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Beyond Advanced · Sets, Relations and Functions

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1
Let X={1,2,…,10}X=\{1,2,\dots,10\}. How many subsets S⊆XS\subseteq X (the empty set included) are such that any two elements of SS differ by at least 33 and SS contains at least one even number?
Not attempted
Your answer: —
Correct answer: 47
2
Let A={ 1, {1}, ϕ, {1,{1}} }A=\{\,1,\ \{1\},\ \phi,\ \{1,\{1\}\}\,\}, where 11 is a number (not a set) and ϕ\phi is the empty set. Let P(A)\mathcal P(A) denote the power set of AA. Which of the following statements is/are correct?
Not attempted
A
{1}∈A∩P(A)\{1\}\in A\cap\mathcal P(A) and also 1∈A∩P(A)1\in A\cap\mathcal P(A)
B
{ϕ}⊂A\{\phi\}\subset A but {ϕ}∉A\{\phi\}\notin A
C
A∩P(A)A\cap\mathcal P(A) has exactly 33 elements
D
{{1}}∈P(A)\{\{1\}\}\in\mathcal P(A) and {{1}}∈A\{\{1\}\}\in A
3
How many non-empty subsets of {1,2,3,…,9}\{1,2,3,\dots,9\} have a sum of elements that is divisible by 33?
Not attempted
Your answer: —
Correct answer: 175
4
Finite sets AA and BB satisfy n(B)<n(A)n(B)<n(A),   n(P(A))−n(P(B))=960\;n\big(\mathcal P(A)\big)-n\big(\mathcal P(B)\big)=960 and n(P(A∩B))=8n\big(\mathcal P(A\cap B)\big)=8. How many subsets of A∪BA\cup B are contained in neither AA nor BB?
Not attempted
A
71047104
B
71687168
C
71127112
D
70967096
5
Let X={1,2,3,4,5}X=\{1,2,3,4,5\}. Find the number of ordered triples (A1,A2,A3)(A_1,A_2,A_3) of subsets of XX such that ϕ≠A1⊊A2⊊A3⊊X\phi\neq A_1\subsetneq A_2\subsetneq A_3\subsetneq X.
Not attempted
Your answer: —
Correct answer: 240
6
List-IList-IIP)Non-empty subsets of {1,2,…,10} with 3 elements and no two consecutive1)16Q)Subsets of {1,2,…,6} whose elements add up to an even number (empty set included)2)27R)Ordered pairs (A,B) of subsets of {1,2,3} with A⊆B3)32S)Subsets of {1,…,6} meeting both {1,2,3} and {4,5,6}4)495)56\begin{array}{ll} \textbf{List-I} & \textbf{List-II} \\ \text{P)} \text{Non-empty subsets of }\{1,2,\dots,10\}\text{ with }3\text{ elements and no two consecutive} & \text{1)} 16 \\ \text{Q)} \text{Subsets of }\{1,2,\dots,6\}\text{ whose elements add up to an even number (empty set included)} & \text{2)} 27 \\ \text{R)} \text{Ordered pairs }(A,B)\text{ of subsets of }\{1,2,3\}\text{ with }A\subseteq B & \text{3)} 32 \\ \text{S)} \text{Subsets of }\{1,\dots,6\}\text{ meeting both }\{1,2,3\}\text{ and }\{4,5,6\} & \text{4)} 49 \\ & \text{5)} 56 \end{array}
Not attempted
A
P-4, Q-3, R-2, S-5
B
P-5, Q-3, R-2, S-4
C
P-5, Q-2, R-3, S-4
D
P-3, Q-5, R-1, S-2
7
Let X={1,2,3,4,5}X=\{1,2,3,4,5\}. The number of ordered pairs (A,B)(A,B) of subsets of XX with n(A∩B)=2n(A\cap B)=2 and A∪B≠XA\cup B\neq X is
Not attempted
A
190190
B
270270
C
8080
D
210210
8
How many positive integers not exceeding 10001000 are divisible by exactly two of the three numbers 44, 66 and 1010?
Not attempted
Your answer: —
Correct answer: 118
9
How many integers from 11 to 10001000 are divisible by none of 66, 1010 and 1515?
Not attempted
Your answer: —
Correct answer: 734
10
In the Venn diagram of three sets A,B,CA,B,C inside a universal set UU, the regions carrying a solid dot together form the region RR. Which of the following expressions represent RR? (Each of the seven parts of A∪B∪CA\cup B\cup C is a separate region.)
