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.
| 651. |
If tan f + cot f = 5, then find the value of (tan2 f + cot2 f) |
| A. | 21 |
| B. | 27 |
| C. | 25 |
| D. | 23 |
| Answer» E. | |
| 652. |
If \(\cos θ = \dfrac{5}{13}, \dfrac{3\pi}{2}< θ < 2\pi\), then tan 2θ is |
| A. | \(-\dfrac{120}{119}\) |
| B. | \(-\dfrac{120}{169}\) |
| C. | \(\dfrac{120}{169}\) |
| D. | \(\dfrac{120}{119}\) |
| Answer» E. | |
| 653. |
From the top of a cliff 90 m high, the angle of depression of the top and bottom of a tower are 30∘ and 60∘ respectively, then the height of the tower is - |
| A. | 30 m |
| B. | 30√3 m |
| C. | (90 - 30√3) m |
| D. | 60 m |
| Answer» E. | |
| 654. |
If \(\sqrt {\frac{{1 - cos\theta }}{{1\; + \;cos\theta }}} \times \sqrt {\frac{{cosec\theta - cot\theta }}{{cosec\theta \; + \;cot\theta }}} \) = \(\frac{{1 - r}}{{1\; + \;r}}\) , then the value of r is: |
| A. | sinθ |
| B. | cosecθ |
| C. | secθ |
| D. | cosθ |
| Answer» E. | |
| 655. |
If 2 cos θ = 2 - sin θ, then what is the value of cos θ? |
| A. | 1 or 3/5 |
| B. | 1 or -1/2 |
| C. | -1 or -1/2 |
| D. | -1 or 3/5 |
| Answer» B. 1 or -1/2 | |
| 656. |
If sec θ + tan θ = p, (p > 1), then (cosec θ + 1)/(cosec θ – 1) = ? |
| A. | \(\frac{{p\; + \;1}}{{p - 1}}\) |
| B. | p2 |
| C. | \(\frac{{p - 1}}{{p\; + \;1}}\) |
| D. | 2P2 |
| Answer» C. \(\frac{{p - 1}}{{p\; + \;1}}\) | |
| 657. |
If θ is a positive acute angle and tan2θ tan3θ = 1, then the value of θ is: |
| A. | 60° |
| B. | 36° |
| C. | 45° |
| D. | 18° |
| Answer» E. | |
| 658. |
\(\frac{{sin\theta [\left( {1 - tan\theta } \right)tan\theta + {{\sec }^2}\theta ]}}{{\left( {1 - \sin \theta } \right)\tan \theta \left( {1 + tan\theta } \right)\left( {sec\theta + tan\theta } \right)}}\) is equal to: |
| A. | sinθ cosθ |
| B. | - 1 |
| C. | cosecθ secθ |
| D. | 1 |
| Answer» E. | |
| 659. |
If cot θ = 5x and cosec θ = 5/x (x ≠ 0), then the value of 5(x2 – 1/x2) is: |
| A. | 1/5 |
| B. | -1/5 |
| C. | -1/4 |
| D. | 1/2 |
| Answer» C. -1/4 | |
| 660. |
In ΔXYZ measure of angle Y is 90°. If cosecX = 17/15, and XY = 4cm, then what is the length (in cm) of side YZ? |
| A. | 7.5 |
| B. | 8.5 |
| C. | 5 |
| D. | 6 |
| Answer» B. 8.5 | |
| 661. |
If m cot A = n, then the value of\(\dfrac{m.Sin~A - n.Cos~A}{n.Cos~A + m.Sin~A}\) is |
| A. | \(\dfrac{n^2 + m^2}{n - m}\) |
| B. | \(\dfrac{m^2 - n^2}{n^2 - m^2}\) |
| C. | \(\dfrac{m^2 - n^2}{m^2 + n^2}\) |
| D. | \(m + \dfrac{n^2}{m^2}\) |
| Answer» D. \(m + \dfrac{n^2}{m^2}\) | |
| 662. |
A 15 m tall house on the bank of a river is facing a chimney directly opposite to it on the other bank. The angle of depression from the roof of the house to the base of the base of the chimney is 30 degree and the angle of elevation from the base of the house to the top of the chimney is 60 degree. Find the approximate width of the river and the height of the chimney. |
| A. | 24 m and 47 m |
| B. | 20 m and 25 m |
| C. | 25 m and 41 m |
| D. | 26 m and 45 m |
| Answer» E. | |
| 663. |
