Q. What is the equation of the line with slope 3 that passes through the point (1, 2)?
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A.
y = 3x + 2
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B.
y = 3x - 1
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C.
y - 2 = 3(x - 1)
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D.
y = 2x + 1
Solution
Using point-slope form: y - y1 = m(x - x1) => y - 2 = 3(x - 1).
Correct Answer: C — y - 2 = 3(x - 1)
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Q. What is the equation of the line with slope 5 that passes through the point (1, 2)?
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A.
y = 5x - 3
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B.
y = 5x + 2
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C.
y = 5x + 1
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D.
y = 5x - 2
Solution
Using point-slope form: y - 2 = 5(x - 1) gives y = 5x - 3.
Correct Answer: C — y = 5x + 1
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Q. What is the equation of the parabola that opens upwards with vertex at the origin and passes through the point (2, 8)?
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A.
y = 2x^2
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B.
y = x^2
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C.
y = 4x^2
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D.
y = 8x^2
Solution
The vertex form of a parabola is y = ax^2. Since it passes through (2, 8), we have 8 = a(2^2) => 8 = 4a => a = 2. Thus, the equation is y = 4x^2.
Correct Answer: C — y = 4x^2
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Q. What is the equation of the parabola with focus at (0, 2) and directrix y = -2?
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A.
x^2 = 8y
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B.
x^2 = -8y
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C.
y^2 = 8x
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D.
y^2 = -8x
Solution
The distance from the focus to the directrix is 4, so the equation is y = (1/4)(x - 0)^2 + 0, which simplifies to x^2 = 8y.
Correct Answer: A — x^2 = 8y
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Q. What is the equation of the parabola with focus at (0, 3) and directrix y = -3?
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A.
x^2 = 12y
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B.
y^2 = 12x
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C.
y = 3x^2
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D.
x = 3y^2
Solution
The distance from the focus to the directrix is 6, so p = 3. The equation is y^2 = 4px = 12y.
Correct Answer: A — x^2 = 12y
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Q. What is the equation of the tangent line to the curve y = x^2 + 2x at the point where x = 1?
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A.
y = 3x - 2
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B.
y = 2x + 1
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C.
y = 2x + 2
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D.
y = x + 3
Solution
f'(x) = 2x + 2. At x = 1, f'(1) = 4. The point is (1, 3). The tangent line is y - 3 = 4(x - 1) => y = 4x - 1.
Correct Answer: A — y = 3x - 2
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Q. What is the equation of the tangent line to the curve y = x^2 + 2x at the point (1, 3)?
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A.
y = 2x + 1
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B.
y = 2x + 2
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C.
y = 3x
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D.
y = x + 2
Solution
f'(x) = 2x + 2. At x = 1, f'(1) = 4. The tangent line is y - 3 = 4(x - 1) => y = 4x - 1.
Correct Answer: A — y = 2x + 1
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Q. What is the family of curves represented by the equation xy = c?
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A.
Hyperbolas
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B.
Parabolas
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C.
Ellipses
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D.
Circles
Solution
The equation xy = c represents a family of hyperbolas with varying constant c.
Correct Answer: A — Hyperbolas
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Q. What is the family of curves represented by the equation x^2/a^2 + y^2/b^2 = 1?
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A.
Ellipses
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B.
Hyperbolas
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C.
Parabolas
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D.
Circles
Solution
The equation x^2/a^2 + y^2/b^2 = 1 represents a family of ellipses with semi-major axis a and semi-minor axis b.
Correct Answer: A — Ellipses
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Q. What is the family of curves represented by the equation y = a sin(bx + c)?
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A.
Sine waves
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B.
Cosine waves
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C.
Linear functions
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D.
Quadratic functions
Solution
The equation y = a sin(bx + c) represents a family of sine waves with amplitude a and phase shift c.
Correct Answer: A — Sine waves
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Q. What is 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 with varying growth rates
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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 where 'k' determines the growth rate.
Correct Answer: B — Exponential functions with varying growth rates
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Q. What is the family of curves represented by the equation y = k/x?
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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 = k/x represents a family of hyperbolas with varying asymptotes depending on the value of 'k'.
Correct Answer: B — Hyperbolas
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Q. What is the first derivative of f(x) = e^(2x)?
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A.
2e^(2x)
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B.
e^(2x)
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C.
e^(x)
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D.
2x*e^(2x)
Solution
Using the chain rule, f'(x) = 2e^(2x).
Correct Answer: A — 2e^(2x)
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Q. What is the general form of the family of curves for the equation x^2 + y^2 = r^2?
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A.
Ellipses
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B.
