10th Class Mathematics Introduction to Trigonometry Question Bank

done Introduction to Trigonometry Total Questions - 25

• 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)

  P	Q	R
  1	3	1

                  done clear
  B)

  P	Q	R
  4	1	2

                  done clear
  C)

  P	Q	R
  2	2	1

                  done clear
  D)

  P	Q	R
  1	1	1

  done clear
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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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