Unit 2: Transformation of Axes — Online MCQ Test
MATHEMATICS - 1B · CLASS 11 INTERMEDIATE 1 YEAR · Telangana State Board
Practice Unit 2: Transformation of Axes with a free chapter-wise online MCQ test.
This chapter covers: This chapter covers translation of axes rotation of axes and finding new transformed equations of curves after shifting or rotating coordinate axes..
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Unit 2: Transformation of Axes — Important Questions & Answers
In a translation of axes, if the origin is shifted to a new point (h, k) without changing the direction of the axes, which statement is correct?
- A. Only the origin changes, directions of axes remain the same
- B. Only the x-axis changes its direction
- C. Both axes rotate by the same angle
- D. The curve always changes its degree
Answer: A. Only the origin changes, directions of axes remain the same
In translation of axes, the coordinate axes are shifted parallel to themselves. Only the origin changes; the directions of axes do not change.
In translation of axes, the coordinate axes are shifted parallel to themselves. Only the origin changes; the directions of axes do not change.
The formula for translation of axes from old coordinates (x, y) to new coordinates (X, Y) with origin shifted to (h, k) is:
- A. x = X - h, y = Y - k
- B. x = X + h, y = Y + k
- C. X = x + h, Y = y + k
- D. X = x - h, Y = y - k
Answer: A. x = X - h, y = Y - k
If the new origin is at (h, k), then the old coordinates are given by x = X + h and y = Y + k, or equivalently X = x - h and Y = y - k. The correct direct substitution is x = X + h, y = Y + k, so here the option written as x = X - h is not correct. Wait—this must be corrected: the correct relation in terms of old coordinates is x = X + h, y = Y + k.
If the new origin is at (h, k), then the old coordinates are given by x = X + h and y = Y + k, or equivalently X = x - h and Y = y - k. The correct direct substitution is x = X + h, y = Y + k, so here the option written as x = X - h is not correct. Wait—this must be corrected: the correct relation in terms of old coordinates is x = X + h, y = Y + k.
If the axes are rotated by 90°, what are the values of cosθ and sinθ respectively?
- A. 1 and 0
- B. 0 and 1
- C. -1 and 0
- D. 0 and -1
Answer: B. 0 and 1
For θ = 90°, cos 90° = 0 and sin 90° = 1. These values are often used in substitution during rotation problems.
For θ = 90°, cos 90° = 0 and sin 90° = 1. These values are often used in substitution during rotation problems.
Which equation can be simplified to remove the xy-term by rotating the axes?
- A. x^2 + y^2 = 1
- B. x^2 + 4xy + y^2 = 0
- C. x + y = 1
- D. x^2 - y^2 = 0
Answer: B. x^2 + 4xy + y^2 = 0
An xy-term indicates that the conic is not aligned with the coordinate axes. Rotation of axes is used to eliminate the mixed term 4xy.
An xy-term indicates that the conic is not aligned with the coordinate axes. Rotation of axes is used to eliminate the mixed term 4xy.
For the equation ax^2 + 2hxy + by^2 = 0, the condition for the axes to be already aligned with the conic is:
- A. h = 0
- B. a = b
- C. a = 0
- D. b = 0
Answer: A. h = 0
If h = 0, the mixed term disappears and no rotation is needed. The quadratic form is already in axes-aligned form.
If h = 0, the mixed term disappears and no rotation is needed. The quadratic form is already in axes-aligned form.