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A line which passes through $P(4,5)$ and making an angle of ${30}^{\circ }$ with positive direction of $x-$axis. Then coordinates of point which is at a distance $4$ unit's from the line on either side of $P$, is

Let $f:R\to R$ be a continuous function satisfying

$f\left(x\right)+{\int }_0^{x}tf\left(t\right)dt+x^2=0$
$forallxϵR.Then$

The probability that atleast one of the two events A and B occurs is 0.6. If A and B occur simultaneously with probability 0.3, evaluate $P\left(\overline{A}\right)+P\left(\overline{B}\right)$.

A quadrilateral has vertices (4, 1), (1, 7), (-6, 0) and (-1, -9). Show that the mid-points of the sides of this quadrilateral form a parallelogram.

The area bounded by the curves $y=\ln x$ and $y=(\ln x{)}^2$ is

The angle between the lines represented by the equation $\lambda x^2+(1-\lambda {)}^{2}xy-\lambda y^2=0$, is

Let $\overrightarrow{a}=-\hat{i}+4\hat{k},\overrightarrow{b}=5\hat{i}+2\hat{k}$ and $\overrightarrow{c}=-3\hat{i}+\hat{k}.$ If $\overrightarrow{a}=x\overrightarrow{b}+y\overrightarrow{c},$ then the value of $(x,y)$ is

The common solution set of $3x-7<5+x$ and $11-5x\le 1$ is

Let $L_1$ be a tangent to the parabola $y^2=4(x+1)$ and $L_2$ be a tangent to the parabola $y^2=8(x+2)$ such that $L_1$ and $L_2$ intersect at right angles. Then $L_1$ and $L_2$ meet on the straight line:

The sum of the series $1\times n+2(n-1)+3(n-2)+\dots \dots +(n-1)\times 2+n\times 1$ is

The value of $\underset{n\to \infty }{lim}[\frac{n}{(n+1)(n+2)}+\frac{n}{(n+2)(n+4)}+\cdots +\frac{1}{6n}]$ is:

$\begin{array}{cccc}& Column-I& & Column-II\\ \left(P\right)& Theshortestdistancebetween& \left(1\right)& 6\\ & originandthecurveis{x}^2+y^2+xy=60is& & \\ \left(Q\right)& Thevalueof{\int }_{-2011}^{2011}\frac{dx}{1+x^9+\sqrt{1+x^{18}}}-2011is& \left(2\right)& 0\\ \left(R\right)& On[0,2]themaximumvalueof& \left(3\right)& 3\\ & f\left(x\right)=max\{x^{,}x-1,3x\}is& & \\ \left(S\right)& Letf:R\to Rbegivenby& \left(4\right)& \sqrt{40}\\ & f\left(x\right)=\{\begin{array}{cccc}|x-\left[x\right]|& when\left[x\right]isodd& & \\ |x-\left[x\right]-1& when\left[x\right]iseven& & \end{array}& & \\ & Thenthevalueof{\int }_{-2}^4& & \\ & f\left(x\right)dx-3is& & \end{array}$

cos⁴A - sin⁴A +1=.

If A is a n x n matrix, then adj(adj A) =

The derivative of $\ln (1+x^2)$ with respect to ${\tan }^{-1}x$ is

If $x,y,z$ are in A.P. and ${\tan }^{-1}x,{\tan }^{-1}y,{\tan }^{-1}z$ are also in A.P., then

Find the equation of the straight line at a distance of 3 units from the origin such that the perpendicular from the origin to the line makes an angle $\alpha$ given by $ta{n}^{-1}\left(\frac{5}{12}\right)$ with the positive direction of x-axis.

Let $z$ be a complex number such that $|2z+\frac{1}{z}|=1$ and $\arg \left(z\right)=\theta$, then minimum value of $8{\sin }^2\theta$ is

Angle between two planes $a_{1}x+b_{1}x+c_{1}x+d_1=0\&a_{2}x+b_{2}x+c_{2}x+d_2=0$ is given by -

If $a+b+c=3$ and $a>0,b>0,c>0$, then the greatest value of $a^2{b}^3{c}^2$ is

If vectors ${\overrightarrow{a}}_1=x\hat{i}-\hat{j}+\hat{k}$ and ${\overrightarrow{a}}_2=\hat{i}+y\hat{j}+2\hat{k}$ are collinear, then a possible unit vector parallel to the vector $x\hat{i}+y\hat{j}+z\hat{k}$ is :

If $a+ib=\frac{(x+i{)}^2}{(2x^2+1)}$, prove that $a^2+b^2=\frac{(x^2+1{)}^2}{(2x^2+1{)}^2}$

For the segment of circuit shown in figure find $V_L$, where $V_C=4sin2t$

A bag contains 7 black and 4 white balls two balls are drawn at a time from the bag. The probability at least one white ball is selected is

Let A and B be square matrices of order 3×3. Is (AB)² = A^{​​​​​​2} B^{​​​​​​2} ? Give reason.

The value of $\underset{x\to \infty }{lim}\frac{2{\left(x\right)}^{1/2}+3{\left(x\right)}^{1/3}+\cdots +n{\left(x\right)}^{1/n}}{{(2x-3)}^{1/2}+{(2x-3)}^{1/3}+\cdots +{(2x-3)}^{1/n}}$ is

In a group of tourists, $40\%$ liked Goa, $30\%$ liked Kerala and $30\%$ liked Bangalore. $7\%$ liked both Goa and Kerala, $5\%$ liked both Kerala and Bangalore, $10\%$ liked both Bangalore and Goa. If $86\%$ of these liked at least one of the places, then what percentage of people liked all three?

The value of the parameter ${}^{\prime }{a}^{\prime }$ so that the line $(3-a)x+ay+(a^2-1)=0$ is a normal to the curve $xy=1,$ may lies in the interval

If the tangents drawn to the hyperbola $4y^2=x^2+1$ intersect the co-ordinate axes at the distinct points $A$ and $B$, then the locus of the mid point of $AB$ is :

The number of quadratic equation(s), with real roots which remain unchanged even after squaring its roots, is

Which among the following won't represent a function ?

If $x=2\cos t-\cos 2t$ and $y=2\sin t-\sin 2t,$ then $\frac{dy}{dx}$ is equal to

If the normal to the parabola $y^2=4ax$ at the point $(a{t}^2,2at)$ cuts the parabola again at $(a{T}^2,2aT)$, then