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Let and \( g(x) = f(|x|) + |f(x)| \). If the number of points where \( g \) is not continuous and is not differentiable are \( \alpha \) and \( \beta \) respectively, then \( \alpha + \beta \) is equal to ______.

If the function \( f(x)=\dfrac{\cos(\sin x)-\cos x}{x^{4}} \) is continuous at each point in its domain and \( f(0)=\dfrac{1}{k} \), then \( k \) is __________.

If \( f \) is continuous at \( x=\dfrac{\pi}{2} \), then the value of \( \displaystyle \int\limits_{0}^{3b-6} \left|x^2+2x-3\right|\,dx \) is:

Let \( a > 0 \) be a root of the equation \( 2x^{2} + x - 2 = 0 \). If \(\displaystyle \lim_{x \to \frac{1}{a}} \frac{16\left(1 - \cos\big(2 + x - 2x^{2}\big)\right)}{(1 - a x)^{2}} = \alpha + \beta \sqrt{17} \), where \( \alpha, \beta \in \mathbb{Z} \), then \( \alpha + \beta \) is equal to \( \_\_\_\_\_\_ \).

The minimum value of the function \( f(x) = \int_{0}^{2} e^{|x - t|}\, dt \) is:

is differentiable on \( \mathbb{R} \), then \( 48(a + b) \) is equal to \( \_\_\_\_ \).

Let $\{x\}$ denote the fractional part of $x$ and $f(x) = \frac{\cos^{-1}(1 - \{x\}^2)\sin^{-1}(1 - \{x\})}{\{x\} - \{x\}^3}, x \neq 0$. If $L$ and $R$ respectively denotes the left hand limit and the right hand limit of $f(x)$ at $x = 0$, then $\frac{32}{\pi^2}(L^2 + R^2)$ is equal to _______.

\( \displaystyle \lim_{x \to \frac{1}{\sqrt{2}}} \frac{\sin(\cos^{-1} x) - x}{1 - \tan(\cos^{-1} x)} \) is equal to:

If the function

is continuous at \( x=0 \), then \( k_1^2+k_2^2 \) is equal to:

If \( \displaystyle \lim_{x \to 1} \frac{(5x+1)^{1/3} - (x+5)^{1/3}}{(2x+3)^{1/2} - (x+4)^{1/2}} = \frac{m\sqrt{5}}{n(2n)^{2/3}} \), where \( \gcd(m,n) = 1 \), then \( 8m + 12n \) is equal to ________.

Let \( a \) be an integer such that \( \displaystyle \lim_{x \to 7} \frac{18 - [ 1 - x ]}{[x - 3a]} \) exits, where \( [ t ] \) is greatest integer \( \le t \). Then \( a \) is equal to:

\( \displaystyle \lim_{x \to 0} \frac{\cos(\sin x) - \cos x}{x^4} \) is equal to:

Let \( \lim_{x\to 2}\frac{(\tan(x-2))(r x^{2}+(p-2)x-2p)}{(x-2)^{2}}=5 \) for some \( r,p\in \mathbb{R} \). If the set of all possible values of \( q \), such that the roots of the equation \( r x^{2}-p x+q=0 \) lie in \( (0,2) \), be the interval \( (\alpha,\beta] \), then \( 4(\alpha+\beta) \) equals:

Let $\alpha,\beta\in\mathbb{R}$ be such that the functionbe differentiable for all $x\in\mathbb{R}$. Then $34(\alpha+\beta)$ is equal to:

Let \( f , g : R \to R \) be functions defined by

Where \( [x] \) denote the greatest integer less than or equal to \( x \). Then, the function \( fog \) is discontinuous at exactly.

Let $[t]$ denote the greatest integer $\leq t$ and $\{t\}$ denote the fractional part of $t$. The integer value of $\alpha$ for which the left hand limit of the function $f(x) = [1+x] + \frac{\alpha^{2[x]+\{x\}} + [x] - 1}{2[x]+\{x\}}$ at $x=0$ is equal to $\alpha - \frac{4}{3}$, is _________.

Let a function \( f:\mathbb{R}\to\mathbb{R} \) be defined as
Where \( [ x ] \) is the greatest integer less than or equal to \( x \). If \( f \) is continuous on \( \mathbb{R} \), then \( (a+b) \) is equal to:

Let \( f : \left[ -\dfrac{\pi}{2}, \dfrac{\pi}{2} \right] \to \mathbb{R} \) be a differentiable function such that \( f(0) = \dfrac{1}{2} \). If the \( \displaystyle \lim_{x \to 0} \dfrac{x \int_{0}^{x} f(t)\,dt}{e^{x^{2}} - 1} = \alpha \), then \( 8\alpha^{2} \) is equal to:

If \( \displaystyle \lim_{x \to \infty}\left(\sqrt{x^2 - x + 1} - a x\right) = b \), then the ordered pair \( (a, b) \) is:

Let \( a, b \in \mathbb{R},\ b \neq 0 \). Define a function

If \( f \) is continuous at \( x = 0 \), then \( 10 - ab \) is equal to ________.

