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.
| 1. |
If A = 30°, B = 60° and C = 135°, then what is the value of sin3A + cos3B + tan3C – 3sin A cos B tan C? |
| A. | 0 |
| B. | 1 |
| C. | 8 |
| D. | 9 |
| Answer» B. 1 | |
| 2. |
If sec2 θ - sec θ = 1, then what is the value of tan12 θ - 3 tan10 θ + 3 tan8 θ - tan6 θ ? |
| A. | -2 |
| B. | -1 |
| C. | 0 |
| D. | 1 |
| Answer» E. | |
| 3. |
If \(\tan α = \frac {m}{m + 1};\;(m \ne -1)\) and \(\tan β = \frac {1}{2m + 1};\) then α + β is equal to: |
| A. | π / 4 |
| B. | π / 3 |
| C. | π / 6 |
| D. | None of these |
| Answer» B. π / 3 | |
| 4. |
If Cosec θ = 25/7, then what is the value of Cot θ? |
| A. | 24/25 |
| B. | 7/24 |
| C. | 7/25 |
| D. | 24/7 |
| Answer» E. | |
| 5. |
If x = cosecA + cosA and y = cosec A - cosA, then find the value of \((\frac{2}{x+y})^2 + (\frac{x-y}{2})^2 - 1\) |
| A. | 2 |
| B. | 1 |
| C. | 3 |
| D. | 0 |
| Answer» E. | |
| 6. |
If cos θ + sin θ = m and sec θ + cosec θ = n, then what is the value of \(\frac{{\rm{n}}}{2}\left( {{{\rm{m}}^2} - 1} \right)?\) |
| A. | m |
| B. | 2m |
| C. | mn |
| D. | 2n |
| Answer» B. 2m | |
| 7. |
cos2 90° + cosec2 90° - cot2 45° = ? |
| A. | 1 |
| B. | 0 |
| C. | ½ |
| D. | √3/2 |
| Answer» C. ½ | |
| 8. |
For 0° < θ < 90°, tanθ + cotθ = 2. θ is equal to: |
| A. | 0° |
| B. | 30° |
| C. | 60° |
| D. | 45° |
| Answer» E. | |
| 9. |
Find the value of cos(A + B), if tan A = 3/4 and tan B = 5/12 and A and B are in quadrant I. |
| A. | 1 |
| B. | 33/65 |
| C. | 16/65 |
| D. | 15/48 |
| Answer» C. 16/65 | |
| 10. |
In ∆PQR, measure of angle Q is 90°. If cosecP = 17/15, and PQ = 0.8 cm, then what is the length (in cm) of side QR? |
| A. | 1.7 |
| B. | 2 |
| C. | 2.5 |
| D. | 1.5 |
| Answer» E. | |
| 11. |
From the top of a lamp post of height x metres, two objects on the ground on the same side of it (and in line with the foot of the lamp post) are observed at angles of depression of 30° and 60°, respectively. The distance between the objects is 32√3 m. The value of x is: |
| A. | 54 |
| B. | 48 |
| C. | 45 |
| D. | 36 |
| Answer» C. 45 | |
| 12. |
If \(\left( {\frac{{{\rm{tan\theta }} - {\rm{sec\theta \;}} + {\rm{\;}}1}}{{{\rm{tan\theta \;}} + {\rm{\;sec\theta }} - 1}}} \right){\rm{sec\theta \;}} = {\rm{\;}}\frac{1}{{\rm{k}}}\) then k = ? |
| A. | 1 – sinθ |
| B. | 1 + cosθ |
| C. | 1 + sinθ |
| D. | 1 – cosθ |
| Answer» D. 1 – cosθ | |
| 13. |
A kite is flying in the sky. The length of a string between a point on the ground and the kite is 420 m. The angle of elevation of the string with the ground is 30°. Assuming that there is no slack in the string, then what is the height (in metres) of the kite? |
