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If y = $sec\left(ta{n}^{-1}x\right)$, then $\frac{dy}{dx}$ at x = 1 is equal to:

What should be the relation between range and codomain of a function, for a function to be an into function -

Numbers are selected at random, one at a time, from the two-digit numbers 00, 01, 02,..., 99 with replacement. An event E occurs if and only if the product of the two digits of a selected number is 18. If four numbers are selected, find probability that the event E occurs at least 3 times.

If ${\left(\frac{1+i}{1-i}\right)}^m$ = 1 then the least integral value of m is

The determinant $\left|\begin{array}{ccc}1& ab& \frac{1}{a}+\frac{1}{b}\\ 1& bc& \frac{1}{b}+\frac{1}{c}\\ 1& ca& \frac{1}{c}+\frac{1}{a}\end{array}\right|$ is equal to

The range of the function $f\left(x\right)=\frac{2+x}{2-x},x\ne 2$ is

The possible values of a for which the point $(a,a^2)$ lies inside the triangle formed by the straight lines 2x + 3y – 1 = 0, x + 2y = 3 and 5x – 6y – 1 = 0 is

Value of ${\int }_0^5(\sqrt{x+2\sqrt{x+1}}+\sqrt{x-2\sqrt{x-1}})dx$ is

If p is the probability that a man aged x years will die in a year, find the probability that out of n men $A_1,A_2,...A_n$ each aged x years, $A_1$ will die in a year and will be the first to die.

p + iq > r + it is meaningful only when

A(0,6), B(8,12), C(8,0) are the Co-ordinate of vertices of triangle ABC. Then...... $\begin{array}{cc}1.Co-ordinateofcentroid& p.(20,6)\\ 2.Co-ordinateofIncentre& q.(0,16)\\ 3.Co-ordinateofExcentre& r.\left(\frac{16}{3}\right),6)\\ & s.(0,-4)\\ & t.(5,6)\end{array}$

Let z and w be the two non-zero complex numbers such that $|z|=|w|$ and $argz+argw=\pi$ . Then z is equal to

Find the intervals in which the function f given by $f\left(x\right)=2x^3-3x^2-36x+7$ is

a) strictly increasing

b) strictly decreasing

${lim}_{x\to 0}\frac{(a+h{)}^{2}sin(a+h)-a^{2}sina}{h}$

$\mathrm{If}f=\left(\begin{array}{ccc}3& 1& 2\\ x& 2& a^2\\ \frac{x^2}{2}& 2x& \frac{x^3}{2}\end{array}\right],\mathrm{Then}\mathrm{the}\mathrm{value}\mathrm{of}f\text{'}\mathrm{at}x=a\mathrm{is}\mathrm{given}\mathrm{as}\mathrm{Where},f\text{'}=\frac{df}{dx}$

Compute mean deviation from mean of the following distribution :

$\begin{array}{ccccccccc}\text{Marks}& 10-20& 20-30& 30-40& 40-50& 50-60& 60-70& 70-80& 80-90\\ \text{No. of students}& 8& 10& 15& 25& 20& 18& 9& 5\end{array}$

$\int e^x\left(\frac{1-cotx}{sinx}\right)dx$

Tangents $PA$ and $PB$ are drawn to the circle $(x-4{)}^2+(y-5{)}^2=4$ from the point $P$ on the curve $y=\sin x$, where $A$ and $B$ lie on the circle. Consider the function y = $f\left(x\right)$ represented by the locus of the centre of the circumcircle of triangle $PAB$, then
Range of $y=f\left(x\right)$ is

${lim}_{n\to \infty }\frac{1^2+2^2+3^2\cdots n^2}{n^3}$ is equal to

The last non-zero digit in 20! is ___

In a fraction twice the numerator is 2 more than the denominator if 3 is added to the numerator and to the denominator the new fraction is 2/ 3 find the original number.

$Iftanp\theta -tanq\theta =0,thenthevaluesof\theta formaseriesin$

The vertex of the parabola $(y-2{)}^2=16(x-1)$ is

Number of integers k for which the equation $x^3–27x+k=0$ has at least two distinct integer roots is

Match the following :

$\begin{array}{cc}coloumnI& coloumnII\\ A)arg\frac{z+1}{z-1}=\frac{\pi }{4}& P)Parabola\\ B)|z-2|=4& Q)Partofacircle\\ C)argz=\frac{\pi }{4}& R)Fullcircle\\ D)z=t+i{t}^2(tϵR)& S)Straightline\end{array}$

Let $C_1$ and $C_2$ be the centres of the circles $x^2+y^2-2x-2y-2=0$ and $x^2+y^2-6x-6y+14=0$ respectively. If $P$ and $Q$ are the points of intersection of these circles, then the area (in sq. units) of the quadrilateral $P{C}_{1}Q{C}_2$ is :

If three lines whose equations are $y=m_{1}x+c_1,y=m_{2}x+c_2$ and $y=m_{3}x+c_3$ are concurrent, then show that $m_1(c-2-c_3)+m_2(c_3-c_1+m_3(c_1-c_2)=0$.

