This quiz works best with JavaScript enabled. Home > Cbse > Class 12 > Science > Mathematics > Class 12 Mathematics Chapter 5 Continuity And Differentiability – Quiz 1 🏠 Homepage 📘 Download PDF Books 📕 Premium PDF Books Class 12 Mathematics Chapter 5 Continuity And Differentiability Quiz 1 (60 MCQs) Quiz Instructions Select an option to see the correct answer instantly. 1. Differential coefficient of sec $\left(ten^{-1}x\right)$ A) $x\sqrt{1+x^2}$. B) $\frac{1}{\sqrt{1+x^2}}$. C) $\frac{x}{\sqrt{1+x^2}}$. D) $\frac{x}{1+x^2}$. Show Answer Correct Answer: C) $\frac{x}{\sqrt{1+x^2}}$. 2. Define a removable discontinuity. A) A removable discontinuity is a point where a function is undefined and cannot be redefined. B) A removable discontinuity is a point on a graph where a function is undefined but can be made continuous by redefining the function at that point. C) A removable discontinuity is a point where a function is always continuous. D) A removable discontinuity is a point where a function is defined but cannot be made continuous. Show Answer Correct Answer: B) A removable discontinuity is a point on a graph where a function is undefined but can be made continuous by redefining the function at that point. 3. If y= $\sin^{-1}\left(\frac{1-x^2}{1+x^2}\right)$ $\frac{\text{d}y}{dx\text{}}$ A) $\frac{2}{1+x^2}$. B) $\frac{-2}{1-x^2}$. C) $\frac{-2}{1+x^2}$. D) None of the above. Show Answer Correct Answer: C) $\frac{-2}{1+x^2}$. 4. Find dy/dx at the given pointx$^{3}$ +2xy-y$^{2}$=11 at (2, 3) A) -4/7. B) 9. C) -9. D) 12. Show Answer Correct Answer: B) 9. 5. F(x) = $\frac{\sin\left(\pi x\right)}{5x}$ $\ne$ A) $\frac{\pi}{5}$. B) 1. C) $\frac{5}{\pi}$. D) 0. Show Answer Correct Answer: A) $\frac{\pi}{5}$. 6. If y=a(1-cos $\theta$ $\left(\theta+\sin\theta\right)$ $\theta$ $\frac{\text{d}y}{\text{d}x}$ A) $-\tan\theta$. B) $\tan\\frac{\theta}{2}$. C) $\cot\\frac{\theta}{2}$. D) $\tan\\theta$. Show Answer Correct Answer: B) $\tan\\frac{\theta}{2}$. 7. If f(x)=[kx+1 if $x\le5$ $x>5$ A) K=5. B) K=9/5. C) K=9. D) None of the above. Show Answer Correct Answer: B) K=9/5. 8. Trigonometric and inverse-trigonometric functions are differentiable in their respective domain. A) False. B) True. C) All the above. D) None of the above. Show Answer Correct Answer: B) True. 9. Is $f\left(x\right)=\frac{x^2-9}{x-3}$ $x=3?$ A) Connot be determined. B) No. C) Does not exist. D) Yes. Show Answer Correct Answer: B) No. 10. Y = |x-1| is a continuous function. A) False. B) True. C) All the above. D) None of the above. Show Answer Correct Answer: B) True. 11. Find the slope of the tangent line of y= t$^{3}$-4t$^{2}$ +1 at the point where t=3 A) 2. B) 1. C) 3. D) 6. Show Answer Correct Answer: C) 3. 12. If a function is differentiable at a point then A) It is continuous at a point. B) It is neccessarily continuous at that point. C) It is continuous at any other point. D) None of these. Show Answer Correct Answer: B) It is neccessarily continuous at that point. 13. Number of points of discontinuity of a function f(x) = [x] in the interval (-1, 2) where [x] represent greatest integer function less than or equal to x A) 4. B) 1. C) 3. D) 2. Show Answer Correct Answer: D) 2. 14. Which one of the following is not true? A) Every differential function is continuous. B) Every rational function is continuous. C) Every polynomial function is continuous. D) Every continuous function is differentiable. Show Answer Correct Answer: D) Every continuous function is differentiable. 15. The slope of a function is described by its ..... A) Expression. B) Third derivative. C) First derivative. D) Second derivative. Show Answer Correct Answer: C) First derivative. 16. What is the derivative of the function f(x) = sin(x)? A) -cos(x). B) Sin(x). C) Cos(x). D) 1. Show Answer Correct Answer: C) Cos(x). 17. $f\left(x\right)=\frac{\left(1-\cos x\right)}{x\sin x}, $ $\ne$ A) Continuous and differentiable. B) Continuous but not differentiable. C) Differentiable but not continuous. D) Neither Continuous nor differentiable. Show Answer Correct Answer: A) Continuous and differentiable. 18. Find the derivative:y=5x$^{2}$e$^{3x}$ A) Y'=5xe$^{3x}$(3x+2). B) Y'=10xe$^{3x}$(2x+3). C) Y'=5xe$^{3x}$(2x+3). D) Y'=10ex$^{3x}$(3x+2). Show Answer Correct Answer: A) Y'=5xe$^{3x}$(3x+2). 19. What is the relationship between continuity and differentiability? A) Continuity implies differentiability, but differentiability does not imply continuity. B) Differentiability always leads to discontinuity. C) Continuity and differentiability are completely unrelated concepts. D) Differentiability implies continuity, but continuity does not imply differentiability. Show Answer Correct Answer: D) Differentiability implies continuity, but continuity does not imply differentiability. 20. Which of the following points is not the point of discontinuity of $f\left(x\right)=\frac{x-7}{x^3-x^2-11x+3}?$ A) $x=-2$. B) $x=2-\sqrt[]{3}$. C) $x=2+\sqrt[]{3}$. D) $x=-3$. Show Answer Correct Answer: A) $x=-2$. 21. F(x) = $\sqrt[]{1-\sqrt[]{1-x^2}}$ A) Continuous on [-1, 1] and differentiable on (-1, 1). B) Continuous on [-1, 1] and differentiable on (-1, 0) U (0, 1). C) Continuous and differentiable on [-1, 1]. D) None. Show Answer Correct Answer: B) Continuous on [-1, 1] and differentiable on (-1, 0) U (0, 1). 22. The function f given by f(x) = $\left|x+3\right|$ $\in$ A) -2. B) 3. C) 0. D) -3. Show Answer Correct Answer: D) -3. 23. True or False:Differentiability implies continuity A) True. B) False. C) All the above. D) None of the above. Show Answer Correct Answer: A) True. 24. The second order derivative of log x with respect to x is A) $\frac{1}{x}$. B) 0. C) $\frac{1}{x^2}$. D) $-\frac{1}{x^2}$. Show Answer Correct Answer: D) $-\frac{1}{x^2}$. 25. Which of the following statements is true regarding the existence of limits? A) Limits can only be evaluated from both sides. B) If a limit exists, the function must be defined at that point. C) Limits can be evaluated at points of discontinuity. D) None of the above. Show Answer Correct Answer: C) Limits can be evaluated at points of discontinuity. 26. Define continuity of a function at a point. A) Continuity at a point means the function is defined at that point. B) A function is continuous if it has a jump discontinuity at that point. C) A function is continuous if it has a removable discontinuity at that point. D) A function is continuous at a point if the limit of the function as x approaches that point exists and is equal to the value of the function at that point. Show Answer Correct Answer: D) A function is continuous at a point if the limit of the function as x approaches that point exists and is equal to the value of the function at that point. 27. If y=log x then $\frac{d^2y}{dx^2}$ A) $\frac{-1}{x^2}$. B) 0. C) X. D) None of the above. Show Answer Correct Answer: A) $\frac{-1}{x^2}$. 28. Find $\frac{dy}{dx}$ $y=x^2e^x$ A) $xe^x\left(x+2\right)$. B) $xe^{-x}\left(x+2\right)$. C) $xe^x\left(x-2\right)$. D) $-xe^x\left(x+2\right)$. Show Answer Correct Answer: A) $xe^x\left(x+2\right)$. 29. Let f(x) = 1 for x <=-1f(x) = |x|, -1 < x < 1f(x) = 0 for x >= 1 A) Continuous everywhere. B) Differentiable everywhere. C) Differentiable at x =-1. D) Continuous at x =-1. Show Answer Correct Answer: D) Continuous at x =-1. 