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Evaluate the following limit:
$\underset{x\to 2}{\text{lim}}\frac{3x^2-x-10}{x^2-4}$

If $y=\frac{a{x}^2}{(x-a)(x-b)(x-c)}+\frac{bx}{(x-b)(x-c)}+\frac{c}{x-c}+1,$ then $\frac{y^{\prime }}{y}$ is equal to
$(\text{Here},y^{\prime }=\frac{dy}{dx})$

If f(x) has a derivative at x=a, then ${lim}_{x\to a}\frac{xf\left(a\right)-af\left(x\right)}{x-a}$ is equal to

Let $\omega$ be a complex cube root of unity with $\omega \ne 1$ and $P=\left[p_{ij}\right]$ be a $n\times n$ matrix with $p_{ij}={\omega }^{i+j}$. Then $P^2\ne 0$, when $n=$

The length of line segment joining -1-i and 2+3i ?

If a function $f:R\to R$ be defined by
f(x) =3x-2, x<0
1,x = 0
4x+1, x>0
Find : f(1), f(-1), f(0), f(2)

If $f(x+y)=f\left(x\right)+f\left(y\right)-xy-3$ and $f\left(1\right)=3$, then the number of solutions for $f\left(n\right)=3n$, where $n\in N$ is

The equation of locus of a point where distance from the y-axis is equal to its distance from the point A(2,1,-1) is-

A square is inscribed in the circle $x^2+y^2-2x+4y-93=0$ with the sides parallel to the coordinate axes. The coordinate of the vertices are-

Find the distance of
the point (−1, −5, −­10) from the point of
intersection of the line
and
the plane.

Out of the given equations, is not a quadratic equation.

If $(x-2{)}^2-5|x-2|+6=0$, then $x\in$

If $c$ is a point at which Rolle's theorem holds for the function, $f\left(x\right)={\log }_e\left(\frac{x^2+\alpha }{7x}\right)$ in the interval $[3,4]$, where $\alpha \in R$, then $f^{\prime \prime }\left(c\right)$ is equal to:

The points A(2,0),B(9,1),C(11,6) and D(4,4) ar the vertices of a quadrilateral ABCD.Determine whether ABCD is a rhombus or not.

If the centroid and a vertex of an equilateral triangle are $(2,3)$ and $(4,3)$ respectively, then the other two vertices of the triangle are

How is Sonia related to the man in the photograph ?

Statement I. Man in the photograph is the only son of Sonia's grandfather

Statement II. The man in the photograph has no brothers or sisters and his father is Sonia’s grandfather.

Number of ways in which 15 indistinguishable oranges can be distributed in 3 different boxes so that every box R have atmost 8 oranges, are

Three poles $A,B$ and $C$ are in a straight line, apart by $10$ metres each. The height of pole $A$ is $20$ metres and the angle of depression from the top of pole $A$ to the top of pole $B$ is ${60}^{\circ }$ and the angle of elevation from the top of pole $B$ to the top of pole $C$ is ${30}^{\circ }.$ The height (in metres) of pole $C$ is

If the ordinates of the points $P$ and $Q$ on the parabola $y^2=12x$ are in the ratio $1:2,$ then the locus of the point of intersection of normals to the parabola at $P$ and $Q$ is

If $\alpha ,\beta$ are the roots of the equation $2x^2-3x-6=0$, then the equation whose roots are ${\alpha }^2-1$ and ${\beta }^2-1$ is

The area bounded by the y-axis,
y = cos x and y = sin x when

A.

B.

C.

D.

Let $M$ and $m$ respectively be the maximum and minimum values of the function $f\left(x\right)={\tan }^{-1}(\sin x+\cos x)\text{in}[0,\frac{\pi }{2}].$ Then the value of $\tan (M-m)$ is equal to

Which of the following condition is true for a monotonically increasing function f(x), which is differentiable in its domain?

Let $PQR$ be a triangle in which $PQ=3$. From the vertex $R$, draw the altitude $RS$ to meet $PQ$ at $S$. Assume that $RS=\sqrt{3}$ and $PS=QR$. Then $PR$ equals

If $I_n={\int }_0^{\frac{\pi }{4}}{\tan }^{n}x\mathrm{dx}$, then the value of $I_9+I_{11}$ is

The solution of the equation tan 2θ = tan 2/θ is

If $\tan \theta =\sqrt{3}$ and 'θ' lies in the third quadrant, then $\sin \theta +\cos \theta$ is

If $n$ is the number of irrational terms in the expansion of ${(3^{\frac{1}{4}}+5^{\frac{1}{8}})}^{60}$, then $(n-1)$ is divisible by:

$\sin 2A=2\sin A$ is true when $A=$