Relations & Functions Practice Questions - DPP for JEE

 Page 1


PART-I (Single Correct MCQs)
1. The domain of the function  +  is
(a) [2, 3]
(b) [–2, 4]
(c) [–2, 2] ? [3, 4]
(d) [–2, 1] ? [2, 4]
2. If 3f(x) – f  = log x
4
, then f(e
–x
) is
(a) 1+ x
(b) 1/x
(c) x
(d) – x
3. The domain of the function  is
(a) (0, )
Page 2


PART-I (Single Correct MCQs)
1. The domain of the function  +  is
(a) [2, 3]
(b) [–2, 4]
(c) [–2, 2] ? [3, 4]
(d) [–2, 1] ? [2, 4]
2. If 3f(x) – f  = log x
4
, then f(e
–x
) is
(a) 1+ x
(b) 1/x
(c) x
(d) – x
3. The domain of the function  is
(a) (0, )
(b) (– , 0)
(c) (– , ) – {0}
(d) (– , )
4. f(x) =  +  is real for all x in
(a) [–4, –3]
(b) [–3, –2]
(c) [–2, 2]
(d) [3, 4]
5. What is the value off (p) + f (q) ?
(a) f (p – q)
(b) f (p + q)
(c) f (p (p + q))
(d) f (q (p – q))
6. A real valued function f (x) satisfies the functional equation
f (x – y) = f (x) f (y) – f (a – x) f (a + y)
where a is a given  constant and f (0) = 1, f (2a – x) is equal to
(a) – f (x)
(b) f (x)
(c) f (a) + f (a – x)
(d) f (– x)
7. Domain of definition of the function
 , is
(a) (b)  
(c) (d)  .
8. A relation R is defined in the set Z of integers as follows (x, y) ? R iff
x
2
 + y
2
 = 9. Which of the following is false?
(a) R = {(0, 3), (0, –3), (3, 0), (–3, 0)}
Page 3


PART-I (Single Correct MCQs)
1. The domain of the function  +  is
(a) [2, 3]
(b) [–2, 4]
(c) [–2, 2] ? [3, 4]
(d) [–2, 1] ? [2, 4]
2. If 3f(x) – f  = log x
4
, then f(e
–x
) is
(a) 1+ x
(b) 1/x
(c) x
(d) – x
3. The domain of the function  is
(a) (0, )
(b) (– , 0)
(c) (– , ) – {0}
(d) (– , )
4. f(x) =  +  is real for all x in
(a) [–4, –3]
(b) [–3, –2]
(c) [–2, 2]
(d) [3, 4]
5. What is the value off (p) + f (q) ?
(a) f (p – q)
(b) f (p + q)
(c) f (p (p + q))
(d) f (q (p – q))
6. A real valued function f (x) satisfies the functional equation
f (x – y) = f (x) f (y) – f (a – x) f (a + y)
where a is a given  constant and f (0) = 1, f (2a – x) is equal to
(a) – f (x)
(b) f (x)
(c) f (a) + f (a – x)
(d) f (– x)
7. Domain of definition of the function
 , is
(a) (b)  
(c) (d)  .
8. A relation R is defined in the set Z of integers as follows (x, y) ? R iff
x
2
 + y
2
 = 9. Which of the following is false?
(a) R = {(0, 3), (0, –3), (3, 0), (–3, 0)}
(b) Domain of R = {–3, 0, 3}
(c) Range of R = {–3, 0, 3}
(d) None of these
9. Let f (x) = , then
(a) f (xy) = f (x) . f (y)
(b) f (xy) >  f (x) . f ( y)
(c) f (xy) < f (x) . f (y)
(d) None of these
10. The domain of the function f (x) =  is
(a)
(b) [–1, 1]
(c)
(d)
11. The function , is
(a) neither an even  nor an odd function
(b) an even function
(c) an odd function
(d) a periodic function
12. The domain of the function
 is
(a) (0, 1)
(b) (0, 1]
(c) [1, 8)
Page 4


PART-I (Single Correct MCQs)
1. The domain of the function  +  is
(a) [2, 3]
(b) [–2, 4]
(c) [–2, 2] ? [3, 4]
(d) [–2, 1] ? [2, 4]
2. If 3f(x) – f  = log x
4
, then f(e
–x
) is
(a) 1+ x
(b) 1/x
(c) x
(d) – x
3. The domain of the function  is
(a) (0, )
(b) (– , 0)
(c) (– , ) – {0}
(d) (– , )
4. f(x) =  +  is real for all x in
(a) [–4, –3]
(b) [–3, –2]
(c) [–2, 2]
(d) [3, 4]
5. What is the value off (p) + f (q) ?
(a) f (p – q)
(b) f (p + q)
(c) f (p (p + q))
(d) f (q (p – q))
6. A real valued function f (x) satisfies the functional equation
f (x – y) = f (x) f (y) – f (a – x) f (a + y)
where a is a given  constant and f (0) = 1, f (2a – x) is equal to
(a) – f (x)
(b) f (x)
(c) f (a) + f (a – x)
(d) f (– x)
7. Domain of definition of the function
 , is
(a) (b)  
(c) (d)  .
8. A relation R is defined in the set Z of integers as follows (x, y) ? R iff
x
2
 + y
2
 = 9. Which of the following is false?
(a) R = {(0, 3), (0, –3), (3, 0), (–3, 0)}
(b) Domain of R = {–3, 0, 3}
(c) Range of R = {–3, 0, 3}
(d) None of these
9. Let f (x) = , then
(a) f (xy) = f (x) . f (y)
(b) f (xy) >  f (x) . f ( y)
(c) f (xy) < f (x) . f (y)
(d) None of these
10. The domain of the function f (x) =  is
(a)
(b) [–1, 1]
(c)
(d)
11. The function , is
(a) neither an even  nor an odd function
(b) an even function
(c) an odd function
(d) a periodic function
12. The domain of the function
 is
(a) (0, 1)
(b) (0, 1]
(c) [1, 8)
(d) (1, 8)
13. The domain of the function f (x) = is
(a) ( – 8, 1)
(b) ( – 8, 1) ? (2, 8)
(c) ( – 8, 1] ? [2, 8)
(d) (2, 8)
14. If f(x) = ln  then range of f(x) is
(a) (0, 1)
(b) (0, 1]
(c) [0, 1)
(d) {0, 1}
15. The function f (x) =  satisfies the equation
(a) f (x + 2) – 2f (x + 1) + f (x) = 0
(b) f (x + 1) + f (x) = f (x (x + 1))
(c) f (x
1
) · f (x
2
) = f (x
1
 + x
2
)
(d) f (x
1
) + f (x
2
) = 
16. If { } denotes the fractional part of x, the range of the function f (x) = 
 is
(a) f
(b) [0, 1/2]
(c) {0, 1/2}
(d) {0}
17. If f (x) =  then f (2x) is equal to
(a)
Page 5


PART-I (Single Correct MCQs)
1. The domain of the function  +  is
(a) [2, 3]
(b) [–2, 4]
(c) [–2, 2] ? [3, 4]
(d) [–2, 1] ? [2, 4]
2. If 3f(x) – f  = log x
4
, then f(e
–x
) is
(a) 1+ x
(b) 1/x
(c) x
(d) – x
3. The domain of the function  is
(a) (0, )
(b) (– , 0)
(c) (– , ) – {0}
(d) (– , )
4. f(x) =  +  is real for all x in
(a) [–4, –3]
(b) [–3, –2]
(c) [–2, 2]
(d) [3, 4]
5. What is the value off (p) + f (q) ?
(a) f (p – q)
(b) f (p + q)
(c) f (p (p + q))
(d) f (q (p – q))
6. A real valued function f (x) satisfies the functional equation
f (x – y) = f (x) f (y) – f (a – x) f (a + y)
where a is a given  constant and f (0) = 1, f (2a – x) is equal to
(a) – f (x)
(b) f (x)
(c) f (a) + f (a – x)
(d) f (– x)
7. Domain of definition of the function
 , is
(a) (b)  
(c) (d)  .
8. A relation R is defined in the set Z of integers as follows (x, y) ? R iff
x
2
 + y
2
 = 9. Which of the following is false?
(a) R = {(0, 3), (0, –3), (3, 0), (–3, 0)}
(b) Domain of R = {–3, 0, 3}
(c) Range of R = {–3, 0, 3}
(d) None of these
9. Let f (x) = , then
(a) f (xy) = f (x) . f (y)
(b) f (xy) >  f (x) . f ( y)
(c) f (xy) < f (x) . f (y)
(d) None of these
10. The domain of the function f (x) =  is
(a)
(b) [–1, 1]
(c)
(d)
11. The function , is
(a) neither an even  nor an odd function
(b) an even function
(c) an odd function
(d) a periodic function
12. The domain of the function
 is
(a) (0, 1)
(b) (0, 1]
(c) [1, 8)
(d) (1, 8)
13. The domain of the function f (x) = is
(a) ( – 8, 1)
(b) ( – 8, 1) ? (2, 8)
(c) ( – 8, 1] ? [2, 8)
(d) (2, 8)
14. If f(x) = ln  then range of f(x) is
(a) (0, 1)
(b) (0, 1]
(c) [0, 1)
(d) {0, 1}
15. The function f (x) =  satisfies the equation
(a) f (x + 2) – 2f (x + 1) + f (x) = 0
(b) f (x + 1) + f (x) = f (x (x + 1))
(c) f (x
1
) · f (x
2
) = f (x
1
 + x
2
)
(d) f (x
1
) + f (x
2
) = 
16. If { } denotes the fractional part of x, the range of the function f (x) = 
 is
(a) f
(b) [0, 1/2]
(c) {0, 1/2}
(d) {0}
17. If f (x) =  then f (2x) is equal to
(a)
(b)
(c)
(d)
18. The domain of the function is
(a) [3/2, 8)
(b) [1, 3/2]
(c) (–8, 1]
(d) (1, 3/2)
19. If f(x + y) = f (x) + 2y
2
 + kxy and f(a) = 2, f(2) = 8, then f(x) is of the
form
(a) 2x
2
(b) 2x
2
 + 1
(c) 2x
2
 – 1
(d) x
2
20. Which of the following relation is NOT a function
(a) f = {(x, x) | x ? R}
(b) g = {(x, 3) | x ? R}
(c) h = { | n ? I}
(d) t = {(n, n
2
) | n ? N}
PART-II (Numeric/Integer Type Questions)
21. Let A = {1, 2, 3, 4, 5}; B = {2, 3, 6, 7}. Then the number of elements in
(A × B) n (B × A) is
22. If n(A) = 4, n(B) = 3, n(A × B × C) = 24, then n(C) =
23. Let f (x) =  and ‘a’ be a real number.  If x
0
 = a,