30. A function whose graph is otherwise continuous will fail to have a derivative at a point where the graph has a ..... A) Corner, cusp, vertical tangent, continuity. B) Corner, cusp, vertical tangent, discontinuity. C) Corner, curve, vertical tangent, continuity. D) Corner, cusp, horizontal tangent, discontinuity. Show Answer Correct Answer: B) Corner, cusp, vertical tangent, discontinuity. 31. The composition of two continuous functions is a continuous function. A) True. B) False. C) All the above. D) None of the above. Show Answer Correct Answer: A) True. 32. Which of these is NOT a hypothesis of the Mean Value Theorem (MVT)? A) Function is continuous on a closed, bounded interval. B) Function is differentiable on the interior of a bounded interval. C) Function gets same values at the ends of an interval. D) None of the above. Show Answer Correct Answer: C) Function gets same values at the ends of an interval. 33. Which theorem states that the sum, difference, product, and quotient of continuous functions are continuous? A) Fundamental Theorem of Calculus. B) Algebra of Continuous Functions. C) Theorem of Limits. D) Chain Rule. Show Answer Correct Answer: B) Algebra of Continuous Functions. 34. At what point is $f\left(x\right)=\frac{x^2-25}{x-5}$ A) -2. B) 2. C) 5. D) -5. Show Answer Correct Answer: C) 5. 35. All functions below are continuous and differentiable EXCEPT ONE. Which is it? A) Rational function. B) Absolute value function. C) Linear function. D) Quadratic function. Show Answer Correct Answer: B) Absolute value function. 36. If a function is differentiable, it is also continuous. A) Yes. B) No. C) It all depends on the function in question. D) None of the above. Show Answer Correct Answer: A) Yes. 37. Derivative of cos$^{-1}$(2x) A) $-\frac{1}{\sqrt{1\-4x^2}}$. B) $-\frac{1}{\sqrt{4x^2\-1}}$. C) $-\frac{2}{\sqrt{1\-4x^2}}$. D) $-\frac{2}{\sqrt{4x^2\-1}}$. Show Answer Correct Answer: C) $-\frac{2}{\sqrt{1\-4x^2}}$. 38. On what value/s of X will make the function $f\left(x\right)=x-4$ A) 4. B) 0. C) -4. D) All values of X. Show Answer Correct Answer: D) All values of X. 39. The function f(x) = x$^{2}$-2x+1 is continuous and differentiable at all real values of x. A) TRUE. B) FALSE. C) All the above. D) None of the above. Show Answer Correct Answer: A) TRUE. 40. Which of the following is a property of logarithmic functions? A) They have a range of positive real numbers. B) They are ever increasing. C) They are defined for all real numbers. D) They are discontinuous at x = 0. Show Answer Correct Answer: B) They are ever increasing. 41. Rolle's theorem is applicable for the function f (x) = |x-1| in [0, 2]. A) True. B) False. C) All the above. D) None of the above. Show Answer Correct Answer: B) False. 42. The function m(x) = |2x-3| is A) Discontinuous at x = 1.5. B) Continuous everywhere. C) Discontinuous at x = 0. D) Continuous at x = 3 but not at x = 0. Show Answer Correct Answer: B) Continuous everywhere. 43. The funciton f(x) = x$^{2}$-2x+1, continuous to the interval [-7, 7] A) TRUE. B) FALSE. C) All the above. D) None of the above. Show Answer Correct Answer: A) TRUE. 44. The function $f\left(x\right)=\frac{5}{x+2}$ A) 5. B) 2. C) 1. D) -2. Show Answer Correct Answer: D) -2. 45. Explain the concept of a piecewise function. A) A piecewise function is a function that is continuous everywhere. B) A piecewise function is a function that has only one sub-function. C) A piecewise function is a function that is defined by multiple sub-functions, each applying to a certain interval of the function's domain. D) A piecewise function is a function that has a fixed number of sub-functions. Show Answer Correct Answer: C) A piecewise function is a function that is defined by multiple sub-functions, each applying to a certain interval of the function's domain. 