Straight Lines · JEE Main previous year questions

If the angles of elevation of the top of a tower from three collinear points $A,B$ and $C$, on a line leading to the foot of the tower, are $30^{\circ },45^{\circ }$ and $60^{\circ }$ respectively, then the ratio, $AB:BC$, is :

1. A.$1:\sqrt{3}$
2. B.$2:3$
3. C.$\sqrt{3}:1$
4. D.$\sqrt{3}:\sqrt{2}$

The angle between the lines whose direction cosines satisfy the equations $l+m+n=0$ and $l^2=m^2+n^2$ is :

1. A.$\frac{\pi }{6}$
2. B.$\frac{\pi }{2}$
3. C.$\frac{\pi }{3}$
4. D.$\frac{\pi }{4}$

Consider the straight lines $L_1:x-y=1$ $L_2:x+y=1$ $L_3:2x+2y=5$ $L_4:2x-2y=7$ The correct statement is

1. A.$L_1\parallel L_4,L_2\parallel L_3,L_1\text{ intersect }{L}_4$.
2. B.$L_1\perp L_2,L_1\parallel L_3,L_1\text{ intersect }{L}_2$.
3. C.$L_1\perp L_2,L_2\parallel L_3,L_1\text{ intersect }{L}_4$.
4. D.$L_1\perp L_2,L_1\perp L_3,L_2\text{ intersect }{L}_4$.

If the point $(1,a)$ lies between the straight lines $x+y=1$ and $2(x+y)=3$ then $a$ lies in interval

1. A.$\left(\frac{3}{2},\infty \right)$
2. B.$\left(1,\frac{3}{2}\right)$
3. C.$(-\infty ,0)$
4. D.$\left(0,\frac{1}{2}\right)$

If two vertices of a triangle are $(5,-1)$ and $(-2,3)$ and its orthocentre is at $(0,0)$, then the third vertex is

1. A.$(4,-7)$
2. B.$(-4,-7)$
3. C.$(-4,7)$
4. D.$(4,7)$

If the straight lines $x+3y=4,3x+y=4$ and $x+y=0$ form a triangle, then the triangle is

1. A.scalene
2. B.equilateral triangle
3. C.isosceles
4. D.right angled isosceles

The line $L$ given by $\frac{x}{5}+\frac{y}{b}=1$ passes through the point $(13,32)$. The line $\mathrm{K}$ is parrallel to $L$ and has the equation $\frac{x}{c}+\frac{y}{3}=1$. Then the distance between $L$ and $K$ is :

1. A.$\sqrt{17}$
2. B.$\frac{17}{\sqrt{15}}$
3. C.$\frac{23}{\sqrt{17}}$
4. D.$\frac{23}{\sqrt{15}}$

A straight line through the point $A(3,4)$ is such that its intercept between the axes is bisected at $A$. Its equation is :

1. A.$x+y=7$
2. B.$3x-4y+7=0$
3. C.$4x+3y=24$
4. D.$3x+4y=25$

If a vertex of a triangle is $(1,1)$ and the mid points of two sides through this vertex are $(-1,2)$ and $(3,2)$ then the centroid of the triangle is :

1. A.$\left(-1,\frac{7}{3}\right)$
2. B.$\left(\frac{-1}{3},\frac{7}{3}\right)$
3. C.$\left(1,\frac{7}{3}\right)$
4. D.$\left(\frac{1}{3},\frac{7}{3}\right)$

The angle between the lines $2x=3y=-z$ and $6x=-y=-4z$ is :

1. A.$0^{\circ }$
2. B.$90^{\circ }$
3. C.$45^{\circ }$
4. D.$30^{\circ }$

A lizard, at an initial distance of $21\mathrm{cm}$ behind an insect moves from rest with an acceleration of $2cm/{\mathrm{s}}^2$ and pursues the insect which is crawling uniformly along a straight line at a speed of $20cm/s$. Then the lizard will catch the insect after :

1. A.$20\mathrm{s}$
2. B.$1\mathrm{s}$
3. C.$21\mathrm{s}$
4. D.$24\mathrm{s}$

Two points $A$ and $B$ move from rest along a straight line with constant acceleration $f$ and $f^{\prime }$ respectively. If $A$ takes $m$ sec. more than $B$ and describes '$n$' units more than $B$ in acquiring the same speed then

1. A.$(f-f^{\prime })m^2=f{f}^{\prime }n$
2. B.$(f+f^{\prime })m^2=f{f}^{\prime }n$
3. C.$\frac{1}{2}(f+f^{\prime })m=f{f}^{\prime }{n}^2$
4. D.$(f^{\prime }-f)n=\frac{1}{2}f{f}^{\prime }{m}^2$

$ABC$ is a triangle. Forces $\vec{P},\vec{Q},\vec{R}$ acting along $IA,IB$ and $IC$ respectively are in equilibrium, where $I$ is the incentre of $\Delta ABC$. Then $P:Q:R$ is

1. A.$\sin A:\sin B:\sin C$
2. B.$\sin \frac{A}{2}:\sin \frac{B}{2}:\sin \frac{C}{2}$
3. C.$\cos \frac{A}{2}:\cos \frac{B}{2}:\cos \frac{C}{2}$
4. D.$\cos A:\cos B:\cos C$

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 the vertex $C$ is the line :

1. A.$3x-2y=3$
2. B.$2x-3y=7$
3. C.$3x+2y=5$
4. D.$2x+3y=9$

The equation of the straight line passing through the point $(4,3)$ and making intercepts on the co-ordinate axes whose sum is $-1$ is :

1. A.$\frac{x}{2}-\frac{y}{3}=1$ and $\frac{x}{-2}+\frac{y}{1}=1$
2. B.$\frac{x}{2}-\frac{y}{3}=-1$ and $\frac{x}{-2}+\frac{y}{1}=-1$
3. C.$\frac{x}{2}+\frac{y}{3}=1$ and $\frac{x}{2}+\frac{y}{1}=1$
4. D.$\frac{x}{2}+\frac{y}{3}=-1$ and $\frac{x}{-2}+\frac{y}{1}=-1$

If $y=f(x)$ makes $+ve$ intercept of $2$ and $0$ unit on $x$ and $y$ axes and encloses an area of $3/4$ square unit with the axes then ${\int }_0^{2}x{f}^{\prime }(x)dx$ is

1. A.$3/2$
2. B.$1$
3. C.$5/4$
4. D.$-3/4$

A triangle with vertices $(4,0),(-1,-1),(3,5)$ is :

1. A.isosceles and right angled
2. B.isosceles but not right angled
3. C.right angled but not isosceles
4. D.neither right angled nor isosceles

The pair of lines represented by $3a{x}^2+5xy+(a^2-2)y^2=0$ are perpendicular to each other for :

1. A.two values of $a$
2. B.$\forall a$
3. C.for one value of $a$
4. D.for no values of $a$