Chapter-22

FUNCTION

PRACTICE SHEET
1. What is the equivalent definition of the function (c) Both I nor II (d) Neither I nor II
2x, x  0 9. If f: R→, g : R→ and g(x) = x + 3 and (fog) (x) =
given by f (x) =  ? (x+3)2, then what is the value of f(−3)?
 0, x  0 (a) −9 (b) 0
(a) f (x) = |x| (b) f (x) = 2x (c) 9 (d) 3
(c) f (x) = |x| + x (d) f (x) = 2 |x|
10. Consider the function f:R→{0,1} such that:
2. If f : R → R+ such that f(x) = (1/3)x, then what is
1 if x is rational
the value of f−1 (x)? f(x) = 
(a) (1 / 3)
x
(b) 3
x 0 if x is irrational
Which one of the following is correct?
(c) log1/3 x (d) log x (1 / 3) (a) The function is one-one into
(b) The function is many-one into
3. Consider the following statements:
(c) The function is one-one onto
The function f (x) = greatest integer  x, x  R is
(d) The function is many-one onto
a continuous function
1.All trigonometric functions are continuous on Directions (for next three): Each item under
R. List I is associated with one or more items under
2.Which of the statements given above is /are List II.
correct? List I (Function)
(a) 1 only (b) 2 only A. sin x
(c) Both 1 and 2 (d) Neither 1 nor 2 B. cos x
C. tan x
 1+ x   2x  List II (Property)
4. If f (x) = log   , then what is f  
 1− x   1− x2  (i)Periodic function
equal to? (ii)Non-periodic function
(a) (f(x))2 (b) 1 (iii)Continuous at every point on (−∞, ∞)
(iv)Discontinuous function
 1− x  (v)Differentiable at every point on (−∞, ∞)
(c) 2f(x) (d) f  
 1+ x  (vi)Not differentiable at every point on (−∞, ∞)
(vii)Has period 
5. If − x2 + 3x + 4 > 0, then which one of the
(viii)Has period 2
following is correct?
(a) x  (−1, 4)  
(ix)Increases on  0, 
(b) x  [−1, 4]  2
(c) x  (, −1)  (4, )  
(d) x  (−, −1]  [4, ) (x)Decreases on  0, 
 2
6. Let f : R → R be a function defined as f(x) = x
 
|x|; for each x  R, - being the set of real (xi)Increases on  ,  
numbers. Which of the following is correct? 2 
(a) f is onto but not onto  
(b) f is onto but not one − one (xii)Decreases on  ,  
(c) f is both one − one and onto
2 
(d) f is neither one − one nor onto 11. A is associated with
(a) (i), (iii), (v), (viii), (ix), (xxii)
7. A mapping f : R → R which is defined as f (x) = (b) (ii), (iv), (vi), (viii), (x), (xxi)
cos x ; x  R is (c) (i), (iii), (v), (vii), (x), (xxi)
(a) One − one only (d) None of these
(b) Onto only
12. B is associated with
(c) One − one onto (a) (ii), (iii), (v), (viii), (ix), (xxii)
(d) Neither one − one nor onto (b) (i), (iii), (v), (viii), (x), (xxii)
8. Consider the following statements: (c) (i), (iii), (v), (viii), (ix), (xxii)
I. Every function has a primitive (d) None of these
II.A primitive of a function is unique 13. C is associated with
Which of the statements given above is/are (a) (i), (iv), (vi), (vii), (ix), (xxi)
correct? (b) (ii), (iv), (vi), (viii), (ix)
(a) Only I (b) Only II (c) (i), (iv), (vi), (vii), (ix)

SANDEEP SINGH BRAR Ph:- +91 9700900034 - 236 -

 (d) None of these 15. Let f:R→R be a function defined as f(x) = x |x|;
for each xR. R being the set of real numbers.
14. A mapping f:R→R which is defined as f(x) = cos
Which one of the following is correct?
x; xR is:
(a) f is one-one but not onto
(a) Only one-one
(b) f is onto but not one-one
(b) Only onto
(c) f is both one-one and onto
(c) One-one onto
(d) f is neither one-one nor onto
(d) Neither one-one nor onto

ANSWER KEY
1. c 2. c 3. d 4. c 5. a 6. c 7. d 8. c 9. c 10. d
11. a 12. b 13. a 14. d 15. c

