List of top Mathematics Questions asked in JEE Main

Let
\(f(x) = 2+|x|-|x-1|+|x+1|,x∈R.\)
Consider \(f'\left(-\frac{3}{2}\right) + f'\left(-\frac{1}{2}\right) + f'\left(\frac{1}{2}\right) + f'\left(\frac{3}{2}\right) = 2\)
(\((S2):\int_{-2}^{2} f(x) \,dx = 12\)
Then,

\(\int_{0}^{20\pi} (|\sin x| + |\cos x|)^2 \,dx\)
is equal to

The angle of elevation of the top P of a vertical tower PQ of height 10 from a point A on the horizontal ground is 45°, Let R be a point on AQ and from a point B, vertically above R, the angle of elevation of P is 60°. If
\(∠BAQ = 30°\)
, AB = d and the area of the trapezium PQRB is α, then the ordered pair (d, α) is :

If the length of the latus rectum of a parabola, whose focus is (a, a) and the tangent at its vertex is x + y = a, is 16, then |a| is equal to :

The area of the region enclosed by
\(y≤4x^2, x2≤9y\ and\ y≤4,\)
is equal to

Consider a curve y = y(x) in the first quadrant as shown in the figure. Let the area A1 is twice the area A2. Then the normal to the curve perpendicular to the line 2x – 12y = 15 does NOT pass through the point.

Fig. 

Let
\(β = \lim_{x →0} \frac{αx-(e^{3x}-1)}{αx(e^{3x}-1) }\)for some\( α \in R.\)
Then the value of α+β is

Let P and Q be any points on the curves (x – 1)2 + (y + 1)2 = 1 and y = x2, respectively. The distance between P and Q is minimum for some value of the abscissa of P in the interval

If z = x + iy satisfies | z | – 2 = 0 and |z – i| – | z + 5i| = 0, then
The slope of normal at any point \((x, y), x > 0, y > 0\) on the curve \(y = y(x)\) is given by \(\frac{x^2}{xy-x^2y2^-1}\) If the curve passes through the point (1, 1), then \(e · y(e)\) is equal to
Let \(A = \begin{bmatrix} 0 & -2 \\[0.3em] 2 & 0 \end{bmatrix}\). If \(M\) and \(N\) are two matrices given by \(M = \displaystyle\sum_{k=1}^{10} A^{2k}\) and \(N = \displaystyle\sum_{k=1}^{10} A^{2k-1}\)
then \(MN^2 \) is :

Let \(X = \begin{bmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \\ \end{bmatrix}\),
Y = αI + βX + γX2 and
Z = α²l - αβX + (β² - αϒ)X² ,α,β,ϒ ∈ R.
If \(Y^{-1} = \begin{bmatrix} \frac{1}{5} & -\frac{2}{5} & \frac{1}{5} \\ 0 & \frac{1}{5} & -\frac{2}{5} \\ 0 & 0 & \frac{1}{5} \\ \end{bmatrix}\),
then ( α - β + ϒ )² is equal to ________.

If \(\frac{dy}{dx} + \frac{2x-y(2^y-1)}{2x-1} = 0, x,y > 0,y(1) = 1\), then \(y(2)\) is equal to:
The greatest integer less than or equal to the sum of first \(100\) terms of the sequence \(\frac 13, \frac 59, \frac {19}{27}, \frac {65}{81},…...\) is equal to ______.
Let the abscissae of the two points \(P\) and \(Q\) be the roots of \(2x^2 – rx + p = 0\) and the ordinates of \(P\) and \(Q\) be the roots of \(x^2 – sx – q = 0\). If the equation of the circle described on \(PQ\) as diameter is \(2(x^2 + y^2) – 11x – 14y – 22 = 0\), then \(2r + s – 2q + p\) is equal to _________.
The number of values of x in the interval \((\frac \pi 4, \frac {7\pi }{4})\) for which \(14cosec^2x – 2sin^2x = 21 – 4cos^2x\) holds, is ______.
Let \(θ\) be the angle between the vectors \(\vec a\) and \(\vec b\), where \(|\vec a|=4\), \(|\vec b|=3\) and \(θ(\frac \pi 4,\ \frac \pi3)\) . Then \(|(\vec a−\vec b)×(\vec a+\vec b)|^2+4(\vec a.\vec b)^2\) is equal to ______.
Let the lines
\(L_1:\vec r=λ(\hat i+2\hat j+3 \hat k), \ λ∈R\)
\(L_2:\vec r=(\hat i+3\hat j+\hat  k)+μ(\hat i+\hat j+\hat 5k), \ μ∈R\)
intersect at the point \(S\). If a plane \(ax + by – z + d = 0\) passes through \(S\) and is parallel to both the lines \(L_1\) and \(L_2\) then the value of \(a + b + d\) is equal to _______.
Let \(ƒ : N→R\) be a function such that \(ƒ(x + y) = 2ƒ(x) ƒ(y)\) for natural numbers x and y. If \(ƒ(1) = 2\), then the value of α for which \(\displaystyle\sum_{k=1}^{10} f(α+k)=\frac {512}{3} (2^{20}−1)\) holds, is:
Let \(λ^*\) be the largest value of \(λ\) for which the function \(f_λ(x) = 4λx^3 – 36λx^2 + 36x + 48\) is increasing for all \(x ∈ ℝ\). Then \(f_λ^* (1) + f_λ^* (– 1)\) is equal to :
Let f: ℝ → ℝ satisfy f(x + y) = 2x f(y) + 4y f(x), ∀ x, y∈ ℝ. If f(2) = 3, then 14. f'(4)/f'(2) is equal to ___.
If the solution of the differential equation
\(\frac{dy}{dx}\) \(+ e^x(x² - 2)y = (x^2 - 2x)(x^2 - 2)e^{2x}\)
satisfies y(0) = 0, then the value of y(2) is ______.
If the solution curve \(y = y(x)\) of the differential equation\( y^2dx + (x^2 – xy + y^2)dy = 0\), which passes through the point \((1,1)\) and intersects the line \(y=\sqrt 3x\) at the point \((α,\sqrt 3α)\), then value of \(log_e(\sqrt 3α)\) is equal to :
Let \(E_1\) and \(E_2\) be two events such that the conditional probabilities \(P(E_1|E_2)=\frac 12\), \(P(E_2|E_1)=\frac 34\) and \(P(E_1∩E_2)=\frac 18\)⋅ Then:
For a natural number \(n\), let \(α_n = 19n – 12n\). Then, the value of \(\frac {31α_9−α_{10}}{57α_8}\) is _____.