Not attempted
ABCU
A
A △ (B △ C)A\,\triangle\,(B\,\triangle\,C), where X△Y=(X−Y)∪(Y−X)X\triangle Y=(X-Y)\cup(Y-X)
B
(A∪B∪C)−[(A∩B)∪(B∩C)∪(C∩A)](A\cup B\cup C)-\big[(A\cap B)\cup(B\cap C)\cup(C\cap A)\big]
C
[(A−B−C)∪(B−A−C)∪(C−A−B)]∪(A∩B∩C)\big[(A-B-C)\cup(B-A-C)\cup(C-A-B)\big]\cup(A\cap B\cap C)
D
[((A∪B)−C)−(A∩B)]∪[(A∩B∩C)∪(C−(A∪B))]\big[((A\cup B)-C)-(A\cap B)\big]\cup\big[(A\cap B\cap C)\cup(C-(A\cup B))\big]
11
Let A,B,CA,B,C be the sets of perfect squares, perfect cubes and perfect fifth powers of positive integers not exceeding 10001000. Then n(A∪B∪C)n(A\cup B\cup C) equals
Not attempted
A
4444
B
3939
C
4141
D
4040
12
In a group of 100100 people exactly 55 like none of the three games chess, ludo and carrom. Exactly 8080 like chess, 8585 like ludo and 9090 like carrom. Let mm be the least possible number of people who like all three games. Find mm.
Not attempted
Your answer: —
Correct answer: 65
13
Let X={1,2,3,4}X=\{1,2,3,4\}. The number of ordered triples (A,B,C)(A,B,C) of subsets of XX with A∪B∪C=XA\cup B\cup C=X, A∩B=ϕA\cap B=\phi, A∩C≠ϕA\cap C\neq\phi and B∩C≠ϕB\cap C\neq\phi is
Not attempted
A
625625
B
194194
C
256256
D
210210
14
Let A,B,CA,B,C be subsets of a universal set UU with n(U)=100n(U)=100, n(A)=60n(A)=60, n(B)=70n(B)=70, n(C)=80n(C)=80. Which of the following statements is/are correct?
Not attempted
A
n(A∩B∩C)≥10n(A\cap B\cap C)\ge 10 for every such choice of A,B,CA,B,C
B
n(A∪B∪C)≥85n(A\cup B\cup C)\ge 85 for every such choice of A,B,CA,B,C
C
It is possible that n((A∩B)−C)=20n\big((A\cap B)-C\big)=20
D
It is possible that n(A∩B)=70n(A\cap B)=70
15
For arbitrary sets A,B,C,DA,B,C,D (empty sets allowed), which of the following is/are correct?
Not attempted
A
(A×B)∩(C×D)=(A∩C)×(B∩D)(A\times B)\cap(C\times D)=(A\cap C)\times(B\cap D)
B
(A×B)∪(C×D)=(A∪C)×(B∪D)(A\times B)\cup(C\times D)=(A\cup C)\times(B\cup D)
C
A×B=B×AA\times B=B\times A if and only if A=BA=B
D
A×(B−C)=(A×B)−(A×C)A\times(B-C)=(A\times B)-(A\times C)
16
A relation RR from A={1,2,3,4}A=\{1,2,3,4\} to B={a,b,c}B=\{a,b,c\} is a subset of A×BA\times B. How many relations RR from AA to BB have domain equal to AA and range equal to BB (every element of AA is related to at least one element of BB, and every element of BB is related to at least one element of AA)?
Not attempted
Your answer: —
Correct answer: 2161
17
Let N={1,2,3,… }\mathbb N=\{1,2,3,\dots\} and R={(x,y):x,y∈N, 2x+3y=60}R=\{(x,y):x,y\in\mathbb N,\ 2x+3y=60\}. If R−1={(y,x):(x,y)∈R}R^{-1}=\{(y,x):(x,y)\in R\}, then n(R)+n(R∩R−1)+n(Dom(R)∩Range(R))n(R)+n(R\cap R^{-1})+n\big(\mathrm{Dom}(R)\cap\mathrm{Range}(R)\big) equals
Not attempted
A
1313
B
1212
C
1414
D
1010
18
How many symmetric relations on the set {1,2,3,4}\{1,2,3,4\} contain exactly 66 ordered pairs?
