List of practice Questions

In a triangle \(ABC\), if \[ r-r_1+r_2+r_3=2\sqrt2R,\qquad r+r_1-r_2+r_3=0 \] and \[ b=2\sqrt2 \] then \[ a+c= \]

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

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 \[ 4x(4x+2)= \]

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

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

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

Two real roots of \[ 3x^4+ax^3+55x^2-52x+12=0 \] are positive and equal. Product of other two roots is 1. If roots belong to natural numbers then \[ a\beta-a+\frac{\gamma}{\delta}= \]

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 \(m:n=\)

If \[ \sqrt{-4x+2i\sqrt{x^{4}+2x^{2}+9}} = \pm(a+ib) \] then \[ a^2+b^2-6= \]

Product of all the five values of \[ (1-i)^{4/5} \] is

If \[ A= \begin{bmatrix} 1& 1\\ 0& 1 \end{bmatrix} \] and \[ S=A+A^{2}+A^{3}+...+A^{12} \] then sum of all elements of matrix S is

If \[ A= \begin{bmatrix} 0& \alpha& \beta\\ \beta& \alpha& 0\\ \alpha& 0& \beta \end{bmatrix} \] where \(\beta& gt;\alpha& gt;0\) and \[ AA^T= \begin{bmatrix} 25& a& b\\ a& 25& 12\\ b& a& 25 \end{bmatrix} \] then \[ a+b+\alpha-\beta= \]

Let \[ A= \begin{bmatrix} x& 2& -1\\ -2& 1& 2x\\ 3x& 2& 1 \end{bmatrix} \] and \[ det(A)=f(x) \] If \(f(x)\) attains minimum value \(m\) at \(x=n\), then \[ \left|\frac{m}{n}\right|= \]

If the system of equations \[ ax+y-2z=3,\qquad 2x-y+3z=b,\qquad x+2y-z=3 \] has infinitely many solutions, then \(3a-2b=\)

If \[ z=i^i \] then \[ z^i= \]

A real valued function \(f\) defined by \[ f(x)=|x|-x \] is

Consider the following Assertion and Reason Assertion: \[ \frac{1}{5\cdot9}+\frac{1}{9\cdot13}+\frac{1}{13\cdot17}+... \text{ to 10 terms }=\frac{9}{41} \] Reason: \[ \text{For all }n\in N, \quad \frac{1}{5\cdot9}+\frac{1}{9\cdot13}+... =\frac{n}{5(4n+5)} \]