List of top Questions asked in JEE Main

Let $Q$ be the mirror image of the point $P(1, 0, 1)$ with respect to the plane $S$: $x + y + z = 5$. If a line L passing through $(1, –1, –1)$, parallel to the line $PQ$ meets the plane $S$ at $R$, then $QR_2$ is equal to :

Let $x=2t,\ y=\frac {t^2}{3}$ be a conic. Let S be the focus and B be the point on the axis of the conic such that SA⊥BA, where A is any point on the conic. If k is the ordinate of the centroid of the ΔSAB, then $\lim\limits_{t \to 1} k$ is equal to

A ray of light passing through the point P(2, 3) reflects on the x-axis at point A and the reflected ray passes through the point Q(5, 4). Let R be the point that divides the line segment AQ internally into the ratio 2:1. Let the co-ordinates of the foot of the perpendicular M from R on the bisector of the angle PAQ be (α, β). Then, the value of $7α + 3β$ is equal to ________ .

Let l be a line which is normal to the curve $y = 2x^2 + x + 2$ at a point P on the curve. If the point Q(6, 4) lies on the line l and O is origin, then the area of the triangle OPQ is equal to _________ .

Let the lines
$y + 2x = \sqrt {11} + 7\sqrt 7$ and $2y + x = 2\sqrt {11} + 6\sqrt 7$
be normal to a circle $C : (x – h)^2 + (y – k)^2 = r^2$. If the line
$\sqrt {11}y - 3x =\frac { 5\sqrt {17}}{3} + 11$
is tangent to the circle C, then the value of $(5h – 8k)^2 + 5r^2$ is equal to _______.

A circle of radius 2 unit passes through the vertex and the focus of the parabola y² = 2x and touches the parabola $y=\left (x−\frac{1}{4}\right)^2+α,$ where $α > 0$. Then $(4α – 8)^2$ is equal to ___________.

Let the common tangents to the curves 4(x² + y²) = 9 and y² = 4x intersect at the point Q. Let an ellipse, centered at the origin O, has lengths of semi-minor and semi-major axes equal to OQ and 6, respectively. If e and I respectively denote the eccentricity and the length of the latus rectum of this ellipse, then
$\frac{1}{e^2}$ is equal to

Let the tangent drawn to the parabola y² = 24x at the point (α, β) is perpendicular to the line 2x + 2y = 5. Then the normal to the hyperbola
$\frac{x^2}{α^2}−\frac{y^2}{β^2}=1$
at the point (α + 4, β + 4) does NOT pass through the point

An ellipse
$E:\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$
passes through the vertices of the hyperbola
$H:\frac{x^2}{49} - \frac{y^2}{64} = -1$
Let the major and minor axes of the ellipse E coincide with the transverse and conjugate axes of the hyperbola H, respectively. Let the product of the eccentricities of E and H be 1/2. If the length of the latus rectum of the ellipse E, then the value of 113l is equal to _____.

Let $O$ be the origin and $A$ be the point $z _1=1+2 i$. If $B$ is the point $z _2, \operatorname{Re}\left( z _2\right)<0$, such that $OAB$ is a right angled isosceles triangle with $OB$ as hypotenuse, then which of the following is NOT true ?

Let z₁ and z₂ be two complex numbers such that

$z_1=iz_2 \,and \,arg(\frac{z_1}{z_2})=π.$

If z² + z + 1 = 0,
$z ∈ C$, then $\left| \sum_{n=1}^{15} \left( z_n + (-1)^n \frac{1}{z_n} \right)^2 \right|$
is equal to ________.

Let $S = z ∈ C: |z-3| <= 1$ and $z (4+3i)+z(4-3)≤24.$
If α + iβ is the point in S which is closest to 4i, then 25(α + β) is equal to ______.

The number of elements in the set { $z = a + ib ∈ C : a,b ∈ Z$ and $1 < | z - 3 + 2i | < 4$ } is _______ .

Let O be the origin and A be the point $z_1 = 1 + 2i$. If B is the point $z_2, Re(z_2) < 0$, such that OAB is a right angled isosceles triangle with OB as hypotenuse, then which of the following is NOT true?

Let α be a root of the equation 1 + x² + x⁴ = 0. Then the value of α¹⁰¹¹ + α²⁰²² – α³⁰³³ is equal to

Let arg(z) represent the principal argument of the complex number z. Then, |z| = 3 and arg(z – 1) – arg(z + 1) = π/4 intersect

Let
$A = \{z \in \mathbb{C} : |\frac{z+1}{z-1}| < 1\}$
and
$B = \{z \in \mathbb{C} : \text{arg}(\frac{z-1}{z+1}) = \frac{2\pi}{3}\}$
Then $A∩B$ is :

The number of points of intersection of $|z - (4 + 3i)| = 2$ and $|z| + |z - 4| = 6$, $z \in \mathbb{C}$, is

Sum of squares of modulus of all the complex numbers z satisfying
$\overline{z}=iz^2+z^2–z $
is equal to ________.

Let S be the set of (α,β),π<α,β<2π,for which the complex number
$\frac{1-i\sinα}{1+2i\sinα}$ is purely imaginary and $\frac{1+i\cosβ}{1-2i\cosβ}$ is purely real,
Let $Zαβ = \sin2α+i\cos2β, (α,β) ∈ S$. Then
$\sum_{(\alpha, \beta) \in S} \left(iZ_{\alpha\beta} + \frac{1}{iZ_{\alpha\beta}}\right)$
is equal to

If z = x + iy satisfies | z | – 2 = 0 and |z – i| – | z + 5i| = 0, then

Let
$S = \left\{z∈C : z^2+\overline{z} = 0 \right\}$. Then $∑_{z∈S}(Re(z)+Im(z))$
is equal to____.

Let a function ƒ : N →N be defined by
$f(n) = \left\{ \begin{array}{ll} 2n & n = 2,4,6,8,\ldots \\ n - 1 & n = 3,7,11,15,\ldots \\ \frac{n+1}{2} & n = 1,5,9,13 \end{array} \right.$
then, ƒ is

Let f: ℝ → ℝ satisfy f(x + y) = 2^x f(y) + 4^y f(x), ∀ x, y∈ ℝ. If f(2) = 3, then 14. f'(4)/f'(2) is equal to ___.