Prove that:

$\left(i\right)\frac{cos(2\pi +\theta )cosec(2\pi +\theta )tan(\pi /2+\theta )}{sec(\pi /2+\theta )cos\theta cot(\pi +\theta )}=1$

$\left(ii\right)\frac{cosec({90}^{\circ }+\theta )+cot({450}^{\circ }+\theta )}{cosec({90}^{\circ }-\theta )+tan({180}^{\circ }-\theta )}+\frac{tan({180}^{\circ }+\theta )+sec({180}^{\circ }-\theta )}{tan({360}^{\circ }+\theta )-sec(-\theta )}=2$

$\left(iii\right)\frac{sin({180}^{\circ }+\theta )cos({90}^{\circ }+\theta )tan({270}^{\circ }-\theta )cot({360}^{\circ }-\theta )}{sin({360}^{\circ }-\theta )cos({360}^{\circ }+\theta )cosec(-\theta )sin({270}^{\circ }+\theta )}\left(iv\right)1+cot\theta -sec(\frac{\pi }{2}+\theta )\}1+cot\theta +sec(\frac{\pi }{2}+\theta \left)\right\}=2cot\theta$

$\left(v\right)\frac{tan({90}^{\circ }-\theta )sec({180}^{\circ }-\theta )sin(-\theta )}{sin({180}^{\circ }+\theta )cot({360}^{\circ }-\theta )cosec({90}^{\circ }-\theta )}=1$

If $ysin\varphi =xsin(2\theta +\varphi )$, prove that $(x+y)cot(\theta +\varphi )=(y-x)cot\theta$

If $a,b,c\in R$ and $a^2+b^2+c^2=1$, then $ab+bc+ca$ lies in the nterval

A and B are partners sharing profits and losses in the ratio of 3:2. A surrenders $\frac{1}{2}$ of his share and B surrenders $\frac{1}{4}$ of his share in favour of C. Calculate new ratio.

Let $f\left(n\right)=\left(\sin 1\right)\left(\sin 2\right)\left(\sin 3\right)\cdots \left(\sin \right(n\left)\right)\forall n\in N$ where $n$ is in radians. Then the number of elements in the set $A=\left\{f\right(1),f(2),\dots ,f(6\left)\right\}$ that are positive, is

One mapping is selected at random from all the mapping of the set A = {1, 2, 3, ......, n} into itself. The probability that the mapping selected is one to one is

If $\left[\begin{array}{cc}x-y& z\\ 2x-y& w\end{array}\right]=\left[\begin{array}{cc}-1& 4\\ 0& 5\end{array}\right]$, find the value of x+y

The function f is defined by
$f\left(x\right)=\{\begin{array}{cc}1-x,& x<0\\ 1,& x=0\\ x+1,& x>0\end{array}$ Draw the graph of f(x).

The value of the expression $\frac{(1+\tan \frac{\pi }{6})(1-\cot \frac{\pi }{6})}{(1+\cos \frac{\pi }{3})(1-\sec \frac{\pi }{3})}$ is

Q. You are a sixth-year graduate student at a large university in the final months of your dissertation research on novel photonic materials. You are worried about your next appointment and have applied for several postdoctoral positions in this field plus a few tenure-track assistant professorships at universities where you would like to work. To your surprise and pleasure, you are invited for an interview for a tenure-track appointment at your undergraduate alma mater, a prestigious research institution in a city where you already have connections and would love to live.

In the question-and-answer period following your seminar on your research, the department chair asks for detailed information about the novel material- preparation technique developed in your graduate research and used extensively in your experiments. Your group is working on a patent application and its members have agreed not to provide details until a paper currently being prepared is submitted for publication. Your thesis advisor will be giving the first major presentation on the technique at a major international conference in a couple of months.

You answer that you and your colleagues are in the process of writing it up for publication and a patent application, and you would be glad to send them an early preprint when it is available. The question and answer period continues and concludes uneventfully and pleasantly. After the seminar, in your private interview with the Chair, he pushes harder for this information, remarking that the Department seeks team players, willing to share information with department colleagues, and referring to your undergraduate roots and the need to prove you are one of them to be a viable candidate for the position.

Questions:

1. What are the interests of the various players?
2. Where are there conflicts of interest?
3. What are your options? And What should you do?

If X = {a, b, c, d} and Y = {f, b, d, g}, find :

(i) X - Y (ii) Y - X (iii) $X\cap Y$

Equation of the plane containing the straight line $\frac{x}{2}=\frac{y}{3}=\frac{z}{4}$ and perpendicular to the plane containing the staight lines $\frac{x}{3}=\frac{y}{4}=\frac{z}{2}$ and $\frac{x}{4}=\frac{y}{2}=\frac{z}{3}$ is

The middle term of expansion of ${(\frac{10}{x}+\frac{x}{10})}^{10}$

∆ area/ area = 1/5 .Then is the error limit in this is 1 and surface area is 5 . It is true or false