Solutions
Sol.1. (c) Therefore, neither (1) nor (2) are true Sol.7. (d)
2x, x  0 Sol.4. (c) Let x1, x2  R
The given function is f (x) =  Given that Then, f (x1) = f (x2)
 0, x  0  cos x1 = cosx2
The equation can be re − written as 1+ x   x1 = 2n  x2
x(x) = log  
 1− x  So, x1  x2
 x + x, x  0 Hence, cos x is not one − one function.
f (x) =   
 0, x  0
2x Now, let y = cos x
1+ 
 2x   + x2  We know, −1,  cos x  1
Hence equivalent definition of given so, f   = log 1
2  2 x   y  [−1, 1]
function is f(x) = |x| + x 1+ x   1− 
 1 + x2  [−1, 1]  R, so, cos x is into function,
Sol.2. (c)
not onto.
 = log  (1 + x ) 
1
x  1 + x 2 + 2x   2
Hence, f (x) = cos x is neither one − one
=log 
Given function is f (x) =  
3
 1 + x 2 − 2x 
   (
 1 − x2  )

nor onto.
Sol.8. (c)
x 2 primitive means preimage of elements of
1 1+ x 
=log 
1+ x 
 = 2 log  
Let f (x) = y, so, y =    1− x   1− x 
range.
3 Sol.9. (c)
Taking log1/3 on both sides 1+ x  g(x) = x + 3 and f(g(x)) = (x + 3)2
=2f [sin cef (x) = log  ]
1  1− x  then it is clear that f(x) = x2
 x  log1/3   =log(1/3) y f(−3) = (−3)2 = 9
3 Sol.5. (a)
Sol.10. (d)
x = log(1/3) y −x2 + 3x + 4 > 0
many one onto
f–1 (x)=log(1/3) x  x2 − 3x − 4 < 0  0 (x − 4) (x + 1) < 0
Sol.11. (a)
Sol.3. (d)  x  (−1, 4) by graph
Here, greatest integer function [x] is Sol.6. (c) Sol.12. (b)
discontinuous at its integral value of x, Given f (x) = x | x |
by graph
cot x and cosec x are discontinuous at If f (x1)= f (x2)
Sol.13. (a)
0, , 2 etc. and tan x and sec x are  x1 | x1 | = x2 | x2|
by graph
 3 5  x1 : x2
Sol.14. (d)
discontinuous at x= , , etc.  f (x) is one − one.
2 2 2 Sol.15. (c)
Also, rango of f(x) = co − domain of f (x)
Therefore the greatest integer function f(x) = x2 when x > 0
 f (x) is onto.
and all trigonometric functions are not Hence, f (x) is both one − one and onto.
continuous for x  R
f(x) = −x2 when x < 0

