CBSE Class 12 Sample Paper for 2022 Boards (MCQ Based - for Term 1)

Question 2 - CBSE Class 12 Sample Paper for 2022 Boards (MCQ Based - for Term 1) - Solutions of Sample Papers and Past Year Papers - for Class 12 Boards

Last updated at Sept. 4, 2021 by

The value of k (k < 0) for which the function
f
defined as
f (x)={((1 -cosβ‘kx)/(x sinβ‘x ),xβ 0Β 1/2,x=0)β€ is continuous at x = 0 is :

(a) Β±Β 1Β (b) β1Β Β Β (c) Β± Β 1/2Β (d) 1/2

This question is
inspired from -
Β
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)

CBSE Class 12 Sample Paper for 2022 Boards (MCQ Based - for Term 1)

Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 10 years. He provides courses for Maths and Science at Teachoo.