LFM Stats And Pure

jee-main 2020 Q61 Recover a Function from a Composition or Functional Equation View

If $g(x) = x ^ { 2 } + x - 1$ and $(g \circ f)(x) = 4x ^ { 2 } - 10x + 5$, then $f \left( \frac { 5 } { 4 } \right)$ is equal to
(1) $\frac { 3 } { 2 }$
(2) $- \frac { 1 } { 2 }$
(3) $\frac { 1 } { 2 }$
(4) $- \frac { 3 } { 2 }$

jee-main 2020 Q61 Determine Domain or Range of a Composite Function View

Let $f:(1,3) \rightarrow R$ be a function defined by $f(x) = \frac{x[x]}{1 + x^{2}}$, where $[x]$ denotes the greatest integer $\leq x$. Then the range of $f$ is
(1) $\left(\frac{2}{5}, \frac{3}{5}\right] \cup \left(\frac{3}{4}, \frac{4}{5}\right)$
(2) $\left(\frac{2}{5}, \frac{1}{2}\right) \cup \left(\frac{3}{5}, \frac{4}{5}\right]$
(3) $\left(\frac{2}{5}, \frac{4}{5}\right]$
(4) $\left(\frac{3}{5}, \frac{4}{5}\right)$

jee-main 2020 Q62 Determine Domain or Range of a Composite Function View

The domain of the function $f(x) = \sin^{-1}\left(\frac{|x| + 5}{x^{2} + 1}\right)$ is $(-\infty, -a] \cup [a, \infty)$, then $a$ is equal to
(1) $\frac{\sqrt{17}}{2}$
(2) $\frac{\sqrt{17} - 1}{2}$
(3) $\frac{1 + \sqrt{17}}{2}$
(4) $\frac{\sqrt{17}}{2} + 1$

jee-main 2020 Q64 Derivative of an Inverse Function View

Let $f$ and $g$ be differentiable functions on $R$ such that $f \circ g$ is the identity function. If for some $a , b \in R , g ^ { \prime } ( a ) = 5$ and $g ( a ) = b$, then $f ^ { \prime } ( b )$ is equal to:
(1) $\frac { 1 } { 5 }$
(2) 1
(3) 5
(4) $\frac { 2 } { 5 }$

jee-main 2020 Q72 Counting Functions with Composition or Mapping Constraints View

Let $A = \{a, b, c\}$ and $B = \{1, 2, 3, 4\}$. Then the number of elements in the set $C = \{f: A \rightarrow B \mid 2 \in f(A)$ and $f$ is not one-one$\}$ is ...

jee-main 2021 Q51 Injectivity, Surjectivity, or Bijectivity Classification View

If $f: R \rightarrow R$ is a function defined by $f(x) = e^{|x|} - e^{-x}$ / $e^{x} + e^{-x}$, then $f$ is:
(1) bijective
(2) $f$ is monotonically increasing on $(0, \infty)$
(3) $f$ is monotonically decreasing on $(0, \infty)$
(4) not differentiable at $x = 0$

jee-main 2021 Q69 Counting Functions with Composition or Mapping Constraints View

Let $A = \{ 1,2,3 , \ldots , 10 \}$ and $f : A \rightarrow A$ be defined as $$f ( k ) = \left\{ \begin{array} { c l } k + 1 & \text { if } k \text { is odd } \\ k & \text { if } k \text { is even } \end{array} \right.$$ Then the number of possible functions $g : A \rightarrow A$ such that $g o f = f$ is:
(1) ${ } ^ { 10 } \mathrm { C } _ { 5 }$
(2) $5 ^ { 5 }$
(3) 5 !
(4) $10 ^ { 5 }$

jee-main 2021 Q70 Injectivity, Surjectivity, or Bijectivity Classification View

Let $f : R \rightarrow R$ be defined as $f(x) = 2x - 1$ and $g : R - \{1\} \rightarrow R$ be defined as $g(x) = \frac { x - \frac { 1 } { 2 } } { x - 1 }$. Then the composition function $f(g(x))$ is:
(1) neither one-one nor onto
(2) onto but not one-one
(3) both one-one and onto
(4) one-one but not onto

jee-main 2021 Q70 Determine Domain or Range of a Composite Function View

Let $f ( x ) = \sin ^ { - 1 } x$ and $g ( x ) = \frac { x ^ { 2 } - x - 2 } { 2 x ^ { 2 } - x - 6 }$. If $g ( 2 ) = \lim _ { x \rightarrow 2 } g ( x )$, then the domain of the function $f o g$ is
(1) $( - \infty , - 1 ] \cup [ 2 , \infty )$
(2) $( - \infty , - 2 ] \cup \left[ - \frac { 3 } { 2 } , \infty \right)$
(3) $( - \infty , - 2 ] \cup \left[ - \frac { 4 } { 3 } , \infty \right)$
(4) $( - \infty , - 2 ] \cup [ - 1 , \infty )$

