The minimum value of 4-1tan 2-2- 4-2tan 2-1- wherever defined- is-correct answer - 1- wrong answer - 0-25

Let nbsp;tan ^2θ_1=a, tan ^2θ_2=b, so given equation becomes (a+1)^2b+(b+1)^2a ⇒ 2(ab+ba)+(a^2b+b^2a+1a+1b) nbsp; Using A.M. ≥ G.M., so ab+ba≥ 2 a^2b+b^2 ...

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Standard XII

Mathematics

General Solution of Trigonometric Equation

The minimum v...

Question

The minimum value of $\frac{{sec}^{4} θ_{1}}{{tan}^{2} θ_{2}} + \frac{{sec}^{4} θ_{2}}{{tan}^{2} θ_{1}}$ , wherever defined, is
(correct answer + 1, wrong answer - 0.25)

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Solution

The correct option is B $8$
Let ${tan}^{2} θ_{1} = a, {tan}^{2} θ_{2} = b$ , so given equation becomes
$\frac{(a + 1)^{2}}{b} + \frac{(b + 1)^{2}}{a} \Rightarrow 2 (\frac{a}{b} + \frac{b}{a}) + (\frac{a^{2}}{b} + \frac{b^{2}}{a} + \frac{1}{a} + \frac{1}{b})$

Using $A . M . \geq G . M .$ , so
$\frac{a}{b} + \frac{b}{a} \geq 2 \frac{a^{2}}{b} + \frac{b^{2}}{a} + \frac{1}{a} + \frac{1}{b} \geq 4 \Rightarrow 2 (\frac{a}{b} + \frac{b}{a}) + (\frac{a^{2}}{b} + \frac{b^{2}}{a} + \frac{1}{a} + \frac{1}{b}) \geq 8$

Hence, the minimum value of the expression is $8$ .
Equality holds when $θ_{1} = θ_{2} = \frac{π}{4} .$

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Similar questions

Q. The minimum value of

\frac{{sec}^{4} θ_{1}}{{tan}^{2} θ_{2}} + \frac{{sec}^{4} θ_{2}}{{tan}^{2} θ_{1}}

, wherever defined, is
(correct answer + 1, wrong answer - 0.25)

Q. Let

θ_{1}, θ_{2}, \dots, θ_{10}

be positive valued angles (in radian) such that

θ_{1} + θ_{2} + \dots + θ_{10} = 2 π

. Define the complex numbers

z_{1} = e^{i θ_{1}}, z_{k} = z_{k - 1} e^{i θ_{k}}

for

k = 2, 3 \dots 10,

where

i = \sqrt{- 1}

. Consider the statements

P

and

Q

given below:

P : | z_{2} - z_{1} | + | z_{3} - z_{2} | + \dots + | z_{10} - z_{9} | + | z_{1} - z_{10} | \leq 2 π

Q : | z_{2}^{2} - z_{1}^{2} | + | z_{3}^{2} - z_{2}^{2} | + \dots + | z_{10}^{2} - z_{9}^{2} | + | z_{1}^{2} - z_{10}^{2} | \leq 4 π

Q. If

θ_{1}

and

θ_{2}

are arguments of two non-zero complex numbers

z_{1}

and

z_{2}

respectively such that

3 | z_{1} | = 4 | z_{2} |

. If

z = \frac{3 z_{1}}{2 z_{2}} + \frac{2 z_{2}}{3 z_{1}}

, then

x^{2} + y^{2} + x y = 1

for all

x, y \in R,

the minimum value of

x^{3} y + x y^{3} + 4

is
(correct answer + 1, wrong answer - 0.25)

x^{2} + y^{2} + x y = 1

for all

x, y \in R,

the minimum value of

x^{3} y + x y^{3} + 4

is
(correct answer + 1, wrong answer - 0.25)

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The minimum value of sec4θ1tan2θ2+sec4θ2tan2θ1, wherever defined, is ​​​​​​​(correct answer + 1, wrong answer - 0.25)

The correct option is B 8Let tan2θ1=a, tan2θ2=b, so given equation becomes (a+1)2b+(b+1)2a⇒2(ab+ba)+(a2b+b2a+1a+1b) Using A.M.≥G.M., so ab+ba≥2a2b+b2a+1a+1b≥4⇒2(ab+ba)+(a2b+b2a+1a+1b)≥8 Hence, the minimum value of the expression is 8. Equality holds when θ1=θ2=π4.

The minimum value of $\frac{{sec}^{4} θ_{1}}{{tan}^{2} θ_{2}} + \frac{{sec}^{4} θ_{2}}{{tan}^{2} θ_{1}}$ , wherever defined, is
(correct answer + 1, wrong answer - 0.25)