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

Answer: Option A

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Text Explanation :

The function f(x) = $\frac{\log (3x-7)}{\sqrt{2x^2-7x+6}}$  is only defined when both the numerator and the denominator of the function are defined are the denominator is not equal to zero.

The logarithm of the function is only defined for positive values :

Hence 3x - 7 is greater than zero. Hence x > 7/3.

The value inside square root are defined for positive values. The value of the quadratic equation in the square root must be positive.

Hence 2x² - 7x + 6 = 0 has the roots:

$\frac{(7+\sqrt{49-48})}{4}$, $\frac{(7-\sqrt{49-48})}{4}$ : 2, 3/2

The quadratic equation is positive for:

$\left(-\infty , \frac{3}{2}\right)$ ∪ (2, ∞)

Since in order to be a part of the domain the values of x must be greater than 7/3  and 7/3 is greater than 2 all values of x which are greater than 7/3 must be a part of the domain for x.

Answer: Option D

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Text Explanation :

$\frac{f\left(x\right)}{g\left(x\right)}$ = $\frac{(x^2+1)}{x^2-1}$ . $\frac{(x-1)}{x+1}$ = $\frac{(x^2+1)}{{(x+1)}^2}$

This function is definitely greater than 0

let y = $\frac{(x^2+1)}{{(x+1)}^2}$

⇒ x² (y − 1) + 2yx + (y − 1) = 0 which is quadratic in x

Disctiminant should be greater than 0

4y² - 4(y - 1)² ≥ 0

⇒ y >= 1/2

When x =1, f(x)/g(x) = 1/3
Hence either the value should be greater than 1/2 or should be equal to 1/3

Answer: Option D

Video Explanation

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Text Explanation :

(x + 4)(x + 6)(x + 8) ⋯ (x + 98) < 0

Critical points are -98, -96, -94, … -8. -6 and -4

For x > -4, the given expression will be positive. Hence, we will reject all values of x > -4.

For -6 < x < -4, the given expression will be negative. The only integral value for x in this range is -5.

For -8 < x < -6, the given expression will be positive.

For -10 < x < -8, the given expression will be negative. The only integral value for x in this range is -9.

The given expression will be negative for x = -5, -9, -13, and so on

Now the only integral value of x between -98 and -96 is -97
∴ f(-97) = -93 × - 91 × -89 × … × -1 × 1

Here we have product of 47 negative terms and 1 positive term, hence the product will be negative.

∴ For -98 < x < -96, the given expression will be negative. The only integral value for x in this range is -97.

Finally, for x < -98, f(x) is product of 48 negative terms, which will give us a positive number. Hence, we will reject all value of x < -98.

The given expression will be negative for x = -5, -9, -13, and so on till -97

Number of possible values of x for which f(x) < 0 = (-5 + 97)/4 + 1 = 24

Hence, option (d).

Answer: Option B

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Text Explanation :

Q(9, 4) = (9)² – (9)(4) + (4)²

= 81 – 36 + 16 = 61

∴ P(7,Q(9,4)) = P(7,61)

= (7)² + (7)(61) + (61)²

= 49 + 427 + 3721 = 4197

Hence, option (b).

Answer: Option C

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Text Explanation :

f(x) = ax + b

∴ f(f(x) = a(ax + b) + b

f(f(x) = a²x + ab + b 

⇒ a²x + b(a + 1) = 9x + 8

∴ a² = 9 ⇒ a = 3. and

b(a + 1) = 8

⇒ b(3 + 1) = 8

⇒ b = 2

Thus, a + b = 5 

Hence, option (c).

Answer: Option B

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Text Explanation :

f(5) = 1/(1 + 5) = 1/6 

f[f(5)] = f(1/6) = 1/[1 + (1/6)] = 6/7

f[f{f(5)}] = f(6/7) = 1/[1 + (6/7)] = 7/13

Hence, option (b).

Answer: Option D

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Text Explanation :

Given function can be rearranged as,

f(x) – f(x – 1) = 2x

⇒ f(2) – f(1) = 2 × 2
⇒ f(3) – f(2) = 2 × 3
⇒ f(4) – f(3) = 2 × 4
…
⇒ f(31) – f(30) = 2 × 31

Adding all these equations we get

f(31) – f(1) = 2 × (2 + 3 + 4 + … + 31)

f(31) – 1 = 2 × ((31×32)/2-1) = 990

⇒ f(31) = 991

Alternately,
From the given function ;

f(2) = 4 + f(1) = 5 = 22 + 1

f(3) = 6 + f(2) = 6 + 5 = 11 = 32 +2

f(4) = 8 + f(3) = 8 + 11 = 19 = 42 + 3

Hence we can see that f(x) = x2 + (x-1)

Hence f(31) = 312 + 30 = 991.

Hence, option (d).

Answer: Option E

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Text Explanation :

From the graph, observe that at y = 0, x
= –19
Only x = 2y² + 3y – 19 satisfies the above criteria.
Hence, option (e).

Answer: Option C

Video Explanation

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Text Explanation :

Substitute x = 0 in f(x² – 1) = x⁴ – 7x² + k₁
⇒ f(–1) = k₁ … (i)
Substitute x = 1 in f(x₃ – 2) = x⁶ – 9x³ + k₂
⇒ f(–1) = 1 – 9 + k₂ … (ii)
From (i) and (ii), k₁ = –8 + k₂
∴ k₂ – k₁ = 8
Hence, option (c).

Answer: Option E

Video Explanation

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Text Explanation :

f(1) = 4
f(2) = f(1+1) = f(1 × 1) = f(1) = 4
…
Assume that f(n – 1) = 4
f(n) = f(1 + (n – 1)) = f(1 × (n – 1)) = f(n – 1) = 4
Thus, by induction principle, f(n) = 4, for all n.
∴ f(1003) = k = 4
Hence, option (e).