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1) Let A and B be two finite sets having m and n elements, respectively. then the total number of mappings from A to B is..
A) mn B) 2ᵐⁿ C) ⁿ D) nᵐ
2) in the above question if A is the empty set, then the total number of mappings from A to B is
A) m B) n C) 1 D) none
3) then total number of injective mappings from a set with m elements to a set with n elements, m≤ n, is
A) mⁿ B) nᵐ C) n!/(n-m)! D) n!
4) The total number of injective mappings from finite set with element to a set with n elements for m > n is...
A) n!/(n-m)! B) m!/(m-n)!
C) nᵐ. D) none
5) which of the four statements given below, is different from the other ?
A) f: A --> B B) f: x--> f(x)
C) f is mapping from A to B
D) f is a function from A to B
6) If f(x)=cos(log x), then f(x) f(y) - 1/2[f(x/y)+ f(xy) is equal to
A) 0 B) 1/2f(x)f(y) C) f(x+y) D) n
7) If A= {x : π/6≤ x≤π/3} and f(x)= cos x - x(1+x), then f(A) is
A) (π/6,π/3) B)(-π/3, -π/6)
C) [1/2-π/3(1+π/3),√3/2-π/6 (1+π/6)}
D) [1/2+π/3(1-π/3),√3/2+π/6(1-π/6)}
8) let f be an injective map with domain {x, y, z} and range {1,2,3} such that exactly one of the following statements is correct and the remaining are false
f(x)= 1, f(y)≠ 2, f(z)≠ 2. The value of f⁻(1) is
A) x B) y C) z D) none
9) f(x)= (a-xⁿ)¹⁾ⁿ where a> 0 and n belongs to N, then fof (x) is
A) a B) x C) xⁿ D) aⁿ
10) let f(x) be defined on [-2,2] and is given by
f(x)= -1, -2≤ x ≤0
x-1, 0< x ≤ 2 and g(x)= f(|x|) + |f(x)|. Then g(x) is equal to
A) - x, -2 ≤x <0 B) -x, -2≤ x < 0
0, 0 ≤x< 1 0, 0≤x<1
x-1, 1≤ x ≤2 2(x-1), 1≤x≤2
C) -x, -2≤ x<0
x-1, 0≤x≤2 D) none
11) The image of the interval [-1,3] under the mapping f: R --> R given by f(x)= 4x³ - 12x is
A) [8,72] B) [-8,72]
C) [0,8] D) none
12) If A= {1,2,3,4}, then which of the following are functions from A to itself ?
A) f₁ ={(x,y) : y= x+1}
B) f₂={(x,y): x+y > 4}
C) f₃ = {(x,y): y< x}
D) f₄={(x,y): x+ y= 5}
13) which of the following function from Z to itself are bijections?
A) f(x)=x³ B) f(x)=x+ 2
C) f(x)= 2x+1 D) f(x)= x²+ x
14) which of the following functions from A={x: -1≤ x ≤ 1} to itself are bijections?
