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JEE MAIN Previous Year Questions (PYQs)

JEE MAIN Straight Line PYQ

JEE MAIN PYQ
Let the three sides of a triangle be 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





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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2025 (4 April Morning Shift) PYQ

Solution

JEE MAIN PYQ
Let the area of $\triangle PQR$ with vertices $P(5,4),\ Q(-2,4)$ and $R(a,b)$ be $35$ square units. If its orthocenter and centroid are $O\!\left(2,\dfrac{14}{5}\right)$ and $C(c,d)$ respectively, then $c+2d$ is equal to:





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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2025 (23 January Morning Shift) PYQ

Solution

JEE MAIN PYQ
A ray of light coming from the point (2, $2\sqrt 3 $) is incident at an angle 30o on the line x = 1 at thepoint A. The ray gets reflected on the line x = 1 and meets x-axis at the point B. Then, the line ABpasses through the point :





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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 6 September 2020 (Morning) PYQ

Solution

JEE MAIN PYQ
A straight line $L$ at a distance of $4$ units from the origin makes positive intercepts on the coordinate axes and the perpendicular from the origin to this line makes an angle of $60^\circ$ with the line $x + y = 0$. Then an equation of the line $L$ is:





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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2019 (12 April Evening Shift) PYQ

Solution

JEE MAIN PYQ
Let $a,b,c$ and $d$ be non-zero numbers. If the point of intersection of the lines $4ax+2ay+c=0$ and $5bx+2by+d=0$ lies in the fourth quadrant and is equidistant from the two axes then :





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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2014 (Offline) PYQ

Solution

JEE MAIN PYQ
Let $PS$ be the median of the triangle with vertices $P(2,2)$, $Q(6,-1)$ and $R(7,3)$. The equation of the line passing through $(1,-1)$ and parallel to $PS$ is :





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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2014 (Offline) PYQ

Solution

JEE MAIN PYQ
Let L denote the line in the xy-plane with x and y intercepts as 3 and 1 respectively. Then the image of the point (–1, –4) in this line is :





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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 6 September 2020 (Evening) PYQ

Solution

JEE MAIN PYQ
Let $A$ be the point of intersection of the lines $L_{1}:\ \dfrac{x-7}{1}=\dfrac{y-5}{0}=\dfrac{z-3}{-1}$ and $L_{2}:\ \dfrac{x-1}{3}=\dfrac{y+3}{4}=\dfrac{z+7}{5}$. Let $B$ and $C$ be the points on the lines $L_{1}$ and $L_{2}$ respectively such that $AB=AC=\sqrt{15}$. Then the square of the area of the triangle $ABC$ is:





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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2025 (4 April Evening Shift) PYQ

Solution

JEE MAIN PYQ
Two sides of a parallelogram are along the lines, $x+y=3$ and $x-y+3=0$. If its diagonals intersect at $(2,4)$, then one of its vertices is :





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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2019 (10 January Evening Shift) PYQ

Solution

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





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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2014 (Offline) PYQ

Solution

JEE MAIN PYQ
The straight lines $l_1$ and $l_2$ pass through the origin and trisect the line segment of the line $L : 9x + 5y = 45$ between the axes. If $m_1$ and $m_2$ are the slopes of the lines $l_1$ and $l_2$, then the point of intersection of the line $y = (m_1 + m_2)x$ with $L$ lies on :





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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2023 (6 April Morning Shift) PYQ

Solution

JEE MAIN PYQ
Two straight lines through the origin $O$ intersect the line $3x+4y=12$ at points $P$ and $Q$ such that $\triangle OPQ$ is isosceles and $\angle POQ=90^\circ$. If $I=OP^2+PQ^2+QO^2$, then the greatest integer $\le I$ is:





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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2024 (5 April Morning Shift) PYQ

Solution

JEE MAIN PYQ
Let $\triangle ABC$ be the triangle such that the equations of lines $AB$ and $AC$ are $3y-x=2$ and $x+y=2$, respectively, and the points $B$ and $C$ lie on the $x$-axis. If $P$ is the orthocentre of $\triangle ABC$, then the area of $\triangle PBC$ is equal to





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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2025 (7 April Morning Shift) PYQ

Solution

JEE MAIN PYQ
Let a, b$ \in $R. If the mirror image of the point P(a, 6, 9) with respect to the line ${{x - 3} \over 7} = {{y - 2} \over 5} = {{z - 1} \over { - 9}}$ is (20, b, $-$a$-$9), then | a + b |, is equal to :





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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2021 (24 February Evening Shift) PYQ

Solution

JEE MAIN PYQ
Let $A(1,1)$, $B(-4,3)$, $C(-2,-5)$ be vertices of a triangle $ABC$, $P$ be a point on side $BC$, and $\Delta_1$ and $\Delta_2$ be the areas of triangles $APB$ and $ABC$, respectively. If $\Delta_1:\Delta_2=4:7$, then the area enclosed by the lines $AP$, $AC$ and the $x$-axis is:





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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2022 (27 July Morning Shift) PYQ

Solution

JEE MAIN PYQ
A ray of light along $x+\sqrt{3}\,y=\sqrt{3}$ gets reflected upon reaching $X$-axis, the equation of the reflected ray is :





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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2013 (Offline) PYQ

