Q. The coordinates of the centroid of a triangle with vertices at (0, 0), (6, 0), and (3, 6) are:
A.
(3, 2)
B.
(3, 3)
C.
(2, 3)
D.
(0, 0)
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Solution
Centroid = ((x1+x2+x3)/3, (y1+y2+y3)/3) = (9/3, 6/3) = (3, 2).
Correct Answer: B — (3, 3)
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Q. The coordinates of the centroid of a triangle with vertices at (2, 3), (4, 5), and (6, 1) are:
A.
(4, 3)
B.
(4, 4)
C.
(3, 3)
D.
(5, 3)
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Solution
Centroid = ((2+4+6)/3, (3+5+1)/3) = (4, 3).
Correct Answer: A — (4, 3)
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Q. The coordinates of the centroid of the triangle with vertices (0, 0), (6, 0), and (3, 6) are:
A.
(3, 2)
B.
(2, 3)
C.
(3, 3)
D.
(0, 0)
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Solution
Centroid = ((0+6+3)/3, (0+0+6)/3) = (3, 2).
Correct Answer: A — (3, 2)
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Q. The coordinates of the centroid of the triangle with vertices (2, 3), (4, 5), and (6, 7) are:
A.
(4, 5)
B.
(3, 4)
C.
(5, 6)
D.
(6, 5)
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Solution
Centroid = ((2+4+6)/3, (3+5+7)/3) = (4, 5).
Correct Answer: B — (3, 4)
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Q. The critical points of the function f(x) = x^3 - 6x^2 + 9x + 1 are:
A.
x = 1, 3
B.
x = 0, 2
C.
x = 2, 4
D.
x = 1, 2
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Solution
Finding f'(x) = 3x^2 - 12x + 9 and solving gives critical points at x = 1 and x = 3.
Correct Answer: A — x = 1, 3
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Q. The distance from the point (1, 2) to the line 2x + 3y - 6 = 0 is:
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Solution
Distance = |2(1) + 3(2) - 6| / √(2² + 3²) = |2 + 6 - 6| / √13 = 2/√13.
Correct Answer: B — 2
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Q. The distance from the point (3, 4) to the line 2x + 3y - 6 = 0 is:
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Solution
Distance = |2(3) + 3(4) - 6| / √(2² + 3²) = |6 + 12 - 6| / √13 = 12/√13.
Correct Answer: B — 2
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Q. The eccentricity of an ellipse is defined as e = c/a. If a = 10 and c = 6, what is the eccentricity?
A.
0.6
B.
0.8
C.
0.4
D.
0.5
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Solution
Eccentricity e = c/a = 6/10 = 0.6.
Correct Answer: B — 0.8
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Q. The equation of a line parallel to y = 2x + 3 and passing through (1, 1) is?
A.
y = 2x - 1
B.
y = 2x + 1
C.
y = 2x + 3
D.
y = 2x - 3
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Solution
Parallel lines have the same slope. Using point-slope form: y - 1 = 2(x - 1) => y = 2x - 1.
Correct Answer: A — y = 2x - 1
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Q. The equation of a line passing through (1, 2) and (3, 6) is:
A.
y = 2x
B.
y = 3x - 1
C.
y = x + 1
D.
y = 4x - 2
Show solution
Solution
Slope = (6-2)/(3-1) = 2. Using point-slope form: y - 2 = 2(x - 1) => y = 2x.
Correct Answer: A — y = 2x
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Q. The equation of a line passing through the points (1, 2) and (3, 6) is:
A.
y = 2x
B.
y = 3x - 1
C.
y = x + 1
D.
y = 4x - 2
Show solution
Solution
Slope = (6-2)/(3-1) = 2. Using point-slope form: y - 2 = 2(x - 1) => y = 2x.
Correct Answer: A — y = 2x
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Q. The equation of a parabola is given by x^2 = 16y. What is the length of the latus rectum?
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Solution
The length of the latus rectum for the parabola x^2 = 4py is given by 4p. Here, 4p = 16, so p = 4. Thus, the length of the latus rectum is 4p = 16.
Correct Answer: B — 8
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Q. The equation of an ellipse is given by 4x^2 + 9y^2 = 36. What is the eccentricity of the ellipse?
A.
0.5
B.
0.6
C.
0.7
D.
0.8
Show solution
Solution
Rewriting gives x^2/9 + y^2/4 = 1. Here, a^2 = 9, b^2 = 4, c = √(a^2 - b^2) = √(9 - 4) = √5. Eccentricity e = c/a = √5/3 ≈ 0.6.
Correct Answer: B — 0.6
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Q. The equation of an ellipse with foci at (0, ±c) and major axis along the y-axis is given by?
A.
x^2/a^2 + y^2/b^2 = 1
B.
y^2/a^2 + x^2/b^2 = 1
C.
x^2/b^2 + y^2/a^2 = 1
D.
y^2/b^2 + x^2/a^2 = 1
Show solution
Solution
The equation of an ellipse with foci at (0, ±c) and major axis along the y-axis is y^2/a^2 + x^2/b^2 = 1.
Correct Answer: B — y^2/a^2 + x^2/b^2 = 1
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Q. The equation of the directrix of the parabola y^2 = 8x is?
A.
x = -2
B.
x = 2
C.
y = -4
D.
y = 4
Show solution
Solution
The directrix of the parabola y^2 = 8x is given by x = -2.