A) f(x)= x/2 B) g(x)=sin(πx/2)
C) h(x)= |x| D) k(x)= x²
15) If f: R--> R be mapping defined by f(x)=x³+ 5, then f⁻¹(x) is equal to
A) (x+5)¹⁾³ B) (x-5)¹⁾³
C) (5-x)¹⁾³ D) 5 - x
16) If f: A--> B and g: B--> C be bijections, then (fog)⁻¹=
A) f⁻¹o g⁻¹ B) fog
C) g⁻¹o f⁻¹ . D) gof
17) Let f: R--> R and g: R--> R be two function, given by f(x)=2x-3, g(x)= x³+5. then (fog)⁻¹(x) is equal to
A) {(x+7)/2}¹⁾³ B) (x-7)3)¹⁾³
C) {(x-2)/7)}¹⁾³ D) (x-7)/2)}¹⁾³
18) let f: R--> R be a function defined by f(x)= cos(5x+2). Then f is
A) injective B) surjective
C) bijective D) none of these
19) If f: A--> B is a bijection and g: B--> A is the inverse of f, then fog is equal to
A) Iᴬ B) Iᴮ C) f D) g
20) Let A={x | -1≤ x ≤ 1} and f: A--A such that f(x)= x|x|, then f is
A) bijection
B) Injective but not surjective
C) surjective but not injective
D) neither Injective nor surjective
21) If f(x)=(3x+2)/(5x-3), then
A) f⁻¹(x)= f(x)
B) f⁻¹(x)=- f(x)
C) (fof)(x)= -x
D) f⁻¹(x)= -1/19 f(x)
22) If f(x)= 2ˣ, then f(0), f(1), f(2), ...are in
A) AP B) GP C) HP D) arbitrary
23) If the function f: R--> A given by f(x)= x²/(x²+1) is a surjection, then A=
A) R B)[0,1] C) (0,1] D) [0,1]
24) Which of the following function is inverse of itself
A) f(x)= (1-x)/(1+x)
B) f(x)= 5ˡᵒᵍ ˣ
C) f(x)= 2ˣ⁽ˣ⁻¹⁾. D) none
25) If f(x)= (x-1)/(1+x), then f(2x) is
A) (f(x)+1)/(f(x)+3)
B) (3f(x)+1)/(f(x)+3)
C) (f(x)+3)/(f(x)+1)
D) (f(x)+3)/(3f(x)+1)
26) if f(x)= log{(1+x)/(1-x)} and g(x)= (3x+ x³)/(1+3x²), then fog(x) equals
A) - f(x) B) 3f(x) C) [f(x)]³ D) n
27) If f(x)= aˣ, which of the following equalities do not hold?
A) f(x+2)- 2 f(x+1)+f(x)= (a-1)²f(x)
B) f(-x) f(x)- 1= 0
C) f(x+y)= f(x)f(y)
D) f(x+3) - 2f(x +2)+f(x+1) = (a-2)² f(x+1)
28) The interval in which the function y=(x-1)/(x²-3x+3) transforms the real line is.
A) (0,∞) B)(-∞,∞) C)[0,1] D) [-1/3,1]
29) Let f(x)= |x - 1|.then
A) f(x²)= [f(x)]² C) f(|x|)=|f(x)|
C) f(x+y)=f(x)+ f(y) D) none
30) If f(x)=ax + b and g(x)= cx+ d, then f(g(x)) =g(f(x)) is equivalent to
A) f(x)= g(x) B) f(b)= g(b)
C) f(d)= g(b) D) f(c)= g(a)
31) which of the following function(s) are injective map(s)
A) f(x)= |x+1|, x belongs to[-1,∞)
B) g(x) = x + 1/x, x belongs to (0,∞)
C) h(x) = x² + 4x -5, x belongs to (0,∞)
D) k(x) = 1/eˣ, x belongs to [0,∞)
32) If f(x) is defined in [0,1] by the rule
f(x)= x, if is rational
1- x, if x is irrational. then for constant for all x belongs to [0,1], f(f(x)) is
A) constant B) 1+x C) x D) none
33) Let f(x)= x and g(x)=| x| for all x belongs to R. then the function €(x) satisfying [€(x) -f(x)]²+ [€(x) - g(x)]²= 0 is
A) €(x)=x, x belongs to [0,∞)
B) €(x)=x, x belongs to R
C) €(x)= - x, x belongs to (-∞,0]
D) €(x)=x + |x|, x belongs to R
34) let f(x)={(ax+b)/(cx+d)}, then fof (x)= x provided that
A) d= - a B) d= a
C) a=b=c=d=1 D) a=b=1
35) if f(x)=(ax²+b)³, then the function g such that f(g(x))= g(f(x)) is given by
A) g(x)= √(b- ³√x)/a}
B) g(x)= 1/(ax²+ b)³
C) g(x)= ³√(ax²+b)
D) √{(³√x - b)/a}
36) If a function f: [2,∞)--> defined by f(x)= x²- 4x+ 5 is a bijection, then B=
A) R B) [2,∞) C) [4,∞) D)[5,∞)
