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Find the value of k for which the lines $3x+y=2,k+2y=3$ and $2x-y=3$ may intersect at a point.

What is the distance between the point A$(0,7,10)$ and C$(-4,9,6)$ in space?

Find the value of $\underset{x\to \frac{\pi }{4}}{lim}\left[\right(sinx{)}^{\frac{-8x}{\pi }}]$

Consider a circle with its centre lying on the focus of the parabola $y^2=2px$ such that it touches the directrix
of the parabola. Then, a point of intersection of the circle and the parabola is

What is the range of f(x) = $\frac{x^2-5x+6}{(x-3)}$ ( R $\to$ set of all real numbers )

Let $f,g,h$ be the length of the perpendiculars from the circumcentre of the $\Delta ABC$ on the sides $a$, $b$, and $c$, respecitively then the value of $k$ for which $\frac{a}{f}+\frac{b}{g}+\frac{c}{h}=k\frac{abc}{fgh},$ is

If $S_n$ = $\sum _{r=0}^n\frac{1}{{}^n{C}_r}$ and $t_n$ = $\sum _{r=0}^n\frac{r}{{}^n{C}_r}$, then $\frac{t_n}{s_n}$, when n =100 equals to

Prove that:

$2{\text{sin}}^2\frac{3\pi }{4}+2{\text{cos}}^2\frac{\pi }{4}+2{\text{sec}}^2\frac{\pi }{3}=10$

Find the principal solution of $3se{c}^{2}x=secx$

Reduce the equation $3x-2y+4=0$ to intercepts form and find the length of the segment intercepted between the axes.

Find the value of $a_1$ for which the coefficient of the middle term in the expansions of $(1+ax{)}^4$and $(1-ax{)}^6$ are equal.

Find the distance between the points $A(-2,1,-3)$ and $B(4,3,-6)$.

If $x\ne \frac{n\pi }{2}and(cosx{)}^{si{n}^{2}x-3sinx+2}=1$ then all solutions of x are given by

The points (-a, -b), (0, 0), (a, b) and ($a^2$, ab) are

Find the least value of p + q if ( $p^2$ - 2p) $x^2$ + ( $q^2$ + q - 2) x = 0 is an identity.

In triangle ABC, right angled at B, if one angle is ${45}^o$, find the value of sin A, cos C, cot A and tan C respectively.

The probability that length of a randomly selected chord of a circle lies between $\frac{1}{2}$ and $\frac{3}{2}$ of its radius is
(correct answer + 1, wrong answer - 0.25)

If $x$ and $R$ stands for distance, then which of the following is dimensionally same as $\int \frac{Rdx}{x^2}$?

$\underset{x\to 1}{lim}f\left(x\right)=5$, $\underset{x\to 2}{lim}g\left(x\right)=6$ and $\underset{x\to 1}{lim}g\left(x\right)=2$ find the value of $\underset{x\to 1}{lim}$ $\left(\right[g\left(x\right){]}^{f\left(x\right)})$ ___

Find the middle term(s) in the expansion of $(a+b{)}^{20}$

Two successive terms in the expansion of $(1+x{)}^{24}$, whose coefficients are in the ratio 4:1 are $(r+1{)}^{th}$ and $(r{)}^{th}$,r <15. Find the value of r

If the circles $x^2+y^2+2ax+cy+a=0and{x}^2+y^2-3ax+dy-1=0$ interesect in two distinct points P andn Q then the line 5x + by - a = 0 passes through P and Q for
(2005)

If ${\log }_{2}x+{\log }_{8}x+{\log }_{64}x=3$, then the value of $x$ is

The function : $R\to [-\frac{1}{2},\frac{1}{2}]$ defined as $f\left(x\right)=\frac{x}{1+x^2}$ is

The set of values of $x$ for which the function ​​$f\left(x\right)=\log \left(\frac{x^2-5x+6}{x^2+x+1}\right)+\sqrt{\frac{1}{[x^2-1]}}$ is defined, is
(where $[.]$ denotes the greatest integer function)

Find the range of $x$ for the following expression:

$\left|\frac{x^2-3x+2}{x^2+3x+2}\right|\ge 1$

Solve $|3x-2|\le \frac{1}{2},xϵR$

The random error in the arithmetic mean of $100$ observations of a physical quantity is $x$, then random error in the arithmetic mean of $400$ observations of the same physical quantity will be

Find the area of the region bounded by $x^2=16y,y=1,y=4$ and the $y$-axis in the first quadrant.

If $x\in (-2,6]$, then $(x^2-2)$ lies in

The statement $(p\Rightarrow \sim p)\wedge (\sim p\Rightarrow p)$ is a:

Prove that the following statement is true : If x, $y\in Z$ such that x and y are odd, then xy is odd.

Three numbers form an increasing G.P. If the middle number is doubled, then the new numbers are in A.P. The common ratio of the G.P. is

Find the coordinates of the foci, the vertices, the eccentricity and the length of the latus rectum of the hyperbola,

$9y^2-4x^2=36$

If α, β, γ and δ are the roots of $x^4$ + 2 $x^3$ + 3 $x^2$ + 4x + 5 = 0. Find the equation whose roots are $\frac{1}{\alpha }$, $\frac{1}{\beta }$, $\frac{1}{\gamma }$, $\frac{1}{\delta }$

If x,y ∈ C ,

statement 1 : |$\frac{2x-4y}{6\overline{y}-3\overline{x}}$| = $\frac{2}{3}$

statement 2 : If z ∈ C, then |z|= |$\overline{z}$| = |-z| = |$\overline{-z}$|

Find the value of 1 + $\alpha$ + ${\alpha }^2$ + ${\alpha }^3$ +.............. ${\alpha }^{n-1}$. If ${\alpha }^k$ = $cos\frac{2\pi k}{n}+isin\frac{2\pi k}{n}$,

K = 0,1,2,..........n-1

If the line joining origin to the points of intersection of the line fx - gy = $\lambda$ and the curve $x^2+hxy-y^2+gx+fy=0$ be mutually perpendicular, then

The value of x for the maximum value of $\sqrt{3}cosx+sinx$ is

If the coefficient of x in the expansion of $(x^2+\frac{k}{x}{)}^5$ is 270, then k =

$\begin{array}{cc}Linepassesthroughthepoints& Slopeoftheline\\ p.(1,6)and(-4,2)& 1.0\\ q.(5,9)and(2,9)& 2.-3\\ r.(-2,-1)and(-3,2)& 3.\frac{4}{5}\\ s.(4,0)and(3,3)& 4.\frac{5}{3}\end{array}$

If P = (1, 0), Q = (-1, 0) and R = (2, 0) are three given points, then the locus of a point S satisfying the relation

$S{Q}^2+S{R}^2=2S{P}^2$ is

Find the value of $\frac{se{c}^{2}x-cose{c}^{2}x}{ta{n}^{2}x-co{t}^{2}x}$. (x $ϵ$ (0,$\frac{\Pi }{2}$),x $\ne$ $\frac{\Pi }{4}$)