List of top Mathematics Questions asked in UP Board XII

Three vectors \(\vec a,\vec b,\vec c\) satisfy the condition \(\vec a+\vec b+\vec c=0\). If \(|\vec a|=3,|\vec b|=4\) and \(|\vec c|=2\), then find the value of \(\mu=\vec a\cdot\vec b+\vec b\cdot\vec c+\vec c\cdot\vec a\).

Find the value of the determinant \(\begin{vmatrix}18&22&-13\\7&-9&11\\-11&7&17\end{vmatrix}\).

Prove that the height of a cylinder of maximum volume, inscribed in a sphere of radius \(R\), is \(\dfrac{2R}{\sqrt3}\).

If \((\vec a+\vec b)\cdot(\vec a-\vec b)=8\) and \(|\vec a|=8|\vec b|\), then find \(|\vec a|\) and \(|\vec b|\).

Show that the points \(A(2,3,-4)\), \(B(1,-2,3)\) and \(C(3,8,-11)\) are collinear.

Simplify: \(\tan^{-1}\!\left(\dfrac{3a^{2}x-x^{3}}{a^{3}-3ax^{2}}\right)\).

Find the minimum value of the objective function \(Z=3x+5y\) under the following constraints by the graphical method: \(x+3y\ge3,\ x+y\ge2,\ x\ge0,\ y\ge0\).

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

Simplify: \(\cos\theta\begin{bmatrix}\cos\theta & \sin\theta\\-\sin\theta & \cos\theta\end{bmatrix}+\sin\theta\begin{bmatrix}\sin\theta & -\cos\theta\\\cos\theta & \sin\theta\end{bmatrix}\).

If two vectors \(\vec a=5\hat i-\hat j-3\hat k\) and \(\vec b=\hat i+3\hat j-5\hat k\), then show that \((\vec a+\vec b)\) and \((\vec a-\vec b)\) are perpendicular.

If \(A\) and \(B\) are independent events and \(P(A)=0.3\), \(P(B)=0.4\), then find (i) \(P(A\cap B)\), (ii) \(P(A\cup B)\).

If \(\sin^{-1}(1-x)-2\sin^{-1}x=\dfrac{\pi}{2}\), then find the value of \(x\).

Prove that the relation given by \(R=\{(1,1),(2,2),(3,3),(1,2),(2,3)\}\) in the set \(\{1,2,3\}\) is reflexive but neither symmetric nor transitive.

Show that the function \(f(x)=x^{3}-6x^{2}+12x,\ x\in R\), is an increasing function on \(R\).

Find the angle between the vectors \(\vec a=\hat i+\hat j-\hat k\) and \(\vec b=\hat i-\hat j+\hat k\).

If the direction ratios of a line are \(3,-2,-6\), then find its direction cosines.

Find the value of \(\cos^{-1}\Big(\dfrac12\Big)+2\sin^{-1}\Big(\dfrac12\Big)\).

Write \(\cot^{-1}\!\Big(\dfrac{1}{\sqrt{x^{2}-1}}\Big)\), \(x>1\) in the simplest form.

The probability of impossible events is:

A relation \(R\) is defined in the set \(N\) as \(R=\{(x,y):y=x+5,\,y>5\}\). Then which of the following is correct?

If \(f:R\to R\) is given by \(f(x)=(5-x^{5})^{1/5}\), then \(f\circ f(x)\) is equal to:

Find \(\displaystyle\int e^x\left(\dfrac{1+\sin x}{1+\cos x}\right)dx\).

Find the particular solution of the differential equation \(\dfrac{dy}{dx}+y\cot x=4x\csc x\), \((x\neq0)\), given that \(y=0\) when \(x=\dfrac{\pi}{2}\).

Find \(\displaystyle\int\dfrac{x\sin^{-1}x}{\sqrt{1-x^2}}\,dx\).

Find \(\displaystyle\int\left[\sqrt{\tan x}+\sqrt{\cot x}\right]dx\).