Case Study Chapter 8 Trigonometry Mathematics

In ΔABC, right angled at B

AB + AC = 9 cm and BC = 3cm.

Question. The value of cot C is
(a) 3/ 4
(b) 1/ 4
(c) 5/ 4
(d) None of these

Answer: A

Question. The value of sec C is
(a) 4/ 3
(b) 5/ 3
(c) 1/ 3
(d) None of these

Answer: B

Question. sin²C + cos²C =
(a) 0
(b) 1
(c) –1
(d) None of these

Answer: B

I. Application of Trigonometry—Height of Tree/Tower: Mr. Suresh is an electrician. He receives a call regarding a fault on a pole from three different colonies A, B and C. He reaches one-by-one to each colony to repair that fault. He needs to reach a point 1.3 m below the top of each pole to undertake the repair work. Observe the following diagrams.

Question. What is the distance of the point where the ladder is placed on the ground if the height of pole is 4 m?
(a) 2.5 m
(b) 3.8 m
(c) 1.56 m
(d) 5.3 m

Answer: C

Question. What should be the length of ladder DQ that enable him to reach the required position if the height of the pole is 4 m?
(a) 5√3/7 m
(b) 9√3/5 m
(c) 7√2/5 m
(d) 4√3/5 m

Answer: B

Question. The distance of the point where the ladder lies on the ground is
(a) 3 √5 m
(b) 4 √2 m
(c) 4 m
(d) 4 √7 m

Answer: C

Question. Given that the length of ladder is 4 √2 m . What is height of pole?
(a) 4,1/2 m
(b) 4 √5 m
(c) 5 √5 m
(d) 5.3 m

Answer: D

Question. The angle of elevation of reaching point of ladder at pole, i.e., H, if the height of the pole is 8.3 m and the distance GF is 7√3 m, is
(a) 30°
(b) 60°
(c) 45°
(d) None of these.

Answer: A

II. A group of students of class X visited India Gate on an educational trip. The teacher and students had interest in history as well. The teacher narrated that India Gate, official name Delhi Memorial, originally called All-India War Memorial, monumental sandstone arch in New Delhi, dedicated to the troops of British India who died in wars fought between 1914 and 1919.The teacher also said that India Gate, which is located at the eastern end of the Rajpath (formerly called the Kingsway), is about 138 feet (42 metres) in height.

Question. They want to see the tower at an angle of 60°. The distance where they should stand will be
(a) 25.24 m
(b) 20.12 m
(c) 42 m
(d) 24.25 m

Answer: D

Question. The ratio of the length of a rod and its shadow is 1:1 . The angle of elevation of the Sun is
(a) 30°
(b) 45°
(c) 60°
(d) 90°

Answer: B

Question. What is the angle of elevation if they are standing at a distance of 42 m away from the monument?
(a) 30°
(b) 45°
(c) 60°
(d) 0°

Answer: B

Question. The angle formed by the line of sight with the horizontal when the object viewed is below the horizontal level is
(a) corresponding angle
(b) angle of elevation
(c) angle of depression
(d) complete angle

Answer: C

Question. If the altitude of the Sun is at 60°, then the height of the vertical tower that will cast a shadow of length 20 m is
(a) 20√3 m
(b) 20/√3m
(c) 15/√3m
(d) 15√3 m

Answer: A

III. A satellite flying at a height h is watching the top of the two tallest mountains in Uttarakhand and Karnataka, they are being Nanda Devi (height 7,816 m) and Mullayanagiri (height 1,930 m). The angles of depression from the satellite, to the top of Nanda Devi and Mullayanagiri are 30° and 60° respectively. If the distance between the peaks of two mountains is 1937 km, and the satellite is vertically above the mid-point of the distance between the two mountains.

Question. The distance of the satellite from the ground is
(a) 1139.4 km
(b) 566.96 km
(c) 1937 km
(d) 1025.36 km

Answer: B

Question. The distance of the satellite from the top of Mullayanagiri is
(a) 1139.4 km
(b) 577.52 km
(c) 1937 km
(d) 1025.36 km

Answer: C

Question. What is the angle of elevation if a man is standing at a distance of 7816 m away from Nanda Devi?
(a) 30°
(b) 45°
(c) 60°
(d) 0°

Answer: B

Question. The distance of the satellite from the top of Nanda Devi is
(a) 1118.29 km
(b) 577.52 km
(c) 1937 km
(d) 1025.36 km

Answer: A

Question. If a mile stone very far away from, makes 45° to the top of Mullayangiri mountain. So, find the distance of this mile stone from the mountain.
(a) 1118.327 km
(b) 566.976 km
(c) 1937 km
(d) 1025.36 km

Answer: C

Question. If tan (A – B) = \( \frac{1}{\sqrt{3}} \) and tan (A + B) = \( \sqrt{3} \) ,0° < A + B \(\le\) 90°, A > B. Then the value of A and B is
(a) 45°, 30°
(b) 45°, 15
(c) 60°, 30°
(d) None of these
Answer: B

Question. If A, B, C are the interior angles of a triangle ABC, then cos \( \left(\frac{A + B}{2}\right) \) equals to
(a) cos \( \frac{C}{2} \)
(b) sec \( \frac{C}{2} \)
(c) cosec \( \frac{C}{2} \)
(d) sin \( \frac{C}{2} \)
Answer: D