Solution

JEE MAIN PYQ
If for $\theta\in\left[-\dfrac{\pi}{3},0\right]$, the points $(x,y)=\big(3\tan(\theta+\tfrac{\pi}{3}),,2\tan(\theta+\tfrac{\pi}{6})\big)$ lie on $xy+\alpha x+\beta y+\gamma=0$, then $\alpha^{2}+\beta^{2}+\gamma^{2}$ is:





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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2025 (7 April Morning Shift) PYQ

Solution

JEE MAIN PYQ
Let $d$ be the distance of the point of intersection of the lines $\dfrac{x+6}{3}=\dfrac{y}{2}=\dfrac{z+1}{1}$ and $\dfrac{x-7}{4}=\dfrac{y-9}{3}=\dfrac{z-4}{2}$ from the point $(7,8,9)$. Then $d^{2}+6$ is equal to:





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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2024 (5 April Morning Shift) PYQ

Solution

JEE MAIN PYQ
If the lines ${{x - 2} \over 1} = {{y - 3} \over 1} = {{z - 4} \over { - k}}$ and ${{x - 1} \over k} = {{y - 4} \over 2} = {{z - 5} \over 1}$ are coplanar, then $k$ can have :





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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2013 (Offline) PYQ

Solution

JEE MAIN PYQ
The portion of the line $4x+5y=20$ in the first quadrant is trisected by the lines $L_1$ and $L_2$ passing through the origin. The tangent of the angle between the lines $L_1$ and $L_2$ is:





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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2024 (27 January Morning Shift) PYQ

Solution

JEE MAIN PYQ
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 $\left(\dfrac{57}{13}, -\dfrac{40}{13}\right)$, then $|\alpha \lambda|$ is equal to





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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2025 (24 January Morning Shift) PYQ

Solution

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





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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2024 (5 April Evening Shift) PYQ

Solution

JEE MAIN PYQ
$ \text{The shortest distance between the lines } \dfrac{x-4}{4}=\dfrac{y+2}{5}=\dfrac{z+3}{3} \text{ and } \dfrac{x-1}{3}=\dfrac{y-3}{4}=\dfrac{z-4}{2} \text{ is:} $





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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2023 (8 April Morning Shift) PYQ

Solution

JEE MAIN PYQ
The foot of the perpendicular drawn from the origin, on the line, $3x + y = \lambda\ (\lambda \ne 0)$ is $P$. If the line meets $x$-axis at $A$ and $y$-axis at $B$, then the ratio $BP : PA$ is :





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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2018 (15 April Evening Shift) PYQ

Solution

JEE MAIN PYQ
The sides of a rhombus $ABCD$ are parallel to the lines, $x - y + 2 = 0$ and $7x - y + 3 = 0$. If the diagonals of the rhombus intersect $P(1,2)$ and the vertex $A$ (different from the origin) is on the $y$-axis, then the coordinate of $A$ is :





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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2018 (15 April Evening Shift) PYQ

Solution

JEE MAIN PYQ
Let the area of the triangle with vertices $A(1,\alpha)$, $B(\alpha,0)$ and $C(0,\alpha)$ be $4$ sq. units. If the points $(\alpha,-\alpha)$, $(-\alpha,\alpha)$ and $(\alpha^2,\beta)$ are collinear, then $\beta$ is equal to:





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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2022 (24 June Evening Shift) PYQ

Solution

JEE MAIN PYQ
Let the image of the point $(1,0,7)$ in the line $\dfrac{x}{1}=\dfrac{y-1}{2}=\dfrac{z-2}{3}$ be the point $(\alpha,\beta,\gamma)$. Then which one of the following points lies on the line passing through $(\alpha,\beta,\gamma)$ and making angles $\dfrac{2\pi}{3}$ and $\dfrac{3\pi}{4}$ with the $y$-axis and $z$-axis respectively, and an acute angle with the $x$-axis?





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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2024 (27 January Evening Shift) PYQ

Solution

JEE MAIN PYQ
If in a parallelogram $ABDC$, the coordinates of $A, B$ and $C$ are respectively $(1,2)$, $(3,4)$ and $(2,5)$, then the equation of the diagonal $AD$ is :





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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2019 (11 January Evening Shift) PYQ

Solution

JEE MAIN PYQ
If the orthocenter of the triangle formed by the lines y = x + 1, y = 4x - 8 and y = mx + c is at (3, -1), then m - c is :





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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2025 (7 April Evening Shift) PYQ

Solution

JEE MAIN PYQ
Let the points $\left(\dfrac{11}{2},,\alpha\right)$ 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:





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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2025 (24 January Evening Shift) PYQ

Solution

JEE MAIN PYQ
Let $R$ be the interior region between the lines $3x - y + 1 = 0$ and $x + 2y - 5 = 0$ containing the origin. The set of all values of $a$, for which the points $(a^2,\,a+1)$ lie in $R$, is:





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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2024 (27 January Evening Shift) PYQ

Solution

JEE MAIN PYQ
Let a variable line of slope $m>0$ passing through $(4,-9)$ intersect the coordinate axes at points $A$ and $B$. The minimum value of the sum of the distances of $A$ and $B$ from the origin is:





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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2024 (6 April Morning Shift) PYQ

