JEE Advanced · Mathematics · 8. Trigonometric Equations

Let $P=\{\theta: \sin \theta-\cos \theta=\sqrt{2} \cos \theta\}$ and $Q=\{\theta: \sin \theta+\cos \theta=\sqrt{2} \sin \theta\}$ be two sets. Then,

A $P \subset Q$ and $Q-P \neq \Phi$
B $Q \not \subset P$
C $P \not \subset Q$
D $P=Q$

Verified Solution

Answer & Solution

Correct Answer

(D) $P=Q$

Step-by-step Solution

Detailed explanation

$P=\{\theta: \sin \theta-\cos \theta=\sqrt{2} \cos \theta\} $
$ \Rightarrow \cos \theta(\sqrt{2}+1)=\sin \theta $
$ \Rightarrow \tan \theta=\sqrt{2}+1 $
$ Q=\{\theta: \sin \theta+\cos \theta=\sqrt{2} \sin \theta\} $
$ \Rightarrow \sin \theta(\sqrt{2}-1)=\cos \theta $
$ \Rightarrow \tan \theta=\frac{1}{\sqrt{2}-1} \times \frac{\sqrt{2}+1}{\sqrt{2}+1} $
$ =(\sqrt{2}+1) $
$ \therefore P=Q$

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

Same subject

Match the Statements/Expressions in Column I with the Statements/Expressions in Column II.

Column I	Column II
(A) The minimum value of $\frac{x^2+2 x+4}{x+2}$ is	(p) 0
(B) Let $A$ and $B$ be $3 \times 3$ matrices of real numbers, where $A$ is symmetric, $B$ is skew-symmetric and $(A+B)(A-B) =$ $(A-B)(A+B)$. If $(A B)^t=(-1)^k$ $A B$, where $(A B)^t$ is the transpose of the matrix $A B$, then possible values of $k$ are	(q) 1
(C) Let $a=\log _3 \log _3 2$. An integer $k$ satisfying $1<2^{\left(-k+3^{-a}\right)}<2$, must be less than	(r) 2
(D) If $\sin \theta=\cos \phi$, then the possible values of $\frac{1}{\pi}\left(\theta \pm \phi-\frac{\pi}{2}\right)$ are	(s) 3

JEE Advanced 2008 Hard

For any positive integer $n$ , define $f_{n} : (0, \infty) \to R$ as $f_{n} (x) = \sum_{i = 1}^{n} \tan^{- 1} (\frac{1}{1 + (x + j) (x + j - 1)})$ for all $x \in (0, \infty)$ . $($ Here, the inverse trigonometric function $\tan^{- 1} x$ assumes values in $(- \frac{π}{2}, \frac{π}{2})$ $)$ Then, which of the following statement(s) is (are) TRUE?

JEE Advanced 2018 Medium

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Let \(P=\{\theta: \sin \theta-\cos \theta=\sqrt{2} \cos \theta\}\) and \(Q=\{\theta: \sin \theta+\cos \theta=\sqrt{2} \sin \theta\}\) be two sets. Then,

Answer & Solution

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See the Complete Solution

Column I	Column II
(A) The minimum value of \(\frac{x^2+2 x+4}{x+2}\) is	(p) 0
(B) Let \(A\) and \(B\) be \(3 \times 3\) matrices of real numbers, where \(A\) is symmetric, \(B\) is skew-symmetric and \((A+B)(A-B) =\) \((A-B)(A+B)\). If \((A B)^t=(-1)^k\) \(A B\), where \((A B)^t\) is the transpose of the matrix \(A B\), then possible values of \(k\) are	(q) 1
(C) Let \(a=\log _3 \log _3 2\). An integer \(k\) satisfying \(1<2^{\left(-k+3^{-a}\right)}<2\), must be less than	(r) 2
(D) If \(\sin \theta=\cos \phi\), then the possible values of \(\frac{1}{\pi}\left(\theta \pm \phi-\frac{\pi}{2}\right)\) are	(s) 3