List of top Questions asked in JEE Main

Let a > 0, b > 0. Let e and l respectively be the eccentricity and length of the latus rectum of the hyperbola 
\(\frac{x^2}{a^2}−\frac{y^2}{b^2}=1\)
Let e′ and l′ respectively be the eccentricity and length of the latus rectum of its conjugate hyperbola. If 
\(e^2=\frac{11}{14}l\) and \((e^′)^2=\frac{11}{8}l^′\)
then the value of 77a + 44b is equal to :

Let A = [aij] be a square matrix of order 3 such that aij = 2j–i, for all i, j = 1, 2, 3. Then, the matrix A2 + A3 + … + A10 is equal to 

Let a biased coin be tossed 5 times. If the probability of getting 4 heads is equal to the probability of getting 5 heads, then the probability of getting atmost two heads is

Let\( Δ,▽∈{∧,∨} \)
be such that \(p▽q⇒((pΔq)▽r) \)
is a tautology. Then \((p▽q)Δr \)
is logically equivalent to:

Let 
\(f(x)=2cos^{−1}⁡x+4cot^{−1}⁡x−3x^2−2x+10,X∈[−1,1]\)
If [a, b] is the range of the function, f then 4a – b is equal to :

Let
\(f(x)=\frac{x−1}{x+1},x∈R− \left\{0,−1,1\right\}\)
If ƒn+1(x) = ƒ(ƒn(x)) for all n∈N, then ƒ6(6) + ƒ7(7) is equal to :

Let PQ be a focal chord of the parabola y2 = 4x such that it subtends an angle of π/2 at the point (3, 0). Let the line segment PQ be also a focal chord of the ellipse \(E:\frac{x^2}{a^2}+\frac{y^2}{b^2}=1,a^2>b^2.\) If e is the eccentricity of the ellipse E, then the value of \(\frac{1}{e^2}\) is equal to

Let the plane 2x + 3y + z + 20 = 0 be rotated through a right angle about its line of intersection with the plane x – 3y + 5z = 8. If the mirror image of the point 
\((2,−\frac{1}{2},2) \)
in the rotated plane is B( a, b, c),then

The angle of elevation of the top of a tower from a point A due north of it is α and from a point B at a distance of 9 units due west of A is \(\cos^{-1}\left(\frac{3}{\sqrt{13}}\right)\)
If the distance of the point B from the tower is 15 units, then cot α is equal to :

The area bounded by the curve 
\(y=|x^2−9| \)
and the line y = 3 is

If \(\vec{a}⋅\vec{b}=1,\vec{b}⋅\vec{c}=2 and\) \(\vec{c}⋅\vec{a}=3,\)
then the value of
[\([\vec{a}×(\vec{b}×\vec{c}),\vec{b}×(\vec{c}×\vec{a}),\vec{c}×(\vec{b}×\vec{a})]\)
 is

The function \(f : R→R\) defined by \(f(x)=lim_{n→∞}\frac{cos2πx-x^{2n}sinx-1}{1+x^{2n+1}-x^{2n}}\) is continuous for all x in
The shortest distance between the lines \(\frac{x−3}{2}=\frac{y−2}{3}=\frac{z−1}{−1}\) and \(\frac{x+3}{2}=\frac{y-6}{1}=\frac{z−5}{3}\) is

The statement 
\((∼(p ⇔∼q))∧q \)
is :

The probability that a randomly chosen one-one function from the set {a, b, c, d} to the set {1, 2, 3, 4, 5} satisfies f(a) + 2f(b) – f(c) = f(d) is :

The area of the bounded region enclosed by the curve 
\(y=3−|x−\frac{1}{2}|−|x+1| \)
and the x-axis is

Let Δε{∧,∨,⇒,⇔} be such that (p∧q)Δ((p∨q)⇒q) is a tautology. Then ∆is equal to 

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

A common tangent T to the curves
\(C_1:\frac{x^2}{4}+\frac{y^2}{9} = 1\)
and
\(C_2:\frac{x^2}{4^2}\frac{-y^2}{143} = 1\)
does not pass through the fourth quadrant. If T touches C1 at (x1, y1) and C2 at (x2, y2), then |2x1 + x2| is equal to ______.

Let a circle C of radius 5 lie below the x-axis. The line L1 : 4x + 3y + 2 = 0 passes through the centre P of the circle C and intersects the line L2 : 3x – 4y – 11 = 0 at Q. The line L2 touches C at the point Q. Then the distance of P from the line 5x – 12y + 51 = 0 is _____

In an examination, there are 10 true-false type questions. Out of 10, a student can guess the answer of 4 questions correctly with probability 3/4 and the remaining 6 questions correctly with probability ¼. If the probability that the student guesses the answers of exactly 8 questions correctly out of 10 is \(\frac{27k}{4^{10}}\),then k is equal to

Let f be a differential function satisfying
f(x) =\(\frac{ 2}{√3} \)\(∫^{√30} f(\frac{λ2x}{3})dλ,x>0 and f(1) = √3.\)
If y = f(x) passes through the point (α, 6), then α is equal to _____

Let [t] denote the greatest integer ≤ t and {t} denote the fractional part of t. The integral value of α for which the left-hand limit of the function \(f(x) = \left\lfloor 1 + x \right\rfloor + \frac{\alpha^{2\left\lfloor x \right\rfloor + \left\{ x \right\}} + \left\lfloor x \right\rfloor - 1}{2\left\lfloor x \right\rfloor + \left\{ x \right\}} \) at x=0 is equal to  \(\alpha -\frac{4}{3}\), is

Let \(S = z ∈ C: |z-3| <= 1\) and \(z (4+3i)+z(4-3)≤24.\)
If α + iβ is the point in S which is closest to 4i, then 25(α + β) is equal to ______.

Let 
\(S = (0, 2\pi) - \left\{\frac{\pi}{2}, \frac{3\pi}{4}, \frac{3\pi}{2}, \frac{7\pi}{4}\right\}\)
. Let y = y(x), x∈S, be the solution curve of the differential equation 
\(\frac{dy}{dx}=\frac{1}{1+sin⁡2x},y(\frac{π}{4})=\frac{1}{2}\)
 .If the sum of abscissas of all the points of intersection of the curve y = y(x) with the curve 
\(y=\sqrt2sin⁡x\) is \(\frac{kπ}{12}\),
 then k is equal to _________.