LFM Stats And Pure

jee-advanced 2020 Q9 Existence or Properties of Functions and Inverses (Proof-Based) View

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ and $g: \mathbb{R} \rightarrow \mathbb{R}$ be functions satisfying $$f(x + y) = f(x) + f(y) + f(x)f(y) \text{ and } f(x) = xg(x)$$ for all $x, y \in \mathbb{R}$. If $\lim_{x \rightarrow 0} g(x) = 1$, then which of the following statements is/are TRUE?
(A) $f$ is differentiable at every $x \in \mathbb{R}$
(B) If $g(0) = 1$, then $g$ is differentiable at every $x \in \mathbb{R}$
(C) The derivative $f'(1)$ is equal to 1
(D) The derivative $f'(0)$ is equal to 1

jee-advanced 2023 Q1 4 marks Existence or Properties of Functions and Inverses (Proof-Based) View

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?
(A) There are infinitely many functions from $S$ to $T$
(B) There are infinitely many strictly increasing functions from $S$ to $T$
(C) The number of continuous functions from $S$ to $T$ is at most 120
(D) Every continuous function from $S$ to $T$ is differentiable

jee-advanced 2024 Q8 4 marks Custom Operation or Property Verification View

Let $f : \mathbb { R } \rightarrow \mathbb { R }$ be a function such that $f ( x + y ) = f ( x ) + f ( y )$ for all $x , y \in \mathbb { R }$, and $g : \mathbb { R } \rightarrow ( 0 , \infty )$ be a function such that $g ( x + y ) = g ( x ) g ( y )$ for all $x , y \in \mathbb { R }$. If $f \left( \frac { - 3 } { 5 } \right) = 12$ and $g \left( \frac { - 1 } { 3 } \right) = 2$, then the value of $\left( f \left( \frac { 1 } { 4 } \right) + g ( - 2 ) - 8 \right) g ( 0 )$ is $\_\_\_\_$ .

jee-advanced 2025 Q6 4 marks Injectivity, Surjectivity, or Bijectivity Classification View

Let $\mathbb { N }$ denote the set of all natural numbers, and $\mathbb { Z }$ denote the set of all integers. Consider the functions $f : \mathbb { N } \rightarrow \mathbb { Z }$ and $g : \mathbb { Z } \rightarrow \mathbb { N }$ defined by
$$f ( n ) = \begin{cases} ( n + 1 ) / 2 & \text { if } n \text { is odd } \\ ( 4 - n ) / 2 & \text { if } n \text { is even } \end{cases}$$
and
$$g ( n ) = \begin{cases} 3 + 2 n & \text { if } n \geq 0 \\ - 2 n & \text { if } n < 0 \end{cases}$$
Define $( g \circ f ) ( n ) = g ( f ( n ) )$ for all $n \in \mathbb { N }$, and $( f \circ g ) ( n ) = f ( g ( n ) )$ for all $n \in \mathbb { Z }$.
Then which of the following statements is (are) TRUE?

(A)	$g \circ f$ is NOT one-one and $g \circ f$ is NOT onto
(B)	$f \circ g$ is NOT one-one but $f \circ g$ is onto
(C)	$g$ is one-one and $g$ is onto
(D)	$f$ is NOT one-one but $f$ is onto

jee-advanced 2025 Q14 4 marks Derivative of an Inverse Function View

Let $\mathbb { R }$ denote the set of all real numbers. Let $f : \mathbb { R } \rightarrow \mathbb { R }$ and $g : \mathbb { R } \rightarrow ( 0,4 )$ be functions defined by
$$f ( x ) = \log _ { e } \left( x ^ { 2 } + 2 x + 4 \right) , \text { and } g ( x ) = \frac { 4 } { 1 + e ^ { - 2 x } }$$
Define the composite function $f \circ g ^ { - 1 }$ by $\left( f \circ g ^ { - 1 } \right) ( x ) = f \left( g ^ { - 1 } ( x ) \right)$, where $g ^ { - 1 }$ is the inverse of the function $g$.
Then the value of the derivative of the composite function $f \circ g ^ { - 1 }$ at $x = 2$ is $\_\_\_\_$.

