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Functions

30 questions · Mathematics · JEE Advanced

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JEE Advanced
Mathematics
Functions

Functions

30 questions · Mathematics · JEE Advanced

Let ℕ denote the set of all natural numbers, and ℤ denote the set of all integers. Consider the functions f: ℕ → ℤ and g: ℤ → ℕ defined by $f (n) = {\frac{( n + 1 )}{2} \frac{( 4 - n )}{2} if n is odd, if n is even,$ …2025 · Shift 1 · Q22 · Multiple correct
Let ℝ denote the set of all real numbers. Let f: ℝ → ℝ be a function such that f(x) > 0 for all x ∈ ℝ, and f(x+y) = f(x)f(y) for all x, y ∈ ℝ. Let the real numbers a₁, a₂, ..., a₅₀ be in an arithmetic progression. If f(a₃₁) = 64f(a₂₅),…2025 · Shift 1 · Q28 · Numerical
Let $R$ denote the set of all real numbers. Let $f : R \to R$ and $g : R \to (0, 4)$ be functions defined by $f (x) = lo g_{e} (x^{2} + 2 x + 4), and g (x) = \frac{4}{1 + e ^{- 2 x}}$ …2025 · Shift 2 · Q30 · Numerical
Let $f : R \to R$ be a function such that $f (x + y) = f (x) + f (y)$ for all $x, y \in R$ , and $g : R \to (0, \infty)$ be a function such that $g (x + y) = g (x) g (y)$ for all $x, y \in R$ . If $f (\frac{- 3}{5}) = 12$ …2024 · Shift 2 · Q25 · Numerical
Let the function $f : R \to R$ be defined by $f (x) = \frac{s i n x}{e ^{π x}} \frac{( x ^{2023} + 2024 x + 2025 )}{( x ^{2} - x + 3 )} + \frac{2}{e ^{π x}} \frac{( x ^{2023} + 2024 x + 2025 )}{( x ^{2} - x + 3 )} .$ …2024 · Shift 2 · Q27 · Numerical
Let $S = (0, 1) \cup (1, 2) \cup (3, 4)$ and $T = {0, 1, 2, 3}$ . Then which of the following statements is(are) true?2023 · Shift 1 · Q18 · Multiple correct
Let $f : [0, 1] \to [0, 1]$ be the function defined by $f (x) = \frac{x ^{3}}{3} - x^{2} + \frac{5}{9} x + \frac{17}{36}$ . Consider the square region $S = [0, 1] \times [0, 1]$ . Let $G = {(x, y) \in S : y > f (x)}$ be called the green region and $R = {(x, y) \in S : y < f (x)}$ …2023 · Shift 1 · Q20 · Multiple correct
Let $∣ M ∣$ denote the determinant of a square matrix $M$ . Let $g : [0, \frac{π}{2}] \to R$ be the function defined by $g (θ) = f (θ) - 1 + f (\frac{π}{2} - θ) - 1$ where $f (θ) = \frac{1}{2} 1 - sin θ - 1 sin θ 1 - sin θ 1 sin θ 1 + sin π sin (θ - \frac{π}{4}) cot (θ + \frac{π}{4}) cos (θ + \frac{π}{4}) - cos \frac{π}{2} lo g_{e} (\frac{π}{4}) tan (θ - \frac{π}{4}) lo g_{e} (\frac{4}{π}) tan π .$ …2022 · Shift 1 · Q32 · Multiple correct
If the function f : R $\to$ R is defined by f(x) = |x| (x $-$ sin x), then which of the following statements is TRUE?2020 · Shift 1 · Q20 · MCQ
Let f : [0, 2] $\to$ R be the function defined by $f (x) = (3 - sin (2 π x)) sin (π x - \frac{π}{4}) - sin (3 π x + \frac{π}{4})$ If $α, β \in [0, 2]$ are such that ${x \in [0, 2] : f (x) \geq 0} = [α, β]$ …2020 · Shift 1 · Q33 · Numerical
For a polynomial g(x) with real coefficients, let mg denote the number of distinct real roots of g(x). Suppose S is the set of polynomials with real coefficients defined by $S = {(x^{2} - 1)^{2} (a_{0} + a_{1} x + a_{2} x^{2} + a_{3} x^{3}) : a_{0}, a_{1}, a_{2}, a_{3} \in R}$ …2020 · Shift 1 · Q35 · Numerical
Let the function f : [0, 1] $\to$ R be defined by $f (x) = \frac{4 ^{x}}{4 ^{x} + 2}$ Then the value of $f (\frac{1}{40}) + f (\frac{2}{40}) + f (\frac{3}{40}) + ... + f (\frac{39}{40}) - f (\frac{1}{2})$ …2020 · Shift 2 · Q34 · Numerical
