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Let \( f:R \to R \) be a function defined by:

Where \( [t] \) is the greatest integer less than or equal to \( t \). Let \( m \) be the number of points where \( f \) is not differentiable and \( I=\int_{-2}^{2} f(x) \, dx \). Then the ordered pair \( (m, I) \) is equal to:

Let a function \( f : \mathbb{R} \to \mathbb{R} \) be defined as:

Where \( b \in \mathbb{R} \). If \( f \) is continuous at \( x = 4 \), then which of the following statements is NOT true?

If the function
is continuous at \( x=0 \), then \( k \) is equal to:

The number of points, where the function \(f:\mathbb{R}\to\mathbb{R}\), \( f(x)=|x-1|\cos|x-2|\sin|x-1|+(x-3)|x^{2}-5x+4| \), is NOT differentiable, is:

\( \displaystyle \lim_{x \to \frac{\pi}{2}} \tan^{2} x \left( \left(2 \sin^{2} x + 3 \sin x + 4 \right)^{\frac{1}{2}} - \left(\sin^{2} x + 6 \sin x + 2 \right)^{\frac{1}{2}} \right) \) is equal to

Let \( f : \mathbb{R} \to \mathbb{R} \) be a continuous function such that \( f(3x) - f(x) = x \). If \( f(8) = 7 \), then \( f(14) \) is equal to:

If \( y(x)=(x^{x})^{x} \), \( x>0 \), then \( \frac{d^{2}x}{dy^{2}}+20 \) at \( x=1 \) is equal to ______.

Let \( f \) be a differentiable function satisfying \( f(x)=\dfrac{2}{\sqrt{3}}\displaystyle\int\limits_{0}^{\sqrt{3}} f\!\left(\frac{\lambda^{2}x}{3}\right)\,d\lambda,\, x>0 \) and \( f(1)=\sqrt{3} \). If \( y=f(x) \) passes through the point \( (\alpha,6) \), then \( \alpha \) is equal to _________.

\(\displaystyle \lim_{x\to \dfrac{\pi}{4}} \frac{8\sqrt{2}-(\cos x+\sin x)^{7}}{\sqrt{2}-\sqrt{2}\sin 2x} \) is equal to:

If $y = \tan^{-1}\left(\sec x^3 - \tan x^3\right), \frac{\pi}{2} < x^3 < \frac{3\pi}{2}$, then:

Let \( \beta = \displaystyle\lim_{x \to 0} \dfrac{\alpha x - \left(e^{3x}-1\right)}{\alpha x \left(e^{3x}-1\right)} \) for some \( \alpha \in R \). Then the value of \( \alpha + \beta \) is:

Let \( f(x) \) be a polynomial function such that \( f(x) + f'(x) + f''(x) = x^{5} + 64 \). Then, the value of \( \lim\limits_{x \to 1} \frac{f(x)}{x-1} \) is equal to:

If \( \displaystyle\lim_{n \to \infty} \left( \sqrt{n^{2}-n-1} + n\alpha  + \beta \right)=0 \), then \( 8(\alpha+\beta) \) is equal to :

, where $[t]$ is the greatest integer $\leq t$. Then, in the open interval $(-1, 1)$, the number of points where $fog$ is discontinuous is equal to __________.

Let \( f : \mathbb{R} \to \mathbb{R} \) be defined as

Where \( a, b, c \in \mathbb{R} \) and \( [t] \) denotes greatest integer less than or equal to \( t \). Then, which of the following statements is true?

The value of \( \displaystyle \lim_{x \to 1} \frac{(x^2-1)\sin^2(\pi x)}{x^4-2x^3+2x-1} \) is equal to:

The function \( f:\mathbb{R}\to \mathbb{R} \) defined by \( f(x)=\displaystyle\lim_{n\to\infty}\frac{\cos(2\pi x)-x^{2n}\sin(x-1)}{1+x^{2n+1}-x^{2n}} \) is continuous for all \( x \) in:

Let \( f(x) = \min \{ 1, 1 + x \sin x \} \), \( 0 \leq x \leq 2\pi \). If \( m \) is the number of points, where \( f \) is not differential and \( n \) is the number of points, where \( f \) is not continuous, then the ordered pair \( (m, n) \) is equal to:

Let $f, g : \mathbb{R} \rightarrow \mathbb{R}$ be two real valued functions defined as

where $k_1$ and $k_2$ are real constants. If $(gof)$ is differentiable at $x = 0$, then $(gof)(-4) + (gof)(4)$ is equal to:

If \( \displaystyle \lim_{x\to 0}\dfrac{\alpha e^{x}+\beta e^{-x}+\gamma \sin x}{x\sin^{2}x}=\dfrac{2}{3} \), where \( \alpha,\beta,\gamma\in \mathbb{R} \), then which of the following is NOT correct?

where \( [t ] \) denotes greatest integer \( \le t \). If \( m \) is the number of points where \( f \) is not continuous and \( n \) is the number of points where \( f \) is not differentiable, then the ordered pair \( (m, n) \) is:

Let \( x(t)=2\sqrt{2}\cos t\,\sqrt{\sin 2t} \) and \( y(t)=2\sqrt{2}\sin t\,\sqrt{\sin 2t} \), \( t\in\left(0,\dfrac{\pi}{2}\right) \). Then \(\displaystyle \dfrac{1 + \left(\dfrac{dy}{dx}\right)^2}{\dfrac{d^2y}{dx^2}}\) at \( t=\dfrac{\pi}{4} \) is equal to:

If \( \cos^{-1}\!\left(\dfrac{y}{2}\right)=\log_{e}\!\left(\dfrac{x}{5}\right)^{5},\ |y|<2 \), then:

are continuous on \( \mathbb{R} \), then \( (g\circ f)(2)+(f\circ g)(-2) \) is equal to: