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The eccentricity the hyperbola whose centre is (6,2) one focus is (4,2) and of eccentricity 2 is

If 0 < A < $\frac{\pi }{2}$ the value of the expression $\frac{tanA}{1-cotA}+\frac{cotA}{1-tanA}$ -sec A cos ecA is equal to

What type of function is the sine function in R ?

If f(x) is differential function satisfying $6x{\int }_0^{1}f\left(xt\right)dt=2x^3-3x^2+6x+5$, then

In a triangle OAC, if B is the mid-point of side AC and $\underset{OA}{\to }=\underset{a}{\to },\underset{OB}{\to }=\underset{b}{\to }$, then what is $\underset{OC}{\to }$ ?

If x co-ordinates of a point P of line joining the points Q(2, 2, 1) and R(5, 2, -2) is 4, then the z-coordinates of P is
[RPET 2000]

The argument of $\frac{1-i\sqrt{3}}{1+i\sqrt{3}}$ is

$cos\frac{\pi }{15}cos\frac{2\pi }{15}cos\frac{4\pi }{15}cos\frac{7\pi }{15}=\frac{1}{16}$

A curve is represented parametrically by $x=t+e^{at}$ and $y=-t+e^{at},t\in R$ and $a>0$. Then, the curve touches the $x-$ axis at

The common tangents to the circles $x^2+y^2-6x=0$ and $x^2+y^2+2x=0$ is/are

If $sin\alpha +sin\beta =a$ and $cos\alpha -cos\beta =b$, then $tan\frac{\alpha -\beta }{2}$=

Let y = f(x) defined on R satisfies $(1+x^2)\frac{dy}{dx}$ = 2x - 2xy and f(0) = 2, then

${lim}_{x\to 0}\frac{\sqrt{1+x^2}-\sqrt{1+x}}{\sqrt{1+x^3}-\sqrt{1+x}}$

Find $\frac{dy}{dx}$ if $y=log\left(cos{e}^x\right)$ .

If $(1+ax+b{x}^2{)}^4=a_0+a_{1}x+a_2{x}^2+\dots +a_8{x}^8;a,b,a_0,a_1\dots a_8ϵR$ and are such that $a_0+a_1+a_2\ne 0$ and $\left|\begin{array}{ccc}{a}_0& a_1& a_2\\ a_1& a_2& a_0\\ a_2& a_0& a_1\end{array}\right|=0$, then

A man walks a distance of $3$ units from the origin towards the north -east $\left(N{45}^{\circ }E\right)$ direction. From there, he walks a distance of $4$ units towards the north -west $\left(N{45}^{\circ }W\right)$direction to reach a point $B$, then the position of $B$ in the Argand plane is

If ${}^n{C}_5={}^n{C}_{10}$, then the value of ${}^{16}{C}_n$ is

In a survey of 600 students in a school, 150 students were found to be taking tea and 225 taking coffee, 100 were taking both tea and coffee. Find how many students were taking neither tea nor coffee?

If |A| =n then | p ( P(A))| = ? Justify your answer.

Evaluate the following limits:

${lim}_{x\to \pi }\frac{1+cosx}{ta{n}^{2}x}$

If $a=cos\theta +isin\theta$, then $\frac{1+a}{1-a}$

What is the approximate value of $\sqrt[5]{5242.999}?$

Let $P=\left[\begin{array}{ccc}3& -1& -2\\ 2& 0& \alpha \\ 3& -5& 0\end{array}\right]$, where $\alpha \in R$. Suppose $Q=\left[q_{ij}\right]$ is a matrix such that $PQ=kI$, where $k\in R,k\ne 0$ and $I$ the identity matrix of order $3$. If $q_{23}=-\frac{k}{8}$ and $det\left(Q\right)=\frac{k^2}{2}$, then:

Prove that : 1 P(1,1) + 2.P(2,2) +3.P(3,3)+...+ n.P (n,n)= P(n+1, n+1)-1.

The angles A, B and C of a triangle ABC are in AP, if b : c = : $\sqrt{3}:\sqrt{2}$, then the angle A is

Solve the following system of linear equations, using matrix method

$5x-2y=3,3x+2y=5$

If functions $f:A\to B$ and $g:B\to A$ satisfy $gof=I_A$, then show that f is one-one and g is onto.

The circle $C_1:x^2+y^2=3$ having centre at origin,O intersects the parabola $x^2=2y$ at the point P in the first quadrant. Let the tangent to the circle $C_1$ at P touches other two circles $C_2$ and $C_3$ at $R_2$ and $R_3$, respectively. Suppose $C_2$ and $C_3$ have equal radii $2\sqrt{3}$ and centres $Q_2$ and $Q_3$, respectively. If $Q_2$ and $Q_3$ lie on the Y-axis then

If $A=\{y:|y|=3x,x\le 2,x\in N\},$
$B=\{x:x\text{is a positive even number},x<14\}$ and $C=\{1,4,8\},$ then $n\left(\right(A\times B)\cap (A\times C\left)\right)$ is

Three fair and unbiased dice and rolled at a time. The probability that the numbers shown are totally different is.

For the curve $y=4x^3-2x^5$ find all the points at which the tangent passes throught the origin.

If the derivative of the function $f\left(x\right)=\{\begin{array}{cc}b{x}^2+ax+4;x\ge -1& \\ a{x}^2+b;x<-1^{\prime }& \end{array}$ is continuous everywhere. Then

The equation of the tangent at the vertex of the parabola $x^2+4x+2y=0$ is

Out of $60$ students in a class, anyone who has chosen to study maths elects to do physics as well. But no one does maths and chemistry, $16$ do physics and chemistry. All the students do at least one of the three subjects and the number of people who do exactly one of the three is more than the number who do more than one of the three. Then the range of cardinal number of students who could have done only chemistry is

Suppose X is a binomial variable B (5, p) and P(X =2) = P (X = 3), then p is equal to

The solution of differential equation $\frac{xdy}{dx}+y=xlogx$ is

If 1 is considered almost perfect. How many minimum value qualities are required for the expression ${\left(\frac{1+i}{1-i}\right)}^m$ to be almost perfect? Mention four values of qulities.

A natural number is chosen at random from the first 100 natural numbers. Then the probability, for the in-equation $x+\frac{100}{x}>50$ satisfied, is

Can our answer vary while calculating inverse of a matrix

Prove that $\frac{\text{sin}(x+y)}{\text{sin}(x-y)}=\frac{\text{tan}x+\text{tan}y}{\text{tan}x-\text{tan}y}$

The number of solutions to
$\sin \left(\pi {\sin }^2\right(\theta \left)\right)+\sin \left(\pi {\cos }^2\right(\theta \left)\right)=2\cos \left(\frac{\pi }{2}\cos \right(\theta \left)\right)$
satisfying $0\le \theta \le 2\pi$ is