If A-2 cos2 - sin- 2 and B-2-3 -2- then A - B is equal to

2 cos^2 θ+sinθ≤ 2 ⇒ 2(1 sin^2 θ)+sinθ≤ 2 ⇒ 2 sin^2 θ sinθ≥0 ⇒sinθ (2 sinθ 1) ≥ 0 ⇒sinθ∈ [ 1,0] ∪[ 12,1] From the diagram, we can conclude A ∩ B ={θ: π2≤θ≤5 π6 ...

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Standard XII

Mathematics

Intersection

If A=θ:2 cos2...

Question

If $A = {θ : 2 {cos}^{2} θ + sin θ \leq 2}$ and $B = {θ : \frac{π}{2} \leq θ \leq \frac{3 π}{2}},$ then $A \cap B$ is equal to

{θ : \frac{π}{2} \leq θ \leq \frac{5 π}{6}}

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{θ : π \leq θ \leq \frac{3 π}{2}}

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{θ : \frac{π}{2} \leq θ \leq \frac{5 π}{6}} \cup {θ : π \leq θ \leq \frac{3 π}{2}}

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{θ : \frac{π}{2} \leq θ \leq \frac{5 π}{6}} \cap {θ : π \leq θ \leq \frac{3 π}{2}}

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Solution

The correct option is C ${θ : \frac{π}{2} \leq θ \leq \frac{5 π}{6}} \cup {θ : π \leq θ \leq \frac{3 π}{2}}$
$2 {cos}^{2} θ + sin θ \leq 2$
$\Rightarrow 2 (1 - {sin}^{2} θ) + sin θ \leq 2$
$\Rightarrow 2 {sin}^{2} θ - sin θ \geq 0$
$\Rightarrow sin θ (2 sin θ - 1) \geq 0$
$\Rightarrow sin θ \in [- 1, 0] \cup [\frac{1}{2}, 1]$

From the diagram, we can conclude
$A \cap B = {θ : \frac{π}{2} \leq θ \leq \frac{5 π}{6}} \cup {θ : π \leq θ \leq \frac{3 π}{2}}$

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Similar questions

Q. Let

A = {θ : 2 {cos}^{2} + θ + sin θ \leq 2}

and

B = {θ : π / 2 \leq θ \leq 3 π / 2} .

Then

A \cap B = {θ : π / 2 \leq θ \leq 5 π / 6 o r π \leq θ \leq 3 π / 4}

If true enter 1 else 0

Q. Fill in the blanks :
A =

θ : 2 c o s^{2} θ + s i n θ \leq 2

B =

θ : \frac{π}{2} \leq \frac{3 π}{2}

then A

\cap

B =........

Q. Solve :

2 {cos}^{2} θ + sin θ \leq 2

, where

\frac{π}{2} \leq θ \leq \frac{3 π}{2}

Q. The solution set for

2 {cos}^{2} θ + sin θ \leq 2,

where

\frac{π}{2} \leq θ \leq \frac{3 π}{2}

, is

Q. If

\frac{π}{2} < θ < \frac{3 π}{2}

, then the value of

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is equal to

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If A={θ:2cos2θ+sinθ≤2} and B={θ:π2≤θ≤3π2}, then A∩B is equal to

The correct option is C {θ:π2≤θ≤5π6}∪{θ:π≤θ≤3π2}2cos2θ+sinθ≤2 ⇒2(1−sin2θ)+sinθ≤2 ⇒2sin2θ−sinθ≥0 ⇒sinθ(2sinθ−1)≥0 ⇒sinθ∈[−1,0]∪[12,1] From the diagram, we can conclude A∩B={θ:π2≤θ≤5π6}∪{θ:π≤θ≤3π2}

If $A = {θ : 2 {cos}^{2} θ + sin θ \leq 2}$ and $B = {θ : \frac{π}{2} \leq θ \leq \frac{3 π}{2}},$ then $A \cap B$ is equal to