Q 21 :
Let R = {a, b, c, d, e} and S = {1, 2, 3, 4}. Total number of onto functions $f : R \to S$ such that $f (a) \neq 1,$ is equal to ________ . [2023]

0%

Enter Answer here

Q 22 :
Let a, b, c be three distinct positive real numbers such that ${(2 a)}^{\log_{e} a} = {(b c)}^{\log_{e} b}$ and $b^{\log_{e} 2} = a^{\log_{e} c}$ .

Then 6a + 5bc is equal to _____________ . [2023]

0%

(8)

${(2 a)}^{\log_{e} a} = {(b c)}^{\log_{e} b}, where 2 a > 0 and b c > 0$

$\Rightarrow \log_{e} a (\log_{e} 2 + \log_{e} a) = \log_{e} b (\log_{e} b + \log_{e} c)$ ...(i)

Also, $b^{\log_{e} 2} = a^{\log_{e} c}$

$\Rightarrow \log_{e} 2 \log_{e} b = \log_{e} c \log_{e} a$ ...(ii)

Let $\log_{e} 2 = α, \log_{e} a = x, \log_{e} b = y, and \log_{e} c = z$

$∴ From (i) and (ii), we get x (α + x) = y (y + z) and α y = x z$

$\Rightarrow x (\frac{x z}{y} + x) = y (y + z)$ $[∵ α = \frac{x z}{y}]$

$\Rightarrow x^{2} (z + y) = y^{2} (y + z)$

$\Rightarrow y + z = 0 or x^{2} = y^{2} \Rightarrow x = - y$ $[∵ x = y is not possible]$

$\Rightarrow b c = 1 or a b = 1$

Now, $b c = 1 \Rightarrow 2 a \log_{e} a = 1 \Rightarrow a = 1 or \frac{1}{2}$

$Case - 1: Distinct possible values of a, b, c can be (\frac{1}{2}, p, \frac{1}{p}), where p \neq 1, 2, \frac{1}{2}$

$∴ 6 a + 5 b c = 6 (\frac{1}{2}) + 5 (p) (\frac{1}{p}) = 3 + 5 = 8$

$Case - 2: (a, b, c) can be (p, \frac{1}{p}, \frac{1}{2}), where p \neq (1, 2, \frac{1}{2})$

So, in this case infinite answers are possible.

Q 23 :
Let A = {1, 2, 3, 4, 5} and B = {1, 2, 3, 4, 5, 6}. Then the number of functions $f : A \to B$ satisfying $f (1) + f (2) = f (4) - 1$ is equal to __________ . [2023]

0%

(360)

We have, $A = {1, 2, 3, 4, 5}; B = {1, 2, 3, 4, 5, 6}$

Also, $f (1) + f (2) = f (4) - 1 \Rightarrow f (1) + f (2) + 1 = f (4)$

Here, $f (4) \leq 6$

$f (1) + f (2) + 1 \leq 6$

$f (1) + f (2) \leq 5$

At $f (1) = 1, then, f (2) = 1, 2, 3, 4 \Rightarrow 4 mappings,$

$f (1) = 2, then, f (2) = 1, 2, 3 \Rightarrow 3 mappings,$

$f (1) = 3, then, f (2) = 1, 2 \Rightarrow 2 mappings$

$f (1) = 4, then, f (2) = 1 \Rightarrow 1 mapping$

$f (3) and f (5) have 6 mappings each.$

$∴ Total number of functions = 10 \times 6 \times 6 = 360$

Q 24 :
Let $[α]$ denote the greatest integer $\leq α$ . Then $[\sqrt{1}] + [\sqrt{2}] + [\sqrt{3}] + . . . + [\sqrt{120}]$ is equal to __________ . [2023]

0%

(825)

$\sum_{r = 1}^{120} [\sqrt{r}] = \sum_{r = 1}^{3} 1 + \sum_{r = 4}^{8} 2 + \sum_{r = 9}^{15} 3 + \sum_{r = 16}^{25} 4 + \dots + \sum_{r = 81}^{99} 9 + \sum_{r = 100}^{120} 10$

$= 3 (1) + 5 (2) + 7 (3) + 9 (4) + \dots + 21 (10)$

$= \sum_{r = 1}^{10} (2 r + 1) r = 2 \times \frac{10 \times 11 \times 21}{6} + \frac{10 \times 11}{2}$