Hyperbolas
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C.
Circles
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D.
Parabolas
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. What is the general form of the family of curves represented by the equation x^2 + y^2 = r^2?
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A.
Circles with radius r
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B.
Ellipses with semi-major axis r
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C.
Hyperbolas with transverse axis r
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D.
Straight lines with slope r
Solution
The equation x^2 + y^2 = r^2 represents a family of circles with radius r centered at the origin.
Correct Answer: A — Circles with radius r
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Q. What is the general form of the family of curves represented by the equation y = mx^2 + c?
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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 = mx^2 + c represents a family of parabolas that open upwards or downwards depending on the sign of m.
Correct Answer: A — Parabolas
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Q. What is the general form of the family of curves represented by y^2 = 4ax?
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A.
Parabolas opening to the right
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B.
Circles with varying centers
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C.
Ellipses with varying foci
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D.
Hyperbolas with varying asymptotes
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 opening to the right
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Q. What is the general solution of the differential equation dy/dx = 3y?
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A.
y = Ce^(3x)
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B.
y = Ce^(-3x)
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C.
y = 3x + C
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D.
y = Cx^3
Solution
The differential equation is separable. Integrating both sides gives ln|y| = 3x + C, hence y = Ce^(3x).
Correct Answer: A — y = Ce^(3x)
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Q. What is the inradius of a triangle with sides 7 cm, 8 cm, and 9 cm?
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A.
3 cm
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B.
4 cm
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C.
5 cm
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D.
6 cm
Solution
Using the formula r = A/s, where A is the area and s is the semi-perimeter. Area = 26 cm², s = 12 cm, so r = 26/12 = 4 cm.
Correct Answer: B — 4 cm
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Q. What is the integral of cos(x)dx?
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A.
sin(x) + C
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B.
cos(x) + C
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C.
-sin(x) + C
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D.
-cos(x) + C
Solution
The integral of cos(x) is sin(x) + C.
Correct Answer: A — sin(x) + C
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Q. What is the integral of f(x) = 2x from 0 to 3?
Solution
∫(2x)dx from 0 to 3 = [x^2] from 0 to 3 = 9 - 0 = 9.
Correct Answer: B — 6
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Q. What is the integral of f(x) = 2x?
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A.
x^2 + C
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B.
2x^2 + C
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C.
x^2 + 2C
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D.
2x + C
Solution
∫2x dx = x^2 + C.
Correct Answer: A — x^2 + C
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Q. What is the integral of f(x) = cos(x)?
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A.
sin(x) + C
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B.
-sin(x) + C
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C.
tan(x) + C
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D.
sec(x) + C
Solution
The integral is sin(x) + C.
Correct Answer: A — sin(x) + C
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Q. What is the integrating factor for the equation dy/dx + 2y = 3x?
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A.
e^(2x)
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B.
e^(-2x)
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C.
e^(3x)
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D.
e^(-3x)
Solution
The integrating factor is e^(∫2dx) = e^(2x).
Correct Answer: A — e^(2x)
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Q. What is the integrating factor for the equation dy/dx + 3y = 6x?
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A.
e^(3x)
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B.
e^(-3x)
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C.
e^(6x)
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D.
e^(-6x)
Solution
The integrating factor is e^(∫3dx) = e^(3x).
Correct Answer: A — e^(3x)
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Q. What is the interquartile range (IQR) of the data set {1, 3, 5, 7, 9, 11, 13, 15}?
Solution
Q1 = 4, Q3 = 10. IQR = Q3 - Q1 = 10 - 4 = 6.
Correct Answer: B — 6
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Q. What is the interquartile range (IQR) of the data set {1, 3, 7, 8, 9, 10}?
Solution
IQR = Q3 - Q1; Q1 = 3, Q3 = 8; IQR = 8 - 3 = 5.
Correct Answer: A — 5
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Q. What is the interquartile range (IQR) of the data set {1, 3, 7, 8, 9}?
Solution
Q1 = 3, Q3 = 8. IQR = Q3 - Q1 = 8 - 3 = 5.
Correct Answer: A — 6
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Q. What is the interquartile range (IQR) of the data set: 1, 2, 3, 4, 5, 6, 7, 8, 9?
Solution
Q1 = 3, Q3 = 7; IQR = Q3 - Q1 = 7 - 3 = 4.
Correct Answer: A — 4
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Q. What is the interquartile range (IQR) of the data set: 1, 3, 5, 7, 9, 11, 13?
Solution
Q1 = 3, Q3 = 9. IQR = Q3 - Q1 = 9 - 3 = 6.
Correct Answer: B — 6
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