jee-main 2007 Q97 Find or Apply an Inverse Function Formula View

The function $f : R \sim \{ 0 \} \rightarrow R$ given by $f ( x ) = \frac { 1 } { x } - \frac { 2 } { e ^ { 2 x } - 1 }$ can be made continuous at $x = 0$ by defining $f ( 0 )$ as
(1) 2
(2) - 1
(3) 0
(4) 1

jee-main 2007 Q103 Determine Domain or Range of a Composite Function View

The largest interval lying in $\left( - \frac { \pi } { 2 } , \frac { \pi } { 2 } \right)$ for which the function $\left[ f ( x ) = 4 ^ { - x ^ { 2 } } + \cos ^ { - 1 } \left( \frac { x } { 2 } - 1 \right) + \log ( \cos x ) \right]$ is defined, is
(1) $[ 0 , \pi ]$
(2) $\left( - \frac { \pi } { 2 } , \frac { \pi } { 2 } \right)$
(3) $\left[ - \frac { \pi } { 4 } , \frac { \pi } { 2 } \right)$
(4) $\left[ 0 , \frac { \pi } { 2 } \right)$

jee-main 2012 Q75 Recover a Function from a Composition or Functional Equation View

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

jee-main 2012 Q78 Injectivity, Surjectivity, or Bijectivity Classification View

Let $A$ and $B$ be non empty sets in $\mathbb{R}$ and $f : A \rightarrow B$ is a bijective function. Statement 1: $f$ is an onto function. Statement 2: There exists a function $g : B \rightarrow A$ such that $f \circ g = I _ { B }$.
(1) Statement 1 is true, Statement 2 is false.
(2) Statement 1 is true, Statement 2 is true; Statement 2 is a correct explanation for Statement 1.
(3) Statement 1 is false, Statement 2 is true.
(4) Statement 1 is true, Statement 2 is true, Statement 2 is not the correct explanation for Statement 1.

jee-main 2014 Q74 Evaluate Composition from Algebraic Definitions View

If $f ( x )$ is continuous and $f \left( \frac { 9 } { 2 } \right) = \frac { 2 } { 9 }$, then $\lim _ { x \rightarrow 0 } f \left( \frac { 1 - \cos 3 x } { x ^ { 2 } } \right)$ equals to
(1) $\frac { 8 } { 9 }$
(2) 0
(3) $\frac { 2 } { 9 }$
(4) $\frac { 9 } { 2 }$

jee-main 2014 Q78 Injectivity, Surjectivity, or Bijectivity Classification View

Let $f : R \rightarrow R$ be defined by $f ( x ) = \frac { | x | - 1 } { | x | + 1 }$, then $f$ is
(1) one-one but not onto
(2) neither one-one nor onto
(3) both one-one and onto
(4) onto but not one-one

jee-main 2014 Q79 Derivative of an Inverse Function View

If $g$ is the inverse of a function $f$ and $f ^ { \prime } ( x ) = \frac { 1 } { 1 + x ^ { 5 } }$, then $g ^ { \prime } ( x )$ is equal to
(1) $\frac { 1 } { 1 + \{ g ( x ) \} ^ { 5 } }$
(2) $1 + \{ g ( x ) \} ^ { 5 }$
(3) $1 + x ^ { 5 }$
(4) $5 x ^ { 4 }$

jee-main 2014 Q79 Find or Apply an Inverse Function Formula View

If the function $f ( x ) = \left\{ \begin{array} { c l } \frac { \sqrt { 2 + \cos x } - 1 } { ( \pi - x ) ^ { 2 } } , & x \neq \pi \\ k , & x = \pi \end{array} \right.$ is continuous at $x = \pi$, then $k$ equals
(1) $\frac { 1 } { 4 }$
(2) 0
(3) 2
(4) $\frac { 1 } { 2 }$

jee-main 2015 Q71 Evaluate Composition from Algebraic Definitions View

Let $f(x) = x^2$, $g(x) = \sin x$ for all $x \in \mathbb{R}$ and $h(x) = (gof)(x) = g(f(x))$. Statement I: $h$ is not differentiable at $x = 0$. Statement II: $(hog)(x) = \sin^2(\sin x)$. Which of the following is correct?
(1) Statement I is false, Statement II is true
(2) Statement I is true, Statement II is false
(3) Both Statement I and Statement II are true
(4) Both Statement I and Statement II are false

jee-main 2017 Q69 Evaluate Composition from Algebraic Definitions View

For $x \in \mathbb { R }$, $f ( x ) = | \log 2 - \sin x |$ and $g ( x ) = f ( f ( x ) )$, then:
(1) $g$ is not differentiable at $x = 0$
(2) $g ^ { \prime } ( 0 ) = \cos ( \log 2 )$
(3) $g ^ { \prime } ( 0 ) = - \cos ( \log 2 )$
(4) $g$ is differentiable at $x = 0$ and $g ^ { \prime } ( 0 ) = - \sin ( \log 2 )$