SANDEEP SINGH BRAR Ph:- +91 9700900034 - 237 -

 NDA PYQ
1. If f(xy) = f(x)f(y), then f(t) may be of the form: 9. Let N denote the set of all non-negative integers
(a) t + k (b) ct + k and Z denote the set of all integers. The function
(c) tk + c (d) tk f:Z→N given by f(x) = |x| is:
[NDA (I) - 2012] (a) One-one but not onto
2. Which of the following statements is correct? (b) Onto but not one-one
(a)ex is an increasing function (c) Both one-one and onto
(b)ex is a decreasing function (c) Neither one-one nor onto
(c)ex is neither increasing nor decreasing function [NDA (I) - 2014]
(d)ex is a constant function 10. Consider the following relations from A to B
[NDA (I) - 2012] where A= {u, v, w, x, y, z} and B = {p, q, r, s}
x +5 1.{(u,p), (v,p), (w,p), (x,q), (y,q), (z,q)}
3. If f:R→ be a function whose inverse is , then 2.{(u,p), (v,q), (w,r), (z,s)}
3 3.{(u,w), (v,r), (w,q), (u,p), (v,q), (z,q)}
what is the value of f(x) ? 4.{(u,q), (v,p), (w,s), (x,r), (y,q), (z,s)}
(a) f(x) = 3x + 5 (b) f(x) = 3x − 5 Which of the above relations are not functions?
(c) f(x) = 5x −3 (d) Does not exist (a) 1 and 2 (b) 1 and 4
[NDA (II) - 2012] (c) 2 and 3 (d) 3 and 4
4. Consider the following statement: [NDA-2014(1)]
1.If f(x) = x3 and g(y) = y3 then f = g 11. Consider the following statements:
2.Identity function is not always a bisection 1.The function f(x)=sin x decreases on the interval
Which of the following statements is/are true (0, /2)
(a) 1 only (b) 2 only 2.The function f(x)=cos x increases on the interval
(c) both 1 and 2 (d) Neither 1 nor 2 (0, /2)
[NDA (II) - 2012] Which of the above statements is/are correct?
5. Let A = {x ∈ R | x ≥ 0} A function f:A→A is defined (a) 1 only (b) 2 only
by f(x) = x2, which of the following is correct ? (c) both 1 and 2 (d) neither 1 nor 2
(a) The function does not have inverse [NDA-2014(1)]
(b) f is its own inverse 12. The function f:N →N, where N being the set of
(c)The function has an inverse but f is not its own natural numbers, defined by f(x) = 2x + 3 is:
inverse (a) Injective and subjective
(d) None of above (b) Injective but not subjective
[NDA (II) - 2012] (c) Not injective but subjective
|x| (d) Neither injective nor subjective
6. What is the range of the function f(x) = ,
x [NDA (II) - 2014]
where x ≠ 0? Direction (for next two): Read the following
(a) Set of all real numbers information carefully and answer the questions
(b) Set of all integers given below:
(c) {−1, 1} x −1
Consider the function f(x) =
(d) {−1, 0, 1} x +1
[NDA (I) - 2013] f (x ) +1
7. Let N be the set of natural numbers and f:N→N 13. What is + x equal to?
be a function given by f(x) = x+1 for xN. Which f ( x ) −1
one of the following is correct? (a) 0 (b) 1
(a) f is one-one and onto (c) 2x (d) 4x
(b) f is one-one but not onto [NDA (II) - 2014]
(c) f is only onto 14. What is f(2x) equal to?
(d) f is neither one-one nor onto f (x) +1 f (x ) +1
[NDA (I) - 2013] (a) (b)
8. If f be a function from the set of natural numbers f (x) + 3 3f ( x ) + 1
to the set of even natural numbers given by f(x) = 3f ( x ) + 1 f (x) + 3
2x. Then, f is: (c) (d)
(a) One-one but not onto f (x) + 3 3f ( x ) + 1
(b) Onto but not one-one [NDA (II) - 2014]
(c) Both one-one and onto 15. What is f[f(x)] equal to?
(d) Neither one-one nor onto (a) x (b) −x
[NDA (II) - 2013] 1
(c) − (d) None of these
x
[NDA (II) - 2014]

SANDEEP SINGH BRAR Ph:- +91 9700900034 - 238 -

16. Let f(x) = ax + b and g(x) = cx + d. Then f (g(x)) = g (b)Both the statements are true and statement II
(f(x) is equivalent to: is not the correct explanation of statement I.
(a) f(c) = g(a) (b) f(a) = g(c) (c)Statement I is true but statement II is false
(c) f(c) = g(d) (d) f(d) = g(b) (d)Statement I is false but statement II is true.
[NDA (II) - 2014] [NDA (II) - 2015]
17. Consider the following functions: 24. If f:R→R, g:R→R be two functions given by f(x) =
I.f(x) = x3, xR 2x −3 and g(x) = x3 + 5, then (fog)−1 (x) is equal to:
II.f(x) = sinx, 0 < x < 2 1
x − 7 3
1

(a)  x + 7  3
(b) 
III.f(x) = ex, xR    
Which of the above functions have inverse defined  2   2 
1
on their ranges?
1

(c)  7 3
(d)  7 3
(a) I and II (b) II and III x −  x + 
 2  2
(c) I and III (d) I, II and III
[NDA (I) - 2015] [NDA (II) - 2015]
2
3x + x 3 x
 1+ x 
18. If f(x) = loge   ,g(x) = and g of (t) = g 25. What is the range of the function y = 1 + x 2 where
 1− x  1 + 3x 2
xR?
 e −1  (a) [0, 1) (b) [0,1]
(f(t)), then what is gof   equal to?
 e +1 (c) (0,1) (d) (0, 1]
(a) 2 (b) 1 [NDA (I) - 2016]
 
