List of top Mathematics Questions

The condition that \(f(x)=ax^3+bx^2+cx+d\) has no extreme value is

If
\[ f(x)=\max\{3-x,\;3+x,\;6\} \] is not differentiable at \(x=a\), and \(x=b\), then \(|a|+|b|=\)

If the curves \(y=x^3-3x^2-8x-4\) and \(y=3x^2+7x+4\) touch each other at a point \(P\), then the equation of common tangent at \(P\) is

If
\[ f(x)=\cot^{-1}\left(\frac{x^x+x^{-x}}{2}\right), \] then \(f'(1)=\)

The maximum value of
\[ f(x)=\frac{x}{1+4x+x^2} \] is

If \(ax+by=1\) is a normal to the parabola
\[ y^2=4px, \] then the condition is

If
\[ x^3-2x^2y^2+5x+y-5=0, \] then at \((1,1)\), \(y''(1)=\)

If \((a,b,c)\) are the direction ratios of a line joining the points \((4,3,-5)\) and \((-2,1,-8)\), then the point \(P(a,3b,2c)\) lies on the plane

The \(x\)-intercept of a plane \(\pi\) passing through the point \((1,1,1)\) is \(\dfrac{5}{2}\) and the perpendicular distance from the origin to the plane \(\pi\) is \(\dfrac{5}{7}\). If the \(y\)-intercept of the plane \(\pi\) is negative and the \(z\)-intercept is positive, then its \(y\)-intercept is

Let \(f:\mathbb{R}^+\to\mathbb{R}^+\) be a function satisfying
\[ f(x)-x=\lambda \text{ constant},\quad \forall x\in\mathbb{R}^+ \] and
\[ f(f(y))=f(xy)+x,\quad \forall x,y\in\mathbb{R}^+. \] Then
\[ \lim_{x\to0}\frac{(f(x))^{1/3}-1}{(f(x))^{1/2}-1}= \]

If
\[ \lim_{x\to0}\frac{|x|}{\sqrt{x^4+4x^2+5}}=k, \] and
\[ \lim_{x\to0}x^4\sin\left(\frac{1}{3\sqrt{x}}\right)=l, \] then \(k+l=\)

If
\[ \lim_{x\to\infty}x^{\log_e x}=0, \] then \(\log_x12=\)

If the tangent at the point \(P\) on the circle
\[ x^2+y^2+6x+6y=2 \] meets the straight line
\[ 5x-2y+6=0 \] at a point \(Q\) on the \(y\)-axis, then the length of \(PQ\) is

The eccentric angle of a point on the ellipse \(x^2+3y^2=6\) lying at a distance of \(2\) units from its centre is

Let origin be the centre, \((\pm 3,0)\) be the foci and \(\dfrac{3}{2}\) be the eccentricity of a hyperbola. Then the line \(2x-y-1=0\)

The locus of a variable point whose chord of contact with respect to the hyperbola
\[ \frac{x^2}{a^2}-\frac{y^2}{b^2}=1 \] subtends a right angle at the origin is

If the point \((a,8,-2)\) divides the line segment joining the points \((1,4,6)\) and \((5,2,10)\) in the ratio \(m:n\), then
\[ \frac{2m}{n}-\frac{a}{3}= \]

Suppose a parabola with focus at \((0,0)\) has
\[ x-y+1=0 \] as its tangent at the vertex. Then the equation of its directrix is

If the lines represented by
\[ ax^2+2hxy+by^2+2gx+2fy+c=0 \] intersect on the \(x\)-axis, which of the following is in general incorrect?

The length of the intercept on the line \(4x-3y-10=0\) by the circle \(x^2+y^2-2x+4y-26=0\) is

The locus of centers of the circles, passing the same area and having \(3x-4y+4=0\) and \(6x-8y+7=0\) as their common tangent, is

For any two nonzero real numbers \(a\) and \(b\), if the line
\[ \frac{x}{a}+\frac{y}{b}=1 \] is a tangent to the circle
\[ x^2+y^2=1, \] then which of the following is true?

The pole of the line \(\dfrac{x}{a}+\dfrac{y}{b}=1\) with respect to the circle \(x^2+y^2=c^2\) is

For \(\alpha \in \left[0,\dfrac{\pi}{2}\right]\), the angle between the lines represented by \[ [x\cos\theta-y]\left[(\cos\theta+\tan\alpha)x-(1-\cos\theta \tan\alpha)y\right]=0 \] is

Suppose a triangle is formed by \(x+y=10\) and the coordinate axes. Then the number of points \((x,y)\), where \(x\) and \(y\) are natural numbers, lying inside the triangle is