LFM Stats And Pure

jee-main 2022 Q72 Derivative of an Inverse Function View

Let $f : R \rightarrow R$ be defined as $f(x) = x ^ { 3 } + x - 5$. If $g(x)$ is a function such that $f( g(x) ) = x , \forall x \in R$, then $g ^ { \prime } (63)$ is equal to
(1) 49
(2) $\frac { 1 } { 49 }$
(3) $\frac { 43 } { 49 }$
(4) $\frac { 3 } { 49 }$

jee-main 2022 Q72 Evaluate Composition from Algebraic Definitions View

If $f ( x ) = \left\{ \begin{array} { l l } x + a , & x \leq 0 \\ | x - 4 | , & x > 0 \end{array} \right.$ and $g ( x ) = \left\{ \begin{array} { l l } x + 1 , & x < 0 \\ ( x - 4 ) ^ { 2 } + b , & x \geq 0 \end{array} \right.$ are continuous on $R$, then $( g \circ f ) ( 2 ) + ( f \circ g ) ( - 2 )$ is equal to:
(1) $- 10$
(2) 10
(3) 8
(4) $- 8$

jee-main 2022 Q72 Find or Apply an Inverse Function Formula View

If for $p \neq q \neq 0$, then function $f ( x ) = \frac { \sqrt [ 7 ] { p ( 729 + x ) } - 3 } { \sqrt [ 3 ] { 729 + q x } - 9 }$ is continuous at $x = 0$, then
(1) $7 p q f ( 0 ) - 1 = 0$
(2) $63 q f ( 0 ) - p ^ { 2 } = 0$
(3) $21 q f ( 0 ) - p ^ { 2 } = 0$
(4) $7 p q f ( 0 ) - 9 = 0$

jee-main 2022 Q73 Evaluate Composition from Algebraic Definitions View

Let $f : R \rightarrow R$ and $g : R \rightarrow R$ be two functions defined by $f(x) = \log _ { \mathrm { e } } ( x ^ { 2 } + 1 ) - e ^ { - x } + 1$ and $g(x) = \frac { 1 - 2 e ^ { 2 x } } { e ^ { x } }$. Then, for which of the following range of $\alpha$, the inequality $f\left( g\left( \frac { ( \alpha - 1 ) ^ { 2 } } { 3 } \right) \right) > f\left( g\left( \alpha - \frac { 5 } { 3 } \right) \right)$ holds?
(1) $( - 2 , - 1 )$
(2) $(2, 3)$
(3) $(1, 2)$
(4) $( - 1, 1 )$

jee-main 2022 Q73 Determine Domain or Range of a Composite Function View

The domain of the function $\cos ^ { - 1 } \left( \frac { 2 \sin ^ { - 1 } \left( \frac { 1 } { 4 x ^ { 2 } - 1 } \right) } { \pi } \right)$ is
(1) $\left( - \infty , - \frac { 1 } { \sqrt { 2 } } \right] \cup \left[ \frac { 1 } { \sqrt { 2 } } , \infty \right) \cup \{ 0 \}$
(2) $\left( - \infty , - \frac { 1 } { \sqrt { 2 } } \right] \cup \left[ \frac { 1 } { \sqrt { 2 } } , \infty \right)$
(3) $\left( - \infty , - \frac { 1 } { \sqrt { 2 } } \right) \cup \left( \frac { 1 } { 2 } , \infty \right) \cup \{ 0 \}$
(4) $R - \left\{ - \frac { 1 } { 2 } , \frac { 1 } { 2 } \right\}$

jee-main 2022 Q85 Counting Functions with Composition or Mapping Constraints View

The number of functions $f$, from the set $A = \left\{ x \in N : x ^ { 2 } - 10 x + 9 \leq 0 \right\}$ to the set $B = \left\{ n ^ { 2 } : n \in N \right\}$ such that $f ( x ) \leq ( x - 3 ) ^ { 2 } + 1$, for every $x \in A$, is $\_\_\_\_$ .

jee-main 2022 Q86 Evaluate Composition from Diagram or Mapping View

Let $S = \{ 1,2,3,4,5,6,7,8,9,10 \}$. Define $f : S \rightarrow S$ as $f ( n ) = \left\{ \begin{array} { c l } 2 n , & \text { if } n = 1,2,3,4,5 \\ 2 n - 11 & \text { if } n = 6,7,8,9,10 \end{array} \right.$ Let $g : S \rightarrow S$ be a function such that $f \circ g ( n ) = \left\{ \begin{array} { l l } n + 1 & , \text { if } n \text { is odd } \\ n - 1 & , \text { if } n \text { is even } \end{array} \right.$, then $g ( 10 ) ( g ( 1 ) + g ( 2 ) + g ( 3 ) + g ( 4 ) + g ( 5 ) )$ is equal to

jee-main 2023 Q62 Injectivity, Surjectivity, or Bijectivity Classification View

Let $f: \mathbb{R} \to \mathbb{R}$ be a function defined by $f(x) = \frac{x^2 + 2}{x^2 + 1}$. Then which of the following is NOT true?
(1) $f(x)$ has a minimum at $x = 0$
(2) $f(x)$ is an even function
(3) $f(x)$ is strictly increasing for $x > 0$
(4) $f(x)$ is onto

jee-main 2023 Q71 Evaluate Composition from Algebraic Definitions View

Let $f$, $g$ and $h$ be the real valued functions defined on $\mathbb{R}$ as $f(x) = \left\{ \begin{array}{cc} \frac{x}{|x|}, & x \neq 0 \\ 1, & x = 0 \end{array} \right.$, $\quad g(x) = \left\{ \begin{array}{cc} \frac{\sin(x+1)}{(x+1)}, & x \neq -1 \\ 1, & x = -1 \end{array} \right.$ and $h(x) = 2[x] - f(x)$, where $[x]$ is the greatest integer $\leq x$. Then the value of $\lim_{x \rightarrow 1} g(h(x-1))$ is
(1) 1
(2) $\sin(1)$
(3) $-1$
(4) 0

jee-main 2023 Q72 Determine Domain or Range of a Composite Function View

The range of $f(x) = 4 \sin ^ { - 1 } \left( \frac { x ^ { 2 } } { x ^ { 2 } + 1 } \right)$ is
(1) $[ 0,2 \pi ]$
(2) $[ 0 , \pi ]$
(3) $[ 0,2 \pi )$
(4) $[ 0 , \pi )$

jee-main 2023 Q72 Determine Domain or Range of a Composite Function View

If the domain of the function $f ( x ) = \log _ { e } \left( 4 x ^ { 2 } + 11 x + 6 \right) + \sin ^ { - 1 } ( 4 x + 3 ) + \cos ^ { - 1 } \left( \frac { 10 x + 6 } { 3 } \right)$ is $( \alpha , \beta ]$, then $36 | \alpha + \beta |$ is equal to
(1) 54
(2) 72
(3) 63
(4) 45

jee-main 2023 Q72 Determine Domain or Range of a Composite Function View

If the domain of the function $f(x) = \frac{x}{1+\lfloor x \rfloor^2}$, where $\lfloor x \rfloor$ is greatest integer $\leq x$, is $[2,6)$, then its range is
(1) $\left\{\frac{5}{26}, \frac{2}{5}\right\} \cup \left\{\frac{9}{29}, \frac{27}{109}, \frac{18}{89}, \frac{9}{53}\right\}$
(2) $\left[\frac{5}{26}, \frac{2}{5}\right]$
(3) $\left\{\frac{5}{37}, \frac{2}{5}\right\} \cup \left\{\frac{9}{29}, \frac{27}{109}, \frac{18}{89}, \frac{9}{53}\right\}$
(4) $\left[\frac{5}{37}, \frac{2}{5}\right]$

jee-main 2023 Q76 Determine Domain or Range of a Composite Function View

The domain of $f ( x ) = \frac { \log _ { ( x + 1 ) } ( x - 2 ) } { e ^ { 2 \log _ { e } x } - ( 2 x + 3 ) } , x \in R$ is
(1) $\mathbb { R } - \{ - 1,3 \}$
(2) $( 2 , \infty ) - \{ 3 \}$
(3) $( - 1 , \infty ) - \{ 3 \}$
(4) $\mathbb { R } - \{ 3 \}$

jee-main 2023 Q77 Counting Functions with Composition or Mapping Constraints View

