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The straight lines whose direction cosines are given by al + bm + cn = 0, fmn + gnl + hlm = 0 are perpendicular, if

The conjugate of $\frac{(2+i{)}^2}{3+i}$ , in the form of a+ib, is

If the graph of $f\left(x\right)$ is given by


then which of the following graph represents $-f(x-1)$

If $3ta{n}^{-1}\frac{1}{2+\sqrt{3}}-ta{n}^{-1}\frac{1}{x}=ta{n}^{-1}\frac{1}{3}$, then $x=$

If $\int \frac{se{c}^{2}x-2010}{si{n}^{2010}x}dx=\frac{P\left(x\right)}{(sinx{)}^{2010}}+C$

Let F(x) = f(x) + $f\left(\frac{1}{x}\right)$, where f(x) = ${\int }_1^x\frac{logt}{1+t}dt$ . Then F(e) =

Complete set of values of $k$ for which the equation $4^x-(k+2)2^x+2k=0,$ has exactly one positive root is

If f(x) = |cos x|, then the value of $f\text{'}\left(\frac{3\pi }{4}\right)$ is

Find the equation of the chord of the ellipse $4x^2+25y^2=100$ whose middle point is (1,1).

The value of $\int \frac{e^x+9\cos x-2\sin x+7}{e^x+7\sin x+11\cos x+14}dx$ is
(where $c$ is the constant of integration)

Area bounded by curve xy = c, x - axis between x=1 and x=4, is

${lim}_{x\to \infty }[\sqrt{x^2+1}-\sqrt{x^2-1}]$

$\text{Solve the equation}|z|=z+1+2i.$

A box contains 10 red balls and 15 blue balls. One ball from the box is lost. Two balls are then drawn from the box and are found to be red ones. Find the chance that the missing ball is a blue ball.

Two dice are rolled simultaneously. The probability that the sum of the two numbers on the top faces will be at least 10 is

$$\begin{array}{cc}\text{Trigonometric Equation}& \text{Principal Solutions}\\ 1.\sin x=\frac{\sqrt{3}}{2}& A.\frac{5\pi }{6}\\ 2.\tan x=\frac{-1}{\sqrt{3}}& B.\frac{5\pi }{12}\\ 3.\sec 2x=\frac{-2}{\sqrt{3}}& C.\frac{7\pi }{12}\\ 4.\cos 3x=\left(\frac{-1}{2}\right)& D.\frac{\pi }{3}\\ & E.\frac{2\pi }{9}\\ & F.\frac{4\pi }{9}\\ & G.\frac{11\pi }{6}\\ & H.\frac{2\pi }{3}\end{array}$$

What is the value of y so that the line through (3, y) and (2, 7) is parallel to the line through (-1, 4) and (0, 6) ?

The function $f\left(x\right)=\left[x\right]\cos \left(\left(\frac{(2x-1)}{2}\right)\pi \right)$, (where [.] denotes the greatest integer function) is discontinuous at

The area of the region bounded by the curve $y^2=4x$ , y-axis and the line y = 3 is

Verify that area of the triangle with vertices (4,6) (7,10) and(1,-2) remains invariant under the translation of axes when the origin is shifted to the point (-2,1).

Let $\overrightarrow{a},\overrightarrow{b}$ and $\overrightarrow{c}$ be three unit vectors such that $\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c}=\overrightarrow{0}$. If $\lambda =\overrightarrow{a}\cdot \overrightarrow{b}+\overrightarrow{b}\cdot \overrightarrow{c}+\overrightarrow{c}\cdot \overrightarrow{a}$ and $\overrightarrow{d}=\overrightarrow{a}\times \overrightarrow{b}+\overrightarrow{b}\times \overrightarrow{c}+\overrightarrow{c}\times \overrightarrow{a}$, then the ordered pair $(\lambda ,\overrightarrow{d})$ is equal to:

