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If the equation $x^3-8x^2+cx+d=0\forall c,d\in R$ has one complex root and one positive root, then select the correct statement.

A, B, C in order cut a pack of cards, replacing them after each cut, on the condition that the first who cuts a spade shall win a prize; find their respective chances.

The separate equations of the asymptotes of rectangular hyperbola $x^2+2xy\cot 2\alpha -y^2=a^2$ are :

The set of all points where the function $f\left(x\right)=\frac{x}{1+|x|}$ is differentiable is

If $\int \frac{dx}{2\sin x-\cos x+5}=A{\tan }^{-1}\frac{f\left(x\right)}{\sqrt{5}}+C,$ for a fixed value of $A$ and function $f\left(x\right)$. Then

(where $C$ is integration constant)

$\frac{d}{dx}\left(\frac{a+bsinx}{b+asinx}\right)=$

Two dice are thrown. The probability that the sum of numbers coming up on them is 9, if it is known that the number 5 always occurs on the first die, is

If $\sum _{i=1}^n(x_i-a)$ and $\sum _{i=1}^n(x_i-a{)}^2=na,(n,a>1)$ then the standard deviation of $n$ observations $x_1,x_2,\cdots ,x_n$ is :

Six non collinear points in a plane be joined in all possible ways by indefinite straight lines, and if no two of them be coincident or parallel, and no three pass through the same point (with the exception of the original 6 points). The number of distinct points of intersection is equal to ___________.

Find the equation of the circle concentric with
$x^2+y^2-4x-6y-3=0$
and which touches the y-axis.

$se{c}^2\theta =\frac{4xy}{(x+y{)}^2}$ is true if and only if

Two straight line intersect at a point $O.$ Points $A_1,A_2,...A_n$ are taken on a line and points $B_1,B_2,...B_n$ are taken on the other line. If the point $O$ is not to be used, then number of triangles that can be drawn using these points as vertices, is

If the ellipse $\frac{x^2}{4}+y^2=1$ meets the ellipse $x^2+\frac{y^2}{a^2}=1$ in four distinct points and $a=b^2-5b+7,$ then $b$ does not lie in

The least positive integer n for which $\sqrt[3]{3n+1}-\sqrt[3]{3n}<\frac{1}{12}$ is

The area bounded between the parabolas $x^2=\frac{y}{4}$ and $x^2=9y$ and the straight line $y=2,$ is

If the line $x+y=1$ touches the parabola $y^2-y+x=0$, then the coordinates of the point of contact are

If $\int \frac{\cos \theta }{5+7\sin \theta -2{\cos }^2\theta }d\theta =A{\log }_e|B\left(\theta \right)|+C,$ where $C$ is a constant of integration, then $\frac{B\left(\theta \right)}{A}$ can be:

Let f(x) = 2x + 5 and $g\left(x\right)=x^2+x$. Describe
(i) f + g
(ii) f - g
(iii) $\frac{f}{g}$. Find the domain in each case.

Find the matrix X so that X $\left[\begin{array}{ccc}1& 2& 3\\ 4& 4& 6\end{array}\right]=\left[\begin{array}{ccc}-7& -8& -9\\ 2& 4& 6\end{array}\right]$.

The values of $x^2$ for x > -3 is set of

Prove the
following by using the principle of mathematical induction for all

(2n +7)
< (n + 3)²

If $\alpha ,\beta$ be the roots of the equation $u^2-2u+2=0$ and if $\cot \theta =x+1$, then $\frac{\left[\right(x+\alpha {)}^n-(x+\beta {)}^n]}{[\alpha -\beta ]}$ is equal to

If $\overrightarrow{x},\overrightarrow{y}$ are two non-zero and non-collinear vectors satisfying $\left[\right(a-2){\alpha }^2+(b-3)\alpha +c]\overrightarrow{x}+\left[\right(a-2){\beta }^2+(b-3)\beta +c]\overrightarrow{y}+\left[\right(a-2){\gamma }^2+(b-3)\gamma +c](\overrightarrow{x}\times \overrightarrow{y})=\overrightarrow{0},$ where $\alpha ,\beta ,\gamma$ are three distinct real numbers, then which of the following statement(s) is/are correct ?

The expansion of $\frac{e^{7x}-e^x}{e^{4x}}$ is equal to