Limits, Continuity & Differentiability · JEE Advanced previous year questions

Let $\alpha$ and $\beta$ be the real numbers such that $$\underset{x\to 0}{\lim }\frac{1}{x^3}\left(\frac{\alpha }{2}{\int }_0^x\frac{1}{1-t^2}dt+\beta x\cos x\right)=2.$$ Then the value of $\alpha +\beta$ is ____.

For all $x>0$, let $y_1(x),y_2(x)$, and $y_3(x)$ be the functions satisfying $$\frac{d{y}_1}{dx}-(\sin x{)}^2{y}_1=0,y_1(1)=5,$$ $$\frac{d{y}_2}{dx}-(\cos x{)}^2{y}_2=0,y_2(1)=\frac{1}{3},$$ $$\frac{d{y}_3}{dx}-\left(\frac{2-x^3}{x^3}\right)y_3=0,y_3(1)=\frac{3}{5e},$$ respectively. Then $$\underset{x\to 0^{+}}{\lim }\frac{y_1(x)y_2(x)y_3(x)+2x}{e^{3x}\sin x}$$ is equal to ____.

Let $\mathbb{R}$ denote the set of all real numbers. For a real number $x$, let $[x]$ denote the greatest integer less than or equal to $x$. Let $n$ denote a natural number. Match each entry in List-I to the correct entry in List-II and choose the correct option. List-I (P) The minimum value of $n$ for which the function $f(x)=\left[\frac{10x^3-45x^2+60x+35}{n}\right]$ is continuous on the interval $[1,2]$, is (Q) The minimum value of $n$ for which $g(x)=(2n^2-13n-15)(x^3+3x)$, $x\in \mathbb{R}$, is an increasing function on $\mathbb{R}$, is (R) The smallest natural number $n$ which is greater than 5, such that $x=3$ is a point of local minima of $h(x)=(x^2-9{)}^n(x^2+2x+3)$, is (S) Number of $x_0\in \mathbb{R}$ such that $l(x)={\sum }_{k=0}^4\left(\sin |x-k|+\cos \left|x-k+\frac{1}{2}\right|\right)$, $x\in \mathbb{R}$, is $\text{NOT}$ differentiable at $x_0$, is List-II (1) 8 (2) 9 (3) 5 (4) 6 (5) 10

1. A.(P) → (1) (Q) → (3) (R) → (2) (S) → (5)
2. B.(P) → (2) (Q) → (1) (R) → (4) (S) → (3)
3. C.(P) → (5) (Q) → (1) (R) → (4) (S) → (3)
4. D.(P) → (2) (Q) → (3) (R) → (1) (S) → (5)

Let $x_0$ be the real number such that $e^{x_0}+x_0=0$. For a given real number $\alpha$, define $$g(x)=\frac{3x{e}^x+3x-\alpha e^x-\alpha x}{3(e^x+1)}$$ for all real numbers $x$. Then which one of the following statements is TRUE?

1. A.For $\alpha =2$, ${\lim }_{x\to x_0}\left|\frac{g(x)+e^{x_0}}{x-x_0}\right|=0$
2. B.For $\alpha =2$, ${\lim }_{x\to x_0}\left|\frac{g(x)+e^{x_0}}{x-x_0}\right|=1$
3. C.For $\alpha =3$, ${\lim }_{x\to x_0}\left|\frac{g(x)+e^{x_0}}{x-x_0}\right|=0$
4. D.For $\alpha =3$, ${\lim }_{x\to x_0}\left|\frac{g(x)+e^{x_0}}{x-x_0}\right|=\frac{2}{3}$

Let $f(x)$ be a continuously differentiable function on the interval $(0,\infty )$ such that $f(1)=2$ and $$\underset{t\to x}{\lim }\frac{t^{10}f(x)-x^{10}f(t)}{t^9-x^9}=1$$ for each $x>0$. Then, for all $x>0$, $f(x)$ is equal to

1. A.$\frac{31}{11x}-\frac{9}{11}{x}^{10}$
2. B.$\frac{9}{11x}+\frac{13}{11}{x}^{10}$
3. C.$\frac{-9}{11x}+\frac{31}{11}{x}^{10}$
4. D.$\frac{13}{11x}+\frac{9}{11}{x}^{10}$

Let $k\in \mathbb{R}$. If ${\lim }_{x\to 0+}(\sin (\sin kx)+\cos x+x{)}^{\frac{2}{x}}=e^6$, then the value of $k$ is

1. A.1
2. B.2
3. C.3
4. D.4

Let $S$ be the set of all $(\alpha ,\beta )\in \mathbb{R}\times \mathbb{R}$ such that $$\underset{x\to \infty }{\lim }\frac{\sin (x^2)({\log }_{e}x{)}^{\alpha }\sin \left(\frac{1}{x^2}\right)}{x^{\alpha \beta }({\log }_e(1+x){)}^{\beta }}=0.$$ Then which of the following is (are) correct?

1. A.$(-1,3)\in S$
2. B.$(-1,1)\in S$
3. C.$(1,-1)\in S$
4. D.$(1,-2)\in S$

Let the function $f:[1,\infty )\to \mathbb{R}$ be defined by $$f(t)=\{\begin{array}{ll}(-1{)}^{n+1}2,& \text{if }t=2n-1,n\in \mathbb{N},\\ \frac{(2n+1-t)}{2}f(2n-1)+\frac{(t-(2n-1))}{2}f(2n+1),& \text{if }2n-1<t<2n+1,n\in \mathbb{N}.\end{array}$$ Define $g(x)={\int }_1^{x}f(t)dt,x\in (1,\infty )$. Let $\alpha$ denote the number of solutions of the equation $g(x)=0$ in the interval $(1,8]$ and $\beta ={\lim }_{x\to 1+}\frac{g(x)}{x-1}$. Then the value of $\alpha +\beta$ is equal to ____.

Let $S=(0,1)\cup (1,2)\cup (3,4)$ and $T=\{0,1,2,3\}$. Then which of the following statements is(are) true?

1. A.There are infinitely many functions from $S$ to $T$
2. B.There are infinitely many strictly increasing functions from $S$ to $T$
3. C.The number of continuous functions from $S$ to $T$ is at most 120
4. D.Every continuous function from $S$ to $T$ is differentiable

Let $f:(0,1)\to \mathbb{R}$ be the function defined as $f(x)=\sqrt{n}$ if $x\in \left[\frac{1}{n+1},\frac{1}{n}\right)$ where $n\in \mathbb{N}$. Let $g:(0,1)\to \mathbb{R}$ be a function such that ${\int }_{x^2}^x\sqrt{\frac{1-t}{t}}dt<g(x)<2\sqrt{x}$ for all $x\in (0,1)$. Then $$\underset{x\to 0}{\lim }f(x)g(x)$$

1. A.does NOT exist
2. B.is equal to 1
3. C.is equal to 2
4. D.is equal to 3

Let $f:(0,1)\to \mathbb{R}$ be the function defined as $f(x)=[4x]{\left(x-\frac{1}{4}\right)}^2\left(x-\frac{1}{2}\right)$, where $[x]$ denotes the greatest integer less than or equal to $x$. Then which of the following statements is(are) true?

1. A.The function $f$ is discontinuous exactly at one point in $(0,1)$
2. B.There is exactly one point in $(0,1)$ at which the function $f$ is continuous but NOT differentiable
3. C.The function $f$ is NOT differentiable at more than three points in $(0,1)$
4. D.The minimum value of the function $f$ is $-\frac{1}{512}$