Let \( f \) and \( g \) be twice differentiable functions on such that
\( f''(x)=g''(x)+6x \)
\( f'(1)=4g'(1)-3=9 \)
\( f(2)=3g(2)=12 \)
Then which of the following is NOT true?

If the function \( f(x)=\dfrac{\sin 3x+\alpha x\sin x-\beta \cos 3x}{x^{3}},\, x\in \mathbb{R}, \) is continuous at \( x=0 \), then \( f(0) \) is equal to:

Let \( A=[2,3,6,8,9,11] \) and \( B=[1,4,5,10,15] \). Let \( R \) be a relation on \( A\times B \) defined by \( (a,b)R(c,d) \) if and only if \( 3ad-7bc \) is an even integer. Then the relation \( R \) is:

Let \( f : [0,3] \to \mathbb{R} \) be defined by \( f(x) = \min\{x-[x],\, 1+[x]-x\} \) where \( [x] \) is the greater integer less than or equal to \( x \). Let \( P \) denote the set containing all \( x \in [0,3] \) where \( f \) is discontinuous, and \( Q \) denote the set containing all \( x \in (0,3) \) where \( f \) is not differentiable. Then the sum of number of elements in \( P \) and \( Q \) is equal to ________.

Let \( m \) and \( n \) be the number of points at which the function \( f(x) = \max \{ x, x^{3}, x^{5}, \ldots , x^{21} \} , \; x \in \mathbb{R}, \) is not differentiable and not continuous, respectively. Then \( m + n \) is equal to ________.

Let \( f:[0,\infty)\to [0,3] \) be a function defined by Then which of the following is true?

For some \( a,b \), let
\( x\ne 0,\; \displaystyle\lim_{x\to 0} f(x)=\lambda+\mu a+\nu b. \) Then \(( \lambda+\mu+\nu)^{2} \) is equal to:

where \([.]\) denotes greatest integer function. If \( \alpha \) and \( \beta \) are the number of points, where \( f \) is not continuous and is not differentiable, respectively, then \( \alpha + \beta \) equals ______.

Consider the function \( f : (0,2) \to \mathbb{R} \) defined by \( f(x)=\frac{x}{2}+\frac{2}{x} \) and the function \( g(x) \) defined by

Let \( f : S \to S \) where \( S = (0,\infty) \) be a twice differentiable function such that \( f(x+1)=x f(x) \). If \( g:S \to \mathbb{R} \) be defined as \( g(x)=\log_e f(x) \), then the value of \( \left|\, g''(5)-g''(1)\,\right| \) is equal to:

For \( \alpha, \beta, \gamma \in \mathbb{R} \), if \( \lim_{x \to 0} \dfrac{x^{2}\sin(\alpha x) + (\gamma-1)e^{\ x^{2}}}{\sin 2x - \beta x} = 3 \), then \( \beta + \gamma - \alpha \) is equal to:

If \( \lim_{x \to 0} \left( \frac{\tan x}{x} \right)^{\frac{1}{x^{2}}} = p \), then \( 96 \log_{e} p \) is equal to __________.

\( \displaystyle \lim_{x\to\frac{\pi}{2}} \left( \dfrac{\displaystyle\int_{x^{3}}^{\left({x}/{2}\right)^{3}} \left(\sin\!\left(2t^{1/3}\right)+\cos\!\left(t^{1/3}\right)\right)\\,dt}{\left(x-\frac{\pi}{2}\right)^{2}} \right) \) is equal to

For \( a,b>0 \), let be a continuous function at \( x=0 \). Then \( \dfrac{b}{a} \) is equal to:

For the function \( f(x) = (\cos x) - x + 1, \ x \in \mathbb{R} \), between the following two statements:
Statement-1 : \( f(x) = 0 \) for only one value of \( x \) in \( [0, \pi] \).
Statement-2 : \( f(x) \) is decreasing in \( [0, \frac{\pi}{2}] \) and increasing in \( [\frac{\pi}{2}, \pi] \).

Let \( f \) be a differentiable function on \( \mathbb{R} \) such that \( f(2)=1 \), \( f'(2)=4 \). Let \( \displaystyle \lim_{x\to 0}\left(f(2+x)\right)^{3/x}=e^{\alpha} \). Then the number of times the curve \( y=4x^{3}-4x^{2}-4(\alpha-7)x-\alpha \) meets x-axis is:

The number of points, at which the function \( f(x) = \max \{ 6x,\; 2 + 3x^2 + |x-1|\cos|x^2 - \tfrac{1}{4}| \} \), \( x \in (-\pi,\pi) \), is not differentiable is, is ________.

