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question_answer1)
If \[{{(\sin \alpha +\cos ec\alpha )}^{2}}+{{(\cos \alpha +\sec \alpha )}^{2}}=k\]\[+{{\tan }^{2}}\alpha +{{\cot }^{2}}\alpha ,\] then \[k=\]________.
A)
9 done
clear
B)
7 done
clear
C)
5 done
clear
D)
3 done
clear
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question_answer2)
\[{{\left( \frac{\sqrt{3}+2\cos A}{1-2\operatorname{sinA}} \right)}^{-3}}+{{\left( \frac{1+2\sin A}{\sqrt{3}-2\cos A} \right)}^{-3}}=\_\_\_\_\_\_.\]
A)
1 done
clear
B)
\[\sqrt{3}\] done
clear
C)
0 done
clear
D)
\[-1\] done
clear
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question_answer3)
If \[\sin \theta +\cos \theta =a\] and \[\sec \theta +\cos ec\theta =b,\] then the value of \[b({{a}^{2}}-1)\]is _____.
A)
\[2a\] done
clear
B)
\[3a\] done
clear
C)
\[0\] done
clear
D)
\[2ab\] done
clear
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question_answer4)
If \[\sec \theta +\tan \theta =x,\] then \[\sec \theta =\_\_\_\_\_.\]
A)
\[\frac{{{x}^{2}}+1}{x}\] done
clear
B)
\[\frac{{{x}^{2}}+1}{2x}\] done
clear
C)
\[\frac{{{x}^{2}}-1}{2x}\] done
clear
D)
\[\frac{{{x}^{2}}-1}{x}\] done
clear
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question_answer5)
If \[\sin A:\cos A=4:7,\] then the value of \[\frac{7\sin A-3\cos A}{7\sin A+2\cos A}\] is __________.
A)
\[\frac{3}{14}\] done
clear
B)
\[\frac{3}{2}\] done
clear
C)
\[\frac{1}{3}\] done
clear
D)
\[\frac{1}{6}\] done
clear
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question_answer6)
If \[A+B={{90}^{o}},\] then \[\frac{\tan A\operatorname{tanB}+tanAcotB}{\sin A\sec B}-\frac{{{\sin }^{2}}B}{{{\cos }^{2}}A}\] is equal to _______.
A)
\[{{\cot }^{2}}A\] done
clear
B)
\[{{\cot }^{2}}B\] done
clear
C)
\[-ta{{n}^{2}}A\] done
clear
D)
\[-{{\cot }^{2}}A\] done
clear
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question_answer7)
If \[\sin \theta =\cos \theta ,\] then \[2{{\tan }^{2}}\theta +{{\sin }^{2}}\theta -1=\_\_\_\_\_.\]
A)
\[\frac{-3}{2}\] done
clear
B)
\[\frac{3}{2}\] done
clear
C)
\[\frac{2}{3}\] done
clear
D)
\[\frac{-2}{3}\] done
clear
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question_answer8)
\[\frac{\sin \theta -2{{\sin }^{2}}\theta }{2{{\cos }^{3}}\theta -\cos \theta }\] is equal to _______.
A)
\[\frac{{{\sin }^{2}}\theta }{\cos \theta }\] done
clear
B)
\[\tan \theta \] done
clear
C)
\[\frac{{{\cos }^{2}}\theta }{\sin \theta }\] done
clear
D)
\[\cot \theta \] done
clear
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question_answer9)
If a \[\cos \theta +b\sin \theta =m\] and a \[\sin \theta -b\cos \theta =n,\] then \[{{a}^{2}}+{{b}^{2}}\] is equal to ______.
A)
\[{{m}^{2}}-{{n}^{2}}\] done
clear
B)
\[{{m}^{2}}{{n}^{2}}\] done
clear
C)
\[{{n}^{2}}-{{m}^{2}}\] done
clear
D)
\[{{m}^{2}}+{{n}^{2}}\] done
clear
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question_answer10)
\[\sqrt{\frac{\sec \theta -1}{\sec \theta +1}}+\sqrt{\frac{\sec \theta +1}{\sec \theta -1}}=\_\_\_\_\_\_\_.\]
A)
\[2\,\cos ec\,\theta \] done
clear
B)
\[\frac{2\,\sin \theta }{\sqrt{\sec \theta }}\] done
clear
C)
\[2\,\cos \theta \] done
clear
D)
\[2\,sec\theta \] done
clear
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question_answer11)
If \[\sin (A+B+C)=1,\] then \[\tan (A-B)=\frac{1}{\sqrt{3}}\] and \[\sec (A+C)=2,\] find A, B and C respectively when they are acute.
