JEE Advanced Past Papers PDF

# BHISHMA for JEE ADVANCED ## One Shot Mathematics - Limit, Continuity, Differentiability & MOD **By - Ashish Agarwal** ### Topics to be covered 1. Problem Practice ## JEE Advanced 2023 ### Question Let $S = (0,1) \cup (1,2) \cup (3,4)$ and $T = \{0,1,2,3\}$. Then which of the following stat...

# BHISHMA for JEE ADVANCED ## One Shot Mathematics - Limit, Continuity, Differentiability & MOD **By - Ashish Agarwal** ### Topics to be covered 1. Problem Practice ## JEE Advanced 2023 ### Question 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? * There are infinitely many functions from $S$ to $T$ * There are infinitely many strictly increasing functions from $S$ to $T$ * The number of continuous functions from $S$ to $T$ is at most 120 * Every continuous function from $S$ to $T$ is differentiable **Answer: A, C, D** ## JEE Advanced 2023 ### Question Let $f: (0,1) \to R$ be the functions defined as $f(x) = \sqrt{n}$ if $x \in [\frac{1}{n+1},\frac{1}{n})$ where $n \in N$. Let $g: (0,1) \to R$ be a function such that $\int_{x^2}^{x} \frac{\sqrt{1-t}}{t} dt < g(x) < 2\sqrt{x}$ for all $x \in (0,1)$. Then $lim_{x \to 0} f(x)g(x)$ * does NOT exist * is equal to 1 * is equal to 2 * is equal to 3 **Answer: C** ## JEE Advanced 2023 (Paper 2) ### Question Let $f: (0,1) \to R$ be the function defined as $f(x) = [4x](x - \frac{1}{2})(x - \frac{1}{4})^2$, where $[x]$ denotes the greatest integer less than or equal to $x$. Then which of the following statements is(are) true? * The function $f$ is discontinuous exactly at one point in $(0,1)$ * There is exactly one point in $(0,1)$ at which the function $f$ is continuous but NOT differentiable * The function $f$ is NOT differentiable at more than three points in $(0,1)$ * The function $f$ is NOT differentiable at exactly two points in $(0,1)$ **Answer: D** ## JEE Advanced 2022 (Paper 1) ### Question Let $\alpha$ be a positive real number. Let $f: R \to R$ and $g: (\alpha, \infty) \to R$ be the functions defined by $f(x) = sin(\frac{\pi x}{12})$ and $g(x) = \frac{2log_e(\sqrt{x} - \sqrt{\alpha})}{log_e(e^{\sqrt{x}} - e^{\sqrt{\alpha}})}$. Then the value of $lim_{x \to \alpha^+} f(g(x))$ is * does NOT exist * is equal to 1 * is equal to 2 * is equal to $\frac{\sqrt{2}}{2}$ **Answer: 0.5** ## JEE Advanced 2020 (Paper 2) ### Question Let the functions $f: (-1,1) \to R$ and $g: (-1,1) \to (-1,1)$ be defined by $f(x) = |2x - 1| + |2x + 1|$ and $g(x) = x - [x]$, where $[x]$ denotes the greatest integer less than or equal to $x$. Let $fog: (-1, 1) \to R$ be the composite function defined by $(fog)(x) = f(g(x))$. Suppose $c$ is the number of points in the interval $(-1,1)$ at which $fog$ is NOT continuous, and suppose $d$ is the number of points in the interval $(-1,1)$ at which $fog$ is NOT differentiable. Then the value of $c + d$ is * 1 * 2 * 3 * 4 **Answer: 4** ## JEE Advanced 2016 ### Question Let $f: [\frac{-1}{2}, 2] \to R$ and $g: [\frac{-1}{2}, 2] \to R$ be functions defined by $f(x) = [x^2 -3]$ and $g(x) = |x|f(x) + 4x - 7|f(x)|$, where $[y]$ denotes the greatest integer less than or equal to $y$ for $y \in R$. Then * $f$ is discontinuous exactly at three points in $[\frac{-1}{2}, 2]$ * $f$ is discontinuous exactly at four points in $[\frac{-1}{2}, 2]$ * $g$ is NOT differentiable exactly at four points in $[\frac{-1}{2}, 2]$ * $g$ is NOT differentiable exactly at five points in $[\frac{-1}{2}, 2]$ **Answer: D** ## JEE Advanced 2012 ### Question For every integer $n$, let $a_n$ and $b_n$ be real numbers. Let function $f: R \to R$ be given by $f(x) = \left\{ \begin{array}{ll} a_n + sin(nx) & \text{for } x \in [2n,2n+1]\\ b_n + cos(nx) & \text{for } x \in (2n-1, 2n) \end{array} \right.