List of top Mathematics Questions asked in JEE Main

Let \([\,\cdot\,]\) denote the greatest integer function. Then \[ \int_{-\pi/2}^{\pi/2} \frac{12(3+[x])}{3+[\sin x]+[\cos x]}\,dx \] is equal to:
Let \[ f(x)=\lim_{\theta\to 0} \frac{\cos(\pi x-\theta)\,\sin(x-1)} {1+x^{\theta/2}(x-1)},\qquad x\in\mathbb{R}. \] Consider the following statements:
[(I)] \(f(x)\) is continuous at \(x=1\).
[(II)] \(f(x)\) is continuous at \(x=-1\).
Then:
Given below are two statements: Statement I: The function \(f:\mathbb{R}\to\mathbb{R}\) defined by \[ f(x)=\frac{x}{1+|x|} \] is one-one. Statement II: The function \(f:\mathbb{R}\to\mathbb{R}\) defined by \[ f(x)=\frac{x^2+4x-30}{x^2-8x+18} \] is many-one. In the light of the above statements, choose the correct answer.
An ellipse has its centre at \((1,-2)\), one focus at \((3,-2)\) and one vertex at \((5,-2)\). Then the length of its latus rectum is:
Let \(y=y(x)\) be the solution of the differential equation \[ x\frac{dy}{dx}=y-x^2\cot x,\quad x\in(0,\pi) \] If \(y\!\left(\frac{\pi}{2}\right)=\frac{\pi^2}{2}\), then \[ 6y\!\left(\frac{\pi}{6}\right)-8y\!\left(\frac{\pi}{4}\right) \] is equal to:
The sum of the coefficients of \(x^{499}\) and \(x^{500}\) in \[ (1+x)^{1000}+x(1+x)^{999}+x^2(1+x)^{998}+\cdots+x^{1000} \] is:
Let the arithmetic mean of \(\frac{1}{a}\) and \(\frac{1}{b}\) be \(\frac{5}{16}\), where \(a>2\). If \(a,4,b\) are in A.P., then the equation \[ ax^2-ax+2(a-2b)=0 \] has:
Let the ellipse \[ E:\ \frac{x^2}{144}+\frac{y^2}{169}=1 \] and the hyperbola \[ H:\ \frac{x^2}{16}-\frac{y^2}{2^2}=1 \] have the same foci. If \(e\) and \(L\) respectively denote the eccentricity and the length of the latus rectum of \(H\), then the value of \(24(e+L)\) is:
Given below are two statements: Statement I: \[ 25^{13}+20^{13}+31^{13} \text{ is divisible by } 7 \] Statement II: The integral part of \(\left(7+4\sqrt3\right)^{25}\) is an odd number. In the light of the above statements, choose the correct answer:
Let \[ f(x)=\int \frac{dx}{2\left(\frac{3}{2}\right)^x+2x\left(\frac12\right)^x} \] such that \(f(0)=-26+24\log_e(2)\). If \(f(1)=a+b\log_e(3)\), where \(a,b\in\mathbb{Z}\), then \(a+b\) is equal to:
Let \(P_1 : y = 4x^2\) and \(P_2 : y = x^2 + 27\) be two parabolas. If the area of the bounded region enclosed between \(P_1\) and \(P_2\) is six times the area of the bounded region enclosed between the line \(y = x\), the line \(x = 0\), and \(P_1\), then the required value is:
The minimum value of \( \cos^2\theta + 6\sin\theta\cos\theta + 3\sin^2\theta \) is:
A complex number 'z' satisfy both \(|z-6|=5\) & \(|z+2-6i|=5\) simultaneously. Find the value of \(z^3 + 3z^2 - 15z + 141\).

If \[ \frac{\cos^2 48^\circ - \sin^2 12^\circ}{\sin^2 24^\circ - \sin^2 6^\circ} = \frac{\alpha + \beta\sqrt{5}}{2}, \] where \( \alpha, \beta \in \mathbb{N} \), then the value of \( \alpha + \beta \) is ___________.

