Practice focused topic-wise MCQs with answers and explanations for quick revision and exam preparation.
Difficulty select karke sirf is topic ke focused MCQs practice karein. Score, timer aur explanations student-friendly flow me milenge.
Showing 331-340 of 350 questions.
In which interval does the value of \(\sin x\) always lie?
C. ( [-1,1] )
किसी भी वास्तविक (x) के लिए \(-1\leq \sin x\leq 1\) होता है। परीक्षा में sine और cosine का basic range सबसे पहले लगाएं। / For any real (x), \(-1\leq \sin x\leq 1\). In exams, apply the basic range of sine and cosine first.
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At which values is \(\tan x\) not defined?
D. \(x=\frac{\pi}{2}+n\pi\)
\(\tan x=\frac{\sin x}{\cos x}\) है, इसलिए \(\cos x=0\) पर यह अपरिभाषित होता है। अतः \(x=\frac{\pi}{2}+n\pi\)। / Since \(\tan x=\frac{\sin x}{\cos x}\), it is undefined when \(\cos x=0\). Thus \(x=\frac{\pi}{2}+n\pi\).
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Where is \(\cosec x\) not defined?
B. \(x=n\pi\)
\(\cosec x=\frac{1}{\sin x}\) है, इसलिए \(\sin x=0\) पर यह अपरिभाषित होता है। अतः \(x=n\pi\)। / Since \(\cosec x=\frac{1}{\sin x}\), it is undefined when \(\sin x=0\). Thus \(x=n\pi\).
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If \(\sin x+\cos x=\sqrt{2}\), what is (x) in \(0\leq x<2\pi\)?
C. \( \frac{\pi}{4} \)
\(\sin x+\cos x=\sqrt{2}\sin\left(x+\frac{\pi}{4}\right)\) है। अधिकतम \(\sqrt{2}\) तब मिलता है जब \(x=\frac{\pi}{4}\)। / We use \(\sin x+\cos x=\sqrt{2}\sin\left(x+\frac{\pi}{4}\right)\). The maximum \(\sqrt{2}\) occurs at \(x=\frac{\pi}{4}\).
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What is \(2\sin x\cos x\) equal to?
B. \(\sin 2x\)
दुगुना कोण सूत्र के अनुसार \(2\sin x\cos x=\sin 2x\)। परीक्षा में यह identity बार-बार काम आती है। / By the double angle formula, \(2\sin x\cos x=\sin 2x\). This identity is used frequently in exams.
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What is \(\cos^2 x-\sin^2 x\) equal to?
C. \(\cos 2x\)
दुगुना कोण पहचान \(\cos 2x=\cos^2 x-\sin^2 x\) है। परीक्षा में \(\cos 2x\) के तीनों रूप याद रखें। / The double angle identity is \(\cos 2x=\cos^2 x-\sin^2 x\). In exams, remember all three forms of \(\cos 2x\).
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A. \(\sec x\)
\(\sec x=1/\cos x\) है, इसलिए \(\cos x=0\) होने पर यह अपरिभाषित है; ये मान \(x=\frac{(2n+1)\pi}{2}\) हैं। साथ ही \(|\sec x|\ge1\), इसलिए दिया गया परास मिलता है। \(\csc x\) के लिए निषिद्ध मान \(n\pi\) होते हैं। परीक्षा टिप: व्युत्क्रम फलनों में हर के शून्य होने वाले कोण पहले जाँचें। / Since \(\sec x=1/\cos x\), it is undefined when \(\cos x=0\), i.e. at \(x=\frac{(2n+1)\pi}{2}\). Also, \(|\sec x|\ge1\), giving the stated range. For \(\csc x\), the excluded values are \(n\pi\). Exam tip: first locate angles that make the denominator zero in reciprocal functions.
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What is \(\tan x+\cot x\) equal to?
A. \( \sec x\cosec x \)
\(\tan x+\cot x=\frac{\sin x}{\cos x}+\frac{\cos x}{\sin x}\) है। सरलीकरण से \(\frac{1}{\sin x\cos x}=\sec x\cosec x\) मिलता है। / We have \(\tan x+\cot x=\frac{\sin x}{\cos x}+\frac{\cos x}{\sin x}\). Simplifying gives \(\frac{1}{\sin x\cos x}=\sec x\cosec x\).
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What is the fundamental period of (f(x)=\tan 3x)?
B. \( \frac{\pi}{3} \)
\(\tan ax\) का आवर्त \( \frac{\pi}{|a|} \) होता है। यहाँ (a=3), इसलिए आवर्त \( \frac{\pi}{3} \) है। / The period of \(\tan ax\) is \( \frac{\pi}{|a|} \). Here (a=3), so the period is \( \frac{\pi}{3} \).
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What is the period of (f(x)=\cos 4x)?
C. \( \frac{\pi}{2} \)
\(\cos ax\) का आवर्त \( \frac{2\pi}{|a|} \) होता है। इसलिए \(\cos 4x\) का आवर्त \( \frac{\pi}{2} \) है। / The period of \(\cos ax\) is \( \frac{2\pi}{|a|} \). Hence the period of \(\cos 4x\) is \( \frac{\pi}{2} \).
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