Given below are two statements:
Statement I: \( \displaystyle \lim_{x \to 0} \left( \frac{ \tan^{-1} x + \log_{e}\!\sqrt{\frac{1+x}{1-x}} - 2x }{ x^{5} } \right) = \frac{2}{5} \)
Statement II: \( \displaystyle \lim_{x \to 1}(x^{\frac{2}{1-x}}) = \frac{1}{e^{2}} \)
In the light of the above statements, choose the correct answer from the options given below:

\(\displaystyle \lim_{x \to 0} \left( \left( \frac{(1 - \cos^{2}(3x))}{\cos^{3}(4x)} \right) \left( \frac{(\sin^{3}(4x)}{(\log_{e}(2x+1))^{5}} \right) \right) \) is equal to:

is continuous at \( x=0 \), then \( a+b \) is equal to:

The value of \( \displaystyle \lim_{x \to 0^+} \frac{\cos^{-1}(x - [x^2]) - \sin^{-1}(x - [x^2])}{x - x^3} \), where \( [x] \) denotes the greatest integer \( \le x \) is:

Let \( f:\mathbb{R}\to\mathbb{R} \) be a function defined as .
If \( f \) is continuous at \( x=0 \), then the value of \( a+b \) is equal to:

\( \displaystyle\lim_{x \to \infty} \frac{(2x^2 - 3x + 5)(3x - 1)^{\frac{x}{2}}}{(3x^2 + 5x + 4)\sqrt{(3x + 2)^x}} \) is equal to:

is continuous at $x = \frac{\pi}{2}$, then $9\lambda + 6 \log_e \mu + \mu^6 - e^{6\lambda}$ is equal to:

If for \( p\neq q\neq 0 \), the function \( f(x)=\dfrac{\sqrt[7]{\,p(729+x)\,}-3}{\sqrt[3]{\,729+qx\,}-9} \) is continuous at \( x=0 \), then:

If \( y(x) = x^x, \, x > 0 \), then \( y''(2) - 2y'(2) \) is equal to:

Let $[t]$ be the greatest integer less than or equal to $t$. Then the least value of $p \in \mathbb{N}$ for which

$\lim_{x \to 0^+} \left( x \left( \left[\frac{1}{x}\right] + \left[\frac{2}{x}\right] + \dots + \left[\frac{p}{x}\right] \right) - x^2 \left( \left[\frac{1}{x^2}\right] + \left[\frac{2^2}{x^2}\right] + \dots + \left[\frac{9^2}{x^2}\right] \right) \right) \geq 1$

is equal to ______.

is continuous at \( x=0 \), then \( \frac{1}{a}+\frac{1}{b}+\frac{4}{k} \) is equal to:

Let \( [t] \) denote the greatest integer less than or equal to \( t \).
Let \( f(x)=x-[x] \), \( g(x)=1-x+[x] \), and \( h(x)=\min\{f(x),g(x)\} \), \( x \in [-2,2] \). Then \( h \) is:

Let \( f(x) = x^6 + 2x^4 + x^3 + 2x + 3, \ x \in \mathbb{R} \). Then the natural number \( n \) for which \( \displaystyle \lim_{x \to 1} \dfrac{x^n f(1) - f(x)}{x - 1} = 44 \) is:

If \( \displaystyle\lim_{x\to 0}\frac{\alpha x e^{x}-\beta \log_{\alpha}(1+x)+\gamma x^{2}e^{-x}}{x\sin^{2}x}=10, \; \alpha,\beta,\gamma \in \mathbb{R}, \) then the value of \( \alpha+\beta+\gamma \) is ____________.

\( \displaystyle \lim_{x \to \infty} \dfrac{\left(\sqrt{3x+1} + \sqrt{3x-1}\right)^{6} + \left(\sqrt{3x+1} - \sqrt{3x-1}\right)^{6}}{\left(x + \sqrt{x^{2} - 1}\right)^{6} + \left(x - \sqrt{x^{2} - 1}\right)^{6}} \, x^{3} \)

Let \( x = 2 \) be a root of the equation \( x^2 + px + q = 0 \) and

Then\( \displaystyle \lim_{x \to 2p^+} [f(x)] \), where [.] denotes greatest integer function, is:

Let \( f : (0,\infty) \to \mathbb{R} \) and \(\displaystyle F(x) = \int_{0}^{x} t\,f(t)\,dt \). If \( F(x^{2}) = x^{4} + x^{5} \), then \(\displaystyle \sum_{r=1}^{12} f(r^{2}) \) is equal to ________.

