JEE (Advanced) 2018 Paper-1

Let $$f:R \rightarrow R$$ and $$g:R\rightarrow R$$ be two non-constant differentiable functions. If $$f'(x) = (e^{(f(x)-g(x))}) g'(x)$$ for all $$x \in R$$, and $$f(1) = g(2) = 1$$, then which of the following statement(s) is (are) TRUE?

Let $$f:[0, \infty)$$ \rightarrow R be a continuous function such that $$f(x)=1-2x+ \int_{0}^{x} e^{x-t} f(t) dt$$ for all $$x \in [0,\infty]$$ Then, which of the following statement(s) is (are) TRUE?

The value of $$((\log_{2} 9)^{2})^{\frac{1}{\log_{2}(log_{2} 9)\times(\sqrt{7})^{\frac{1}{\log_{4} 7}}}}$$ ..........

The number of 5 digit numbers which are divisible by 4, with digits from the set {1, 2, 3, 4, 5} and the repetition of digits is allowed, is _____

Let 𝑋 be the set consisting of the first 2018 terms of the arithmetic progression 1, 6, 11, … , and 𝑌 be the set consisting of the first 2018 terms of the arithmetic
progression 9, 16, 23, … . Then, the number of elements in the set $$X \cup Y$$ is ..............

The number of real solutions of the equation
$$\sin^{-1}\left(\sum_{i=1}^\infty X^{i+1} -x\sum_{i=1}^\infty \left(\frac{x}{2}\right)^{i} \right )=\frac{\pi}{2}-\cos^{-1}\left(\sum_{i=1}^\infty \left(\frac{-x}{2}\right)^{i}-\sum_{i=1}^\infty \left(-x\right)^{i}\right)$$ lying in the interval $$(- \frac{1}{2},\frac{1}{2})$$(Here, the inverse trigonometric functions $$\sin^{−1}x and \cos^{−1}x$$ assume values in [$$- \frac{\pi}{2},\frac{\pi}{2}$$] and and $$[0, \pi]$$, respectively.)

For each positive integer n, let
$$Y_{n} = \frac{1}{n}\left((n+1)(n+2).......(n+n)\right)^{\frac{1}{n}}$$
for $$x \in R$$, let [x] be the greatest integer less than or equal to x. If $$\lim_{n \rightarrow \infty} Y_{n} = L$$, then the value of [L] is ..........

let $$\overrightarrow{a}$$ and $$\overrightarrow{b}$$ be two unit vectors such that $$\overrightarrow{a} .\overrightarrow{b}$$ =0 For some $$X,y \in R$$ let $$\overrightarrow{C}$$=X$$\overrightarrow{a}$$+Y$$\overrightarrow{b}$$+$$(\overrightarrow{a} \times \overrightarrow{a})$$.If $$ \mid \overrightarrow{c} \mid$$=2 and the $$\overrightarrow{c}$$ is inclined at the same angle $$\alpha$$ to both $$\overrightarrow{a}$$ and $$\overrightarrow{b}$$,then the value of $$8 \cos^{2} \alpha$$ is .......

Let a, b, c be three non-zero real numbers such that the equation $$\sqrt{3}a \cos x+2b \sin x=C,X \in [-\frac{\pi}{2},\frac{\pi}{2}]$$ has two distinct real roots $$\alpha$$ and $$\beta$$ with$$\alpha+\beta=\frac{\pi}{3}$$, then the value of $$\frac{b}{a}$$ is ....... .

A farmer $$𝐹_{1}$$ has a land in the shape of a triangle with vertices at 𝑃(0, 0), 𝑄(1, 1) and 𝑅(2, 0). From this land, a neighbouring farmer $$𝐹_{2}$$ takes away the region which lies between the side 𝑃𝑄 and a curve of the form y =$$ 𝑥^{n}(n >1)$$ If the area of the region taken away by the farmer $$𝐹_{2}$$ is exactly 30% of the area of $$\triangle$$ 𝑃𝑄𝑅, then the value of 𝑛 is .......