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Consider the lines given by

$L_1:x+3y-5=0$

$L_2:3x-ky-1=0$

$L_3:5x+2y-12=0$



Match the Statements/Exp[ressions in Column I with the Statements/Expressions in Column II.

$\begin{array}{cc}\text{Column I}& \text{Column II}\\ L_1,L_2,L_3\text{are concurrent, if}& \text{k = -9}\\ \text{One of}{L}_1,L_2,L_3\text{is parallel to at least one of the other two, if}& \text{k}=\frac{6}{5}\\ L_1,L_2,L_3\text{form a triangle, if}& k=\frac{5}{6}\\ L_1,L_2,L_3\text{do not form a triangle, if}& \text{k = 5}\end{array}$

The value(s) of $x$ for which the expression $y=|x+2|+|x-7|$ is minimum is/are

The sum of $x-$intercept and $y-$intercept of the common tangent to the parabola $y^2=16x$ and $x^2=128y$ is

All the points on the x- axis have

[MP PET 1988]

The equation of a tangent to the parabola $y^2=8x$ which makes an angle ${45}^{\circ }$ with the line $y=3x+5$ is

If $20$ persons are invited for a party then the number of different ways the persons and the host can be seated around a circular table, if two particular persons are to be seated on either side of the host is

There are three values of t for which the following system of equations has non-trivial solutions.

(a-t) x+by + cz=0

bx + (c - t) y +az=0

cx + ay + (b-t) z=0

We can express the product of the three values of t in the form of determainant as

In a factory 70% of the workers like oranges and 64% likes apples. If each worker likes at least one fruit, What is the minimum percentage of workers who like both the fruits?

The sum of the series $1^2$.2 + $2^2$.3 + $3^2$.4 + ........ to n terms is

If the parabolas $y^2=4b(x-c)$ and $y^2=8ax$ have a common normal except the axis of symmetry of the parabolas, then the range of $\frac{c}{2a-b}$ is

You are given $8$ balls of different colours (black, white, ...). The number of ways in which these balls can be arranged in a row so that the two balls of particular colour (say red and white) may never come together, is

If all the roots of the equation $p{x}^4+q{x}^2+r=0,p\ne 0,q^2\ge 9pr$ are real, then which of the following option(s) is/are correct?

In $\Delta ABC,if\angle C={90}^{\circ },\angle A={30}^{\circ },c=20$, then the values of a and b are

The angle between the vectors $\overline{u}=<3,0>and\overline{v}=<5,5>$ is ___

Suppose that $20$ pillars of the same height have been erected along the boundry of a circular stadium. If the top of each pillar has been connected by beams with the top of all its non-adjacent pillars, then the total number of beams is :

Let $A(2,-3)$ and $B(-2,1)$ be vertices of a triangle $ABC.$ If the centroid of this triangle moves on the line $2x+3y=1,$ then the locus of vertex $C$ is

$LetA=\left[\begin{array}{cc}0& \alpha \\ 0& 0\end{array}\right]and(A+I{)}^{50}-50A=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right],Then,thevalueofa+b+c+dis$

The range of the function $y=\frac{x-1}{(x^2-3x+3)}$ is [a, b] where a, b are respectively

The vector$\overrightarrow{V}$ directed along the internal bisector of the angle between the vectors

$\overrightarrow{A}=3\hat{i}-4\hat{j},\overrightarrow{B}=4\hat{i}+2\hat{j}-4\hat{k}$ and $\left(\overrightarrow{V}\right|=2$ is

If a point in argand plane $A(2,3)$ rotated through origin by $\frac{\pi }{4}$ in anticlockwise direction, then the new coordinates of the point will be

Let $f:[a,b]\to R$ be a function such that for

$c\epsilon (a,b),f^1\left(c\right)=f^{11}\left(c\right)=f^{111}\left(c\right)=f^{tv}\left(c\right)=f^v\left(c\right)=0$ then

Find the values of y for which the following will be positive, negative or zero. $y=x-6\sqrt{x}+8$

A five digit number divisible by 3 has to be formed using the numerals 0, 1, 2, 3, 4 and 5 without repetition. The total number of ways in which this can be done is

Let $P$ be a plane passing through the points $(2,1,0),(4,1,1)$ and $(5,0,1)$ and $R$ be any point $(2,1,6)$. Then the image of $R$ in the plane $P$ is:

If the line $ax+by=2$ is a normal to the circle $x^2+y^2-4x-4y=0$ and a tangent to the circle $x^2+y^2=1,$ then

If a tangent to the parabola $y^2=4ax$ meets the x-axis at T and the tangent at the vertex A, at P. Let the rectangle TAPQ be completed. Then the locus of the point Q is

The number of ways of selecting two squares from a chess board so that they have exactly one common corner is

The complete solution set of $\sin x-\sqrt{3}\cos x=0$ is

If one root is $n$ times the other for the equation $a{x}^2+bx+c=0,$ then

Let $p,q,r$ be roots of cubic equation $x^3+2x^2+3x+3=0$, then

Two tangents are drawn to end points of the latus rectum of the parabola $y^2=4x$. The equation of the parabola which touches both the tangents as well as the latus rectum is

If a, b, c be three real numbers of the same sign then the value of $\frac{a}{b}+\frac{b}{c}+\frac{c}{a}$ lies in the interval

The value of $\sum _{i=1}^n\sum _{j=1}^i\sum _{k=1}^{j}1=220,$ then the value of $n$ equals

Distance between the points (1, 3, 2) and (2, 1, 3) is

[MP PET 1988]

The general solution of $y^{2}dx+(x^2-xy+y^2)dy=0$ is

[EAMCET 2003]

The equation of the common tangent to $x^2=6y$ and $2x^2-4y^2=9$ can be

The lines joining the points of intersection of line x + y = 1 and curve $x^2+y^2-2y+\lambda =0$ to the origin are perpendicular, then the value of $\lambda$ will be

If $(p^2-4p+5,2)=(2p-3,|p-2|)$, then the number of values of $p$ is

If $A$ and $B$ are disjoint non-empty sets, then $A–(A–B)$ is

Let $P(asec\theta ,btan\theta )$ and $Q(asec\varphi ,btan\varphi )$, where $\theta +\varphi =\frac{\pi }{2}$, be two points on the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$.

If (h, k) is the point of the intersection of the normals at P and Q, then k is equal to

A circle of radius $7$ units touches the coordinate axes in the second quadrant. If the circle makes five complete rolls along the positive direction of $x-$axis, then the equation of circle in new position is

$(\text{Assume}\pi =\frac{22}{7})$

The value of $2({\cos }^{2}73°+{\cos }^{2}47°)-\cos 154°$ is

The number of solutions of ${\cos }^{2}x+\frac{\sqrt{3}+1}{2}\sin x-\frac{\sqrt{3}}{4}-1=0$, where $x\in [-\pi ,\pi ]$ is

If $(2x-1{)}^{20}-(ax+b{)}^{20}={(x^2+px+q)}^{10}$ holds true $\forall x\in R$ where $a,b,p$ and $q$ are real numbers, then which of the following is (are) CORRECT?

Find the locus of the point P if $A{P}^2-B{P}^2=18$. Where $A\equiv (1,2,-3)anaB\equiv (3,-2,1)$

If $\overrightarrow{a},\overrightarrow{b}$ and $\overrightarrow{c}$ are three non – coplanar vectors, then $(\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c}).\left(\right(\overrightarrow{a}+\overrightarrow{b})\times (\overrightarrow{a}+\overrightarrow{c}\left)\right)$ equals