Sequences and Series jee mains questions | Sequences and Series Previous Year Question

Q 1 :

Let $a, b, c$ and $d$ be positive real numbers such that $a + b + c + d = 11 .$ If the maximum value of $a^{5} b^{3} c^{2} d$ is $3750 β$ , then the value of $β$ is [2023]

55
108
90
110

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4

(3)

$Given: a + b + c + d = 11$

$\frac{5 (\frac{a}{5}) + 3 (\frac{b}{3}) + 2 (\frac{c}{2}) + d}{11} \geq {(\frac{a^{5} b^{3} c^{2} d}{5^{5} 3^{3} 2^{2}})}^{\frac{1}{11}}$

$\Rightarrow 1 \geq {(\frac{a^{5} b^{3} c^{2} d}{5^{5} 3^{3} 2^{2}})}^{\frac{1}{11}} \Rightarrow a^{5} b^{3} c^{2} d \leq 5^{5} 3^{3} 2^{2}$

$∴$ $Max of a^{5} b^{3} c^{2} d = 5^{5} 3^{3} 2^{2}$

$\Rightarrow 3750 β = 5^{5} 3^{3} 2^{2} \Rightarrow β = 90$

Q 2 :

Let $A_{1}$ and $A_{2}$ be two arithmetic means and $G_{1}, G_{2}, G_{3}$ be three geometric means of two distinct positive numbers. Then $G_{1}^{4} + G_{2}^{4} + G_{3}^{4} + G_{1}^{2} G_{3}^{2}$ is equal to [2023]

$2 (A_{1} + A_{2}) G_{1}^{2} G_{3}^{2}$
${(A_{1} + A_{2})}^{2} G_{1} G_{3}$
$(A_{1} + A_{2}) G_{1}^{2} G_{3}^{2}$
$2 (A_{1} + A_{2}) G_{1} G_{3}$

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(2)

$Given: a, A_{1}, A_{2}, b be in an A.P.$

$∴$ $d = \frac{b - a}{3}$

$\Rightarrow$ $A_{1} = a + d = a + (\frac{b - a}{3}) = \frac{2 a + b}{3}$

and $A_{2} = a + 2 d = a + 2 \frac{(b - a)}{3} = \frac{a + 2 b}{3}$

$∴$ $A_{1} + A_{2} = \frac{2 a + b}{3} + \frac{a + 2 b}{3} = a + b ...(i)$

$Also, let a, G_{1}, G_{2}, G_{3}, b be in a G.P.$

$Now, a r^{4} = b \Rightarrow r = {(\frac{b}{a})}^{1 / 4}$

$∴$ $G_{1} = a^{3 / 4} b^{1 / 4}, G_{2} = \sqrt{a b}, G_{3} = a^{1 / 4} b^{3 / 4}$

$Now, G_{1}^{4} + G_{2}^{4} + G_{3}^{4} + G_{1}^{2} G_{3}^{2}$

$= a^{3} b + a^{2} b^{2} + a b^{3} + a^{3 / 2} b^{1 / 2} a^{1 / 2} b^{3 / 2}$

$= a^{3} b + a b^{3} + a^{2} b^{2} + a^{2} b^{2}$

$= a b (a^{2} + b^{2}) + 2 a^{2} b^{2}$

$= a b (a^{2} + b^{2} + 2 a b) = {(A_{1} + A_{2})}^{2} \cdot G_{1} G_{3} [Using (i)]$

Q 3 :

Let three real numbers $a, b, c$ be in arithmetic progression and $a + 1, b, c + 3$ be in geometric progression. If $a > 10$ and the arithmetic mean of $a, b$ and $c$ is 8, then the cube of the geometric mean of $a, b$ and $c$ is [2024]

316
128
120
312

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(3)

$2 b = a + c$ $[∵ a, b, c$ are in A.P. $]$ ...(i)

$b^{2} = (a + 1) (c + 3)$ $[∵ a + 1, b, c + 3$ are in G.P. $]$ ...(ii)

Also, $\frac{a + b + c}{3} = 8$ $[∵$ A.M. of $a, b, c]$ ...(iii)

$\Rightarrow \frac{3 b}{3} = 8 \Rightarrow b = 8$ [Using (i)]

By (ii), $64 = (a + 1) (c + 3)$ and by (iii), $c = 16 - a$

$\Rightarrow 64 = (a + 1) (16 - a + 3)$

$\Rightarrow a^{2} - 18 a + 45 = 0 \Rightarrow a = 15, 3$

$∴$ $a = 15$ $(a \neq 3, as a > 10)$

Also, $c = 1$

$∴$ ${[{(a \cdot b \cdot c)}^{1 / 3}]}^{3} = 15 \times 8 \times 1 = 120$

Q 4 :

Let the range of the function $f (x) = \frac{1}{2 + \sin 3 x + \cos 3 x},$ $x \in ℝ$ be $[a, b]$ . If $α$ and $β$ are respectively the A.M. and the G.M. of $a$ and $b,$ then $\frac{α}{β}$ is equal to [2024]

$π$
$2$
$\sqrt{π}$
$\sqrt{2}$

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(4)

$f (x) = \frac{1}{2 + \sin 3 x + \cos 3 x}$

Now, $- \sqrt{2} \leq \sin 3 x + \cos 3 x \leq \sqrt{2}$

$\frac{1}{2 + \sqrt{2}} \leq \frac{1}{2 + \sin 3 x + \cos 3 x} \leq \frac{1}{2 - \sqrt{2}}$

$[a, b] = [\frac{1}{2 + \sqrt{2}}, \frac{1}{2 - \sqrt{2}}]$

Now, A.M. of $a$ and $b$ i.e., $α = \frac{1}{2} [\frac{1}{2 + \sqrt{2}} + \frac{1}{2 - \sqrt{2}}]$

$\frac{1}{2} [\frac{4}{4 - 2}] = 1$

$β = G.M. of a and b = \sqrt{\frac{1}{2 + \sqrt{2}} \cdot \frac{1}{2 - \sqrt{2}}} = \sqrt{\frac{1}{4 - 2}} = \frac{1}{\sqrt{2}}$

So, $\frac{α}{β} = \frac{\frac{1}{1}}{\sqrt{2}} = \sqrt{2}$

Topic Question Set

Q 1 :

Let $a, b, c$ and $d$ be positive real numbers such that $a + b + c + d = 11 .$ If the maximum value of $a^{5} b^{3} c^{2} d$ is $3750 β$ , then the value of $β$ is [2023]

Q 2 :

Let $A_{1}$ and $A_{2}$ be two arithmetic means and $G_{1}, G_{2}, G_{3}$ be three geometric means of two distinct positive numbers. Then $G_{1}^{4} + G_{2}^{4} + G_{3}^{4} + G_{1}^{2} G_{3}^{2}$ is equal to [2023]

Q 3 :

Let three real numbers $a, b, c$ be in arithmetic progression and $a + 1, b, c + 3$ be in geometric progression. If $a > 10$ and the arithmetic mean of $a, b$ and $c$ is 8, then the cube of the geometric mean of $a, b$ and $c$ is [2024]

Q 4 :

Let the range of the function $f (x) = \frac{1}{2 + \sin 3 x + \cos 3 x},$ $x \in ℝ$ be $[a, b]$ . If $α$ and $β$ are respectively the A.M. and the G.M. of $a$ and $b,$ then $\frac{α}{β}$ is equal to [2024]

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