jee-main 2021 Q70 Determine Domain or Range of a Composite Function View

Let $\alpha \in R$ be such that the function $f ( x ) = \left\{ \begin{array} { l l } \frac { \cos ^ { - 1 } \left( 1 - \{ x \} ^ { 2 } \right) \sin ^ { - 1 } ( 1 - \{ x \} ) } { \{ x \} - \{ x \} ^ { 3 } } , & x \neq 0 \\ \alpha , & x = 0 \end{array} \right.$ is continuous at $x = 0$, where $\{ x \} = x - [ x ] , [ x ]$ is the greatest integer less than or equal to $x$. Then :
(1) $\alpha = \frac { \pi } { \sqrt { 2 } }$
(2) $\alpha = 0$
(3) no such $\alpha$ exists
(4) $\alpha = \frac { \pi } { 4 }$

jee-main 2021 Q70 Find or Apply an Inverse Function Formula View

The number of real roots of the equation $\tan ^ { - 1 } \sqrt { x ( x + 1 ) } + \sin ^ { - 1 } \sqrt { x ^ { 2 } + x + 1 } = \frac { \pi } { 4 }$ is:
(1) 1
(2) 2
(3) 4
(4) 0

jee-main 2021 Q71 Determine Domain or Range of a Composite Function View

Let $f : R \rightarrow R$ be defined as $f ( x ) = \begin{cases} 2 \sin \left( - \frac { \pi x } { 2 } \right) , & \text { if } x < - 1 \\ \left| a x ^ { 2 } + x + b \right| , & \text { if } - 1 \leq x \leq 1 \\ \sin ( \pi x ) , & \text { if } x > 1 \end{cases}$ If $f ( x )$ is continuous on $R$, then $a + b$ equals :
(1) 1
(2) 3
(3) - 3
(4) - 1

jee-main 2021 Q71 Determine Domain or Range of a Composite Function View

Let $[ x ]$ denote the greatest integer $\leq x$, where $x \in R$. If the domain of the real valued function $f ( x ) = \sqrt { \frac { | [ x ] | - 2 } { | [ x ] | - 3 } }$ is $( - \infty , a ) \cup [ b , c ) \cup [ 4 , \infty ) , a < b < c$, then the value of $a + b + c$ is:
(1) 8
(2) 1
(3) $- 2$
(4) $- 3$

jee-main 2021 Q71 Recover a Function from a Composition or Functional Equation View

Let $f : R - \left\{ \frac { \alpha } { 6 } \right\} \rightarrow R$ be defined by $f ( x ) = \left( \frac { 5 x + 3 } { 6 x - \alpha } \right)$. Then the value of $\alpha$ for which $( f \circ f ) ( x ) = x$, for all $x \in R - \left\{ \frac { \alpha } { 6 } \right\}$, is
(1) No such $\alpha$ exists
(2) 5
(3) 8
(4) 6

jee-main 2021 Q72 Determine Domain or Range of a Composite Function View

Let a function $f : R \rightarrow R$ be defined as, $f ( x ) = \begin{cases} \sin x - e ^ { x } & \text { if } x \leq 0 \\ a + [ - x ] & \text { if } 0 < x < 1 \\ 2 x - b & \text { if } x \geq 1 \end{cases}$
Where $[ x ]$ is the greatest integer less than or equal to $x$. If $f$ is continuous on $R$, then ( $a + b$ ) is equal to:
(1) 4
(2) 3
(3) 2
(4) 5

jee-main 2021 Q72 Injectivity, Surjectivity, or Bijectivity Classification View

Let $f , g : N \rightarrow N$ such that $f ( n + 1 ) = f ( n ) + f ( 1 ) \forall n \in N$ and $g$ be any arbitrary function. Which of the following statements is NOT true?
(1) If $f$ is onto, then $f ( n ) = n \forall n \in N$
(2) If $g$ is onto, then $f o g$ is one-one
(3) $f$ is one-one
(4) If $f \circ g$ is one-one, then $g$ is one-one

jee-main 2021 Q73 Determine Domain or Range of a Composite Function View

Let the functions $f : R \rightarrow R$ and $g : R \rightarrow R$ be defined as : $f ( x ) = \left\{ \begin{array} { l l } x + 2 , & x < 0 \\ x ^ { 2 } , & x \geq 0 \end{array} \right.$ and $g ( x ) = \begin{cases} x ^ { 3 } , & x < 1 \\ 3 x - 2 , & x \geq 1 \end{cases}$ Then, the number of points in $R$ where $( f \circ g ) ( x )$ is NOT differentiable is equal to :
(1) 3
(2) 1
(3) 0
(4) 2