If \[ \int (\sin x)^{-\frac{11}{2}} (\cos x)^{-\frac{5}{2}} \, dx \] is equal to \[ -\frac{p_1}{q_1}(\cot x)^{\frac{9}{2}} -\frac{p_2}{q_2}(\cot x)^{\frac{5}{2}} -\frac{p_3}{q_3}(\cot x)^{\frac{1}{2}} +\frac{p_4}{q_4}(\cot x)^{-\frac{3}{2}} + C, \] where \( p_i, q_i \) are positive integers with \( \gcd(p_i,q_i)=1 \) for \( i=1,2,3,4 \), then the value of \[ \frac{15\,p_1 p_2 p_3 p_4}{q_1 q_2 q_3 q_4} \] is ___________.

Let \( \alpha = \dfrac{-1 + i\sqrt{3}}{2} \) and \( \beta = \dfrac{-1 - i\sqrt{3}}{2} \), where \( i = \sqrt{-1} \). If 
\[ (7 - 7\alpha + 9\beta)^{20} + (9 + 7\alpha - 7\beta)^{20} + (-7 + 9\alpha + 7\beta)^{20} + (14 + 7\alpha + 7\beta)^{20} = m^{10}, \] then the value of \( m \) is ___________.

If \( A = \{1,2,3,4,5,6\} \) and \( B = \{1,2,3,\ldots,9\} \), then the number of strictly increasing functions \( f : A \to B \) such that \( f(i) \neq i \) for all \( i = 1,2,3,4,5,6 \) is:
Let \(O\) be the vertex of the parabola \(y^{2}=16x\). The locus of the centroid of \(\triangle OPA\), when point \(P\) lies on the parabola and point \(A\) lies on the \(x\)-axis such that \(\angle OPA = 90^\circ\), is:
If the domain of the function
\[ f(x)=\sin^{-1}\!\left(\frac{5-x}{3+2x}\right)+\frac{1}{\log_e(10-x)} \]
is \( (-\infty,\alpha] \cup [\beta,\gamma) - \{\delta\} \), then \( 6(\alpha+\beta+\gamma+\delta) \) is equal to
If \( O \) is the vertex of the parabola \( x^2 = 4y \), \( Q \) is a point on the parabola. If \( C \) is the locus of the point which divides \( OQ \) in the ratio \( 2:3 \), then the equation of the chord of \( C \) which is bisected at the point \( (1,2) \) is:
Let one end of focal chord of the parabola \( y^2 = 16x \) be \( (16,16) \).
If \( P(\alpha,\beta) \) divides this focal chord internally in the ratio \( 5:2 \), then the minimum value of \( \alpha + \beta \) is equal to
Image of point P (1, 2, a) with respect to line mirror \(\frac{x-6}{3} = \frac{y-7}{2} = \frac{z-7}{-2}\) is point Q (5, b, c), then value of \((a^2 + b^2 + c^2)\) is :
Ellipse \( E:\; \dfrac{x^2}{36} + \dfrac{y^2}{16} = 1 \). A hyperbola is confocal with the ellipse and the eccentricity of the hyperbola is equal to \(5\). If the principal axis of the hyperbola is the \(x\)-axis, then the length of the latus rectum of the hyperbola is:
The area enclosed by \[ x^2 + 4y^2 \le 4,\qquad y \le |x| - 1,\qquad y \ge 1 - |x| \] is equal to:
Rhombus \( ABCD \) is given with vertices \( A(1,2) \), \( C(-3,-6) \) and sides \( AD \) and \( BC \) are parallel to the line \( 7x - y = 14 \). If coordinates of \( B \) and \( D \) are \( (\alpha,\beta) \) and \( (\gamma,\delta) \) respectively, then find \( \alpha + \beta + \gamma + \delta \):