Solution

JEE MAIN PYQ
A straight line through a fixed point $(2,3)$ intersects the coordinate axes at distinct points $P$ and $Q$. If $O$ is the origin and the rectangle $OPRQ$ is completed, then the locus of $R$ is :





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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2018 (Offline) PYQ

Solution

JEE MAIN PYQ
The intersection of three lines x - y = 0, x + 2y = 3 and 2x + y = 6 is a





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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2021 (26 February Morning Shift) PYQ

Solution

JEE MAIN PYQ
The shortest distance between the lines $\dfrac{x+2}{1}=\dfrac{y}{-2}=\dfrac{z-5}{2}$ and $\dfrac{x-4}{1}=\dfrac{y-1}{2}=\dfrac{z+3}{0}$ is:





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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2023 (10 April Morning Shift) PYQ

Solution

JEE MAIN PYQ
An ordered pair ($\alpha $, $\beta $) for which the system of linear equations
(1 + $\alpha $) x + $\beta $y + z = 2
$\alpha $x + (1 + $\beta $)y + z = 3
$\alpha $x + $\beta $y + 2z = 2
has a unique solution, is :





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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2019 (12 January Morning Shift) PYQ

Solution

JEE MAIN PYQ
$\text { If } f(x)=\left\{\begin{array}{ll} x^3 \sin \left(\frac{1}{x}\right), & x \neq 0 \\ 0 & , x=0 \end{array}\right. \text {, then }$





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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2024 (6 April Morning Shift) PYQ

Solution

JEE MAIN PYQ
In $\triangle ABC$, suppose $y=x$ is the equation of the bisector of the angle $B$ and the equation of the side $AC$ is $2x-y=2$. If $2AB=BC$ and the points $A$ and $B$ are respectively $(4,6)$ and $(\alpha,\beta)$, then $\alpha+2\beta$ is equal to:





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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2024 (29 January Morning Shift) PYQ

Solution

JEE MAIN PYQ
If the straight line $2x-3y+17=0$ is perpendicular to the line passing through the points $(7,17)$ and $(15,\beta)$, then $\beta$ equals :





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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2019 (12 January Morning Shift) PYQ

Solution

JEE MAIN PYQ
If a circle of radius $R$ passes through the origin $O$ and intersects the coordinate axes at $A$ and $B$, then the locus of the foot of the perpendicular from $O$ on $AB$ is:





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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2019 (12 January Evening Shift) PYQ

Solution

JEE MAIN PYQ
Let $A$ be the point of intersection of the lines $3x+2y=14$ and $5x-y=6$, and $B$ be the point of intersection of the lines $4x+3y=8$ and $6x+y=5$. The distance of the point $P(5,-2)$ from the line $AB$ is:





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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2024 (29 January Evening Shift) PYQ

Solution

JEE MAIN PYQ
The square of the distance of the point $\left(\dfrac{15}{7},,\dfrac{32}{7},,7\right)$ from the line $\dfrac{x+1}{3}=\dfrac{y+3}{5}=\dfrac{z+5}{7}$ in the direction of the vector $\hat{i}+4\hat{j}+7\hat{k}$ is:





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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2025 (28 January Evening Shift) PYQ

Solution

JEE MAIN PYQ
The distance of the point $(2,3)$ from the line $2x-3y+28=0$, measured parallel to the line $\sqrt{3}\,x-y+1=0$, is equal to:





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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2024 (29 January Evening Shift) PYQ

Solution

JEE MAIN PYQ
Two equal sides of an isosceles triangle are along $-x+2y=4$ and $x+y=4$. If $m$ is the slope of its third side, then the sum of all possible distinct values of $m$ is:





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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2025 (28 January Evening Shift) PYQ

Solution

JEE MAIN PYQ
Let $m_1, m_2$ be the slopes of two adjacent sides of a square of side $a$ such that $a^{2}+11a+3\left(m_{1}^{2}+m_{2}^{2}\right)=220.$ If one vertex of the square is $\big(10(\cos\alpha-\sin\alpha),\,10(\sin\alpha+\cos\alpha)\big)$, where $\alpha\in(0,\tfrac{\pi}{2})$, and the equation of one diagonal is $(\cos\alpha-\sin\alpha)x+(\sin\alpha+\cos\alpha)y=10$, then $ 72\left(\sin^{4}\alpha+\cos^{4}\alpha\right)+a^{2}-3a+13 $ is equal to:





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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2022 (29 July Evening Shift) PYQ

Solution

JEE MAIN PYQ
If a straight line passing through the point $P(-3,4)$ is such that its intercepted portion between the coordinate axes is bisected at $P$, then its equation is:





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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2019 (12 January Evening Shift) PYQ

Solution

JEE MAIN PYQ
Let $P(\alpha,\beta,\gamma)$ be the image of the point $Q(3,-3,1)$ in the line $\dfrac{x-0}{1}=\dfrac{y-3}{1}=\dfrac{z-1}{-1}$ and let $R$ be the point $(2,5,-1)$. If the area of $\triangle PQR$ is $\lambda$ and $\lambda^{2}=14K$, then $K$ is:





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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2024 (6 April Evening Shift) PYQ

Solution

JEE MAIN PYQ
If the angle between the lines $\dfrac{x}{2}=\dfrac{y}{2}=\dfrac{z}{1}$ and $\dfrac{5-x}{-2}=\dfrac{7y-14}{p}=\dfrac{z-3}{4}$ is $\cos^{-1}\left(\dfrac{2}{3}\right)$, then $p$ is equal to :





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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2018 (16 April Morning Shift) PYQ

Solution

JEE MAIN PYQ
Let $R$ be a rectangle given by the lines $x = 0$, $x = 2$, $y = 0$ and $y = 5$. Let $A(\alpha, 0)$ and $B(0, \beta)$, $\alpha \in [0, 2]$ and $\beta \in [0, 5]$, be such that the line segment $AB$ divides the area of the rectangle $R$ in the ratio $4 : 1$. Then, the mid-point of $AB$ lies on a:





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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2023 (11 April Morning Shift) PYQ

Solution

JEE MAIN PYQ
Let a be the length of a side of a square OABC with O being the origin. Its side OA makes an acute angle $\alpha $ with the positive x-axis and the equations of its diagonals are $(\sqrt{3}+1)x+(\sqrt{3}-1)y=0$ and $(\sqrt{3}-1)x-(\sqrt{3}+1)y+8\sqrt{3}=0$. Then $a$2 is equal to :





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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2025 (8 April Evening Shift) PYQ

Solution

JEE MAIN PYQ
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 $\dfrac{\alpha}{2}$ 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 $\dfrac{1}{\sqrt{2}}$, then the value of $3a^2 \tan^2 \alpha - 2\sqrt{3}$ is:





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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2025 (8 April Evening Shift) PYQ

Solution

JEE MAIN PYQ
If the foot of the perpendicular from point (4, 3, 8) on the line ${L_1}:{{x - a} \over l} = {{y - 2} \over 3} = {{z - b} \over 4}$, l $\ne$ 0 is (3, 5, 7), then the shortest distance between the line L1 and line ${L_2}:{{x - 2} \over 3} = {{y - 4} \over 4} = {{z - 5} \over 5}$ is equal to :





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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2021 (16 March Evening Shift) PYQ

Solution

JEE MAIN PYQ
A line passing through the point \(A(9,0)\) makes an angle of \(30^\circ\) with the positive direction of the \(x\)-axis. If this line is rotated about \(A\) through an angle of \(15^\circ\) in the clockwise direction, then its equation in the new position is: C





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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2024 (30 January Morning Shift) PYQ

Solution

JEE MAIN PYQ
The set of all possible values of $\theta $ in the interval (0, $\pi $) for which the points (1, 2) and (sin $\theta $, cos $\theta $) lie on the same side of the line x + y =1 is :





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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2 September 2020 (Evening) PYQ

Solution

JEE MAIN PYQ
A point on the straight line $3x+5y=15$ which is equidistant from the coordinate axes will lie only in:





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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2019 (8 April Morning Shift) PYQ

Solution

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





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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2024 (8 April Morning Shift) PYQ

Solution

JEE MAIN PYQ
Let A($-$1, 1), B(3, 4) and C(2, 0) be given three points. A line y = mx, m > 0, intersects lines AC and BC at point P and Q respectively. Let A1 and A2 be the areas of $\Delta$ABC and $\Delta$PQC respectively, such that A1 = 3A2, then the value of m is equal to :





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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2021 (16 March Evening Shift) PYQ

Solution

JEE MAIN PYQ
In an isosceles triangle ABC, the vertex A is (6, 1) and the equation of the base BC is 2x + y = 4. Let the point B lie on the line x + 3y = 7. If ($\alpha$, $\beta$) is the centroid of $\Delta$ABC, then 15($\alpha$ + $\beta$) is equal to :





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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2022 (27 June Morning Shift) PYQ

Solution

JEE MAIN PYQ
If the shortest distance between the lines $\dfrac{x-\lambda}{2}=\dfrac{y-4}{3}=\dfrac{z-3}{4}$ and $\dfrac{x-2}{4}=\dfrac{y-4}{6}=\dfrac{z-7}{8}$ is $\dfrac{13}{\sqrt{29}}$, then a value of $\lambda$ is:1





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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2024 (8 April Evening Shift) PYQ

Solution

JEE MAIN PYQ
Let the lines \[ \ell_1:\ \frac{x+5}{3}=\frac{y+4}{1}=\frac{z-\alpha}{-2} \quad\text{and}\quad \ell_2:\ 3x+2y+z-2=0\;=\;x-3y+2z-13 \] be coplanar. If the point $P(a,b,c)$ on $\ell_1$ is nearest to the point $Q(-4,-3,2)$, then $|a|+|b|+|c|$ is equal to:





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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2023 (12 April Morning Shift) PYQ

Solution

JEE MAIN PYQ
If the line segment joining the points $(5,2)$ and $(2,a)$ subtends an angle $\dfrac{\pi}{4}$ at the origin, then the absolute value of the product of all possible values of $a$ is:





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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2024 (8 April Evening Shift) PYQ

Solution

JEE MAIN PYQ
The shortest distance between the lines ${{x - 3} \over 2} = {{y - 2} \over 3} = {{z - 1} \over { - 1}}$ and ${{x + 3} \over 2} = {{y - 6} \over 1} = {{z - 5} \over 3}$, is :





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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2022 (27 June Evening Shift) PYQ

Solution

JEE MAIN PYQ
Let $A$ be the point of intersection of the lines $3x+2y=14$ and $5x-y=6$, and $B$ be the point of intersection of the lines $4x+3y=8$ and $6x+y=5$. The distance of the point $P(5,-2)$ from the line $AB$ is:





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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2025 (29 January Evening Shift) PYQ

Solution

JEE MAIN PYQ
The locus of the point of intersection of the straight lines, $tx-2y-3t=0$, $x-2ty+3=0\ (t\in\mathbb{R}),$ is :





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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2017 (8 April Morning Shift) PYQ

Solution

JEE MAIN PYQ
The distance of the point $(2,3)$ from the line $2x-3y+28=0$, measured parallel to the line $\sqrt{3},x-y+1=0$, is equal to:





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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2025 (29 January Evening Shift) PYQ

Solution

JEE MAIN PYQ
If the system of linear equations
$x-2y+kz=1$
$2x+y+z=2$
$3x-y-kz=3$
has a solution $(x,y,z)$ with $z\ne0$, then $(x,y)$ lies on the straight line whose equation is:





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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2019 (8 April Evening Shift) PYQ

Solution

JEE MAIN PYQ
Suppose the points $(h,k)$, $(1,2)$ and $(-3,4)$ lie on the line $L_1$. If a line $L_2$ passing through the points $(h,k)$ and $(4,3)$ is perpendicular to $L_1$, then $\dfrac{k}{h}$ equals:





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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2019 (8 April Evening Shift) PYQ

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JEE MAIN PYQ
The shortest distance between the lines $x+1=2y=-12z$ and $x=y+2=6z-6$ is:





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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2023 (25 January Evening Shift) PYQ

Solution

JEE MAIN PYQ
A man is walking on a straight line. The arithmetic mean of the reciprocals of the intercepts of this line on the coordinate axes is $\frac{1}{4}$ . Three stones A, B and C are placed at the points (1, 1), (2, 2) and (4, 4) respectively. Then, which of these stones is / are on the path of the man?





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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2021 (24 February Morning Shift) PYQ

Solution

JEE MAIN PYQ
Let $\alpha,\beta,\gamma,\delta\in\mathbb{Z}$ and let $A(\alpha,\beta),\ B(1,0),\ C(\gamma,\delta),\ D(1,2)$ be the vertices of a parallelogram $ABCD$. If $AB=\sqrt{10}$ and the points $A$ and $C$ lie on the line $3y=2x+1$, then $2(\alpha+\beta+\gamma+\delta)$ is equal to:





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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2024 (31 January Morning Shift) PYQ

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JEE MAIN PYQ
The number of integral values of m so that the abscissa of point of intersection of lines 3x + 4y = 9 and y = mx + 1 is also an integer, is :





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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2021 (18 March Morning Shift) PYQ

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JEE MAIN PYQ
The equation of one of the straight lines which passes through the point (1, 3) and makes an angles ${\tan ^{ - 1}}\left( {\sqrt 2 } \right)$ with the straight line, y + 1 = 3${\sqrt 2 }$ x is :





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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2021 (18 March Morning Shift) PYQ

Solution

JEE MAIN PYQ
Let the vertices $Q$ and $R$ of the triangle $PQR$ lie on the line $\dfrac{x + 3}{5} = \dfrac{y - 1}{2} = \dfrac{z + 4}{3}$, $QR = 5$ and the coordinates of the point $P$ be $(0, 2, 3)$. If the area of the triangle $PQR$ is $\dfrac{m}{n}$, then:





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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2025 (2 April Morning Shift) PYQ

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JEE MAIN PYQ
Let $B$ and $C$ be the two points on the line $y+x=0$ such that $B$ and $C$ are symmetric with respect to the origin. Suppose $A$ is a point on $y-2x=2$ such that $\triangle ABC$ is an equilateral triangle. Then, the area of the $\triangle ABC$ is:





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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2023 (29 January Morning Shift) PYQ

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JEE MAIN PYQ
A variable line $L$ passes through the point $(3,5)$ and intersects the positive coordinate axes at the points $A$ and $B$. The minimum area of the triangle $OAB$, where $O$ is the origin, is:





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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2024 (9 April Morning Shift) PYQ

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JEE MAIN PYQ
The line that is coplanar to the line $\dfrac{x+3}{-3}=\dfrac{y-1}{1}=\dfrac{z-5}{5}$ is:






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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2023 (13 April Evening Shift) PYQ

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JEE MAIN PYQ
Let a triangle be bounded by the lines L1 : 2x + 5y = 10; L2 : $-$4x + 3y = 12 and the line L3, which passes through the point P(2, 3), intersects L2 at A and L1 at B. If the point P divides the line-segment AB, internally in the ratio 1 : 3, then the area of the triangle is equal to :





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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2022 (28 June Evening Shift) PYQ

Solution

Let $A=(a,b)\in L_2\Rightarrow -4a+3b=12.$ Since $AP:PB=1:3$, by section formula $P=\dfrac{B+3A}{4}\Rightarrow B=4P-3A=(8-3a,\;12-3b).$ Because $B\in L_1$, $2(8-3a)+5(12-3b)=10\Rightarrow 2a+5b=22.$ Solve \[ \begin{cases} 2a+5b=22,\\ -4a+3b=12 \end{cases} \Rightarrow a=\dfrac{3}{13},\quad b=\dfrac{56}{13}. \] Thus \[ A=\left(\dfrac{3}{13},\dfrac{56}{13}\right),\quad B=\left(\dfrac{95}{13},-\dfrac{12}{13}\right). \] Intersection $C=L_1\cap L_2$: \[ \begin{cases} 2x+5y=10,\\ -4x+3y=12 \end{cases} \Rightarrow C=\left(-\dfrac{15}{13},\dfrac{32}{13}\right). \] Area \[ \Delta=\frac12\left| \begin{vmatrix} x_A&y_A&1\\ x_B&y_B&1\\ x_C&y_C&1 \end{vmatrix}\right| =\frac12\left|(B-A)\times(C-A)\right| =\frac12\left|(92)(-24)-(-68)(-18)\right| =\frac{1716}{169} =\boxed{\dfrac{132}{13}}. \] Answer: $\boxed{\dfrac{132}{13}}$.
JEE MAIN PYQ
Let $(\alpha,\beta)$ be the centroid of the triangle formed by the lines $15x-y=82$, $6x-5y=-4$ and $9x+4y=17$. Then $\alpha+2\beta$ and $2\alpha-\beta$ are the roots of the equation:





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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2023 (13 April Evening Shift) PYQ

Solution

JEE MAIN PYQ
A light ray emits from the origin making an angle $30^\circ$ with the positive $x$-axis. After getting reflected by the line $x+y=1$, if this ray intersects the $x$-axis at $Q$, then the abscissa of $Q$ is:





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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2023 (29 January Morning Shift) PYQ

Solution

$y=\tan30^\circ,x=\dfrac{x}{\sqrt3}$ hits the mirror $x+y=1$ at $P\left(\dfrac{\sqrt3}{\sqrt3+1},,\dfrac{1}{\sqrt3+1}\right)$. 
The mirror’s normal is along $(1,1)$, so reflecting the unit direction $u=(\cos30^\circ,\sin30^\circ)=\left(\dfrac{\sqrt3}{2},\dfrac12\right)$ about the line gives $u'=u-2(u\cdot \hat n)\hat n=\left(-\dfrac12,-\dfrac{\sqrt3}{2}\right)$, 
i.e. slope $m'=\sqrt3$. 
The reflected ray through $P$ is $y-y_0=\sqrt3(x-x_0)$. 
Intersecting $y=0$ gives $x=x_0-\dfrac{y_0}{\sqrt3} $
$=\dfrac{\sqrt3}{\sqrt3+1}-\dfrac{1}{\sqrt3(\sqrt3+1)}$
$=\dfrac{2}{3+\sqrt3}$
JEE MAIN PYQ
Slope of a line passing through $P(2, 3)$ and intersecting the line $x + y = 7$ at a distance of $4$ units from $P$, is:





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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2019 (9 April Morning Shift) PYQ

Solution

JEE MAIN PYQ
If p and q are the lengths of the perpendiculars from the origin on the lines,:- x cosec $\alpha$ $-$ y sec $\alpha$ = k cot 2$\alpha$ and, x sin$\alpha$ + y cos$\alpha$ = k sin2$\alpha$ respectively, then k2 is equal to :





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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2021 (31 August Morning Shift) PYQ

Solution

JEE MAIN PYQ
Two sides of a rhombus are along the lines, $x - y + 1 = 0$ and $7x - y - 5 = 0$. If its diagonals intersect at $(-1, -2)$, then which one of the following is a vertex of this rhombus?





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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2016 (Offline) PYQ

Solution

JEE MAIN PYQ
The shortest distance, between lines $L_1$ and $L_2$, where $L_1: \frac{x-1}{2}=\frac{y+1}{-3}=\frac{z+4}{2}$ and $L_2$ is the line, passing through the points $\mathrm{A}(-4,4,3), \mathrm{B}(-1,6,3)$ and perpendicular to the line $\frac{x-3}{-2}=\frac{y}{3}=\frac{z-1}{1}$, is





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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2024 (31 January Evening Shift) PYQ

Solution

JEE MAIN PYQ
If the image of the point $P(1, 0, 3)$ in the line joining the points $A(4, 7, 1)$ and $B(3, 5, 3)$ is $Q(\alpha, \beta, \gamma)$, then $\alpha + \beta + \gamma$ is equal to:





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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2025 (2 April Evening Shift) PYQ

Solution

JEE MAIN PYQ
Consider the line $L$ passing through the points $(1,2,3)$ and $(2,3,5)$. The distance of the point $\left(\dfrac{11}{3},\dfrac{11}{3},\dfrac{19}{3}\right)$ from the line $L$ along the line $\dfrac{3x-11}{2}=\dfrac{3y-11}{1}=\dfrac{3z-19}{2}$ is equal to:





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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2024 (9 April Evening Shift) PYQ

Solution

JEE MAIN PYQ
If the two lines $x+(a-1)y=1$ and $2x+a^{2}y=1$ $(a\in\mathbb{R}\setminus{0,1})$ are perpendicular, then the distance of their point of intersection from the origin is:





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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2019 (9 April Evening Shift) PYQ

Solution

JEE MAIN PYQ
The shortest distance between the lines \[ \frac{x-1}{2}=\frac{y+8}{-7}=\frac{z-4}{5} \quad\text{and}\quad \frac{x-1}{2}=\frac{y-2}{1}=\frac{z-6}{-3} \] is:





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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2023 (29 January Evening Shift) PYQ

Solution

JEE MAIN PYQ
Let a variable line passing through the centre of the circle $x^{2}+y^{2}-16x-4y=0$ meet the positive coordinate axes at the points $A$ and $B$. Then the minimum value of $OA+OB$, where $O$ is the origin, is:





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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2024 (31 January Evening Shift) PYQ

Solution

JEE MAIN PYQ
If the length of the perpendicular from the point $(\beta,0,\beta)\ (\beta\ne0)$ to the line, $\dfrac{x}{1}=\dfrac{y-1}{0}=\dfrac{z+1}{-1}$ is $\sqrt{\dfrac{3}{2}}$, then $\beta$ is equal to:





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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2019 (10 April Morning Shift) PYQ

Solution

JEE MAIN PYQ
The lines x = ay $-$ 1 = z $-$ 2 and x = 3y $-$ 2 = bz $-$ 2, (ab $\ne$ 0) are coplanar, if :





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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2021 (20 July Evening Shift) PYQ

Solution

JEE MAIN PYQ
The region represented by $|x-y|\le 2$ and $|x+y|\le 2$ is bounded by a:





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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2019 (10 April Morning Shift) PYQ

Solution

JEE MAIN PYQ
The point $(2,1)$ is translated parallel to the line $L : x - y = 4$ by $2\sqrt{3}$ units. If the new point $Q$ lies in the third quadrant, then the equation of the line passing through $Q$ and perpendicular to $L$ is:





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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2016 (9 April Morning Shift) PYQ

Solution

JEE MAIN PYQ
If a variable line drawn through the intersection of the lines $\dfrac{x}{3} + \dfrac{y}{4} = 1$ and $\dfrac{x}{4} + \dfrac{y}{3} = 1$ meets the coordinate axes at $A$ and $B$ $(A \ne B)$, then the locus of the midpoint of $AB$ is:





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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2016 (9 April Morning Shift) PYQ

Solution

JEE MAIN PYQ
Let a line passing through the point $(4,1,0)$ intersect the line $\mathrm{L}_1: \frac{x-1}{2}=\frac{y-2}{3}=\frac{z-3}{4}$ at the point $A(\alpha, \beta, \gamma)$ and the line $\mathrm{L}_2: x-6=y=-z+4$ at the point $B(a, b, c)$. Then $\left|\begin{array}{lll}1 & 0 & 1 \\ \alpha & \beta & \gamma \\ a & b & c\end{array}\right|$ is equal to





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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2025 (3 April Morning Shift) PYQ

Solution

JEE MAIN PYQ
Line $L_1$ passes through the point $(1, 2, 3)$ and is parallel to the $z$-axis. Line $L_2$ passes through the point $(\lambda, 5, 6)$ and is parallel to the $y$-axis. Let for $\lambda = \lambda_1, \lambda_2,$ $\lambda_2 < \lambda_1,$ the shortest distance between the two lines be $3$. Then the square of the distance of the point $(\lambda_1, \lambda_2, 7)$ from the line $L_1$ is





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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2025 (3 April Morning Shift) PYQ

Solution

JEE MAIN PYQ
If the perpendicular bisector of the line segment joining the points P(1 ,4) and Q(k, 3) has y-intercept equal to –4, then a value of k is :





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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 4 September 2020 (Evening) PYQ

Solution

JEE MAIN PYQ
If the lines $x=ay+b,\ z=cy+d$ and $x=a'z+b',\ y=c'z+d'$ are perpendicular, then:





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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2019 (9 January Evening Shift) PYQ

Solution

JEE MAIN PYQ
Let $\alpha_1, \alpha_2 ; (\alpha_1 < \alpha_2)$ be the values of $\alpha$ for the points $(\alpha, -3), (2, 0)$ and $(1, \alpha)$ to be collinear. Then the equation of the line, passing through $(\alpha_1, \alpha_2)$ and making an angle of $\frac{\pi}{3}$ with the positive direction of the x-axis, is :





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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2022 (30 June Morning Shift) PYQ

Solution

JEE MAIN PYQ
A line passes through the origin and makes equal angles with the positive coordinate axes. It intersects the lines $L_1: 2x+y+6=0$ and $L_2: 4x+2y-p=0,; p>0$ at the points $A$ and $B$, respectively. If $|AB|=\dfrac{9}{\sqrt{2}}$ and the foot of the perpendicular from the point $A$ on the line $L_2$ is $M$, then $\dfrac{AM}{BM}$ is equal to





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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2025 (3 April Morning Shift) PYQ

Solution

JEE MAIN PYQ
If the shortest distance between the straight lines $3(x - 1) = 6(y - 2) = 2(z - 1)$ and $4(x - 2) = 2(y - \lambda ) = (z - 3),\lambda \in R$ is ${1 \over {\sqrt {38} }}$, then the integral value of $\lambda$ is equal to :





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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2021 (22 July Evening Shift) PYQ

Solution

JEE MAIN PYQ
Consider the lines $x(3\lambda+1)+y(7\lambda+2)=17\lambda+5$, $\lambda$ being a parameter, all passing through a point $P$. One of these lines (say $L$) is farthest from the origin. If the distance of $L$ from the point $(3,6)$ is $d$, then the value of $d^2$ is:





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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2025 (3 April Evening Shift) PYQ

Solution

JEE MAIN PYQ
Lines are drawn parallel to the line $4x-3y+2=0$, at a distance $\dfrac{3}{5}$ from the origin. Then which one of the following points lies on any of these lines?