The number of functions $f : \{ 1,2,3,4 \} \rightarrow \{ \mathrm { a } \in \mathbb { Z } : | \mathrm { a } | \leq 8 \}$ satisfying $f ( \mathrm { n } ) + \frac { 1 } { \mathrm { n } } f ( \mathrm { n } + 1 ) = 1 , \forall \mathrm { n } \in \{ 1,2,3 \}$ is
(1) 3
(2) 4
(3) 1
(4) 2

jee-main 2023 Q77 Injectivity, Surjectivity, or Bijectivity Classification View

Let $f : R \rightarrow R$ be a function such that $f ( x ) = \frac { x ^ { 2 } + 2 x + 1 } { x ^ { 2 } + 1 }$. Then
(1) $f ( x )$ is many-one in $( - \infty , - 1 )$
(2) $f ( x )$ is many-one in $( 1 , \infty )$
(3) $f ( x )$ is one-one in $[ 1 , \infty )$ but not in $( - \infty , \infty )$
(4) $f ( x )$ is one-one in $( - \infty , \infty )$

jee-main 2023 Q77 Determine Domain or Range of a Composite Function View

The range of the function $f(x) = \sqrt{3 - x} + \sqrt{2 + x}$ is
(1) $[\sqrt{5}, \sqrt{10}]$
(2) $[2\sqrt{2}, \sqrt{11}]$
(3) $[\sqrt{5}, \sqrt{13}]$
(4) $[\sqrt{2}, \sqrt{7}]$

jee-main 2023 Q77 Determine Domain or Range of a Composite Function View

Let $D$ be the domain of the function $f ( x ) = \sin ^ { - 1 } \left( \log _ { 3 x } \left( \frac { 6 + 2 \log _ { 3 } x } { - 5 x } \right) \right)$. If the range of the function $g : D \rightarrow \mathbb { R }$ defined by $g ( x ) = x - [ x ]$, ([x] is the greatest integer function), is $( \alpha , \beta )$, then $\alpha ^ { 2 } + \frac { 5 } { \beta }$ is equal to
(1) 135
(2) 45
(3) 46
(4) 136

jee-main 2023 Q78 Evaluate Composition from Algebraic Definitions View

For some $a , b , c \in \mathbb { N }$, let $f ( x ) = a x - 3$ and $g ( x ) = x ^ { b } + c , x \in \mathbb { R }$. If $( f \circ g ) ^ { - 1 } ( x ) = \left( \frac { x - 7 } { 2 } \right) ^ { \frac { 1 } { 3 } }$, then $( f \circ g ) ( a c ) + ( g \circ f ) ( b )$ is equal to $\_\_\_\_$ .

jee-main 2023 Q78 Find or Apply an Inverse Function Formula View

If the function $f ( x ) = \left\{ \begin{array} { c l } ( 1 + | \cos x | ) \frac { \lambda } { | \cos x | } , & 0 < x < \frac { \pi } { 2 } \\ \mu , & x = \frac { \pi } { 2 } \\ e ^ { \frac { \cot 6 x } { \cot 4 x } } , & \frac { \pi } { 2 } < x < \pi \end{array} \right.$ is continuous at $x = \frac { \pi } { 2 }$, then $9 \lambda + 6 \log _ { e } \mu + \mu ^ { 6 } - e ^ { 6 \lambda }$ is equal to
(1) 11
(2) 8
(3) $2 e ^ { 4 } + 8$
(4) 10

jee-main 2023 Q78 Recover a Function from a Composition or Functional Equation View

Let $f : \mathbb { R } \rightarrow \mathbb { R }$ be a differentiable function that satisfies the relation $f ( x + y ) = f ( x ) + f ( y ) - 1 , \forall x , y \in \mathbb { R }$. If $f ^ { \prime } ( 0 ) = 2$, then $| f ( - 2 ) |$ is equal to

jee-main 2023 Q78 Counting Functions with Composition or Mapping Constraints View

