List of top Mathematics Questions

If \(f(x)\) is a differentiable function and \[ y=e^{f(x)+e^{f(x)+e^{f(x)+\cdots \infty}}}, \] then \[ \frac{dy}{dx}= \ ? \]

The area of the region bounded by the curve \[ y=|x-2|+|x-8|, \] the X-axis and the lines \(x=0\) and \(x=10\) is:

If \[ \frac{dy}{dx} = (x^3-x) -(1-3x^2)\tan x +(x^3-x)\tan^2x \] and \[ y(1)=0, \] then \[ \frac{64}{\pi}y\!\left(\frac{\pi}{4}\right) = ? \]

If \([t]\) denotes the greatest integer function, then \[ \int_{-2}^{2} \left[ \frac{x^2+[x+1]} {1+x^2} \right]dx = \ ? \]

If \[ \frac{dy}{dx} - \frac{2x}{x^2+b}y = -2x(x^2+b), \] with \[ y(0)=12,\qquad y(1)=10, \] then the sum of all possible values of \(b\) is:

A rectangle is inscribed in the parabola \[ y=9-x^{2} \] such that two of its vertices are on the X-axis and another two on the parabola. The dimensions of such rectangle lying above the X-axis and having maximum area is:

Evaluate: \[ \int \sin^{-1}x\,dx-\int \cos^{-1}x\,dx. \]

If \[ \int\frac{dx}{\sqrt{4x^{2}+11x+6}} = \frac12\cosh^{-1}\!\left(\frac{f(x)}{5}\right)+C \] and \[ f(1)=19, \] then \(f(2)=\) ?

If \[ \int \log x\sqrt{\left(\frac{\log x}{x}\right)^2+\frac1{x^2}}\,dx = \frac{f(x)}{3}\sqrt{1+(\log x)^2}+C \] and \(f(1)=1\), then \(f(e)=\) ?

If \[ \int\frac1{1+\cos x}\,dx = \frac1{f\!\left(\frac x2\right)}+C_1, \] then \[ \int f(x)\,dx= \ ? \]

Let \[ f:[0,1]\rightarrow\mathbb R \] be a function defined by \[ f(x)+f(1-x)=1. \] Then \[ \int_0^1 f(x)\,dx= \ ? \]

If the angle between the curves \[ y^2=4x \] and \[ y=ax^2-5 \] at the point \((1,2)\) is \(\alpha\), then \[ (a-2)|\tan\alpha| = ? \]

In the interval \([-5,5]\), if \[ f(x)=(x+3)^2(x-2)^3 \] is increasing on \[ S=\{x\mid -5\le x<\alpha \text{ and } \beta<x\le5\}, \] then \(f(\alpha)-f(\beta)=\) ?

Statement-I: The equation of the tangent to the curve \[ y=3x^2-5 \] drawn through the point \((1,2)\) is \[ y=6x-4. \] Statement-II: If \(L,M,N\) are respectively the lengths of tangent, normal and subnormal drawn to a curve at a point \((a,b)\), then \[ \frac{(L)(N)}{M}=b^2. \] Choose the correct option.

A function is defined as \[ f(x)= \begin{cases} 3x-1, & 0\le x\le 2,\\ \sqrt{25(x-1)}, & 2\le x<\infty. \end{cases} \] For \(f(x)\) in the interval \(\left[\frac13,3\right]\), choose the correct statement.

If \[ \frac{d}{dx} \left( \frac{\sec x+\tan x} {\sec x-\tan x} \right) =k \] at \[ x=\frac{\pi}{4}, \] then \[ \frac{k}{2\sqrt2}-2\sqrt2= \ ?} \]

If \[ f(x)= \frac{\lambda e^{\frac1x}+3e^{-\frac1x}} {(\lambda+2)e^{\frac1x}-e^{-\frac1x}}, \qquad x\neq0 \] and \(f(0)=k\), \(k\in\mathbb R\), is a continuous function at \(x=0\), then \(2\lambda=\) ?

If the direction cosines of the line common to the planes \[ x+2y-z-1=0 \] and \[ 3x-4y+z-5=0 \] are \((l,m,n)\), then \(|l+m-n|=\)

\(f(x)\) is an \(n^{th}\) degree polynomial and \(\alpha_1,\alpha_2,\ldots,\alpha_n\) are distinct zeros of \(f(x)\). \(g(x)\) is a polynomial having three zeros common with the zeros of \(f(x)\). Assertion (A): \(|f(x)|g(x)\) is continuous and differentiable at all \(\alpha_i\).
Reason (R): \[ \lim_{x\to a}\frac{|x-a|}{x-a} \] does not exist and \[ \lim_{x\to a}|x-a|=0. \] Choose the correct option.}

If \(P(1,2,5)\), \(Q(3,0,7)\), \(R(6,-3,10)\) are collinear and \((\alpha,\beta,\gamma)\) is a point at distance 3 from \(P\) on the same line, then the value of \(\alpha+\beta+\gamma\) lying between 6 and 7 is:

The centre of the ellipse lies on the lines \(2x+3y=5\) and \(x+3y=4\). If the eccentricity of the ellipse is \(\frac23\), length of its major axis is 4 and its minor axis is parallel to Y-axis, then the equation of the ellipse is:

\(y=4\) is the directrix of the parabola \[ x^{2}+8x+12y+k=0. \] If \(l\) is the length of its latus rectum, then \(l-k=\) ?

For a hyperbola \[ \frac{x^2}{a^2}-\frac{y^2}{b^2}=1, \] the distance between its vertex and focus lying on the positive X-axis is 2. If the length of its latus rectum is 13, then the eccentricity is:

The area of the rectangle formed by the tangents drawn at the ends of both major and minor axes of an ellipse is 24. If the eccentricity of the ellipse is \(\frac{1}{4}\), then the equation of the ellipse is:

\((1,1)\) is the focus of the parabola \[ y^{2}-4ax-2ay+a^{2}=0. \] If the circles \[ (x-\alpha)^2+(y-\beta)^2=r^2 \] touch the X-axis and the axis of the given parabola, then \[ \{(\alpha,\beta)\} \] is: