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JEE Advanced · Mathematics · Sets, Relations and Functions

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1
Let A={1,2,3,4,5}A=\{1,2,3,4,5\} and B={1,2,3,4,5,6,7}B=\{1,2,3,4,5,6,7\}. Let SS be the set of functions f:A→Bf:A\to B with f(2)≠2,f(4)≠4f(2)\ne2, f(4)\ne4. Let T={f∈S:there exists g:B→N with g(f(x))=2x for all x∈A}T=\{f\in S: \text{there exists } g:B\to\mathbb N \text{ with } g(f(x))=2^x \text{ for all } x\in A\}. Find ∣T∣|T|.
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Your answer: —
Correct answer: 1860
2
Let S={1,2,…,10}S=\{1,2,\dots,10\}. Consider X={R:R is an equivalence relation on S with exactly 42 elements}X=\{R : R\text{ is an equivalence relation on }S\text{ with exactly }42\text{ elements}\} (as a set of ordered pairs). Find the number of elements in XX.
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Your answer: —
Correct answer: 2520
3
Let R\mathbb{R} denote the set of all real numbers. Let ai,bi∈Ra_i, b_i \in \mathbb{R} for i∈{1,2,3}i \in \{1, 2, 3\}. Define the functions f:R→Rf:\mathbb{R}\to\mathbb{R}, g:R→Rg:\mathbb{R}\to\mathbb{R}, and h:R→Rh:\mathbb{R}\to\mathbb{R} by f(x)=a1+10x+a2x2+a3x3+x4f(x) = a_1 + 10x + a_2x^2 + a_3x^3 + x^4, g(x)=b1+3x+b2x2+b3x3+x4g(x) = b_1 + 3x + b_2x^2 + b_3x^3 + x^4, h(x)=f(x+1)−g(x+2)h(x) = f(x+1) - g(x+2). If f(x)≠g(x)f(x) \neq g(x) for every x∈Rx \in \mathbb{R}, then the coefficient of x3x^3 in h(x)h(x) is
Not attempted
A
88
B
22
C
−4-4
D
−6-6
4
Let N\mathbb{N} denote the set of all natural numbers, and Z\mathbb{Z} denote the set of all integers. Consider the functions f:N→Zf:\mathbb{N}\to\mathbb{Z} and g:Z→Ng:\mathbb{Z}\to\mathbb{N} defined by f(n)={(n+1)/2if n is odd,(4−n)/2if n is even,g(n)={3+2nif n≥0,−2nif n<0.f(n) = \begin{cases} (n+1)/2 & \text{if } n \text{ is odd}, \\ (4-n)/2 & \text{if } n \text{ is even}, \end{cases} \qquad g(n) = \begin{cases} 3+2n & \text{if } n\ge 0, \\ -2n & \text{if } n<0. \end{cases} Define (g∘f)(n)=g(f(n))(g\circ f)(n) = g(f(n)) for all n∈Nn\in\mathbb{N}, and (f∘g)(n)=f(g(n))(f\circ g)(n)=f(g(n)) for all n∈Zn\in\mathbb{Z}. Then which of the following statements is (are) TRUE?
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A
g∘fg\circ f is NOT one-one and g∘fg\circ f is NOT onto
B
f∘gf\circ g is NOT one-one but f∘gf\circ g is onto
C
gg is one-one and gg is onto
D
ff is NOT one-one but ff is onto
5
Let the set of all relations RR on the set {a,b,c,d,e,f}\{a,b,c,d,e,f\}, such that RR is reflexive and symmetric, and RR contains exactly 1010 elements, be denoted by S\mathcal{S}. Then the number of elements in S\mathcal{S} is _________________.
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Your answer: —
Correct answer: 105
6
Let f:R→Rf:\mathbb{R}\to\mathbb{R} be a function such that f(x+y)=f(x)+f(y)f(x+y)=f(x)+f(y) for all x,y∈Rx,y\in\mathbb{R}, and g:R→(0,∞)g:\mathbb{R}\to(0,\infty) be a function such that g(x+y)=g(x)g(y)g(x+y)=g(x)g(y) for all x,y∈Rx,y\in\mathbb{R}. If f(−35)=12f\left(-\dfrac35\right)=12 and g(−13)=2g\left(-\dfrac13\right)=2, then the value of (f(14)+g(−2)−8)g(0)\left(f\left(\dfrac14\right)+g(-2)-8\right)g(0) is ________.
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Your answer: —
Correct answer: 51
7
Paragraph\textbf{Paragraph}: Let S={1,2,3,4,5,6}S=\{1,2,3,4,5,6\} and XX be the set of all relations RR from SS to SS that satisfy both the following properties: (i) RR has exactly 66 elements. (ii) For each (a,b)∈R(a,b)\in R, we have ∣a−b∣≥2|a-b|\ge2. If n(X)=mC6n(X)={}^{m}C_6, then the value of mm is __________.
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Your answer: —
Correct answer: 20
8
Paragraph\textbf{Paragraph}: Let S={1,2,3,4,5,6}S=\{1,2,3,4,5,6\} and XX be the set of all relations RR from SS to SS that satisfy both the following properties: (i) RR has exactly 66 elements. (ii) For each (a,b)∈R(a,b)\in R, we have ∣a−b∣≥2|a-b|\ge2. Let Y={R∈X: the range of R has exactly one element}Y=\{R\in X:\text{ the range of }R\text{ has exactly one element}\} and Z={R∈X:R is a function from S to S}Z=\{R\in X: R\text{ is a function from }S\text{ to }S\}. Let n(A)n(A) denote the number of elements in a set AA. If the value of n(Y)+n(Z)n(Y)+n(Z) is k2k^2, then ∣k∣|k| is __________.
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Your answer: —
Correct answer: 36
9
Let S={a+b2:a,b∈Z}S=\{a+b\sqrt2 : a,b\in\mathbb Z\}, T1={(−1+2)n:n∈N}T_1=\{(-1+\sqrt2)^n : n\in\mathbb N\}, and T2={(1+2)n:n∈N}T_2=\{(1+\sqrt2)^n : n\in\mathbb N\}. Then which of the following statements is (are) TRUE?
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A
Z∪T1∪T2⊂S\mathbb Z\cup T_1\cup T_2\subset S
B
T1∩(0,12024)=ϕT_1\cap\left(0,\dfrac1{2024}\right)=\phi
C
T2∩(2024,∞)≠ϕT_2\cap(2024,\infty)\ne\phi
D
For any a,b∈Za,b\in\mathbb Z, cos⁡(π(a+b2))+isin⁡(π(a+b2))∈Z\cos(\pi(a+b\sqrt2))+i\sin(\pi(a+b\sqrt2))\in\mathbb Z if and only if b=0b=0
10
Let a=32a=3\sqrt2 and b=151/66b=\dfrac{1}{5^{1/6}\sqrt6}. If x,y∈Rx,y\in\mathbb R are such that 3x+2y=log⁡a(18)5/43x+2y=\log_a\left(18\right)^{5/4} and 2x−y=log⁡b(1080)2x-y=\log_b\left(\sqrt{1080}\right), then 4x+5y4x+5y is equal to ________.