Not attempted
Your answer: —
Correct answer: 116
19
Let A={x∈Z:∣x∣≤4}A=\{x\in\mathbb Z:|x|\le4\} and B={y∈Z:∣y−1∣≤3}B=\{y\in\mathbb Z:|y-1|\le3\}. The number of ordered pairs (x,y)∈A×B(x,y)\in A\times B such that x2+y2≤10x^2+y^2\le10 and x+yx+y is even is
Not attempted
A
21
B
17
C
34
D
19
20
Let A={1,2,…,12}A=\{1,2,\dots,12\} and let R={(a,b)∈A×A: a divides b}R=\{(a,b)\in A\times A:\ a \text{ divides } b\}. Which of the following statements is/are correct?
Not attempted
A
n(R)=35n(R)=35
B
The composite relation R∘RR\circ R (pairs (a,c)(a,c) for which some bb has (a,b)∈R(a,b)\in R and (b,c)∈R(b,c)\in R) has exactly 3535 elements
C
Exactly 55 elements of AA occur as second coordinate in exactly two ordered pairs of RR
D
Exactly 22 elements of AA occur as second coordinate in exactly four ordered pairs of RR
21
A relation from a set AA to a set BB is any subset of A×BA\times B. There are exactly 2122^{12} relations from AA to BB. Also A⊊BA\subsetneq B and n(A)≥2n(A)\ge2. Find the sum of all possible values of n(P(B))n\big(\mathcal P(B)\big).
Not attempted
Your answer: —
Correct answer: 80
22
How many equivalence relations on the set {1,2,3,4,5,6}\{1,2,3,4,5,6\} contain the pair (3,4)(3,4) but do not contain the pair (1,2)(1,2)?
Not attempted
Your answer: —
Correct answer: 37
23
Which of the following is/are equivalence relations?
Not attempted
A
On Z\mathbb Z: a R ba\,R\,b if and only if abab is a perfect square (the number 00 counts as a perfect square)
B
On N\mathbb N: a R ba\,R\,b if and only if a−ba-b is divisible by 33 or divisible by 55
C
On R\mathbb R: x R yx\,R\,y if and only if x−yx-y is a rational number
D
On N×N\mathbb N\times\mathbb N: (a,b) R (c,d)(a,b)\,R\,(c,d) if and only if a+d=b+ca+d=b+c
24
A relation RR on A={1,2,3,4,5,6,7}A=\{1,2,3,4,5,6,7\} is drawn as a directed graph: an arrow from aa to bb means (a,b)∈R(a,b)\in R, and a small loop at a node aa means (a,a)∈R(a,a)\in R. The relation consists of the ordered pairs (1,2),(2,3),(3,1),(4,5),(5,6),(6,6),(7,7)(1,2),(2,3),(3,1),(4,5),(5,6),(6,6),(7,7) shown in the figure. What is the least number of further ordered pairs that must be added to RR so that it becomes an equivalence relation?
Not attempted
1234567
A
1414
B
1919
C
1212
D
1010
25
How many equivalence relations on {1,2,3,4,5,6,7}\{1,2,3,4,5,6,7\} have exactly 33 equivalence classes, with 1,2,31,2,3 lying in three different classes and with (4,5)(4,5) belonging to the relation?
Not attempted
Your answer: —
Correct answer: 27
26
On A={1,2,…,12}A=\{1,2,\dots,12\} let R={(a,b)∈A×A: ∣a−b∣=4 or ∣a−b∣=6}R=\{(a,b)\in A\times A:\ |a-b|=4 \text{ or } |a-b|=6\}. The smallest number of ordered pairs that must be added to RR to obtain an equivalence relation on AA is
Not attempted
A
7272
B
4444
C
4040
D
5858
27
Let X={1,2,3}X=\{1,2,3\} and consider relations on the power set P(X)\mathcal P(X). Which of the following statements is/are correct?
