100 Super Revision Test – CBSE Class 12 Maths
Chapter: Inverse Trigonometric Functions
CBSE 12 Matrices Click Here
Q1. \(\sin^{-1}(\sqrt3/2)\).
Ans: \(\pi/3\)
Q2. \(\tan^{-1}(1)\).
Ans: \(\pi/4\)
Q3. \(\sin^{-1}(\sin 5\pi/3)\).
Ans: \(-\pi/3\)
Q4. \(\cos^{-1}(-1/2)\).
Ans: \(2\pi/3\)
Q5. \(\sin^{-1}x+\cos^{-1}x\).
Ans: \(\pi/2\)
Q6. \(\tan^{-1}x+\cot^{-1}x\).
Ans: \(\pi/2\)
Q7. \(\tan^{-1}2+\tan^{-1}3\).
Ans: \(-\pi/4\)
Q8. \(\cot^{-1}(\sqrt3)\).
Ans: \(\pi/6\)
Q9. \(\sin^{-1}(-x)\).
Ans: \(-\sin^{-1}x\)
Q10. \(\cos^{-1}(-x)\).
Ans: \(\pi-\cos^{-1}x\)
Q11. \(\tan^{-1}(-x)\).
Ans: \(-\tan^{-1}x\)
Q12. \(\sin^{-1}x+\sin^{-1}\sqrt{1-x^2}\) for 0≤x≤1.
Ans: \(\pi/2\)
Q13. \(\tan^{-1}(1/x)\) for x>0.
Ans: \(\pi/2-\tan^{-1}x\)
Q14. \(\sin^{-1}(\sin\pi/4)\).
Ans: \(\pi/4\)
Q15. \(\cos^{-1}(\cos7\pi/6)\).
Ans: \(5\pi/6\)
Q16. \(\tan^{-1}(\tan5\pi/4)\).
Ans: \(-3\pi/4\)
Q17. \(\sin^{-1}(1)\).
Ans: \(\pi/2\)
Q18. \(\sin^{-1}(0)\).
Ans: 0
Q19. \(\cos^{-1}(1)\).
Ans: 0
Q20. \(\tan^{-1}(\sqrt3)\).
Ans: \(\pi/3\)
Q21. \(\tan^{-1}\frac{2x}{1-x^2}=\)
Ans: \(2\tan^{-1}x\)
Q22. \(\sin^{-1}(2x\sqrt{1-x^2})=\)
Ans: \(2\sin^{-1}x\)
Q23. \(\cos^{-1}(2x^2-1)=\)
Ans: \(2\cos^{-1}x\)
Q24. \(\sin^{-1}\left(\frac{3x-4x^3}{1-3x^2+4x^4}\right)=\)
Ans: \(3\sin^{-1}x\)
Q25. Domain of \(\sin^{-1}x\).
Ans: [−1,1]
Q26. Range of \(\sin^{-1}x\).
Ans: [−\pi/2,\pi/2]
Q27. Range of \(\cos^{-1}x\).
Ans: [0,\pi]
Q28. Range of \(\tan^{-1}x\).
Ans: (−\pi/2,\pi/2)
Q29. \(\cot^{-1}1\).
Ans: \(\pi/4\)
Q30. \(\tan^{-1}(\tan\pi)\).
Ans: 0
Q31. If y=\(\sin^{-1}x\), find \(\cos y\).
Ans: \(\sqrt{1-x^2}\)
Q32. If y=\(\tan^{-1}x\), find \(\sin y\).
Ans: \(x/\sqrt{1+x^2}\)
Q33. \(\cos^{-1}x+\cos^{-1}(-x)\).
Ans: \(\pi\)
Q34. True/False: \(\sin^{-1}x=\tan^{-1}\frac{x}{\sqrt{1-x^2}}\).
Ans: True
Q35. Add formula validity condition for tan inverse sum.
Ans: valid if denominator ≠0; choose branch depending on xy
Q36. \(\sin^{-1}x+\sin^{-1}y\) formula.
