Level 72 • 50/50 questions • 40 seconds per question.
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What is the period of the function \(\sin x\)?
C. \(2\pi\)
\(\sin x\) हर \(2\pi\) के बाद अपना मान दोहराता है। परीक्षा में काल पूछने पर सबसे पहले मूल फलन पहचानें। / \(\sin x\) repeats its value after every \(2\pi\). In exams first identify the basic function.
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What is the range of the function \(\cos x\)?
B. ([-1,1])
\(\cos x\) का मान हमेशा (-1) और (1) के बीच रहता है। परिसर में अंतिम मान भी शामिल होते हैं। / The value of \(\cos x\) always lies between (-1) and (1). The end values are included in the range.
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Which of the following functions is odd?
C. \(\sin x\)
क्योंकि (\sin(-x)=-\sin x), इसलिए \(\sin x\) विषम फलन है। विषम फलन में (f(-x)=-f(x)) होता है। / Since (\sin(-x)=-\sin x), \(\sin x\) is an odd function. For an odd function, (f(-x)=-f(x)).
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Which of the following functions is even?
D. \(\cos x\)
क्योंकि (\cos(-x)=\cos x), इसलिए \(\cos x\) सम फलन है। सम फलन में (f(-x)=f(x)) होता है। / Since (\cos(-x)=\cos x), \(\cos x\) is an even function. For an even function, (f(-x)=f(x)).
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What is the value of the identity \(\sin^2 x+\cos^2 x\)?
B. (1)
मूल पहचान \(\sin^2 x+\cos^2 x=1\) होती है। इसे याद रखना कई प्रश्नों में काम आता है। / The basic identity is \(\sin^2 x+\cos^2 x=1\). Remembering it helps in many questions.
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What is the period of the function \(\tan x\)?
A. \(\pi\)
\(\tan x\) हर \(\pi\) के बाद अपना मान दोहराता है। \(\tan x\) और \(\cot x\) का काल \(\pi\) होता है। / \(\tan x\) repeats its value after every \(\pi\). The period of \(\tan x\) and \(\cot x\) is \(\pi\).
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The function \(\sec x\) is the reciprocal of which function?
C. \(\cos x\)
\(\sec x=\frac{1}{\cos x}\) होता है। व्युत्क्रम संबंधों को जोड़ी बनाकर याद करें। / \(\sec x=\frac{1}{\cos x}\). Learn reciprocal relations in pairs.
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The function \(\cosec x\) is the reciprocal of which function?
B. \(\sin x\)
\(\cosec x=\frac{1}{\sin x}\) होता है। ध्यान रखें कि \(\cosec x\) का संबंध \(\sin x\) से है। / \(\cosec x=\frac{1}{\sin x}\). Remember that \(\cosec x\) is related to \(\sin x\).
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How is \(\tan x\) written in terms of \(\sin x\) and \(\cos x\)?
C. \(\frac{\sin x}{\cos x}\)
\(\tan x=\frac{\sin x}{\cos x}\) मूल अनुपात पहचान है। ऐसे प्रश्नों में अंश और हर को ध्यान से देखें। / \(\tan x=\frac{\sin x}{\cos x}\) is a basic quotient identity. In such questions, check numerator and denominator carefully.
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How is \(\cot x\) written in terms of \(\sin x\) and \(\cos x\)?
B. \(\frac{\cos x}{\sin x}\)
\(\cot x=\frac{\cos x}{\sin x}\) होता है। \(\tan x\) और \(\cot x\) एक-दूसरे के व्युत्क्रम हैं। / \(\cot x=\frac{\cos x}{\sin x}\). \(\tan x\) and \(\cot x\) are reciprocals of each other.
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What is the range of the function \(\sin x\)?
A. ([-1,1])
\(\sin x\) का मान (-1) से (1) तक होता है। ग्राफ याद रखने से परिसर जल्दी मिलता है। / The value of \(\sin x\) lies from (-1) to (1). Remembering the graph helps find the range quickly.