$= \frac{10 \times 11}{2} \times 15 = 825$

Q 25 :
For some a, b, c $\in$ N, let $f (x) = a x - 3$ and $g (x) = x^{b} + c, x \in R .$ If ${(f o g)}^{- 1} (x) = {(\frac{x - 7}{2})}^{1 / 3},$ then (fog)(ac) + (gof)(b) is equal to ___________ . [2023]

0%

(2039)

Given, $f (x) = a x - 3, g (x) = x^{b} + c$ and ${(f o g)}^{- 1} (x) = {(\frac{x - 7}{2})}^{\frac{1}{3}}$

Let $f o g (x) = h^{- 1} (x) .$ So, $h^{- 1} (x) = {(\frac{x - 7}{2})}^{1 / 3}$

Let $y = {(\frac{x - 7}{2})}^{1 / 3}$ $\Rightarrow$ $y^{3} = \frac{x - 7}{2} \Rightarrow x = 2 y^{3} + 7$

$∴$ $h (x) = f o g (x) = 2 x^{3} + 7$

Now, $f o g (x) = a (x^{b} + c) - 3 \Rightarrow 2 x^{3} + 7 = a (x^{b} + c) - 3$

Comparing the coefficients of like powers, we get

$a = 2, b = 3$ and $c = 5$

So, $f o g (a c) = f o g (10) = 2 {(10)}^{3} + 7 = 2007$

$g [f (x)] = {(a x - 3)}^{b} + c = {(2 x - 3)}^{3} + 5$

$g o f (b) = (2 {(3) - 3)}^{3} + 5 = 27 + 5 = 32$

$∴ f o g (a c) + g o f (b) = 2007 + 32 = 2039$

Q 26 :
Suppose $f$ is a function satisfying $f (x + y) = f (x) + f (y)$ for all $x, y \in ℕ$ and $f (1) = \frac{1}{5} .$ If $\sum_{n = 1}^{m} \frac{f (n)}{n (n + 1) (n + 2)} = \frac{1}{12},$ then $m$ is equal to ___________ . [2023]

0%

(10)

Given $f (x + y) = f (x) + f (y) and f (1) = \frac{1}{5}$

So, $f (2) = f (1) + f (1) = \frac{2}{5}; f (3) = f (2) + f (1) = \frac{3}{5}$

$∴ \sum_{n = 1}^{m} \frac{f (n)}{n (n + 1) (n + 2)} = \frac{1}{5} \sum_{n = 1}^{m} (\frac{1}{n + 1} - \frac{1}{n + 2})$

$= \frac{1}{5} (\frac{1}{2} - \frac{1}{3} + \frac{1}{3} - \frac{1}{4} + \dots + \frac{1}{m + 1} - \frac{1}{m + 2})$

$= \frac{1}{5} (\frac{1}{2} - \frac{1}{m + 2}) = \frac{m}{10 (m + 2)} = \frac{1}{12}$

$∴ m = 10$

Q 27 :
Let S = {1, 2, 3, 4, 5, 6}. Then the number of one-one functions $f : S \to P (S),$ where $P (S)$ denote the power set of S, such that $f (n) \subset f (m)$ where $n < m$ is __________ . [2023]

0%

(3240)

Q 28 :
Let $f^{1} (x) = \frac{3 x + 2}{2 x + 3}, x \in R - {\frac{- 3}{2}}$

For $n \geq 2,$ define $f^{n} (x) = f^{1} o f^{n - 1} (x) .$

If $f^{5} (x) = \frac{a x + b}{b x + a}, g c d (a, b) = 1,$ then a + b is equal to _________ . [2023]

0%

(3125)

We have, $f^{1} (x) = \frac{3 x + 2}{2 x + 3}, x \in ℝ - {\frac{- 2}{3}}$

$f^{2} (x) = \frac{13 x + 12}{12 x + 13}; f^{3} (x) = \frac{63 x + 62}{62 x + 63}$

$∴$ $f^{5} (x) = \frac{1563 x + 1562}{1562 x + 1563} = \frac{a x + b}{b x + a}$

$∴$ $a + b = 1563 + 1562 = 3125$

Q 29 :
Let A = {1, 2, 3, 5, 8, 9}. Then the number of possible functions $f : A \to A$ such that $f (m \cdot n) = f (m) \cdot f (n)$ for every $m, n \in A$ with $m \cdot n \in A$ is equal to __________________ . [2023]