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Your answer: —
Correct answer: 8
11
Let f:R→Rf:\mathbb R\to\mathbb R and g:R→Rg:\mathbb R\to\mathbb R be functions defined by f(x)={x∣x∣sin⁡(1x),x≠00,x=0f(x)=\begin{cases}x|x|\sin\left(\dfrac1x\right), & x\ne0\\ 0, & x=0\end{cases} and g(x)={1−2x,0≤x≤120,otherwiseg(x)=\begin{cases}1-2x, & 0\le x\le\dfrac12\\ 0, & \text{otherwise}\end{cases} Let a,b,c,d∈Ra,b,c,d\in\mathbb R. Define the function h:R→Rh:\mathbb R\to\mathbb R by h(x)=a f(x)+b(g(x)+g(12−x))+c(x−g(x))+d g(x), x∈Rh(x)=a\,f(x)+b\left(g(x)+g\left(\dfrac12-x\right)\right)+c\left(x-g(x)\right)+d\,g(x),\ x\in\mathbb R. Match each entry in List-I to the correct entry in List-II. List-I: (P) If a=0,b=1,c=0,d=0a=0,b=1,c=0,d=0, then (Q) If a=1,b=0,c=0,d=0a=1,b=0,c=0,d=0, then (R) If a=0,b=0,c=1,d=0a=0,b=0,c=1,d=0, then (S) If a=0,b=0,c=0,d=1a=0,b=0,c=0,d=1, then List-II: (1) hh is one-one. (2) hh is onto. (3) hh is differentiable on R\mathbb R. (4) the range of hh is [0,1][0,1]. (5) the range of hh is {0,1}\{0,1\}. The correct option is
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A
(P)→\to(4) (Q)→\to(3) (R)→\to(1) (S)→\to(2)
B
(P)→\to(5) (Q)→\to(2) (R)→\to(4) (S)→\to(3)
C
(P)→\to(5) (Q)→\to(3) (R)→\to(2) (S)→\to(4)
D
(P)→\to(4) (Q)→\to(2) (R)→\to(1) (S)→\to(3)
12
Let S=(0,1)∪(1,2)∪(3,4)S = (0,1) \cup (1,2) \cup (3,4) and T={0,1,2,3}T = \{0,1,2,3\}. Then which of the following statements is(are) true?
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A
There are infinitely many functions from SS to TT
B
There are infinitely many strictly increasing functions from SS to TT
C
The number of continuous functions from SS to TT is at most 120120
D
Every continuous function from SS to TT is differentiable
13
In a study about a pandemic, data of 900900 persons was collected. It was found that 190190 persons had symptom of fever, 220220 persons had symptom of cough, 220220 persons had symptom of breathing problem, 330330 persons had symptom of fever or cough or both, 350350 persons had symptom of cough or breathing problem or both, 340340 persons had symptom of fever or breathing problem or both, 3030 persons had all three symptoms (fever, cough and breathing problem). If a person is chosen randomly from these 900900 persons, then the probability that the person has at most one symptom is __________.
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Your answer: —
Correct answer: 0.8
14
Let the function f:[0,1]→Rf:[0,1] \to \mathbb{R} be defined by f(x)=4x4x+2f(x) = \dfrac{4^x}{4^x+2}. Then the value of f(140)+f(240)+f(340)+⋯+f(3940)−f(12)f\left(\dfrac1{40}\right)+f\left(\dfrac2{40}\right)+f\left(\dfrac3{40}\right)+\cdots+f\left(\dfrac{39}{40}\right)-f\left(\dfrac12\right) is ____
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Your answer: —
Correct answer: 19
15
If the function f:R→Rf: \mathbb{R} \to \mathbb{R} is defined by f(x)=∣x∣(x−sin⁡x)f(x) = |x|(x - \sin x), then which of the following statements is TRUE?
