MCQOPTIONS
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MCQOPTIONS
This section includes 1153 Mcqs, each offering curated multiple-choice questions to sharpen your UPSEE knowledge and support exam preparation. Choose a topic below to get started.
| 51. |
If cos 45° - sec 60° = x, then the value of x is |
| A. | \(\frac{{\left( {\sqrt 3 - 4} \right)}}{{2\sqrt 3 }}\) |
| B. | 1 |
| C. | \(\frac{{\left( {1 - 2\sqrt 2 } \right)}}{{\sqrt 2 }}\) |
| D. | \(\frac{{\left( {\sqrt 3 \; + \;\sqrt 2 } \right)}}{{\sqrt 2 }}\) |
| Answer» D. \(\frac{{\left( {\sqrt 3 \; + \;\sqrt 2 } \right)}}{{\sqrt 2 }}\) | |
| 52. |
If the angles of a triangle are in the ratio A: B: C = 1: 2: 3 and the circum-radius is 10 cm. Find the value of cos B? |
| A. | 1 |
| B. | 1 / 2 |
| C. | √3 / 2 |
| D. | None of these |
| Answer» C. √3 / 2 | |
| 53. |
If sin θ = 2/3, find the value of sec θ and cot θ? |
| A. | √5/2, 2/√5 |
| B. | 2/√5, 3/5 |
| C. | 3√5/5, √5/2 |
| D. | 3/5, 3√5/5 |
| Answer» D. 3/5, 3√5/5 | |
| 54. |
If sinθ = 0.96, then find cotθ |
| A. | 0.2916 |
| B. | 0.4563 |
| C. | 0.4876 |
| D. | 0.3456 |
| Answer» B. 0.4563 | |
| 55. |
ΔABC is right angled at B. If sec A = 5/3, then what is the value of cosec C? |
| A. | 5/3 |
| B. | 3/4 |
| C. | 4/5 |
| D. | 4/3 |
| Answer» B. 3/4 | |
| 56. |
If cosec θ = 17/8, then what is the value of cos θ? |
| A. | 15/8 |
| B. | 15/17 |
| C. | 8/15 |
| D. | 17/15 |
| Answer» C. 8/15 | |
| 57. |
If x = cosecθ – sinθ and y = secθ – cosθ, then the relation between x and y is |
| A. | x2 + y2 + 3 = 1 |
| B. | x2y2 (x2 + y2 + 3) = 1 |
| C. | x2(x2 + y2– 5) = 1 |
| D. | y2(x2 + y2– 5) = 1 |
| Answer» C. x2(x2 + y2– 5) = 1 | |
| 58. |
If 7(cosec257° − tan233°) + 2 sin90° − 4 tan252° y tan238° = y/2, then the value of y is: |
| A. | 2 |
| B. | 4 |
| C. | 1 |
| D. | 3 |
| Answer» B. 4 | |
| 59. |
If sin x – cos x = 0, then the value of (sin3 x – cos3 x) is: |
| A. | 1 |
| B. | 2 |
| C. | 0 |
| D. | 4 |
| Answer» D. 4 | |
| 60. |
If \(\frac{{1\; + \;\sin \phi }}{{1 - \sin \phi }} = \frac{{{p^2}}}{{{q^2}}}\), then sec ϕ is equal to∶ |
| A. | \(\frac{1}{{{p^2}}} + \frac{1}{{{q^2}}}\) |
| B. | \(\frac{1}{2}\left( {\frac{q}{p} + \frac{p}{q}} \right)\) |
| C. | \(\frac{{{p^2}{q^2}}}{{{p^2} + {q^2}}}\) |
| D. | \(\frac{{2{p^2}{q^2}}}{{{p^2} + {q^2}}}\) |
| Answer» C. \(\frac{{{p^2}{q^2}}}{{{p^2} + {q^2}}}\) | |
| 61. |
\(\frac{{\left( {\sec \theta + \tan \theta } \right)\left( {1 - \sin \theta } \right)}}{{cosec\theta \left( {1\; + \;\cos \theta } \right)\left( {cosec\;\theta - \cot \theta } \right)}}\) is equal to |
| A. | sin θ |
| B. | cosec θ |
| C. | cos θ |
| D. | sec θ |
| Answer» D. sec θ | |
| 62. |
∆XYZ is right angled at Y. If tanX = 15/8, then what is the value of sinZ? |
