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Evaluate the following limits:

${lim}_{x\to 0}\frac{sin3x}{5x}$

If the tangent to the ellipse $x^2+4y^2=16$ at $P\left(\theta \right)$ is normal to circle $x^2+y^2-8x-4y=0$. Then $\theta$ equals to

How many of following are matched correctly

Let $C$ be the circle with centre at $(1,1)$ and radius = $1$. If $T$ is the circle centred at $(0,y)$, passing through origin and touching the circle $C$ externally, then the radius of $T$ is equal to:

Suppose the probability for A to win a game against B is 0.4. If A has an option of playing either a "best of 3 games" or a "best of 5 games" match against B, which option should A choose so that the probability of his winning the match is higher ? (No game ends in a draw)

If a relation $R$ defined on $A=\{1,3,5,7\}$, then which of the following is/are void relation?

${lim}_{x\to }\frac{e^x-x-1}{x}$

For a positive integer n, find the value of $(1-i{)}^n{(1-\frac{1}{i})}^n$

${lim}_{x\to \frac{\pi }{4}\frac{\sqrt{cosx}-\sqrt{sinx}}{x-\frac{\pi }{4}}}$

Let $f:R\to R$ be a function defined by $f\left(x\right)=\{\begin{array}{cc}{x}^2+2mx-1,& x\le 0\\ mx-1,& x>0\end{array}.$ If $f$ is one-one, then $m$ can be

The tangent and normal to the ellipse $3x^2+5y^2=32$ at the point $P(2,2)$ meet the $x-$axis at $Q$ and $R$, respectively. Then the area (in sq. units) of the triangle $PQR$ is :

The solution set of $x^2+1<10$ is

When a polynomial $p\left(x\right)$ is divided by $x-2,$ the remainder is $7.$ When $p\left(x\right)$ is divided by $x-3,$ the remainder is $9.$ If $r\left(x\right)$ is the remainder when $p\left(x\right)$ is divided by $(x-2)(x-3),$ then the value of $r(-1)$ is

The equation of the normal to the curve $y=-\sqrt{x}+2$ at the point of its intersection with the bisector of the first quadrant is :

Find the absolue maximum value and the absolute minimum value of the following function in the given intervals:

$f\left(x\right)=sinx+cosx,xϵ[0,\pi ]$

Find the value of λ if the point (3, 5) lies inside the circle $x^2+y^2+6x+\lambda y+5=0$

If $x^2-x-2$ is factor of $x^4-\lambda x^2-\mu$, then $\sqrt{({\lambda }^2-{\mu }^2)}$ equals.

Solve: $t^2+2t+1=0$

A particle is projected over a triangle from one extremity of its horizontal base. Grazing over the vertex, it falls on the other extremity of the base. If $\alpha$ and $\beta$ be the base angles of the triangle and $\theta$ the angle of projection, then find the relation between the angles.

If x, y, z > =, then find the value of xy + yz + zx.

What is the maximum value of sin(sinx)

If $\alpha$ and $\beta$ are the roots of the equation $x^2-2x+2=0$, then least value of $n$ for which ${\left(\frac{\alpha }{\beta }\right)}^n=1$ is:

Let $F_1(x_1,0)$ and $F_2(x_2,0),$ for $x_1<0$ and $x_2>0,$ be the foci of the ellipse $\frac{x^2}{9}+\frac{y^2}{8}=1.$ Suppose a parabola having vertex at the origin and focus at $F_2$ intersects the ellipse at point $M$ in the first quadrant and at point $N$ in the fourth quadrant.

The orthocentre of the triangle $F_{1}MN$ is

The upper part of a tree broken over by the wind makes an angle of ${30}^{\circ }$ with the ground and the distance from the root to the point where the top of the tree touches the ground is 15 m. Using sine rule, find the height of the tree.

Why is 666^{2 is called devil's number?}

If all permutations of the letters of the word APPLE are arranged as in the dictionary, what is the 35^th word?

Number of words that can be formed by taking $4$ letters at a time out of the letters of the word $MATHEMATICS$ is

The least value of the expression $2{\log }_{10}x-{\log }_x\left(0.01\right)$, for $x>1$, is

The value of $\frac{\cos 15\pi +\tan \left(\frac{5\pi }{4}\right)}{\sin 16\pi +\sec 6\pi }$ is

Let $\alpha ,\beta$ be the roots of $x^2+3x+5=0$ then the equation whose roots are $-\frac{1}{\alpha }$ and $-\frac{1}{\beta }$ is :

If $A,B,C$ are the three angles in a triangle such that $2\sin B\sin (A+B)-\cos A=1$ and $2\sin C\sin (B+C)-\cos B=0$, then

If a vertex of a triangle is (1, 1) and the middle points of two sides passing through it are (–2, 3) and (5, 2), then find coordinates of centroid of the triangle.

$\text{At what point the origin be shifted so that the equation}{x}^2+xy-3x-y+2=0\text{does not contain any first degree term and constant term ?}$

Let f and g be two functions given by f = {(2, 4), (5, 6), (8, -1), (10, -3)} and g = {(2, 5), (7, 1), (8, 4), (10, 13), (11, -5)}
Find the domain of f + g.

A fair die is rolled. The probability that the first time 1 occurs at an even throw, is