Sets, Relations & Functions PYQs Set 21 (PDF)

Set 21 of 25 · Questions 601-630 of 734

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JEE Main Mathematics · Sets, Relations & Functions · Set 21 (Q601-630)

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601.
Two newspapers A and B are published in a city. It is known that 25% of the city populations reads A and 20% reads B while 8% reads both A and B. Further, 30% of those who read A but not B look into advertisements and 40% of those who read B but not A also look into advertisements, while 50% of those who read both A and B look into advertisements. Then the percentage of the population who look into advertisement is:-
1. (1) $13.9$
2. (2) $12.8$
3. (3) $13$
4. (4) $13.5$
602.
Consider the following relations
R : {(x, y)| x ,y are real numbers and x = wy for some rational number w}
S = ${(\frac{m}{n}, \frac{p}{q})$ | m, n, p and q are integers such that n, q ≠ 0 and qm = pn}
Then
1. (1) $neither R nor S is an equivalence relation$
2. (2) $S is an equivalence relation but R is not an equivalence relation$
3. (3) $R and S both are equivalence relations$
4. (4) $R is an equivalence relation but S is not an equivalence relation$
603.
If $[x]$ denotes the greatest integer $\leq x$ , then the domain of the function $f (x) = \frac{4 - x ^{2}}{[ x ] + 2}$ is
1. (1) $(- \infty, - 2] \cup [- 1, 2)$
2. (2) $(- \infty, - 2) \cup [- 1, 2]$
3. (3) $(- \infty, - 2) \cup (- 1, 2)$
4. (4) $(- \infty, - 1] \cup [1, 2]$
604.
Let $f : R \to R$ be defined as $f (x) = 2 x - 1$ and $g : R - 1 \to R$ be defined as $g (x) = \frac{x - \frac{1}{2}}{x - 1}$ . Then the
composition function $f (g (x))$ is:
1. (1) $both one-one and onto$
2. (2) $onto but not one-one$
3. (3) $neither one-one nor onto$
4. (4) $one-one but not onto$
605.
The domain of the function, $f (x) = log_{10} (\frac{5 x - x ^{2}}{4})$ is
1. (1) $[0, 1]$
2. (2) $[1, 4]$
3. (3) $[4, 5]$
4. (4) $(- \infty, \infty)$
606.
Let f(x) = ${- 1, x^{2} - 1, - 2 \leq x < 0 0 \leq x \leq 2$ and g(x) = |f(x)| + f(|x|). Then, in the interval (–2, 2), g is :
1. (1) $not differentiable at two point$
2. (2) $not continuous$
3. (3) $not differentiable at one point$
4. (4) $differentiable at all points$
607.
If the domain of the function $f (x) = log_{7} (1 - log_{4} (x^{2} - 9 x + 18))$ is $(α, β) \cup (γ, o)$ , then $α + β + γ + o$ is equal to
1. (1) $17$
2. (2) $15$
3. (3) $16$
4. (4) $18$
608.
If the function $f : (- \infty, - 1] \to (a, b]$ defined by $f (x) = e^{x^{3} - 3 x + 1}$ is one-one and onto, then the distance of the point $P (2 b + 4, a + 2)$ from the line $x + e^{- 3} y = 4$ is :
1. (1) $2 1 + e^{6}$
2. (2) $4 1 + e^{6}$
3. (3) $3 1 + e^{6}$
4. (4) $1 + e^{6}$
609.
The function $f : R \to R$ defined by $f (x) = \frac{x}{1 + x ^{2}}$ is
1. (1) $surjective but not injective$
2. (2) $bijective$
3. (3) $injective but not surjective$
4. (4) $neither injective nor surjective$
610.
Let $f : R \to R$ be defined as f(x) = x-1 and $g : R -_{1, 1} \to R$ be defined as $g (x) = \frac{x ^{2}}{x ^{2} - 1}$ . Then the function fog is :
1. (1) $one-one but not onto function$
2. (2) $onto but not one-one function$
3. (3) $both one-one and onto function$
4. (4) $neither one-one nor onto function$
611.
Let a function f: N $\to$ N be defined by
$f (n) = ⎩ ⎨ ⎧ 2 n, n - 1, \frac{n + 1}{2}, n = 2, 4, 6, 8... n = 3, 7, 11, 15.... n = 1, 5, 9, 13, ....$
then f is
1. (1) $one-one but not onto$
2. (2) $onto but not one-one$
3. (3) $neither one-one nor onto$
4. (4) $one-one and onto$
612.
