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Reduce the equation 3 x - 2 y + 6 = 0 to the intercept form and find the x and y intercepts.

If the system of linear equations
$x+ay+z=3$
$x+2y+2z=6$
$x+5y+3z=b$
has no solution, then:

The approximate value of $(1.0002{)}^{3000}$ is

Tangents are drawn from different points on the line $x-y+10=0$ to the parabola $y^2=4x$. If the chords of contact pass through a fixed point, then the coordinates of the fixed point is

For $n\in I,$ the line $x=n\pi +\frac{\pi }{2}$ does not intersect the graph of

In a $△ABC$, if $\cos A\cos B\cos C=\frac{\sqrt{3}-1}{8}$ and $\sin A\sin B\sin C=\frac{3+\sqrt{3}}{8}$, then the angles of the triangle are

If -1+$\sqrt{-3}$=r$e^{i\theta }$, then $\theta$ is equal to

Find the roots of the quadratic equation $a{x}^2+bx+c=0$ in terms of a, b, c.

If $f^{\prime \prime }\left(0\right)=k,k\ne 0,$ then the value of
$\underset{x\to 0}{lim}\frac{2f\left(x\right)-3f\left(2x\right)+f\left(4x\right)}{x^2}$is

Find the numerically greatest term in the expansion of $(7-5x{)}^{11}$ when $x=\frac{2}{3}$.

$tan\alpha +2tan2\alpha +4tan4\alpha +8cot8\alpha$=



[IIT 1988; MP PET 1991]

n( > 3) persons are sitting in a row. Two of them are selected. Write the probability that they are together.

The angle of intersection of the curves $y=2si{n}^{2}xandy=cos2xatx=\frac{\pi }{6}$ is

Consider the following system of linear equations
$\left[\begin{array}{ccc}2& 1& -4\\ 4& 3& -12\\ 1& 2& -8\end{array}\right]\left[\begin{array}{c}x\\ y\\ z\end{array}\right]=\left[\begin{array}{c}a\\ 5\\ 7\end{array}\right]$
Number of values of $a$ for which system has infinitely many solutions.

AB is a chord of the parabola $y^2=4ax$ with vertex at A. BC is drawn perpendicular to AB meeting the axis at C. The projection of BC on the x-axis is-

The value of the expression :

${\cos }^{-1}\left(\cos \frac{7\pi }{6}\right)$ is equal to

From the point (1,-2,3) lines are drawn to meet the sphere $x^2+y^2+z^2=4$ and they are divided internally in the ratio 2:3. The locus of the point of division is-

If $\alpha ,\beta$ are the roots of $2x^2+5x+1=0$, then the equation whose roots are $2\alpha +1,2\beta +1$ is

Match $\text{List I}$ with the $\text{List II}$ and select the correct answer using the code given below the lists :



$\begin{array}{cccc}& \text{List - I}& & \text{List - II}\\ \text{(A)}& \text{Possible integral value(s) of}k\text{for which the point}M(0,k)\text{lies on or inside the triangle formed by the lines}y+3x+2=0,& \text{(P)}& 0\\ & 3y-2x-5=0\text{and}4y+x-14=0.& & \\ \text{(B)}& \text{If}\hat{a},\hat{b}\text{and}\hat{c}\text{are non-coplanar vectors, then the vectors}\overrightarrow{V_1}=\hat{a}+2\hat{b}+3\hat{c},\overrightarrow{V_2}=\lambda \hat{b}+4\hat{c}\text{and}\overrightarrow{V_3}=(2\lambda -1)\hat{c}\text{where}\lambda & \text{(Q)}& 1\\ & \text{is a scalar, can be non-coplanar, for}\lambda \text{equals}& & \\ \text{(C)}& \text{The distance of the}z-\text{axis from the image of the point}A(2,–3,3)\text{in the plane}x-2y-z+1=0,\text{is}& \text{(R)}& 2\\ & \text{The figure given below shows a pyramid}DOABC\text{(where O is the origin) with a square base whose sides are}1\text{unit long.}& & \\ \text{(D)}& \text{The pyramid's height is also}1\text{unit and the point}D\text{stands directly above the mid point of the diagonal}OB\text{. If the angle}& \text{(S)}& 3\\ & \text{between}\overrightarrow{OB}\text{and}\overrightarrow{OD}\text{is}{\tan }^{-1}\sqrt{K},\text{then}K\text{is equal to}& & \\ & & \text{(T)}& 4\end{array}$





Which of the following is a CORRECT combination?

${\int }_0^{\frac{\pi }{4}}ta{n}^{2}xdx=$ [Roorkee 1983, Pb. CET 2000]

For $x>0$, let $f\left(x\right)={\int }_1^x\frac{\text{log}t}{1+t}dt$. Then $f\left(x\right)+f\left(\frac{1}{x}\right)$ is equal to

The maximum sum of the series $20+19\frac{1}{3}+18\frac{2}{3}+.....$is

Let $f:A\to B$ be a function, where $A=\{x_1,x_2,x_3...,x_6\}$ and $B=\{y_1,y_2,y_3...,y_{10}\}$ given by $f\left(x\right)=y.$ Then the number of functions from $A$ to $B$ such that $f\left(x_1\right)<f\left(x_2\right)<f\left(x_3\right)=f\left(x_4\right)<f\left(x_5\right)<f\left(x_6\right)$ is

The solution set of the inequality $\left({\cot }^{-1}x\right)\left({\tan }^{-1}x\right)+(2-\frac{\pi }{2}){\cot }^{-1}x-3{\tan }^{-1}x-3(2-\frac{\pi }{2})>0,$ is

If $1,d_1,d_2,d_3,d_4$ are roots of $x^5=1$ then the value of expression :$E=\frac{\omega -d_1}{{\omega }^2-d_1}\cdot \frac{\omega -d_2}{{\omega }^2-d_2}\cdot \frac{\omega -d_3}{{\omega }^2-d_3}\cdot \frac{\omega -d_4}{{\omega }^2-d_4}$ is

[Here $\omega$ is the cube root of unity]

The equation of the circle passing through the foci of the ellipse

$\frac{x^2}{9}+\frac{y^2}{16}=1$ and having the centre at

$(0,3)$ is

The sum of integral values of $n$ for which $n^2+17n+75$ is a perfect square is

The value of the expression ${}^{47}{C}_4+{\sum }_{j=1}^5{}^{52-j}{C}_3$ is

In a triangle $ABC,$ points $D,E$ and $F$ are taken on the sides $BC,CA$ and $AB$ respectively, such that $\frac{BD}{DC}=\frac{CE}{EA}=\frac{AF}{FB}=n.$ Then Area of $△DEF$ is equal to