Q. For the parabola y = x^2 - 4x + 3, find the coordinates of the vertex.
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A.
(2, -1)
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B.
(1, 2)
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C.
(2, 1)
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D.
(1, -1)
Solution
To find the vertex, use x = -b/(2a). Here, a = 1, b = -4, so x = 2. Substitute x = 2 into the equation to find y = -1. Thus, the vertex is (2, -1).
Correct Answer: A — (2, -1)
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Q. For the parabola y^2 = 16x, what is the coordinates of the vertex?
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A.
(0, 0)
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B.
(4, 0)
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C.
(0, 4)
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D.
(0, -4)
Solution
The vertex of the parabola y^2 = 4px is at (0, 0).
Correct Answer: A — (0, 0)
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Q. For the parabola y^2 = 20x, what is the coordinates of the vertex?
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A.
(0, 0)
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B.
(5, 0)
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C.
(0, 5)
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D.
(10, 0)
Solution
The vertex of the parabola y^2 = 4px is at (0, 0). Here, p = 5, but the vertex remains at (0, 0).
Correct Answer: A — (0, 0)
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Q. Identify the family of curves represented by the equation x^2 + y^2 = r^2.
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A.
Straight lines
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B.
Ellipses
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C.
Circles
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D.
Hyperbolas
Solution
The equation x^2 + y^2 = r^2 represents a family of circles with varying radii 'r'.
Correct Answer: C — Circles
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Q. Identify the family of curves represented by the equation y = a(x - h)^2 + k.
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A.
Parabolas
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B.
Circles
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C.
Ellipses
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D.
Hyperbolas
Solution
The equation y = a(x - h)^2 + k represents a family of parabolas that open upwards or downwards.
Correct Answer: A — Parabolas
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Q. Identify the family of curves represented by the equation y = ax^2 + bx + c.
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A.
Linear functions
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B.
Quadratic functions
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C.
Cubic functions
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D.
Exponential functions
Solution
The equation y = ax^2 + bx + c represents a family of quadratic functions with varying coefficients (a, b, c).
Correct Answer: B — Quadratic functions
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Q. Identify the family of curves represented by the equation y = a^x.
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A.
Exponential functions
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B.
Logarithmic functions
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C.
Polynomial functions
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D.
Trigonometric functions
Solution
The equation y = a^x represents a family of exponential functions where a is a positive constant.
Correct Answer: A — Exponential functions
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Q. Identify the family of curves represented by the equation y = c/x, where c is a constant.
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A.
Linear functions
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B.
Hyperbolas
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C.
Parabolas
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D.
Circles
Solution
The equation y = c/x represents a family of hyperbolas with varying asymptotes.
Correct Answer: B — Hyperbolas
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Q. Identify the family of curves represented by the equation y = e^(kx), where k is a constant.
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A.
Linear functions
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B.
Exponential functions
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C.
Logarithmic functions
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D.
Polynomial functions
Solution
The equation y = e^(kx) represents a family of exponential functions with varying growth rates determined by k.
Correct Answer: B — Exponential functions
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Q. Identify the family of curves represented by the equation y = e^(kx).
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A.
Linear functions
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B.
Exponential functions
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C.
Logarithmic functions
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D.
Polynomial functions
Solution
The equation y = e^(kx) represents a family of exponential functions with varying growth rates determined by 'k'.
Correct Answer: B — Exponential functions
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Q. Identify the family of curves represented by the equation y = mx^3 + c.
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A.
Cubic functions
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B.
Quadratic functions
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C.
Linear functions
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D.
Exponential functions
Solution
The equation y = mx^3 + c represents a family of cubic functions with varying coefficients m and c.
Correct Answer: A — Cubic functions
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Q. Identify the family of curves represented by the equation y = sin(kx) for varying k.
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A.
Sine waves
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B.
Cosine waves
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C.
Straight lines
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D.
Parabolas
Solution
The equation y = sin(kx) represents a family of sine waves with varying frequencies determined by k.
Correct Answer: A — Sine waves
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Q. Identify the family of curves represented by the equation y^2 = 4ax.
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A.
Parabolas
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B.
Hyperbolas
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C.
Ellipses
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D.
Straight lines
Solution
The equation y^2 = 4ax represents a family of parabolas that open to the right with varying values of 'a'.
Correct Answer: A — Parabolas
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Q. If a circle has a diameter of 10, what is its circumference?
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A.
10π
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B.
20π
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C.
5π
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D.
15π
Solution
Circumference C = πd = π(10) = 10π.
Correct Answer: B — 20π
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Q. If a circle has the equation x² + y² - 4x + 6y + 9 = 0, what is the center of the circle?
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A.
(2, -3)
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B.
(2, 3)
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C.
(-2, 3)
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D.
(-2, -3)
Solution
Rearranging the equation to standard form gives (x - 2)² + (y + 3)² = 0, thus the center is (2, -3).
Correct Answer: A — (2, -3)
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Q. If a line passes through the points (1, 1) and (2, 3), what is its equation in slope-intercept form?
