Q.1. The function is
(a) continuous at x = 1
(b) differentiable at x =1
(c) continuous at x=3
(d) differentiable at x = 3
Correct Answer is options (a, b and c)
Apply 2 hospital rule and simplifying we get
∴ (λ , μ) can be (5, 3)
Q.2. Let L = If L is finite, then
(a) a = 1
(b) a = 2
(c) L = (1/8)
(d) L = 1/16
Correct Answer is options (a and c)
Since L is finite
Also,
Q.3. Let f(x) = then
(a)
(b)
(c) f(x) does not exist
(d) none of these
Correct Answer is options (a and b)
For x < 1, x^{2n} → 0 as n → ∞ and for x > 1, 1/x^{m}→ 0 as n → ∞
Q.4. If f(x) = , then
(a) f(x) = e^{2}
(b) f(x) = e^{–2}
(c) f(x) = 0
(d) f(x) = 1
Correct Answer is options (a and b)
f(x) does not exists.
Q.5. The function f (x) = where u =
(a) has a removable discontinuous at x = 1
(b) has irremovable discontinuous at x = 0, (3/2)
(c) discontinuous at u = 2, 1
(d) discontinuous at u = 1, 2
Correct Answer is options (a, b and d)
The function is discontinuous at x = 1
is discontinuous at u = 1, 2
i.e. at x = 0, (3/2) also we have
Q.6. If f(x) = and f (1) = 7, f(x) , g(x) and h(x) are all continuous function at x =1 . Then which of the following statement(s) is/are correct.
(a) g (1) + h (1)= 70
(b) g (1)  h (1)= 28
(c) g (1) + h (1)= 60
(d) g (1)  h (1) = 28
Correct Answer is options (a and b)
When x <1 When x > 1
h (1) = 21 ∴ g (1)= 49
∴ g (1)  h (1) = 28
g (1) + h (1)= 70
Q.7. Let f:(0, ∞) → R and F(x) = If F(x^{2}) = x^{4}+ x^{5}, then
(a) f(4) = 7
(b) f(x) is continuous everywhere
(c) f(x) increases for all x > 0
(d) f(x) is onto.
Correct Answer is options (a, b and c)
Differentiating both sides, we get x^{2}f (x^{2}) 2x = 4x^{3}+ 5x^{4}
Clearly, f(x) is continuous and increasing and f (4) = 2 + (5/2) x 2 = 7
Range of the function (2, ∞) ⊂ R so, f(x) is into.
Q.8. f(x) = min {1, cos x, 1sin x}, – π ≤ x ≤ π, then
(a) f(x) is not differentiable at 0
(b) f(x) is differentiable at π/2
(c) f(x) has local maxima at 0
(d) none of these
Correct Answer is options (a and c)
We have, f (x) = min {1, cos x, 1 sin x}
∴ f (x ) can be rewritten as
∵ f'(0) = 0
Hence, f(x) has local maxima at 0 and f(x) is not differentiable at x = 0.
Q.9. Let [x] denote the greatest integer less than or equal to x. If f(x) =[x sinπx], then f(x) is
(a) Continuous at x = 0
(b) Continuous in (1, 0)
(c) Differentiable at x =1
(d) Differentiable in (1,1)
Correct Answer is options (a, b and d)
x sinπx is an even function of x
Also 0 ≤ x < 1 ⇒ x sinπ x ≤ x = x < 1
⇒ [x sinπx]= 0
Since x sinπx is even, 1 < x < 1 ⇒ f (x) = 0
∴ f is differentiable for all x ∈ (1, 1) and f'(x) =0
∴ f is not continuous at x = 1 and therefore not differentiable at x = 1
Q.10. If F (x) = f(x) g(x) and f'(x) g'(x) = c, then
(a)
(b)
(c)
(d)
Correct Answer is options (a, b and c)
Given, F(x) = f(x).g (x)
Differentiating both sides w.r.t. x, we get
F'(x) = f'(x).g (x)+ g(x).f (x)
Again differentiating both sides w.r.t. x, we get
F"(x) = f"(x)g(x)+g"(x).f(x)+ 2f"(x).g(x)
F"(x) = f"(x).g(x) + g"(x).f(x) + 2c ....(ii)
Dividing both sides by F(x) = f x).g(x)
{∵ f'(x) .g'(x) = c}
Then
Again given, f'(x) g'(x) = c
Differentiating both sides w.r.t. x, we get
f'(x) g"( x ) + g(x) f"(x)= 0
From equa. (ii)
F"(x) = f"(x).g (x) + g"(x).f(x) + 2c
Differentiating both sides w.r.t x, we get
F"'(x) = f"(x).g' (x)+ f"'(x).g (x) +g"(x).f'(x) + f (x).g"'(x) + 0
= f"'(x).g (x) + g"'(x).f (x) + 0
Now, dividing both sides by F(x) = f(x) g(x)
Then,
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