$ for all integers $n$. If $f$ is continuous, then which of the following holds(s) for all $n$? * $a_{n-1} - b_{n-1} = 0$ * $a_n - b_n = 1$ * $a_n - b_{n+1} = 1$ * $a_{n-1} - b_n = -1$ **Answer: D** ## JEE Advanced 2015 (Paper 1) ### Question Let $g: R \to R$ be a differentiable function with $g(0) = 0, g'(0) = 0$ and $g'(1) \neq 0$. Let $f(x) = \left\{ \begin{array}{ll} \frac{x}{|x|}g(x) & \text{for } x \neq 0 \\ 0 & \text{for } x = 0 \end{array} \right.$ and $h(x) = e^{|x|}$ for all $x \in R$. Let $(foh)(x)$ denote $f(h(x))$ and $(hof)(x)$ denote $f(f(x))$. Then which of the following is (are) true? * $f$ is differentiable at $x = 0$ * $h$ is differentiable at $x = 0$ * $foh$ is differentiable at $x = 0$ * $hof$ is differentiable at $x = 0$ **Answer: A, D** ## JEE Advanced 2015 (Paper 1) ### Question Let $f(x) = sin(\frac{\pi}{6}sin(\frac{\pi}{2}sin(x)))$ for all $x \in R$ and $g(x) = \frac{\pi}{2} sin(x)$ for all $x \in R$. Let $(fog)(x)$ denote $f(g(x))$ and $(gof)(x)$ denote $g(f(x))$. Then which of the following is/are true? * Range of $f$ is $[\frac{-1}{2}, \frac{1}{2}]$ * Range of $fog$ is $[\frac{-1}{2}, \frac{1}{2}]$ * $lim_{x \to 0} \frac{f(x)}{g(x)} = \frac{\pi}{6}$ * There is an $x \in R$ such that $(gof)(x) = 1$ **Answer: A, B, C ** ## JEE Advanced 2020 (Paper 1) ### Question Let the function $f: R \to R$ be defined by $f(x) = x^3 - x^2 + (x - 1)sin(x)$ and let $g: R \to R$ be an arbitrary function. Let $fg: R \to R$ be the product function defined by $(fg)(x) = f(x)g(x)$. Then which of the following statements is/are TRUE ? * If $g$ is continuous at $ x = 1$, then $fg$ is differentiable at $x = 1$ * If $fg$ is differentiable at $x = 1$, then $g$ is continuous at $x = 1$ * If $g$ is differentiable at $x = 1$, then $fg$ is differentiable at $x = 1$ * If $fg$ is differentiable at $x = 1$, then $g$ is differentiable at $x = 1$ **Answer: A, C** ## JEE Advanced 2023 (Paper 2) ### Question Let $S$ be the set of all twice differentiable functions $f$ from $R$ to $R$ such that $\frac{d^2f}{dx^2}(x) > 0$ for all $x \in (-1,1)$. For $f \in S$, let $X_f$ be the number of points $x \in (-1,1)$ for which $f(x) = x$. Then which of the following statements is(are) true? * There exists a function $f \in S$ such that $X_f = 0$ * For every function $f \in S$, we have $X_f \le 2$ * There exists a function $f \in S$ such that $X_f = 2$ * There does NOT exist any function $f$ in $S$ such that $X_f = 1$ **Answer: A, B, C** ## JEE Advanced 2016 (Paper 1) ### Question Let $f: R \to R$, $g: R \to R$ and $h: R \to R$ be differentiable functions such that $f(x) = x^3 + 3x + 2$, $g(f(x)) = x$ and $h(g(g(x))) = x$ for all $x \in R$. Then * $g'(2) = \frac{1}{15}$ * $h'(1) = 666$ * $h(0) = 16$ * $h(g(3)) = 36$ **Answer: B, C**