List of top Questions asked in JEE Main

Let f(x)=\(\left\{\begin{matrix} |4x^2-8x+5|, & if\,8x^2-6x+1\geq0 \\ |4x^2-8x+5|, & if\,8x^2-6x+1<0 \end{matrix}\right.\) where [α] denotes the greatest integer less than or equal to α.Then the number of points in R where f is not differentiable is

Let,
\(\vec{a}=a\hat{i}+2\hat{j}−\hat{k}\) and \(\vec{b}=−2\hat{i}+α\hat{j}+\hat{k}\)
, where α ∈ R. If the area of the parallelogram whose adjacent sides are represented by the vectors
\(\vec{a}\) and \(\vec{b}\) is \(\sqrt{15(α^2+4)}\) , then the value of \(2|\vec{a}|^2+(\vec{a}⋅\vec{b})|\vec{b}|^2 \)
is equal to :

Let f(x) = [2x² + 1] and \(g(x)=\left\{\begin{matrix} 2x-3,\,x<0&\\2x+3, x≥0 &\end{matrix}\right.\)where [t] is the greatest integer ≤ t. Then, in the open interval (–1, 1), the number of points where fog is discontinuous is equal to _______.

Let p and q be two real numbers such that p + q = 3 and p⁴ + q⁴ = 369. Then
\((\frac{1}{p} + \frac{1}{q} )^{-2}\)
is equal to _______.

Let z = a + ib, b≠ 0 be complex numbers satisfying

\(\begin{array}{l} z^2=\overline{z}\cdot 2^{1-\left|z\right|}.\end{array}\)

Then the least value of n∈ N, such that zⁿ= (z + 1)ⁿ, is equal to _____.

\(\begin{array}{l} I_n\left(x\right)=\int_0^x\frac{1}{\left(t^2+5\right)^n}dt, n=1, 2, 3,\cdots\end{array}\)

Then

Let \(S=2+\frac{6}{7}+\frac{12}{7 ^2}+\frac{20}{7 ^3}+\frac{30}{7 ^4}+…\) Then \(4S\) is equal to

The area bounded by the curves y = |x² – 1| and y = 1 is

If α, β are the roots of the equation
\(x^2-(5+3^{\sqrt{log_35}}-5^{\sqrt{log_53}})+3(3^{(log_35)^{\frac{1}{3}}}-5^{(log_53)^{\frac{2}{3}}}-1) = 0\)
then the equation, whose roots are α + 1/β and β + 1/α , is

Let \(x(t)=2\sqrt2costsin\sqrt2t \) and \(y(t)=2\sqrt2sintsin\sqrt2t\) ,\(t ∈(0,\frac{π}{2})\)

Then \(\frac{1+(\frac{dy}{dx})^2}{\frac{d^2y}{dx^2}}\) at \(t=\frac{π}{4}\) is equal to

If the function
\(f(x) = \begin{cases} \frac {log_e(1−x+x^2)+log_e(1+x+x_2)}{sec⁡\ x−cos⁡\ x}, & \quad x∈(−\frac \pi2,\frac \pi2)−{0}\\ k, & \quad x=0 \end{cases}\)
is continuous at \(x = 0\), then k is equal to

The sum of the absolute maximum and absolute minimum values of the function \(\begin{array}{l} f\left(x\right)=\tan^{-1}\left(\sin x-\cos x\right) \end{array}\)in the interval\( [0, π]\) is

1L aqueous solution of \(H_2SO_4\) contains 0.02 m mol \(H_2SO_4\). 50% of this solution is diluted with deionized water to give 1L solution (A). In solution (A), 0.01 m mol of \(H_2SO_4\) are added. Total m mols of \(H_2SO_4\) in the final solution is ______ \(× 10^3\) m mols.

Let the normal at the point P on the parabola y² = 6x pass through the point (5, –8). If the tangent at P to the parabola intersects its directrix at the point Q, then the ordinate of the point Q is

The equations of the sides AB, BC and CA of a triangle ABC are 2x + y = 0, x + py = 15a and x – y = 3, respectively. If its orthocentre is
\((2, a),−\frac{1}{2}<a<2 \)
then p is equal to _______.

If for some p, q, r ∈ R, not all have same sign, one of the roots of the equation (p² + q²)x² – 2q(p + r)x + q² + r² = 0 is also a root of the equation x² + 2x – 8 = 0, then (q² + r²)/p² is equal to _______ .

Choose the correct answer:

1. Let A = {x ∈ R : | x + 1 | < 2} and B = {x ∈ R : | x – 1| ≥ 2}. Then which one of the following statements is NOT true?

Let a, b ∈ R be such that the equation ax² – 2bx + 15 = 0 has a repeated root α. If α and β are the roots of the equation x² – 2bx + 21 = 0, then α² + β² is equal to

If n arithmetic means are inserted between a and 100 such that the ratio of the first mean to the last mean is 1 : 7 and a + n = 33, then the value of n is:

Let
A = \(\begin{bmatrix} 1 \\ 1 \\ 1 \\ \end{bmatrix}\) and B = \(\begin{bmatrix} 9^2 & -10^2 & 11^2 \\ 12^2 & 13^2 & 14^2 \\ -15^2 & 16^2 & 17^2 \\ \end{bmatrix}\)
then the value of A'BA is

The equations of the sides AB, BC and CA of a triangle ABC are 2x + y = 0, x + py = 39 and x – y = 3, respectively and P(2, 3) is its circumcentre. Then which of the following is NOT true?

The integral
\(\frac{24}{\pi} \int_{0}^{\sqrt{2}} \frac{2 - x^2}{(2 + x^2) \sqrt{4 + x^4}} \, dx\)
is equal to _______.

The depression in the freezing point observed for a formic acid solution of concentration 0.5 mL L^–1 is 0.0405°C. Density of formic acid is 1.05 g mL^–1. The Van’t Hoff factor of the formic acid solution is nearly (Given for water k_f = 1.86 k kg mol^–1)

When 800 mL of 0.5 M nitric acid is heated in a beaker, its volume is reduced to half and 11.5 g of nitric acid is evaporated. The molarity of the remaining nitric acid solution is \(x × 10^{–2} M\). (Nearest integer)
(Molar mass of nitric acid is 63 g mol^–1)

Let the coefficients of the middle terms in the expansion of\(\begin{array}{l} \left(\dfrac{1}{\sqrt 6}+ \beta x\right)^4, \left(1 – 3\beta x\right)^2 \text{and}\left(1 – \dfrac{\beta}{2}x\right )^6, \beta > 0,\end{array}\)respectively form the first three terms of an A.P. If d is the common difference of this A.P., then \(\begin{array}{l} 50-\frac{2d}{\beta^2} \end{array}\)is equal to _______.