If \( [t] \) denotes the greatest integer \( \le t \), then the number of points, at which the function \( f(x)=4|2x+3|+9\left[x+\dfrac{1}{2}\right]-12[x+20] \) is not differentiable in the open interval \( (-20,20) \), is ________.

If \( \log_{e} y = 3\sin^{-1}x \), then \( (1-x^{2})y''-xy' \) at \( x=\dfrac{1}{2} \) is equal to

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

where \( \alpha, \beta \in R \). If \( f \) is continuous at \( x = 0 \), then \( \alpha^2 + \beta^2 \) is equal to:

Let \( y = \log_e\!\left(\dfrac{1-x^2}{1+x^2}\right) ,-1 < x < 1\). Then at \(x = \dfrac{1}{2}\), the value of \(225(y'-y'')\) is equal to:

Consider the function

where \( [x] \) denotes the greatest integer less than or equal to \( x \). If \( S \) denotes the set of all ordered pairs \( (a, b) \) such that \( f(x) \) is continuous at \( x = 3 \), then the number of elements in \( S \) is:

Let \( f:\mathbb{R} \to \mathbb{R} \) be a function such that \( f(2) = 4 \) and \( f'(2) = 1 \). Then, the value of \( \displaystyle \lim_{x \to 2} \frac{x^2 f(2) - 4 f(x)}{x - 2} \) is equal to:

Let \( f:\left[-\frac{\pi}{4},\frac{\pi}{4}\right]\to \mathbb{R} \) be defined as
If \( f \) is continuous at \( x=0 \), then the value of \( 6a+b^{2} \) is equal to:

If \( \lim_{x \to 0} \dfrac{\cos(2x) + a \cos(4x) - b}{x^{4}} \) is finite, then \( (a + b) \) is equal to:

Let \([t]\) denote the greatest integer \( \le t \). The number of points where the function \( f(x) = [x] \, |x^2 - 1| + \sin\!\left( \dfrac{\pi}{[x] + 3} \right) - [x + 1], \ x \in (-2, 2) \) is continuous is:

If ‘\( R \)’ is the least value of ‘\( a \)’ such that the function \( f(x) = x^{2} + ax + 1 \) is increasing on \( [1, 2] \) and ‘\( S \)’ is the Greatest value of ‘\( a \)’ such that the function \( f(x) = x^{2} + ax + 1 \) is decreasing on \( [1, 2] \), then the value of \( |R - S| \) is __________ .

then \( \dfrac{1}{5} f'(0) \) is equal to:

The function \( f(x) = x^{3} - 6x^{2} + ax + b \) is such that \( f(2) = f(4) = 0 \). Consider two statements, (\( S1 \)) there exists \( x_{1}, x_{2} \in (2, 4) \), \( x_{1} < x_{2} \), such that \( f'(x_{1}) = -1 \) and \( f'(x_{2}) = 0 \). (\( S2 \)) there exists \( x_{3}, x_{4} \in (2, 4) \), \( x_{3} < x_{4} \), such that \( f \) is decreasing in \( (2, x_{4}) \), increasing in \( (x_{4}, 4) \) and \( 2\, f'(x_{3}) = \sqrt{3}\, f(x_{4}) \).

$\lim_{t \to 0} \left( 1^{\frac{1}{\sin^2 t}} + 2^{\frac{1}{\sin^2 t}} + \dots + n^{\frac{1}{\sin^2 t}} \right)^{\sin^2 t} \text{ is equal to:}$

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:

If \( f(x) = x^2 + g'(1)x + g''(2) \) and \( g(x) = f(1)x^2 + x f'(x) + f''(x) \), then the value of \( f(4) - g(4) \) is equal to ________.

The value of the limit \( \displaystyle\lim_{\theta \to 0} \dfrac{\tan\!\left(\pi \cos^{2}\theta\right)}{\sin\!\left(2\pi \sin^{2}\theta\right)} \) is equal to:

be continuous at \( x = -\dfrac{3}{2} \). If \( f of (x) = \dfrac{7}{5} \), then \( x \) is equal to:

is differentiable at every point of the domain, then the values of \( a \) and \( b \) are respectively:

The number of real roots of the equation \( e^{4x}+2e^{3x}-e^{x}-6=0 \) is/are:

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:

Let \( f:(0,\pi)\to \mathbb{R} \) be a function given by

Let \( f: \mathbb{R} - \{ 0 \} \to \mathbb{R} \) be a function such that \( f(x) - 6 f\!\left( \frac{1}{x} \right) = \frac{35}{3x} - \frac{5}{2} \). If the \( \lim_{x \to 0} \left( \frac{1}{\alpha x} + f(x) \right) = \beta \) ; \( \alpha, \beta \in \mathbb{R} \), then \( \alpha + 2\beta \) is equal to:

