300 Most Important Algebra Questions for CAT
Functions
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Q2. The number of integers ‘n’ that satisfy the inequalities |n – 33| < |n – 50| < |n – 13| is:
10
11
12
13
Q3. A real valued function f(x) is such that f(x + y) = f(x) + f(y) + 6xy + 3 for all real values of x and y. Find the value of f(3) if f(–1) = 6.
0
-6
6
-3
16
21
19
25
Q6. Let |m – 3| + |n – 4| = 6, where m, n are one digit whole numbers. What is the maximum value of m × n.
Q7. Let |x + y| + |x – y| = 4, then what is the maximum possible value of x2 – 8x + y2 – 3y?
Q9. If f(x) = x2 – 10x and g(x) = 2x + 5, where x is real, then the minimum value of f(g(x)) – 4x is:
-34
-30
-26
-24
Q10. For any real number ‘x’, the condition, |7x – 70| + |7x – 140| ≥ k, necessarily holds true. The maximum possible value of k is:
60
50
75
70
32/21
21/12
12/21
21/32
Q12. f(x) = x2 + px + q and f(x) ≥ 0, for all real values of x. If F(x) = f(2x + 3) – f(2x – 3) and F(10) = 204, then the smallest possible value of q is:
2
9
0
1
Q13. The function f(x) = |x – 1| + |2.8 – x| + |x – 3|, where x is a real number, can attain a minimum value of:
1
2
3
2.5
Q14. If 4|3x + 1| = 162x – 4, find the sum of all the possible values of x.
8
9
10
11
Q15. If f (x) + f (y) = f (x + y), where f (t) > 0 for any t > 0, find the value off f(1) + f (3) + f (5) + f (7) +…. f(19), given f (10) = 1/8.
4/5
3/7
5/4
8/3
Q16. How many integral values of x satisfy the equation x = |2x – |120 – 3x||?
1
2
4
3
Q17. Find the value of (x – y), if |x + 2017| – |x – 2017| = 2 and |y – 2017| – |y + 2017| = 2
0
2
3
Cannot be determined
Q18. For all integers x, f(f(x)) = f(x + 2) – 3. If f(1) = 4 and f(4) = 3 then find the value of f(5).
10
11
14
12
Q19. Let f(x) + f(2x) + f(1 + x) + f(2 – x) = x for all x. What is the value of 16 × f(0)?
-6
-8
-4
0
Q20. What is the area of the closed region bounded by the equation |x| + |y| = 8?
32
60
64
128
Q21. If |x – 1| – |x| + |2x + 3| ≥ 2x + 4, where x ≥ –10, then the possible integer values of x is:
11
12
14
10
Q22. If f(x) = |x – 4| and g(x) = x2, find the number of values of x for which f(g(x)) = g(f(x)).
0
1
2
3
Q23. Find the solution set for [x] + [2x] + [3x] + [4x] = 14, where x is a real number and [x] is the greatest integer less than or equal to x.
x < 5/3
3/2 ≤ x < 5/3
1 ≤ x < 4/3
1/2 ≤ x < 2
Q25. If y = |x – 2.5| + |x – 3.6| + |x – 4.7| + ……… + |x – 16.8|, for how many real values of x does y attain its minimum possible value?
1
3
10
Infinite
Q26. For a real number k, f(x) = 2kx + 9. If 3f(3) = f(6) and f(9) – f(3) = N, where N is a natural number. Find the sum of the digits of N.
2
3
5
6
Q27. If f(x2 – 2) = 5x4 – 3bx2 + 5c and f(x – 3) = 2x3 + 3cx – 2b, then b + c is:
-11
-17/2
-80/7
Cannot be determined
Q28. Find the number of integer values of x for the function f(x) = |x| + |x – 1| where the value of f(x) achieves its lowest possible value.
0
1
2
More than 2
10
8
7
12
Q30. A function f(p) is defined as f(p) + 2f(1 – p) = p + 3. The value of f(2) will be:
1/3
-1/3
-7/2
3/2
Q31. It is given that 3 < x < 4 where x is a real number, and |x ‒ a| + |x ‒ b| = 4 where both ‘a’ and ‘b’ are real numbers such that a ≤ 3 and b ≥ 4. Find the minimum value of ‘a’ and the maximum value of ‘b’, respectively.
0, 5
0, 4
1, 6
0, 7
Q32. Find the number of integer values of x for the function f(x) = |x| + |x – 1| where f(x) = f(f(x)), given that x is a real number.
0
1
2
4
Q33. It is given that |15 – a| = |15 – b| and |10 – b| = |10 – c|, where a, b, c are distinct natural numbers. How many distinct sets of {a, b, c} are possible?
17
18
20
19
10
14
12
18
Q35. Let f(n) and g(n) be functions defined on all positive integers such that f(n) = n3 and g(n + 1) = 10 – g(n). If g(n3 – n) = 7, then find f(g(1)) – g(f(2)).
Q36. Find the value f(f(f(f(30)))) if for all integers ‘x’, there is a function f such that f(x2 + x) = 20.
7
9
11
13
Q39. A function ‘f’ is defined on all natural numbers ‘a’ and ‘b’ such that f(a × b) = f(a) × f(b). If the output of the function is also a natural number and f(3) > f(2) > f(1), then find the value of f(48), given that f(54) = 375.
405
415
390
420
5
4
6
7
Q41. If f(xy) = f(x) × f(y) for all positive integers, ‘x’ and ‘y’, and f(2) ≠ f(3), then the value of f(1) is:
Q42. If y = |x + 1| – |x – 2|, then which of the following is true of y?
– 3 ≤ y ≤ 0
– 3 ≤ y ≤ 3
y ≤ - 3
y ≥ 3
Q43. If f(x) minimum of (3x + 5, 10 – 2x), what is the maximum possible value of f(x)?
1
4
5
8
Q44. If f(x) = x – a, then f(f(f(x))) is:
x3 - a3
x - 3a
3(x -a)
3x - a
1
54
55
27
Q46. If [x] denotes the greatest integer function less than or equal to x, then the value of x for which 5(x – 1) [x – 1] = 259 is:
7.4
8.6
8.4
7.6
Q47. For all natural numbers x, f(1) + f(2) + f(3) + … + f(x) = x3 f(x) and f(8) = 91, then which of the following numbers is the possible factor of f(8) – f(9)?
2
3
5
7
Q48. If f(x) + f(x – 1) = x2 for all real values of x such that f(29) = 80, then find f(80).
Q49. Let f(n) be a function defined as f(n + 2) = f(n + 1) + f(n) for all positive real values of ‘n’. If f(1) = f(2) = 1, then find the highest common factor of f(8) and f(12).
1
2
5
3
Q50. Let f(x + y) = f(x)f(y), for all x, y. If f(6) = 5, then f(–6) + f(–12) is:
6/5
25
6/25
24
Q51. A function f is defined such that f(1) = 2, f(2) = 5, and f(n) = f(n – 1) – f(n – 2) for all integer values of n > 2. What is the value of f(100)?
Q52. If f(x) is a real function such that 3f(x) = f(x + 1) + 2f(x – 1) for all x ≥ 1 and f(0) = 1, f(1) = 2 then f(7) is equal to:
Q53. Let for all real values of x, f(2x) = 4f(x) + 6, f(x + 2) = f(x) + 12x + 12 and f(1) = 1. What is the value of f(42)?
5290
6468
4240
4678
Q54. Let f be a function such that f(x) = f(x – 2) – f(x – 1) where x ≥ 3. If x is a natural number and f(1) = 0, f(2) = 1, then what is the value of f(9)?
Q55. If f(x + y) = f(x) f(y) and f(0) ≠ 0 for any value of ‘x’ and ‘y’. f(–10) = 5, what is f(20)?
10
25
1/10
1/25
Q56. Let f(x) = min {182 – 11x, 30 + x2}, where x is a positive real number. Find the maximum possible value of f(x).
90
92
91
94
Q57. A function is defined as f(n) = 2(n – 1) + f(n – 1) for positive integral values of n. If f(0) = 1, then find the value of f(n) × f(m), where n × m = 9 and n ≠ m.
73
49
273
Cannot be determined
Q58. Let f(x) = [x], where [x] denotes the greatest integer less than or equal to x. If ‘a’ and ‘b’ are two real numbers such that f(3b – 2) = a – 2 and f(a + 2) = b + 6, then find the sum of a and b.
15
16
17
18
1/4
1/2
1
1/8
Q63. A function f(x) is defined as f(x) = x3 – bx2 – a2x – 24c such that f(a) = f(b) = f(c) = 0 and c = – 12. Find the value of f(3).
10
-120
-135
40
Q64. A function f(x) is such that, f(x) + f(1 – x) + f(1 + x) + f(2 + x) = 2x for all real values of x, and f(0) = 1, then find the value of f(4).
5
0
6
4
50
25
49
Cannot be determined
260
280
290
300
Q67. For how many integral values of x will the function f(x) = |x – 1!| + |x – 2!| + |x – 3!| + |x – 4!| assumes the minimum value?
