List of practice Questions

Let \[ \vec a=\vec i+2\vec j+\vec k \] and \[ \vec b=2\vec i-\vec j+\vec k \] be two vectors. If vector \[ \vec r=x\vec i+y\vec j+2\vec k \] is along the bisector of angle between \(\vec a\) and \(\vec b\), then \[ |\vec r|= \]

If the points with position vectors \[ x\vec i+2\vec j+y\vec k \] \[ \vec i-2\vec j+2x\vec k \] and \[ 2\vec i+3\vec j-\vec k \] are collinear, then \[ 10x-25y= \]

Let \[ \vec a=2\vec i-\vec j-3\vec k,\qquad \vec b=\vec i+3\vec j-2\vec k,\qquad \vec c=3\vec i-2\vec j+\vec k \] If magnitude of projection of \[ \vec a+\lambda\vec b \] on \(\vec c\) is \[ \frac{10}{\sqrt{14}} \] then sum of squares of magnitudes of all such vectors is

Let \[ \vec a=\vec i-2\vec j+2\vec k \] and \[ \vec b=2\vec i+3\vec j-6\vec k \] If \[ \alpha\vec i+\beta\vec j+\gamma\vec k \] is perpendicular to plane of \[ 2\vec a+\vec b \] and \[ \vec b-\vec a \] such that \[ \alpha+\beta+\gamma=46 \] then \[ \alpha-2\beta+3\gamma= \]

Let \[ \vec a=3\vec i-2\vec j+5\vec k,\qquad \vec b=\vec i+3\vec j-2\vec k \] If \(\vec c\) is a vector such that \[ \vec b\times\vec c=\vec a \] and \[ \vec b\cdot\vec c=5 \] then \[ 14\vec c\times\vec a= \]

If \(\alpha,\beta\in\left(-\frac{\pi}{2},\frac{\pi}{2}\right)\), \[ \cos^4\alpha=\frac1{16},\qquad \sin^4\beta=\frac1{16} \] then \[ \cos\alpha+\cos\beta= \]

Number of solutions of equation \[ 3^{2\sin^2x}+3^{2\cos^2x}=6 \] lying in interval \[ [-\pi,\pi] \] is

If \[ (2\sin^{-1}x)^3=\pi^3-(2\cos^{-1}x)^3 \] then one value of \[ \cos(2\sin^{-1}x-3\cos^{-1}x) \] is

Evaluate \[ e^{Sinh^{-1}(2\sqrt2)}+e^{Cosh^{-1}(3)} \]

In triangle ABC, if \[ \frac{a}{b+c}+\frac{c}{a+b}=1 \] and \[ s=r+a \] then \[ \sin A+\sin B+\sin C= \]

Evaluate \[ 4\sin\frac{\pi}{6}\sin\frac{2\pi}{6}\sin\frac{3\pi}{6}\sin\frac{4\pi}{6}\sin\frac{5\pi}{6} \]

If \[ x=\sin18^\circ \] and \[ y=\tan22\frac12^\circ \] then \[ 4x(4x+2)= \]

The rank of the word ‘NEEDED’, when all letters of this word are permuted in all possible ways to form different 6-letter words and arranged in dictionary order, is

If number of circular permutations of 10 distinct things taken 5 at a time is \(m\) and number of linear permutations of 9 distinct things taken 4 at a time is \(n\), then \(m:n=\)

A student found 6 Mathematics books, 5 Physics books and 4 Chemistry books. If he buys at least one book of each subject, total number of ways is

The term independent of \(x\) in expansion of \[ \left(\frac{\sqrt{x}}{2}-\frac{3}{x}\right)^{12} \] is

Coefficient of \(x^3\) in the expansion of \[ \frac{(1-2x^2)^{\frac13}}{(2+x)^{\frac12}} \] is

If \[ \frac{2x^3+x-3}{x^4-5x^2+4} \] then partial fraction form is

If all roots of \[ x^5-3x^4+2x^3-3x^2+5x-2=0 \] are increased by real value h so that term containing \(x^3\) vanishes in transformed equation and h is integer, then h=

The rank of the word ‘NEEDED’, when all letters of this word are permuted in all possible ways to form different 6-letter words and arranged in dictionary order, is

If number of circular permutations of 10 distinct things taken 5 at a time is \(m\) and number of linear permutations of 9 distinct things taken 4 at a time is \(n\), then \(m:n=\)

A student found 6 Mathematics books, 5 Physics books and 4 Chemistry books. If he buys at least one book of each subject, total number of ways is

The term independent of \(x\) in expansion of \[ \left(\frac{\sqrt{x}}{2}-\frac{3}{x}\right)^{12} \] is

If \[ \sqrt[3]{i}=cis~\alpha,\qquad \alpha \text{ belongs to second quadrant} \] and \[ \sqrt[3]{-i}=cis~\beta,\qquad \beta \text{ belongs to third quadrant} \] then \[ cis~\alpha+cis~\beta= \]

If \(\alpha\in R\) and equation \[ (x-\alpha)(x-3)+1=0 \] has equal roots, then sum of squares of all values of \(\alpha\) is