| A. | 210 |
| B. | 140√3 |
| C. | 210√3 |
| D. | 150 |
| Answer» B. 140√3 | |
| 14. |
From the top of a 10 m high building, the angle of elevation of the top of a tower is 60° and the angle of depression of the foot of the tower is θ, such that tan θ = 2/3. What is the height of the tower to nearest metres? |
| A. | 34 m |
| B. | 35 m |
| C. | 36 m |
| D. | 33 m |
| Answer» D. 33 m | |
| 15. |
If Secθ = 17/15, then Tanθ = ? |
| A. | 15/8 |
| B. | 8/15 |
| C. | 17/8 |
| D. | 15/17 |
| Answer» C. 17/8 | |
| 16. |
If cot θ = 3/4, then sinθ + cosθ – tanθ is equal to∶ |
| A. | 2/15 |
| B. | 1/20 |
| C. | -1/20 |
| D. | 1/15 |
| Answer» E. | |
| 17. |
If \((\frac {cosα}{sinα + cosβ}) + (\frac {cosβ}{sinβ – cosα}) = (\frac{x}{sinα – cosβ}) + (\frac{cosβ}{sinβ + cosα})\) then x is equal to: |
| A. | sinβ |
| B. | cosβ |
| C. | sin∝ |
| D. | cos∝ |
| Answer» E. | |
| 18. |
If √3cotθ = 1, 0 ≤ θ ≤ 90°, then the value of cos2θ - sin2θ is |
| A. | 1/2 |
| B. | -1/2 |
| C. | 1/√2 |
| D. | None of the above |
| Answer» C. 1/√2 | |
| 19. |
ΔABC is right angled at B. If m∠C = 60°, then find the value of (cot A - 1/2). |
| A. | -1/√3 |
| B. | (3 – √2)/3√2 |
| C. | (2√3 – 1)/2 |
| D. | (√2 – √3)/√6 |
| Answer» D. (√2 – √3)/√6 | |
| 20. |
If tan x = 1/√3 such that x ∈ [0, π /2], then which of the following is the solution of the equation? |
| A. | 2π/3 |
| B. | π/6 |
| C. | 3π/4 |
| D. | 5π/6 |
| Answer» C. 3π/4 | |
| 21. |
If sin (A - B) = 1/2 and cos (A + B) = 1/2, then what is the value of sin A cos A + sin2 A sin B cos B + cos3 A cos B tan A? |
| A. | 1 / 2 |
| B. | (6 + √3)/8 |
| C. | 1 / 4 |
| D. | 3 / 2 |
| Answer» C. 1 / 4 | |
| 22. |
If \(\sin {\rm{\theta }} = \frac{{{{\rm{P}}^2} - 1}}{{{{\rm{P}}^2} + 1}},\) then cos θ is equal to: |
| A. | \(\frac{{2{\rm{P}}}}{{1 + {{\rm{P}}^2}}}\) |
| B. | \(\frac{{\rm{P}}}{{{{\rm{P}}^2} - 1}}\) |
| C. | \(\frac{{\rm{P}}}{{1 + {{\rm{P}}^2}}}\) |
| D. | \(\frac{{2{\rm{P}}}}{{{{\rm{P}}^2} - 1}}\) |
| Answer» B. \(\frac{{\rm{P}}}{{{{\rm{P}}^2} - 1}}\) | |
| 23. |
If 5cotθ = 3, find the value of \(\frac{{6sin{\rm{\theta }} - 3{\rm{cos\theta }}}}{{7sin{\rm{\theta }} + 3{\rm{cos\theta }}}}\) |
| A. | 44/21 |
| B. | 21/44 |
| C. | 11/40 |
| D. | 20/41 |
| Answer» C. 11/40 | |
| 24. |
If sin α + sin β = a and cos α - cos β = b, then the value of \(\tan \left( {\frac{{\alpha - \beta }}{2}} \right)\) will be |
| A. | a + b |
| B. | a - b |
| C. | \(- \frac{a}{b}\) |
| D. | \( - \frac{b}{a}\) |
| Answer» E. | |
| 25. |
A ladder is placed against a wall such that its foot is at a distance of 5 m from the wall and its top reaches a window 12 in above the ground. Find the length of the ladder. |
| A. | 13 m |
