## Question 5 At which of the following intervals will f(x) be increasing? (a) (-∞", "  (-1)/2) ∪ (1/2 ", " ∞) (b) (1/2,0) ∪ (1/2 ", " ∞) (c) (0", "  1/2) ∪ (1/2 ", " ∞) (d) (-∞", "  (-1)/2) ∪ (0", "  1/2)

1. Chapter 6 Class 12 Application of Derivatives (Term 1)
2. Serial order wise
3. Case Based Questions (MCQ)

Transcript

Question The shape of a toy is given as π(π₯)=6(2π₯^4βπ₯^2). To make the toy beautiful 2 sticks which are perpendicular to each other were placed at a point (2, 3), above the toy. Question 1 Which value from the following may be abscissa of critical point? (a) Β± 1/4 (b) Β± 1/2 (c) Β± 1 (d) None Critical point is point where πβ²(π)=π π(π₯)=6(2π₯^4βπ₯^2 ) Finding πβ²(π) π^β² (π₯)=π(6(2π₯^4βπ₯^2 ) )/ππ₯ =6(2 Γ 4π₯^3β2π₯) =6(8π₯^3β2π₯) =6 Γ 2π₯(4π₯^2β1) =πππ(ππ^πβπ) Putting π^β² (π)= 0 12π₯(4π₯^2β1)=0 12π₯ = 0 π = 0 4π₯^2β1= 0 4π₯^2=1 π₯^2=1/4 π = Β± π/π β΄ x = 0, Β± π/π So, the correct answer is (b) Question 2 Find the slope of the normal based on the position of the stick. (a) 360 (b) β360 (c) 1/360 (d) (β1)/360 We need to find Slope of Normal at (2, 3) First, letβs find Slope of Tangent at (2, 3) Slope of tangent = π^β² (π₯) = 12π₯(4π₯^2β1) Putting x = 2 = 12 Γ 2 Γ (4(2)2 β 1) = 24(4 Γ 4 β 1) = 24(16 β 1) = 24 Γ 15 = 360 Thus, Slope of Normal = (β1)/(πππππ ππ πππππππ‘) = (βπ)/πππ So, the correct answer is (d) Question 3 What will be the equation of the tangent at the critical point if it passes through (2, 3)? (a) x + 360y = 1082 (b) y = 360x β 717 (c) x = 717y + 360 (d) none We found the slope of tangent at (2, 3) Slope of tangent at (2, 3) = 360 Finding equation of line passing through (2, 3) with slope 360 y β y1 = Slope Γ (x β x1) y β 3 = 360 (x β 2) y β 3 = 360x β 720 y = 360x β 720 + 3 y = 360x β 717 So, the correct answer is (b) Question 4 Find the second order derivative of the function at x = 5. (a) 598 (b) 1176 (c) 3588 (d) 3312 We know that π(π₯)=6(2π₯^4βπ₯^2 ) And, π^β² (π) = 12π₯(4π₯^2β1) = πππ^πβπππ Now, π^β²β² (π₯)=(π(48π₯^3 β 12π₯))/ππ₯ = 48 Γ 3π₯^2β12 = ππππ^πβππ Putting x = 5 = 144 Γ 5^2β12 = 144 Γ 25β12 = 3600β12 = ππππ So, the correct answer is (c) Question 5 At which of the following intervals will f(x) be increasing? (a) (ββ", " (β1)/2) βͺ (1/2 ", " β) (b) (1/2,0) βͺ (1/2 ", " β) (c) (0", " 1/2) βͺ (1/2 ", " β) (d) (ββ", " (β1)/2) βͺ (0", " 1/2) π(π₯) is increasing where π^β² (π)>π Putting fβ(x) = 0 12π₯(4π₯^2β1)=0 12π₯((2π₯)^2β1^2 )=0 12π₯(2π₯β1)(2π₯+1)=0 So, x = 0, π/π , (βπ)/π Hence, β΄ f is strictly increasing in ((βπ)/π,π) and (π/π ", " β) So, the correct answer is (b)

Case Based Questions (MCQ)