Ans: \(\sin^{-1}(x\sqrt{1-y^2}+y\sqrt{1-x^2})\)
Q37. \(\tan^{-1}x+\tan^{-1}y=\tan^{-1}\frac{x+y}{1-xy}\) when?
Ans: when \(xy<1 div="">
Q38. \(\tan^{-1}2+\tan^{-1}3\).
Ans: \(-\pi/4\)
Q39. \(\tan^{-1}\frac{1}{2}+\tan^{-1}\frac{1}{3}=\)
Ans: \(\pi/4\)
Q40. \(\tan^{-1}\frac{1}{5}+\tan^{-1}\frac{1}{8}=\)
Ans: \(\pi/6\)
Q41. \(\tan^{-1}1+\tan^{-1}2+\tan^{-1}3\).
Ans: \(\pi\)
Q42. \(\sin^{-1}\frac{3}{5}\) in radians approx.
Ans: \(\approx0.6435\)
Q43. \(\tan^{-1}\frac{3x-x^3}{1-3x^2}=\)
Ans: \(3\tan^{-1}x\)
Q44. \(\sin^{-1}x+\cos^{-1}x\) value.
Ans: \(\pi/2\)
Q45. \(\tan(\sin^{-1}x)\).
Ans: \(x/\sqrt{1-x^2}\)
Q46. \(\sin(\tan^{-1}x)\).
Ans: \(x/\sqrt{1+x^2}\)
Q47. \(\cos(\tan^{-1}x)\).
Ans: \(1/\sqrt{1+x^2}\)
Q48. \(\tan^{-1}1+\tan^{-1}2+\tan^{-1}3\) repeat.
Ans: \(\pi\)
Q49. \(\tan^{-1}\frac{1}{2}+\tan^{-1}\frac{1}{3}=\)
Ans: \(\pi/4\)
Q50. \(\tan^{-1}a-\tan^{-1}b\).
Ans: \(\tan^{-1}\frac{a-b}{1+ab}\)
Q51. Principal value: \(\sin^{-1}(-1/2)\).
Ans: \(-\pi/6\)
Q52. Principal value: \(\cos^{-1}(-1/2)\).
Ans: \(2\pi/3\)
Q53. Principal value: \(\tan^{-1}(-1)\).
Ans: \(-\pi/4\)
Q54. \(\tan^{-1}x+\tan^{-1}(1/x)\) for x>0.
Ans: \(\pi/2\)
Q55. \(\tan^{-1}x+\tan^{-1}(1/x)\) for x<0 .="" class="answer" div="">Ans: \(-\pi/2\)
Q56. \(\cos^{-1}x-\sin^{-1}x\).
Ans: \(\pi/2-2\sin^{-1}x\)
Q57. If \(\sin^{-1}x=\pi/6\), x= ?
Ans: 1/2
Q58. \(\tan^{-1}(\sqrt3)+\tan^{-1}(1/\sqrt3)\).
Ans: \(\pi/2\)
Q59. \(\sin(\cos^{-1}x)\).
Ans: \(\sqrt{1-x^2}\)
Q60. \(\tan(\cos^{-1}(12/13))\).
Ans: \(5/12\)
Q61. \(\sin(\sin^{-1}x)\).
Ans: x
Q62. \(\cos(\cos^{-1}x)\).
Ans: x
Q63. \(\tan(\tan^{-1}x)\).
Ans: x
Q64. \(\cos(\sin^{-1}x)\).
Ans: \(\sqrt{1-x^2}\)
Q65. \(\sin(\cos^{-1}x)\).
Ans: \(\sqrt{1-x^2}\)
Q66. Simplify: \(\tan(\cot^{-1}x)\).
Ans: \(1/x\)
Q67. \(\cot(\tan^{-1}x)\).
Ans: \(1/x\)
Q68. \(\sin^{-1}x+\sin^{-1}y=\) given formula.
Ans: see Q36
Q69. \(\cos^{-1}x+\cos^{-1}y\) formula.
Ans: \(\cos^{-1}(xy-\sqrt{(1-x^2)(1-y^2)})\)
Q70. \(\tan^{-1}x-\tan^{-1}y\).