Not attempted
A
A R1 B  ⟺  A⊆BA\,R_1\,B\iff A\subseteq B or B⊆AB\subseteq A: R1R_1 is reflexive and symmetric but not transitive
B
A R2 B  ⟺  n(A)−n(B)A\,R_2\,B\iff n(A)-n(B) is even: R2R_2 is an equivalence relation with exactly 22 equivalence classes
C
A R3 B  ⟺  A∩B≠ϕA\,R_3\,B\iff A\cap B\neq\phi: R3R_3 is reflexive
D
A R4 B  ⟺  A∪B=XA\,R_4\,B\iff A\cup B=X: R4R_4 is symmetric but neither reflexive nor transitive
28
The number of relations on {1,2,3,4}\{1,2,3,4\} that are symmetric and transitive but not reflexive is
Not attempted
A
5252
B
1515
C
3737
D
4949
29
Let f(x)=3x2+9x+173x2+9x+7f(x)=\dfrac{3x^2+9x+17}{3x^2+9x+7} for real xx. Find the sum of all integers that lie in the range of ff.
Not attempted
Your answer: —
Correct answer: 860
30
The domain of f(x)=log⁡(x−1) ⁣(x2−5x+6)+4−xf(x)=\log_{(x-1)}\!\big(x^2-5x+6\big)+\sqrt{4-x} is
Not attempted
A
(1,2]∪(3,4](1,2]\cup(3,4]
B
(1,2)∪(3,4)(1,2)\cup(3,4)
C
(1,4](1,4]
D
(1,2)∪(3,4](1,2)\cup(3,4]
31
For f(x)=x2−3x+4x2+3x+4f(x)=\dfrac{x^2-3x+4}{x^2+3x+4}, x∈Rx\in\mathbb R, how many integers kk are there for which the equation f(x)=kf(x)=k has two distinct real solutions?
Not attempted
Your answer: —
Correct answer: 5
32
Which of the following pairs of functions f,gf,g (each with its natural, largest domain) are equal functions?
Not attempted
A
f(x)=x−1 x−2 f(x)=\sqrt{x-1}\,\sqrt{x-2}\ and  g(x)=(x−1)(x−2)\ g(x)=\sqrt{(x-1)(x-2)}
B
f(x)=∣x∣ f(x)=|x|\ and  g(x)=x2\ g(x)=\sqrt{x^2}
C
f(x)=ln⁡(ex) f(x)=\ln\left(e^{x}\right)\ and  g(x)=x\ g(x)=x
D
f(x)=x3+xx2+1 f(x)=\dfrac{x^3+x}{x^2+1}\ and  g(x)=x\ g(x)=x
33
A function ff defined for all real x≠0,1x\ne0,1 satisfies f(x)+f ⁣(11−x)=xf(x)+f\!\left(\dfrac{1}{1-x}\right)=x for every such xx. Then f(2)f(2) equals
Not attempted
A
74\dfrac74
B
32\dfrac32
C
54\dfrac54
D
22
34
Let f(x)=1⌊x⌋2−7⌊x⌋−18f(x)=\dfrac{1}{\sqrt{\lfloor x\rfloor^{2}-7\lfloor x\rfloor-18}}, where ⌊x⌋\lfloor x\rfloor is the greatest integer not exceeding xx. How many integers in the closed interval [−10,12][-10,12] belong to the domain of ff?
Not attempted
Your answer: —
Correct answer: 11
35
The domain of f(x)=1(k−2)x2+2(k−2)x+4f(x)=\dfrac{1}{\sqrt{(k-2)x^2+2(k-2)x+4}} is all of R\mathbb R. How many integers kk are possible?
Not attempted
Your answer: —
Correct answer: 4
36
Find the number of onto functions f:{1,2,3,4,5,6,7}→{a,b,c,d}f:\{1,2,3,4,5,6,7\}\to\{a,b,c,d\} satisfying f(1)=f(2)f(1)=f(2) and f(3)≠f(4)f(3)\ne f(4).
Not attempted
Your answer: —
Correct answer: 1320
37
The number of bijections f:{1,2,3,4,5,6}→{1,2,3,4,5,6}f:\{1,2,3,4,5,6\}\to\{1,2,3,4,5,6\} such that f(i)≠if(i)\ne i and f(f(i))≠if(f(i))\ne i for every ii is
Not attempted
A
265265
B
120120
C
160160
D
200200
38
Which of the following statements is/are correct?
