List of top Questions asked in JEE Main- 2021

The set of all values of k \(>\) -1, for which the equation
(3x\(^2\)+4x+3)\(^2\) - (k+1)(3x\(^2\)+4x+3)(3x\(^2\)+4x+2) + k(3x\(^2\)+4x+2)\(^2\) = 0 has real roots, is :
Let [\(\lambda\)] be the greatest integer less than or equal to \(\lambda\). The set of all values of \(\lambda\) for which the system of linear equations x+y+z=4, 3x+2y+5z=3, 9x + 4y + (28 + [\(\lambda\)])z = [\(\lambda\)] has a solution is :
Let \( A = \begin{pmatrix} \lfloor x+1 \rfloor & \lfloor x+2 \rfloor & \lfloor x+3 \rfloor \\ \lfloor x \rfloor & \lfloor x+3 \rfloor & \lfloor x+3 \rfloor \\ \lfloor x \rfloor & \lfloor x+2 \rfloor & \lfloor x+4 \rfloor \end{pmatrix} \), where \( \lfloor t \rfloor \) denotes the greatest integer less than or equal to \( t \). If \( \det(A) = 192 \), then the set of values of \( x \) is the interval:
If 0 \(<\) x \(<\) 1 and y = \(\frac{1}{2}x^2 + \frac{2}{3}x^3 + \frac{3}{4}x^4 + ...\), then the value of e\(^{1+y}\) at x = \(\frac{1}{2}\) is :
If \( \lim_{x \to \infty} (\sqrt{x^2 - x + 1} - ax) = b \), then the ordered pair (a, b) is :
If \(y(x) = \cot^{-1}\left(\frac{\sqrt{1 + \sin x} + \sqrt{1 - \sin x}}{\sqrt{1 + \sin x} - \sqrt{1 - \sin x}}\right), x \in \left(\frac{\pi}{2}, \pi\right)\), then \(\frac{dy}{dx}\) at \(x = \frac{5\pi}{6}\) is :
Let M and m respectively be the maximum and minimum values of the function \(f(x) = \tan^{-1}(\sin x + \cos x)\) in \([0, \pi/2]\). Then the value of \(\tan(M-m)\) is equal to :
A box open from top is made from a rectangular sheet of dimension a \(\times\) b by cutting squares each of side x from each of the four corners and folding up the flaps. If the volume of the box is maximum, then x is equal to:
The value of the integral \( \int_{0}^{1} \frac{\sqrt{x} dx}{(1+x)(1+3x)(3+x)} \) is :
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 :
A differential equation representing the family of parabolas with axis parallel to y-axis and whose length of latus rectum is the distance of the point (2, -3) form the line 3x + 4y = 5, is given by :
If the solution curve of the differential equation \((2x - 10y^3)dy + y dx = 0\), passes through the points (0, 1) and (2, \(\beta\)), then \(\beta\) is a root of the equation :
Let A(a, 0), B(b, 2b + 1) and C(0, b), b \(\neq\) 0, |b| \(\neq\) 1, be points such that the area of triangle ABC is 1 sq. unit, then the sum of all possible values of a is :
If two tangents drawn from a point P to the parabola \(y^2 = 16(x-3)\) are at right angles, then the locus of point P is :
The angle between the straight lines, whose direction cosines are given by the equations 2l + 2m - n = 0 and mn + nl + lm = 0, is :
The equation of the plane passing through the line of intersection of the planes \( \vec{r} \cdot (\hat{i} + \hat{j} + \hat{k}) = 1 \) and \( \vec{r} \cdot (2\hat{i} + 3\hat{j} - \hat{k}) + 4 = 0 \) and parallel to the x-axis is :
Each of the persons A and B independently tosses three fair coins. The probability that both of them get the same number of heads is :
The Boolean expression \((p \land q) \Rightarrow ((r \land q) \land p)\) is equivalent to :
Two poles, AB of length a metres and CD of length a + b (b \(\neq\) a) metres are erected at the same horizontal level with bases at B and D. If BD = x and \(\tan(\angle ACB) = \frac{1}{2}\), then :
Let \(z_1\) and \(z_2\) be two complex numbers such that \( \arg(z_1 - z_2) = \frac{\pi}{4} \) and \(z_1, z_2\) satisfy the equation \(|z-3| = \text{Re}(z)\). Then the imaginary part of \(z_1+z_2\) is equal to _________.

Let S = 1, 2, 3, 4, 5, 6, 9. Then the number of elements in the set T={A \(\subseteq\) S: A \(\neq \emptyset\) and the sum of all the elements of A is not a multiple of 3} is _________. 
 

\(3 \times 7^{22} + 2 \times 10^{22} - 44\) when divided by 18 leaves the remainder _________.
If \( \int \frac{2e^x + 3e^{-x}}{4e^x + 7e^{-x}} dx = \frac{1}{14}(ux + v \log_e(4e^x + 7e^{-x})) + C \), where C is a constant of integration, then u + v is equal to _________.
Two circles each of radius 5 units touch each other at the point (1, 2). If the equation of their common tangent is 4x + 3y = 10, and C\(_1\)(\(\alpha\), \(\beta\)) and C\(_2\)(\(\gamma\), \(\delta\)), C\(_1\) \(\neq\) C\(_2\) are their centres, then \(|(\alpha+\beta)(\gamma+\delta)|\) is equal to _________.
Let A(sec\(\theta\), 2tan\(\theta\)) and B(sec\(\phi\), 2tan\(\phi\)), where \(\theta+\phi=\pi/2\), be two points on the hyperbola \(2x^2-y^2=2\). If (\(\alpha\), \(\beta\)) is the point of the intersection of the normals to the hyperbola at A and B, then \((2\beta)^2\) is equal to _________.