Straight Lines · JEE Main previous year questions

If a straight line drawn through the point of intersection of the lines $4x+3y-1=0$ and $3x+4y-1=0$, meets the co-ordinate axes at the points P and Q, then the locus of the mid point of PQ is :

1. A.$x+y-7=0$
2. B.$x+y-14xy=0$
3. C.$2x+y+14xy=0$
4. D.$x+2y-14xy=0$

Given below are two statements : Statement I : 3-phenylpropene reacts with HBr and gives secondary alkyl bromide having a chiral carbon atom as the major product. Statement II : Aryl chlorides and aryl cyanides can be prepared by Sandmeyer reaction as well as Gattermann reaction. In the light of the above statements, choose the correct answer from the options given below

1. A.Both Statement I and Statement II are true
2. B.Both Statement I and Statement II are false
3. C.Statement I is true but Statement II is false
4. D.Statement I is false but Statement II is true

In an equilateral triangle $\mathrm{PQR}$, let the vertex $\mathrm{P}$ be at $(3,5)$ and the side $\mathrm{QR}$ be along the line $x+y=4$. If the orthocentre of the triangle $\mathrm{PQR}$ is $(\alpha ,\beta )$, then $9(\alpha +\beta )$ is equal to:

1. A.16
2. B.27
3. C.36
4. D.48

Let a triangle $\mathrm{PQR}$ be such that $\mathrm{P}$ and $\mathrm{Q}$ lie on the line $\frac{x+3}{8}=\frac{y-4}{2}=\frac{z+1}{2}$ and are at a distance of 6 units from $\mathrm{R}(1,2,3)$. If $(\alpha ,\beta ,\gamma )$ is the centroid of $\Delta \mathrm{PQR}$, then $\alpha +\beta +\gamma$ is equal to :

1. A.$4$
2. B.$5$
3. C.$6$
4. D.$8$

From the point $(-1,-1)$, two rays are sent making angles of $45^{\circ }$ with the line $x+y=0$. These rays get reflected from the mirror $x+2y=1$. If the equations of the reflected rays are $ax+by=9$ and $cx+dy=7$, $a,b,c,d\in \mathbf{Z}$, then the value of $ad+bc$ is ____.

Let the line $L_1:x+3=0$ intersect the lines $L_2:x-y=0$ and $L_3:3x+y=0$ at the points A and B, respectively. Let the bisector of the obtuse angle between the lines $L_2$ and $L_3$ intersect the line $L_1$ at the point C. Then ${\mathrm{BC}}^2:{\mathrm{AC}}^2$ is equal to:

1. A.$5:1$
2. B.$1:5$
3. C.$2:3$
4. D.$3:2$

Let the vertex A of a triangle ABC be $(1,2)$, and the mid-point of the side AB be $(5,-1)$. If the centroid of this triangle is $(3,4)$ and its circumcenter is $(\alpha ,\beta )$, then $21(\alpha +\beta )$ is equal to:

1. A.309
2. B.403
3. C.497
4. D.524

Given below are two statements : Statement (I) : The first ionisation enthalpy of the elements Na, Mg, Cl and Ar follows the order Na > Mg > Cl > Ar. Statement (II) : Among Ca, Al, Fe and B, the third ionisation enthalpy is very high for Ca. In the light of the above statements, choose the correct answer from the options given below :

1. A.Both Statement I and Statement II are true
2. B.Both Statement I and Statement II are false
3. C.Statement I is true but Statement II is false
4. D.Statement I is false but Statement II is true

Let $A(1,0),B(2,-1)$ and $C\left(\frac{7}{3},\frac{4}{3}\right)$ be three points. If the equation of the bisector of the angle ABC is $\alpha x+\beta y=5$, then the value of ${\alpha }^2+{\beta }^2$ is

1. A.5
2. B.10
3. C.8
4. D.13

Let A(1, 2) and C(-3, -6) be two diagonally opposite vertices of a rhombus, whose sides AD and BC are parallel to the line $7x-y=14$. If $\mathrm{B}(\alpha ,\beta )$ and $\mathrm{D}(\gamma ,\delta )$ are the other two vertices, then $|\alpha +\beta +\gamma +\delta |$ is equal to :

1. A.3
2. B.6
3. C.1
4. D.9

A line passing through the point $P(a,0)$ makes an acute angle $\alpha$ with the positive x-axis. Let this line be rotated about the point $P$ through an angle $\frac{\alpha }{2}$ in the clockwise direction. If in the new position, the slope of the line is $2-\sqrt{3}$ and its distance from the origin is $\frac{1}{\sqrt{2}}$, then the value of $3a^2{\tan }^2\alpha -2\sqrt{3}$ is :

1. A.$8$
2. B.$4$
3. C.$5$
4. D.$6$

Let the area of the triangle formed by the lines $x+2=y-1=z$, $\frac{x-3}{5}=\frac{y}{-1}=\frac{z-1}{1}$ and $\frac{x}{-3}=\frac{y-3}{3}=\frac{z-2}{1}$ be $A$. Then $A^2$ is equal to $\underline{}$.

Let the three sides of a triangle lie on the lines $4x-7y+10=0,x+y=5$ and $7x+4y=15$. Then the distance of its orthocentre from the orthocentre of the triangle formed by the lines $x=0,y=0$ and $x+y=1$ is

1. A.$\sqrt{20}$
2. B.$20$
3. C.$\sqrt{5}$
4. D.$5$

A line passes through the origin and makes equal angles with the positive coordinate axes. It intersects the lines ${\mathrm{L}}_1:2x+y+6=0$ and ${\mathrm{L}}_2:4x+2y-p=0$, $p>0$, at the points A and B , respectively. If $AB=\frac{9}{\sqrt{2}}$ and the foot of the perpendicular from the point $A$ on the line $L_2$ is $M$, then $\frac{AM}{BM}$ is equal to

1. A.5
2. B.3
3. C.2
4. D.4

If $z_1,z_2,z_3\in \mathbb{C}$ are the vertices of an equilateral triangle whose centroid is $z_0$, then ${\sum }_{k=1}^3(z_k-z_0{)}^2$ is equal to

1. A.$0$
2. B.$1$
3. C.$\mathrm{i}$
4. D.$-\mathrm{i}$

Let the area of the triangle formed by a straight line $L:x+by+c=0$ with co-ordinate axes be 48 square units. If the perpendicular drawn from the origin to the line L makes an angle of $45^{\circ }$ with the positive $x$-axis, then the value of $b^2+c^2$ is :

1. A.90
2. B.83
3. C.93
4. D.97

Let the lines $3x-4y-\alpha =0,8x-11y-33=0$, and $2x-3y+\lambda =0$ be concurrent. If the image of the point $(1,2)$ in the line $2x-3y+\lambda =0$ is $\left(\frac{57}{13},\frac{-40}{13}\right)$, then $|\alpha \lambda |$ is equal to

1. A.$91$
2. B.$113$
3. C.$101$
4. D.$84$

Let the points $\left(\frac{11}{2},\alpha \right)$ lie on or inside the triangle with sides $x+y=11,x+2y=16$ and $2x+3y=29$. Then the product of the smallest and the largest values of $\alpha$ is equal to :

1. A.22
2. B.33
3. C.55
4. D.44

Let the triangle PQR be the image of the triangle with vertices $(1,3)$, $(3,1)$ and $(2,4)$ in the line $x+2y=2$. If the centroid of $△\mathrm{PQR}$ is the point $(\alpha ,\beta )$, then $15(\alpha -\beta )$ is equal to :

1. A.21
2. B.19
3. C.22
4. D.24

The equations of two sides AB and AC of a triangle ABC are $4x+y=14$ and $3x-2y=5$, respectively. The point $(2,-\frac{4}{3})$ divides the third side BC internally in the ratio $2:1$, the equation of the side BC is

1. A.$x+6y+6=0$
2. B.$x-3y-6=0$
3. C.$x+3y+2=0$
4. D.$x-6y-10=0$

If the orthocentre of the triangle formed by the lines $2x+3y-1=0,x+2y-1=0$ and $ax+by-1=0$, is the centroid of another triangle, whose circumcentre and orthocentre respectively are $(3,4)$ and $(-6,-8)$, then the value of $|a-b|$ is ________.

Let a ray of light passing through the point $(3,10)$ reflects on the line $2x+y=6$ and the reflected ray passes through the point $(7,2)$. If the equation of the incident ray is $ax+by+1=0$, then $a^2+b^2+3ab$ is equal to ____.

Let $\text{A}(-1,1)$ and $\text{B}(2,3)$ be two points and P be a variable point above the line AB such that the area of $△\text{PAB}$ is 10. If the locus of P is $ax+by=15$, then $5a+2b$ is :

1. A.$-\frac{12}{5}$
2. B.$-\frac{6}{5}$
3. C.6
4. D.4

Let $\mathrm{P}$ and $\mathrm{Q}$ be the points on the line $\frac{x+3}{8}=\frac{y-4}{2}=\frac{z+1}{2}$ which are at a distance of 6 units from the point $\mathrm{R}(1,2,3)$. If the centroid of the triangle $\mathrm{PQR}$ is $(\alpha ,\beta ,\gamma )$, then ${\alpha }^2+{\beta }^2+{\gamma }^2$ is :

1. A.18
2. B.24
3. C.26
4. D.36

Let $A(a,b),B(3,4)$ and $C(-6,-8)$ respectively denote the centroid, circumcentre and orthocentre of a triangle. Then, the distance of the point $P(2a+3,7b+5)$ from the line $2x+3y-4=0$ measured parallel to the line $x-2y-1=0$ is

1. A.$\frac{17\sqrt{5}}{6}$
2. B.$\frac{15\sqrt{5}}{7}$
3. C.$\frac{17\sqrt{5}}{7}$
4. D.$\frac{\sqrt{5}}{17}$