Not attempted
A
f:R→(−1,1)f:\mathbb R\to(-1,1), f(x)=x1+∣x∣f(x)=\dfrac{x}{1+|x|}, is a bijection
B
f:R→Rf:\mathbb R\to\mathbb R, f(x)=x3+x2+x+1f(x)=x^3+x^2+x+1, is a bijection
C
f:R→Rf:\mathbb R\to\mathbb R, f(x)=x3−3xf(x)=x^3-3x, is onto but not one-one
D
f:[1,∞)→[1,∞)f:[1,\infty)\to[1,\infty), f(x)=x+1xf(x)=x+\dfrac1x, is a bijection
39
How many one-one functions f:{1,2,3,4,5}→{1,2,3,4,5,6,7}f:\{1,2,3,4,5\}\to\{1,2,3,4,5,6,7\} satisfy f(i)≠if(i)\ne i for every i∈{1,2,3,4,5}i\in\{1,2,3,4,5\}?
Not attempted
Your answer: —
Correct answer: 1214
40
The number of functions f:{1,2,3,4}→{1,2,3,4}f:\{1,2,3,4\}\to\{1,2,3,4\} satisfying f(f(x))=f(x)f(f(x))=f(x) for all xx is
Not attempted
A
2424
B
4444
C
6565
D
4141
41
The arrow diagram shows a function f:A→Bf:A\to B with A={1,2,3,4,5,6,7}A=\{1,2,3,4,5,6,7\} and B={p,q,r,s}B=\{p,q,r,s\}: f(1)=f(2)=f(3)=pf(1)=f(2)=f(3)=p, f(4)=qf(4)=q, f(5)=f(6)=rf(5)=f(6)=r, f(7)=sf(7)=s. The number of functions h:A→Ah:A\to A satisfying f(h(a))=f(a)f(h(a))=f(a) for every a∈Aa\in A is
Not attempted
1234567pqrsAB
A
5454
B
108108
C
8181
D
1212
42
For how many integers kk is f:R→Rf:\mathbb R\to\mathbb R, f(x)=x3+kx2+3x+7f(x)=x^3+kx^2+3x+7, one-one?
Not attempted
Your answer: —
Correct answer: 7
43
Let f(x)=11−xf(x)=\dfrac{1}{1-x} and define f1=ff_1=f, fn+1=f∘fnf_{n+1}=f\circ f_n for n≥1n\ge1. The value of f2026(3)f_{2026}(3) is
Not attempted
A
−12-\dfrac12
B
23\dfrac23
C
33
D
14\dfrac14
44
Let f(x)=x2−2f(x)=x^2-2. Find the number of distinct real solutions of f(f(f(x)))=xf(f(f(x)))=x.
Not attempted
Your answer: —
Correct answer: 8
45
Let f:A→Bf:A\to B and g:B→Cg:B\to C be functions. Which of the following statements is/are correct?
Not attempted
A
If g∘fg\circ f is one-one then gg must be one-one
B
If g∘fg\circ f is onto then gg must be onto
C
If g∘fg\circ f is onto then ff must be onto
D
For n(A)=2n(A)=2, n(B)=3n(B)=3, C=AC=A, the number of pairs (f,g)(f,g) with g∘f=idAg\circ f=\mathrm{id}_A is 1313
46
How many bijections f:{1,2,3,4,5}→{1,2,3,4,5}f:\{1,2,3,4,5\}\to\{1,2,3,4,5\} satisfy f∘f=idf\circ f=\mathrm{id} and f(1)≠1f(1)\ne1?
Not attempted
Your answer: —
Correct answer: 16
47
A function gg is defined on the largest possible domain by g ⁣(x+1x)=x3+1x3g\!\left(x+\dfrac1x\right)=x^3+\dfrac1{x^3}, where xx is a non-zero real. Which statement is correct?
Not attempted
A
The domain of gg is R∖{0}\mathbb R\setminus\{0\} and g(3)=18g(3)=18
B
The domain of gg is (−∞,−2)∪(2,∞)(-\infty,-2)\cup(2,\infty) and g(3)=18g(3)=18
C
The domain of gg is (−∞,−2]∪[2,∞)(-\infty,-2]\cup[2,\infty) and g(3)=18g(3)=18
D
The domain of gg is (−∞,−2]∪[2,∞)(-\infty,-2]\cup[2,\infty) and g(3)=24g(3)=24
48
Let f(x)=2x−1x+1f(x)=\dfrac{2x-1}{x+1} and let fnf^{n} denote the nn-fold composite f∘f∘⋯∘ff\circ f\circ\cdots\circ f. Find the smallest positive integer nn for which fn(x)=xf^{n}(x)=x for all admissible xx.
