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For a quadratic equation $a{x}^2+bx+c=0,a\ne 0$, if $c=0$ then the roots of the equation are

Solve the given inequality for real x: 3x – 7 > 5x – 1

A circle with radius $2$ units passing through origin, cuts the $x-$ axis and $y-$ axis at $A$ and $B$ respectively. The locus of centroid of the triangle $OAB$ is

Find the principal values of the following questions:

$co{s}^{-1}(-\frac{1}{\sqrt{2}})$

Let $\alpha ,\beta$ (a < b) be the roots of the equation .If $a{x}^2+bx+c=0.If{lim}_{x\to m}\frac{|a{x}^2+bx+c|}{a{x}^2+bx+c}=1,then$

An electronic assembly consists of two subsystems, say A and B. From previous testing procedures, the following probabilities are assumed to be known

P(A fails)= 0.2, P(B fails alone) =0.15, P(A and B fail)=0.15

Evaluate the following probabilities

P (A fails/B has failed )

P(A fails alone)

In a $\Delta ABC$. If $\tan \frac{A}{2},\tan \frac{B}{2},\tan \frac{C}{2}$ are in $H.P$, then $a,b,c$ are in:

The set of all values of $k>-1,$ for which the equation $(3x^2+4x+3{)}^2-(k+1\left)\right(3x^2+4x+3\left)\right(3x^2+4x+2)+k(3x^2+4x+2{)}^2=0$ has real roots, is

Number of binary operations on the set {a,b} are
(a)10
(b)16
(c)20
(d)8

If a sequence $\left\{a_n\right\}$ is defined as

$a_n=\{\begin{array}{c}\frac{1}{n},\text{when}n\text{is odd}\\ -\frac{1}{n^2},\text{when}n\text{is even}\end{array}$

where $n\in N$, then the sum of first four terms is

Three collinear vectors $\overrightarrow{a},\overrightarrow{b}$ and $\overrightarrow{c}$ are such that $x^2\cdot \overrightarrow{a}-5x\cdot \overrightarrow{b}+6\overrightarrow{c}=\overrightarrow{0},$ then the value(s) of $x$ is(are)

If the circle $C_1:x^2+y^2=16$ intersects another circle $C_2$ of radius $6$ in such a manner that their common chord is of maximum length and has slope $=\frac{1}{2}$. Then the co-ordinates of the centre of the circle(s) $C_2$ is/are

In a $△ABC$ , if $\tan A\tan B=3$, then the value of $\frac{\cos C}{\cos A\cos B}$ is

The product of non-real roots of the equation $(x^2+x-2)(x^2+x-3)=12$ is

If ${\log }_3{\left(\frac{2\sin 2x}{3}\right)}^2+1=0,x\in [0,2\pi ],$
then which among the following value(s) of $x$ satisfying the above equation

If $I=\int \frac{8x-11}{\sqrt{5+2x-x^2}}dx=p\sqrt{5+2x-x^2}+q{\sin }^{-1}\left(\frac{x-1}{\sqrt{6}}\right)+C,$ then the value of $|p+q|$ ;$(p,q\in R)$ is

(where $C$ is integration constant)

Find the angle between
the planes whose vector equations are

and
.

Evaluate the definite integrals.
${\int }_0^1\frac{2x+3}{(5x^2+1)}dx.$

Let $f\left(x\right),g\left(x\right)$ and $h\left(x\right)$ be polynomials of degree $4$ and satisfy the equation $\left|\begin{array}{ccc}f\left(x\right)& g\left(x\right)& h\left(x\right)\\ a& b& c\\ p& q& r\end{array}\right|=l{x}^4+m{x}^3+n{x}^2+4x+1$ where $a,b,c,p,q,r,l,m,n$ are real constants. Then the value of $\left|\begin{array}{ccc}{f}^{\prime \prime \prime }\left(0\right)-f^{\prime \prime }\left(0\right)& g^{\prime \prime \prime }\left(0\right)-g^{\prime \prime }\left(0\right)& h^{\prime \prime \prime }\left(0\right)-h^{\prime \prime }\left(0\right)\\ a& b& c\\ p& q& r\end{array}\right|$ is

For the curve y = 3sin $\theta cos\theta ,x=e^{\theta }sin\theta ,0\le \theta \le \pi$; tangent is parallel to x-axis, then $\theta$ =

Which of the following identifier names are invalid and why?

If $\alpha =\underset{x\to \pi /4}{lim}\frac{{\tan }^{3}x-\tan x}{\cos (x+\frac{\pi }{4})}$ and $\beta =\underset{x\to 0}{lim}{\left(\cos x\right)}^{\cot x}$

are the roots of the equation, $a{x}^2+bx-4=0$, then the ordered pair $(a,b)$ is

$Ifsecxcos5x+1=0,where0<x\le \frac{\pi }{2},\text{find the value of}$

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If $X=1+a+a^2+....\infty$, where |a| < 1 and $y=1+b+b^2+.....\infty$ where |b| <1,

prove that $1+ab+a^2{b}^2+.....\infty \frac{xy}{x+y-1}$

In a $△ABC$, if $\cos A\cos B\cos C=\frac{\sqrt{3}-1}{8}$ and $\sin A\sin B\sin C=\frac{3+\sqrt{3}}{8}$, then the angles of the triangle are

Let $x$ denote the total number of one-one functions from a set $A$ with $3$ elements to a set $B$ with $5$ elements and $y$ denote the total number of one-one functions from the set $A$ to the set $A\times B$. Then:

The equation of the normals to the curve $y=2x^3+2x$ which are parallel to $2x+16y=7$ is