Title 12 Maths AR Ch 6.pdf ✅

Class 12 Maths Chapter 6 Applications of Derivatives Assertion and Reason Questions Directions: Each of these questions contains two statements, Assertion and Reason. Each of these questions also has four alternative choices, only one of which is the correct answer. You have to select one of the codes (a), (b), (c) and (d) given below. (a) Assertion is correct, Reason is correct; Reason is a correct explanation for assertion. (b) Assertion is correct, Reason is correct; Reason is not a correct explanation for Assertion (c) Assertion is correct, Reason is incorrect (d) Assertion is incorrect, Reason is correct. 1. Assertion: Let f: R → R be a function such that f(x) = x 3 + x 2 + 3x + sin x. Then f is one- one. Reason: f(x) neither increasing nor decreasing function. 2. Assertion: f(x) = 2x 3 − 9x 2 + 12x − 3 is increasing outside the interval (1,2). Reason: f ′ (x) < 0 for x ∈ (1,2). 3. Assertion: The curves x = y 2 and xy = k cut at right angle, if 8k 2 = 1. Reason: Two curves intersect at right angle, if the tangents to the curves at the point of intersection are perpendicular to each other i.e., product of their slope is -1. 4. Assertion: If the radius of a sphere is measure as 9 m with an error of 0.03 m, then the approximate error in calculating its surface area is 2.16πm2 . Reason: We have, ΔS = ( ds dr) Δr where, ΔS = Approximate error in calculating the surface area, Δr = Error in measuring radius r. 5. Assertion: If the length of three sides of a trapezium other than base are equal to 10 cm, then the area of trapezium when it is maximum, is 75√3 cm2 . Reason: Area of trapezium is maximum at x = 5. 6. Assertion: If two positive numbers are such that sum is 16 and sum of their cubes is minimum, then numbers are 8,8. Reason: If f be a function defined on an interval I and c ∈ I and let f be
twice differentiable at c, then x = c is a point of local minima if f ′ (c) = 0 and f ′′(c) > 0 and f(c) is local minimum value of f. 7. Assertion: Let f: R → R be a function such that f(x) = x 3 + x 2 + 3x + sin x. Then f is one- one. Reason: f(x) neither increasing nor decreasing function. 8. Assertion: f(x) = cos2 x + cos3 (x + π 3 ) − cos xcos3 (x + π 3 ) then f ′ (x) = 0 Reason: Derivative of constant function is zero. 9. Assertion: The function f(x) = ae x+be −x ce x+de−x is increasing function of x, then bc > ad. Reason: f(x) is increasing if f ′ (x) > 0 for all x. 10. Assertion: The ordinate of a point describing the circle x 2 + y 2 = 25 decreases at the rate of 1.5 cm/s. The rate of change of the abscissa of the point when ordinate equals 4 cm is 2 cm/s. Reason: xdx + ydy = 0. 11. Assertion: If f ′ (x) = (x − 1) 3 (x − 2) 8 , then f(x) has neither maximum nor minimum at x = 2. Reason: f ′ (x) changes sign from negative to positive at x = 2. 12. Consider the function f(x) = { |sin x| for 0 < |x| ≤ π 2 1 2 for x = 0 Assertion: f has a local maximum value at x = 0. Reason: f ′ (0) = 0 and f ′′(0) < 0 13. Assertion: The maximum value of the function y = sin x in [0,2π] is at x = π 2 . Reason: The first derivative of the function is zero at x = π 2 and second derivative is negative at x = π 2 .
14. Assertion: The minimum value of the function y = cos x in [0,2π] is at x = π. Reason: The first derivative of the function is zero at x = π and second derivative is negative at x = π. 15. Assertion: The function y 2 = 4x has no absolute maximum or minimum. Reason: In the graph of the function the value of increases unboundedly and decreases unboundedly as x increases. Answers 1. (c) 2. (b) 3. (a) 4. (a) 5. (a) 6. (a) 7. (c) 8. (a) 9. (d) 10. (b) 11. (c) 12. (c) 13. (a) 14. (c) 15. (a)