Not attempted
Your answer: —
Correct answer: 6
49
The graph of a function f:[0,6]→Rf:[0,6]\to\mathbb R is the polygonal line joining, in order, the points (0,1),(1,2),(2,52),(3,3),(4,72),(5,112),(6,6)(0,1),(1,2),(2,\tfrac52),(3,3),(4,\tfrac72),(5,\tfrac{11}{2}),(6,6) (see the figure; the dashed line is y=xy=x). The function is strictly increasing, so it has an inverse f−1f^{-1} defined on its range. The number of real values of xx for which f(x)=f−1(x)f(x)=f^{-1}(x) is
Not attempted
xyO123456123456y = x
A
22
B
44
C
55
D
33
50
Find the value of ∑k=11000⌊k3 ⌋\displaystyle\sum_{k=1}^{1000}\left\lfloor\sqrt[3]{k}\,\right\rfloor, where ⌊⋅⌋\lfloor\cdot\rfloor denotes the greatest integer function.
Not attempted
Your answer: —
Correct answer: 6985
51
Let SS be the set of all real solutions of x2−6⌊x⌋+5=0x^2-6\lfloor x\rfloor+5=0, where ⌊x⌋\lfloor x\rfloor is the greatest integer not exceeding xx. The value of ∑x∈Sx2\sum_{x\in S}x^2 is
Not attempted
A
6565
B
9090
C
5656
D
7575
52
Which of the following statements is/are true for the greatest integer function ⌊⋅⌋\lfloor\cdot\rfloor and fractional part {x}=x−⌊x⌋\{x\}=x-\lfloor x\rfloor?
Not attempted
A
⌊2x⌋=2⌊x⌋\lfloor2x\rfloor=2\lfloor x\rfloor for every real xx
B
⌊x⌋+⌊x+13⌋+⌊x+23⌋=⌊3x⌋\lfloor x\rfloor+\left\lfloor x+\tfrac13\right\rfloor+\left\lfloor x+\tfrac23\right\rfloor=\lfloor3x\rfloor for every real xx
C
⌊x⌋ ⌊y⌋≤⌊xy⌋\lfloor x\rfloor\,\lfloor y\rfloor\le\lfloor xy\rfloor for all real x,yx,y
D
For every real xx: ⌊x⌋+⌊−x⌋∈{0,−1}\lfloor x\rfloor+\lfloor -x\rfloor\in\{0,-1\} and {x}+{−x}∈{0,1}\{x\}+\{-x\}\in\{0,1\}
53
Find the number of real solutions of the equation ⌊x⌋ {x}=x−3\lfloor x\rfloor\,\{x\}=x-3 in the interval [0,10)[0,10), where {x}=x−⌊x⌋\{x\}=x-\lfloor x\rfloor is the fractional part.
Not attempted
Your answer: —
Correct answer: 7
54
How many distinct values does f(x)=⌊x⌋+⌊2x⌋+⌊3x⌋f(x)=\lfloor x\rfloor+\lfloor2x\rfloor+\lfloor3x\rfloor take as xx ranges over [0,5)[0,5)?
Not attempted
Your answer: —
Correct answer: 20
55
The greatest integer function is denoted ⌊⋅⌋\lfloor\cdot\rfloor. Which graph shows y=∣ x−⌊x+12⌋ ∣y=\big|\,x-\lfloor x+\tfrac12\rfloor\,\big| for −2≤x≤2-2\le x\le2?
Not attempted
A
Graph shown (curve 2)
xyO-2-1120.51
B
Graph shown (curve 3)
xyO-2-1120.51
C
Graph shown (curve 4)
xyO-2-1120.51
D
Graph shown (curve 1)
xyO-2-1120.51
56
Evaluate ∑k=199⌊98k100⌋\displaystyle\sum_{k=1}^{99}\left\lfloor\frac{98k}{100}\right\rfloor.
Not attempted
Your answer: —
Correct answer: 4802
57
A function f:R→Rf:\mathbb R\to\mathbb R satisfies the given condition for all real xx (wherever the expressions are defined). In which cases is ff necessarily periodic with the stated period?