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What is the range of the function \(\tan x\)?
C. (\(-\infty,\infty\))
\(\tan x\) सभी वास्तविक मान ले सकता है। इसलिए इसका परिसर (\(-\infty,\infty\)) है। / \(\tan x\) can take all real values. Therefore, its range is (\(-\infty,\infty\)).
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What is the value of \(\sec^2 x-\tan^2 x\)?
D. (1)
क्योंकि \(\sec^2 x=1+\tan^2 x\), इसलिए \(\sec^2 x-\tan^2 x=1\)। पहचान को सही दिशा में प्रयोग करें। / Since \(\sec^2 x=1+\tan^2 x\), \(\sec^2 x-\tan^2 x=1\). Use the identity in the correct direction.
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What is the value of \(\cosec^2 x-\cot^2 x\)?
B. (1)
क्योंकि \(\cosec^2 x=1+\cot^2 x\), अंतर (1) होगा। यह पहचान \(\sin^2 x+\cos^2 x=1\) से जुड़ी है। / Since \(\cosec^2 x=1+\cot^2 x\), the difference is (1). This identity is linked to \(\sin^2 x+\cos^2 x=1\).
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If \(\sin x=0\), what is the value of \(\cos x\) at (x=0)?
C. (1)
(x=0) पर \(\cos 0=1\) होता है। मानक कोणों के मान याद रखना आसान प्रश्नों में मदद करता है। / At (x=0), \(\cos 0=1\). Remembering standard angle values helps in easy questions.
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What is the value of \(\sin 0\)?
B. (0)
\(\sin 0=0\) एक मानक मान है। परीक्षा में (0), \(\frac{\pi}{2}\) और \(\pi\) के मान जरूर याद रखें। / \(\sin 0=0\) is a standard value. In exams, remember the values for (0), \(\frac{\pi}{2}\), and \(\pi\).
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What is the value of \(\tan 0\)?
C. (0)
\(\tan 0=\frac{\sin 0}{\cos 0}=\frac{0}{1}=0\)। अनुपात पहचान से मान आसानी से निकलेगा। / \(\tan 0=\frac{\sin 0}{\cos 0}=\frac{0}{1}=0\). The quotient identity gives the value easily.
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What is the value of \(\sin \frac{\pi}{2}\)?
B. (1)
\(\frac{\pi}{2}\) पर \(\sin x\) का मान (1) होता है। यूनिट सर्कल से यह मान जल्दी याद होता है। / At \(\frac{\pi}{2}\), the value of \(\sin x\) is (1). The unit circle helps remember this value quickly.
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What is the value of \(\cos \frac{\pi}{2}\)?
C. (0)
\(\frac{\pi}{2}\) पर \(\cos x\) का मान (0) है। मानक कोणों की तालिका से गलती कम होती है। / At \(\frac{\pi}{2}\), the value of \(\cos x\) is (0). A standard angle table reduces mistakes.
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What is (\sin\(\pi-x\)) equal to?
C. \(\sin x\)
\(\pi-x\) दूसरे चतुर्थांश में होता है और वहाँ \(\sin x\) धनात्मक रहता है। इसलिए (\sin\(\pi-x\)=\sin x)। / \(\pi-x\) lies in the second quadrant and \(\sin x\) remains positive there. Hence (\sin\(\pi-x\)=\sin x).
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What is (\cos\(\pi-x\)) equal to?
B. \(-\cos x\)
दूसरे चतुर्थांश में \(\cos x\) ऋणात्मक होता है। इसलिए (\cos\(\pi-x\)=-\cos x)। / In the second quadrant, \(\cos x\) is negative. Therefore, (\cos\(\pi-x\)=-\cos x).
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What is (\tan\(\pi+x\)) equal to?
