List of top Mathematics Questions

Let \(\mathbb R\) denote the set of all real numbers. Let \[ f:\mathbb R\to\mathbb R \] be an arbitrary function and let \[ g:\mathbb R\to\mathbb R \] be the function defined by \[ g(x)=xf(x),\qquad \forall x\in\mathbb R \] Then which of the following statements is (are) TRUE?

At a point $P(x,y)$ on a curve $x=f(y)$, the x-intercept of the tangent is always equal to the y-coordinate of the point of contact, then $f(y)=$

If $f(x)=\left|\frac{x}{3}-3^{-x}\right|$ and $\int_{0}^{5}f(x)dx=K+\frac{3^{n}+1}{3^{5}\log 3}$, then $nK=$

The area of the region bounded by the curve $y=x^{2}+x$, the lines $y=x$, $x=1$ and $y=2$ is

_n=1^_0^1x^3(1-x^2)^ndx=

If $I_{n}=\int\sin^{n}x dx$, then $I_{6}-\frac{5}{6}I_{4}=$

If $y=f(x)$ is a solution of $\frac{dy}{dx}=(y-kx)^{2}+K$ when $x=0$ and $y=1$, then $f(2)=$

If $f(x)$ is a twice differentiable function and $f^{\prime}(0)=0$, then $\int_{0}^{\pi/2}(f(x)+f^{\prime\prime}(x))\cos x dx=$

If the degree of the differential equation $\left(\frac{d^{2}y}{dx^{2}}\right)^{3/2}+5\left(\frac{d^{2}y}{dx^{2}}\right)^{5/2}=7y$ is $m$ and its order is $n$, then $y=Ae^{mx}+Be^{nx}$ is solution of the differential equation

Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be such that $f(2+x)=f(2-x)\,\,\forall x\in\mathbb{R}$. If $f(x)$ is twice differentiable such that $f^{\prime}(1)=0$, then which one of the following is true?

Let $f:\mathbb{R}^{+}\rightarrow\mathbb{R}$ be such that $f(e)=\frac{7}{4}$ and $f^{\prime}(x^{2})=\frac{3\log x}{x^{2}}$, then $f(e^{2})=$

If the local maximum 'M' and local minimum 'm' of the function $f(x)=x-\frac{x^{2}}{2}-xe^{2-x}$ exist at $x=\alpha$ and $x=\beta$ respectively, then $2\alpha m+\beta M=$

If $\int(4\sin\theta+\cos\theta)\cot\theta\cos\theta d\theta=2\theta+\sin2\theta+\log(\sin\theta)+f(2\theta)+c$ and $f(0)=\frac{1}{2}$, then $f(x)=$

If $\int\frac{1}{x^{2}+4x+\alpha}dx=\frac{1}{2\sqrt{2}}\tan^{-1}\left(\frac{x+2}{2\sqrt{2}}\right)+c$, then $\int\frac{1}{x^{2}+4x-\alpha}dx=$

If $\int\frac{2\sin x+a\cos x}{b\sin x+4\cos x}dx=\frac{2}{5}x-\frac{1}{5}\log(b\sin x+4\cos x)+c$, then $a+b=$

If the acute angle between the parabolas $y=8x-x^{2}$ and $y=x^{2}-4x$ is $\theta$, then $\tan\theta=$

If the function $f(x)=\begin{cases}\frac{2h(x)-g(x)}{(h(x)+7)^{2/3}}, & x\ne0 \\ \frac{7}{4}, & x=0\end{cases}$ is continuous at $x=0$ and $\lim_{x\rightarrow0}h(x)=1$, then $\lim_{x\rightarrow0}g(x)=$

Let $[y]$ represent the greatest integer less than or equal to $y$. Then the set of all $x$ at which $f(x)=\cos^{-1}[4x+3]$ is differentiable is

If the length of the tangent at a point on the parabola $y^{2}=4ax$ is $4a\sqrt{5}$, then the length of the sub-normal at that point is

If $f(x)=\begin{cases}\frac{1-\sin^{3}x}{3\cos^{2}x}, & x\ne\frac{\pi}{2} \\ \frac{1}{2}, & x=\frac{\pi}{2}\end{cases}$, then $f^{\prime}\left(\frac{\pi}{2}\right)=$

If $\tan^{-1}x^{2}+\tan^{-1}y^{2}=\frac{\pi}{2}$, then $\left(\frac{dy}{dx}\right)_{(-1,2)}=$

If $\lim_{n\rightarrow5}(\frac{[n]}{2})^{3}-(\frac{[n]^{3}}{2^{4}})=k$, then $\lim_{n\rightarrow k^{+}}(\frac{[n]}{2})^{3}-(\frac{[n]^{3}}{2^{4}})=$

$\lim_{x\rightarrow1}\frac{(9x-1)(\sqrt{x}-1)}{3x^{2}+2x-5}=$

The points (0, $\lambda$, 1), ($\mu$, 3, -1), ($\lambda$, 5, 0), ($\mu$, 6, $\mu$) taken in that order, form a square. If $\lambda$, $\mu$ are positive real numbers, then the length of its side is

If $\theta$ is the acute angle between a line $\frac{x-1}{1}=\frac{y-1}{-1}=\frac{z-1}{1}$ and a normal to the plane $2x+3y+4z=0$ then $\tan^{2}\theta+sec^{2}\theta=$