(c) 0 (d) 1/2 26. If f(x1) − f(x2) = f  x1 − x 2  for, x1, x2 (−1, 1), then
[NDA (I) - 2015]  1 − x1 x 2 
x what is f(x) equal to?
19. For each non zero real number x, let f (x) = ,
|x|  1− x   2+x 
(a) In   (b) In  
the range of function is  1+ x   1− x 
(a)a null set
 1− x   1+ x 
(b)a set consisting of only one element (c) tan−1   (d) tan−1  
(c)a set consisting of two element  1+ x   1− x 
(d)a set consisting of infinitely many elements [NDA (I) - 2016]
[NDA (I) - 2015] Direction (for next two): Let f(x) be the greatest
integer function and g(x) be the modulus
1 function.
20. If g(x) = and f(x) = x, x ≠0, then which one of
f (x) 27. What is the value of (gof)  − 5  − (fog)  − 5 
   
the following is correct?  3  3
(a) f(f(f(g(g(f(x)))))) = g(g(f(g(f(x))))) (a) – 1 (b) 0
(b) f(f(g(g(g(f(x)))))) = g(g(f(g(f(x)))))
(c) 1 (d) 2
(c) f(g(g(f(g(g(f(x)))))) = g(g(f(g(f(x)))))
[NDA (I) - 2016]
(d) f(f(f(g(g(f(x)))))) = f(f(f(g(f(x)))))
[NDA (II) - 2015] 28. What is the value of (fof)  − 9  + (gog) ( −2 )
 
1  5
21. The domain of the function f(x) = is
| x | −x (a) – 1 (b) 0
(a) [0, ∞) (b) (−∞, 0) (c) 1 (d) 2
(c) [1, ∞) (d) (−∞, 0] [NDA (I) - 2016]
[NDA (II) - 2015] 1
22. f(xy) = f(x) + f(y) is true for all 29. What is the domain of the function f(x) =
(a) Polynomial function
|x| − x
(b) Trigonometric functions ?
(c) Exponential functions (a) (−∞, 0) (b) (0, ∞)
(d) Logarithmic functions (c) 0 < x < 1 (d) x > 1
[NDA (II) - 2015] [NDA (II) - 2016]
23. Consider the following statements. x f (a )
Statement I: The function f:R→R such that f(x) = 30. If f(x) = x − 1 , then what is equal to:
f ( a + 1)
x for all xR is one-one.
3

Statement II: f(a) =f(b) a = b for all a, b R if  a 

(a) f  −  (b) f(a2)
the function f is one-one  a +1
Which one of the following is correct in respect of
the above statements? 1
(c) f   (d) f(−a)
(a)Both the statements are true and statement II a
is the correct explanation of statement I. [NDA (I) - 2017]

SANDEEP SINGH BRAR Ph:- +91 9700900034 - 239 -

 a −1  1
31. Let f(a) = ,Consider the following: (c) 0,  (d) [0, 1]
a +1 2 
1 [NDA (II) - 2017]
1. f(2a) = f(a) + 1 2. f   = − f(a) 39. The inverse of the function y = 5In x is:
a 1
Which of the above is / are correct? (a) x = y In5 , y  0 (b) x = y , y  0
In 5

(a) Only 1 (b) Only 2 1

(c) Both 1 and 2 (d) neither 1 nor 2
(c) x = y In 5 , y  0 (d) x = y Iny, y > 0
[NDA (I) - 2017]
32. The function f : X→Y defined by f(x) = cos x, [NDA (II) - 2017]
where xX, is one-one and onto if X and Y are 40. Which of the following graphs represents the
respectively equal to: x
function f (x) = , x  0
(a) [0, ] and [−1, 1] x
  
(b)  − ,  and [−1,1] y
 2 2
   (a)
(c)  − ,  and [0, 1] 1
 2 2
(d) [0, ] and [0, 1]
[NDA (I) - 2017] x
0
33. Let f(x) = px + q and g(x) = mx + n. Then f (g(x)) =
g (f(x) is equivalent to:
(a) f(p) = g(m) (b) f(q) = g(n) y
(c) f(n) = g(q) (d) f(m) = g(p)
[NDA (I) - 2017] 1
34. Let f:[−6,6]→R be defined by f(x) = x2 − 3.
Consider the following: (b)
I. (fofof) (−1) = (fofof) (1) x
II (fofof) (−1) − 4 (fofof) (1) = (fof) (o) 0
Which of the above is/are correct?
(a) Only I (b) Only II
(c) Both I and II (d) Neither I nor II y
[NDA (I) - 2017]
x , x is rational
35. Let f(x) :  1
0, x is irrational (c)
0, x is rational
and g(x) : 
x , x is irrational x
0
If f : R→R and g:R → R, then (f–g) is:
(a) one-one and into
(b) neither one-one nor onto
(c) many-one and onto (d) None of above
(d) one-one and onto NDA (II) - 2017]
[NDA-2017(1)] 41. Let f(n) =  1 n  where [x] denotes the greatest
 4 + 1000  ,
36. Which one of the following functions is neither  
even nor odd? 1000

(a) x2–1 (b) x + 3/x integer function . Then the value of

n =1

f (n) is
(c) |x| (d) x2(x–3)
[NDA-2017(1)] (a) 251 (b) 250
37. The function f (x) = |x| − x is:
3 (c) 1 (d) 0
(a) Odd NDA (II) - 2017]
(b) Even 4x + x 4 1+ x 
(c) Both even and odd 42. If f(x) = and g(x) = In   , then what is
1 + 4x 3  1− x 
(d) Neither even nor odd
[NDA (II) - 2017]  e −1 
the value of fog   equal to:
x2  e +1
38. If x is any real number, then belongs to
1+ x4 (a) 2 (b) 1
which one of the following intervals? (c) 0 (d) 1/2
[NDA (II) - 2017]
 1
(a) (0, 1) (b)  0, 
 2 

SANDEEP SINGH BRAR Ph:- +91 9700900034 - 240 -

43. Let [x] denote the greatest integer function. What x−2
is the number of solutions of the equation x2 - 4x 53. If (x) = , x  -2 is then f-1(x) equal to?
+ [x] = 0 in the interval [0,2] ?
x+2
(x + 2)
(a) Zero (no solution) (b) One (a) 4(x + 2) (b)
(c) Two (d) Three x−2 4(x − 2)
[NDA (I) - 2018]
44. What is the period of the function f(x) = sin x? (c) (x + 2) (d) 2(x + 1)
x−2 1− x
(a) /4 (b) /2 [NDA (I) - 2019]
(c)  (d) 2 54. For r > 0, f(r) is the ratio of perimeter to area of a
[NDA (I) - 2018] circle of radius r. Then f(1)+f(2) is equal to
45. For f to be a function, what is the domain of f, if (a) 1 (b) 2
1 (c) 3 (d) 4
f(x) = ?
| x | −x [NDA (I) - 2019]
1+ x
55. If f (x) = 3 , then f(x).f(y).f(z) is equal to
(a) (−, 0) (b) (0, )
(c) (−, ) (d) (−, 0] (a) f(x+y+z) (b) f(x+y+z+1)
[NDA (I) - 2018] (c) f(x+y+z+2) (d) f(x+y+z+3)
46. If f : R→S defined by f(x) = 4 sin x −3 cos x + 1 is [NDA (I) - 2019]
onto, then what is S equal to? 56. The domain of the function (2 − x)(x − 3)
(a) [−5,5] (b) (−5,5)
(c) (−4, 6) (d) [−4, 6] (a) (0,∞) (b) [0,∞)
[NDA (I) - 2018] (c) [2,3] (d) (2,3)
[NDA (I) - 2019]
x2
47. Suppose f : R → R is defined by f(x) = What 57. A function f defined by f(x) = ln( x + 1 − x) is
2
1+ x2
is the range of the function? (a) An even function
(a) [0, 1) (b) [0, 1] (b) An odd function
(c) (0, 1] (d) (0, 1) (c) Both even and odd function
[NDA (I) - 2018] (d) Neither even nor odd function
48. Which one of the following is correct in respect of [NDA (I) - 2019]
the function f:R→R+ defined as f(x) = |x+1|? Direction (for next three) :
(a) f(x2) = [f(x)]2 (b) f(|x|) = |f(x)| Read the following information and answer the
(c) f(x+y) = f(x) + f(y) (d) None of the above three items that follow:
[NDA (I) - 2018] Let f(x) = x2 + 2x −5 and g(x) = 5x + 30
49. If f(x) is an even function, where f(x) ≠ 0, then 58. What are the roots of the equation g[f(x)] = 0
which of the following is correct? (a) 1, −1 (b) −1, −1
(a) f'(x) is even function (c) 1, 1 (d) 0, 1
(b) f'(x) is odd function [NDA (II) - 2019]
(c)f'(x) may be an even or odd function depending 59. Consider the following statements:
on the type of function 1.f[g(x)] is a polynomial of degree 3
(d) f'(x) is a constant function 2. g[g(x)] is a polynomial of degree 2
[NDA (I) - 2018] Which of the above is/are correct?
x −1 (a) Only I (b) Only II
50. If f(x) = (c) Both I and II (d) Neither I nor II
x − 4 defines a function of R, then what is
[NDA (II) - 2019]
domain?
60. If h(x) = 5f(x) − xg(x), then what is the derivative
(a) (-∞,4)∪(4,∞) (b) (4,∞)
of h(x)?
(c) (1,4)∪(4,∞) (d) [1,4)∪(4,∞)
(a) −40 (b) −20
[NDA (II) - 2018]
(c) −10 (d) 0
51. A function f: A → R is defined by the equation f(x)
[NDA (II) - 2019]
= x2 − 4x + 5 where A = (1,4). What is the range of
61. Which one of the following is the second degree
the function?
polynomial function f(x) where f(0)= 5, f(−1)= 10
(a) (2, 5) (b) (1, 5)
and f(1) = 6?
(c) [1, 5) (d) [1, 5]
(a) 5x2 −2x + 5 = 0 (b) 3x2 −2x − 5 = 0
[NDA (II) - 2018]
(c) 3x −2x + 5 = 0
2 (d) 3x2 −10x + 5 = 0
52. Let A = (xR: −1  x  1) and S be the subset of A [NDA (II) - 2019]
 B, defined by: S = [(x,y) A  B : x2 + y2 = 1] Direction for next two:
Which one of the following is correct? Let f(x) = x2 , g(x) = tanx and h(x) = lnx.
(a) S is a one-one function from A into B
(b) S is a many one function from A into B 
62. For x = , what is the value of [ho(gof)](x)?
(c) S is a objective mapping form A into B 2
(d) S is not a function (a) 0 (b) 1
[NDA (II) - 2018] (c) /4 (d) /2
[NDA (II) - 2019]

SANDEEP SINGH BRAR Ph:- +91 9700900034 - 241 -

63. What is [fo(fof)](2) = (a) x2 − 5x + 4 (b) x2 − 5x + 6
(a) 2 (b) 8 (c) x2 + 3x + 3 (d) x2 − 3x + 1
(c) 16 (d) 256 [NDA (I) - 2021]
[NDA (II) - 2019] 74. What is the period of the function f(x) = In(2 +
Direction for next three: sin2 x)

Consider the function f(x) = g(x) + h(x) (a) (b) 
2
x  4x  (c)2 (d) 3
where g(x) = sin   and h(x) = cos  
4  5  [NDA (II) - 2021]
64. What is the period of function g(x)? 75. What is the range of the function f(x)=1–sinx
(a) π (b) 2π defined on entire real line?
(c) 4π (d) 8π (a) (0,2) (b) [0,2]
[NDA (II) - 2019] (c) (–1,1) (d) [–1,1]
65. What is the period of function h(x)? [NDA (II) - 2021]
(a) π (b) 4π/5 76. Consider the following statements in respect of
(c) 5π/2 (d) 3π/2 relations and functions:”
[NDA (II) - 2019] 1.all relations are functions but all functions are
66. What is the period of function f(x)? not relation.
(a) 10π (b) 20π 2.A relation from A to B is a subset of Cartesian
(c) 40π (d) 80π product A  B
[NDA (II) - 2019] 3.A relation in A is a subset of Cartesian product
67. If f(x) = 3x2 − 5x + p and f(0) and f(1) are AA
opposite sign , then which of the following is Which of the above statements are correct?
correct ? (a)1 and 2 only (b) 2 and 3 only
(a) −2 < p < 0 (b) −2 < p < 2 (c)1 and 3 only (d) 1, 2 and 3
(c) 0 < p < 2 (d) 3 < p < 5 [NDA (II) 2021]
[NDA 2020] 1  1  1 
77. If 4f(x) –f   =  2x +  2x −  , then what is f(2)
68. What is the domain of the function f(x) = cos-1 x  x  x
(x−2)? equal to?
(a) [-1, 1] (b) [1, 3] (a) 0 (b) 1
(c) [0, 5] (d) [-2, 1] (c) 2 (d) 4
[NDA 2020] [NDA (I) - 2022]
69. If f(x) = 2x −x2, then what is the value of f(x + 2) 78. If f(x) = 4x+3, then what is fofof (–1)equal to?
+ f(x−2) when x = 0? (a) –1 (b) 0
(a) −8 (b) −4 (c) 1 (d) 2
(c) 8 (d) 4 [NDA (I) - 2022]
[NDA 2020] 79. Consider the following in respect of the function
70. Which one of the following is correct in respect f(x)=10x.
of the graph of y = 1 ? 1.Its domain is (–, )
x −1 2.It is a continuous function
(a)The domain is {x∈ R|x≠1} and the range of the 3.It is differentiable at x = 0
set of reals. Which of the above statement are correct?
(b)The domain is {x∈ R|x≠1}, the range is {y∈ (a) 1 and 2 only (b) 2 and 3 only
R|y≠0}, and the graph intersects y-axis at (0,-1) (c) 1 and 3 only (d) 1, 2 and 3
(c)The domain is the set of reals and the range is [NDA (I) - 2022]
singleton set {0}. 80. What is the domain of the function f(x) =
(d)The domain is {x∈ R|x≠1}, the range is the set
1 − (x − 1)2 ?
of points on the y axis.
[NDA 2020] (a) (0, 1) (b) [–1, 1]
71. Consider the following statements: (c) (0, 2) (d) [0, 2]
1. A function f: Z → Z, defined by f(x) = x + 1, is [NDA (I) - 2022]
one as well as onto. 81. Let A = {7,8,9,10,11,12,13,14,15,16} and let
2. A function f: N → N, defined by f(x) = x + 1, is f:A→N be defined by f(x)=the highest prime factor
one but onto. of x. How many elements are there in the range
Which of the above is/are correct? of f?
(a) Only I (b) Only II (a) 4 (b) 5
(c) Both I and II (d) Neither I nor II (c) 6 (d) 7
[NDA 2021 (I)] [NDA 2022 (II)]
72. What is the domain of the function f(x) = 3x 82. Let z=[y] and y =[x] – x, where [.] is the greatest
(a) (−∞,∞) (b) (0,∞) integer function. If x is not an integer but
(c) [0,∞) (d) (−∞,∞) − {0} positive, then what is the value of z?
[NDA (I) - 2021] (a) –1 (b) 0
73. If f(x + 1) = x2 −3x +2, then what is f(x) equal to ? (c) 1 (d) 2

SANDEEP SINGH BRAR Ph:- +91 9700900034 - 242 -

 [NDA 2022 (II)] [NDA – 2023 (1)]
83. If f(x) = 4x + 1 and g(x) = kx + 2 such the fog(x) = 2x + 3
gof(x), then what is the value of k? 92. A mapping f:A→B defined as f(x) = , xA.
3x + 5
(a) 7 (b) 5
(c) 4 (d) 3 If f is to be onto, then what are A and B equal
[NDA 2022 (II)] to?
 5  2
84. If f() = sec 2  − 1 , then what is f ( ) + f () equal (a) A = R\ −  and B = R − 
1 − f ()f ()  3  3
to?  5
(a) f(–) (b) f(+) (b) A = R \and B = R\ − 
(c) f()f() (d) f()  3
[NDA 2022 (II)]  3
(c) A = R\ −  and B = R\(0)
85. If f(x) = ln  x + 1 + x 2 , then which one of the  2
 
following is correct?  5
(d) A = R\ −  and B = R\  
2
(a) f(x)+f(–x) = 0 (b) f(x)–f(–x)=0
 3 3
(c) 2f(x) = f(–x) (d) f(x) = 2f(–x)
[NDA 2022 (II)] [NDA – 2023 (1)]
86. Consider the following statements: 93. Consider the following statements:
1.The relation f defined by
1.If f is the subset of X  Z defined by f =
{(xy, x–y); x, y Z}, then f is a function from Z to 
x 3 , 0  x  2
f(x)=  is a function
Z. 
4x, 2  x  8
2.If f is the subset of N  N defined by f = {(xy, 2.The relation g defined by
x+y); x, y N}, then f is a function from N to N. x 2 , 0  x  4
Which of the statements given above is/are g(x)=  is a function
correct? 3x, 4  x  8
(a)1 only (b) 2 only which of the statements given above is/are
(c) Both 1 and 2 (d) Neither 1 nor 2 correct?
[NDA 2022 (II)] (a) 1 only (b) 2 only
87. If f(x) = x2 + 2 and g(x) = 2x – 3, then what is (c) both 1 and 2 (d) neither 1 nor 2
(fg)(1) equal to? [NDA-2023 (2)]
(a) 3 (b) 1 f (x)
(c) –2 (d) –3 94. A function satisfies f(x–y)= , where f(y)0. If
f ( y)
[NDA – 2023 (1)]
f(1)=0.5, then what is f(2) + f(3) +f(4) + f(5) + f(6)
88. What is the range of the function f(x) = x + |x| if
equal to?
the domain is the set of real numbers?
(a) 15/32 (b) 17/32
(a)(0, ) (b) [0, )
(c) 29/64 (d) 31/64
(c)(–, ) (d) [1, ) [NDA-2023 (2)]
[NDA – 2023 (1)] Consider the following for the next (02) items
89. If f(x) = x(4x2–3), then what is f(sin) equal to? that follow
(a) –sin3 (b) – cos3 Let f(x) = x2–1 and gof(x)=x–x+1
(c) sin3 (d) –sin4 95. Which one of the following is a possible
[NDA – 2023 (1)] expression for g(x)?
Consider the following for the next two (02)
(a) x + 1 − 4 x + 1 (b) x + 1 − 4 x + 1 + 1
items that follow:
Let f(x) = sin [2] x + cos [–2]x where [.] is a (c) x +1 + 4 x +1 (d) x+1– x + 1 + 1
greatest integer function. [NDA-2023 (2)]
96. What is g(15) equal to?

90. What is f   equal to? (a) 1 (b) 2
2 (c) 3 (d) 4
(a) –1 (b) 0 [NDA-2023 (2)]
(c) 1 (d) 2 Consider the following for the next (02) items
[NDA – 2023 (1)] that follow
 Let a function f be defined on R–(0) and
91. What is f   equal to? 2f(x)+f(1/x)= x+3.
4 97. What is f(0.5) equal to?
1 (a) 1/2 (b) 2/3
(a) – (b) –1 (c) 1 (d) 2
2 [NDA-2023 (2)]
1 98. If f is differentiable, then what is f’(0.5) equal to?
(c) 1 (d) (a) 1/4 (b) 2/3
2
(c) 2 (d) 4

SANDEEP SINGH BRAR Ph:- +91 9700900034 - 243 -

 [NDA-2023 (2)] (c) /2
Consider the following for the next (02) items (d) the function is non-periodic
that follow [NDA-2023 (2)]
A function is defined by f(x)=+sin2x 101. What is the domain of the function f(x) =
99. What is the range of the function? 2−x + 2+x ?
(a) [0,1] (b) [,+1] (a) (–2,2) (b) [–2,2]
(c) [–1, +1] (d) [–1, –1] (c) R–(–2,2) (d) R–[–2,2]
[NDA-2023 (2)] [NDA-2023 (2)]
100. What is the period of the function?
(a) 2
(b) 

ANSWER KEY

1. d 2. a 3. b 4. c 5. c 6. c 7. b 8. c 9. b 10. c
11. d 12. b 13. a 14. c 15. c 16. d 17. c 18. b 19. c 20. b

21. b 22 d 23. a 24. b 25. a 26. a 27. c 28. b 29. a 30. b

31. b 32. a 33. c 34. c 35. a 36. d 37. d 38. c 39. b 40. c

41. a 42. b 43. b 44. d 45. d 46. d 47. a 48. d 49. b 50. d

51. b 52. d 53. d 54. c 55. c 56. c 57. b 58. b 59. d 60. b

61. c 62. a 63. d 64. d 65. c 66. c 67. c 68. b 69. a 70. b

71. c 72. a 73. b 74. c 75. b 76. b 77. d 78. a 79. d 80. d

81. b 82. a 83. a 84. b 85. a 86. d 87. a 88. b 89. a 90. b

91. d 92. d 93. a 94. d 95. b 96. c 97. b 98. c 99. b 100. b

101. b

SANDEEP SINGH BRAR Ph:- +91 9700900034 - 244 -