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Q 30 :
Let $f (x) = x^{5} + 2 e^{x / 4}$ for all $x \in R$ . Consider a function $g (x)$ such that $(g o f)$ $(x) = x$ for all $x \in R$ . Then the value of $8 g' (2)$ is [2024]

4
16
8
2

1

2

3

4

(B)

We have, $(g o f) (x) = x \Rightarrow g (f (x)) = x$

Differentiating w.r.t. $x$ , we get

$g' (f (x)) \cdot f' (x) = 1 \Rightarrow g' (f (x)) = \frac{1}{f' (x)}$

Now, $f (x) = x^{5} + 2 e^{x / 4} \Rightarrow f' (x) = 5 x^{4} + \frac{1}{2} e^{x / 4}$

$∴$ $f' (0) = \frac{1}{2}, f (0) = 2 \Rightarrow g' (2) = 2$

Hence, $8 g' (2) = 16$

Topic Question Set

Q 21 :
Let R = {a, b, c, d, e} and S = {1, 2, 3, 4}. Total number of onto functions $f : R \to S$ such that $f (a) \neq 1,$ is equal to ________ . [2023]

0%

Q 22 :
Let a, b, c be three distinct positive real numbers such that ${(2 a)}^{\log_{e} a} = {(b c)}^{\log_{e} b}$ and $b^{\log_{e} 2} = a^{\log_{e} c}$ .

Then 6a + 5bc is equal to _____________ . [2023]

0%

Q 23 :
Let A = {1, 2, 3, 4, 5} and B = {1, 2, 3, 4, 5, 6}. Then the number of functions $f : A \to B$ satisfying $f (1) + f (2) = f (4) - 1$ is equal to __________ . [2023]

0%

Q 24 :
Let $[α]$ denote the greatest integer $\leq α$ . Then $[\sqrt{1}] + [\sqrt{2}] + [\sqrt{3}] + . . . + [\sqrt{120}]$ is equal to __________ . [2023]

0%

Q 25 :
For some a, b, c $\in$ N, let $f (x) = a x - 3$ and $g (x) = x^{b} + c, x \in R .$ If ${(f o g)}^{- 1} (x) = {(\frac{x - 7}{2})}^{1 / 3},$ then (fog)(ac) + (gof)(b) is equal to ___________ . [2023]

0%

Q 26 :
Suppose $f$ is a function satisfying $f (x + y) = f (x) + f (y)$ for all $x, y \in ℕ$ and $f (1) = \frac{1}{5} .$ If $\sum_{n = 1}^{m} \frac{f (n)}{n (n + 1) (n + 2)} = \frac{1}{12},$ then $m$ is equal to ___________ . [2023]

0%

Q 27 :
Let S = {1, 2, 3, 4, 5, 6}. Then the number of one-one functions $f : S \to P (S),$ where $P (S)$ denote the power set of S, such that $f (n) \subset f (m)$ where $n < m$ is __________ . [2023]

0%

(3240)

Q 28 :
Let $f^{1} (x) = \frac{3 x + 2}{2 x + 3}, x \in R - {\frac{- 3}{2}}$

For $n \geq 2,$ define $f^{n} (x) = f^{1} o f^{n - 1} (x) .$

If $f^{5} (x) = \frac{a x + b}{b x + a}, g c d (a, b) = 1,$ then a + b is equal to _________ . [2023]

0%

Q 29 :
Let A = {1, 2, 3, 5, 8, 9}. Then the number of possible functions $f : A \to A$ such that $f (m \cdot n) = f (m) \cdot f (n)$ for every $m, n \in A$ with $m \cdot n \in A$ is equal to __________________ . [2023]

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Q 30 :
Let $f (x) = x^{5} + 2 e^{x / 4}$ for all $x \in R$ . Consider a function $g (x)$ such that $(g o f)$ $(x) = x$ for all $x \in R$ . Then the value of $8 g' (2)$ is [2024]

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Topic Question Set

Q 21 : Let R = {a, b, c, d, e} and S = {1, 2, 3, 4}. Total number of onto functions f:R→S such that f(a)≠1, is equal to ________ . [2023]

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Q 22 : Let a, b, c be three distinct positive real numbers such that (2a)logea=(bc)logeb and bloge2=alogec. Then 6a + 5bc is equal to _____________ . [2023]

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Q 23 : Let A = {1, 2, 3, 4, 5} and B = {1, 2, 3, 4, 5, 6}. Then the number of functions f:A→B satisfying f(1)+f(2)=f(4)-1 is equal to __________ . [2023]

