Match the followingColumn - IColumn - IIA)If f(x)={sin{x},x<1cosx+a;x≥1 where {.} denotes the fractional part function, such that f(x) is continuous at x=1. If |k|=a2sin(4−π)4 then k isp)1B)If the function f(x)=(1−cos(sinx))x2 is continuous at x=0, then f(0) isq)0C)f(x)=[x,x∈Q1−x,x∉Q, then the values of x at whichf(x) is continuousr)−1D)If f(x)=x+{−x}+[x], where [x] and {x} represents integral and fractional part of x, then the values of at which f(x) is discontinuouss)12

A) limh→0sin{1−h}=cos1+a ⇒ limh→0sin{1−h}−cos1=a ⇒ a=sin1−cos1 Now |k|=sin1−cos12(sin1.12−cos1.12)=1 k=±1B) f(0)=limx→02sin2(sinx2)x2(sinx2)2×(sinx2)2 ⇒ f(0)=12C) Function should have same rule for Q & Q' ⇒x=1−x⇒x=12D) f(x)=x+{−x}+[x]x is continuous at x∈RCheck x=I (where is integer), f(I+)=2I+1 f(I−)=2I−1 so f(x) is discontinuous at every integeri.e., 1, 0, - 1

Match the followingColumn - IColumn - IIA)If f(x)={sin{x},x<1cosx+a;x≥1 where {.} denotes the fractional part function, such that f(x) is continuous at x=1. If |k|=a2sin(4−π)4 then k isp)1B)If the function f(x)=(1−cos(sinx))x2 is continuous at x=0, then f(0) isq)0C)f(x)=[x,x∈Q1−x,x∉Q, then the values of x at whichf(x) is continuousr)−1D)If f(x)=x+{−x}+[x], where [x] and {x} represents integral and fractional part of x, then the values of at which f(x) is discontinuouss)12

A) limh→0sin{1−h}=cos1+a ⇒ limh→0sin{1−h}−cos1=a ⇒ a=sin1−cos1 Now |k|=sin1−cos12(sin1.12−cos1.12)=1 k=±1B) f(0)=limx→02sin2(sinx2)x2(sinx2)2×(sinx2)2 ⇒ f(0)=12C) Function should have same rule for Q & Q' ⇒x=1−x⇒x=12D) f(x)=x+{−x}+[x]x is continuous at x∈RCheck x=I (where is integer), f(I+)=2I+1 f(I−)=2I−1 so f(x) is discontinuous at every integeri.e., 1, 0, - 1

AI Mentor

Check Your IQ

Free Expert Demo

Try Test

Courses

Dropper NEET Course Dropper JEE Course Class - 12 NEET Course Class - 12 JEE Course Class - 11 NEET Course Class - 11 JEE Course Class - 10 Foundation NEET Course Class - 10 Foundation JEE Course Class - 10 CBSE Course Class - 9 Foundation NEET Course Class - 9 Foundation JEE Course Class -9 CBSE Course Class - 8 CBSE Course Class - 7 CBSE Course Class - 6 CBSE Course

Offline Centres

Match the following
Column - I Column - II
A) If $f (x) = {\begin{matrix} \sin {x}, & x < 1 \\ \cos x + a; & x \geq 1 \end{matrix}$ where {.} denotes the fractional part function, such that $f (x)$ is continuous at $x = 1.$ If $| k | = \frac{a}{\sqrt{2} \sin \frac{(4 - π)}{4}}$ then $k$ is p) 1
B) If the function $f (x) = \frac{(1 - \cos (\sin x))}{x^{2}}$ is continuous at $x = 0,$ then $f (0)$ is q) 0
C) $f (x) = [\begin{matrix} x & , & x \in Q \\ 1 - x & , & x \notin Q \end{matrix},$ then the values of $x$ at which $f (x)$ is continuous r) $- 1$
D) If $f (x) = x + {- x} + [x],$ where $[x]$ and ${x}$ represents integral and fractional part of $x,$ then the values of at which $f (x)$ is discontinuous s) $\frac{1}{2}$

see full answer

Your Exam Success, Personally Taken Care Of

1:1 expert mentors customize learning to your strength and weaknesses – so you score higher in school , IIT JEE and NEET entrance exams.

An Intiative by Sri Chaitanya

A-pr, B-s, C-s, D-pqr

A-rs, B-pqrs, C-p, D-r

A-p, B-s, C-pq, D-prs

A-r, B-p, C-q, D-s

answer is A.

(Unlock A.I Detailed Solution for FREE)

Best Courses for You

JEE

NEET

Foundation JEE

Foundation NEET

CBSE

Detailed Solution

A) $\lim_{h \to 0} \sin {1 - h} = \cos 1 + a$
$\Rightarrow \lim_{h \to 0} \sin {1 - h} - \cos 1 = a \Rightarrow a = \sin 1 - \cos 1$
Now $| k | = \frac{\sin 1 - \cos 1}{\sqrt{2} (\sin 1. \frac{1}{\sqrt{2}} - \cos 1. \frac{1}{\sqrt{2}})} = 1$
$k = \pm 1$
B) $f (0) = \lim_{x \to 0} \frac{2 \sin^{2} (\frac{\sin x}{2})}{x^{2} {(\frac{\sin x}{2})}^{2}} \times {(\frac{\sin x}{2})}^{2}$
$\Rightarrow f (0) = \frac{1}{2}$
C) Function should have same rule for $Q & Q'$
$\Rightarrow x = 1 - x \Rightarrow x = \frac{1}{2}$
D) $f (x) = x + {- x} + [x]$
$x$ is continuous at $x \in R$
Check $x = I$ (where is integer),
$f (I^{+}) = 2 I + 1 f (I^{-}) = 2 I - 1$
so $f (x)$ is discontinuous at every integer
i.e., 1, 0, - 1

Watch 3-min video & get full concept clarity

courses

No courses found

Result Producing online coaching for JEE/NEET of south India

Largest All India Test Series for JEE/NEET

Best Online Course for IIT JEE

Best Online Course for NEET

Best Online Course for CBSE

Ready to Test Your Skills?

Check your Performance Today with our Free Mock Test used by Toppers!

Take Free Test

Get Expert Academic Guidance – Connect with a Counselor Today!