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If $si{n}^2$ x + $co{s}^2$y = 2 $se{c}^2$ z, find the value of $co{s}^2$ x + $si{n}^2$ y + 2 $si{n}^2$ z.

S = ${\sum }_{r=0}^{10}co{s}^3\frac{r\pi }{3}$ find the value of 16S

If I < x < I +1 , find [-x] where I is an integer.

If ${\alpha }^2$ - (11 - i)$\alpha$ + 24 + 3$\beta$i = 0.Find the values of $\beta$.

Find the cente and radius of the following circle.
$x^2+y^2-8x-10y-12=0$

The number of terms in the expansion of ${(x^2+1+\frac{1}{x^2})}^n,n\in$ N, is

From the employees of a company, 5 persons are selected to represents them in the managing committee of the company. Particulars of five persons are as follows:
$$\begin{array}{cccc}\text{S. No.}& \text{Name}& \text{Sex}& \text{Age in years}\\ 1& \text{Harish}& M& 30\\ 2& \text{Rohan}& M& 33\\ 3& \text{Sheetal}& F& 46\\ 4& \text{Alice}& F& 28\\ 5& \text{Salim}& M& 41\end{array}$$
A person is selected at random from this group to act as a spokesperson. What is the probability that the spokesperson will be either male or over 35 years?

For every natural number n

The equation of the bisectors of the angle between the lines represented by the equation $x^2-y^2=0$, is

A solution is to be kept between ${68}^{\circ }F$ and ${77}^{\circ }F$. What is the range of temperature in degree Celsius (C) if the Celsius /Fahrenheit (F) convension formula is given by $F=\frac{9}{5}C+32$?

If S(p,q,r)=$\sim p\vee \sim (q\vee r)$ is a compund statement, then S(~p,~q,~r) is

Find the $(r+1{)}^{th}$ term in the expansion of $(x+y{)}^n$

If in a triangle ABC, a = 15, b = 36, c = 39, then sin C/2 is equal to

A rod of length 12 cm moves with its ends always touching the coordinate axes. Determine the equation of the path of a moving point P on the rod which is 3 cm from the end in contact with the x-axis.

Find the coordinates of the points which trisect the line segment AB, if two points are A(2, 1, -3) and B (5, -8, 3).

${}^n{C}_0$ + ${}^n{C}_2$ + ${}^n{C}_4$...................=

Which one of the following represents an impossible arrangement for the values of n, l, mand m_s?

On the parabola $y=x^2$, the point least distance from the straight line$y=2x-4$ is

$li{m}_{x\to 0}\frac{(e^{1/x}-1)}{\left(e^{1/x}\right)+1}$

P: Arjun is the fastest. Q: Azad is the captain. denotes the statement "Arjun is the fastest OR Azad is not the captain"

If $S_{2n}=3S_n$ and $S_{5n}=k{S}_{3n}$, where $S_n$ is the sum of $n$ terms of an A.P., then the value of $k$ is

The distance between the directrices of the hyperbola $x^2-y^2=9$ is ___ .

Find the equation of the circle with radius 5 whose centre lies on x - axis and passes through the point (2,3)

Find the value of $se{c}^{2}x$ - $cose{c}^{2}x$.

For x > 0, Let $A=\left[\begin{array}{ccc}x+\frac{1}{x}& 0& 0\\ 0& x& 0\\ 0& 0& 16\end{array}\right]B=\left[\begin{array}{ccc}\frac{5x}{x^2+1}& 0& 0\\ 0& \frac{3}{x}& 0\\ 0& 0& \frac{1}{4}\end{array}\right]$
$X=(AB{)}^{-1}+(AB{)}^{-2}+(AB{)}^{-3}+...\infty$
$Z=X^{-1}-2I$ (I is identity matrix of order 3)
$\begin{array}{cc}\text{(P) minimum value of [Tr(Ax)]}is& \left(1\right)24\\ \text{(when [.])}\to \text{represent integer function}& \\ \text{(Q) det}\left(X^{-1}\right)is& \left(2\right)12\\ \text{(R) If}Tr(z+z^2+---+z^{10})=2^a+b,& \\ (a,b\in N)\text{then a + b is}& \left(3\right)6\\ \text{(S) If value of}|adj\left(\sqrt{5}{X}^{-1}\right)|=k& \\ \text{then number of positive divisors}& \left(4\right)19\\ \text{of k which are odd is}& \end{array}$