37) The function f: R--> R defined by f(x)=(x-1((x-2)(x-3) is
A) one-one but not into onto
B) onto but not one-one
C) both one-one and onto
D) neither one-one nor onto
38) Let A={x,y,z}, B={u,v,w} and f: A-->B be defined by f(x)=u, f(y)= v, f(z)= w. Then f is
A) surjective but not injective
B) Injective but not surjective
C) bijective D) none
39) If f: R --> R, defined by f(x)= x²+1, then the values of f⁻¹(17) and f⁻¹(-3) respectively are
A) ¢,{4,-4}. B){3,-3},¢
C) {4,-4},¢ D) {4,-4},{2,-2}
40) the function f: N--> N(N is a set of natural numbers) defined by f(n)= 2n+3, is
A) surjective B) injective
C) bijective D) none
41) the composite mapping of fog of the maps f: R --> R, f(x)= sinx and g: R--> R, g(x)= x², is
A) x² sinx B) (sinx)²
C) sin x². D) (sinx)/x²
42) let f: R--> R be defined by f(x)=3x-4. then f⁻¹(x) is
A) (x+4)/3. B) x/3 - 4
C) 3x+4 D) none
43) the number of surjections from A={1, 2,....n}, n≥ 2 onto B={a,b} is
A) ⁿP₂ B) 2ⁿ-2 C) 2ⁿ -1 D) none
44) set A has three elements and set B has 4 elements. the number of injection that can be defined from A to B is
A) 144 B) 12 C) 24 D) 64
45)f: R--> R is a function defined f(x)= 10x - 7. If g= f⁻¹ then g(x)=
A) 1/(10x -7)
B) 1/(10x +7)
C) (x +7)/10 D) (x -7)/10
46) the number of bijective functions from set A to itself when A contains 106 elements is
A) 106 B) 106² C) 106! D) 2¹⁰⁶
47) f(x)= |sin x| has an inverse if its domain is
A) [0,π] B) [0, π/2] C) [-π/4,π/4] D) n
48) let A be a set of containing 10 distinct elements, then the total number of distinct functions from A to A is
A) 10! B) 10¹⁰ C) 2¹⁰ D) 2¹⁰ - 1
49) let f(x)= (x+1)² - 1, (x≥-1). then the the set S= {x: f(x)= f⁻¹(x)} is
A) {0,-1, (-3+i√3)/2, -3-i√3))2)}
B) {0,1,-1}
C) {0,-1} D) empty
50) The function f: [-1/2, 1/2] --> [-π,π/2] define by f(x)=sin⁻¹ (3x- 4x³) is
A) bijection
B) injection but not surjection
C) surjection but not an injection
D) neither an injection nor a surjection
51) Let f: R --> R be a functions defined by f(x)= (ₑ|x| - ₑ-x)/(ₑx + ₑ-x) then.
A) f is a bijection
B) f is an injection only
C) f is a subject only
D) f is neither an injection nor a surjection.
52) let f: (e, ∞) --> R be defined by f(x)= log [log(log x)], then
A) f is one one but not onto
B) f is onto but not one-one
C) f is both one one and onto
D) the range of f is equal to its co-domain
53) let f: R -{n} --> R be a function defined by
f(x)= (x-m)/(x-n), where m ≠ n, then.
A) f is one-one onto
B) f is one-one into
C) f is many one onto
D) f is many one into
EXERCISE --2
1) Let A= {1,2,3}, B={2,3,4}, then which of the following is a function from A to B ?
A) {(1,2),(1,3),(2,3),(3,3)}
B) {(1,3),(2,4){
C) {(1,3),(2,2),(3,3)}
D) {(1,2),(2,3),(3,2),(3,4)}
2) if f: Q--> Q is defined as f(x)= x², then f⁻¹(9) is equal to
A) 3 B)-3 C) {-3,3} D) ¢
3) If f: R--> R is given by f(x)= x³ + 3, then f⁻¹(x) is equal to
A) ³√x - 3 B) ³√x + 3
C) ³√(x - 3) D) x + ³√3
4) The domain and range are same for
A) a constant function
B) identity function
C) an injective map
D) a surjective map
5) Let f(x)= x³ be a function with domain {0,1,2,3}. Then domain of f⁻¹is..
A) {3,2,1,0} B) {0,-1,-2,-3}
C) {0,1,8,27} D) {0,-1,-8,-27}
6) which one of the following is not a function ?
A) {(x,y): x,y∈ R, x²= y}
B) {(x,y): x, y∈ R, y²= x}
C) {(x,y): x,y∈ R, x= y³}
D) {(x,y): x, y∈ R, x³= y}
7) let f: R--> R be such that f(x)= x³ + 3, then f is
A) a bijection B) an injection
C) a surjection D) none
8) let A={x: -1≤ x ≤ 1} and f: A--> A, f(x)= sin πx, , then f is
A) one-one onto
B) one-one into
C) many one onto
D) many one into
9) let f: R--> R be such that f(x)= x² - 3, then f⁻¹ is given by
A) √(x+3) B) √x + 3
C) x+ √3 D) none
10) let A= {x,y)| y= log x, x>0, a>1} and B={(x,y) : y= logₐ x, 0< a<1, x> 0}. then A ∩ B is
A) R x R B) {(1,0)} C) {(0,1)} D) n
11) let f(x) be bijection such that the curve y= f(x) passes through (2,-3). Then the curve y=f⁻¹(x) passes through
A) (2,-3) B) (-3,2) C) (-2,3) D)(2,3)
12) If f(x)= cos (log x), then
f(x²)f(y²)- 1/2 [ f(x²/y²) + f(x²y²)] has the value
A) -2 B) -1 C) 1/2 D) none
13) if f(x)= cos (log x), then
f(x)f(y)- 1/2 [ f(x/y) + f(xy)] has the value
A) -1 B) 1/2 C) -2 D) none
14) Let f(x)= |x -1|, then
A) f(x²)=[ f(x)]² B) f(x+y)=f(x)f(y)
C) f(|x)|=|f(x)| D) none
15) the range of f(x)=cos [x], for _ -π/2< x<π/2 is
A) {-1,1,0} B) {cos 1, cos 2, 1}
C) { cos 1, - cos 1, 1 D) [-1,1]
16) Let f: R--> R be a function defined by f(x)=|x| for all x belongs to R, and let A={0,1}. then f⁻¹A is
A) (-1,1) B)[0,1) C) (-1,0) D) none
17) which of the following function is not invertible ?
A) f: R --> R, f(x)= 4x+5
B) f: R --> R⁺ U {0}, f(x)= x²
C) f: R⁺--> R⁺, f(x)= 1/x² D) none
18) Which of the following functions are bijections?
A) f: R --> R, f(x)= sinx
B) f: R⁺ --> R, f(x)= 2√x + 1
C) f: [0,π]-->[-1,1], f(x)= cosx D) n
19) which of the following functions?
A) {(x,y): y²= x, x, y belongs to R}
B) {(x,y): y=| x| , x, y belongs to R}
C) {(x,y): x²+y²=1,x,y belongs to R}
A) {(x,y): x²-y²=1,x,y belongs to R}
20) Let f be a function with domain X and range Y. Let A, B is the subset or equal to Y and C, D is the subset or equal to X. Which of the following is not true?
A) f⁻¹(A U B)= f⁻¹(A)U f⁻¹(B)
B) f⁻¹(A ∩ B)= f⁻¹(A) ∩f⁻¹(B)
C) f(C U D)= f(C) U f(D)
D) f(C ∩ D)= f(C) ∩ f(D)
ANSWERS:
1. C 2. C 3.C 4.D
5. C. 6. B 7.A. 8. C
9. D 10. B 11.B 12. D
13. D 14. D 15.B 16. D
17. B. 18. C 19. B 20. C
EXERCISE- 3
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