JEE MATH APEX

Functions-JEE-Mains-PYQ’s

Statement I : The function f : \mathbb{R} \to \mathbb{R} defined by f(x) = \frac{x}{1+|x|} is one-one.
Statement II : The function f : \mathbb{R} \to \mathbb{R} defined by f(x) = \frac{x^2+4x-30}{x^2-8x+18} is many-one.
In the light of the above statements, choose the correct answer from the options given below :
(A) Statement I is true but Statement II is false
(B) Both Statement I and Statement II are false
(C) Both Statement I and Statement II are true
(D) Statement I is false but Statement II is true
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The sum of all the elements in the range of
f(x) = \text{Sgn}(\sin x) + \text{Sgn}(\cos x) + \text{Sgn}(\tan x) + \text{Sgn}(\cot x),
x \neq \frac{n\pi}{2}, n \in \mathbb{Z}, where
\text{Sgn}(t) = \begin{cases} 1, & \text{if } t > 0 \\ -1, & \text{if } t < 0 \end{cases}
is :
(A) 4
(B) 0
(C) 2
(D) -2
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If g(x) = 3x^2 + 2x - 3, f(0) = -3 and 4g(f(x)) = 3x^2 - 32x + 72, then f(g(2)) is equal to:
(A) \frac{7}{2}
(B) -\frac{25}{6}
(C) \frac{25}{6}
(D) -\frac{7}{2}
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Let f be a function such that 3f(x) + 2f\left(\frac{m}{19x}\right) = 5x, x \neq 0, where m = \sum_{i=1}^{9} (i)^2. Then f(5) - f(2) is equal to
(A) 36
(B) 9
(C) -9
(D) 18
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Let f(x) = [x]^2 - [x+3] - 3, x \in \mathbb{R}, where [.] is the greatest integer function. Then
(A) f(x) = 0 for finitely many values of x
(B) f(x) < 0 only for x \in [-1, 3)
(C) \int_{0}^{2} f(x)dx = -6
(D) f(x) > 0 only for x \in [4, \infty)
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Let the domain of the function
f(x) = \log_3 \log_5 (7 - \log_2 (x^2 - 10x + 85)) + \sin^{-1} \left( \left| \frac{3x-7}{17-x} \right| \right) be (\alpha, \beta].
Then \alpha + \beta is equal to :
(A) 12
(B) 8
(C) 10
(D) 9
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Let f and g be functions satisfying f(x+y) = f(x)f(y), f(1) = 7 and g(x+y) = g(xy), g(1) = 1, for all x, y \in \mathbb{N}. If \sum_{x=1}^{n} \left( \frac{f(x)}{g(x)} \right) = 19607, then n is equal to :
(A) 6
(B) 7
(C) 4
(D) 5
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If the domain of the function f(x) = \sin^{-1} \left( \frac{5-x}{3+2x} \right) + \frac{1}{\log_e(10-x)} is (-\infty, \alpha] \cup [\beta, \gamma) - \{\delta\}, then 6(\alpha + \beta + \gamma + \delta) is equal to
(A) 66
(B) 68
(C) 70
(D) 67
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If the range of the function
f(x) = \frac{5 - x}{x^2 - 3x + 2}, x \neq 1, 2, is (-\infty, \alpha] \cup [\beta, \infty), then \alpha^2 + \beta^2 is equal to:
(A) 188
(B) 192
(C) 190
(D) 194
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Let the domains of the functions
f(x) = \log_4 \log_3 \log_7 (8 - \log_2 (x^2 + 4x + 5)) and
g(x) = \sin^{-1} \left( \frac{7x + 10}{x - 2} \right) be (\alpha, \beta) and [\gamma, \delta] respectively. Then \alpha^2 + \beta^2 + \gamma^2 + \delta^2 is equal to:
(A) 15
(B) 13
(C) 16
(D) 14
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Correct Answer: [ ]

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