List of top Questions asked in JEE Main

Let $\alpha, \beta$ be two roots of the equation $x^2 + (20)^{1/4}x + (5)^{1/2} = 0$. Then $\alpha^8 + \beta^8$ is equal to :
Let
A = $\{(x, y) \in R \times R | 2x^2 + 2y^2 - 2x - 2y = 1\}$,
B = $\{(x, y) \in R \times R | 4x^2 + 4y^2 - 16y + 7 = 0\}$ and
C = $\{(x, y) \in R \times R | x^2 + y^2 - 4x - 2y + 5 \le r^2\}$.
Then the minimum value of $|r|$ such that $A \cup B \subseteq C$ is
Let $y=y(x)$ be solution of the differential equation $\log_e\left(\frac{dy}{dx}\right) = 3x+4y$, with $y(0)=0$. If $y\left(-\frac{2}{3}\log_e 2\right) = \alpha \log_e 2$, then the value of $\alpha$ is equal to :
If $\sin\theta + \cos\theta = \frac{1}{2}$, then $16(\sin(2\theta) + \cos(4\theta) + \sin(6\theta))$ is equal to :
The probability that a randomly selected 2-digit number belongs to the set $\{n \in N : (2^n - 2) \text{ is a multiple of } 3\}$ is equal to :
The compound statement $(P \lor Q) \land (\sim P) \Rightarrow Q$ is equivalent to :
The value of $\lim_{n \to \infty} \frac{1}{n}\sum_{j=1}^{n}\frac{(2j-1)+8n}{(2j-1)+4n}$ is equal to :
Let a plane P pass through the point (3, 7, -7) and contain the line, $\frac{x-2}{-3} = \frac{y-3}{2} = \frac{z+2}{1}$. If distance of the plane P from the origin is d, then $d^2$ is equal to _________.

Let 

, $x \in [0, \pi]$. Then the maximum value of $f(x)$ is equal to _________. 
 

Let F : [3, 5] $\rightarrow$ R be a twice differentiable function on (3, 5) such that
$F(x) = e^{-x} \int_3^x (3t^2 + 2t + 4F'(t))dt$.
If $F'(4) = \frac{\alpha e^\beta - 224}{(e^\beta-4)^2}$, then $\alpha+\beta$ is equal to _________.
Let $\vec{a}=\hat{i}+\hat{j}+\hat{k}$, $\vec{b}$ and $\vec{c}=\hat{j}-\hat{k}$ be three vectors such that $\vec{a} \times \vec{b} = \vec{c}$ and $\vec{a} \cdot \vec{b} = 1$. If the length of projection vector of the vector $\vec{b}$ on the vector $\vec{a} \times \vec{c}$ is $l$, then the value of $3l^2$ is equal to _________.
Let the domain of the function $f(x) = \log_4(\log_5(\log_3(18x-x^2-77)))$ be $(a, b)$. Then the value of the integral $\int_a^b \frac{\sin^3 x}{\sin^3 x + \sin^3(a+b-x)} dx$ is equal to _________.
If $\log_3 2, \log_3(2^x-5), \log_3(2^x-\frac{7}{2})$ are in an arithmetic progression, then the value of x is equal to _________.
For real numbers $\alpha$ and $\beta$, consider the following system of linear equations:
$x+y-z=2$
$x+2y+\alpha z = 1$
$2x-y+z = \beta$
If the system has infinite solutions, then $\alpha+\beta$ is equal to _________.

Let S = 1, 2, 3, 4, 5, 6, 7. Then the number of possible functions $f: S \rightarrow S$ such that $f(m \cdot n) = f(m) \cdot f(n)$ for every $m, n \in S$ and $m \cdot n \in S$ is equal to _________. 
 

If $y=y(x)$, $y \in [0, \pi/2)$ is the solution of the differential equation $\sec y \frac{dy}{dx} - \sin(x+y) - \sin(x-y) = 0$, with $y(0)=0$, then $5y'(\pi/2)$ is equal to _________.
Let $f:[0, 3] \rightarrow R$ be defined by $f(x) = \min\{x-[x], 1+[x]-x\}$ where $[x]$ is the greatest integer less than or equal to x. Let P denote the set containing all $x \in [0, 3]$ where f is discontinuous, and Q denote the set containing all $x \in [0, 3]$ where f is not differentiable. Then the sum of number of elements in P and Q is equal to _________.
A correct experiment is performed to measure the speed of sound in air using a resonance column tube of diameter 6 cm. The frequency of the tuning fork is 504 Hz. Speed of the sound is given as 336 m/s. The zero of the metre scale coincides with the top end of the resonance column tube. The reading of the water level in the column tube, when the first resonance occurs is :

In an octagon ABCDEFGH of equal side, what is the sum of $\vec{AB} + \vec{AC} + \vec{AD} + \vec{AE} + \vec{AF} + \vec{AG} + \vec{AH}$, if $\vec{AO} = 2i + 3j - 4k$?

Two satellites A and B of masses 200 kg and 400 kg are revolving round the earth at height of 600 km and 1600 km respectively. If $T_A$ and $T_B$ are the time periods of A and B respectively then the value of $T_B - T_A$ is : [$R_e = 6400$ km, $M_e = 6 \times 10^{24}$ kg]
 

A 5V battery is connected across the points X and Y. Assume D1 and D2 to be normal silicon diodes. Find the current supplied by the battery if the +ve terminal of the battery is connected to point X and -ve to point Y. 

The pitch of the screw gauge is 1 mm and there are 100 divisions on the circular scale. When nothing is put in between the jaws, the zero of the circular scale lies 8 divisions below the reference line. When a wire is placed between the jaws, the first linear scale division is clearly visible while 72nd division on circular scale coincides with the reference line. The radius of the wire is :
An $\alpha$ particle and a proton are accelerated from rest by a potential difference of 200 V. After this, their de Broglie wavelengths are $\lambda_\alpha$ and $\lambda_p$ respectively. The ratio $\lambda_p/\lambda_\alpha$ is :
Two coherent light sources having intensity in the ratio $2x$ produce an interference pattern. The ratio $(I_{max} - I_{min}) / (I_{max} + I_{min})$ will be :
An engine of a train, moving with uniform acceleration, passes the signal-post with velocity $u$ and the last compartment with velocity $v$. The velocity with which middle point of the train passes the signal post is :