jee-main 2017 Q70 Injectivity, Surjectivity, or Bijectivity Classification View

The function $f : \mathbb { R } \to \left[ - \frac { 1 } { 2 } , \frac { 1 } { 2 } \right]$ defined as $f ( x ) = \frac { x } { 1 + x ^ { 2 } }$, is:
(1) Surjective but not injective
(2) Neither injective nor surjective
(3) Invertible
(4) Injective but not surjective

jee-main 2017 Q77 Injectivity, Surjectivity, or Bijectivity Classification View

The function $f : \mathbb{R} \rightarrow \left(-\dfrac{1}{2}, \dfrac{1}{2}\right)$ defined as $f(x) = \dfrac{x}{1 + x^2}$, is:
(1) Invertible
(2) Injective but not surjective
(3) Surjective but not injective
(4) Neither injective nor surjective

jee-main 2017 Q79 Recover a Function from a Composition or Functional Equation View

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

jee-main 2019 Q61 Determine Domain or Range of a Composite Function View

Let $f : \mathbb{R} \rightarrow \mathbb{R}$ be defined by $f(x) = \frac{x}{1+x^2}$, $x \in \mathbb{R}$. Then the range of $f$ is:
(1) $\mathbb{R} - [-1, 1]$
(2) $(-1, 1) - \{0\}$
(3) $\left[-\frac{1}{2}, \frac{1}{2}\right]$
(4) $\left(-\frac{1}{2}, \frac{1}{2}\right)$

jee-main 2019 Q76 Injectivity, Surjectivity, or Bijectivity Classification View

If the function $f : R - \{ 1 , - 1 \} \rightarrow A$ defined by $f ( x ) = \frac { x ^ { 2 } } { 1 - x ^ { 2 } }$, is surjective, then $A$ is equal to
(1) $[ 0 , \infty )$
(2) $R - \{ - 1 \}$
(3) $R - [ - 1,0 )$
(4) $R - ( - 1,0 )$

jee-main 2019 Q78 Evaluate Composition from Algebraic Definitions View

If $f(x) = \log_e\frac{1-x}{1+x}$, $|x| < 1$, then $f\left(\frac{2x}{1+x^2}\right)$ is equal to
(1) $f(x^2)$
(2) $2f(x^2)$
(3) $-2f(x)$
(4) $2f(x)$

jee-main 2019 Q78 Find or Apply an Inverse Function Formula View

If the function $f$ defined on $\left( \frac { \pi } { 6 } , \frac { \pi } { 3 } \right)$ by $f ( x ) = \left\{ \begin{array} { l l } \frac { \sqrt { 2 } \cos x - 1 } { \cot x - 1 } , & x \neq \frac { \pi } { 4 } \\ k , & x = \frac { \pi } { 4 } \end{array} \right.$ is continuous, then $k$ is equal to
(1) $\frac { 1 } { 2 }$
(2) 1
(3) 2
(4) $\frac { 1 } { \sqrt { 2 } }$

jee-main 2019 Q79 Determine Domain or Range of a Composite Function View

The domain of the definition of the function $f ( x ) = \frac { 1 } { 4 - x ^ { 2 } } + \log _ { 10 } \left( x ^ { 3 } - x \right)$ is:
(1) $( - 1,0 ) \cup ( 1,2 ) \cup ( 2 , \infty )$
(2) $( 1,2 ) \cup ( 2 , \infty )$
(3) $( - 2 , - 1 ) \cup ( - 1,0 ) \cup ( 2 , \infty )$
(4) $( - 1,0 ) \cup ( 1,2 ) \cup ( 3 , \infty )$

jee-main 2019 Q81 Injectivity, Surjectivity, or Bijectivity Classification View

Let $A = \{x \in R : x$ is not a positive integer$\}$. Define a function $f: A \rightarrow R$ as $f(x) = \frac{2x}{x-1}$, then $f$ is:
(1) Injective but not surjective
(2) Not injective
(3) Surjective but not injective
(4) Neither injective nor surjective

jee-main 2020 Q59 Find or Apply an Inverse Function Formula View

If $R = \{(x, y) : x, y \in Z, x^{2} + 3y^{2} \leq 8\}$ is a relation on the set of integers $Z$, then the domain of $R^{-1}$ is
(1) $\{-2, -1, 1, 2\}$
(2) $\{0, 1\}$
(3) $\{-2, -1, 0, 1, 2\}$
(4) $\{-1, 0, 1\}$

1
2
3
4
5
6
7
8
9

Load questions...

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