C. \(\tan x\)
\(\tan x\) का काल \(\pi\) है, इसलिए (\tan\(\pi+x\)=\tan x)। काल का उपयोग करके उत्तर जल्दी मिलता है। / The period of \(\tan x\) is \(\pi\), so (\tan\(\pi+x\)=\tan x). Using period gives the answer quickly.
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B. \(-\sin x\)
\(\sin x\) विषम फलन है, इसलिए (\sin(-x)=-\sin x)। सम-विषम गुण जल्दी पहचानें। / \(\sin x\) is an odd function, so (\sin(-x)=-\sin x). Identify even-odd properties quickly.
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C. \(\cos x\)
\(\cos x\) सम फलन है, इसलिए (\cos(-x)=\cos x)। ऋण कोण वाले प्रश्नों में सम-विषम गुण लगाएँ। / \(\cos x\) is an even function, so (\cos(-x)=\cos x). Use even-odd properties for negative angle questions.
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B. \(-\tan x\)
\(\tan x\) विषम फलन है, इसलिए (\tan(-x)=-\tan x)। इसे \(\frac{\sin x}{\cos x}\) से भी समझ सकते हैं। / \(\tan x\) is an odd function, so (\tan(-x)=-\tan x). This can also be understood from \(\frac{\sin x}{\cos x}\).
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What is the sign of \(\sin x\) in the first quadrant?
C. धनात्मकPositive
पहले चतुर्थांश में सभी त्रिकोणमितीय फलन धनात्मक होते हैं। चतुर्थांश संकेत तालिका याद रखें। / In the first quadrant, all trigonometric functions are positive. Remember the quadrant sign table.
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What is the sign of \(\cos x\) in the second quadrant?
B. ऋणात्मकNegative
दूसरे चतुर्थांश में केवल \(\sin x\) और \(\cosec x\) धनात्मक होते हैं। इसलिए \(\cos x\) ऋणात्मक होता है। / In the second quadrant, only \(\sin x\) and \(\cosec x\) are positive. Therefore, \(\cos x\) is negative.
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What is the sign of \(\tan x\) in the third quadrant?
C. धनात्मकPositive
तीसरे चतुर्थांश में \(\tan x\) और \(\cot x\) धनात्मक होते हैं। संकेत नियम से उत्तर सीधे मिलता है। / In the third quadrant, \(\tan x\) and \(\cot x\) are positive. The sign rule gives the answer directly.
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What is the sign of \(\sin x\) in the fourth quadrant?
B. ऋणात्मकNegative
चौथे चतुर्थांश में \(\cos x\) और \(\sec x\) धनात्मक होते हैं। इसलिए \(\sin x\) ऋणात्मक होता है। / In the fourth quadrant, \(\cos x\) and \(\sec x\) are positive. Hence \(\sin x\) is negative.
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What is (\sin\(2\pi+x\)) equal to?
A. \(\sin x\)
\(\sin x\) का काल \(2\pi\) है, इसलिए (\sin\(2\pi+x\)=\sin x)। काल वाली पहचान याद रखें। / The period of \(\sin x\) is \(2\pi\), so (\sin\(2\pi+x\)=\sin x). Remember periodic identities.
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What is (\cos\(2\pi+x\)) equal to?
C. \(\cos x\)
\(\cos x\) हर \(2\pi\) पर दोहरता है। इसलिए (\cos\(2\pi+x\)=\cos x)। / \(\cos x\) repeats after every \(2\pi\). Therefore, (\cos\(2\pi+x\)=\cos x).
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What is (\sin\(\frac{\pi}{2}-x\)) equal to?
C. \(\cos x\)
पूरक कोण पहचान से (\sin\(\frac{\pi}{2}-x\)=\cos x) होता है। \(\frac{\pi}{2}\) के साथ फलन बदलता है। / By the complementary angle identity, (\sin\(\frac{\pi}{2}-x\)=\cos x). With \(\frac{\pi}{2}\), the function changes.
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What is (\cos\(\frac{\pi}{2}-x\)) equal to?
