JEE Advanced · Mathematics · 23. C&D

Let $f : [- \frac{1}{2}, 2] \to R$ and $g : [- \frac{1}{2}, 2] \to R$ be functions defined by $f (x) = [x^{2} - 3]$ and $g (x) = |x| f (x) + |4 x - 7| f (x)$ , where $[y]$ denotes the greatest integer less than or equal to $y$ for $y \in R$ . Then

A $f$ is discontinuous exactly at three points in $[- \frac{1}{2}, 2]$
B $f$ is discontinuous exactly at four points in $[- \frac{1}{2}, 2]$
C $g$ is NOT differentiable exactly at four points in $(- \frac{1}{2}, 2)$
D $g$ is NOT differentiable exactly at five points in $(- \frac{1}{2}, 2)$

Verified Solution

Answer & Solution

Correct Answer

(C) $g$ is NOT differentiable exactly at four points in $(- \frac{1}{2}, 2)$

Step-by-step Solution

Detailed explanation

f (x) = [x^{2}] - 3

g (x) = (|x| + |4 x - 7|) ([x^{2}] - 3)

∵

f

is discontinuous at (when x² is an integer)

x = 1, \sqrt{2}, \sqrt{3}, 2 i n [- \frac{1}{2}, 2]

and

|x| + |4 x - 7| \neq 0

x = 1, \sqrt{2}, \sqrt{3}, 2

\Rightarrow

g (x)

is discontinuous at

x = 1, \sqrt{2}, \sqrt{3}

(- \frac{1}{2}, 2)

In

(0 - δ, 0 + δ)

(

δ

is very small)

⇒ g (x) = (|x| + |4 x - 7|) . (- 3)

\Rightarrow

'g' is non derivable at

x = 0

.as

| x |

is there which is not diff. at

x = 0

In

(\frac{7}{4} - δ, \frac{7}{4} + δ)

⇒ g (x) = 0

f (x) = 0

\Rightarrow

Derivable at

x = \frac{7}{4}

∴

'g' is non - derivable at

0, 1, \sqrt{2}, \sqrt{3}

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

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From JEE Advanced

An ideal gas is undergoing a cyclic thermodynamic process in different ways as shown in the corresponding P-V diagrams in column 3 of the table. Consider only the path from state 1 to state 2. W denotes the corresponding work done on the system. The equations and plots in the table have standard notations as used in thermodynamic process. Here

γ

is the ratio of heat capacities at constant pressure and constant volume. The number of moles in the gas is n.

Column – 1	Column – 2	Column – 3
(I) $W_{1 \rightarrow 2}=\frac{1}{\gamma-1}$ $\left(P_2 V_2-P_1 V_1\right)$	(i) Isothermal	(P)
(II) $W_{1 \rightarrow 2}=$ $-P V_2+P V_1$	(ii) Isochoric	(Q)
(III) $W_{1 \rightarrow 2}=0$	(iii) Isobaric	(R)
(IV) $W_{1 \rightarrow 2}=$ $-n R T \ln \left(\frac{V_2}{V_1}\right)$	(iv) Adiabatic	(S)

Which one of the following options is the correct combination?

JEE Advanced 2017 Medium

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Let f :-12, 2→R and g:-12,2→R be functions defined by fx=x2-3 and gx=xfx+4x-7fx , where y denotes the greatest integer less than or equal to y for y∈R . Then

Step-by-step Solution

Let $f : [- \frac{1}{2}, 2] \to R$ and $g : [- \frac{1}{2}, 2] \to R$ be functions defined by $f (x) = [x^{2} - 3]$ and $g (x) = |x| f (x) + |4 x - 7| f (x)$ , where $[y]$ denotes the greatest integer less than or equal to $y$ for $y \in R$ . Then