A)
\[{{60}^{o}},{{0}^{o}},{{30}^{o}}\] done
clear
B)
\[{{30}^{o}},{{60}^{o}},{{90}^{o}}\] done
clear
C)
\[{{60}^{o}},{{30}^{o}},{{0}^{o}}\] done
clear
D)
\[{{0}^{o}},{{60}^{o}},{{30}^{o}}\] done
clear
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question_answer12)
If \[x=r\sin \theta \,cos\phi ,\] \[y=r\,\sin \theta \,\sin \phi \]and \[z=r\,\cos \theta ,\] then _______.
A)
\[{{x}^{2}}+{{y}^{2}}+{{z}^{2}}={{r}^{2}}\] done
clear
B)
\[{{x}^{2}}+{{y}^{2}}-{{z}^{2}}={{r}^{2}}\] done
clear
C)
\[{{x}^{2}}-{{y}^{2}}+{{z}^{2}}={{r}^{2}}\] done
clear
D)
\[{{z}^{2}}+{{y}^{2}}-{{x}^{2}}={{r}^{2}}\] done
clear
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question_answer13)
\[(1+{{\tan }^{2}}A)+\left( 1+\frac{1}{{{\tan }^{2}}A} \right)\] is equal to
A)
\[\frac{1}{{{\sin }^{2}}A-{{\sin }^{4}}A}\] done
clear
B)
\[\frac{1}{{{\sin }^{2}}A+{{\sin }^{4}}A}\] done
clear
C)
\[\frac{{{\cos }^{2}}A}{\sin A+{{\sin }^{2}}A}\] done
clear
D)
\[\frac{{{\cos }^{2}}A}{\sin A-{{\sin }^{2}}A}\] done
clear
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question_answer14)
If \[\frac{x}{a}\cos \theta +\frac{y}{b}\sin \theta =1,\frac{x}{a}\sin \theta -\frac{y}{b}\cos \theta =1,\]then _________.
A)
\[{{x}^{2}}+{{y}^{2}}={{a}^{2}}+{{b}^{2}}\] done
clear
B)
\[\frac{{{x}^{2}}}{{{a}^{2}}}+\frac{{{y}^{2}}}{{{b}^{2}}}=2\] done
clear
C)
\[{{a}^{2}}{{x}^{2}}+{{b}^{2}}{{y}^{2}}=1\] done
clear
D)
\[{{x}^{2}}-{{y}^{2}}={{a}^{2}}-{{b}^{2}}\] done
clear
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question_answer15)
\[{{\left( \frac{\tan {{20}^{o}}}{\text{cosec 7}{{\text{0}}^{o}}} \right)}^{2}}+{{\left( \frac{\cot {{20}^{o}}}{\sec {{70}^{o}}} \right)}^{2}}+2\tan {{15}^{o}}\tan {{37}^{o}}\]\[\tan {{53}^{o}}\,\tan {{60}^{o}}\,\tan {{75}^{o}}\] is equal to ______.
A)
\[1+2\sqrt{3}\] done
clear
B)
\[-1+2\sqrt{3}\] done
clear
C)
\[1+\sqrt{3}\] done
clear
D)
\[1-\sqrt{3}\] done
clear
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question_answer16)
If \[X\,{{\sin }^{3}}\theta +Y\,{{\cos }^{3}}\theta =\sin \theta \,\cos \theta \] and \[X\sin \theta =Ycos\theta ,\]then _______.
A)
\[{{X}^{3}}+{{Y}^{3}}=1\] done
clear
B)
\[{{X}^{2}}-{{Y}^{2}}=1\] done
clear
C)
\[{{X}^{2}}+{{Y}^{2}}=1\] done
clear
D)
\[{{X}^{4}}+{{Y}^{4}}=1\] done
clear
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question_answer17)
If \[\cot \theta =\frac{15}{8}.\] then evaluate \[\frac{(2+2\sin \theta )\,(1-\sin \theta )}{(1+\cos \theta )\,(2-2\cos \theta )}\]
A)
\[\frac{8}{15}\] done
clear
B)
\[\frac{15}{8}\] done
clear
C)
\[\frac{64}{225}\] done
clear
D)
\[\frac{225}{64}\] done
clear
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question_answer18)
If \[\sin x+{{\sin }^{2}}x=1,\] then \[{{\cos }^{8}}x+2{{\cos }^{6}}x+{{\cos }^{4}}x=\_\_\_\_\_.\]
A)
0 done
clear
B)
\[-1\] done
clear
C)
1 done
clear
D)
2 done
clear
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question_answer19)
If \[a\,\sec \theta +b\,\tan \theta =1\] and \[{{a}^{2}}{{\sec }^{2}}\theta -{{b}^{2}}{{\tan }^{2}}\theta =5,\] then \[{{a}^{2}}{{b}^{2}}+4{{a}^{2}}\]is equal to ______.
