NCERT Exemplar - MCQs

Chapter 5 Class 12 Continuity and Differentiability
Serial order wise

(D) 1.5

This question is similar to Ex 5.1, 18 - Chapter 5 Class 12 - Continuity and Differentiability

Transcript

Question 1 The function f (x) = {β 8(sinβ‘π₯/π₯ " + cos x, if x " β " 0" @π ", if x " =" 0" )β€ is continuous at x = 0, then the value of k is (A) 3 (B) 2 (C) 1 (D) 1.5 At π = 0 f(x) is continuous at π₯=0 if L.H.L = R.H.L = π(0) if if limβ¬(xβ0^β ) π(π₯) = limβ¬(xβ0^+ ) π(π₯) = π(0) LHL at x β 0 (πππ)β¬(π±βπ^β ) f(x) = limβ¬(hβ0) f(0 β h) = (πππ)β¬(π‘βπ) f(βh) = limβ¬(hβ0) sinβ‘γ(ββ)γ/((ββ)) " + cos (β h)" = (πππ)β¬(π‘βπ) γβπππγβ‘π/(βπ) " +" (πππ)β¬(π‘βπ) " cos h" = limβ¬(hβ0) sinβ‘β/β " + " (πππ)β¬(ββ0) " cos h" Using limβ¬(xβ0) sinβ‘π₯/π₯=1 = 1 + "cos 0" = 1 + 1 = 2 RHL at x β 0 (πππ)β¬(πβπ^+ ) f(x) = limβ¬(hβ0) f(0 + h) = (πππ)β¬(π‘βπ) f(h) = limβ¬(hβ0) sinβ‘β/β " + cos h" = (πππ)β¬(π‘βπ) πππβ‘π/π " + " (πππ)β¬(π‘βπ) " cos h" Using limβ¬(xβ0) sinβ‘π₯/π₯=1 = 1 + "cos 0" = 1 + 1 = 2 At π=π π(π₯)=π π(π)=π Since f(x) is continuous at x = 0. L.H.L = R.H.L = π(0) π(0)=2 π=π So, the correct answer is (B)