Engineering Mathematics Quiz

1. If f(a+h) = f(a) + hf(a + θh), then

(A) 0 < θ < 1 (B) 0 ≤ θ ≤ 1 (C) 1 < θ < 0 (D) None of these

Answer ⇒ (A) 0 < θ < 1

2. If Rn be the Lagrange's form of reminder in the expansion of Cauchy's mean value theorem, then Rn is equal to

(A) h/n! f'(a + θh) (B) 1/n! f(a + θh) (C) f'(a + θh) (D) None of these

Answer ⇒ (A) h/n! f'(a + θh)

3. If Rolle's theorem is applied to f(x) = x(x²-1) in [0,1], then c is equal to

(A) 1 (B) 0 (C) 1/√3 (D) 1/3

Answer ⇒ (C) 1/√3

4. If Cauchy's mean value theorem is applied to the function f(x) = x and g(x) = x², then the value of c is

(A) 3/2 (B) 0 (C) 1/2 (D) 3/2

Answer ⇒ (C) 1/2

5. If f(x) satisfied the conditions of Rolle's theorem in [a,b], then there exists a point on the curve between (a,f(a)) and (b,f(b)) at which the tangent is

(A) Parallel to x-axis (B) Parallel to y-axis (C) Parallel to chord AB (D) Perpendicular to chord AB

Answer ⇒ (A) Parallel to x-axis

6. \( \lim_{x \to 0} \frac{\log x}{\cot x} \) is equal to

(A) 0 (B) 1 (C) 2 (D) None of these

Answer ⇒ (D) None of these

7. \( \lim_{x \to 0} (1+x)^{\frac{1}{x}} \) is equal to

(A) e (B) e² (C) 0 (D) None of these

Answer ⇒ (A) e

8. \( \lim_{x \to 0} \tan(x \log x) \) is equal to

(A) 0 (B) 1 (C) 2 (D) 3

Answer ⇒ (A) 0

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