A pole of length 7 m is fixed vertically on the top of a tower. The angle of elevation of the top of the pole observed from a point on the ground is 60° and the angle of depression of the same point on the ground from the top of the tower is 45° The height (in m) of the tower is: |
| A. | 7(2√3 – 1) |
| B. | 7√3/2+ 2 |
| C. | 7√3 |
| D. | 7(√3 + 1)/2 |
| Answer» E. | |
| 664. |
If tan α = √5 – 2, then the value of tan α – cot α = ? |
| A. | -2 |
| B. | √5 + 2 |
| C. | -4 |
| D. | 2√5 |
| Answer» D. 2√5 | |
| 665. |
If 2 sin 3θ = 1, then the value of θ is∶ |
| A. | 10° |
| B. | 45° |
| C. | 20° |
| D. | 30° |
| Answer» B. 45° | |
| 666. |
Let AB represents a building of height h meter with A being its top, B being its bottom. Let A’B’ represents a tower of height (h + x) meter (x > 0) with A’ being its top and B’ being its bottom. Let BB’ = d meter. Let the angle of elevation A’ as seen from A be 45°.Consider the following statements:Statement I: h + x > dStatement II: The angle of depression of B as seen from A’ is less than 45°Which one of the following is correct in respect to the above statements? |
| A. | Both Statement I and Statement II are true and Statement II is the correct explanation of Statement I |
| B. | Both Statement I and Statement II are true and Statement II is not the correct explanation of Statement I |
| C. | Statement I is true but Statement II is false |
| D. | Statement I is false but Statement II is true |
| Answer» D. Statement I is false but Statement II is true | |
| 667. |
If sin 2A = cos (A - 18°), then the value of A is - |
| A. | 18° |
| B. | 36° |
| C. | 54° |
| D. | 72° |
| Answer» C. 54° | |
| 668. |
∆PQR is right angled at Q. If ∠R = 60°, what is the length of PR (in cm), if RQ = 4√3 cm? |
| A. | 8 |
| B. | 4 |
| C. | 8√3 |
| D. | 8/√3 |
| Answer» D. 8/√3 | |
| 669. |
If secA – tanA = x, then the value of x is |
| A. | 1/(sec2A – tan2A) |
| B. | 1/(sec2A + tan2A) |
| C. | \(1/\sqrt {\left[ {\left( {sec2A\; - \;tan2A} \right)} \right]}\) |
| D. | 1/(secA + tanA) |
| Answer» E. | |
| 670. |
If 2sinθ = 5cosθ, then \(\frac{{\sin \theta \; + \;\cos \theta }}{{\sin \theta - \cos \theta }}\) is equal to |
| A. | 9/5 |
| B. | 7/3 |
| C. | 2/3 |
| D. | 5/3 |
| Answer» C. 2/3 | |
| 671. |
If 3cos2A + 6sin2A = 3, 0° ≤ A ≤ 90°, then the value of A is: |
| A. | 0° |
| B. | 30° |
| C. | 90° |
| D. | 45° |
| Answer» B. 30° | |
| 672. |
If 2y cosθ = x sinθ and 2x secθ – y cosecθ = 3, then what is x2 + 4y2 equal to? |
| A. | 1 |
| B. | 2 |
| C. | 4 |
| D. | 8 |
| Answer» D. 8 | |
| 673. |
A person from the top of a hill observes a vehicle moving towards him at a uniform speed. It takes 10 minutes for the angle of depression to change from 45° to 60°. After this the time required by the vehicle to reach the bottom of the hill is |
| A. | 12 min 20 sec |
| B. | 13 min |
| C. | 13 min 40 sec |
| D. | 14 min 24 sec |
| Answer» D. 14 min 24 sec | |
| 674. |
