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If $z=\frac{1}{(1-i)(2+3i)},$, then |z| =

Find the locus of a point equidistant from the point (2,4) and the y-axis.

A sequence (x_0, x_1, x_2, x_3, ......) is defined by letting $x_0=5$ and $x_k=4+x_{k-1}$ for all natural numbers k. Show that xn = 5 + 4n for all n $ϵ$ N using mathematical induction.

The locus of the mid point of the chord of the circle $x^2+y^2-2x-2y-2=0$ which makes an angle of ${120}^0$ at the centre is.

Prove that:

(i) If $cosx=-\frac{3}{5}$ and x lies in the IIIrd quadrant, find the values of $cos\frac{x}{2},sin\frac{x}{2}$ and sin 2x.

(ii) If $cosx=-\frac{3}{5}$ and x lies in the IInd quadrant, find the values of sin 2x and sin $\frac{x}{2}$

The sum of the series 3.6 + 4.7 + 5.8 + ........ upto (n - 2) terms

Find the equation of a circle passing through the point $(7,3)$ having radius $3$ units and whose centre lies on the line $y=x-1$.

If 12$co{t}^2\theta$ - 31 cosec $\theta$ + 32 = 0, then the value of sin θ is

If the sum of the two roots of the equation $4x^3+16x^2-9x-36=0$ is zero, then the roots are

It is given that the events A and B are such that $P\left(A\right)=\frac{1}{4},P\left(\frac{A}{B}\right)=\frac{1}{2}andP\left(\frac{B}{A}\right)=\frac{2}{3}.$ Then P(B) is

A and B are square matrices and A is non-singular matrix, $(A^{-1}BA{)}^n,nϵI^{+}$ is equal to

If $\alpha ,\beta ,\gamma ,\delta$ are the smallest positive angles in ascending order of magnitude which have their sines equal to the positive quantity k, then the value of $4sin\frac{\alpha }{2}+3sin\frac{\beta }{2}+2sin\frac{\gamma }{2}+sin\frac{\delta }{2}$ is equal to

If $f\left(x\right)=ln\left(\frac{x^2+e}{x^2+1}\right),$ then range of f(x) is

Find the range of rational expression $y=\frac{x^2+34x-71}{x^2+2x-7}$ if x is real

The value of $\frac{tanx}{tan3x}$ whenever defined never lie between

Find the value of the $\sum _{r=1}^{100}\frac{\tan \left(2^{r-1}\right)}{\cos 2^r}$ is

Find the projection of the point $(1,0)$ on the line joining the points $(-1,2)$ and $(5,4).$

If the lines $2x-py+1=0,3x-qy+1=0$ and $4x-ry+1=0$ are concurrent, then $p,q,r$ are in

The range of the function $f\left(x\right)=4-\sqrt{x^2-10x+25}$ is

If $x\in (\frac{-\pi }{2},\frac{\pi }{2}),$ then $\sqrt{\frac{1-\sin x}{1+\sin x}}$ is equal to

$C_1+2C_2+3C_3+4C_4+......+n{C}_n$ =

A line passing through the point A with position vector $\overrightarrow{a}=4\hat{i}+2\hat{j}+2\hat{k}$ is parallel to the vector $\overrightarrow{b}=2\hat{i}+3\hat{j}+6\hat{k}$.Find the length of the perpendicular drawn on this line from a point P with vector ${\overrightarrow{r}}_1=\hat{i}+2\hat{j}+3\hat{k}$.

If $z=x+iy,x,y\in R$ and $Im\left(\frac{2z+1}{i\overline{z}+1}\right)=-2$, then

Let $A,G$ and $H$ be the arithmetic mean, geometric mean and harmonic mean, respetively of two distinct positive real numbers. If $\alpha$ is the smallest of the two roots of the equation $A(G–H)x^2+G(H–A)x+H(A–G)=0$, then

Write the equation of the unit circle concentric with
$x^2+y^2-8x+4y-8=0.$

For three non-impossible events $A,B$ and $C$ and $P(A\cap B\cap C)=0$, $P(A\cup B\cup C)=\frac{3}{4}$, $P(A\cap B)=\frac{1}{3}$ and $P\left(C\right)=\frac{1}{6}$. The probability, exactly one of $A$ or $B$ occurs but $C$ doesn't occur is -

Find the least positie integral value of n for which ${\left(\frac{1+i}{1-i}\right)}^n$ is real.

One card is drawn from a pack of 52 cards. The probability that it is the card of a king or spade is

From the given figure what is phase difference between current $i_1$ and $i_2$

Find the derivative of $f\left(x\right)$
$f\left(x\right)=\tan (4+2x)$

The correct statement concerning the following data set:
$2,5,9,3,4,7,3,8,11,15,10$ is

${lim}_{x\to \frac{3\pi }{2}}\frac{1+cose{c}^{3}x}{co{t}^{2}x}$

Prove that the lines $2x-3y+1=0,x+y=3,2x-3y=2$ and $x+y=4$ form a parallelogram.

Find the distance of a point (2, 5, −3) from the plane $\overrightarrow{r}.(6\hat{i}-3\hat{j}+2\hat{k})=4$.

What should be the domain for which the function $y=\frac{x}{\sqrt{x^2-1}}$ is continuous.

4 identical hats are distributed among 4 boys such that any boy can get any number of hats. What is the probability that no boys gets more than 2 hats.

The term independent of x in the expansion of $(1-x{)}^2{(x+\frac{1}{x})}^{10}$ is