Not attempted
A
f(x+1)=f(x)−f(x−1)f(x+1)=f(x)-f(x-1), with period 66
B
f(x+2)=−f(x)f(x+2)=-f(x), with period 44
C
f(x+1)=1+f(x)1−f(x)f(x+1)=\dfrac{1+f(x)}{1-f(x)}, with period 44
D
ff is odd and f(x)=f(2−x)f(x)=f(2-x), with period 44
58
Find the fundamental (smallest positive) period of f(x)=∣sin⁡πx3∣+∣cos⁡πx2∣f(x)=\left|\sin\dfrac{\pi x}{3}\right|+\left|\cos\dfrac{\pi x}{2}\right|.
Not attempted
Your answer: —
Correct answer: 6
59
A function f:{−2,−1,0,1,2}→{−1,0,1}f:\{-2,-1,0,1,2\}\to\{-1,0,1\} is called even if f(−x)=f(x)f(-x)=f(x) for all xx in the domain and odd if f(−x)=−f(x)f(-x)=-f(x) for all xx. How many such functions are neither even nor odd?
Not attempted
Your answer: —
Correct answer: 208
60
An odd function f:R→Rf:\mathbb R\to\mathbb R has period 44, and for 0≤x≤20\le x\le2 it is given by f(x)=x(2−x)f(x)=x(2-x) (see the figure). The value of f ⁣(112)+f ⁣(−294)f\!\left(\tfrac{11}2\right)+f\!\left(-\tfrac{29}4\right) is
Not attempted
xyO121y = f(x), 0 ≤ x ≤ 2
A
−316-\dfrac3{16}
B
2716\dfrac{27}{16}
C
32\dfrac32
D
1516\dfrac{15}{16}
61
For a positive integer nn let fn(x)=cos⁡x+cos⁡ ⁣(n x)f_n(x)=\cos x+\cos\!\big(\sqrt n\,x\big). For how many integers nn with 1≤n≤1001\le n\le100 is fnf_n a periodic function?
Not attempted
Your answer: —
Correct answer: 10
62
List-IList-IIP)f(x)=sin⁡(sin⁡x)1)even and periodicQ)f(x)=⌊x⌋+⌊−x⌋2)odd and periodicR)f(x)={x} (fractional part)3)neither even nor odd, but periodicS)f(x)=cos⁡(x2)4)odd but not periodic5)even but not periodic\begin{array}{ll} \textbf{List-I} & \textbf{List-II} \\ \text{P)} f(x)=\sin(\sin x) & \text{1)} \text{even and periodic} \\ \text{Q)} f(x)=\lfloor x\rfloor+\lfloor -x\rfloor & \text{2)} \text{odd and periodic} \\ \text{R)} f(x)=\{x\}\ \text{(fractional part)} & \text{3)} \text{neither even nor odd, but periodic} \\ \text{S)} f(x)=\cos(x^2) & \text{4)} \text{odd but not periodic} \\ & \text{5)} \text{even but not periodic} \end{array}
Not attempted
A
P-1, Q-2, R-3, S-5
B
P-2, Q-1, R-5, S-3
C
P-2, Q-1, R-3, S-5
D
P-4, Q-3, R-1, S-5
63
Every function gg with a symmetric domain can be written uniquely as g=ge+gog=g_e+g_o with geg_e even and gog_o odd. For g(x)=1x2+x+1g(x)=\dfrac1{x^2+x+1} the value of ge(1)+go(2)g_e(1)+g_o(2) is
Not attempted
A
47\dfrac47
B
1621\dfrac{16}{21}
C
23\dfrac23
D
57\dfrac57
64
For how many integer values of kk does the equation ∣x2−6∣x∣+5∣=k\big|x^2-6|x|+5\big|=k have exactly 88 distinct real solutions?
Not attempted
Your answer: —
Correct answer: 3
65
The graph of y=f(x)y=f(x) is the polygonal line with vertices (−3,2),(−1,−1),(1,1),(3,−2)(-3,2),(-1,-1),(1,1),(3,-2) (see the first figure; ff is defined only on [−3,3][-3,3]). Which of the graphs in the options is the graph of y=f(∣x∣−1)y=f(|x|-1)?
Not attempted
xyO-3-2-1123-2-112
A
Graph shown (curve 2)
xyO-4-224-2-1123
B
Graph shown (curve 1)
xyO-4-224-2-1123
C
Graph shown (curve 3)
xyO-4-224-2-1123
D
Graph shown (curve 4)
xyO-4-224-2-1123
66
Let f(x)=x2−4x+3f(x)=x^2-4x+3. Which of the following statements is/are correct?