A kite is flying at a height of 50 m. If the length of the string is 100 m then the inclination of the string to the horizontal ground in degree measures is:A. 90B. 45C. 60D. 30 |
| A. | B |
| B. | C |
| C. | D |
| D. | A |
| Answer» D. A | |
| 675. |
If cos α + cos β + cos γ = 0, where \(0 < {\rm{\alpha }} \le \frac{{\rm{\pi }}}{2},0 < {\rm{\beta }} \le \frac{{\rm{\pi }}}{2}\:\&\: 0 < {\rm{\gamma }} \le \frac{{\rm{\pi }}}{2}\) sin α + sin β + sin γ? |
| A. | 0 |
| B. | 3 |
| C. | \(\frac{{5\sqrt 2 }}{2}\) |
| D. | \(\frac{{3\sqrt 2 }}{2}\) |
| Answer» C. \(\frac{{5\sqrt 2 }}{2}\) | |
| 676. |
A tower stands on the top of a building which is 40 metres high. The angles of depression of a point situated on the ground from the top and the bottom of the tower are found to be 60° and 45° respectively. What is the height (in metres) of the tower? |
| A. | 20√3 |
| B. | 30(√3 + 1) |
| C. | 40(√3 - 1) |
| D. | 50(√3 - 1) |
| Answer» D. 50(√3 - 1) | |
| 677. |
If cos θ = 4/5, then (sec θ + cosec θ) = |
| A. | 7/5 |
| B. | 15/12 |
| C. | 35/12 |
| D. | 12/5 |
| Answer» D. 12/5 | |
| 678. |
If the angles of elevation of a balloon from two consecutive kilometre-stones along a straight road are 30° and 60° respectively, then the height of the balloon above the ground will be (assume the balloon is in the between the two mile stones) |
| A. | \(√3/2\) km |
| B. | \(4/√3\) km |
| C. | \(2√3\) km |
| D. | \(3√3\) km |
| Answer» B. \(4/√3\) km | |
| 679. |
From the top of 75 m high tower, the angel of depression of two points P and Q on opposite side of the base of the tower on level ground is θ and ϕ, such that tanθ = 3/4 and tanϕ = 5/8. What is the distance between the points P and Q? |
| A. | 190 m |
| B. | 200 m |
| C. | 180 m |
| D. | 220 m |
| Answer» E. | |
| 680. |
If 12tan A = 5, cosec A =?A. 13/5B. 12/13C. 5/13D. 12/5 |
| A. | B |
| B. | C |
| C. | A |
| D. | D |
| Answer» D. D | |
| 681. |
For θ being an acute angle, 4(2sin2 θ + 7cos2 θ) = 13, what is the value of θ? |
| A. | 60° |
| B. | 0° |
| C. | 45° |
| D. | 30° |
| Answer» B. 0° | |
| 682. |
If sec θ + tan θ = 12.5, then sec θ – tan θ = ? |
| A. | 0.4 |
| B. | 1 |
| C. | 0.8 |
| D. | 0.08 |
| Answer» E. | |
| 683. |
If \(\frac{{\sec \theta \; - \;\tan \theta }}{{\sec \theta \; + \;\tan \theta }}\; = \;\frac{3}{5},\) then the value of \(\frac{{cosec\theta \; + \;\cot \theta }}{{cosec\;\theta \; - {\rm{\;cot}}\theta }}\) is: |
| A. | 27 + √15 |
| B. | 33 + 4√15 |
| C. | 31 + 8√15 |
| D. | 24 + √15 |
| Answer» D. 24 + √15 | |
| 684. |
A spherical balloon of radius r subtends an angle α at the eye of an observer, while the angle of elevation of its centre is β. What is the height of the centre of the balloon (neglecting the height of the observer)? |
| A. | \(\frac{{{\rm{r}}\sin {\rm{\beta }}}}{{\sin \left( {\frac{{\rm{\alpha }}}{2}} \right)}}\) |
| B. | \(\frac{{{\rm{r}}\sin {\rm{\beta }}}}{{\sin \left( {\frac{{\rm{\alpha }}}{4}} \right)}}\) |
| C. | \(\frac{{{\rm{r}}\sin \left( {\frac{{\rm{\beta }}}{2}} \right)}}{{\sin {\rm{\alpha }}}}\) |