In Figure I an air column of length $l_1$, is entrapped by a column of $Hg$ of length $8\text{cm}$. In Figure II length of same air column at the same temperature is $l_2$. Then $l_1/l_2$ is?
(1 atm = $76\text{cm of}Hg$)

Which of the following cannot be valid assignment of probabilities for outcomes of sample space
S = $$\left\{\begin{array}{ccccccc}{w}_1,& w_2,& w_3,& w_4,& w_5,& w_6,& w_7,\end{array}\right\}$$

$$\begin{array}{cccccccc}& w_1& w_2& w_3& w_4& w_5& w_6& w_7\\ \left(a\right)& 0.1& 0.1& 0.05& 0.03& 0.01& 0.2& 0.6\\ \left(b\right)& \frac{1}{7}& \frac{1}{7}& \frac{1}{7}& \frac{1}{7}& \frac{1}{7}& \frac{1}{7}& \frac{1}{7}\\ \left(c\right)& 0.1& 0.2& 0.3& 0.4& 0.5& 0.6& 0.7\\ \left(d\right)& -0.1& 0.2& 0.3& 0.4& -0.2& 0.1& 0.3\\ \left(e\right)& \frac{1}{14}& \frac{2}{14}& \frac{3}{14}& \frac{4}{14}& \frac{5}{14}& \frac{6}{14}& \frac{15}{14}\end{array}$$

Express each of the following in exponential form:

$\left(i\right)lo{g}_{2}64=6$

$\left(ii\right)lo{g}_{10}0.01=-2$

If z lies on the curve |z| = 1 such that a $\le$ |z+1| + |1 + $z^2$ - z| $\le$ b then (a,b) can be

Equation of the ellipse with foci ($\pm ,0$) and e = $\frac{1}{4}$ is

A metal crystallises in a face centred cubic structure. If the edge length of its unit cell is 'a', the closest approach between two atoms in metallic crystal will be

Find the ratio in which the line ax + by + c = 0 divides the line segment joining p$(x_1,y_1)$ and Q $(x_2,y_2)$.

You are given t = $\frac{(a{x}_2+b{y}_2+c)}{a{x}_1+b{y}_1+c}$

Differentiate x cos x by first principle.

Or

Evaluate $li{m}_{y\to 0}\frac{(x+y)sec(x+y)-xsecx}{y}$

Evaluate: $\sqrt{6+8i}$

Write the set $G=\{1,3,5,7,9,11,\dots \}$ in the set-builder form.

The value of x $\in (-2\pi ,2\pi )$ such that $\frac{sinx+icosx}{1+i}$, where $i=\sqrt{-1}$, is purely imaginary are given by

If the ratio of 7^th term from the beginning to the seventh term from the end in the expansion of ($\sqrt[3]{32}+\frac{1}{\sqrt{3}}{)}^n$is $\frac{1}{6}$, then n is

The sum of infinite terms of the following series

1 + $\frac{4}{5}$ + $\frac{7}{5^2}$ + $\frac{10}{5^3}$ + ........... will be

The point of intersection of the tangents of the circle $x^2+y^2=10$, drawn at end points of the chord x + y = 2 is

A curve is represented by y=sin x. If x is changed from $\frac{\pi }{3}to\frac{\pi }{3}+\frac{\pi }{100}$, find approximately the change in y.

Find derivative of $f\left(x\right)=x+\frac{1}{x}+1$

Find the $y-$ intercept of the line passing through the points $A(3,-2)$ and $B(-1,3)$.

If the odds in favour of an event be $\frac{3}{3}$, find the probability of the occurrence of the event.

The quadratic equation $\tan \theta x^2+2(\sec \theta +\cos \theta )x+(\tan \theta +3\sqrt{2}\cot \theta )$ always has

How many integers satisfy the condition $|x|\le -5$