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A.
y = 2x - 1
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B.
y = 3x - 2
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C.
y = 2x + 1
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D.
y = x + 2
Solution
Slope m = (3 - 1) / (2 - 1) = 2. Using point-slope form: y - 1 = 2(x - 1) => y = 2x - 1.
Correct Answer: A — y = 2x - 1
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Q. If an ellipse has a semi-major axis of 10 and a semi-minor axis of 6, what is the value of b^2?
Solution
For an ellipse, b is the semi-minor axis. Here, b = 6, so b^2 = 6^2 = 36.
Correct Answer: A — 36
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Q. If the center of a circle is at (0, 0) and it passes through the point (3, 4), what is the radius of the circle?
Solution
The radius is the distance from the center to the point, which is √(3² + 4²) = √25 = 5.
Correct Answer: A — 5
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Q. If the center of a circle is at (0, 0) and it passes through the point (3, 4), what is the equation of the circle?
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A.
x² + y² = 25
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B.
x² + y² = 12
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C.
x² + y² = 7
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D.
x² + y² = 16
Solution
The radius is 5 (distance from (0,0) to (3,4)), so the equation is x² + y² = 5² = 25.
Correct Answer: A — x² + y² = 25
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Q. If the coordinates of the vertices of a triangle are (1, 1), (4, 5), and (7, 2), what is the perimeter of the triangle?
Solution
Calculate distances AB, BC, CA and sum them: AB = 5, BC = 5, CA = 7. Perimeter = 5 + 5 + 7 = 17.
Correct Answer: B — 14
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Q. If the coordinates of the vertices of a triangle are (1, 2), (3, 4), and (5, 2), what is the perimeter of the triangle?
Solution
Perimeter = AB + BC + CA = √[(3-1)² + (4-2)²] + √[(5-3)² + (2-4)²] + √[(1-5)² + (2-2)²] = 2.83 + 2.83 + 4 = 10.
Correct Answer: B — 10
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Q. If the coordinates of the vertices of a triangle are (1, 2), (4, 6), and (7, 2), what is the perimeter of the triangle?
Solution
Perimeter = AB + BC + CA = √[(4-1)² + (6-2)²] + √[(7-4)² + (2-6)²] + √[(1-7)² + (2-2)²] = 3 + √(9 + 16) + 6 = 3 + 5 + 6 = 14.
Correct Answer: B — 14
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Q. If the coordinates of the vertices of a triangle are A(1, 1), B(4, 5), and C(7, 2), what is the area of the triangle?
Solution
Area = 1/2 | x1(y2-y3) + x2(y3-y1) + x3(y1-y2) | = 1/2 | 1(5-2) + 4(2-1) + 7(1-5) | = 1/2 | 3 + 4 - 28 | = 1/2 * 21 = 10.5.
Correct Answer: B — 12
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Q. If the directrix of a parabola is given by the equation y = -p, what is the equation of the parabola?
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A.
y^2 = 4px
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B.
x^2 = 4py
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C.
y^2 = -4px
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D.
x^2 = -4py
Solution
The equation of a parabola with directrix y = -p is y^2 = -4px.
Correct Answer: C — y^2 = -4px
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Q. If the eccentricity of a parabola is e, what is the value of e?
Solution
The eccentricity of a parabola is always equal to 1.
Correct Answer: B — 1
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Q. If the equation of a parabola is given by y^2 = 12x, what is the value of 'p'?
Solution
In the equation y^2 = 4px, p = 3, hence the value of 'p' is 3.
Correct Answer: B — 6
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Q. If the equation of an ellipse is 9x^2 + 16y^2 = 144, what are the lengths of the semi-major and semi-minor axes?
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A.
3, 4
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B.
4, 3
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C.
6, 8
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D.
8, 6
Solution
Rewriting the equation in standard form gives (x^2/16) + (y^2/9) = 1, so semi-major axis a = 4 and semi-minor axis b = 3.
Correct Answer: A — 3, 4
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Q. If the family of curves is given by y = k/x, what type of curves does it represent?
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A.
Linear
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B.
Hyperbolic
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C.
Circular
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D.
Exponential
Solution
The equation y = k/x represents a family of hyperbolas.
Correct Answer: B — Hyperbolic
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Q. If the lengths of the semi-major and semi-minor axes of an ellipse are 5 and 3 respectively, what is the distance between the foci?
Solution
The distance between the foci is given by 2c, where c = √(a^2 - b^2). Here, c = √(5^2 - 3^2) = √16 = 4, so the distance is 2c = 8.
Correct Answer: A — 4
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Q. If the line 3x + 4y = 12 intersects the x-axis, what is the point of intersection?
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A.
(4, 0)
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B.
(0, 3)
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C.
(0, 4)
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D.
(3, 0)
Solution
Set y = 0 in the equation: 3x = 12 => x = 4. The point is (4, 0).
Correct Answer: A — (4, 0)
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