List of practice Questions

Find the value of $k$, if for the complex numbers $z_{1}$ and $z_{2}$, $\left|1-\bar{z}_{1}z_{2}\right|^{2}-\left|z_{1}-z_{2}\right|^{2} =k\left(1-\left|z_{1}\right|^{2}\right)\left(1-\left|z_{2}\right|^{2}\right)$.

Find the ratio in which the line segment joining the points $(2, 4, 5)$ and $(3, 5, -4)$ is divided by the $xz$-plane.

Find the standard deviation for the following data:

Find the sum of the sequence $7, 77, 777,7777,...$ to $n$ terms.

Find the principal value $tan^{-1} \, \left(\sqrt{3}\right)$

Find the principal value $tan^{-1}\left(-\sqrt{3}\right) + sec^{-1}\left(-2\right)-cosec^{-1}\left(\frac{2}{\sqrt{3}}\right)$

Find the range of the data $35$, $50$, $48$, $62$, $27$, $39$, $43$, $72$, $56$, $68$.

Find the ratio in which the line segment joining the points $(4, 8,10)$ and $(6, 10, -8)$ is divided by the $yz$-plane.

Find the principal value $\sin^{-1}\left(\frac{-1}{2}\right)$

Find the principal value $sec^{-1} (2)$

Find the principal value $tan^{-1} \, (-1)$

Find the points of trisection of the segment joining the points $A( 1,0, -6)$ and $B(-5,9,6)$.

Find the principal value $cos^{-1} \left(\frac{1}{2}\right)$

Find the principal value $cot^{-1} (1)$

Find the principal value $cot^{-1} \, \left(\sqrt{-3}\right)$

Find the number of ways of choosing $4$ cards from a pack of $52$ playing cards when cards are of the same colour.

Find the number of ways of choosing $4$ face cards from a pack of $52$ playing cards.

Find the points of local maxima and local minima respectively for the function $f\left(x\right) = sin\,2x-x$, where $-\frac{\pi}{2} \le x \le\frac{\pi }{2}$

Find the number of arrangements of the letters of the word when words begin with $I$ and end in $P$.

Find the number of different $8$ - letter arrangements that can be made from the letters of the word so that (i) all vowels occur together (ii) all vowels do not occur together.

Find the number of different signals that can be generated by arranging at least $2$ flags in order (one below the other) on a vertical staff, if five different flags are available.

Find the multiplicative inverse of $2 - 3i$.

Find the new coordinates of the point $(1, 1)$ if the origin is shifted to the point $(-3, -2)$ by a translation of axes.

Find the new coordinates of the point $(5, 0)$ if the origin is shifted to the point $(-3, -2)$ by a translation of axes

Find the number of $4$ letter words, with or without meaning, which can be formed out of the letters of the word , where the repetition of the letters is not allowed.