| D. | \(\frac{{{\rm{r}}\sin {\rm{\alpha }}}}{{\sin \left( {\frac{{\rm{\beta }}}{2}} \right)}}\) |
| Answer» B. \(\frac{{{\rm{r}}\sin {\rm{\beta }}}}{{\sin \left( {\frac{{\rm{\alpha }}}{4}} \right)}}\) | |
| 685. |
If in a triangle PQR, the sides p, q, r are in A. P., then the value of \(\cot \frac{P}{2}\) \(\cot \frac{R}{2}\) is: |
| A. | 0 |
| B. | 1 |
| C. | 2 |
| D. | 3 |
| Answer» E. | |
| 686. |
If 3cos2x - 2sin2x = -0.75 and 0° ≤ x ≤ 90°, then x = ? |
| A. | 30° |
| B. | 90° |
| C. | 60° |
| D. | 45° |
| Answer» D. 45° | |
| 687. |
If cosθ = 3/5, then find the value of tanθ.cotθ + sinθ |
| A. | \(1\frac{4}{8}\) |
| B. | \(1\frac{4}{6}\) |
| C. | \(1\frac{4}{5}\) |
| D. | \(1\frac{4}{7}\) |
| Answer» D. \(1\frac{4}{7}\) | |
| 688. |
If 0 |
| A. | 6 |
| B. | 8 |
| C. | 9 |
| D. | 7 |
| Answer» D. 7 | |
| 689. |
If sec θ = 13/12, then what is the value of sin θ? |
| A. | 5/13 |
| B. | 12/5 |
| C. | 12/13 |
| D. | 5/12 |
| Answer» B. 12/5 | |
| 690. |
If Tan θ = 9/40, then Sec θ = ? |
| A. | 40/41 |
| B. | 9/41 |
| C. | 41/40 |
| D. | 41/9 |
| Answer» D. 41/9 | |
| 691. |
From the top of a platform 7 m high, the angle of elevation of a tower was 30°. If the tower was 47 m high, how far away from the tower was the platform positioned? |
| A. | 15√3 m |
| B. | 40 m |
| C. | 45√3 m |
| D. | 40√3 m |
| Answer» E. | |
| 692. |
∆ ABC is right angled at B. If m∠A = 30°, then Sec C =? |
| A. | 1/2 |
| B. | 1/√2 |
| C. | 2 |
| D. | 1/√3 |
| Answer» D. 1/√3 | |
| 693. |
If cos 11π/6 = x, then the value of x is |
| A. | -√3/2 |
| B. | 1/2 |
| C. | 2 |
| D. | √3/2 |
| Answer» E. | |
| 694. |
Let A and B be two towers with same base. From the midpoint of the line joining their feet, the angles of elevation of the tops of A and B are 30° and 60°, respectively. The ratio of the heights of B and A is: |
| A. | 3 : 1 |
| B. | \(1: \sqrt 3\) |
| C. | 1 : 2 |
| D. | 1 : 3 |
| Answer» B. \(1: \sqrt 3\) | |
| 695. |
ΔPQR is right angled at Q. If cot P = 5/12, then what is the value of tan R? |
| A. | 5/13 |
| B. | 5/12 |
| C. | 13/5 |
| D. | 13/12 |
| Answer» C. 13/5 | |
| 696. |
If 2(cosec239°- tan251°) -2/3(sin90°) –tan256° y tan234° = y/3, then the value of y is: |
| A. | -1 |
| B. | 1 |
| C. | 23 |
| D. | -23 |
| Answer» C. 23 | |
| 697. |
Consider the following inequalities :1. sin 1° < cos 57°2. cos 60° > sin 57°Which of the above is/are correct? |
| A. | 1 only |
| B. | 2 only |
| C. | Both 1 and 2 |
| D. | Neither 1 nor 2 |
| Answer» B. 2 only | |
| 698. |
If sin x \(= \frac{4}{5},\) then \(\frac{{\tan x}}{{\cot x}} = ?\)A. 13/9B. 3/4C. 9/16D. 16/9 |
| A. | D |
| B. | A |
| C. | B |
| D. | C |
| Answer» B. A | |
| 699. |
If 3sin θ = 2cos θ, then \(\frac{{4\sin \theta - \cos \theta }}{{4\cos \theta \; + \;\sin \theta }}\) is equal to∶ |
| A. | 5/8 |
| B. | 5/11 |
| C. | 5/14 |
| D. | 5/7 |
| Answer» D. 5/7 | |
| 700. |
If A + B = 90° and cos B = 1/3; then the value of sin A is:A. 1/2B. 1/4C. 1/3D. 2/3 |
| A. | C |
| B. | B |
| C. | D |
| D. | A |
| Answer» B. B | |