MCQ Mathematics Chapter 12: LIMITS AND DERIVATIVES , HS 1st year

 

1. What is the average velocity of a body between $t = 0$ and $t = 2$ seconds, if the distance covered is $s = 4.9t^2$?

   • A) $9.8 \, \text{m/s}$
   • B) $14.7 \, \text{m/s}$
   • C) $19.6 \, \text{m/s}$
   • D) $24.5 \, \text{m/s}$
     Answer: A) $9.8 \, \text{m/s}$

2. What does the term "instantaneous velocity" refer to in the context of derivatives?

   • A) Average velocity over an interval
   • B) Velocity at a specific point
   • C) Distance covered at a specific point
   • D) Slope of a secant line
     Answer: B) Velocity at a specific point

3. If $f(x) = x^2$, what is the value of $\lim_{x \to 0} f(x)$?

   • A) 0
   • B) 1
   • C) Does not exist
   • D) $\infty$
     Answer: A) 0

4. For the function $g(x) = |x|$, what is the limit as $x \to 0$?

   • A) $\infty$
   • B) 0
   • C) Undefined
   • D) 1
     Answer: B) 0

5. Which of the following is true about the left-hand and right-hand limits?

   • A) They are always equal
   • B) If unequal, the limit of the function at that point does not exist
   • C) They depend on the derivative
   • D) They are not related to the existence of a limit
     Answer: B) If unequal, the limit of the function at that point does not exist

6. What is the derivative of $f(x) = 3x$ at $x = 2$?

   • A) 2
   • B) 3
   • C) 6
   • D) 0
     Answer: B) 3

7. What is the derivative of a constant function $f(x) = a$?

   • A) 0
   • B) $a$
   • C) 1
   • D) Undefined
     Answer: A) 0

8. Using the binomial theorem, what is the derivative of $f(x) = x^n$?

• A) $nx^{n-1}$
• B) $nx^n$
• C) $n^2x^{n-1}$
• D) None of these
  Answer: A) $nx^{n-1}$

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MCQs

9. What is the right-hand limit of $f(x)$ at $x = 0$, given $f(x) = 1$ for $x \leq 0$ and $f(x) = 2$ for $x > 0$?

   • A) 1
   • B) 2
   • C) Does not exist
   • D) 0
     Answer: B) 2

10. For $f(x) = |x|/x$, what is the left-hand limit as $x \to 0$?

    • A) 0
    • B) 1
    • C) -1
    • D) Does not exist
      Answer: C) -1

11. What is the value of $\lim_{x \to 5} (x + 10)$?

    • A) 10
    • B) 15
    • C) 5
    • D) Does not exist
      Answer: B) 15

12. If $f(x) = x^3$, what is $\lim_{x \to 1} f(x)$?

    • A) 0
    • B) 1
    • C) 3
    • D) 0.5
      Answer: B) 1

13. What is the derivative of $f(x) = x^2$ at $x = 2$?

    • A) 2
    • B) 4
    • C) 6
    • D) 8
      Answer: D) 8

14. What is $\lim_{x \to 2} \frac{(x - 2)}{(x - 2)^2}$?

    • A) 0
    • B) $\infty$
    • C) 1
    • D) -1
      Answer: B) $\infty$

15. If $f(x) = x^2 + x$, what is the derivative at $x = 1$?

    • A) 1
    • B) 2
    • C) 3
    • D) 4
      Answer: C) 3

16. For $f(x) = 3x^2 - x + 4$, what is $f'(x)$?

    • A) $6x - 1$
    • B) $6x + 4$
    • C) $3x - 1$
    • D) $6x - 3$
      Answer: A) $6x - 1$

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More MCQs

17. The instantaneous velocity of a body is:

    • A) The slope of the tangent at a point
    • B) The slope of the secant
    • C) The displacement divided by time
    • D) Constant for any function
      Answer: A) The slope of the tangent at a point

18. What is the derivative of $f(x) = 10x$?

    • A) $10$
    • B) $1$
    • C) $0$
    • D) $x^9$
      Answer: A) $10$

19. What is the derivative of $f(x) = \sin x$ at $x = 0$?

    • A) 1
    • B) 0
    • C) $\infty$
    • D) -1
      Answer: A) 1

20. Which theorem is used to find $\lim_{x \to a} g(x)$, given $f(x) \leq g(x) \leq h(x)$ and $\lim_{x \to a} f(x) = \lim_{x \to a} h(x) = L$?