Not attempted
A
ff is one-one, but NOT onto
B
ff is onto, but NOT one-one
C
ff is BOTH one-one and onto
D
ff is NEITHER one-one NOR onto
16
Let E1={x∈R:x≠1 and xx−1>0}E_1=\left\{x\in\mathbb{R}: x\neq1 \text{ and } \frac{x}{x-1}>0\right\} and E2={x∈E1:sin⁡−1(log⁡e(xx−1)) is a real number}E_2=\left\{x\in E_1: \sin^{-1}\left(\log_e\left(\frac{x}{x-1}\right)\right) \text{ is a real number}\right\}. (Here, the inverse trigonometric function sin⁡−1x\sin^{-1}x assumes values in [−π2,π2]\left[-\frac{\pi}{2},\frac{\pi}{2}\right].) Let f:E1→Rf:E_1\to\mathbb{R} be the function defined by f(x)=log⁡e(xx−1)f(x)=\log_e\left(\frac{x}{x-1}\right) and g:E2→Rg:E_2\to\mathbb{R} be the function defined by g(x)=sin⁡−1(log⁡e(xx−1))g(x)=\sin^{-1}\left(\log_e\left(\frac{x}{x-1}\right)\right). Match each entry in LIST-I to the correct entry in LIST-II. LIST-I\textbf{LIST-I} P. The range of ff is Q. The range of gg contains R. The domain of ff contains S. The domain of gg is LIST-II\textbf{LIST-II} 1. (−∞,11−e]∪[ee−1,∞)\left(-\infty,\frac{1}{1-e}\right]\cup\left[\frac{e}{e-1},\infty\right) 2. (0,1)(0,1) 3. [−12,12]\left[-\frac12,\frac12\right] 4. (−∞,0)∪(0,∞)(-\infty,0)\cup(0,\infty) 5. (−∞,ee−1]\left(-\infty,\frac{e}{e-1}\right] 6. (−∞,0)∪(12,ee−1](-\infty,0)\cup\left(\frac12,\frac{e}{e-1}\right]
Not attempted
A
P→\to4; Q→\to2; R→\to1; S→\to1
B
P→\to3; Q→\to3; R→\to6; S→\to5
C
P→\to4; Q→\to2; R→\to1; S→\to6
D
P→\to4; Q→\to3; R→\to6; S→\to5
17
Let XX be a set with exactly 55 elements and YY be a set with exactly 77 elements. If α\alpha is the number of one-one functions from XX to YY and β\beta is the number of onto functions from YY to XX, then the value of 15!(β−α)\frac{1}{5!}(\beta-\alpha) is _____.
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Your answer: —
Correct answer: 119
18
The value of ((log⁡29)2)1log⁡2(log⁡29)×(7)1log⁡47\left((\log_2 9)^2\right)^{\frac{1}{\log_2(\log_2 9)}} \times (\sqrt7)^{\frac{1}{\log_4 7}} is ______.
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Your answer: —
Correct answer: 8
19
Let f1:R→Rf_1:\mathbb{R}\to\mathbb{R}, f2:[0,∞)→Rf_2:[0,\infty)\to\mathbb{R}, f3:R→Rf_3:\mathbb{R}\to\mathbb{R} and f4:R→[0,∞)f_4:\mathbb{R}\to[0,\infty) be defined by f1(x)={∣x∣if x<0,exif x≥0;f_1(x)=\begin{cases}|x| & \text{if }x<0,\\ e^x & \text{if }x\ge0;\end{cases} f2(x)=x2f_2(x)=x^2; f3(x)={sin⁡xif x<0,xif x≥0;f_3(x)=\begin{cases}\sin x & \text{if }x<0,\\ x & \text{if }x\ge0;\end{cases} and f4(x)={f2(f1(x))if x<0,f2(f1(x))−1if x≥0.f_4(x)=\begin{cases}f_2(f_1(x)) & \text{if }x<0,\\ f_2(f_1(x))-1 & \text{if }x\ge0.\end{cases} Match each entry of List I\textbf{List I} to its value in List II\textbf{List II}. List I\textbf{List I} P.\textbf{P.} f4f_4 is Q.\textbf{Q.} f3f_3 is R.\textbf{R.} f2∘f1f_2\circ f_1 is S.\textbf{S.} f2f_2 is List II\textbf{List II} 1.\textbf{1.} onto but not one-one 2.\textbf{2.} neither continuous nor one-one 3.\textbf{3.} differentiable but not one-one 4.\textbf{4.} continuous and one-one Which of the following is the correct combination?