| A. | 15/17 |
| B. | 17/8 |
| C. | 17/15 |
| D. | 8/17 |
| Answer» E. | |
| 63. |
A Navy captain going away from a lighthouse at the speed of 4(√3 – 1) m/s. He observes that it takes him 1 minute to change the angle of elevation of the top of the lighthouse from 60° to 45°. What is the height (in metres) of the lighthouse? |
| A. | 240√3 |
| B. | 480(√3 – 1) |
| C. | 360√3 |
| D. | 280√2 |
| Answer» B. 480(√3 – 1) | |
| 64. |
If sec θ – tan θ = P, then cosec θ = ? |
| A. | \(\frac{{{{\rm{P}}^2} + 1}}{{1 - {{\rm{P}}^2}}}\) |
| B. | \(\frac{{1 - {{\rm{P}}^2}}}{{1 + {{\rm{P}}^2}}}\) |
| C. | \(\frac{{2{\rm{P}}}}{{1 - {{\rm{P}}^2}}}\) |
| D. | \(\frac{{2{\rm{P}}}}{{1 + {{\rm{P}}^2}}}\) |
| Answer» B. \(\frac{{1 - {{\rm{P}}^2}}}{{1 + {{\rm{P}}^2}}}\) | |
| 65. |
Find the value of sin45°/sin30°. |
| A. | 2 |
| B. | √2 |
| C. | 2√2 |
| D. | 1/√2 |
| Answer» C. 2√2 | |
| 66. |
If Cot θ = 24/7, then what is the value of Sec θ? |
| A. | 7/25 |
| B. | 25/24 |
| C. | 8/25 |
| D. | 9/25 |
| Answer» C. 8/25 | |
| 67. |
Find the value of tan 8° tan 22° cot 60° tan 68° tan 82° |
| A. | √3 |
| B. | 1 |
| C. | 2/√3 |
| D. | 1/√3 |
| Answer» E. | |
| 68. |
In ΔABC measure of angle B is 90°. If cos A = 8/17, and AB = 4 cm, then what is the length (in cm) of side BC? |
| A. | 8.5 |
| B. | 7.5 |
| C. | 5 |
| D. | 6 |
| Answer» C. 5 | |
| 69. |
An aeroplane is flying horizontally at a height of 1.8 km above the ground. The angle of elevation of a plane from point X is 60° and after 20 seconds, its angle of elevation from X becomes 30°. If point X is on the ground, then what is the speed (in km/hr) of the aeroplane? |
| A. | 216√3 |
| B. | 105√3 |
| C. | 201√3 |
| D. | 305√3 |
| Answer» B. 105√3 | |
| 70. |
\(\cot \left[\tan^{-1} \dfrac{1}{2}+\tan^{-1} \dfrac{1}{8}\right]= \ ?\) |
| A. | 3/2 |
| B. | -1 |
| C. | \(\sqrt{2}\) |
| D. | \(-\sqrt{2}\) |
| Answer» B. -1 | |
| 71. |
Let x and y be real numbers such that \(\dfrac{\rm{sin}\ x}{\rm{sin} \ y} = 3 \ \rm{and}\ \dfrac{\rm{cos}\ x}{\rm{cos} \ y} = \dfrac{1}{2}\). The value of \( \dfrac{\rm{sin}\ 2x}{\rm{sin}\ 2y}+\dfrac{\rm{cos}\ 2x}{\rm{cos} \ 2y} =\)is |
| A. | None of these |
| B. | \(\dfrac{51}{58}\) |
| C. | \(\dfrac{49}{58}\) |
| D. | \(\dfrac{41}{58}\) |
| Answer» D. \(\dfrac{41}{58}\) | |
| 72. |
If sec(5θ - 50°) = cosec(θ + 32°), then the value of θ is: (0° |
| A. | 108° |
| B. | 18° |
| C. | 36° |
| D. | 54° |
| Answer» C. 36° | |
| 73. |
If sin x = cos2x, then cos2x (1 + cos2x) is equal to: |
| A. | 1 |
| B. | 2 |
| C. | None of these |
| D. | 0 |
| Answer» B. 2 | |
| 74. |
If \(\rm \tan \alpha = \dfrac{m}{m+1}\) and \(\rm \tan \beta = \dfrac{1}{2m+1}\), then α + β is equal to: |
| A. | \(\dfrac{\pi}{3}\) |
| B. | \(\dfrac{\pi}{4}\) |