A)
\[9{{b}^{2}}\] done
clear
B)
\[\frac{9}{{{a}^{2}}}\] done
clear
C)
\[\frac{-2}{b}\] done
clear
D)
2 done
clear
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question_answer20)
In a \[\Delta ABC,\] it is given that \[\angle C={{90}^{o}}\] and \[\tan A=\frac{1}{\sqrt{3}},\]find the value of \[(\sin A\,\cos B+\cos A\,\sin B)\].
A)
1 done
clear
B)
\[\frac{1}{2}\] done
clear
C)
0 done
clear
D)
3 done
clear
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question_answer21)
If \[\text{cosec}-\sin \theta =l\]and \[\sec \theta -\cos \theta =m,\] then \[{{l}^{2}}{{m}^{2}}({{l}^{2}}+{{m}^{2}}+3)=\_\_\_\_\_.\]
A)
1 done
clear
B)
2 done
clear
C)
\[2\sin \theta \] done
clear
D)
\[sin\theta \,\cos \theta \] done
clear
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question_answer22)
Fill in the blanks.
(i) If \[x=a{{\cos }^{3}}\theta ,\,y=b\,{{\sin }^{3}}\theta \] then\[{{\left( \frac{x}{a} \right)}^{2/3}}+{{\left( \frac{y}{b} \right)}^{2/3}}=\underline{P}.\] |
(ii) If \[x=a\sec \theta \,\,cos\phi ,\] \[y=b\,\sec \theta \sin \phi \] and \[z=c\,\tan \theta ,\] then \[\frac{{{x}^{2}}}{{{a}^{2}}}+\frac{{{y}^{2}}}{{{b}^{2}}}-\frac{{{z}^{2}}}{{{c}^{2}}}=\underline{Q}.\] |
(iii) If \[\cos A+{{\cos }^{2}}a=1,\] then \[{{\sin }^{2}}A+{{\sin }^{4}}A=\underline{R}.\] |
A)
B)
C)
D)
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question_answer23)
. Which of the following is true?
A)
(a) \[\cos \theta \,sin\theta -\frac{\sin \theta \cos ({{90}^{o}}-\theta )\cos \theta }{\sec ({{90}^{o}}-\theta )}\] \[-\frac{\cos \theta sin({{90}^{o}}-\theta )\sin \theta }{\cos ec({{90}^{o}}-\theta )}=0\] (b) If A and B are complementary angles, then \[\sin A=\sqrt{\frac{\cos A}{\sin B}-\cos A\,\sin B}\] Only (a) done
clear
B)
Only (b) done
clear
C)
Neither (a) nor (b) done
clear
D)
Both (a) and (b) done
clear
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question_answer24)
Which of the following is CORRECT statement?
(i) \[3{{(\sin \theta -\cos \theta )}^{4}}+6{{(\sin \theta +\cos \theta )}^{2}}\] \[+4({{\sin }^{6}}\theta +{{\cos }^{6}}\theta )\] is independent of \[\theta \]. |
(ii) If \[\cos ec\,\theta -\sin \theta ={{a}^{3}},\] \[\sec \theta -\cos \theta ={{b}^{3}},\]then \[{{a}^{2}}{{b}^{2}}({{a}^{2}}+{{b}^{2}})=2\] |
A)
Only (i) done
clear
B)
Only (ii) done
clear
C)
Both (i) and (ii) done
clear
D)
Neither (i) or (ii) done
clear
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question_answer25)
Find the value of \[\frac{1}{\tan \theta }+\frac{\sin \theta }{1+\cos \theta },\] if \[1+{{\cot }^{2}}\theta ={{(\sqrt{3+2\sqrt{2}}-1)}^{2}}.\]
A)
2 done
clear
B)
3 done
clear
C)
\[-2\] done
clear
D)
1 done
clear
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