Algebra - Functions & Graphs - Previous Year CAT/MBA Questions

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1. CAT 2024 QA Slot 1 | Algebra - Functions & Graphs CAT Question

Consider two sets A = {2, 3, 5, 7, 11, 13} and B = {1, 8, 27}. Let f be a function from A to B such that for every element b in B, there is at least one element a in A such that f(a) = b. Then, the total number of such function f is

(a)
667
(b)
540
(c)
665
(d)
537

2. CAT 2024 QA Slot 2 | Algebra - Functions & Graphs CAT Question

A function f maps the set of natural numbers to whole numbers, such that f(xy) = f(x)f(y) + f(x) + f(y) for all x, y and f(p) = 1 for every prime number p. Then, the value of f(160000) is

(a)
8191
(b)
2047
(c)
4095
(d)
1023

3. CAT 2024 QA Slot 3 | Algebra - Functions & Graphs CAT Question

For any non-zero real number x, let f(x) + 2f(1/x) = 3x. Then, the sum of all possible values of x for which f(x) = 3, is

(a)
3
(b)
-2
(c)
2
(d)
-3

4. CAT 2023 QA Slot 3 | Algebra - Functions & Graphs CAT Question

Suppose f(x, y) is a real-valued function such that f(3x + 2y, 2x - 5y) = 19x, for all real numbers x and y. The value of x for which f(x, 2x) = 27, is

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5. CAT 2022 QA Slot 1 | Algebra - Functions & Graphs CAT Question

For any real number x, let [x] be the largest integer less than or equal to x. If $\sum_{n = 1}^{N} [\frac{1}{5} + \frac{n}{25}]$ = 25, then N is

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Answer: 44

Video Explanation

Text Explanation :

$[\frac{1}{5} + \frac{n}{25}]$ = 0 when $\frac{1}{5} + \frac{n}{25}$ < 1 ⇒ n < 20

$[\frac{1}{5} + \frac{n}{25}]$ = 1 when 1 ≤ $\frac{1}{5} + \frac{n}{25}$ < 2 ⇒ 20 ≤ n < 45

$\sum_{n = 1}^{44} [\frac{1}{5} + \frac{n}{25}]$ = $\sum_{n = 1}^{19} [\frac{1}{5} + \frac{n}{25}]$ + $\sum_{n = 20}^{44} [\frac{1}{5} + \frac{n}{25}]$ = 0 + (1 + 1 + 1 + … + 1 (25 times)) = 25

∴ N = 44

Hence, 44.

6. CAT 2022 QA Slot 2 | Algebra - Functions & Graphs CAT Question

Suppose for all integers x, there are two functions f and g such that f(x) + f(x - 1) - 1 = 0 and g(x) = x². If f(x² - x) = 5, then the value of the sum f(g(5)) + g(f(5)) is

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7. CAT 2022 QA Slot 3 | Algebra - Functions & Graphs CAT Question

Let r be a real number and f(x) = $\{\begin{matrix} 2 x - r & if x \geq r \\ r & if x < r \end{matrix} .$ Then, the equation f(x) = f(f(x)) holds for all real values of x where

(a)
x ≠ r
(b)
x ≥ r
(c)
x > r
(d)
x ≤ r

8. CAT 2021 QA Slot 1 | Algebra - Functions & Graphs CAT Question

f(x) = $\frac{x^{2} + 2 x - 15}{x^{2} - 7 x - 18}$ is negative if and only if

(a)
X < -5 or 3 < x < 9
(b)
-2 < x < 3 or x > 9
(c)
-5 < x < -2 or 3 < x < 9
(d)
x < -5 or -2 < x < 3

Answer: Option C

Video Explanation

Text Explanation :

Given, f(x) = $\frac{x^{2} + 2 x - 15}{x^{2} - 7 x - 18}$ < 0

⇒ $\frac{(x + 5) (x - 3)}{(x - 9) (x + 2)}$ < 0

Here, the critical points are -5, -2, 3 and 9.

∴ f(x) will be positive when -5 < x < -2 or 3 < x < 9.

Hence, option (c).

9. CAT 2021 QA Slot 3 | Algebra - Functions & Graphs CAT Question

If f(x) = x² – 7x and g(x) = x + 3, then the minimum value of f(g(x)) – 3x is

(a)
-12
(b)
-16
(c)
-15
(d)
20

10. CAT 2020 QA Slot 1 | Algebra - Functions & Graphs CAT Question

Among 100 students, x₁ have birthdays in January, x₂ have birthdays in February, and so on. If x₀ = max(x₁, x₂, …., x₁₂), then the smallest possible value of x₀ is

(a)
10
(b)
8
(c)
12
(d)
9

11. CAT 2020 QA Slot 1 | Algebra - Functions & Graphs CAT Question

If f(5 + x) = f(5 - x) for every real x, and f(x) = 0 has four distinct real roots, then the sum of these roots is

(a)
20
(b)
40
(c)
0
(d)
10

12. CAT 2020 QA Slot 1 | Algebra - Functions & Graphs CAT Question

The area of the region satisfying the inequalities |x| - y ≤ 1, y ≥ 0 and y ≤ 1 is

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13. CAT 2020 QA Slot 2 | Algebra - Functions & Graphs CAT Question

Let f(x) = x² + ax + b and g(x) = f(x + 1) – f(x – 1). If f(x) ≥ 0 for all real x, and g(20) = 72, then the smallest possible value of b is

(a)
16
(b)
1
(c)
4
(d)
0

14. CAT 2020 QA Slot 3 | Algebra - Functions & Graphs CAT Question

If f(x + y) = f(x)f(y) and f(5) = 4, then f(10) – f(-10) is equal to

(a)
0
(b)
15.9375
(c)
3
(d)
14.0625

15. CAT 2019 QA Slot 1 | Algebra - Functions & Graphs CAT Question

For any positive integer n, let f(n) = n(n + 1) if n is even, and f(n) = n + 3 if n is odd. If m is a positive integer such that 8 f(m + 1) − f(m) = 2, then m equals

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16. CAT 2019 QA Slot 1 | Algebra - Functions & Graphs CAT Question

Consider a function f satisfying f(x + y) = f(x) f(y) where x, y are positive integers and f(1) = 2. If f(a + 1) + f(a + 2) +…+ f(a + n) = 16(2ⁿ – 1) then a is equal to

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17. CAT 2019 QA Slot 2 | Algebra - Functions & Graphs CAT Question

Let f be a function such that f(mn) = f(m) × f(n) for every positive integers m and n. If f(1), f(2) and f(3) are positive integers, f(1) < f(2), and f(24) = 54, then f(18) equals

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18. CAT 2018 QA Slot 1 | Algebra - Functions & Graphs CAT Question

If f(x + 2) = f(x) + f(x + 1) for all positive integers x, and f(11) = 91, f(15) = 617, then f(10) equals

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19. CAT 2018 QA Slot 1 | Algebra - Functions & Graphs CAT Question

Let f(x) = min {2x², 52 − 5x}, where x is any positive real number. Then the maximum possible value of f(x) is

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20. CAT 2018 QA Slot 2 | Algebra - Functions & Graphs CAT Question

Let f(x) = max {5x, 52 – 2x²}, where x is any positive real number. Then the minimum possible value of f(x) is

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Answer: 20

Video Explanation

Text Explanation :

The graph of the function 52 – 2x² will be of ‘inverted U’ shape, while the graph of the function y = 5x will be a straight line with a positive slope.

Therefore, the minimum value of the required function will be obtained at a point of intersection of y = 52 – 2x² and y = 5x.

Therefore, 52 – 2x² = 5x or 2x²+ 5x – 52 = 0.

∴ (x - 4)(2x + 13) = 0

∴ x = 4 or x = $- \frac{13}{2}$

Since x is a positive real number, x = 4.

At x = 4, 5x = 52 – 2x² = 20

Hence, 20.

21. CAT 2017 QA Slot 1 | Algebra - Functions & Graphs CAT Question

The shortest distance of the point (1/2,1) from the curve y = |x - 1| + |x + 1| is

(a)
1
(b)
0
(c)
√2
(d)
√32

22. CAT 2017 QA Slot 1 | Algebra - Functions & Graphs CAT Question

If f(x) = (5x+2)/(3x-5) and g(x) = x² – 2x – 1, then the value of g(f(f(3))) is:

(a)
2
(b)
13
(c)
6
(d)
23

Answer: Option A

Video Explanation

Text Explanation :

f(x) = (5x+2)/(3x-5), g(x) = x² – 2x – 1

f(3) = $\frac{5 \times 3 + 2}{3 \times 3 - 5} = \frac{17}{4}$

f(17/4) = $\frac{5 \times \frac{17}{4} + 2}{3 \times \frac{17}{4} - 5}$ = $\frac{85 + 8}{51 - 20}$ = $\frac{93}{31}$ = 3

g(3) = 3² – 2 × 3 – 1 = 2.

Hence, option (a).

23. CAT 2017 QA Slot 2 | Algebra - Functions & Graphs CAT Question

Let f(x) = x² and g(x) = 2x, for all real x. Then the value of f(f(g(x)) + g(f(x))) at x = 1 is

(a)
16
(b)
18
(c)
36
(d)
40

24. CAT 2017 QA Slot 2 | Algebra - Functions & Graphs CAT Question

If f(ab) = f(a)f(b) for all positive integers a and b, then the largest possible value of f(1) is

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25. CAT 2017 QA Slot 2 | Algebra - Functions & Graphs CAT Question

Let f(x) = 2x – 5 and g(x) = 7 – 2x. Then |f(x) + g(x)| = |f(x)| + |g(x)| if and only if

(a)
$\frac{5}{2} < x < \frac{7}{2}$
(b)
$x \leq \frac{5}{2} o r x \geq \frac{7}{2}$
(c)
$x < \frac{5}{2} o r x \geq \frac{7}{2}$
(d)
$\frac{5}{2} \leq x \leq \frac{7}{2}$

Answer: Option D

Video Explanation

Text Explanation :

Now

|f(x) + g(x)| = |f(x)| + |g(x)|

This is true only if both f(x) and g(x) are both negative or both positive or both are zero

Case 1: Now if both f(x) and g(x) are greater than or equal to zero.

f(x) = 2x - 5 ≥ 0 or x ≥ $\frac{5}{2}$

g(x) = 7 - 2x ≥ 0 or x ≤ $\frac{7}{2}$

∴ $\frac{5}{2} \leq x \leq \frac{7}{2}$

Case 2: Now if both f(x) and g(x) are less than or equal to zero.

f(x) = 2x - 5 ≤ 0 or x ≤ $\frac{5}{2}$

g(x) = 7 - 2x ≤ 0 or x ≥ $\frac{7}{2}$

This means x ≥ $\frac{7}{2}$ and x ≤ $\frac{5}{2}$ .

However, this is not possible.

Hence, option (d).

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Algebra - Functions & Graphs - Previous Year CAT/MBA Questions

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