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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2019 (10 April Evening Shift) PYQ

Solution

JEE MAIN PYQ
A straight line through origin $O$ meets the lines $3y = 10 - 4x$ and $8x + 6y + 5 = 0$ at points $A$ and $B$ respectively. Then $O$ divides the segment $AB$ in the ratio :





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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2016 (10 April Morning Shift) PYQ

Solution

JEE MAIN PYQ
A ray of light is incident along a line which meets another line $7x - y + 1 = 0$ at the point $(0,1)$. The ray is then reflected from this point along the line $y + 2x = 1$. Then the equation of the line of incidence of the ray of light is:





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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2016 (10 April Morning Shift) PYQ

Solution

JEE MAIN PYQ
Let $\vec a=2\hat{\imath}+\hat{\jmath}+\hat{k}$, and $\vec b,\vec c$ be two nonzero vectors such that $\left\lvert \vec a+\vec b+\vec c \right\rvert=\left\lvert \vec a+\vec b-\vec c \right\rvert$ and $\vec b\cdot\vec c=0$. Consider the statements: (A) $\left\lvert \vec a+\lambda\vec c \right\rvert \ge \lvert \vec a\rvert \text{ for all } \lambda\in\mathbb{R}$. (B) $\vec a$ and $\vec c$ are always parallel. Then,





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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2023 (31 January Morning Shift) PYQ

Solution

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





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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2024 (1 February Evening Shift) PYQ

Solution

JEE MAIN PYQ
Let the point $P(\alpha, \beta)$ be at a unit distance from each of the two lines  
$L_1 : 3x - 4y + 12 = 0$, and $L_2 : 8x + 6y + 11 = 0$.  

If $P$ lies below $L_1$ and above $L_2$, then $100(\alpha + \beta)$ is equal to :






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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2022 (25 July Evening Shift) PYQ

Solution

JEE MAIN PYQ
Let the shortest distance between the lines $\dfrac{x-3}{3}=\dfrac{y-\alpha}{-1}=\dfrac{z-3}{1}$ and $\dfrac{x+3}{-3}=\dfrac{y+7}{2}=\dfrac{z-\beta}{4}$ be $3\sqrt{30}$. Then the positive value of $5\alpha+\beta$ is





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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2025 (4 April Morning Shift) PYQ

Solution

JEE MAIN PYQ
A point P moves on the line $2x - 3y + 4 = 0.$ If $Q(1, 4)$ and $R(3, -2)$ are fixed points, then the locus of the centroid of $\triangle PQR$ is a line :





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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2019 (10 January Morning Shift) PYQ

Solution

JEE MAIN PYQ
The vertices of a triangle are $A(-1,3)$, $B(-2,2)$ and $C(3,-1)$. A new triangle is formed by shifting the sides of the triangle by one unit inwards. Then the equation of the side of the new triangle nearest to the origin is:





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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2024 (4 April Morning Shift) PYQ

Solution

JEE MAIN PYQ
Locus of the image of the point $(2,3)$ in the line $(2x-3y+4)+k(x-2y+3)=0,\;k\in\mathbb{R},$ is a :





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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2015 (Offline) PYQ

Solution

JEE MAIN PYQ
A $2,\text{m}$ ladder leans against a vertical wall. If the top of the ladder begins to slide down the wall at the rate of $25,\text{cm/sec}$, then the rate (in $\text{cm/sec}$) at which the bottom of the ladder slides away from the wall on the horizontal ground when the top of the ladder is $1,\text{m}$ above the ground is:





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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2019 (12 April Morning Shift) PYQ

Solution

JEE MAIN PYQ
The shortest distance between the lines $\dfrac{x-5}{1}=\dfrac{y-2}{2}=\dfrac{z-4}{-3}$ and $\dfrac{x+3}{1}=\dfrac{y+5}{4}=\dfrac{z-1}{-5}$ is:





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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2023 (1 February Morning Shift) PYQ

Solution

JEE MAIN PYQ
Let $A$ and $B$ be two distinct points on the line $L:\ \dfrac{x-6}{3}=\dfrac{y-7}{2}=\dfrac{z-7}{-2}$. Both $A$ and $B$ are at a distance $2\sqrt{17}$ from the foot of perpendicular drawn from the point $(1,2,3)$ on the line $L$. If $O$ is the origin, then $\overrightarrow{OA}\cdot\overrightarrow{OB}$ is equal to





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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2025 (4 April Morning Shift) PYQ

Solution

JEE MAIN PYQ
The area (in sq. units) of the region A = {(x, y) : |x| + |y| $ \le $ 1, 2y2 $ \ge $ |x|}





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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 6 September 2020 (Morning) PYQ

Solution

JEE MAIN PYQ
The length of the perpendicular from the point $(1,-2,5)$ on the line passing through $(1,2,4)$ and parallel to the line $x+y-z=0 = x-2y+3z-5$ is :





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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2022 (26 July Morning Shift) PYQ

Solution


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