Ans: \(\tan^{-1}\frac{x-y}{1+xy}\)
Q71. \(\tan^{-1}x+\tan^{-1}y+\tan^{-1}z=\pi\) condition.
Ans: \(xyz=x+y+z\)
Q72. Principal value: \(\sin^{-1}(\sin 7\pi/6)\).
Ans: \(-\pi/6\)
Q73. Principal value: \(\cos^{-1}(\cos 5\pi/3)\).
Ans: \(\pi/3\)
Q74. \(\tan^{-1}(\tan 7\pi/6)\).
Ans: \(\pi/6\)
Q75. \(\sin^{-1}(2/3)+\sin^{-1}(\sqrt{5}/3)\).
Ans: \(\pi/2\)
Q76. \(\tan^{-1}(1/2)+\tan^{-1}(1/5)+\tan^{-1}(1/8)\).
Ans: \(\pi/4\)
Q77. \(\tan^{-1}1+\tan^{-1}2+\tan^{-1}3=\)
Ans: \(\pi\)
Q78. \(\cos^{-1}x=\sin^{-1}(\sqrt{1-x^2})\) True/False?
Ans: True
Q79. \(\tan^{-1}\frac{1-x}{1+x}=\)
Ans: \(\pi/4-\tan^{-1}x\)
Q80. \(\cos^{-1}x+\tan^{-1}(\sqrt{1-x^2}/x)=\)
Ans: \(\pi/2\)
Q81. Solve: \(\tan^{-1}x+\tan^{-1}2x=\pi/4\).
Ans: \(x=1/\sqrt3\)
Q82. Solve: \(\tan^{-1}x+\tan^{-1}y=\pi/4\) express relation.
Ans: \( (x+y)/(1-xy)=1\)
Q83. Identity: \(\sin^{-1}(2a/(1+a^2))=\)
Ans: \(2\tan^{-1}a\)
Q84. Identity: \(\tan^{-1}(2a/(1-a^2))=\)
Ans: \(2\tan^{-1}a\)
Q85. Simplify: \(\sin^{-1}(3/5)+\sin^{-1}(4/5)\).
Ans: \(\pi/2\)
Q86. Principal value: \(\sin^{-1}(\sin(-11\pi/6))\).
Ans: \(\pi/6\)
Q87. Principal value: \(\cos^{-1}(\cos(10\pi/3))\).
Ans: \(\pi/3\)
Q88. \(\tan^{-1}(1/3)+\tan^{-1}(1/7)+\tan^{-1}(1/8)=\)
Ans: \(\pi/4\)
Q89. Show: \(\tan^{-1}1+\tan^{-1}2+\tan^{-1}3=\pi\) (short).
Ans: use addition formula → tan sum =0 → sum = π
Q90. Solve: \(\sin^{-1}x+\tan^{-1}(x/\sqrt{1-x^2})=\pi/2\).
Ans: identity holds for |x|≤1
Q101. \(\tan^{-1}(1/2)+\tan^{-1}(1/5)+\tan^{-1}(1/8)=\)
Ans: \(\pi/4\)
Q102. Solve: \(\sin^{-1}x=\tan^{-1}(1/\sqrt3)\).
Ans: x=1/2
Q103. \(\cos^{-1}(\cos9\pi/4)=\) principal value.
Ans: \(\pi/4\)
Q104. \(\tan^{-1}2+\tan^{-1}3+\tan^{-1}6=\)
Ans: \(\pi\)
Q105. If \(\sin^{-1}x+\sin^{-1}y=\pi/2\) then y=?
Ans: \(\sqrt{1-x^2}\)
Q106. Solve: \(\tan^{-1}x+\tan^{-1}2x+\tan^{-1}3x=\pi\).
Ans: x=1
Q107. MCQ: Principal value of \(\cot^{-1}(-1)\) is?
Ans: 3\pi/4
Q108. \(\sin(\sin^{-1}3/5+\sin^{-1}4/5)=\)
Ans: 1
Q109. Solve: \(\cos^{-1}x=2\sin^{-1}(1/2)\).