Not attempted
A
The graph of y=f(∣x∣)y=f(|x|) meets the xx-axis at exactly 22 points
B
The equation ∣f(x)∣=1|f(x)|=1 has exactly 44 real solutions
C
The graph of y=f(∣x∣)y=f(|x|) has exactly one point of local minimum
D
The graph of y=∣f(∣x∣)∣y=\big|f(|x|)\big| has exactly 77 points of local extremum (maxima and minima together)
67
Find the total area (in square units) of the bounded regions enclosed between the graph of y=∣∣x∣−2∣y=\big||x|-2\big| and the line y=1y=1 (see the figure).
Not attempted
xyO-4-3-2-1123412y = 1
Your answer: —
Correct answer: 3
68
The graph of y=f(x)y=f(x) is transformed in the following order: (1) translated 22 units to the right; (2) reflected in the yy-axis; (3) stretched horizontally by a factor 22 away from the yy-axis (each point moves to twice its distance from the yy-axis, with the same yy-coordinate). The equation of the resulting graph is
Not attempted
A
y=f ⁣(−x2+1)y=f\!\left(-\dfrac x2+1\right)
B
y=f(−2x−2)y=f(-2x-2)
C
y=f ⁣(−x2−2)y=f\!\left(-\dfrac x2-2\right)
D
y=f ⁣(−x2+2)y=f\!\left(-\dfrac x2+2\right)
69
The graph of a function f:R→Rf:\mathbb R\to\mathbb R has the symmetries indicated in each statement. Which of the following conclusions is/are correct?
Not attempted
A
Symmetric about the lines x=1x=1 and x=3x=3 ⇒\Rightarrow ff has period 44
B
Symmetric about the points (1,0)(1,0) and (3,0)(3,0) ⇒\Rightarrow ff has period 44
C
Symmetric about the line x=1x=1 and the point (3,0)(3,0) ⇒\Rightarrow ff has period 88
D
Symmetric about the origin and the line x=ax=a (a≠0)(a\neq0) ⇒\Rightarrow ff has period 2∣a∣2|a|
70
The graph of f:[−3,3]→Rf:[-3,3]\to\mathbb R is the polygonal line with vertices (−3,1),(−2,−1),(0,2),(1,0),(2,−2),(3,1)(-3,1),(-2,-1),(0,2),(1,0),(2,-2),(3,1) (see the figure). The number of real solutions of f(∣f(x)∣)=0f\big(|f(x)|\big)=0 is
Not attempted
xyO-3-2-1123-2-112
A
88
B
66
C
55
D
77
First find all zeros of ff. Then ∣f(x)∣|f(x)| must equal one of those zeros - but only the non-negative zeros are possible values of ∣f(x)∣|f(x)|, and ∣f(x)∣|f(x)| can never exceed the maximum of ∣f∣|f|.
Zeros of ff (linear interpolation on each segment): x=−52x=-\tfrac52 (between (−3,1)(-3,1) and (−2,−1)(-2,-1)), x=−43x=-\tfrac43 (between (−2,−1)(-2,-1) and (0,2)(0,2)), x=1x=1 (a vertex) and x=83x=\tfrac83 (between (2,−2)(2,-2) and (3,1)(3,1)). f(∣f(x)∣)=0f(|f(x)|)=0 requires ∣f(x)∣|f(x)| to be a zero of ff and non-negative: ∣f(x)∣=1|f(x)|=1 or ∣f(x)∣=83|f(x)|=\tfrac83. But ∣f(x)∣≤2|f(x)|\le2 on [−3,3][-3,3], so 83\tfrac83 is impossible. Hence ∣f(x)∣=1|f(x)|=1, i.e. f(x)=1f(x)=1 or f(x)=−1f(x)=-1. f(x)=1f(x)=1: x=−3, −23, 12, 3x=-3,\ -\tfrac23,\ \tfrac12,\ 3 (four solutions). f(x)=−1f(x)=-1: x=−2, 32, 73x=-2,\ \tfrac32,\ \tfrac73 (three solutions). Total 4+3=74+3=7. Missing the endpoint solutions x=±3x=\pm3 or the vertex x=−2x=-2 gives smaller counts, and counting the impossible value 83\tfrac83 gives larger ones.