Let \( y(x) = (1 + x)(1 + x^2)(1 + x^4)(1 + x^8)(1 + x^{16}) \). Then \( y' - y'' \) at \( x = -1 \) is equal to:

If \( \displaystyle \lim_{x\to 0} \frac{e^{(a-1)x} + 2\cos bx + (c-2)e^{-x}}{x\cos x - \log_e(1 + x)} = 2 \), then \( a^2 + b^2 + c^2 \) 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:

Let $\displaystyle \lim_{n \rightarrow \infty} \left( \frac{n}{\sqrt{n^4+1}} - \frac{2n}{(n^2+1)\sqrt{n^4+1}} + \frac{n}{\sqrt{n^4+16}} - \frac{8n}{(n^2+4)\sqrt{n^4+16}} \right) + ... + \frac{n}{\sqrt{n^4+n^4}} - \frac{2n \cdot n^2}{(n^2+n^2)\sqrt{n^4+n^4}} $
be $\frac{\pi}{k}$, using only the principal values of the inverse trigonometric functions. Then $k^2$ is equal to _______.

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

If \( a = \displaystyle \lim_{x \to 0} \dfrac{\sqrt{1 + \sqrt{1 + x^4}} - \sqrt{2}}{x^4} \) and \( b = \displaystyle \lim_{x \to 0} \dfrac{\sin^2 x}{\sqrt{2} - \sqrt{1 + \cos x}} \), then the value of \( ab^3 \) is:

Let for a differentiable function \( f : (0, \infty) \to \mathbb{R} \), \( f(x) - f(y) \ge \log_e\!\left(\frac{x}{y}\right) + x - y, \, \forall \, x, y \in (0, \infty) \). Then \( \displaystyle \sum_{n=1}^{20} f'\!\left(\frac{1}{n^2}\right) \) is equal to ____________.

be continuous at \( x = 0 \). Then \( e^{a} \, b \, c \) is equal to:

Let the function \( f(x) = (x^{2}-1)\lvert x^{2}-ax+2 \rvert + \cos \lvert x \rvert \) be not differentiable at the two points \( x = \alpha = 2 \) and \( x = \beta \). Then the distance of the point \( (\alpha,\beta) \) from the line \( 12x + 5y + 10 = 0 \) is equal to:

Let \([t]\) denote the greatest integer less than or equal to \( t \). Let \( f:[0,\infty)\to \mathbb{R} \) be a function defined by \( f(x)=\left[\frac{x}{2}+3\right]-[\sqrt{x}] \). Let \( S \) be the set of all points in the interval \( [0,8] \) at which \( f \) is not continuous. Then \( \sum\limits_{a\in S} a \) is equal to ______.

If the value of \( \displaystyle\lim_{x \to 0}\left(2-\cos x \sqrt{\cos 2x}\right)^\left({\dfrac{x+2}{x^{2}}}\right) \) is equal to \( e^{a} \), then \( a \) is equal to _______.

Consider the function \( f:(0,\infty)\to \mathbb{R} \) defined by \( f(x)=e^{-\lvert \log_{e} x \rvert} \). If \( m \) and \( n \) be respectively the number of points at which \( f \) is not continuous and \( f \) is not differentiable, then \( m+n \) is:

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:

Let a line perpendicular to the line \( 2x - y = 10 \) touch the parabola \( y^{2} = 4(x - 9) \) at the point \( P \). The distance of the point \( P \) from the centre of the circle \( x^{2} + y^{2} - 14x - 8y + 56 = 0 \) is \( \_\_\_\_\_\_\_ \).

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

If \(\displaystyle \lim_{x \to 0} \dfrac{e^{ax} - \cos(bx) - \dfrac{cxe^{-cx}}{2}}{1 - \cos(2x)} = 17\), then \(5a^2+b^2\) is equal to:

Let \([t]\) denote the greatest integer less than or equal to \( t \). If the function

is continuous at \( x=0 \), then \( a^{2}+b^{2} \) is equal to:

Let \( f(x)=\displaystyle\sum_{k=1}^{10}k\,x^{k},\; x\in \mathbb{R} \). If \( 2f(2)+f'(2)=119(2)^{n}+1 \) then \( n \) is equal to __________.

Consider the function Where \( P(x) \) is a polynomial such that \( P''(x) \) is always a constant and \( P(3) = 9 \). If \( f(x) \) is continuous at \( x = 2 \), then \( P(5) \) is equal to ________.

Let \( f \) be any continuous function on \( [0,\,2] \) and twice differentiable on \( (0,\,2) \). If \( f(0)=0 \), \( f(1)=1 \) and \( f(2)=2 \), then:

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 _________.