Ans: x=0
Q110. \(\tan[\tan^{-1}(1/2)+\tan^{-1}(1/3)]=\)
Ans: 1
Q111. Condition for \(\sin^{-1}x+\sin^{-1}y\) formula validity.
Ans: expression under principal branch and |x|,|y|≤1
Q112. Solve: \(\tan^{-1}x+\tan^{-1}2x=\pi/4\).
Ans: x=1/\sqrt3
Q113. \(\sin^{-1}(\sin(-7\pi/6))=\)
Ans: \(\pi/6\)
Q114. Simplify: \(\tan^{-1}(2/3)+\tan^{-1}(3/4)+\tan^{-1}(4/5)\).
Ans: \(\pi/2-\tan^{-1}5\)
Q115. If \(\tan^{-1}a+\tan^{-1}b=\pi/4\) then relation.
Ans: (a+b)/(1-ab)=1
Q116. MCQ: Range of principal \(\cot^{-1}x\) is?
Ans: (0,\pi)
Q117. Solve: \(\sin^{-1}x+\sin^{-1}2x=\pi/2\).
Ans: x=1/\sqrt5
Q118. \(\tan^{-1}1+\tan^{-1}2+\tan^{-1}3+\tan^{-1}4=\)
Ans: 5\pi/4
Q119. \(\cos^{-1}(\cos(-11\pi/6))=\)
Ans: \(\pi/6\)
Q120. \(\tan(\sin^{-1}5/13)=\)
Ans: 5/12
Q121. True/False: if tan inverse sum = π then xyz=x+y+z.
Ans: True
Q122. If \(\sin^{-1}x=\alpha,\cos^{-1}y=\beta,\alpha+\beta=\pi/2\) then y=?
Ans: \(\sqrt{1-x^2}\)
Q123. \(\tan^{-1}1/3+\tan^{-1}1/7+\tan^{-1}1/8=\)
Ans: \(\pi/4\)
Q124. MCQ: \(\sin^{-1}(\sqrt2/2)=\)
Ans: \(\pi/4\)
Q125. Solve: \(\sin^{-1}x=\cos^{-1}(2x)\).
Ans: x=1/\sqrt5
Q126. Sum of small inverse tangents equals \(\pi/4\) pattern (example series).
Ans: many telescoping sets give \(\pi/4\)
Q127. \(\sin^{-1}(\sin11\pi/6)=\)
Ans: -\pi/6
Q128. When add formula gets +π adjustment?
Ans: when 1-xy<0 xy="">1)
Q129. Solve: \(\sin^{-1}x+\sin^{-1}(2x-1)=0\).
Ans: x=1/2
Q130. MCQ: \(\tan^{-1}0=\)
Ans: 0
Q131. \(\sin(\cos^{-1}3/5)=\)
Ans: 4/5
Q132. For x>0 show \(\tan^{-1}x+\tan^{-1}1/x=\pi/2\).
Ans: identity holds
Q133. \(\sin^{-1}2/3+\sin^{-1}\sqrt5/3=\)
Ans: \(\pi/2\)
Q134. \(\cos^{-1}(\cos10\pi/3)=\)
Ans: \(\pi/3\)
Q135. Solve cubic tangent sum eqn — main root.
Ans: x=1
Q136. MCQ: \(\sin^{-1}(-1)=\)
Ans: -\pi/2
Q137. \(\tan(\cos^{-1}12/13)=\)
Ans: 5/12
Q138. Short reason for \(\tan^{-1}1+\tan^{-1}2+\tan^{-1}3=\pi\).
Ans: use tan addition → tan(sum)=0, sum=π
Q139. Solve in [−1,1]: \(\sin^{-1}x=1/2\) (interpretation note).
Ans: if RHS radian value then x=sin(1/2)≈0.4794; if RHS meant π/6 then x=1/2
Q140. \(\sin^{-1}12/13+\cos^{-1}5/13=\)
Ans: \(\pi/2\)
