Tracking Amit's movements: Starting North, he goes 10 km North; turns right (East) and walks 5 km; turns left (North) and walks 8 km; turns right (East) and walks 3 km.
His net displacement is 8 km North and 8 km East (5+3), placing point B in the North-East direction from point A.
Set port X at origin with North as positive y-axis and East as positive x-axis. The ship sails 60 km at 45° from North towards North-East.
The ship then sails 80 km at 45° from South towards South-East. This means 45° East of South direction, or equivalently, the angle is -45° from East (or 315° from North).
Total displacement components from port X:
$$x_{total} = x_1 + x_2 = 30\sqrt{2} + 40
Let Apartment B be at the origin (0, 0). Apartment A is 4 km East, so A is at (4, 0). Apartment C is 3 km North of B, so C is at (0, 3). Apartment D is 2 km West of C, so D is at (-2, 3).
Using the distance formula between points D(-2, 3) and A(4, 0):
The displacement vector from D to A is (6, -3), meaning 6 km East and 3 km South. The angle from East toward South is calculated as:
**The
Priya starts facing North. She turns 90° clockwise.
From East, she turns 45° counter-clockwise, then 135° clockwise.
From South, she turns 90° counter-clockwise.
Priya is now facing East, not South. However, if we verify the total rotation: North → 90° CW → 45° CCW → 135° CW → 90° CCW gives a net rotation of 90° clockwise from North, which is East.
Note: The given answer of South appears to be incorrect based on the step-by-step calculation. The correct answer should be East.
Starting at origin (0, 0), walking 5 km South gives position (0, -5), then 9 km North gives displacement of -5 + 9 = 4 km North.
Walking 12 km West gives position (-12, 4), then 7 km East gives displacement of -12 + 7 = -5 km (or 5 km West).
The final position is 5 km West and 4 km North of the starting point. Since the person is West and North of the origin, the direction is North-West.
The person is now in the North-West direction with respect to the starting point, not South-West as given in option A. However, if the answer key states South-West, there may be an error in either the problem statement or the provided answer.
Let point A be at origin (0, 0). North is positive y-direction, East is positive x-direction.
Starting position: (0, 0)
- Walks 10 km North: position becomes (0, 10)
- Turns right (now facing East) and walks 15 km: position becomes (15, 10)
- Turns left (now facing North) and walks 8 km: position becomes (15, 18)
- Turns right (now facing East) and walks 12 km: position becomes (27, 18)
The net displacement is the straight-line distance from point A to point B using the Pythagorean theorem.
Find the angle from North towards East using trigonometry.
This means the displacement is approximately 56.3° East of
This question asks you to find the 5th term of a geometric progression using the formula for the nth term.
First term: a = 2
Common ratio: r = 6/2 = 3 (or 18/6 = 3, or 54/18 = 3)
The nth term of a GP is given by:
Therefore, the 5th term = 162
The answer is (B) 162
For a 2×2 matrix, the determinant is calculated using the cross-product formula: (top-left × bottom-right) minus (top-right × bottom-left).
Matrix = [[1, 2], [3, 4]]
where a = 1, b = 2, c = 3, d = 4
Therefore, the determinant of [[1,2],[3,4]] = -2
The answer is (A) -2
This question tests your knowledge of standard trigonometric values for common angles.
The trigonometric ratios for 0°, 30°, 45°, 60°, and 90° are fixed values that must be memorized for competitive exams.
For 60°, the cosine value is:
At 60°, if you place a point on the unit circle, the x-coordinate (which represents cosine) equals \(\frac{1}{2}\).
- \(\sin 60° = \frac{\sqrt{3}}{2}\) (option B is sine, not cosine)
- \(\cos 0° = 1\) (option C)
- \(\cos 90° = 0\) (option D)
Therefore, \(\cos 60° = \frac{1}{2}\)
The correct answer is A) 1/2
To find the nth term of an arithmetic progression, we use the formula \(a_n = a + (n-1)d\) where a is the first term and d is the common difference.
From the AP: 2, 5, 8, 11...
First term a = 2
Common difference d = 5 - 2 = 3
We need to find \(a_{10}\) using:
Therefore, the 10th term of the AP is 29.
The answer is (A) 29.