List of top Questions asked in JEE Main

Let $\hat a$ and $\hat b$ be two unit vectors such that $| (a+b)+2(a × b)|$ =$ 2$. If $θ ∈ (0, π)$ is the angle between a and b, then among the statements: $(S1)$: $2|a×b|=|a-b|$ $(S2)$: The projection of a on $(\hat a+\hat b)$ is $\frac{1}{2}$

Let the hyperbola
$H:\frac{x^2}{a^2}−y^2=1$
and the ellipse
$E:3x^2+4y^2=12$
be such that the length of latus rectum of H is equal to the length of latus rectum of E. If eH and eE are the eccentricities of H and E respectively, then the value of
$12 (e^{2}_H+e^{2}_E)$ is equal to _____ .

Let $\overrightarrow a$ and $\overrightarrow b$ be the vectors along the diagonals of a parallelogram having area $2\sqrt2$. Let the angle between $\overrightarrow a$ and $\overrightarrow b$ be acute,$ |\overrightarrow a|=1$, and $|\overrightarrow a⋅\overrightarrow b|=|\overrightarrow a×\overrightarrow b|$ If $\overrightarrow c=2\sqrt2(\overrightarrow a×\overrightarrow b→)−2\overrightarrow b$ then an angle between$ \overrightarrow b$ and $\overrightarrow c$ is

The area of the smaller region enclosed by the curves y² = 8x + 4 and
x²+y²+4√3x-4=0
is equal to

Consider the following statements:
A : Rishi is a judge.
B : Rishi is honest.
C : Rishi is not arrogant.
The negation of the statement “if Rishi is a judge and he is not arrogant, then he is honest” is

The area of the region
$\left\{(x,y) : y² ≤ 8x, y ≥ \sqrt2x, x ≥ 1 \right\}$
is

Let y = y(x) be the solution of the differential equation
$x ( 1 - x² ) \frac{dy}{dx} + ( 3x²y - y - 4x³ ) = 0, x > 1$
with y(2) = –2. Then y(3) is equal to

Let the solution curve y = y(x) of the differential equation
$[ \frac{x}{\sqrt{x² -y²}} + e^\frac{y}{x} ] x \frac{dy}{dx} = x + [ \frac{x}{\sqrt{x² -y²}} + e^\frac{y}{x} ]y$
pass through the points (1, 0) and (2α, α), α> 0. Then α is equal to

If y = y (x) is the solution of the differential equation
$(1 + e^{2x})\frac{dy}{dx} + 2(1 + y^2)e^x = 0$
and y(0) = 0, then
$6 \left( y'(0) + \left( \log_e\left(\sqrt{3}\right) \right)^2 \right)$
is equal to

Let
$\vec{a}=\hat{i} - 2\hat{j} + 3\hat{k}, \vec{b}=\hat{i} - \hat{j} + \hat{k} $ and $\vec{c}$
be a vector such that
$\vec{a} + (\vec{b}×\vec{c}) = \vec{0}$ and $\vec{b}.\vec{c} = 5.$
Then the value of 3($\vec{c}.\vec{a}$) is equal to

For $\alpha \in N$, consider a relation $R$ on $N$ given by $R=\{(x, y): 3 x+\alpha y$ is a multiple of 7$\}$ The relation $R$ is an equivalence relation if and only if :

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

While estimating the nitrogen present in an organic compound by Kjeldahl’s method, the ammonia evolved from 0.25 g of the compound neutralized 2.5 mL of 2 M H₂SO₄. The percentage of nitrogen present in organic compound is ________

Let the mirror image of the point (a, b, c) with respect to the plane 3x – 4y + 12z + 19 = 0 be (a – 6, β, γ). If a + b + c = 5, then 7β – 9γ is equal to __________.

Let E₁, E₂, E₃be three mutually exclusive events such that
$P(E_1)=\frac{2+3p}{6}, P(E_2)=\frac{2−p}{8} and\ P(E_3)=\frac{1−p}{2}.$
If the maximum and minimum values of p are p₁ and p₂, then (p₁ + p₂) is equal to :

Let z₁ and z₂ be two complex numbers such that

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

Choose the correct answer :

1. The probability that a randomly chosen 2 × 2 matrix with all the entries from the set of first 10 primes, is singular, is equal to :

Let the mean and the variance of 5 observations x₁, x₂, x₃, x₄, x₅ be $\frac{24}{5} $ and $\frac{194}{25}$, respectively. If the mean and variance of the first 4 observations are $\frac{7}{2}$ and a, respectively, then (4a + x₅) is equal to

If $∫\frac{1}{x}$ $√{\frac{1-x}{1+x}}$ dx = $g(x) + c,g(1) = 0$ , then g $(\frac{1}{2})$
is equal to

The normal to the hyperbola
$\frac{x²}{a²} - \frac{y²}{9} = 1$
at the point (8, 3√3) on it passes through the point:

If the inverse trigonometric functions take principal values, then
$cos^{-1} ( \frac{3}{10} cos (tan^{-1} (\frac{4}{3})) + \frac{2}{5} sin (tan^{-1} (\frac{4}{3})))$
is equal to :

$\lim_{{x \to \frac{1}{\sqrt{2}}}} \frac{\sin(\cos^{-1}(x)) - x}{1 - \tan(\cos^{-1}(x))}$
is equal to :

g :R→R be two real valued functions defined as
$f(x) = \begin{cases} -|x + 3| & x < 0 \\ e^x, & x \geq 0 \end{cases}$
and
$g(x) = \begin{cases} x^2 + k_1x ,& x < 0 \\ 4x + k_2 ,& x \geq 0 \end{cases}$
where k₁ and k₂ are real constants. If (goƒ) is differentiable at x = 0, then (goƒ) (–4) + (goƒ) (4) is
equal to:

Let R be the point (3, 7) and let P and Q be two points on the line x + y = 5 such that PQR is an equilateral triangle, Then the area of ΔPQR is

The number of ways to distribute 30 identical candies among four children C₁, C₂, C₃ and C₄ so that C₂ receives atleast 4 and atmost 7 candies, C₃ receives atleast 2 and atmost 6 candies, is equal to: