COMPASSLearning

Talk to us Register Login

JEE Main · 2022 · 24 Jun 2022 · Shift 2 · Sequences and Series

Let

x, y > 0

. If

x^{3} y^{2} = 2^{15}

, then the least value of

3 x + 2 y

is

(A)30
(B)32
(C)36
(D)40✓ Correct

Answer: (D)

Solution

Using the AM-GM inequality:

\frac{x + x + x + y + y}{5} \geq (x^{3} \cdot y^{2})^{1/5}

\frac{3 x + 2 y}{5} \geq (2^{15})^{1/5} = 2^{3} = 8

(3 x + 2 y)_{m i n} = 40

Equality holds when

x = y

, which gives

x = y = 4

.

JEE Main · 2022 · 24 Jun 2022 · Shift 2 · Single correct option

Practise more: all JEE Main PYQs on Sequences and Series →

More JEE Main questions on Sequences and Series

Let $a_{1}, a_{2}, a_{3}, \dots$ . be a G.P. of increasing positive terms such that $a_{2} \cdot a_{3} \cdot a_{4} = 64$ and $a_{1} + a_{3} + a_{5} = \frac{813}{7}$ .

Then $a_{3} + a_{5} + a_{7}$ is equal to :
JEE Main 2026View solution →
Let $a_{1} = 1$ and for $n \geq 1, a_{n + 1} = \frac{1}{2} a_{n} + \frac{n ^{2} - 2 n - 1}{n ^{2} ( n + 1 ) ^{2}}$ . Then $\sum_{n = 1}^{\infty} (a_{n} - \frac{2}{n ^{2}})$ is equal to
JEE Main 2026View solution →
The positive integer n, for which the solutions of the equation
$x (x + 2) + (x + 2) (x + 4) + \dots + (x + 2 n - 2) (x + 2 n) = \frac{8 n}{3}$

are two consecutive even integers, is :-
JEE Main 2026View solution →
Let $a_{1}, \frac{a _{2}}{2}, \frac{a _{3}}{2 ^{2}}, \dots, \frac{a _{10}}{2 ^{9}}$ be a G.P. of common ratio $\frac{1}{2}$ . If $a_{1} + a_{2} + \dots + a_{10} = 62$ , then $a_{1}$ is equal to :
JEE Main 2026View solution →
If the sum of the first four terms of an A.P. is 6 and the sum of its first six terms is 4, then the sum of its first twelve terms is
JEE Main 2026View solution →
Suppose $a, b, c$ are in A.P. and $a^{2}, 2 b^{2}, c^{2}$ are in G.P. If $a < b < c$ and $a + b + c = 1$ , then $9 (a^{2} + b^{2} + c^{2})$ is equa …
JEE Main 2026View solution →