| C. | \(\dfrac{\pi}{6}\) |
| D. | π |
| Answer» C. \(\dfrac{\pi}{6}\) | |
| 75. |
Find tan 75° if tan (A + B) \( = \frac{{\tan A + \tan B}}{{1 - \tan A\tan B}}\) |
| A. | 2 + √3 |
| B. | 0 |
| C. | 1 |
| D. | √3 |
| Answer» B. 0 | |
| 76. |
If A = sin2θ + cos4θ where 0 ≤ θ < π/2, then which of the following is correct? |
| A. | 1 ≤ A ≤ 2 |
| B. | 3/4 ≤ A ≤ 1 |
| C. | 13/16 ≤ A ≤ 2 |
| D. | 3/4 ≤ A ≤ 13/16 |
| Answer» C. 13/16 ≤ A ≤ 2 | |
| 77. |
If cos x = 5/13 then cot x will be |
| A. | 12/5 |
| B. | 7/13 |
| C. | 13/5 |
| D. | 5/12 |
| Answer» E. | |
| 78. |
If x = cos α + i sin α and y = cos β + i sin β, then the value of \(\frac{{x - y}}{{x + y}}\) is: |
| A. | \(\tan \left( {\frac{\alpha }{\beta }} \right)\) |
| B. | \(\tan \left( {\frac{{\alpha - \beta }}{2}} \right)\) |
| C. | \(i\tan \left( {\frac{{\alpha - \beta }}{2}} \right)\) |
| D. | \(i\tan \left( {\alpha - \beta } \right)\) |
| Answer» D. \(i\tan \left( {\alpha - \beta } \right)\) | |
| 79. |
Find the value of \(\dfrac{\tan 60^\circ - \tan 15^\circ}{1 + \tan 60^\circ \tan 15^\circ}\) |
| A. | 1 |
| B. | \(\dfrac{1}{\sqrt{2}}\) |
| C. | \(\dfrac{1}{2}\) |
| D. | \(\dfrac{\sqrt{3}}{2}\) |
| Answer» B. \(\dfrac{1}{\sqrt{2}}\) | |
| 80. |
Consider the following statements:1. There exists \({\rm{\theta }} \in \left( { - \frac{{\rm{\pi }}}{2},\frac{{\rm{\pi }}}{2}} \right)\) for which tan-1 (tan θ) ≠ θ2. \({\sin ^{ - 1}}\left( {\frac{1}{3}} \right) - {\sin ^{ - 1}}\left( {\frac{1}{5}} \right) = {\sin ^{ - 1}}\left( {\frac{{2\sqrt 2 \left( {\sqrt 3 - 1} \right)}}{{15}}} \right)\) Which of the above statements is/are correct? |
| A. | 1 only |
| B. | 2 only |
| C. | Both 1 and 2 |
| D. | Neither 1 nor |
| Answer» C. Both 1 and 2 | |
| 81. |
\(\frac{{sin\;A\; + \;sin\;B}}{{cos\;A - \cos B}} + \frac{{\cos A\; + \;\cos B}}{{sin\;A - sin\;B}} = \;?\) |
| A. | sin A cos B |
| B. | tan A tan B |
| C. | 0 |
| D. | cos A cos B |
| Answer» D. cos A cos B | |
| 82. |
ΔABC is right angled at B. BD is its altitude. AD = 4 cm and DC = 12 cm. What is the value of AB (in cm)? |
| A. | 9 |
| B. | 10 |
| C. | 6 |
| D. | 8 |
| Answer» E. | |
| 83. |
If x = sin-1 (sin 10) and y = cos-1 (cos 10), then y – x is equal to: |
| A. | 0 |
| B. | 10 |
| C. | 7π |
| D. | π |
| Answer» E. | |
| 84. |
If (cosθ + sinθ) : (cosθ–sinθ) = (√3 + 1) : (√3 – 1),0° < θ 90°, then what is the value of secθ? |
| A. | 2 |
| B. | √2 |
| C. | 1 |
| D. | (2√3)/3 |
| Answer» E. | |
| 85. |
If \(SecA = \frac{\sqrt{11}}{3}\), then the value of \(\frac{Cosec^2 A + Tan^2 A}{Sin^2 A + Cot^2 A}\) is: |
| A. | \(\frac{4}{9}\) |
| B. | \(\frac{9}{4}\) |
| C. | \(\frac{11}{9}\) |
| D. | \(\frac{2}{11}\) |
| Answer» D. \(\frac{2}{11}\) | |
| 86. |
If tan \({\rm{\theta }} = \frac{8}{{15}}\) and θ is acute, the cosec θ = ? |
