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Let $C_1$ and $C_2$ be the two curves on the complex plane defined as
$C_1:z+\overline{z}=2|z-1|$
$C_2:arg(z+1+i)=\alpha$
Where \alpha belongs to the interval $(0,\frac{\pi }{2})$ such that curves $C_1$ and $C_2$ have exactly one point in common and which is denoted by $P\left(z_0\right)$
The value of $|z_0|$ is

The radius of the circle whose centre is (–7, 0) and which cuts the parabola $y^2=8x$ at A and B such that its common chord AB subtends a right angle at the vertex of the parabola is

The number of pairs (x,y) where both x and y are real satisfying x² + y² + 2 = ( 1+x )(1 + y) is

The remainder where $(25{)}^{15}$ is divided by 13 equal to

The angle between the tangents drawn from the point (1, 4) to the parabola $y^2$ = 4x is

The locus of point of intersection of the two tangents to the ellipse $b^2{x}^2+a^2{y}^2=a^2{b}^2$ which makes an angle ${60}^{\circ }$ with one another is

Let x, y and z be positive integers such that GCD(x, y, z) =1: x<y<z and x²+y²=z². Then which of the following is always true?

Find three numbers in G.P. whose sum is 65 and whose product is 3375.

$(sin\alpha +cosec\alpha {)}^2+(cos\alpha +sec\alpha {)}^2-(ta{n}^2\alpha +co{t}^2\alpha )$ for all values of $\alpha$ is equal to

Mark the correct alternative in each of the following :

L is a variable line such that the algebraic sum of the distance of the points $(1,1),(2,0)$ and $(0,2)$ from the line is equal to zero. The line L will always pass through

If the lines $x+q=0,y-2=0$ and $3x+2y+5=0$ are concurrent, then the value of q will be

Find the acute angle between the two planes 3x – 6y + 2z = 7 and 2x + 2y – 2z =5.

Let $\overrightarrow{a}=2\hat{i}+3\hat{j}-6\hat{k},\overrightarrow{b}=2\hat{i}-3\hat{j}+6\hat{k}$ and $\overrightarrow{c}=-2\hat{i}+3\hat{j}+6\hat{k}$. Let ${\overrightarrow{a}}_1$ be the projection of $\overrightarrow{a}$ on $\overrightarrow{b}$ and ${\overrightarrow{a}}_2$ be the projection of ${\overrightarrow{a}}_1$ on $\overrightarrow{c}$. Then, ${\overrightarrow{a}}_1.\overrightarrow{b}$ is equal to

$\text{Find the equation of the hyperbola,referred to its principal axes as axes of coordinates,in the following cases:}\left(i\right)\text{the distance between the foci =16 and eccentricity}=\sqrt{2}\left(ii\right)\text{conjugate axis is 5 and the distance between foci=13}\left(iii\right)\text{conjugate axis is 7 and passes through the point (3,-2)}$

If X and Y are two sets such that $X\cup Y$ has 18, X has 8 elements and Y has 15 elements; how many elements does $X\cap Y$ have?

The range of the functions $f\left(x\right)=sin\left[x\right]\frac{-\pi }{4}<x<\frac{\pi }{4}$, where [x] denotes the greatest integer ≤ x is

If ${}^{n+2}{C}_8{:}^{n-2}{P}_4=57:16,$ find n.

Differentiate the following functions with respect to x :

$\frac{x^5-cosx}{sinx}$

Find the equation to the straight line which passes through the point (5, 6) and has intercepts on the axes

(i) equal in magnitude and both positive. (ii) equal in magnitude but opposite in sign.

The largest power of $2$ that divides $\frac{200!}{100!}$

Show that the line. through the points (4, 7, 8), (2, 3, 4) is parallel to the line through the points (-1, - 2,1), (1, 2, 5).

If |a+b| = |a-b| then (a,b) =

Evaluate the following:
(i) $sin{78}^{\circ }cos{18}^{\circ }-cos{78}^{\circ }sin{18}^{\circ }$
(ii) $cos{47}^{\circ }cos{13}^{\circ }-sin{47}^{\circ }sin{13}^{\circ }$
(iii) $sin{36}^{\circ }cos{9}^{\circ }+cos{36}^{\circ }sin{9}^{\circ }$
(iv) $cos{80}^{\circ }cos{20}^{\circ }+sin{18}^{\circ }sin{20}^{\circ }$

The mean and standard deviations of height and weights of 50 students of a class are as follows :

$\begin{array}{ccc}& \text{Weights}& \text{Heights}\\ \text{Mean}& 63.2\text{kg}& 63.2\text{inch}\\ \text{Standard deviation}& 5.6\text{kg}& 11.5\text{inch}\end{array}$

Which shows more variability, heights or weights ?

The value of $si{n}^2\left(\frac{\pi }{18}\right)+si{n}^2\left(\frac{\pi }{9}\right)+si{n}^2\left(\frac{7\pi }{18}\right)+si{n}^2\left(\frac{4\pi }{9}\right)$ is

A variable line having negative slope and passing through the point (8, 2) meets the axes at A and B. Find the minimum value of the sum OA+OB where O is the origin.

Let the $n$ terms of series is $\frac{1}{\mathrm{1.2.3.4}}+\frac{1}{\mathrm{2.3.4.5}}+\frac{1}{\mathrm{3.4.5.6}}+\dots \dots$ then

The area bounded by the curves $y=(x-1{)}^2,y=(x+1{)}^2$ and $y=\frac{1}{4}$

If \(a \epsilon {2, 4, 6, 9} and \(b \epsilon \{4, 6, 18, 27\}, then form the set of all ordered pairs (a, b) such that a divides b and a < b.

Write an equation repesenting a pair of lines through the point (a,b) and parallel to the coordinates axes.

If the length of perpendicular from origin to a line is $6$ units and the line makes an angle of ${30}^{\circ }$ with the positive $y-$axis (anti clockwise), then the equation of line is

A tank with rectangular base and rectangular sides, open at the top is to be constructed so that its depth is 2m and volume is $8m^3$. If building of tank costs Rs 70 per sq metre for the base and Rs 45 per sq metre for sides. What is the cost of least expensive tank ?

Find the values of the given expression.

$si{n}^{-1}\left(sin\frac{2\pi }{3}\right)$

For the division statement $65÷5=13$ , match the terms with their respective values :

gauss's law shd be invalid if ?

Without expanding the determinant, prove that

$\left|\begin{array}{ccc}a& a^2& bc\\ b& b^2& ca\\ c& c^2& ab\end{array}\right|=\left|\begin{array}{ccc}1& a^2& a^3\\ 1& b^2& b^3\\ 1& c^2& c^3\end{array}\right|$

Let $A=\left[\begin{array}{ccc}1& -2& 1\\ -2& 3& 1\\ 1& 1& 5\end{array}\right]$, verify that $\left(a\right)[adjA{]}^{-1}=adj(A^{-1})$

$\left(b\right)(A^{-1}{)}^{-1}=A$

Find the interval in which the following function is strictly incerasing or decreasing

$-2x^3-9x^2-12x+1$

Find the value of,

$ta{n}^{-1}(\frac{cosx-sinx}{cosx+sinx}$), for $0<x<\pi$.

Three rods made of same material and having the same cross-section have been joined as shown in figure. Each rod is of the same length. The ends are kept at $0^{\circ }C$, ${90}^{\circ }C$ & ${60}^{\circ }C$. The temperature of the junction of the rods will be