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JIPMAT 2021

Geometry

>

Trigonometry

Conceptual

The minimum value of $(2 \sin^2\theta + 3 \cos^2\theta)$ is

0

1

2

3

Correct Option: 3

1) Given expression:

2\sin^2 \theta + 3\cos^2 \theta

2) Using identity

\sin^2 \theta + \cos^2 \theta = 1

:

\newline

Therefore,

\sin^2 \theta = 1 - \cos^2 \theta

3) Substituting:

\newline

2(1 - \cos^2 \theta) + 3\cos^2 \theta

\newline

= 2 - 2\cos^2 \theta + 3\cos^2 \theta

\newline

= 2 + \cos^2 \theta

4) Since

\cos^2 \theta

varies between 0 and 1:

\newline

- When

\cos^2 \theta = 0

, expression = 2

\newline

- When

\cos^2 \theta = 1

, expression = 3

5) Therefore:

\newline

Minimum value = 2 (when

\cos^2 \theta = 0

)

\newline

Maximum value = 3 (when

\cos^2 \theta = 1

)

The minimum value is 2.

Related questions:

JIPMAT 2022

Two poles of height 6 m and 11 m stand vertically upright on a plane ground. If the distance between their foot is 12 m, then the distance between their tops is

JIPMAT 2022

Given below are two statement:

Statement I: If

\sin (\theta)=\frac{5}{13}

, then the value of

\tan (\theta)=\frac{5}{12}

Statement II: if

\cot (\theta)=\frac{12}{5}

, then the value of

\sin (\theta)=\frac{5}{12}

In the light of the above statements, choose the correct answer form the question below:

JIPMAT 2022

Which of the following trigonometric identities are true?

\begin{aligned} & \sin ^2\left(41^{\circ}\right)+\sin ^2\left(49^{\circ}\right)=1 \\ & \sin ^2\left(60^{\circ}\right)-2 \tan \left(45^{\circ}\right)-\cos ^2\left(30^{\circ}\right)=-1 \\ & \sin ^2(\theta)+\frac{1}{1+\tan ^2(\theta)}=1 \end{aligned}