List of top Questions asked in JEE Main, KCET

If \[ \alpha=\int_{0}^{2\sqrt{3}} \log_2(x^2+4)\,dx + \int_{2}^{4} \sqrt{2^x-4}\,dx, \] then \(\alpha^2\) is equal to _____.
Let a circle \(C\) have its centre in the first quadrant, intersect the coordinate axes at exactly three points and cut off equal intercepts from the coordinate axes. If the length of the chord of \(C\) on the line \(x+y=1\) is \(\sqrt{14}\), then the square of the radius of \(C\) is _____.}
Let \(a,b,c \in \{1,2,3,4\}\). If the probability that \[ ax^2 + 2\sqrt{2}\,bx + c>0 \quad \text{for all } x \in \mathbb{R} \] is \( \frac{m}{n} \), where \(\gcd(m,n)=1\), then \(m+n\) is equal to _____.
If \[ \sum_{k=1}^{n} a_k = 6n^3, \] then} \[ \sum_{k=1}^{6}\left(\frac{a_{k+1}-a_k}{36}\right)^2 \] is equal to _______.
If the domain of the function} \[ f(x)=\sqrt{\log_{0.6}\left(\left|\frac{2x-5}{x^2-4}\right|\right)} \] is \((-\infty,a] \cup \{b\} \cup [c,d) \cup (e,\infty)\), then the value of \(a+b+c+d+e\) is _______.}
Let \(y=y(x)\) be the solution curve of the differential equation} \[ (1+\sin x)\frac{dy}{dx}+(y+1)\cos x=0,\qquad y(0)=0. \] If the curve passes through the point \( \left(\alpha,-\frac12\right) \), then a value of \( \alpha \) is:
Let \([\,]\) denote the greatest integer function. Then the value of} \[ \int_{0}^{3}\left(\frac{e^x+e^{-x}}{[x]!}\right)dx \] is:
If the curve \(y=f(x)\) passes through the point \((1,e)\) and satisfies the differential equation} \[ dy=y(2+\log_e x)\,dx,\quad x>0, \] then \(f(e)\) is equal to:
If \[ \lim_{x\to 2}\frac{\sin(x^3-5x^2+ax+b)}{(\sqrt{x-1}-1)\log_e(x-1)}=m, \] then \(a+b+m\) is equal to:
If \(|\vec a|=2\) and \(|\vec b|=3\), then the maximum value of \[ 3\left|\left(\vec a+2\vec b\right)\right| + 4\left|\left(3\vec a-2\vec b\right)\right| \] is:
If the point of intersection of the lines \[ \frac{x+1}{3}=\frac{y+a}{5}=\frac{z+b+1}{7} \] \[ \frac{x-2}{1}=\frac{y-b}{4}=\frac{z-2a}{7} \] lies on the \(xy\)-plane, then the value of \(a+b\) is:
Let \[ S=\{x\in[-\pi,\pi]:\sin x(\sin x+\cos x)=a,\; a\in\mathbb{Z}\}. \] Then \(n(S)\) is equal to:
If \[ \sin\left(\frac{\pi}{18}\right)\sin\left(\frac{5\pi}{18}\right)\sin\left(\frac{7\pi}{18}\right)=K, \] then the value of \[ \sin\left(\frac{10K\pi}{3}\right) \] is:
Let an ellipse \[ \frac{x^2}{a^2}+\frac{y^2}{b^2}=1,\quad a<b \] pass through the point \((4,3)\) and have eccentricity \( \frac{\sqrt5}{3} \). Then the length of its latus rectum is:
Let the midpoints of the sides of a triangle \(ABC\) be \( \left(\frac{5}{2},7\right), \left(\frac{5}{2},3\right)\) and \( (4,5) \). If its incentre is \((h,k)\), then \(3h+k\) is equal to:
If the mean of the following grouped data is \(21\): then \(k\) is one of the roots of the equation:
The number of elements in the set} \[ S=\left\{(r,k): k\in \mathbb{Z} \text{ and } {^{36}C_{r+1}}=\frac{6\left({^{35}C_r}\right)}{k^2-3}\right\} \] is:
The number of seven-digit numbers that can be formed by using the digits \(1,2,3,5,7\) such that each digit is used at least once, is:
Let \(A\) be the set of first \(101\) terms of an A.P., whose first term is \(1\) and the common difference is \(5\), and let \(B\) be the set of first \(71\) terms of an A.P., whose first term is \(9\) and the common difference is \(7\). Then the number of elements in \(A \cap B\), which are divisible by \(3\), is:
Let \(x\) and \(y\) be real numbers such that} \[ 50\left(\frac{2x}{1+3i} - \frac{y}{1-2i}\right) = 31 + 17i, \qquad i = \sqrt{-1}. \] Then the value of \(10(x-3y)\) is:
Let \( \alpha, \alpha + 2 \in \mathbb{Z} \) be the roots of the quadratic equation} \[ x(x+2) + (x+1)(x+3) + (x+2)(x+4) + \cdots + (x+n-1)(x+n+1) = 4n \] for some \( n \in \mathbb{N} \). Then \( n + \alpha \) is equal to:

Net gravitational force at the center of a square is found to be \( F_1 \) when four particles having masses \( M, 2M, 3M \) and \( 4M \) are placed at the four corners of the square as shown in figure, and it is \( F_2 \) when the positions of \( 3M \) and \( 4M \) are interchanged. The ratio \( \dfrac{F_1}{F_2} = \dfrac{\alpha}{\sqrt{5}} \). The value of \( \alpha \) is 

A rectangle is formed by the lines \( x = 0 \), \( y = 0 \), \( x = 3 \) and \( y = 4 \). Let the line \( L \) be perpendicular to \( 3x + y + 6 = 0 \) and divide the area of the rectangle into two equal parts. Then the distance of the point \( \left(\frac{1}{2}, -5\right) \) from the line \( L \) is equal to :

Two point charges 2q and q are placed at vertex A and centre of face CDEF of the cube as shown in figure. The electric flux passing through the cube is : 

Let $\alpha, \beta$ be the roots of the quadratic equation \[ 12x^2 - 20x + 3\lambda = 0,\ \lambda \in \mathbb{Z}. \] If \[ \frac{1}{2} \le |\beta-\alpha| \le \frac{3}{2}, \] then the sum of all possible values of $\lambda$ is