| A. | 17/8 |
| B. | 1 |
| C. | 8/17 |
| D. | 17/9 |
| Answer» B. 1 | |
| 87. |
If (sin θ + cosec θ)2 + (cos θ + sec θ)2 = k + tan2 θ + cot2 θ, then the value of k is equal to: |
| A. | 5 |
| B. | 7 |
| C. | 2 |
| D. | 9 |
| Answer» C. 2 | |
| 88. |
∆PQR is right angled at Q. If ∠P = 60°, then find the value of (sec R + 1/2). |
| A. | (2√2 + √3)/2 |
| B. | (√3 + 4)/2√3 |
| C. | √3 + 2 |
| D. | 5/6 |
| Answer» C. √3 + 2 | |
| 89. |
ΔABC is right angled at B. If m∠C = 45°, then find the value of (cosec A - √3). |
| A. | (4 – √3)/2 |
| B. | √2 – √3 |
| C. | -√3/2 |
| D. | (√6 – 1)/√3 |
| Answer» C. -√3/2 | |
| 90. |
If tan (11θ) = cot (7θ), then what is the value of sin2 (6θ) + sec2(9θ) + cosec2 (12θ)? |
| A. | 43/12 |
| B. | 31/12 |
| C. | 23/6 |
| D. | 35/12 |
| Answer» B. 31/12 | |
| 91. |
(1 + sec 20° + cot 70°)(1 - cosec 20° + tan70°) is equal to: |
| A. | 0 |
| B. | 1 |
| C. | 2 |
| D. | 3 |
| Answer» D. 3 | |
| 92. |
∆ABC is right angled at B. If m∠A = 45°, then find the value of (cosec C + 1/√3). |
| A. | (3√2 + 1)/3 |
| B. | (2 + √3)/2 |
| C. | 5/√3 |
| D. | (√6 + 1)/√3 |
| Answer» E. | |
| 93. |
A ladder leads to a window which is 48 feet above the ground on one side of the street (one side). From that place, the ladder reaches the window on the other side of the street (second side) 55 feet above the ground. If the length of that ladder is 73 feet, find the width of the street. |
| A. | 103 ft |
| B. | 110 ft |
| C. | 92 ft |
| D. | 100 ft |
| Answer» B. 110 ft | |
| 94. |
If angle A = 30° and angle B = 90°, sin (A + B) = ? |
| A. | \(\frac{2}{{\sqrt 3 }}\) |
| B. | \(\frac{1}{2}\) |
| C. | \(\frac{1}{{\sqrt 2 }}\) |
| D. | \(\frac{{\sqrt 3 }}{2}\) |
| Answer» E. | |
| 95. |
If sec θ - tan θ = 3, then cos θ is equal to: |
| A. | 4/9 |
| B. | 3/7 |
| C. | 2/5 |
| D. | 3/5 |
| Answer» E. | |
| 96. |
If tan α = 1/2, tan β = 1/3, then find α + β. |
| A. | 0° |
| B. | 45° |
| C. | 90° |
| D. | 135° |
| Answer» C. 90° | |
| 97. |
If Cot A = 12/5, then the value of (Sin A + Cos A) × Cosec A is |
| A. | 17/5 |
| B. | 13/5 |
| C. | 14/5 |
| D. | 1 |
| Answer» B. 13/5 | |
| 98. |
If cosx.cosy + sinx.siny = – 1 then cosx + cosy is |
| A. | – 2 |
| B. | 1 |
| C. | 0 |
| D. | 2 |
| Answer» D. 2 | |
| 99. |
If A = (sin45° – sin30°) / (cos45° + cos60°) and B = (sec45° – tan45°) / (cosec45° + cot45°), then which one of the following is correct? |
| A. | A = B |
| B. | A > B > 0 |
| C. | A < B |
| D. | B < A < 0 |
| Answer» B. A > B > 0 | |
| 100. |
If \(\sin θ - \cos θ = \frac{1}{{29}}\) find the value of sinθ + cosθ. |
| A. | \(\frac{22}{{29}}\) |
| B. | \(\frac{41}{{29}}\) |
| C. | \(\frac{42}{{29}}\) |
| D. | \(\frac{2}{{29}}\) |
| Answer» C. \(\frac{42}{{29}}\) | |