).

Find the vector and Cartesian equations of a line which passes through the point \((1,2,3)\) and is parallel to the vector \(2\hat i+3\hat j+2\hat k\).

If \(y=5\cos x-3\sin x\), then prove that \(\dfrac{d^2y}{dx^2}+y=0\).

If \(f(x)=\begin{cases}x+2, & x\ne0\\1, & x=0\end{cases}\), then prove that the function is not continuous at \(x=0\).

Find the value of the determinant \(\begin{vmatrix}x&a&x+a\\y&b&y+b\\z&c&z+c\end{vmatrix}\).

Find the principal value of \(\sin^{-1}\left(\dfrac{1}{\sqrt2}\right)\).

Find the value of \(\displaystyle\int_2^3\dfrac1x\,dx\).

If \(P(A)=\dfrac7{13}\), \(P(B)=\dfrac9{13}\) and \(P(A\cap B)=\dfrac4{13}\), then find \(P\left(\dfrac AB\right)\).

If functions \(f:R\to R\) and \(g:R\to R\) are defined respectively as \(f(x)=\cos x\) and \(g(x)=3x^2\), then find \(gof\) and \(fog\).

Find the angle between vectors \(\hat i-2\hat j+3\hat k\) and \(3\hat i-2\hat j+\hat k\).

Direction ratios of a line are 2, \(-1\) and \(-2\). Find the direction cosines of the line.

If \(x-y=\pi\), then the value of \(\dfrac{dy}{dx}\) is: