JEE MAIN Parabola Previous Year Question Paper and Solution with PDF

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JEE MAIN Previous Year Questions (PYQs)

JEE MAIN Parabola PYQ

Qus : 1 JEE MAIN PYQ

If the line $3x - 2y + 12 = 0$ intersects the parabola $4y = 3x^2$ at the points $A$ and $B$, then at the vertex of the parabola, the line segment $AB$ subtends an angle equal to:

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

Solution

Qus : 2 JEE MAIN PYQ

The area (in sq. units) of the region described by $\{(x,y):y^{2}\le2x,\;y\ge4x-1\}$ is:

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

Solution

Qus : 3 JEE MAIN PYQ

Let $PQ$ be a chord of the parabola $y^{2}=12x$ and the midpoint of $PQ$ be at $(4,1)$. Then, which of the following points lies on the line passing through the points $P$ and $Q$?

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

Solution

Qus : 4 JEE MAIN PYQ

The axis of a parabola is the line $y=x$ and its vertex and focus are in the first quadrant at distances $\sqrt{2}$ and $2\sqrt{2}$ units from the origin, respectively. If the point $(1,k)$ lies on the parabola, then a possible value of $k$ is:

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

Solution

Qus : 5 JEE MAIN PYQ

The length of the chord of the parabola $x^2=4y$ having equation $x-\sqrt{2}\,y+4\sqrt{2}=0$ is –

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

Solution

Qus : 6 JEE MAIN PYQ

Let the shortest distance from $(a,0)$, $a>0$, to the parabola $y^{2}=4x$ be $4$. Then the equation of the circle passing through the point $(a,0)$ and the focus of the parabola, with centre on the axis of the parabola, is:

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

Solution

Qus : 7 JEE MAIN PYQ

Let P be the parabola, whose focus is $(-2,1)$ and directrix is $2 x+y+2=0$. Then the sum of the ordinates of the points on P, whose abscissa is $-$2, is

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

Solution

Qus : 8 JEE MAIN PYQ

$f:R \to R$ be defined as$f(x) = \left\{ {\matrix{ { - 55x,} & {if\,x < - 5} \cr {2{x^3} - 3{x^2} - 120x,} & {if\, - 5 \le x \le 4} \cr {2{x^3} - 3{x^2} - 36x - 336,} & {if\,x > 4,} \cr } } \right.$ Let A = {x $ \in $ R : f is increasing}. Then A is equal to :

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

Solution

Qus : 9 JEE MAIN PYQ

If the shortest distance of the parabola $y^2=4x$ from the centre of the circle $x^2+y^2-4x-16y+64=0$ is $d$, then $d^2$ is equal to:

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

Solution

Qus : 10 JEE MAIN PYQ

Let $S=\{1,2,\ldots,20\}$. A subset $B$ of $S$ is said to be “nice”, if the sum of the elements of $B$ is $203$. Then the probability that a randomly chosen subset of $S$ is “nice” is :

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

Solution

Qus : 11 JEE MAIN PYQ

Tangents drawn from the point $(-8,0)$ to the parabola $y^{2} = 8x$ touch the parabola at $P$ and $Q$. If $F$ is the focus of the parabola, then the area of the triangle $PFQ$ (in sq. units) is equal to :

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

Solution

Qus : 12 JEE MAIN PYQ

A particle is moving in the xy-plane along a curve C passing through the point (3, 3). The tangent to the curve C at the point P meets the x-axis at Q. If the y-axis bisects the segment PQ, then C is a parabola with :

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

Solution

Qus : 13 JEE MAIN PYQ

$ \text{Let } R \text{ be the focus of the parabola } y^{2}=20x \text{ and the line } y=mx+c \text{ intersect the parabola at two points } P \text{ and } Q. $ $ \text{Let the point } G(10,10) \text{ be the centroid of the triangle } PQR. \text{ If } c-m=6, \text{ then } (PQ)^{2} \text{ is:} $

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

Solution

Qus : 14 JEE MAIN PYQ

The locus of the mid points of the chords of the hyperbola x² $-$ y² = 4, which touch the parabola y² = 8x, is :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2021 (26 August Evening Shift) PYQ

Solution

Qus : 15 JEE MAIN PYQ

If the equation of the parabola with vertex $V!\left(\frac{3}{2},,3\right)$ and directrix $x+2y=0$ is $\alpha x^2+\beta y^2-\gamma xy-30x-60y+225=0$, then $\alpha+\beta+\gamma$ is equal to:

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

Solution

Qus : 16 JEE MAIN PYQ

If the area of the triangle whose one vertex is at the vertex of the parabola, $y^{2}+4(x-a^{2})=0$ and the other two vertices are the points of intersection of the parabola and $y$-axis, is $250$ sq. units, then a value of $a$ is :

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

Solution

Qus : 17 JEE MAIN PYQ

A hyperbola passes through the foci of the ellipse ${{{x^2}} \over {25}} + {{{y^2}} \over {16}} = 1$ and its transverse and conjugate axes coincide with major and minor axes of the ellipse, respectively. If the product of their eccentricities is one, then the equation of the hyperbola is :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2021 (25 February Evening Shift) PYQ

Solution

Qus : 18 JEE MAIN PYQ

Let $P(4,-4)$ and $Q(9,6)$ be two points on the parabola $y^{2}=4x$, and let $X$ be any point on the arc $POQ$ of this parabola, where $O$ is the vertex, such that the area of $\triangle PXQ$ is maximum. Then this maximum area (in sq. units) is:

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

Solution

Qus : 19 JEE MAIN PYQ

If two tangents drawn from a point P to the parabola y² = 16(x $-$ 3) are at right angles, then the locus of point P is :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2021 (27 August Evening Shift) PYQ

Solution

Qus : 20 JEE MAIN PYQ

Let $ABCD$ be a trapezium whose vertices lie on the parabola $y^{2}=4x$. Let the sides $AD$ and $BC$ of the trapezium be parallel to the $y$-axis. If the diagonal $AC$ is of length $\dfrac{25}{4}$ and it passes through the point $(1,0)$, then the area of $ABCD$ is

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

Solution

Qus : 21 JEE MAIN PYQ

The area of the region bounded by the parabola (y $-$ 2)² = (x $-$ 1), the tangent to it at the point whose ordinate is 3 and the x-axis is :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2021 (27 August Evening Shift) PYQ

Solution

Qus : 22 JEE MAIN PYQ

The area of the region enclosed between the parabolas y² = 2x $-$ 1 and y² = 4x $-$ 3 is

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

Solution

Qus : 23 JEE MAIN PYQ

$ \text{Let the focal chord of the parabola } P: y^{2}=4x \text{ along the line } L: y=mx+c,; m>0 \text{ meet the parabola at the points } M \text{ and } N. \text{ Let the line } L \text{ be a tangent to the hyperbola } H: x^{2}-y^{2}=4. \text{ If } O \text{ is the vertex of } P \text{ and } F \text{ is the focus of } H \text{ on the positive } x\text{-axis, then the area of the quadrilateral } OMFN \text{ is:} $

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

Solution

Qus : 24 JEE MAIN PYQ

The area (in sq. units) of an equilateral triangle inscribed in the parabola y2 = 8x, with one of its vertices on the vertex of this parabola, is :

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

Solution

Qus : 25 JEE MAIN PYQ

If the equation of the parabola, whose vertex is at (5, 4) and the directrix is $3x + y - 29 = 0$, is ${x^2} + a{y^2} + bxy + cx + dy + k = 0$, then $a + b + c + d + k$ is equal to :

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

Solution

Qus : 26 JEE MAIN PYQ

Let $P$ be a point on the parabola, $y^{2} = 12x$ and $N$ be the foot of the perpendicular drawn from $P$ on the axis of the parabola. A line is now drawn through the mid-point $M$ of $PN$, parallel to its axis which meets the parabola at $Q$. If the $y$-intercept of the line $NQ$ is $\tfrac{4}{3}$, then :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 3 September 2020 (Morning) PYQ