Let $x$ denote the total number of one-one functions from a set $A$ with $3$ elements to a set $B$ with $5$ elementsand $y$ denote the total number of one-one functions from the set $A$ to the set $A \times B$ .
Then:
1. (1) $y = 273 x$
2. (2) $2 y = 91 x$
3. (3) $y = 91 x$
4. (4) $2 y = 273 x$
613.
Let $f (x) = 2 x^{n} + λ$ , $λ \in R$ , $n \in N$ , and $f (4) = 133$ , $f (5) = 255$ . Then the sum of all the positive integer divisors of $(f (3) - f (2))$ is
1. (1) $61$
2. (2) $60$
3. (3) $58$
4. (4) $59$
614.
Let $A = {2, 3, 4}$ and $B = {8, 9, 12}$ . Then the number of elements in the relation $R = {((a_{1}, b_{1}), (a_{2}, b_{2})) \in (A \times B, A \times B) : a_{1}$ divides $b_{2}$ and $a_{2}$ divides $b_{1}}$ is
1. (1) $36$
2. (2) $24$
3. (3) $18$
4. (4) $12$
615.
The relation $R = {(a, b) : g cd (a, b) = 1, 2 a \neq = b, a, b \in Z}$ is
1. (1) $Transitive but not reflexive$
2. (2) $Symmetric but not transitive$
3. (3) $Reflexive but not symmetric$
4. (4) $Neither symmetric nor transitive$
616.
$lim x \to 1 \frac{( 2 x - 3 ) ( x - 1 )}{2 x ^{2} + x - 3} =$
1. (1) $\frac{1}{10}$
2. (2) $- \frac{1}{10}$
3. (3) $\frac{2}{5}$
4. (4) $-$
617.
The absolute minimum value, of the function $f (x) = x^{2} - x + 1 + [x^{2} - x + 1]$ , where $[t]$ denotes the greatest integer function, in the interval $[- 1, 2]$ is
1. (1) $\frac{3}{4}$
2. (2) $\frac{3}{2}$
3. (3) $\frac{1}{4}$
4. (4) $\frac{5}{4}$
618.
$Let R be the set of all real numbers. Let f : R \to R be a function defined by f (x) = {2 x - 5, if x < - 3 x + 2, if - 3 \leq x < 53 x + 1, if x \geq 5 Match the following$
$List-I$ $List-II$
$(A) f (- 5) + f (0) + f (- 1) =$ $(I) 16$
$(B) f (f (5) + 10 f (- 3)) =$ $(II) 40$
$(C) f (∣ f (- 4) ∣) =$ $(III) - 32$
$(D) f (f (f (1))) =$ $(IV) - 12$
$(V) 19$
$The correct answer is (A B C D)$
1. (1) $III II V I$
2. (2) $V IV I III$
3. (3) $IV V II I$
4. (4) $IV V III I$
619.
Let $f : R \to R$ be defined by $f (x) = \frac{x}{1 + x ^{2}}, x \in R$ . Then the range of $f$ is:
1. (1) $[- \frac{1}{2}, \frac{1}{2}]$
2. (2) $R - [- 1, 1]$
3. (3) $R - [- \frac{1}{2}, \frac{1}{2}]$
4. (4) $(- 1, 1) - {0}$
620.
If the domain of the function $f (x) = cos^{- 1} (\frac{2 - ∣ x ∣}{4}) + (lo g_{e} (3 - x))^{- 1}$ is $[- α, β) - {γ}$ , then $α + β + γ$ is equal to :
1. (1) $12$
2. (2) $9$
3. (3) $11$
4. (4) $8$
621.
Let $f : R \to R$ be a function which satisfies $f (x + y) = f (x) + f (y) \forall x, y \in R$ . If $f (1) = 2$ . If $f (1) = 2$ and $g (n) = \sum_{k = 1}^{n - 1} f (k), n \in N$ then thevalue of $n$ , for which $g (n) = 20$ is:
1. (1) $20$
2. (2) $9$
3. (3) $4$
4. (4) $5$
622.