Question. If tan \( \theta \) = \( \frac{12}{5} \), then the value of sin \( \theta \) is :-
(a) \( \frac{12}{13} \)
(b) \( \frac{13}{12} \)
(c) \( \frac{5}{12} \)
(d) \( \frac{5}{13} \)
Answer: A

Question. If the value of sin \( \theta \) = \( \frac{3}{5} \), then the value of cosec \( \theta \) is :-
(a) \( \frac{13}{5} \)
(b) \( \frac{1}{5} \)
(c) \( \frac{5}{3} \)
(d) 3
Answer: C

Question. The value of tan \( \theta = \sqrt{2} – 1 \), then the value of the expression \( \frac{\tan \theta}{1 + \tan^2 \theta} \) is :-
(a) \( \frac{\sqrt{2}}{4} \)
(b) \( \frac{1}{4} \)
(c) \( \sqrt{2} \)
(d) \( \frac{1}{\sqrt{2}} \)
Answer: A

Question. The value of sec 45° is :-
(a) \( \frac{1}{\sqrt{2}} \)
(b) \( \sqrt{2} \)
(c) 1
(d) \( \frac{\sqrt{3}}{2} \)
Answer: B

Question. The value of sin 30° cos 45° + cos 30° sin 45° is :-
(a) \( \frac{\sqrt{2} - \sqrt{6}}{4} \)
(b) \( \frac{\sqrt{6}}{4} \)
(c) \( \frac{\sqrt{2}}{4} \)
(d) \( \frac{\sqrt{2} + \sqrt{6}}{4} \)
Answer: D

Question. The value of \( \sin^2\theta + \cos^2\theta \) is always :-
(a) 1
(b) 0
(c) –1
(d) 2
Answer: A

Question. If cos (40° + x) = sin 30°, then the value of x is :-
(a) 19°
(b) 23°
(c) 22°
(d) 20°
Answer: D

Question. A rhombus of side 20 cm has two angles of 60° each, then the length of the diagonals is (in cm) :-
(a) 20 \( \sqrt{3} \) , 20
(b) 20, \( \sqrt{3} \)
(c) 12 \( \sqrt{3} \) , 12
(d) 15 \( \sqrt{3} \) , 15
Answer: A

Question. The values of tan\(\theta\) and cot\(\theta\) are equal when :-
(a) \( \theta \) = 30°
(b) \( \theta \) = 45°
(c) \( \theta \) = 90°
(d) \( \theta \) = 0°
Answer: B

Question. If cosec \( \alpha = \frac{13}{12} \), then the value of tan \( \alpha \) is
(a) \( \frac{12}{5} \)
(b) \( \frac{12}{13} \)
(c) \( \frac{5}{12} \)
(d) \( \frac{5}{13} \)
Answer: A

Question. If tan \( \theta = \frac{x}{y} \), then cos \( \theta \) is equal to :-
(a) \( \frac{x}{\sqrt{x^2 + y^2}} \)
(b) \( \frac{x}{y} \)
(c) \( \frac{y}{\sqrt{x^2 + y^2}} \)
(d) \( \frac{x^2 + y^2}{x^2 - y^2} \)
Answer: C

Question. Which one of the following is correct?
(a) \( \sec^2 \alpha = 1 – \tan^2 \alpha \)
(b) \( \sin^2 \alpha = 1 + \cos^2 \alpha \)
(c) tan \( \alpha \) cot \( \alpha \) = 1
(d) none of these
Answer: C

Question. Which one of the following is true?
(a) sin (90° – \( \theta \)) = sin \( \theta \)
(b) cos (90° – \( \theta \)) = cos \( \theta \)
(c) sin (90° – \( \theta \)) = cos \( \theta \)
(d) tan (90° + \( \theta \)) = tan \( \theta \)
Answer: C

Question. The value of sinB cos (90° - B) + cos B sin (90° - B) is
(a) 0
(b) 1
(c) sinB cosB
(d) 2 \( \sin^2 \) B
Answer: B

Question. The value of \( \cos^4 \theta + \sin^4 \theta + 2 \cos^2 \theta \sin^2 \theta \), . when \( \theta = 45^\circ \) is
(a) 1
(b) 2
(c) \( \frac{1}{2} \)
(d) 2 \( \sqrt{2} \)
Answer: A

Question. On simplifying \( \sqrt{\frac{1 - \cos \theta}{1 + \cos \theta}} \) two students got the following answers:
I. cosec \( \theta \) – cot \( \theta \) II. \( \frac{1}{\text{cosec } \theta + \text{cot } \theta} \)
What can you say about this?
(a) both I and II are correct
(b) both are wrong
(c) I is wrong, II is correct
(d) I is correct, II is wrong
Answer: A

Question. If sec \( \beta = x + \frac{1}{4x} \), then the value of sec \( \beta + \tan \beta \) is equal to
(a) 2x
(b) \( \frac{x}{2} \)
(c) 3x
(d) \( \frac{x}{3} \)
Answer: A

Question. In an acute angled \( \Delta ABC \), a = 4 cm, b = 6 cm, sin B = \( \frac{3}{4} \), then the value of angle A is
(a) 30°
(b) 45°
(c) 60°
(d) none of these
Answer: A