JEE Mains JEE Advance BITSAT
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Let \( n \ge 2 \) be a natural number and \( f:[0,1]\to \mathbb{R} \) be the function defined by

If \( n \) is such that the area of the region bounded by the curves \( x=0 \), \( x=1 \), \( y=0 \) and \( y=f(x) \) is \( 4 \), then the maximum value of the function \( f \) is _____.

The product of all positive real values of \( x \) satisfying the equation \( x^{\left(16(\log_{5}x)^{3}-68\log_{5}x\right)}=5^{-16} \) is _______.

Let \( S = (0,1) \cup (1,2) \cup (3,4) \) and \( T = \{0,1,2,3\} \). Then which of the following statement is(are) true?

Let \( f:\mathbb{R}\to \mathbb{R} \) be a function such that \( f(x+y)=f(x)+f(y) \) for all \( x,y\in \mathbb{R} \), and \( g:\mathbb{R}\to (0,\infty) \) be a function such that \( g(x+y)=g(x)g(y) \) for all \( x,y\in \mathbb{R} \). If \( f\!\left(\dfrac{-3}{5}\right)=12 \) and \( g\!\left(\dfrac{-1}{3}\right)=2 \), then the value of \( \left( f\!\left(\dfrac{1}{4}\right)+g(-2)-8 \right) g(0) \) is ____________.

Let \( \mathbb{R} \) denote the set of all real numbers. Let \( a_i, b_i \in \mathbb{R} \) for \( i \in \{1,2,3\} \). Define the functions \( f : \mathbb{R} \to \mathbb{R},\ g : \mathbb{R} \to \mathbb{R} \) and \( h : \mathbb{R} \to \mathbb{R} \) by
\( f(x)=a_1+10x+a_2x^2+a_3x^3+x^4 \), \( g(x)=b_1+3x+b_2x^2+b_3x^3+x^4 \), \( h(x)=f(x+1)-g(x+2) \).
If \( f(x)\neq g(x) \) for every \( x\in\mathbb{R} \), then the coefficient of \( x^3 \) in \( h(x) \) is:

Let \( \mathbb{R} \) denote the set of all real numbers. Let \( f:\mathbb{R}\to\mathbb{R} \) and \( g:\mathbb{R}\to(0,4) \) be functions defined by \( f(x)=\log_e\!\left(x^2+2x+4\right) \), and \( g(x)=\dfrac{4}{1+e^{-2x}} \). Define the composite function \( f\circ g^{-1} \) by \( (f\circ g^{-1})(x)=f\!\left(g^{-1}(x)\right) \), where \( g^{-1} \) is the inverse of the function \( g \).
Then the value of the derivative of the composite function \( f\circ g^{-1} \) at \( x=2 \) is ___________.

Let \( f : \mathbb{R} \to \mathbb{R} \) and \( g : \mathbb{R} \to \mathbb{R} \) be functions defined by

Let \( a, b, c, d \in \mathbb{R} \). Define the function \( h : \mathbb{R} \to \mathbb{R} \) by
\( h(x) = a\,f(x) + b\!\left(g(x) + g\!\left(\dfrac{1}{2} - x\right)\right) + c(x - g(x)) + d\,g(x),\ x \in \mathbb{R}. \)
Match each entry in List-I to the correct entry in List-II.

The correct option is:

Let \( \mathbb{R} \) denote the set of all real numbers. Consider the polynomial function \( f:\mathbb{R}\to\mathbb{R} \) defined by
\( f(x)=\dfrac{d^{10}}{dx^{10}}\!\left((x^{2}-1)^{10}\right) \), for all \( x\in\mathbb{R} \).
Here \( \dfrac{d^{10}}{dx^{10}}\!\left((x^{2}-1)^{10}\right) \) is the \( 10^{\text{th}} \) order derivative of the function \( (x^{2}-1)^{10} \).
Then which of the following statements is (are) TRUE?

Let \( f:\mathbb{R}\to \mathbb{R} \) be a function defined by

Then which of the following statements is TRUE?

Let \( \mathbb{N} \) denote the set of all natural numbers, and \( \mathbb{Z} \) denote the set of all integers. Consider the functions \( f:\mathbb{N}\to \mathbb{Z} \) and \( g:\mathbb{Z}\to \mathbb{N} \) defined by
Define \( (g^\circ f)(n)=g(f(n)) \) for all \( n\in \mathbb{N} \), and \( (f^\circ g)(n)=f(g(n)) \) for all \( n\in \mathbb{Z} \).
Then which of the following statements is(are) TRUE?

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JEE Advanced 2023 Paper - 1

Let \( S = (0,1) \cup (1,2) \cup (3,4) \) and \( T = \{0,1,2,3\} \). Then which of the following statement is(are) true?