D. \(\sin x\)
पूरक कोण पहचान से (\cos\(\frac{\pi}{2}-x\)=\sin x) होता है। ऐसे प्रश्नों में \(\sin x\) और \(\cos x\) आपस में बदलते हैं। / By the complementary angle identity, (\cos\(\frac{\pi}{2}-x\)=\sin x). In such questions, \(\sin x\) and \(\cos x\) interchange.
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What is (\tan\(\frac{\pi}{2}-x\)) equal to?
B. \(\cot x\)
\(\frac{\pi}{2}\) के पूरक कोण में \(\tan x\) बदलकर \(\cot x\) हो जाता है। पूरक पहचान याद रखें। / For a complementary angle with \(\frac{\pi}{2}\), \(\tan x\) changes to \(\cot x\). Remember cofunction identities.
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What is the period of the function \(\sec x\)?
C. \(2\pi\)
\(\sec x\) का काल \(\cos x\) जैसा \(2\pi\) होता है। व्युत्क्रम फलन का काल मूल फलन से जुड़ा रहता है। / The period of \(\sec x\) is \(2\pi\), like \(\cos x\). The reciprocal function keeps the related basic period.
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What is the period of the function \(\cot x\)?
A. \(\pi\)
\(\cot x\) हर \(\pi\) के बाद अपना मान दोहराता है। \(\tan x\) और \(\cot x\) दोनों का काल \(\pi\) है। / \(\cot x\) repeats its value after every \(\pi\). Both \(\tan x\) and \(\cot x\) have period \(\pi\).
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What is the period of the function \(\cosec x\)?
C. \(2\pi\)
\(\cosec x\) \(\sin x\) का व्युत्क्रम है और इसका काल \(2\pi\) है। व्युत्क्रम संबंध से काल पहचानें। / \(\cosec x\) is the reciprocal of \(\sin x\) and its period is \(2\pi\). Use reciprocal relation to identify the period.
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What is the range of \(\sec x\)?
D. (\(-\infty,-1]\cup[1,\infty\))
\(\sec x=\frac{1}{\cos x}\) है और \(\cos x\) का मान ([-1,1]) में होता है। इसलिए \(\sec x\) का मान (1) से बड़ा या (-1) से छोटा होता है। / \(\sec x=\frac{1}{\cos x}\) and \(\cos x\) lies in ([-1,1]). Hence \(\sec x\) is greater than or equal to (1) or less than or equal to (-1).
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What is the range of \(\cosec x\)?
A. (\(-\infty,-1]\cup[1,\infty\))
\(\cosec x=\frac{1}{\sin x}\) है। इसलिए इसका परिसर (\(-\infty,-1]\cup[1,\infty\)) होता है। / \(\cosec x=\frac{1}{\sin x}\). Therefore, its range is (\(-\infty,-1]\cup[1,\infty\)).
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Where is \(\tan x\) undefined?
A. \(\cos x=0\)
\(\tan x=\frac{\sin x}{\cos x}\) है, इसलिए \(\cos x=0\) पर यह अपरिभाषित होता है। हर शून्य होने पर भिन्न परिभाषित नहीं रहती। / \(\tan x=\frac{\sin x}{\cos x}\), so it is undefined when \(\cos x=0\). A fraction is not defined when the denominator is zero.
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Where is \(\cot x\) undefined?
B. \(\sin x=0\)
\(\cot x=\frac{\cos x}{\sin x}\) है, इसलिए \(\sin x=0\) पर यह अपरिभाषित होता है। हर को हमेशा ध्यान से देखें। / \(\cot x=\frac{\cos x}{\sin x}\), so it is undefined when \(\sin x=0\). Always check the denominator carefully.
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Where is \(\sec x\) undefined?
C. \(\cos x=0\)
\(\sec x=\frac{1}{\cos x}\) है, इसलिए \(\cos x=0\) पर यह अपरिभाषित होता है। व्युत्क्रम फलनों में हर शून्य नहीं हो सकता। / \(\sec x=\frac{1}{\cos x}\), so it is undefined when \(\cos x=0\). In reciprocal functions, the denominator cannot be zero.
