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The equation of the diameter of the circle $x^2+y^2+2x-4y-4=0$ which is parallel to $3x+5y-4=0$ is

If the point $(\alpha ,{\alpha }^2)$ lies between $x+y-2=0$ and $4x+4y=3,$ then the range of $\alpha$ is

For each $x\in R$, let $\left[x\right]$ be the greatest integer less than or equal to $x$. Then $\underset{x\to 0{}^{-}}{lim}\frac{x\left(\right[x]+|x|)\sin \left[x\right]}{|x|}$ is equal to :

The value of $\frac{2+3i}{4+5i}$ is

If p + q + r = a + b + c = 0, then the determinant

$\Delta =\left|\begin{array}{ccc}pa& qb& rc\\ qc& ra& pb\\ rb& pc& qa\end{array}\right|$ equals

The domain of the function $f\left(x\right)={\left[{\log }_{10}\left(\frac{5x-x^2}{4}\right)\right]}^{1/2}$ is

The abscissa of the point on the curve $yx=(x+a{)}^2$ at which the normal cuts off numerically equal intercepts on the coordinate axes is

If the lines $x+y=|a|$ and $ax-y=1$ intersect each other in the first quadrant, then the range of $a$ is

The angle of intersection of the normals at the point $(-\frac{5}{\sqrt{2}},\frac{3}{\sqrt{2}})$ of the curves $x^2-y^2=8$ and $9x^2+25y^2=225$ is

If $y=\sum _{r=1}^x{\tan }^{-1}\left(\frac{1}{1+r+r^2}\right),$ then $\frac{dy}{dx}$ at $x=2$ is equal to

A variable plane at a distance of $1$ unit from the origin cuts the coordinates axes at $A,B$ and $C.$ If the centroid $D(x,y,z)$ of triangle $ABC,$ satisfies the relation $\frac{1}{x^2}+\frac{1}{y^2}+\frac{1}{z^2}=k,$ then the value of $k$ is

If $a=4\hat{i}+6\hat{j}$ and $b=3\hat{j}+4\hat{k}$, then the vector form of the component of a along b is

If $n$ is an odd integer, then $(1+i{)}^{6n}+(1-i{)}^{6n}=$

If $n\left(A\right)=50,n(A\cup B)=80,n(A\cap B)=30$, then $n(B–A)=$

Number of real solutions of the equation $|||x^2-1|-1|+3|=1$ is

Let $P\left(z\right)=z^3+a{z}^2+bz+c,$ where $a,b,c\in R$. If there exists a complex number $w$ such that the three roots of $P\left(z\right)$ are $w+3i,w+9i$ and $2w-4$, where $i^2=-1$, then the value of $a+b+c$ is

​​​​​​​(correct answer + 1, wrong answer - 0.25)

The circles $z\overline{z}+z\overline{a_1}+a_1\overline{z}+b_1=0,b_1\in R$ and $z\overline{z}+z\overline{a_2}+\overline{z_2}{a}_2+b_2=0,b_2\in R$ will intersect orthogonally if

The centre of the circle $z\overline{z}-(2+3i)z-(2-3i)\overline{z}+9$ = 0 is

The locus of the vertices of the family of parabola $y=\frac{a^3{x}^2}{3}+\frac{a^{2}x}{2}-2a$, $a$ being parameter is :

If $\alpha ,\beta ,\gamma$ are the roots of $x^3+3x^2+4x+5=0$ , then which of the following is/ are true.

Three positive numbers form an increasing GP. If the middle term of the GP is doubled, then new numbers are in AP. Then, the common ratio of the GP is ___.

If $\alpha$ and $\beta (\alpha <\beta )$ are the roots of the equation $x^2+bx+c=0$, where c < 0 < b, then

If $A=\{0,1,2,3,5\},B=\{0,1,2,3\},C=\{0,1,4,5\}$, then $n\left(A\Delta B\right)+n\left(B\Delta C\right)+n\left(C\Delta A\right)$ is equal to

Two sides of a parallelogram are along the lines, $x+y=3$ and $x-y+3=0$. If its diagonals intersect at $(2,4)$ then one of its vertex is :

The sides $AC$ and $AB$ of a $△ABC$ touches the conjugate hyperbola of the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ at $C$ and $B.$ If the vertex $A$ lies on the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1,$ then the side $BC$ always touches

The points (5, – 4, 2), (4, –3, 1), (7, – 6, 4) and (8, –7, 5) are the vertices of [RPET 2002]

The coefficient of $x^3$ in the expansion of $\frac{(1+3x{)}^2}{1-2x}$

will be

In bridge game of playing cards, 4 players are distributed one card each by turn so that each player gets 13 cards. Find out the probability of a specified player getting a black ace and a king.

If ${\log }_{a}x>y$ and $0<a<1$ . Then

The locus of the image of the focus of the ellipse $\frac{x^2}{25}+\frac{y^2}{9}=1$ with respect to any of the tangents to the ellipse is

Let $c_1:x^2+y^2=1;C_2:(x-10{)}^2+y^2=1and{C}_3;x^2+y^2-10x-42y+457=0$ be three circle.A circle C has been drawn to touch circles $C_1$ and $C_2$ externally and $C_3$ internally. Now circles $C_1,C_2$ and $C_3$ start rolling on the circumference of circle C in anticlockwise direction with constant speed. The centroid of the triangle formed by joining the centres of rolling circles $C_1,C_2$ and $C_3$ lies on

If in a triangle a = $\sqrt{3}+1,b=\sqrt{3}-1,C={60}^{\circ }$ then the value of A is ?

If $A$ and $B$ are complemetary angles, then $\sqrt{\frac{\cos A}{\sin B}}-\cos A\sin B=$

The difference between fourth and first term of a $G.P.$ is $52$ and sum of the first three terms is $26$. If $T_n$ and $S_n$ represent $n^{th}$ term and sum to $n$ terms respectively of the $G.P.$, then which of the following is/are true

If $\overrightarrow{a}=\hat{i}+\hat{j}+\hat{k},\overrightarrow{b}=2\hat{i}-\hat{j}+\hat{k}$ and $\overrightarrow{c}=\hat{i}+x\hat{j}+y\hat{k}$, are linearly dependent and $|\overrightarrow{c}|=\sqrt{3}$ then $(x,y)$ is

The circle $x^2+y^2=4$ cuts the line joining the points A(1, 0) and B(3, 4) in two points P and Q. Let $\frac{BP}{PA}=\alpha$ and

$\frac{BQ}{QA}=\beta$. Then $\alpha and\beta$ are roots of the quadratic equation

The shortest distance between the point $(\frac{3}{2},0)$ and the curve $y=\sqrt{x},(x>0),$ is :

If $m$ is the minimum value of $k$ for which the function $f\left(x\right)=x\sqrt{kx-x^2}$ is increasing in the interval $[0,3]$ and $M$ is the maximum value of $f$ in $[0,3]$ when $k=m,$ then the ordered pair $(m,M)$ is equal to :

The maximum value of $2{\cos }^{4}x+2{\cos }^4({90}^{\circ }-x)$ is

If $p=(8+3\sqrt{7}{)}^n$ and $f=p-\left[p\right]$, then

(where [.] denotes the greatest integer function)