If \(\displaystyle \lim_{x \to 0} \frac{3 + \alpha \sin x + \beta \cos x + \log_e(1 - x)}{3 \tan^2 x} = \frac{1}{3} \), then \( 2\alpha - \beta \) is equal to:

$\text{The number of points of discontinuity of the function } f(x) = \left[ \frac{x^2}{2} \right] - [\sqrt{x}], x \in, \text{[0,4] where}$$[.] \text{ denotes the greatest integer function, is __________ .}$

Let \( [x] \) denote the greatest integer function, and let \( m \) and \( n \) respectively be the numbers of the points, where the function \( f(x) = [x] + |x - 2|, \, -2 < x < 3, \) is not continuous and not differentiable. Then \( m + n \) is equal to :

The value of \( \displaystyle \lim_{x \to 0} 2\left( \frac{1-\cos x \sqrt{\cos 2x}\, \sqrt[3]{\cos 3x}\, \ldots \, \sqrt[10]{\cos 10x}}{x^2} \right) \) is ______.

Let \( f \), \( g \) and \( h \) be the real valued functions defined on \( \mathbb{R} \) as:

and \( h(x)=2[x]-f(x) \), where \( [x] \) is the greatest integer \( \leq x \).
Then the value of \( \lim\limits_{x\to 1} g\!\left(h(x-1)\right) \) is:

Let \( [ x] \) denote the greatest integer function and \( f(x)=\max\{\,1+x+[ x ], \ 2+x, \ x+2[ x ]\,\} \), \( 0\le x\le 2 \). Let \( m \) be the number of points in \([0,2]\), where \( f \) is not continuous and \( n \) be the number of points in \( (0,2) \), where \( f \) is not differentiable. Then \( (m+n)^{2}+2 \) 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:

Let \( f(x)=[x^2-x]+|-x+[x]| \), where \( x\in \mathbb{R} \) and \( [t] \) denotes the greatest integer less than or equal to \( t \). Then, \( f \) is:

If the function \( f(x) = \frac{e^{x}\left(e^{\tan x - x}-1\right)+\log_{9}(\sec x + \tan x)-x}{\tan x - x} \) is continuous at \( x = 0 \), then the value of \( f(0) \) is equal to:

Let \( f:[-1,2]\to \mathbb{R} \) be given \( f(x)=2x^{2}+x+[x^{2}]-[x] \) where \( [t] \) denotes the greatest integer less than or equal to \( t \). The number of points, where \( f \) is not continuous is:

Let a function \( g : (0,4] \to \mathbb{R} \) be defined as
, then the number of points in the interval \( (0,4) \) where \( g(x) \) is NOT differentiable, is ____________.

Let \( f(x) = x^{3} + x^{2} f'(1) + x f''(2) + f'''(3), \, x \in \mathbb{R} \). Then \( f'(10) \) is equal to ____________.

If \( 2x^{y} + 3y^{x} = 20 \), then \( \dfrac{dy}{dx} \) at \( (2, 2) \) is equal to:

Let \( g(x) \) be a linear function andis continuous at \( x = 0 \). If \( f'(1)=f(-1) \), then the value \( g(3) \) is:

Let \( f:\mathbb{R}\to\mathbb{R} \) be defined as If \( f \) is continuous at \( x=0 \), then \( \alpha \) is equal to:

If \(\displaystyle y = \frac{(\sqrt{x} + 1)(x^2 - \sqrt{x})}{x\sqrt{x} + x + \sqrt{x}} + \frac{1}{15}\,(3\cos^2 x - 5)\cos^3 x \), then \( 96\,y'\!\left(\dfrac{\pi}{6}\right) \) is equal to ________.

Let \( f:\mathbb{R}\to\mathbb{R} \) be a function defined as
Let \( g:\mathbb{R}\to\mathbb{R} \) be given by \( g(x)=f(x+2)-f(x-2) \). If \( n \) and \( m \) denote the number of points in \( \mathbb{R} \) where \( g \) is not continuous and not differentiable, respectively, then \( n+m \) is equal to ________.

If \(\displaystyle\lim_{x \to \infty} \left( \left( \frac{e}{1 - e} \right) \left( \frac{1}{e} - \frac{x}{1 + x} \right) \right)^x = \alpha \), then the value of \(\displaystyle {\frac{\log_{e}\alpha}{1+\log_{e}\alpha}} \) equals:

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 \( k \) and \( m \) be positive real numbers such that the function is differentiable for all \(x>0\).Then\(\displaystyle \frac{8 f'(8)}{f'\left(\dfrac{1}{8}\right)}\) is equal to ______.