The derivative of $f (x) = x^{t a n^{- 1} x}$ with respect to $g (x) = sec^{- 1} (\frac{1}{2 x ^{2} - 1})$ is
1. (1) $\frac{1}{2} 1 - x^{2} x^{t a n^{- 1} x} [\frac{l o g x}{1 + x ^{2}} + \frac{t a n ^{- 1} x}{x}]$
2. (2) $- \frac{1}{2} 1 - x ˙^{2} x^{t a n^{- 1}} [lo g (tan^{- 1} x) + x (1 + x^{2})] ta$
3. (3) $\frac{- 2 x ^{t a n^{- 1}} [ \frac{l o g x}{1 + x ^{2}} + \frac{t a n ^{- 1} x}{x} ]}{1 - x ^{2}}$
4. (4) $- \frac{1}{2} 1 - x^{2} x^{t a n x^{- 1}} [\frac{l o g x}{1 + x ^{2}} + \frac{t a n ^{- 1} x}{x}]$
623.
If $f (x) = ⎩ ⎨ ⎧ \frac{∣ x - 4∣}{x - 4} + a, a + b, \frac{∣ x - 4∣}{x - 4} + b, x < 4 x = 4 x > 4$ is continuous at $x = 4$ , then
1. (1) $a = 1, b = 1$
2. (2) $a = 1, b = - 1$
3. (3) $a = 0, b = 0$
4. (4) $a = - 1, b = 1$
624.
Consider the following lists.
$List I: (A) f (x) = \frac{∣ x + 2∣}{x + 2}, x \neq = - 2 (B) g (x) = ∣ [x] ∣, x \in R (C) h (x) = ∣ x - [x] ∣, x \in R (D) f (x) = \frac{1}{2 - s i n 3 x}, x \in R List II (ranges): 1. (\frac{1}{3}, 1) 2. Z 3. W 4. [0, 1) 5. {- 1, 1}$
Match List I with List II.
1. (1) $5 3 2 1$
2. (2) $3 2 4 1$
3. (3) $5 3 4 1$
4. (4) $1 2 3 4$
625.
Let N be the set of natural numbers and two functions $f$ and $g$ be defined as $f, g : N \to N$ such that $f (n) = {\frac{n + 1}{2}, \frac{n}{2}, if n is odd if n is even$ and $g (n) = n - (- 1)^{n}$ . Then $f \circ g$ is:
1. (1) $onto both not one-one$
2. (2) $one-one but not onto.$
3. (3) $both one-one and onto.$
4. (4) $neither one-one nor onto.$
626.
The function $f : X \to Y$ defined by $f (x) = sin x$ is one-one but not onto if $X$ and $Y$ are respectively equal to
1. (1) $R and R$
2. (2) $[0, π] and [0, 1]$
3. (3) $[0, \frac{π}{2}] and [- 1, 1]$
4. (4) $[- \frac{π}{2}, \frac{π}{2}] and [- 1, 1]$
627.
$\frac{d}{dx} (lim y \to 2 \frac{1}{y - 2} (\frac{1}{x} - \frac{1}{x + y - 2})) =$
1. (1) $1/ x^{2}$
2. (2) $2/ x^{3}$
3. (3) $- 2/ x^{3}$
4. (4) $1/ x^{3}$
628.
If $[x]$ denotes the greatest integer function, then the domain of the function $f (x) = \frac{x - [ x ]}{l o g ( x ^{2} - x )}$ , is
1. (1) $(1, \infty)$
2. (2) $(1, \infty) - Z$
3. (3) $R - [\frac{1 - 5}{2}, \frac{1 + 5}{2}]$
4. (4) $[\frac{1 - 5}{2}, \frac{5 + 1}{2}]$
629.
If $\frac{d}{d x} (\frac{2 x + 1}{( x + 1 ) ^{2} ( x - 2 )}) = \frac{A}{( x - 2 ) ^{2}} + \frac{B}{( x + 1 ) ^{3}} + \frac{C}{( x + 1 ) ^{2}}$ then $A + B + C =$
1. (1) $\frac{- 2}{3}$
2. (2) $\frac{2}{3}$
3. (3) $\frac{1}{3}$
4. (4) $\frac{- 1}{3}$
630.
$Let f : R \to R and g : R \to R be respectively given by f(x) = |x| + 1 and g(x) = x2 + 1. Define h : R \to R by h (x) = {max {f (x), g (x)} min {f (x), g (x)} if x \leq 0, if x > 0. The number of points at which h(x) is not differentiable is$