| B. | 24 m |
| C. | 15 m |
| D. | 16 m |
| Answer» B. 24 m | |
| 26. |
If sin θ = 3 sin (θ + 2α), then the value of tan (θ + α) + 2 tan α is equal to |
| A. | -1 |
| B. | 0 |
| C. | 1 |
| D. | 2 |
| Answer» C. 1 | |
| 27. |
If P = sin20 θ + cos48 θ, then the inequality that holds for all values of θ is |
| A. | P ≥ 1 |
| B. | 0 < P ≤ 1 |
| C. | 1 < P < 3 |
| D. | 0 ≤ P ≤ 1 |
| Answer» C. 1 < P < 3 | |
| 28. |
If 3 tan θ = cot θ where \(0 \le \theta \le \frac{\pi }{2},\) then what is the value of θ? |
| A. | \(\frac{\pi }{6}\) |
| B. | \(\frac{\pi }{4}\) |
| C. | \(\frac{\pi }{3}\) |
| D. | \(\frac{\pi }{2}\) |
| Answer» B. \(\frac{\pi }{4}\) | |
| 29. |
If x = sin∅, then which among the following correctly represents the range of x? |
| A. | – 1 ≤ x ≤ 1 |
| B. | – 1 < x < 1 |
| C. | 0 < x < 1 |
| D. | 0 ≤ x ≤ 1 |
| Answer» B. – 1 < x < 1 | |
| 30. |
If \(cosec \theta + \cot \theta = 2, then \sin \theta\) is: |
| A. | \(\frac{4}{5}\) |
| B. | \(\frac{2}{5}\) |
| C. | \(\frac{3}{5}\) |
| D. | \(\frac{3}{4}\) |
| Answer» B. \(\frac{2}{5}\) | |
| 31. |
Find the value of tan π |
| A. | π |
| B. | 1 |
| C. | 0 |
| D. | -1 |
| Answer» D. -1 | |
| 32. |
∆DEF is right angled at E. If cot D = 5/12, then what is the value of sin F? |
| A. | 5/12 |
| B. | 13/5 |
| C. | 5/13 |
| D. | 13/12 |
| Answer» D. 13/12 | |
| 33. |
Find the height (in metres) of the building if the sun’s shadow increases by 10 m when the angle of elevation of the sun decreases from 60° to 45°. |
| A. | 10√3/(√3 + 1) |
| B. | 10/(√3 - 1) |
| C. | 10√3 |
| D. | 10√3/(√3 - 1) |
| Answer» E. | |
| 34. |
If cot(A/2) = x, then the value of x is? |
| A. | √[(1 + cosA)/(1 - cosA)] |
| B. | cosecA - cotA |
| C. | √[(1 - cosA)/2] |
| D. | √[(1 + cosA)/2] |
| Answer» B. cosecA - cotA | |
| 35. |
If sin A = ¾, and A is an acute angle, then find the value of tan A. |
| A. | 5/√7 |
| B. | 6/√7 |
| C. | 4/√7 |
| D. | 3/√7 |
| Answer» E. | |
| 36. |
(cosec A – sinA)2 + (secA – cosA)2 – (cotA – tanA)2 is equal to∶ |
| A. | 2 |
| B. | 0 |
| C. | 1 |
| D. | –1 |
| Answer» D. –1 | |
| 37. |
If 2sin2θ + 3sinθ – 2 = 0, (0° < θ < 90°) then the value of θ is: |
| A. | 90° |
| B. | 30° |
| C. | 45° |
| D. | 60° |
| Answer» C. 45° | |
| 38. |
If √[(1 + cosA)/2] = x, then find the value of x. |
| A. | sin(A/2) |
| B. | tan(A/2) |
| C. | cot(A/2) |
| D. | cos(A/2) |
| Answer» E. | |
| 39. |
Find tan 75°, by splitting 75° as (45° + 30° ). |
| A. | 2 + √3 |
| B. | 2√3 + 1 |
| C. | \(\frac{{\sqrt 3\; - \;1}}{{\sqrt 3\; + \;1}}\) |
| D. | 2√3 – 1 |
| Answer» B. 2√3 + 1 | |
| 40. |
If cot A = tan(2A - 45°), A is an acute angle then tan A is equal to: |
| A. | \(\sqrt{3}\) |
| B. | 0 |
| C. | 1 |