46. If Limit exist in a function it always unique A) False. B) True. C) Limit can be infinite. D) Limits can vary based on direction. Show Answer Correct Answer: B) True. 47. Find dy/dx if y$^{4}$ = x A) 4y$^{3}$ dy/dx = 0. B) 4y$^{3}$ dy/dx = x. C) 4y$^{3}$ = x. D) Dy/dx = 1/4y$^{3}$. Show Answer Correct Answer: D) Dy/dx = 1/4y$^{3}$. 48. Which of the following statements is true regarding the continuity of a function? A) A function must be differentiable to be continuous. B) All continuous functions are differentiable. C) A function can be continuous at a point but not differentiable. D) None of the above. Show Answer Correct Answer: C) A function can be continuous at a point but not differentiable. 49. Can a function be continuous but not differentiable? A) Yes. B) No. C) All the above. D) None of the above. Show Answer Correct Answer: A) Yes. 50. Every Differentialable function are continuous but converse is not true A) Every continuous function is differentiable. B) Differentiable functions can be discontinuous. C) Every differentiable function is continuous, but not every continuous function is differentiable. D) All continuous functions are differentiable. Show Answer Correct Answer: C) Every differentiable function is continuous, but not every continuous function is differentiable. 51. For a function, $f\left(x\right)$ $x=c$ $\lim_{x\rightarrow c}f\left(x\right)$ $f\left(c\right)$ $\lim_{x\rightarrow c}f\left(x\right)=f\left(c\right)$ A) I only. B) I and ii only. C) Iii only. D) I, ii, and iii. Show Answer Correct Answer: D) I, ii, and iii. 52. If (a, b) is a local maximum, then what will be true about f'(a)? A) Cannot be determined. B) It's negative. C) It's zero. D) It's positive. Show Answer Correct Answer: C) It's zero. 53. If $f\left(x\right)$ A) True. B) False. C) All the above. D) None of the above. Show Answer Correct Answer: A) True. 54. What is the significance of the derivative of a function? A) The derivative of a function is always equal to zero. B) The derivative of a function has no significance. C) The significance of the derivative of a function lies in understanding how the function is changing and its behavior at specific points. D) The derivative of a function represents the area under the curve. Show Answer Correct Answer: C) The significance of the derivative of a function lies in understanding how the function is changing and its behavior at specific points. 55. |sin x| is a differentiable function for every value of x. A) False. B) True. C) All the above. D) None of the above. Show Answer Correct Answer: A) False. 56. The function $f\left(x\right)=\frac{5}{x^4-16}$ A) True. B) False. C) Maybe. D) Niether. Show Answer Correct Answer: A) True. 57. What is the definition of continuity at a point c for a function f? A) F is continuous at c if lim (x $\rightarrow$ c) f(x) exists. B) F is continuous at c if f is differentiable at c. C) F is continuous at c if f(c) = 0. D) F is continuous at c if lim (x $\rightarrow$ c) f(x) = f(c). Show Answer Correct Answer: D) F is continuous at c if lim (x $\rightarrow$ c) f(x) = f(c). 58. Identify the point(s) of discontinuity. $f\left(x\right)=\frac{x-1}{3x^2-3}$ A) Only x = 1. B) Only x =-1. C) Only x = 3. D) X =-1 and x = 1. E) X = 1 and x = 3. Show Answer Correct Answer: D) X =-1 and x = 1. 59. A function is continuous at a number c if and only if $f\left(c\right)=\lim_{x\rightarrow c}f\left(x\right)$ A) TRUE. B) FALSE. C) All the above. D) None of the above. Show Answer Correct Answer: A) TRUE. 60. The function f (x) = |x| + |x-1| is A) Continuous at x = 1 but not at x = 0. B) Discontinuous at x = 0 as well as at x = 1. C) Continuous at x = 0 as well as at x = 1. D) Continuous at x = 0 but not at x = 1. Show Answer Correct Answer: B) Discontinuous at x = 0 as well as at x = 1. 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