Not attempted
A
P→3, Q→1, R→4, S→2P{\to}3,\ Q{\to}1,\ R{\to}4,\ S{\to}2
B
P→1, Q→3, R→4, S→2P{\to}1,\ Q{\to}3,\ R{\to}4,\ S{\to}2
C
P→3, Q→1, R→2, S→4P{\to}3,\ Q{\to}1,\ R{\to}2,\ S{\to}4
D
P→1, Q→3, R→2, S→4P{\to}1,\ Q{\to}3,\ R{\to}2,\ S{\to}4
20
Let f:(−π2,π2)→Rf:\left(-\frac{\pi}{2},\frac{\pi}{2}\right)\to\mathbb{R} be given by f(x)=(log⁡(sec⁡x+tan⁡x))3f(x)=(\log(\sec x+\tan x))^3. Then:
Not attempted
A
f(x)f(x) is an odd function.
B
f(x)f(x) is a one-one function.
C
f(x)f(x) is an onto function.
D
f(x)f(x) is an even function.
Tip\textbf{Tip}: First show u(x)=log⁡(sec⁡x+tan⁡x)u(x)=\log(\sec x+\tan x) is odd and strictly increasing using u′(x)=sec⁡x>0u'(x)=\sec x>0 and the identity (sec⁡x−tan⁡x)(sec⁡x+tan⁡x)=1(\sec x-\tan x)(\sec x+\tan x)=1; cubing an odd, strictly increasing bijection preserves all three properties.
Deconstruction\textbf{Deconstruction}: Let u(x)=log⁡(sec⁡x+tan⁡x)u(x)=\log(\sec x+\tan x), so f(x)=u(x)3f(x)=u(x)^3. We must check odd/even-ness, injectivity, and surjectivity of ff. Strategy\textbf{Strategy}: Establish the properties of uu first, then transfer them to f=u3f=u^3 using that t↦t3t\mapsto t^3 is an odd, strictly increasing bijection of R\mathbb{R}. Execution\textbf{Execution}: uu is odd: u(−x)=log⁡(sec⁡x−tan⁡x)u(-x)=\log(\sec x-\tan x). Since (sec⁡x−tan⁡x)(sec⁡x+tan⁡x)=sec⁡2x−tan⁡2x=1(\sec x-\tan x)(\sec x+\tan x)=\sec^2x-\tan^2x=1, sec⁡x−tan⁡x=1sec⁡x+tan⁡x\sec x-\tan x = \frac{1}{\sec x+\tan x}, so u(−x)=−u(x)u(-x)=-u(x). uu is strictly increasing: u′(x)=sec⁡x>0u'(x)=\sec x>0 throughout (−π/2,π/2)\left(-\pi/2,\pi/2\right). Since uu is odd, f(−x)=u(−x)3=(−u(x))3=−f(x)f(-x)=u(-x)^3=(-u(x))^3=-f(x): ff is odd — proves (A)\textbf{(A)}, and rules out (D)\textbf{(D)} (an odd function that is also even must be identically 00, which ff is not). Since uu is strictly increasing and t↦t3t\mapsto t^3 is strictly increasing on R\mathbb{R}, the composite f=u3f=u^3 is strictly increasing, hence one-one — proves (B)\textbf{(B)}. Since u:(−π/2,π/2)→Ru:\left(-\pi/2,\pi/2\right)\to\mathbb{R} is continuous, strictly increasing, with u(x)→±∞u(x)\to\pm\infty as x→±π/2x\to\pm\pi/2, uu is onto R\mathbb{R}; and t↦t3t\mapsto t^3 is a bijection of R\mathbb{R}, so ff is also onto R\mathbb{R} — proves (C)\textbf{(C)}. Verification\textbf{Verification}: f(0)=(log⁡1)3=0f(0)=(\log 1)^3=0, consistent with an odd function; and ff being a strictly increasing bijection of an interval onto R\mathbb{R} automatically forces both one-one and onto. Final Answer\textbf{Final Answer}: (A), (B), and (C).