Let $A = \{1, 2, 3, 5, 8, 9\}$. Then the number of possible functions $f: A \rightarrow A$ such that $f(m \cdot n) = f(m) \cdot f(n)$ for every $m, n \in A$ with $m \cdot n \in A$ is equal to

jee-main 2024 Q68 Evaluate Composition from Algebraic Definitions View

Let $f(x) = \begin{cases} x-1, & x \text{ is even,} \\ 2x, & x \text{ is odd,} \end{cases}$ $x \in \mathbb{N}$. If for some $a \in \mathbb{N}$, $f(f(f(a))) = 21$, then $\lim_{x \to a^-} \left(\frac{x^3}{a} - \left\lfloor\frac{x}{a}\right\rfloor\right)$, where $\lfloor t \rfloor$ denotes the greatest integer less than or equal to $t$, is equal to:
(1) 121
(2) 144
(3) 169
(4) 225

jee-main 2024 Q70 Determine Domain or Range of a Composite Function View

If the domain of the function $f(x) = \log_e\frac{2x+3}{4x^2+x-3} + \cos^{-1}\frac{2x-1}{x+2}$ is $(\alpha, \beta]$, then the value of $5\beta - 4\alpha$ is equal to
(1) 10
(2) 12
(3) 11
(4) 9

jee-main 2024 Q71 Determine Domain or Range of a Composite Function View

If $f ( x ) = \left\{ \begin{array} { l } 2 + 2 x , - 1 \leq x < 0 \\ 1 - \frac { x } { 3 } , 0 \leq x \leq 3 \end{array} ; g ( x ) = \left\{ \begin{array} { l } - x , - 3 \leq x \leq 0 \\ x , 0 < x \leq 1 \end{array} \right. \right.$, then range of $( f \circ g ( x ) )$ is
(1) $( 0,1 ]$
(2) $[ 0,3 )$
(3) $[ 0,1 ]$
(4) $[ 0,1 )$

jee-main 2024 Q71 Determine Domain or Range of a Composite Function View

Let $\mathrm { f } : \mathrm { R } - \frac { - 1 } { 2 } \rightarrow \mathrm { R }$ and $\mathrm { g } : \mathrm { R } - \frac { - 5 } { 2 } \rightarrow \mathrm { R }$ be defined as $\mathrm { fx } = \frac { 2 \mathrm { x } + 3 } { 2 \mathrm { x } + 1 }$ and $\mathrm { gx } = \frac { | \mathrm { x } | + 1 } { 2 \mathrm { x } + 5 }$. Then the domain of the function fog is :
(1) $\mathrm { R } - - \frac { 5 } { 2 }$
(2) $R$
(3) $R - \frac { 1 } { 4 }$
(4) $\mathrm { R } - - \frac { 5 } { 2 } , - \frac { 7 } { 4 }$

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LFM Stats And Pure

Completing the square and sketching 61

Inequalities 181

Discriminant and conditions for roots 65

Solving quadratics and applications 131

Curve Sketching 228

Factor & Remainder Theorem 61

Polynomial Division & Manipulation 35

Binomial Theorem (positive integer n) 197

Function Transformations 28

Composite & Inverse Functions 211

Partial Fractions 12

Generalised Binomial Theorem 10

Complex Numbers Arithmetic 207

Complex Numbers Argand & Loci 159

Permutations & Arrangements 255

Combinations & Selection 237

Measures of Location and Spread 181

Probability Definitions 212

Tree Diagrams 18

Principle of Inclusion/Exclusion 33

Independent Events 35

Conditional Probability 167

Discrete Probability Distributions 176

Uniform Distribution 3

Binomial Distribution 101

Geometric Distribution 8

Hypergeometric Distribution 9

Negative Binomial Distribution 1

Modelling and Hypothesis Testing 22

Hypothesis test of binomial distributions 2

Data representation 21

Continuous Probability Distributions and Random Variables 322

Continuous Uniform Random Variables 3

Geometric Probability 16

Normal Distribution 87

Approximating Binomial to Normal Distribution 4

Sign Change & Interval Methods 26

Fixed Point Iteration 9

Newton-Raphson method 6