CBSE Class 12 Sample Paper for 2022 Boards (MCQ Based - for Term 1)
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CBSE Class 12 Sample Paper for 2022 Boards (MCQ Based - for Term 1)
Last updated at April 16, 2024 by Teachoo
This question is inspired from - Question 21 - CBSE Class 12 Sample Paper for 2021 Boards
Question 2 The value of k (k < 0) for which the function f defined as π (π₯)={β((1 βcosβ‘ππ₯)/(π₯ sinβ‘π₯ ),π₯β 0@1/2, π₯=0)β€ is continuous at x = 0 is : (a) Β± 1 (b) β1 (c) Β± 1/2 (d) 1/2 Given that function is continuous at x = 0 π(π₯) is continuous at x = 0 i.e. (π₯π’π¦)β¬(π±βπ) π(π)=π(π) Limit at x β 0 (πππ)β¬(π₯β0) f(x) = (πππ)β¬(ββ0) f(h) = limβ¬(hβ0) (1 β cosβ‘πβ)/(β (sinβ‘β) ) = limβ¬(hβ0) (π γπ¬π’π§γ^πβ‘γππ/πγ)/(β (sinβ‘β)) = limβ¬(hβ0) (2 sin^2β‘γπβ/2γ)/1 Γ1/(β (sinβ‘β)) = (πππ)β¬(π‘βπ) (π γπππγ^πβ‘γππ/πγ)/(ππ/π)^π Γ (ππ/π)^π/(π (πππβ‘π)) = limβ¬(hβ0) (2 sin^2β‘γπβ/2γ)/(πβ/2)^2 Γ (π^2 β^2)/(4β (sinβ‘β)) = limβ¬(hβ0) sin^2β‘γπβ/2γ/(πβ/2)^2 Γ (π^π π)/(π(πππβ‘π)) = π^π/π limβ¬(hβ0) sin^2β‘γπβ/2γ/(πβ/2)^2 Γ β/sinβ‘β = π^2/2 limβ¬(hβ0) sin^2β‘γπβ/2γ/(πβ/2)^2 Γlimβ¬(hβ0) β/sinβ‘β = π^2/2 Γ 1 Γ 1 = π^π/π Now, (π₯π’π¦)β¬(π±βπ) π(π)=π(π) π^2/2 = 1/2 π^2 =1 π =Β±π But, given that k < 0 Thus, only value is k = β1 So, the correct answer is (b)