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Where is \(\cosec x\) undefined?
B. \(\sin x=0\)
\(\cosec x=\frac{1}{\sin x}\) है, इसलिए \(\sin x=0\) पर यह अपरिभाषित होता है। व्युत्क्रम में नीचे वाला फलन शून्य नहीं होना चाहिए। / \(\cosec x=\frac{1}{\sin x}\), so it is undefined when \(\sin x=0\). In a reciprocal, the denominator function must not be zero.
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If \(\sin x=\frac{3}{5}\) and (x) is in the first quadrant, what is the value of \(\cos x\)?
C. \(\frac{4}{5}\)
\(\sin^2 x+\cos^2 x=1\) से \(\cos x=\frac{4}{5}\) मिलता है। प्रथम चतुर्थांश में \(\cos x\) धनात्मक होता है। / Using \(\sin^2 x+\cos^2 x=1\), \(\cos x=\frac{4}{5}\). In the first quadrant, \(\cos x\) is positive.
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If \(\cos x=\frac{12}{13}\) and (x) is in the first quadrant, what is the value of \(\sin x\)?
A. \(\frac{5}{13}\)
\(\sin^2 x=1-\cos^2 x\) से \(\sin x=\frac{5}{13}\) मिलता है। प्रथम चतुर्थांश में मान धनात्मक लिया जाता है। / From \(\sin^2 x=1-\cos^2 x\), \(\sin x=\frac{5}{13}\). In the first quadrant, the positive value is taken.
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If \(\tan x=\frac{7}{24}\) and (x) is in the first quadrant, what is the value of \(\sec x\)?
B. \(\frac{25}{24}\)
\(\sec^2 x=1+\tan^2 x\) लगाने पर \(\sec x=\frac{25}{24}\) मिलता है। प्रथम चतुर्थांश में \(\sec x\) धनात्मक होता है। / Using \(\sec^2 x=1+\tan^2 x\), \(\sec x=\frac{25}{24}\). In the first quadrant, \(\sec x\) is positive.
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If \(\cot x=\frac{8}{15}\) and (x) is in the first quadrant, what is the value of \(\cosec x\)?
B. \(\frac{17}{15}\)
\(\cosec^2 x=1+\cot^2 x\) से \(\cosec x=\frac{17}{15}\) मिलता है। प्रथम चतुर्थांश में धनात्मक मान लें। / From \(\cosec^2 x=1+\cot^2 x\), \(\cosec x=\frac{17}{15}\). Take the positive value in the first quadrant.
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What is (\sin\(\pi+x\)) equal to?
C. \(-\sin x\)
\(\pi+x\) तीसरे चतुर्थांश में आता है और वहाँ \(\sin x\) ऋणात्मक होता है। इसलिए (\sin\(\pi+x\)=-\sin x)। / \(\pi+x\) lies in the third quadrant and \(\sin x\) is negative there. Hence (\sin\(\pi+x\)=-\sin x).
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What is (\cos\(\pi+x\)) equal to?
B. \(-\cos x\)
\(\pi+x\) तीसरे चतुर्थांश में है और वहाँ \(\cos x\) ऋणात्मक होता है। इसलिए (\cos\(\pi+x\)=-\cos x)। / \(\pi+x\) is in the third quadrant and \(\cos x\) is negative there. Therefore, (\cos\(\pi+x\)=-\cos x).
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Which of the following identities is correct?
D. \(1+\tan^2 x=\sec^2 x\)
सही पहचान \(1+\tan^2 x=\sec^2 x\) है। पहचान आधारित प्रश्नों में चिन्ह और भिन्न को सावधानी से देखें। / The correct identity is \(1+\tan^2 x=\sec^2 x\). In identity-based questions, check signs and fractions carefully.
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