    • A) Fundamental Theorem of Calculus
    • B) Mean Value Theorem
    • C) Sandwich Theorem
    • D) Taylor's Theorem
      Answer: C) Sandwich Theorem

21. For $f(x) = x^2 + 3x + 5$, what is the value of $f'(x)$?

    • A) $2x + 3$
    • B) $3x^2 + 5$
    • C) $2x + 5$
    • D) $x^2 + 3$
      Answer: A) $2x + 3$

22. What is $\lim_{x \to 0} \frac{\sin x}{x}$?

    • A) $0$
    • B) $1$
    • C) $\infty$
    • D) $\pi$
      Answer: B) $1$

23. The derivative of $f(x) = a$, where $a$ is a constant, is:

    • A) $0$
    • B) $1$
    • C) $a$
    • D) Undefined
      Answer: A) $0$

24. The right-hand and left-hand limits of $f(x)$ are different. What does this imply?

    • A) The limit exists
    • B) The limit does not exist
    • C) The function is not continuous
    • D) Both B and C
      Answer: D) Both B and C

25. What is the value of $\lim_{x \to 1} \frac{x^3 - 1}{x - 1}$?

• A) 1
• B) 2
• C) 3
• D) 4
  Answer: C) 3

26. What is the derivative of $f(x) = x^3$?

    • A) $x^2$
    • B) $3x^2$
    • C) $2x$
    • D) $3x^3$
      Answer: B) $3x^2$

27. What is the derivative of $f(x) = x^n$, where $n$ is a real number?

    • A) $nx^{n-1}$
    • B) $nx^n$
    • C) $n^2x^{n-1}$
    • D) $nx^{n+1}$
      Answer: A) $nx^{n-1}$

28. What is $\lim_{x \to 2} \frac{x^2 - 4}{x - 2}$?

    • A) 2
    • B) 4
    • C) 6
    • D) Undefined
      Answer: C) 4

29. What is $\lim_{x \to 0} \frac{1 - \cos x}{x^2}$?

    • A) 0
    • B) 1
    • C) $\frac{1}{2}$
    • D) $\infty$
      Answer: C) $\frac{1}{2}$

30. If $f(x) = \sin x$, what is the derivative $f'(x)$?

    • A) $\cos x$
    • B) $-\sin x$
    • C) $\tan x$
    • D) $\sin x$
      Answer: A) $\cos x$

31. What is the derivative of $f(x) = \cos x$?

    • A) $-\sin x$
    • B) $\sin x$
    • C) $\cos x$
    • D) $1$
      Answer: A) $-\sin x$

32. Which rule is used to differentiate $f(x)g(x)$?

    • A) Quotient Rule
    • B) Product Rule
    • C) Chain Rule
    • D) Addition Rule
      Answer: B) Product Rule

33. What is the derivative of $f(x) = \tan x$?

    • A) $\sec^2 x$
    • B) $\csc^2 x$
    • C) $-\sin x$
    • D) $\cos x$
      Answer: A) $\sec^2 x$

34. What is the derivative of $f(x) = e^x$?

    • A) $e^x$
    • B) $xe^{x-1}$
    • C) $e^{x-1}$
    • D) $\ln x$
      Answer: A) $e^x$

35. What is the derivative of $f(x) = \ln x$?

    • A) $\frac{1}{x}$
    • B) $\ln x$
    • C) $\ln(x-1)$
    • D) Undefined
      Answer: A) $\frac{1}{x}$

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36. What is the derivative of $f(x) = \sec x$?

    • A) $\sec x \tan x$
    • B) $\csc x$
    • C) $\tan x$
    • D) $\sin x$
      Answer: A) $\sec x \tan x$

37. What is the derivative of $f(x) = \csc x$?

    • A) $-\csc x \cot x$
    • B) $\cot x$
    • C) $\sin x$
    • D) $-\cos x$
      Answer: A) $-\csc x \cot x$

38. What is the derivative of $f(x) = \cot x$?

    • A) $-\csc^2 x$
    • B) $\sec^2 x$
    • C) $-\sin^2 x$
    • D) $\cos^2 x$
      Answer: A) $-\csc^2 x$

39. If $u(x) = x^2$ and $v(x) = \sin x$, what is $\frac{d}{dx}(u(x)v(x))$?

    • A) $2x \sin x + x^2 \cos x$
    • B) $x^2 \cos x + \sin x$
    • C) $2x \cos x - x^2 \sin x$
    • D) $x^2 \sin x + 2x \cos x$
      Answer: A) $2x \sin x + x^2 \cos x$

40. What is the derivative of $f(x) = \sqrt{x}$?

    • A) $\frac{1}{2\sqrt{x}}$
    • B) $\sqrt{x}$
    • C) $2\sqrt{x}$
    • D) $x^{3/2}$
      Answer: A) $\frac{1}{2\sqrt{x}}$

41. If $f(x) = x^2 + 3x - 5$, find $f'(2)$:

    • A) 7
    • B) 9
    • C) 8
    • D) 11
      Answer: C) 7

42. What is the value of $\lim_{x \to 0} \frac{\tan x}{x}$?

    • A) 1
    • B) 0
    • C) $\infty$
    • D) Undefined
      Answer: A) 1

43. Which of the following represents the slope of the tangent to the curve $y = f(x)$ at a point $x = a$?

    • A) $f'(a)$
    • B) $f(a)$
    • C) $f''(a)$
    • D) None of these
      Answer: A) $f'(a)$

44. For $f(x) = x^4 - 2x^2 + 3$, what is $f'(x)$?

    • A) $4x^3 - 4x$
    • B) $4x^3 + 4x$
    • C) $4x^2 - 2x$
    • D) $4x^3 + 3x$
      Answer: A) $4x^3 - 4x$

45. Which of the following best describes a function with no derivative at $x = c$?

• A) Continuous but not differentiable
• B) Discontinuous
• C) Always differentiable
• D) None of these
  Answer: A) Continuous but not differentiable

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