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.
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What is (\cos\(\frac{3\pi}{2}-x\)) equal to?
C. -\(\sin x\)
\(\frac{3\pi}{2}-x\) तीसरे चतुर्थांश से जुड़ा है। \(\cos\) बदलकर \(\sin\) होता है और चिन्ह ऋणात्मक रहता है। / \(\frac{3\pi}{2}-x\) is related to the third quadrant. \(\cos\) changes to \(\sin\) with a negative sign.
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What is (\tan\(\frac{3\pi}{2}+x\)) equal to?
C. -\(\cot x\)
\(\frac{3\pi}{2}+x\) पर \(\tan\) बदलकर \(\cot\) होता है और चिन्ह ऋणात्मक है। इसलिए (\tan\(\frac{3\pi}{2}+x\)=-\cot x)। / At \(\frac{3\pi}{2}+x\), \(\tan\) changes to \(\cot\) with a negative sign. Hence (\tan\(\frac{3\pi}{2}+x\)=-\cot x).
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What is the simplified value of \(\frac{\sin(\pi-x)}{\cos(\pi+x)}\)?
A. \(-\tan x\)
पहचानें: \(\sin(\pi-x)=\sin x\) तथा \(\cos(\pi+x)=-\cos x\)। अतः \(\frac{\sin(\pi-x)}{\cos(\pi+x)}=\frac{\sin x}{-\cos x}=-\tan x\)। \(\tan x\) निकटतम विचलक है, पर हर में ऋण चिह्न के कारण वह गलत है। परीक्षा टिप: \(\pi\pm x\) वाले कोणों में sin और cos के चिह्न अलग-अलग जाँचें। / Using allied-angle identities, \(\sin(\pi-x)=\sin x\) and \(\cos(\pi+x)=-\cos x\). Therefore, \(\frac{\sin(\pi-x)}{\cos(\pi+x)}=\frac{\sin x}{-\cos x}=-\tan x\). \(\tan x\) is the closest distractor, but it misses the negative sign from the denominator. Exam tip: For angles involving \(\pi\pm x\), check the sign of each trigonometric function separately.
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What is the simplified value of \(\frac{\cos(2\pi-x)}{\sin(\pi+x)}\)?
B. \(-\cot x\)
सर्वसमिकाओं \(\cos(2\pi-x)=\cos x\) तथा \(\sin(\pi+x)=-\sin x\) का प्रयोग करने पर \[\frac{\cos(2\pi-x)}{\sin(\pi+x)}=\frac{\cos x}{-\sin x}=-\cot x.\] अतः सही उत्तर \(-\cot x\) है। \(\cot x\) लेने पर हर के ऋण चिह्न की उपेक्षा हो जाएगी। परीक्षा टिप: \(\pi+x\) वाले कोण में sine का चिह्न ऋणात्मक होता है। / Using the identities \(\cos(2\pi-x)=\cos x\) and \(\sin(\pi+x)=-\sin x\), \[\frac{\cos(2\pi-x)}{\sin(\pi+x)}=\frac{\cos x}{-\sin x}=-\cot x.\] Therefore, the correct answer is \(-\cot x\). Choosing \(\cot x\) would ignore the negative sign in the denominator. Exam tip: for an angle of the form \(\pi+x\), sine has a negative sign.
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How can \(\sec^2 x+\cosec^2 x\) be written in terms of \(\tan x\) and \(\cot x\)?
B. \(2+\tan^2 x+\cot^2 x\)
\(\sec^2 x=1+\tan^2 x\) और \(\cosec^2 x=1+\cot^2 x\) लगाएँ। योग \(2+\tan^2 x+\cot^2 x\) होगा। / Use \(\sec^2 x=1+\tan^2 x\) and \(\cosec^2 x=1+\cot^2 x\). The sum becomes \(2+\tan^2 x+\cot^2 x\).
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What is the simplified value of \(\frac{\tan x+\cot x}{\sec x\cosec x}\)?
A. (1)
\(\tan x+\cot x=\frac{1}{\sin x\cos x}\) और \(\sec x\cosec x=\frac{1}{\sin x\cos x}\) होता है। इसलिए अनुपात (1) है। / \(\tan x+\cot x=\frac{1}{\sin x\cos x}\) and \(\sec x\cosec x=\frac{1}{\sin x\cos x}\). Hence the ratio is (1).
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What is the simplified value of \(\frac{\sec x}{\tan x}\)?
B. \(\cosec x\)
\(\sec x=\frac{1}{\cos x}\) और \(\tan x=\frac{\sin x}{\cos x}\) रखें। अनुपात \(\frac{1}{\sin x}=\cosec x\) बनता है। / Put \(\sec x=\frac{1}{\cos x}\) and \(\tan x=\frac{\sin x}{\cos x}\). The ratio becomes \(\frac{1}{\sin x}=\cosec x\).
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What is the simplified value of \(\frac{\cosec x}{\cot x}\)?
B. \(\sec x\)
\(\cosec x=\frac{1}{\sin x}\) और \(\cot x=\frac{\cos x}{\sin x}\) रखें। अनुपात \(\frac{1}{\cos x}=\sec x\) होगा। / Put \(\cosec x=\frac{1}{\sin x}\) and \(\cot x=\frac{\cos x}{\sin x}\). The ratio becomes \(\frac{1}{\cos x}=\sec x\).
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What is the simplified value of (\sin-2 x\(1+\cot^2 x\))?
B. \(1\)
पहचान \(1+\cot^2 x=\cosec^2 x\) का प्रयोग करें। अतः \(\sin^2 x(1+\cot^2 x)=\sin^2 x\cdot\cosec^2 x=\sin^2 x\cdot\frac{1}{\sin^2 x}=1\)। इसलिए सही उत्तर \(1\) है। \(\sin^2 x\) केवल गुणनखंड है, सरल मान नहीं। परीक्षा टिप: \(1+\cot^2 x\) दिखते ही \(\cosec^2 x\) वाली पहचान याद करें। / Use the identity \(1+\cot^2 x=\cosec^2 x\). Thus, \(\sin^2 x(1+\cot^2 x)=\sin^2 x\cdot\cosec^2 x=\sin^2 x\cdot\frac{1}{\sin^2 x}=1\). Therefore, the correct answer is \(1\). \(\sin^2 x\) is only a factor, not the simplified value. Exam tip: When you see \(1+\cot^2 x\), recall the \(\cosec^2 x\) identity.
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What is the simplified value of (\cos-2 x\(1+\tan^2 x\))?
A. \(1\)
पहचान \(1+\tan^2 x=\sec^2 x\) का प्रयोग करने पर दिया गया व्यंजक \(\cos^2 x\sec^2 x\) बनता है। क्योंकि \(\sec^2 x=1/\cos^2 x\), इसलिए इसका मान \(1\) है। \(\sec^2 x\) केवल कोष्ठक वाले भाग का मान है, पूरे व्यंजक का नहीं। परीक्षा टिप: \(1+\tan^2 x\) दिखते ही इसे \(\sec^2 x\) से बदलें। / Using the identity \(1+\tan^2 x=\sec^2 x\), the expression becomes \(\cos^2 x\sec^2 x\). Since \(\sec^2 x=1/\cos^2 x\), its value is \(1\). \(\sec^2 x\) is only the value of the bracketed part, not of the complete expression. Exam tip: When you see \(1+\tan^2 x\), replace it with \(\sec^2 x\).
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