jee-main 2021 Q74 Injectivity, Surjectivity, or Bijectivity Classification View

Consider function $f : A \rightarrow B$ and $g : B \rightarrow C ( A , B , C \subseteq R )$ such that $( g o f ) ^ { - 1 }$ exists, then:
(1) $f$ and $g$ both are one-one
(2) $f$ and $g$ both are onto
(3) $f$ is one-one and $g$ is onto
(4) $f$ is onto and $g$ is one-one

jee-main 2022 Q69 Determine Domain or Range of a Composite Function View

The function $f : R \rightarrow R$ defined by $f ( x ) = \lim _ { n \rightarrow \infty } \frac { \cos ( 2 \pi x ) - x ^ { 2 n } \sin ( x - 1 ) } { 1 + x ^ { 2 n + 1 } - x ^ { 2 n } }$ is continuous for all $x$ in
(1) $R - \{ - 1 \}$
(2) $R - \{ - 1,1 \}$
(3) $R - \{ 1 \}$
(4) $R - \{ 0 \}$

jee-main 2022 Q70 Injectivity, Surjectivity, or Bijectivity Classification View

Let a function $f : \mathbb{N} \rightarrow \mathbb{N}$ be defined by $$f(n) = \begin{cases} 2n, & n = 2,4,6,8,\ldots \\ n-1, & n = 3,7,11,15,\ldots \\ \frac{n+1}{2}, & n = 1,5,9,13,\ldots \end{cases}$$ then, $f$ is
(1) One-one and onto
(2) One-one but not onto
(3) Onto but not one-one
(4) Neither one-one nor onto

jee-main 2022 Q71 Injectivity, Surjectivity, or Bijectivity Classification View

Let $f : \mathbb { R } \rightarrow \mathbb { R }$ be defined as $f ( x ) = x - 1$ and $g : R \rightarrow \{ 1 , - 1 \} \rightarrow \mathbb { R }$ be defined as $g ( x ) = \frac { x ^ { 2 } } { x ^ { 2 } - 1 }$. Then the function $f o g$ is:
(1) One-one but not onto
(2) onto but not one-one
(3) Both one-one and onto
(4) Neither one-one nor onto

jee-main 2022 Q71 Determine Domain or Range of a Composite Function View

The domain of the function $f(x) = \sin^{-1}\left(\frac{x^2 - 3x + 2}{x^2 + 2x + 7}\right)$ is
(1) $[1, \infty)$
(2) $(-1, 2]$
(3) $[-1, \infty)$
(4) $(-\infty, 2]$

jee-main 2022 Q71 Evaluate Composition from Algebraic Definitions View

Let $f ( x ) = \frac { x - 1 } { x + 1 } , x \in R - \{ 0 , - 1 , 1 \}$. If $f ^ { n + 1 } ( x ) = f \left( f ^ { n } ( x ) \right)$ for all $n \in N$, then $f ^ { 6 } ( 6 ) + f ^ { 7 } ( 7 )$ is equal to
(1) $\frac { 7 } { 6 }$
(2) $- \frac { 3 } { 2 }$
(3) $\frac { 7 } { 12 }$
(4) $- \frac { 11 } { 12 }$

jee-main 2022 Q71 Determine Domain or Range of a Composite Function View

The domain of the function $f ( x ) = \sin ^ { - 1 } \left[ 2 x ^ { 2 } - 3 \right] + \log _ { 2 } \left( \log _ { \frac { 1 } { 2 } } \left( x ^ { 2 } - 5 x + 5 \right) \right)$, where $[ t ]$ is the greatest integer function, is
(1) $\left( - \sqrt { \frac { 5 } { 2 } } , \frac { 5 - \sqrt { 5 } } { 2 } \right)$
(2) $\left( \frac { 5 - \sqrt { 5 } } { 2 } , \frac { 5 + \sqrt { 5 } } { 2 } \right)$
(3) $\left( 1 , \frac { 5 - \sqrt { 5 } } { 2 } \right)$
(4) $\left[ 1 , \frac { 5 + \sqrt { } 5 } { 2 } \right)$

jee-main 2022 Q72 Determine Domain or Range of a Composite Function View

The domain of $f ( x ) = \frac { \cos ^ { - 1 } \left( \frac { x ^ { 2 } - 5 x + 6 } { x ^ { 2 } - 9 } \right) } { \log \left( x ^ { 2 } - 3 x + 2 \right) }$ is
(1) $x \in \left[ \frac { - 1 } { 2 } , 1 \right) \cup ( 2 , \infty ) - \{ 3 \}$
(2) $x \in \left[ \frac { - 1 } { 2 } , 1 \right] \cup ( 2 , \infty ) - \{ 3 \}$
(3) $x \in \left( \frac { - 1 } { 2 } , 1 \right) \cup [ 2 , \infty ) - \{ 3 \}$
(4) $x \in \left[ \frac { - 1 } { 2 } , 1 \right) \cup [ 2 , \infty ) - \{ 3 \}$

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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