List of top Mathematics Questions on Continuity and differentiability asked in BITSAT

If \(y^x = e^{y - x}\), then \(\frac{dy}{dx}\) is equal to
If \(y^x = e^{y - x}\), then \(\frac{dy}{dx}\) is equal to:
The equation of a common tangent to the parabolas \( y = x^2 \) and \( y = -(x - 2)^2 \) is:
The function \( f(x) = \tan^{-1}(\sin x + \cos x) \) is an increasing function in:
If \( \vec{a} = \hat{i} + \hat{j} + \hat{k} \), \( \vec{a} \cdot \vec{b} = 1 \) and \( \vec{a} \times \vec{b} = \hat{j} - \hat{k} \), then \( \vec{b} \) is:
The maximum value of \( z = 5x + 2y \) subject to the constraints: \[ x + y \leq 7, \quad x + 2y \leq 10, \quad x, y \geq 0 \]
The curve given by \( x + y = e^{xy} \) has a tangent parallel to the Y-axis at the point:
If \( y = \sqrt{\frac{1 + \cos 2\theta}{1 - \cos 2\theta}} \), then \( \frac{dy}{d\theta} \) at \( \theta = \frac{3\pi}{4} \) is:
Find: \( \lim_{x \to 0} \frac{| \sin x |}{x} \)
The lines \[ \frac{x - 2}{1} = \frac{y - 3}{1} = \frac{z - 4}{-k} \] and \[ \frac{x - 1}{k} = \frac{y - 4}{2} = \frac{z - 5}{1} \] are coplanar if:
The function f(x)=x-|x-x²|,-1≤ x\le1 is
If f(x)= begincases (xlog(cos x))/(log(1+x²)), & x≠ 0
0, & x=0 endcases then f(x) is
The number of points at which the function f(x)=(1)/(log|x|) is discontinuous is
If f(x)= begincases (xlog(cos x))/(log(1+x²)), & x≠0
0, & x=0 endcases then f(x) is
Let f:RtoR be a function such that f(x+y)=f(x)+f(y). If f(x) is differentiable at x=0, then which one of the following is incorrect?
If
\[ f(x) = \frac{x}{1+x} + \frac{x}{(x+1)(2x+1)} + \frac{x}{(2x+1)(3x+1)} + \cdots \]

then at \(x = 0\), \(f(x)\) is:
If
\( f(x) = \begin{cases} 1, & 0 < x \le \dfrac{3\pi}{4} \\ 2\sin\left(\dfrac{2x}{9}\right), & \dfrac{3\pi}{4} < x < \pi \end{cases} \)
then:
If f(x)= begincases sin x, & when x is rational
cos x, & when x is irrational endcases then the function is:
If
\[ f(x) = \begin{cases} \dfrac{x \log(\cos x)}{\log(1 + x^2)}, & x \ne 0 \\[2mm] 0, & x = 0 \end{cases} \]
then \(f(x)\) is
For any differentiable function \( y \) of \( x \),
\( \dfrac{d^2 x}{dy^2} \left( \dfrac{dy}{dx} \right)^3 + \dfrac{d^2 y}{dx^2} = \)
If f(x)= begincases (xlog(cos x))/(log(1+x²)), & x\ne0
0, & x=0 endcases then f(x) is
The number of points at which the function f(x)=(1)/(log|x|) is discontinuous is
Let \(f:\mathbb R\to\mathbb R\) be a function such that \(f(x+y)=f(x)+f(y)\) for all \(x,y\in\mathbb R\). If \(f(x)\) is differentiable at \(x=0\), then which one of the following is incorrect?
If \(y=\left(x+\sqrt{1+x^2}\right)^n\), then \((1+x^2)\dfrac{d^2y}{dx^2}+x\dfrac{dy}{dx}\) is
If a function \(f(x)\) is given by \[ f(x)=\frac{x}{1+x}+\frac{x}{(x+1)(2x+1)}+\frac{x}{(2x+1)(3x+1)}+\cdots+\infty, \] then at \(x=0\), \(f(x)\)