Q. Find the distance between the points A(2, 3) and B(5, 7).
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Solution
Distance = √[(5-2)² + (7-3)²] = √[3² + 4²] = √[9 + 16] = √25 = 5.
Correct Answer:
C
— 5
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Q. Find the distance from the point (1, 2) to the line 3x + 4y - 12 = 0.
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Solution
Distance = |Ax1 + By1 + C| / sqrt(A^2 + B^2) = |3(1) + 4(2) - 12| / sqrt(3^2 + 4^2) = |3 + 8 - 12| / 5 = 1.
Correct Answer:
A
— 2
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Q. Find the distance from the point (3, 4) to the line 2x + 3y - 6 = 0.
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Solution
Distance = |Ax1 + By1 + C| / sqrt(A^2 + B^2) = |2*3 + 3*4 - 6| / sqrt(2^2 + 3^2) = |6 + 12 - 6| / sqrt(13) = 12 / sqrt(13).
Correct Answer:
B
— 3
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Q. Find the eigenvalues of the matrix A = [[2, 1], [1, 2]].
A.
1, 3
B.
2, 2
C.
3, 1
D.
0, 4
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Solution
The characteristic polynomial is det(A - λI) = (2-λ)(2-λ) - 1 = λ^2 - 4λ + 3 = 0, giving eigenvalues 1 and 3.
Correct Answer:
A
— 1, 3
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Q. Find the eigenvalues of the matrix G = [[2, 1], [1, 2]]. (2020)
A.
1, 3
B.
2, 2
C.
3, 1
D.
0, 4
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Solution
The characteristic polynomial is det(G - λI) = (2-λ)(2-λ) - 1 = λ^2 - 4λ + 3 = 0. The eigenvalues are λ = 1 and λ = 3.
Correct Answer:
A
— 1, 3
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Q. Find the eigenvalues of the matrix G = [[5, 4], [2, 3]]. (2020)
A.
1, 7
B.
2, 6
C.
3, 5
D.
4, 4
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Solution
The eigenvalues are found by solving the characteristic equation det(G - λI) = 0. This gives λ^2 - 8λ + 7 = 0, which factors to (λ - 1)(λ - 7) = 0, hence λ = 1, 7.
Correct Answer:
A
— 1, 7
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Q. Find the equation of the circle with center (2, -3) and radius 5.
A.
(x-2)² + (y+3)² = 25
B.
(x+2)² + (y-3)² = 25
C.
(x-2)² + (y-3)² = 25
D.
(x+2)² + (y+3)² = 25
Show solution
Solution
Equation of circle: (x-h)² + (y-k)² = r² => (x-2)² + (y+3)² = 5².
Correct Answer:
A
— (x-2)² + (y+3)² = 25
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Q. Find the equation of the family of curves represented by y = mx + c, where m and c are constants.
A.
y = mx + c
B.
y = mx^2 + c
C.
y = c/x + m
D.
y = m^2x + c
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Solution
The equation y = mx + c represents a family of straight lines where m is the slope and c is the y-intercept.
Correct Answer:
A
— y = mx + c
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Q. Find the equation of the line parallel to y = 3x + 2 and passing through (4, 5).
A.
y = 3x - 7
B.
y = 3x + 5
C.
y = 3x + 2
D.
y = 3x - 2
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Solution
Since the line is parallel, it has the same slope. Using point-slope form: y - 5 = 3(x - 4) gives y = 3x - 7.
Correct Answer:
A
— y = 3x - 7
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Q. Find the equation of the line parallel to y = 3x + 2 that passes through the point (4, 1).
A.
y = 3x - 11
B.
y = 3x + 1
C.
y = 3x + 2
D.
y = 3x - 2
Show solution
Solution
Since the line is parallel, it has the same slope (3). Using point-slope form: y - 1 = 3(x - 4) gives y = 3x - 11.
Correct Answer:
A
— y = 3x - 11
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Q. Find the equation of the line parallel to y = 5x - 2 that passes through the point (2, 3).
A.
y = 5x - 7
B.
y = 5x + 2
C.
y = 5x - 5
D.
y = 5x + 1
Show solution
Solution
Since the slope is the same (5), using point-slope form: y - 3 = 5(x - 2) gives y = 5x - 7.
Correct Answer:
A
— y = 5x - 7
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Q. Find the equation of the line parallel to y = 5x - 3 that passes through the point (2, 1).
A.
y = 5x - 9
B.
y = 5x + 1
C.
y = 5x - 7
D.
y = 5x + 3
Show solution
Solution
Since the slope is the same (5), using point-slope form: y - 1 = 5(x - 2) gives y = 5x - 9.
Correct Answer:
A
— y = 5x - 9
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Q. Find the equation of the line passing through the points (1, 2) and (3, 4).
A.
y = x + 1
B.
y = 2x
C.
y = x + 3
D.
y = 2x - 1
Show solution
Solution
The slope m = (4-2)/(3-1) = 1. Using point-slope form: y - 2 = 1(x - 1) gives y = x + 1.
Correct Answer:
A
— y = x + 1
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Q. Find the equation of the line passing through the points (2, 3) and (4, 7). (2020)
A.
y = 2x - 1
B.
y = 2x + 1
C.
y = 3x - 3
D.
y = 2x + 3
Show solution
Solution
The slope m = (7 - 3) / (4 - 2) = 2. Using point-slope form: y - 3 = 2(x - 2) gives y = 2x + 1.
Correct Answer:
B
— y = 2x + 1
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Q. Find the equation of the line that is perpendicular to y = 5x + 2 and passes through the origin.
A.
y = -1/5x
B.
y = 5x
C.
y = -5x
D.
y = 1/5x
Show solution
Solution
The slope of the given line is 5. The slope of the perpendicular line is -1/5. Using y = mx + c, we get y = -1/5x.
Correct Answer:
C
— y = -5x
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Q. Find the equation of the line that is perpendicular to y = 5x + 2 and passes through (2, 3).
A.
y = -1/5x + 4
B.
y = 5x - 7
C.
y = -5x + 13
D.
y = 1/5x + 2
Show solution
Solution
The slope of the perpendicular line is -1/5. Using point-slope form: y - 3 = -1/5(x - 2) gives y = -1/5x + 13/5.
Correct Answer:
C
— y = -5x + 13
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Q. Find the equation of the line that passes through (0, 0) and has a slope of 5.
A.
y = 5x
B.
y = x/5
C.
y = 5/x
D.
y = 1/5x
Show solution
Solution
Using the slope-intercept form y = mx + b, with m = 5 and b = 0, we get y = 5x.
Correct Answer:
A
— y = 5x
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Q. Find the equation of the line that passes through (2, 3) and is perpendicular to the line y = 1/3x + 2.
A.
y = -3x + 9
B.
y = 3x - 3
C.
y = -1/3x + 4
D.
y = 1/3x + 1
Show solution
Solution
The slope of the given line is 1/3, so the perpendicular slope is -3. Using point-slope form, y - 3 = -3(x - 2) gives y = -3x + 9.
Correct Answer:
A
— y = -3x + 9
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Q. Find the equation of the line that passes through (2, 3) and is perpendicular to the line y = 4x - 1.
A.
y = -1/4x + 4
B.
y = 4x - 5
C.
y = -4x + 11
D.
y = 1/4x + 2
Show solution
Solution
The slope of the given line is 4, so the perpendicular slope is -1/4. Using point-slope form, we get y - 3 = -1/4(x - 2) which simplifies to y = -1/4x + 11/4.
Correct Answer:
C
— y = -4x + 11
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Q. Find the equation of the line that passes through the origin and has a slope of -2.
A.
y = -2x
B.
y = 2x
C.
y = -x
D.
y = x
Show solution
Solution
Using the slope-intercept form: y = mx + b, where b = 0, we have y = -2x.
Correct Answer:
A
— y = -2x
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Q. Find the equation of the line that passes through the origin and has a slope of -3.
