List of top Mathematics Questions

If \([\,\,]\) represents greatest integer function, then \[ \int_{-1}^{1}\left(x[1+\sin \pi x]+1\right)dx= \]

If \[ \lim_{n\to\infty} \left[ \frac{n}{(n+1)\sqrt{2n+1}} + \frac{n}{(n+2)\sqrt{2(2n+2)}} + \frac{n}{(n+3)\sqrt{3(2n+3)}} +\cdots+n\ \text{terms} \right] = \int_0^1 f(x)\,dx, \] then \(f(x)=\)

Let \(T\gt 0\) be a fixed number. If \(f:\mathbb{R}\to\mathbb{R}\) is a continuous function such that \[ f(x+T)=f(x),\qquad x\in \mathbb{R}. \] If \[ I=\int_0^T f(x)\,dx, \] then \[ \int_0^{5T} f(2x)\,dx= \]

Evaluate \[ \int \frac{e^{\tan^{-1}x}}{1+x^2} \left[ \left(\sec^{-1}\sqrt{1+x^2}\right)^2+ \cos^{-1}\left(\frac{1-x^2}{1+x^2}\right) \right]dx \]

Evaluate \[ \int \frac{dx}{(x-3)^{4/5}(x+1)^{6/5}} \]

If \[ \int \frac{3x+4}{x^3-2x+4}\,dx=\log f(x)+C, \] then \(f(3)=\)

If \(a,b\gt 0\), then minimum value of \[ y=\frac{b^2}{a-x}+\frac{a^2}{x}, \qquad 0\lt x\lt a \] is

The point on the curve \[ y=x^2+4x+3 \] which is closest to the line \[ y=3x+2 \] is

If \[ I_n=\int(\cos^n x+\sin^n x)\,dx \] and \[ I_n-\frac{n-1}{n}I_{n-2} = \frac{\sin x\cos x}{n}f(x), \] then \(f(x)=\)

If \[ y=x^3-ax^2+48x+7 \] is an increasing function for all real values of \(x\), then \(a\) lies in the interval

If \[ 3f(\cos x)+2f(\sin x)=5x, \] then \[ f'(\cos x)+f'(\sin x)= \]

Assertion (A): \[ \frac{d}{dx}\left(\frac{x^2\sin x}{\log x}\right) = \frac{x^2\sin x}{\log x} \left(\cot x+\frac{2}{x}-\frac{1}{x\log x}\right) \] Reason (R): \[ \frac{d}{dx}\left(\frac{uv}{w}\right) = \frac{uv}{w} \left[ \frac{u'}{u}+\frac{v'}{v}-\frac{w'}{w} \right] \]

If the normal drawn at a point \(P\) on the curve \[ 3y=6x-5x^3 \] passes through \((0,0)\), then the positive integral value of the abscissa of the point \(P\) is

The line joining the points \((0,3)\) and \((5,-2)\) is a tangent to the curve \[ y=\frac{C}{x+1} \] then \(C=\)

Evaluate \[ \lim_{x\to 0}\frac{3|x|-x}{|x|-2x} - \lim_{x\to 0}\frac{\log(1+x^3)}{\sin^3 x} \]

If \[ x=f(\theta) \] and \[ y=g(\theta), \] then \[ \frac{d^2y}{dx^2}= \]

If \(e_1\) and \(e_2\) are the eccentricities of the hyperbola \[ 16x^2-9y^2=1 \] and its conjugate respectively, then \(3e_1=\)

If \(P\) divides the line segment joining the points \(A(1,2,-1)\) and \(B(-1,0,1)\) externally in the ratio \(1:2\) and \(Q=(1,3,-1)\), then \(PQ=\)

If the direction cosines of a line are \[ \left( \frac{a}{\sqrt{83}}, \frac{5}{\sqrt{83}}, \frac{c}{\sqrt{83}} \right) \] and \(c-a=4\), then \(ca=\)

Let the plane \(\pi\) pass through the point \((1,0,1)\) and perpendicular to the planes \[ 2x+3y-z=2 \] and \[ x-y+2z=1. \] Let the equation of the plane passing through the point \((11,7,5)\) and parallel to the plane \(\pi\) be \[ ax+by-z+d=0. \] Then \[ \frac{a}{b}+\frac{b}{d}= \]

Evaluate \[ \lim_{x\to -\infty}\log_e(\cosh x)+x \]

If \(a,b,c\) are three distinct real numbers and \[ \lim_{x\to \infty} \frac{(b-c)x^2+(c-a)x+(a-b)} {(a-b)x^2+(b-c)x+(c-a)} = \frac{1}{2} \] then \(a+2c=\)

If the angle between the straight lines joining the foci and the ends of the minor axis of the ellipse \[ \frac{x^2}{a^2}+\frac{y^2}{b^2}=1 \] is \(90^\circ\), then its eccentricity is

The locus of point of intersection of tangents at the ends of normal chord of the hyperbola \[ x^2-y^2=a^2 \] is

The radical centre of the three circles \[ x^2+y^2-1=0, \] \[ x^2+y^2-8x+15=0 \] and \[ x^2+y^2+10y+24=0 \] is