| D. | \(\frac{1}{2}\) |
| Answer» D. \(\frac{1}{2}\) | |
| 41. |
A boy is observing a hot air balloon flying at 692 m above ground. The angle of elevation of balloon from the eyes of the boy was 60°. After sometime, the balloon moves away horizontally from the boy and the angle reduces to 30°. What is the approximate distance travelled by the balloon? Take \(\sqrt 3 = 1.73\) |
| A. | 400 m |
| B. | 800 m |
| C. | 600 m |
| D. | 200 m |
| Answer» C. 600 m | |
| 42. |
If cosecθ + cotθ = 2.5, then cosecθ = ? |
| A. | 1.45 |
| B. | 1.35 |
| C. | 1.5 |
| D. | 1.55 |
| Answer» B. 1.35 | |
| 43. |
If sin3A = cos(30° - A), where A is an acute angle, then what is the value of 2cosecA + tan22A - cot2A |
| A. | 17/2 |
| B. | 5 |
| C. | 4 |
| D. | 7/4 |
| Answer» D. 7/4 | |
| 44. |
Consider the function f(θ) = 4 (\({\sin ^2}\theta + {\cos ^4}\theta\)) Consider the following statements:1. f (θ) = 2 has no solution 2. \({\rm{f\;}}\left( {\rm{\theta }} \right){\rm{}} = {\rm{}}\frac{7}{2}\) has a solution.Which of the above statements is/are correct? |
| A. | 1 only |
| B. | 2 only |
| C. | Both 1 and 2 |
| D. | Neither 1 nor 2 |
| Answer» D. Neither 1 nor 2 | |
| 45. |
If sinθ + cosθ = √3 cos (90 - θ), then what is the value of tanθ? |
| A. | √3 - 1 |
| B. | √3 + 1 |
| C. | (√3 + 1)/2 |
| D. | (√3 - 1)/2 |
| Answer» D. (√3 - 1)/2 | |
| 46. |
Consider the following statements:1. \({\tan ^{ - 1}}{\rm{x}} + {\tan ^{ - 1}}\left( {\frac{1}{{\rm{x}}}} \right) = {\rm{\pi }}\)2. There exist x, y ∈ [-1, 1], where x ≠ y such that sin-1 x + cos-1 \({\rm{y}} = \frac{{\rm{\pi }}}{2}\)Which of the above statements is/are correct? |
| A. | 1 only |
| B. | 2 only |
| C. | Both 1 and 2 |
| D. | Neither 1 nor 2 |
| Answer» E. | |
| 47. |
From the top of a 150 m tall building A, the angle of elevation to the top of the building B is 45 degrees and the angle of depression to the bottom of the building B is 30 degrees. What is the height of the building B? |
| A. | 250 m |
| B. | 450/√3 m |
| C. | 150√3 m |
| D. | 150 (1 + √3) m |
| Answer» E. | |
| 48. |
If cosθ = 4/5, then sin2θ cosθ + cos2θ sinθ is equal to: |
| A. | 84/125 |
| B. | 14/25 |
| C. | 82/125 |
| D. | 16/25 |
| Answer» B. 14/25 | |
| 49. |
If \(\rm \frac{\tan x}{2} = \frac{\tan y}{3} = \frac{\tan z}{5}\) and x + y + z = π, then the value of tan2 x + tan2 y + tan2 z is: |
| A. | \(\dfrac {38}3\) |
| B. | \(\dfrac {29}3\) |
| C. | \(\dfrac {11}3\) |
| D. | None of these |
| Answer» B. \(\dfrac {29}3\) | |
| 50. |
If tan θ + cot θ = x, then what is the value of tan4 θ + cot4 θ? |
| A. | (x3 - 3) 2 + 2 |
| B. | (x4 - 2x) + 4 |
| C. | x(x - 4) + 2 |
| D. | x2(x2 - 4) + 2 |
| Answer» E. | |