Solution

Qus : 27 JEE MAIN PYQ

The locus of the mid-point of the line segment joining the focus of the parabola y²= 4ax to a moving point of the parabola, is another parabola whose directrix is :

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

Solution

Qus : 28 JEE MAIN PYQ

If one end of a focal chord of the parabola $y^{2}=16x$ is at $(1,4)$, then the length of this focal chord is:

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

Solution

Qus : 29 JEE MAIN PYQ

Let the focal chord $PQ$ of the parabola $y^2 = 4x$ make an angle of $60^\circ$ with the positive $x$-axis, where $P$ lies in the first quadrant. If the circle, whose one diameter is $PS$, $S$ being the focus of the parabola, touches the $y$-axis at the point $(0, \alpha)$, then $5\alpha^2$ is equal to:

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

Solution

Qus : 30 JEE MAIN PYQ

Let the latus ractum of the parabola y² = 4x be the common chord to the circles C₁ and C₂each of them having radius 2$\sqrt 5 $. Then, the distance between the centres of the circles C₁and C₂ is :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 3 September 2020 (Evening) PYQ

Solution

Qus : 31 JEE MAIN PYQ

The length of the latus rectum of a parabola, whose vertex and focus are on the positive x-axis at a distance R and S (> R) respectively from the origin, is :

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

Solution

Qus : 32 JEE MAIN PYQ

Let the point P of the focal chord PQ of the parabola $y^2=16 x$ be $(1,-4)$. If the focus of the parabola divides the chord $P Q$ in the ratio $m: n, \operatorname{gcd}(m, n)=1$, then $m^2+n^2$ is equal to :

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

Solution

Qus : 33 JEE MAIN PYQ

Axis of a parabola lies along the x–axis. If its vertex and focus are at distances $2$ and $4$ respectively from the origin on the positive x–axis, then which of the following points does not lie on it?

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

Solution

Qus : 34 JEE MAIN PYQ

Let 'a' be a real number such that the function f(x) = ax² + 6x $-$ 15, x $\in$ R is increasing in $\left( { - \infty ,{3 \over 4}} \right)$ and decreasing in $\left( {{3 \over 4},\infty } \right)$. Then the function g(x) = ax² $-$ 6x + 15, x$\in$R has a :

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

Solution

Qus : 35 JEE MAIN PYQ

Let $A(4,-4)$ and $B(9,6)$ be points on the parabola $y^{2}=4x$. Let $C$ be chosen on the arc $AOB$ of the parabola, where $O$ is the origin, such that the area of $\triangle ACB$ is maximum. Then, the area (in sq. units) of $\triangle ACB$ is:

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

Solution

Qus : 36 JEE MAIN PYQ

The parabolas: $a x^{2}+2 b x+c y=0$ and $d x^{2}+2 e x+f y=0$ intersect on the line $y=1$. If $a,b,c,d,e,f$ are positive real numbers and $a,b,c$ are in G.P., then:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2023 (30 January Evening Shift) PYQ

Solution

Qus : 37 JEE MAIN PYQ

Let the parabola $y=x^2+px-3$ meet the coordinate axes at the points $P,Q,R$. If the circle $C$ with centre at $(-1,-1)$ passes through the points $P,Q$ and $R$, then the area of $\triangle PQR$ is:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2025 (22 January Morning Shift) PYQ

Solution

Qus : 38 JEE MAIN PYQ

Let $P(4,4\sqrt{3})$ be a point on the parabola $y^{2}=4ax$ and $PQ$ be a focal chord of the parabola. If $M$ and $N$ are the feet of perpendiculars drawn from $P$ and $Q$ respectively on the directrix of the parabola, then the area of the quadrilateral $PQMN$ is equal to:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2025 (22 January Evening Shift) PYQ

Solution

Qus : 39 JEE MAIN PYQ

Let $O$ be the vertex and $Q$ be any point on the parabola, $x^{2}=8y$. If the point $P$ divides the line segment $OQ$ internally in the ratio $1:3$, then locus of $P$ is :

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

Solution

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