Let the function $f : (0, π) \to R$ be defined by $f (θ) = (sin θ + cos θ)^{2} + (sin θ - cos θ)^{4}$ Suppose the function f has a local minimum at $θ$ precisely when $θ \in {λ_{1} π, ...., λ_{r} π}$ …2020 · Shift 2 · Q36 · Numerical
Let X be a set with exactly 5 elements and Y be a set with exactly 7 elements. If $α$ is the number of one-one functions from X to Y and $β$ is the number of onto functions from Y to X, then the value of $\frac{1}{5 !} (β - α)$ …2018 · Shift 2 · Q27 · Numerical
Let $E_{1} = {x \in R : x \neq = 1 and \frac{x}{x - 1} > 0}$ and $E_{2} = {x \in E_{1} : sin^{- 1} (lo g_{e} (\frac{x}{x - 1})) is a real number}$ (Here, the inverse…2018 · Shift 2 · Q33 · MCQ
Let S = {1, 2, 3, .........., 9}. For k = 1, 2, .........., 5, let Nk be the number of subsets of S, each containing five elements out of which exactly k are odd. Then N1 + N2 + N3 + N4 + N5 =2017 · Shift 2 · Q23 · MCQ
Let $f (x) = sin (\frac{π}{6} sin (\frac{π}{2} sin x))$ for all $x \in R$ and g(x) = $\frac{π}{2} sin x$ for all x $\in$ R. Let $(f \circ g) (x)$ denote f(g(x)) and $(g \circ f) (x)$ denote g(f(x)).…2015 · Shift 1 · Q40 · Multiple correct
For every pair of continuous function f, g : [0, 1] $\to$ R such that max {f(x) : x $\in$ [0, 1]} = max {g(x) : x $\in$ [0, 1]}. The correct statement(s) is (are)2014 · Shift 1 · Q33 · Multiple correct
Let $f : (- \frac{π}{2}, \frac{π}{2}) \to R$ be given by $f (x) = [lo g (sec x + tan x)]^{3}$ . Then,2014 · Shift 1 · Q36 · Multiple correct
Let f1 : R $\to$ R, f2 : [0, $\infty$ ) $\to$ R, f3 : R $\to$ R, and f4 : R $\to$ [0, $\infty$ ) be defined by $f_{1} (x) = {∣ x ∣ e^{x} i f x < 0, i f x \geq 0;$ … Includes diagram2014 · Shift 2 · Q40 · MCQ
The function $f : [0, 3] \to [1, 29]$ , defined by $f (x) = 2 x^{3} - 15 x^{2} + 36 x + 1$ , is2012 · Shift 1 · Q39 · MCQ
Let $f : (- 1, 1) \to R$ be such that $f (cos 4 θ) = \frac{2}{2 - s e c ^{2} θ}$ for $θ \in (0, \frac{π}{4}) \cup (\frac{π}{4}, \frac{π}{2})$ . Then the value(s) of $f (\frac{1}{3})$ …2012 · Shift 2 · Q40 · Multiple correct
Let f(x) = x2 and g(x) = sin x for all x $\in$ R. Then the set of all x satisfying $(f \circ g \circ g \circ f) (x) = (g \circ g \circ f) (x)$ , where $(f \circ g) (x) = f (g (x))$ , is2011 · Shift 2 · Q34 · MCQ
Let $f : (0, 1) \to R$ be defined by $f (x) = \frac{b - x}{1 - b x}$ , where b is a constant such that $0 < b < 1$ . Then2011 · Shift 2 · Q37 · Multiple correct
Match the statements given in Column I with the intervals/union of intervals given in Column II : Includes diagram2011 · Shift 2 · Q40 · MCQ
Let $h$ and $h$ be real valued functions defined on the interval $[0, 1]$ by $f (x) = e^{x^{2}} + e^{- x^{2}}$ , $g (x) = x e^{x^{2}} + e^{- x^{2}}$ and $h (x) = x^{2} e^{x^{2}} + e^{- x^{2}}$ . If $c$ and $c$ denote, respectively, the absolute maximum of $h$ and $h$ …2010 · Shift 1 · Q52 · MCQ
Let $S = {1, 2, 3, 4}$ . The total number of unordered pairs of disjoint subsets of $S$ is equal to :2010 · Shift 2 · Q33 · MCQ
Consider the polynomial $f (x) = 1 + 2 x + 3 x^{2} + 4 x^{3} .$ Let $s$ be the sum of all distinct real roots of $f (x)$ and let $t = ∣ s ∣ .$ The real numbers lies in the interval2010 · Shift 2 · Q35 · MCQ
Consider the polynomial $f (x) = 1 + 2 x + 3 x^{2} + 4 x^{3} .$ Let $s$ be the sum of all distinct real roots of $f (x)$ and let $t = ∣ s ∣ .$ The function $f^{'} (x)$ is2010 · Shift 2 · Q36 · MCQ
If the function $f (x) = x^{3} + e^{x /2}$ and $g (x) = f^{- 1} (x)$ , then the value of $g^{'} (1)$ is .2009 · Shift 2 · Q29 · Numerical