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Q 24 : Let [α] denote the greatest integer ≤α. Then [1]+[2]+[3]+...+[120] is equal to __________ . [2023]

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Q 25 : For some a, b, c∈N, let f(x)=ax-3 and g(x)=xb+c,x∈R. If (fog)-1(x)=(x-72)1/3, then (fog)(ac) + (gof)(b) is equal to ___________ . [2023]

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Q 26 : Suppose f is a function satisfying f(x+y)=f(x)+f(y) for all x,y∈ℕ and f(1)=15. If ∑n=1mf(n)n(n+1)(n+2)=112, then m is equal to ___________ . [2023]

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Q 27 : Let S = {1, 2, 3, 4, 5, 6}. Then the number of one-one functions f:S→P(S), where P(S) denote the power set of S, such that f(n)⊂f(m) where n<m is __________ . [2023]

0%

(3240)

Q 28 : Let f1(x)=3x+22x+3,x∈R-{-32} For n≥2, define fn(x)=f1ofn-1(x). If f5(x)=ax+bbx+a,gcd(a,b)=1, then a + b is equal to _________ . [2023]

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Q 29 : Let A = {1, 2, 3, 5, 8, 9}. Then the number of possible functions f:A→A such that f(m·n)=f(m)·f(n) for every m,n∈A with m·n∈A is equal to __________________ . [2023]

0%

Q 30 : Let f(x)=x5+2ex/4 for all x∈R. Consider a function g(x) such that (gof) (x)=x for all x∈R. Then the value of 8g'(2) is [2024]

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Q 21 :
Let R = {a, b, c, d, e} and S = {1, 2, 3, 4}. Total number of onto functions $f : R \to S$ such that $f (a) \neq 1,$ is equal to ________ . [2023]

Q 22 :
Let a, b, c be three distinct positive real numbers such that ${(2 a)}^{\log_{e} a} = {(b c)}^{\log_{e} b}$ and $b^{\log_{e} 2} = a^{\log_{e} c}$ .

Then 6a + 5bc is equal to _____________ . [2023]

Q 23 :
Let A = {1, 2, 3, 4, 5} and B = {1, 2, 3, 4, 5, 6}. Then the number of functions $f : A \to B$ satisfying $f (1) + f (2) = f (4) - 1$ is equal to __________ . [2023]

Q 24 :
Let $[α]$ denote the greatest integer $\leq α$ . Then $[\sqrt{1}] + [\sqrt{2}] + [\sqrt{3}] + . . . + [\sqrt{120}]$ is equal to __________ . [2023]

Q 25 :
For some a, b, c $\in$ N, let $f (x) = a x - 3$ and $g (x) = x^{b} + c, x \in R .$ If ${(f o g)}^{- 1} (x) = {(\frac{x - 7}{2})}^{1 / 3},$ then (fog)(ac) + (gof)(b) is equal to ___________ . [2023]

Q 26 :
Suppose $f$ is a function satisfying $f (x + y) = f (x) + f (y)$ for all $x, y \in ℕ$ and $f (1) = \frac{1}{5} .$ If $\sum_{n = 1}^{m} \frac{f (n)}{n (n + 1) (n + 2)} = \frac{1}{12},$ then $m$ is equal to ___________ . [2023]

Q 27 :
Let S = {1, 2, 3, 4, 5, 6}. Then the number of one-one functions $f : S \to P (S),$ where $P (S)$ denote the power set of S, such that $f (n) \subset f (m)$ where $n < m$ is __________ . [2023]

Q 28 :
Let $f^{1} (x) = \frac{3 x + 2}{2 x + 3}, x \in R - {\frac{- 3}{2}}$

For $n \geq 2,$ define $f^{n} (x) = f^{1} o f^{n - 1} (x) .$

If $f^{5} (x) = \frac{a x + b}{b x + a}, g c d (a, b) = 1,$ then a + b is equal to _________ . [2023]

Q 29 :
Let A = {1, 2, 3, 5, 8, 9}. Then the number of possible functions $f : A \to A$ such that $f (m \cdot n) = f (m) \cdot f (n)$ for every $m, n \in A$ with $m \cdot n \in A$ is equal to __________________ . [2023]

Q 30 :
Let $f (x) = x^{5} + 2 e^{x / 4}$ for all $x \in R$ . Consider a function $g (x)$ such that $(g o f)$ $(x) = x$ for all $x \in R$ . Then the value of $8 g' (2)$ is [2024]