Let \( f(x) = x^5 + 2e^{x/4} \) for all \( x \in \mathbb{R} \). Consider a function \( g(x) \) such that \( (gof)(x) = x \) for all \( x \in \mathbb{R} \). Then the value of \( 8g'(2) \) is:

For the differentiable function \( f:\mathbb{R}-\{0\}\to \mathbb{R} \), let \( 3f(x)+2f\!\left(\dfrac{1}{x}\right)=\dfrac{1}{x}-10 \), then \( \left|\,f(3)+f'\!\left(\dfrac{1}{4}\right)\,\right| \) is equal to:

Let \( f \) be a real valued function, defined on \( \mathbb{R}-\{-1,1\} \) and given by \( f(x)=3\log_e \left|\dfrac{x-1}{x+1}\right|-\dfrac{2}{x-1} \). Then in which of the following intervals, function \( f(x) \) is increasing?

If \( \displaystyle \lim_{x \to 1^{+}} \frac{(x-1)\big(6+\lambda \cos(x-1)\big)+\mu \sin(1-x)}{(x-1)^3} = -1 \), where \( \lambda, \mu \in \mathbb{R} \), then \( \lambda + \mu \) is equal to:

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 :

Suppose $f(x) = \frac{(2^x + 2^{-x})\tan x\sqrt{\tan^{-1}(x^2 - x + 1)}}{(7x^2 + 3x + 1)^3}$. Then the value of $f'(0)$ is equal to:

\( \displaystyle \lim_{x \to 0} \frac{e^{2|\sin x|}-2|\sin x|-1}{x^{2}} \)

is equal to:

If \(\displaystyle \lim_{x\to 0}\frac{ax^{2}e^{x}-b\log_{e}(1+x)+cxe^{-x}}{x^{2}\sin x}=1 \), then \( 16\left(a^{2}+b^{2}+c^{2}\right) \) is equal to ________.

Let \( [ x ] \) be the greatest integer \( \leq x \). Then the number of points in the interval \( (-2,1) \), where the function \( f(x)=| [ x ] |+\sqrt{x-[ x ]} \) is discontinuous, is_____.

If \( y=\tan^{-1}\!\left(\dfrac{3\cos x-4\sin x}{4\cos x+3\sin x}\right)+2\tan^{-1}\!\left(\dfrac{x}{1+\sqrt{1-x^{2}}}\right) \), then \( \dfrac{dy}{dx} \) at \( x=\dfrac{\sqrt{3}}{2} \) is equal to:

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

If \( f \) is continuous everywhere in \( \mathbb{R} \) and \( m \) is the number of points where \( f \) is NOT differentiable then \( m+a+b+c \) equals:

Then at \( x=0 \)

, 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 __________.

\( \displaystyle \lim_{n\to\infty}\frac{(1^{2}-1)(n-1)+(2^{2}-2)(n-2)+\cdots+((n-1)^{2}-(n-1))\cdot 1}{(1^{3}+2^{3}+\cdots+n^{3})-(1^{2}+2^{2}+\cdots+n^{2})} \) is equal to:

The value of \( \displaystyle \lim_{x\to 0} \left( \dfrac{x^{2}\sin^{2}x}{x^{2}-\sin^{2}x} \right) \) is:

If \( \displaystyle \int \frac{\sin^{\frac{3}{2}} x + \cos^{\frac{3}{2}} x}{\sqrt{\sin^3 x \cos^3 x \sin(x - \theta)}} dx = A \sqrt{\cos\theta \tan x - \sin\theta} + B \sqrt{\cos\theta - \sin\theta \cot x} + C \), where \( C \) is the integration constant, then \( AB \) is equal to:

The value of \( \displaystyle \lim_{n \to \infty} \frac{[r] + [2r] + \ldots + [nr]}{n^{2}} \), where \( r \) is a non-zero number and \( [r] \) denotes the greatest integer than or equal to \( r \), is equal to:

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

Where \( [x] \) is the greatest integer less than or equal to \( x \). If \( f \) is continuous at \( x=2 \), then \( \lambda+\mu \) 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?

