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If the two equations $x^2-cx+d=0$ and $x^2-ax+b=0$ have one common root and the second has equal roots, then $2(b+d)=$)

In $△ABC,\angle A={\tan }^{-1}2$ and $\angle B={\tan }^{-1}3$ and the length of side opposite to the smallest angle is $2\sqrt{5}.$ If perimeter of the triangle is $\sqrt{l}+\sqrt{m}+\sqrt{n}$ and circumradius is $\sqrt{k},$ then

Which of the following gives the equation of director circle of the ellipse $\frac{x^2}{25}+\frac{y^2}{16}=1$?

The number of solutions of the equation $1+{\sin }^{4}x={\cos }^{2}3x,x\in [-\frac{5\pi }{2},\frac{5\pi }{2}]$ is :

Which among the following is/are skew hermitian matrix.

The equations of the tangents to the ellipse $x^2+16y^2=16,$ each one of which makes an angle of ${60}^{\circ }$ with the $x-$axis, is

If $f\left(x\right)=x^2$ and $g\left(x\right)=2^x$, then the solution set of the equation $f\left(2^x\right)=g\left(x^2\right)$ is

ABC is an isosceles triangle inscribed in a circle of radius r. If AB = AC and h is the altitude from A to BC.The triangle ABC has perimeter $P=2[\sqrt{(2hr-h^2)}+\sqrt{2hr}]$ and A be the area of the triangle .Find $\underset{h\to 0}{lim}\frac{A}{P^3}$

If $4{\sin }^2\theta +2(\sqrt{3}+1)\cos \theta =4+\sqrt{3}$, then the general solution is

Which of the following cases may lead to non trivial solutions in case of system of linear equations according to Cramer's rule Convention?

If $x^2+y^2-2by+ac=0$ is the equation of a point circle, then $a,b,c$ are in

(where $a,b,c$ are positive real numbers)

The parametric equation of parabola $(y-2{)}^2=12(x-4)$ is

If $f:R\to R$ and $g:R\to R$ are given by f(x) = |x| and g(x) = [x], then $g\left(f\right(x\left)\right)\le f\left(g\right(x)$ is true for -

If ${\log }_{x-2}6+{\log }_{x+2}6>{\log }_{x-2}6\cdot {\log }_{x+2}6$, then $x\in$

The locus of the foot of perpendicular drawn from the centre of the ellipse $x^2+3y^2=6$ on any tangent to it is:

The number of ordered pairs $(x,y)$ satisfying the equation $x^2+2x\sin \left(xy\right)+1=0$ is

$($where $y\in [0,2\pi ])$

The minors of - 4 and 9 and the co-factors of - 4 and 9 in determinant $\left|\begin{array}{ccc}-1& -2& 3\\ -4& -5& -6\\ -7& 8& 9\end{array}\right|$ are respectively

The line $\frac{x}{a}+\frac{y}{b}=1$ moves in such a way that $\frac{1}{a^2}+\frac{1}{b^2}=\frac{1}{c^2}$ where $c$ is a constant. The locus of foot of perpendicular from the origin on the given line will be

If both the roots of $x^2+2(a+2)x+9a-1=0$ are negative, then ${}^{\prime }{a}^{\prime }$ lies in

POQis a straight line through the origin O,P and Q represent the complex numbers a+ib andc+id respectively and OP=OQ, then

If the roots of the equation $a{x}^2+bx+c=0$ are reciprocal of the roots of the equation $p{x}^2+qx+r=0$, then which of the following options is always correct?

If $-3<\frac{2x-1}{3}\le 5$, then $x$ lies in the interval

Two sides of a rhombus are along the lines x-y+1=0 and 7x-y-5=0. If its diagonals intersect at (-1, -2), then which one of the following is a vertex of this rhombus?

The differential equation for a family of curves is $\frac{\mathrm{dy}}{\mathrm{dx}}=\frac{y}{2x}.$ What is the differential equation for the orthogonal trajectory of the curves?

The area of the pentagon formed by the vertices $(1,2),(4,1),(5,3),(3,7),(2,6)$ is

If the pair of straight lines $\sqrt{3}xy-x^2=0$ is tangent to the circle at $P$ and $Q$ from origin $O$ such that area of the smaller sector formed by $CP$ and $CQ$ is $3\pi$ sq. unit, where $C$ is the centre of circle, then $OP$ equals to

$\frac{1-i}{1+i}$ is equal to

The value of $\theta$ which satisfy the equation $3{\tan }^2\theta +3\tan \theta -\cot \theta =1$ (where $n\in Z$) can be

The locus of midpoint of the chord of contact of $x^2+y^2=2$ from the points on $3x+4y=10$ is a circle whose centre is

If $(x+2),3,5$ are the lengths of sides of a triangle, then $x$ lies in

If x = $e^{y+e^{y+e^{y+e^{y+...\infty }}}}$, then $\frac{dy}{dx}$ is

For any set $M$ if $M\cup \varnothing =\varnothing ,$ then

The diagram shown below represents the interval

The locus of the centre of the circle which cuts $x^2+y^2-20x+4=0$ orthogonally and touches the line $x=2,$ is

$\text{The values of constants a and b so that}\underset{x\to \infty }{lim}(\frac{x^2+1}{x+1}-ax-b)=\frac{1}{2},are$

If A has 3 elements and P(A) denotes the power set of A, then the number of elements in P(P(A)) is

The coefficient of $x^{18}$ in the product $(1+x)(1-x{)}^{10}(1+x+x^2{)}^9$

The number of ways of selecting $15$ teams from $15$ men and $15$ women such that each team consists of a man and a woman, is

If ${\log }_{10}(x^3+y^3)-{\log }_{10}(x^2-xy+y^2)\le 2\forall x>0,y>0$, then the maximum value of $x+y$ is

If $A=\left|\begin{array}{cc}\alpha & 2\\ 2& \alpha \end{array}\right|and|A^3|=125$, then the value of $\alpha$ is

If $y=mx+c$ touches the parabola $y^2=4a(x+a)$, then