Let f(x)=(1-x)^ 2 ^ 2 x+x^ 2 for all x ∈ I R and let g(x)=∫_ 1 ^ x ((2(t-1))/(t+1)- t) f(t) d t for all x ∈(1, ∞) . Question: Consider the statements: P : There exists some x ∈ R such that f(x)+2 x =2(1+x^ 2 ) Q : There

Correct answer: C. P is false and Q is true For the statement P, f(x)+2 x=2(1+x^ 2 ) (1-x)^ 2 ^ 2 x+x^ 2 +2 x=2(1+x^ 2 ) (1-x)^ 2 ^ 2 x=x^ 2 -2 x+1+1 (1-x)^ 2 ^ 2 x=(1-x)^ 2 +1 (1-x)^ 2 ^ 2 x=-1 , which is not possible for any real value of x . Hence P is not true. Let H(x)=2 f(x)+1-2 x(1+x) H(0)=2 f(0)+1-0=1 and H(1)=2 f(1)+1-4=-3 Hence, H(x) has a solution in (0,1) Therefore, Q is true.

JEE Advanced 2012Paper 2

Exam: JEE Advanced
Year: 2012
Paper: Paper 2
Subject: Mathematics
Chapter: Functions

JEE Advanced 2012 · Paper 2 · Mathematics · Functions

Let f(x)=(1-x)^ 2 ^ 2 x+x^ 2 for all x ∈ I R and let g(x)=∫_ 1 ^ x...

Let

f(x)=(1-x)^{2} \sin ^{2} x+x^{2}

for all

x \in I R and let

g(x)=\int_{1}^{x}\left(\frac{2(t-1)}{t+1}-\ln t\right) f(t) d t

for all

x \in(1, \infty)

. <strong> Question:</strong> Consider the statements:

: There exists some

x \in \mathrm{R}

such that

f(x)+2 x =2\left(1+x^{2}\right)

Q

: There exists some

x \in \mathrm{R}

such that

2 f(x)+1

=2 x(1+x)

Then

Options

Aboth $P$ and $Q$ are true
B $P$ is true and $Q$ is false
C $P$ is false and $Q$ is true
Dboth $P$ and $Q$ are false

Correct answer

C. $P$ is false and $Q$ is true

Step-by-step solution

For the statement

P, f(x)+2 x=2\left(1+x^{2}\right)

\Rightarrow(1-x)^{2} \sin ^{2} x+x^{2}+2 x=2\left(1+x^{2}\right)

\Rightarrow(1-x)^{2} \sin ^{2} x=x^{2}-2 x+1+1

\Rightarrow(1-x)^{2} \sin ^{2} x=(1-x)^{2}+1

\Rightarrow(1-x)^{2} \cos ^{2} x=-1

, which is not possible for any real value of

x

. Hence

P

is not true. Let

H(x)=2 f(x)+1-2 x(1+x)

H(0)=2 f(0)+1-0=1

and

H(1)=2 f(1)+1-4=-3

Hence,

H(x)

has a solution in

(0,1)

Therefore,

Q

is true.

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