If the ellipse $\frac{x^2}{4}+y^2=1$ meets the ellipse $x^2+\frac{y^2}{a^2}=1$ in four distinct points and $a=b^2-5b+7,$ then $b$ does not lie in

List all the elements of the following sets :

(i) A = {$x:x^2\underset{–}{<}10,xϵZ$}

(ii) B = { $x:x=\frac{1}{2n-1},1\underset{–}{<}n\underset{–}{<}5$}

(iii) {$C=x:x\text{is an integer},-\frac{1}{2}<x<\frac{9}{2}$ }

(iv) D = {x : x is a vowel in the word "EQUATION"}

(v) E = {x : x is a month of a year not having 31 days}

(vi) F = {x : x is a letter of the word "MISSISSIPPI" }

Let $y=mx+{\lambda }_i,i=1,2,3,...,n$ be a family of $n$ parallel lines subjected to following conditions.
$\left(1\right)m$ being a constant
$\left(2\right)\sum _{i=1}^n{\lambda }_i=1$
A variable line through origin intersects the lines at $P_i(i=1,2,3,...,n)$ and $Q$ be a point on variable line such that $\sum _{i=1}^{n}O{P}_i=OQ$. If the locus of $Q$ is a straight line which passes through a fixed point $(a,b)\forall m\in R,$ then the value of $(3a+2b)$ is

The area of an equilateral triangle inscribed in the circle $x^2+y^2+2gx+2fy+c=0$ is

Show that the lines with direction cosines $\frac{12}{13},\frac{-3}{13},\frac{-4}{13};\frac{4}{13},\frac{12}{13},\frac{3}{13};\frac{3}{13},\frac{-4}{13},\frac{12}{13}$ are mutually perpendicular.

If xyz=1 ,x+1/z=5 ,y+1/x=29 ,then find z+1/y=

$\text{If}\frac{1-ix}{1+ix}=a+ib,\text{then}{a}^2+b^2=$

A tangent is drawn to the circle $x^2+y^2=9$ at $(3,3)$, then the equation of the line which is parallel to the tangent and passes through $(5,4)$ is

What is the least value of $\frac{25{\sec }^{4}x-50{\sec }^{2}x+74}{{\tan }^{2}x}$ ?

The domain of the function $f\left(x\right)=\frac{\sqrt{{\log }_{0.4}\left(\frac{x-1}{x+5}\right)}}{x^2-36}$ is

The number of solution(s) of $y=||x{|}^2-2|x|-2|$ and $y=1$ is

Let $x$, $y$ be real variables satisfying the equation $x^2+y^2+8x-10y+40=0$. If $a=max\left\{\right(x+2{)}^2+(y-3{)}^2\}$ and $b=min\left\{\right(x+2{)}^2+(y-3{)}^2\}$, then

If $A=\left(\begin{array}{cc}-i& 0\\ 0& i\end{array}\right),then{A}^2=$

Equation of the line of shortest distace between the lines $\frac{x}{2}=\frac{y}{-3}=\frac{z}{1}$ and $\frac{x-2}{3}=\frac{y-1}{-5}=\frac{z+2}{5}$ is

Let a, b, c be real numbers a $\ne$ 0. If $\alpha$ is a root of $a^2{x}^2$ + bx + c = 0, $\beta$ is a root of $a^2{x}^2$ - bx - c = 0 and 0 < $\alpha <\beta$, then the equation $a^2{x}^2$ + 2bx + 2c = 0 has a root $\gamma$ that always satisfies

The area enclosed between the curves $y^2=x$ and $\left|y\right|=\left|x\right|$ is

If $f\left(x\right)=\left[\begin{array}{ccc}cosx& -sinx& 0\\ sinx& cosx& 0\\ 0& 0& 1\end{array}\right]$ then $f(\alpha +\beta )=$

Two subsets A and B of a set ‘S’ consisting of ‘n’ elements are constructed randomly. The Probability that $A⊑B$ is equal to: