Integer Answer Type Questions for JEE: Sets, Relation & Functions | Chapter-wise Tests for JEE Main & Advanced PDF Download

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Q.1. If f is a function such that f(0) = 2, f(1) = 3 and f(x + 2) = 2f(x) – f(x + 1) for every real x then f(5) is

Ans. 13
For x = 0, f(2)  = 2f(0) – f(1)  = 2 × 2 – 3  = 1,
for x = 1, f(3)  = 2f(1) – f(2)  = 6 – 1  = 5,
for x = 2, f(4)  = 2f(2) – f(3)  = 2 × 1 – 5  = -3,
for x = 3, f(5)  = 2f(3) – f(4)  = 2×5 – (-3)= 13.

Q.2. If f''(x) = - f(x) and g(x) = f'(x) and F(x) =  and given that F(5) = 5, then F(10) is equal to 

Ans. 5
f''(x) = -f(x) and f'(x) = g(x)
⇒ f''(x) . f'(x) + f(x) . f'(x) = 0
⇒ f(x)² + (f'(x))² = c
⇒ (f(x)² + (g(x))² = c
⇒ F(x) = c  
⇒ F(10) = 5.

Q.3. If f (x + y) = f(x) + f(y) - xy - 1 for all x, y, and f(1) = 1 then the number of Solutions of f(n) = n, n ∈ N, is

Ans. 1
Putting y = 1, f (x + 1) = f (x) + f (1) - x - 1 = f (x) - x.
∴ f (n + 1) = f (n) - n < f (n).
So, f (n) < f (n - 1) < f (n - 2) < … < f (1) = 1
∴ f (n) = n holds for n = 1 only.

Q.4. The period of the function f(x) =  is

Ans. 24
The period of sinπx/3 is 2π/π/3, i.e. 6. The period of cosπx/4 is  , i.e., 8.
LCM of 6 and 8 is 24. So, the period of f(x) = 24.

Q.5. If the period of the function f(x) =  is 3k then k, is

k, is

Ans. 8
 has period 6
4cos πx/4 has period 8
Net period 24 3k = 24
⇒ k = 8. 

Q.6. If the function f (x) = , then the value of g '(1) is

Ans. 2
Let y = g ( x)= f ^-1 ( x) . Then x = 1
⇒ y = 0(sin ce f (0) = 1)

∴ g ' (1)= 2

Q.7. The inverse of the function y =  is  , m ≠ 0, then the value of m is

Ans. 2

The function is one-one for if y(x₁) = y(x₂)  
then  ⇒ x₁ = x₂
We have,

⇒
⇒
∴ m = 2 .

Q.8. If log₂x + logx² = 5/2 = log₂y + log_y2 and x ≠ y, then the value of x + y - √2 is 

Ans. 4
log₂x + log_x2 = 5/2 = log₂y + log_y2
⇒  where t = log₂x,, s = log₂y
⇒ t = 2, 1/2 & s = 2, 1/2 (as t ≠ s,  x ≠ y)
∴ t = 2 and s = 1/2 or t = 1/2 and s = 3
∴ log₂X = 2 and log₂y = 1/2
x = 4 and y = √2
x + y - √2 = 4 + √2 - √2 = 4.

Q.9. The number of elements in the range of the function f : (-∞,1) → R , defined by f (x) = [9^x - 3^x -+ 1] , where [.] is the greatest integer function, is

Ans. 7
Put 3^x = t . Then 3x → 0 as x → -∞ and 3x → 3 as x → 1

No of elements in the range of [g(t)] is 7.

Q.10. Let A and B be finite sets containing respectively 3 and 2 elements. Find the number of functions that can be defined from A to B 

Ans. 8
Since each of 3 elements of A can associated to an element of B in 2 ways.
Therefore all the 3 elements can be associated with elements of B in 2³ ways.