Correct Answer: A — x = -2
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Q. The equation of the line passing through (1, 2) and (3, 6) is:
A.
y = 2x
B.
y = 3x - 1
C.
y = x + 1
D.
y = 4x - 2
Show solution
Solution
Slope = (6-2)/(3-1) = 2. Using point-slope form: y - 2 = 2(x - 1) => y = 2x.
Correct Answer: A — y = 2x
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Q. The equation of the line passing through the points (1, 2) and (3, 6) is:
A.
y = 2x
B.
y = 3x - 1
C.
y = 4x - 2
D.
y = x + 1
Show solution
Solution
Slope = (6-2)/(3-1) = 2. Using point-slope form: y - 2 = 2(x - 1) => y = 2x.
Correct Answer: A — y = 2x
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Q. The equation of the pair of lines through the origin is given by y = mx. If m1 and m2 are the slopes, what is the condition for them to be perpendicular?
A.
m1 + m2 = 0
B.
m1 * m2 = 1
C.
m1 - m2 = 0
D.
m1 * m2 = -1
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Solution
For two lines to be perpendicular, the product of their slopes must equal -1.
Correct Answer: D — m1 * m2 = -1
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Q. The equation of the pair of lines through the origin with slopes m1 and m2 is given by:
A.
y = mx
B.
y^2 = mx
C.
x^2 + y^2 = 0
D.
x^2 - 2mxy + y^2 = 0
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Solution
The correct form of the equation representing the lines through the origin is x^2 - 2mxy + y^2 = 0.
Correct Answer: D — x^2 - 2mxy + y^2 = 0
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Q. The equation of the pair of lines through the origin with slopes m1 and m2 is:
A.
y = m1x + m2x
B.
y = (m1 + m2)x
C.
y = m1x - m2x
D.
y = m1x * m2x
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Solution
The equation of the lines can be expressed as y = (m1 + m2)x, representing the sum of the slopes.
Correct Answer: B — y = (m1 + m2)x
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Q. The equation of the tangent line to the curve y = x^2 at the point (2, 4) is:
A.
y = 2x
B.
y = 4x - 4
C.
y = 4x - 8
D.
y = x + 2
Show solution
Solution
The slope of the tangent at x = 2 is f'(x) = 2x, so f'(2) = 4. The equation of the tangent line is y - 4 = 4(x - 2), which simplifies to y = 4x - 8.
Correct Answer: C — y = 4x - 8
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Q. The equation of the tangent to the curve y = x^2 at the point (2, 4) is:
A.
y = 2x - 4
B.
y = 2x
C.
y = x + 2
D.
y = x^2 - 2
Show solution
Solution
The derivative f'(x) = 2x. At x = 2, f'(2) = 4. The equation of the tangent line is y - 4 = 4(x - 2), which simplifies to y = 2x - 4.
Correct Answer: A — y = 2x - 4
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Q. The equation x^2 + 2x + 1 = 0 can be factored as:
A.
(x + 1)(x + 1)
B.
(x - 1)(x - 1)
C.
(x + 2)(x + 1)
D.
(x - 2)(x - 1)
Show solution
Solution
This is a perfect square: (x + 1)^2 = 0.
Correct Answer: A — (x + 1)(x + 1)
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Q. The equation x^2 + 4x + 4 = 0 has:
A.
Two distinct roots
B.
One repeated root
C.
No real roots
D.
None of these
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Solution
The discriminant is 0, indicating one repeated root.
Correct Answer: B — One repeated root
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Q. The equation x^2 - 2x + 1 = 0 has:
A.
Two distinct roots
B.
One repeated root
C.
No real roots
D.
Infinitely many roots
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Solution
The discriminant is 0, indicating one repeated root.
Correct Answer: B — One repeated root
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Q. The equation x^2 - 6x + k = 0 has roots that are both positive. What is the range of k?
A.
k < 0
B.
k > 0
C.
k > 9
D.
k < 9
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Solution
For both roots to be positive, k must be greater than the square of half the coefficient of x: k > (6/2)^2 = 9.
Correct Answer: C — k > 9
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Q. The family of curves defined by the equation x^2 + y^2 = r^2 represents:
A.
Ellipses
B.
Hyperbolas
C.
Circles
D.
Parabolas
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Solution
The equation x^2 + y^2 = r^2 represents a circle with radius r.
Correct Answer: C — Circles
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Q. The family of curves defined by the equation y = a(x - h)^2 + k represents which type of function?
A.
Linear
B.
Quadratic
C.
Cubic
D.
Rational
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Solution
The equation y = a(x - h)^2 + k represents a quadratic function in vertex form.
Correct Answer: B — Quadratic
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Q. The family of curves defined by the equation y = a(x - h)^2 + k represents:
A.
Parabolas
B.
Circles
C.
Ellipses
D.
Hyperbolas
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Solution
The equation y = a(x - h)^2 + k represents a family of parabolas with vertex (h, k).
Correct Answer: A — Parabolas
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Q. The family of curves defined by the equation y = ax^2 + bx + c is known as:
A.
Linear functions
B.
Quadratic functions
C.
Polynomial functions
D.
Rational functions
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Solution
The equation y = ax^2 + bx + c represents a quadratic function.
Correct Answer: B — Quadratic functions
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