A.
y = -3x
B.
y = 3x
C.
y = -x/3
D.
y = 1/3x
Show solution
Solution
Using the slope-intercept form, the equation is y = -3x.
Correct Answer:
A
— y = -3x
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Q. Find the equation of the line that passes through the point (1, 2) and has a slope of 3.
A.
y = 3x + 1
B.
y = 3x - 1
C.
y = 3x + 2
D.
y = 3x - 2
Show solution
Solution
Using point-slope form: y - 2 = 3(x - 1) => y = 3x - 1.
Correct Answer:
C
— y = 3x + 2
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Q. Find the equation of the line that passes through the point (2, -3) and has a slope of 4.
A.
y = 4x - 11
B.
y = 4x + 5
C.
y = 4x - 3
D.
y = 4x + 3
Show solution
Solution
Using point-slope form, y + 3 = 4(x - 2) simplifies to y = 4x - 11.
Correct Answer:
A
— y = 4x - 11
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Q. Find the equation of the line that passes through the point (2, 3) and has a slope of -1.
A.
y = -x + 5
B.
y = -x + 3
C.
y = x + 1
D.
y = -x + 1
Show solution
Solution
Using point-slope form: y - 3 = -1(x - 2) => y = -x + 5.
Correct Answer:
A
— y = -x + 5
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Q. Find the equation of the line that passes through the point (4, -1) and is perpendicular to the line y = 3x + 2.
A.
y = -1/3x + 5/3
B.
y = 3x - 13
C.
y = -3x + 11
D.
y = 1/3x - 5/3
Show solution
Solution
The slope of the given line is 3, so the slope of the perpendicular line is -1/3. Using point-slope form, we get y + 1 = -1/3(x - 4), which simplifies to y = -1/3x + 11/3.
Correct Answer:
C
— y = -3x + 11
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Q. Find the equation of the line that passes through the point (4, 5) and is perpendicular to the line y = 1/3x + 2.
A.
y = -3x + 17
B.
y = 3x - 7
C.
y = -3x + 5
D.
y = 1/3x + 5
Show solution
Solution
The slope of the given line is 1/3, so the slope of the perpendicular line is -3. Using point-slope form, we get y - 5 = -3(x - 4), which simplifies to y = -3x + 17.
Correct Answer:
A
— y = -3x + 17
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Q. Find the equation of the line that passes through the points (2, 3) and (4, 7).
A.
y = 2x - 1
B.
y = 2x + 1
C.
y = 3x - 3
D.
y = x + 1
Show solution
Solution
The slope m = (7 - 3) / (4 - 2) = 2. Using point-slope form: y - 3 = 2(x - 2) gives y = 2x + 1.
Correct Answer:
B
— y = 2x + 1
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Q. Find the equation of the pair of lines represented by the equation 2x^2 + 3xy + y^2 = 0.
A.
y = -2x, y = -x/3
B.
y = -3x/2, y = -x/2
C.
y = -x/3, y = -3x
D.
y = -x/2, y = -2x
Show solution
Solution
Using the quadratic formula for the slopes gives m1 = -2 and m2 = -1/3.
Correct Answer:
A
— y = -2x, y = -x/3
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Q. Find the equation of the pair of lines represented by the equation x^2 - 4y^2 = 0.
A.
x = 2y, x = -2y
B.
x = 4y, x = -4y
C.
x = 0, y = 0
D.
x = y, x = -y
Show solution
Solution
Factoring the equation gives (x - 2y)(x + 2y) = 0, which represents the lines x = 2y and x = -2y.
Correct Answer:
A
— x = 2y, x = -2y
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Q. Find the equation of the parabola that opens downwards with vertex at (0, 0) and passes through the point (2, -4).
A.
y = -x^2
B.
y = -2x^2
C.
y = -1/2x^2
D.
y = -4x^2
Show solution
Solution
Using the vertex form and substituting the point (2, -4), we find that the equation is y = -2x^2.
Correct Answer:
B
— y = -2x^2
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