\( \displaystyle \lim_{x\to 0} cosec x \left( \sqrt{2\cos^2 x + 3\cos x} - \sqrt{\cos^2 x + \sin x + 4} \right) \) is:

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:

If for some \( a \in \mathbb{N} \), \( f(f(f(a))) = 21 \), then \(\displaystyle \lim_{x \to a^-} \left\{ \frac{|x|^3}{a} - \left[ \frac{x}{a} \right] \right\} \), where \( [t] \) denotes the greatest integer less than or equal to \( t \), is equal to:

The sum of all the local minimum values of the twice differentiable function \( f:\mathbb{R}\to\mathbb{R} \) defined by \( f(x)=x^{3}-3x^{2}-\dfrac{3\,f''(2)}{2}\,x+f'(1) \) is:

If \( \alpha, \beta \) are the distinct roots of \( x^2+bx+c=0 \), then
\( \displaystyle \lim_{x\to \beta} \frac{e^{2(x^2+bx+c)}-1-2(x^2+bx+c)}{(x-\beta)^2} \)
is equal to:

Let \( f : (-\infty, \infty) - \{0\} \to \) \(\mathbb{R}\) be a differentiable function such that \( f'(1) = \lim_{a \to \infty} a^2 f\!\left(\frac{1}{a}\right) \). Then \( \lim_{a \to \infty} \left[ \dfrac{a(a+1)}{2} \tan^{-1}\!\left(\dfrac{1}{a}\right) \right] + a^2 - 2 \log_e a \) 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:

If the function is continuous at \( x=0 \), then the value of \( a^2 \) is equal to:

where \( a>0 \) and \( g(x):=(f(|x|)-|f(x|))/2 \). Then the function \( g:[-a,a]\to[-a,a] \) is:

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:

The value of \( \displaystyle \lim_{x \to 0} \frac{\log_e \!\big( \sec(e{x}) \cdot \sec(e^{2}x) \cdots \sec(e^{10}x) \big)}{e^{2} - e^{2 \cos x}} \) is equal to:

The product of all possible values of \( \alpha \), for which \( \displaystyle \lim_{x \to 0}\left(\dfrac{1 - \cos(\alpha x)\cos((\alpha + 1)x)\cos((\alpha + 2)x)}{\sin^{2}((\alpha + 1)x)}\right) = 2 \), is:

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:

Let \( f:(a,b)\to \mathbb{R} \) be twice differentiable function such that \( f(x)=\int\limits_{a}^{x} g(t)\,dt \) for a differentiable function \( g(x) \). If \( f(x)=0 \) has exactly five distinct roots in \( (a,b) \) then \( g(x)\,g'(x)=0 \) has at least:

If \( \alpha = \displaystyle \lim_{x \to \pi/4} \dfrac{\tan^{3} x - \tan x}{\cos \left( x + \dfrac{\pi}{4} \right)} \) and \( \beta =\displaystyle \lim_{x \to 0} (\cos x)^{\cot x} \) are the roots of the equation \( ax^{2} + bx - 4 = 0 \) , then the ordered pair \( (a , b) \) is:

Let \( f(x) = \left| 2x^2 + 5|x| - 3 \right| ,\; x \in \mathbb{R} \). If \( m \) and \( n \) denote the number of points where \( f \) is not continuous and not differentiable respectively, then \( m + n \) is equal to:

The value of \( \displaystyle \lim_{n \to \infty} \dfrac{1+2-3+4+5-6+\ldots+(3n-2)+(3n-1)-3n}{\sqrt{2n^{4}+4n+3}-\sqrt{n^{4}+5n+4}} \) is:

Let \( f(x) = \dfrac{\sin x + \cos x - \sqrt{2}}{\sin x - \cos x},\; x \in [0, \pi] - \left\{ \dfrac{\pi}{4} \right\}. \) Then \( f\!\left(\dfrac{7\pi}{12}\right) f''\!\left(\dfrac{7\pi}{12}\right) \) 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?

Let \( g : \mathbb{R} \to \mathbb{R} \) be a non constant twice differentiable function such that \( g'\!\left(\dfrac{1}{2}\right) = g'\!\left(\dfrac{3}{2}\right) \). If a real valued function \( f \) is defined as \( f(x) = \dfrac{1}{2}\,[g(x) + g(2 - x)] \), then:

The value of\( \displaystyle \lim_{n \to \infty}\left( \sum_{k=1}^{n} \frac{k^3+6k^2+11k+5}{(k+3)!} \right) \) is:

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:

If \( \lim_{x\to 2}\frac{\sin(x^3-5x^2+ax+b)}{(\sqrt{x-1}-1)\log_e(x-1)}=m \), then \( a+b+m \) is equal to:

If \( f(x) = x^3 - x^2 f'(1) + x f''(2) - f'''(3), \, x \in \mathbb{R} \), then:

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:

Let \( f : \mathbb{R} \to \mathbb{R} \) be a continuous function.
Then $\displaystyle \lim_{x \to \frac{\pi}{4}} \frac{\frac{\pi}{4} \int_{2}^{\sec^2 x} f(x) \, dx}{x^2 - \frac{\pi^2}{16}}$ is equal to:

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

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JEE Main 2021 31st August Morning Shift

The number of real roots of the equation \( e^{4x}+2e^{3x}-e^{x}-6=0 \) is/are: