Q. What is the distance between the points (3, 7) and (3, 1)?
Solution
Using the distance formula: d = √[(3 - 3)² + (1 - 7)²] = √[0 + 36] = √36 = 6.
Correct Answer:
A
— 6
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Q. What is the distance between the points (5, 5) and (1, 1)?
Solution
Using the distance formula: d = √[(1 - 5)² + (1 - 5)²] = √[16 + 16] = √32 = 4√2.
Correct Answer:
A
— 4√2
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Q. What is the distance between the points (6, 8) and (6, 2)?
Solution
Using the distance formula: d = √[(6 - 6)² + (2 - 8)²] = √[0 + 36] = √36 = 6.
Correct Answer:
A
— 6
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Q. What is the equation of a circle with center at (2, -3) and radius 4? (2022)
-
A.
(x-2)² + (y+3)² = 16
-
B.
(x+2)² + (y-3)² = 16
-
C.
(x-2)² + (y-3)² = 16
-
D.
(x+2)² + (y+3)² = 16
Solution
The standard form of a circle's equation is (x-h)² + (y-k)² = r². Here, h=2, k=-3, r=4. Thus, (x-2)² + (y+3)² = 16.
Correct Answer:
A
— (x-2)² + (y+3)² = 16
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Q. What is the equation of a circle with center at (3, -2) and radius 4? (2023)
-
A.
(x-3)² + (y+2)² = 16
-
B.
(x+3)² + (y-2)² = 16
-
C.
(x-3)² + (y-2)² = 16
-
D.
(x+3)² + (y+2)² = 16
Solution
The standard equation of a circle is (x-h)² + (y-k)² = r². Here, h=3, k=-2, r=4. Thus, (x-3)² + (y+2)² = 16.
Correct Answer:
A
— (x-3)² + (y+2)² = 16
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Q. What is the equation of a circle with center at (3, -2) and radius 5? (2022)
-
A.
(x-3)² + (y+2)² = 25
-
B.
(x+3)² + (y-2)² = 25
-
C.
(x-3)² + (y-2)² = 25
-
D.
(x+3)² + (y+2)² = 25
Solution
The standard form of a circle's equation is (x-h)² + (y-k)² = r². Here, h=3, k=-2, r=5, so (x-3)² + (y+2)² = 25.
Correct Answer:
A
— (x-3)² + (y+2)² = 25
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Q. What is the equation of the circle with center at (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
Solution
Standard form of a circle: (x-h)² + (y-k)² = r². Here, h=2, k=-3, r=5.
Correct Answer:
A
— (x-2)² + (y+3)² = 25
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Q. What is the equation of the directrix of the parabola x^2 = 12y?
-
A.
y = 3
-
B.
y = -3
-
C.
y = 6
-
D.
y = -6
Solution
The directrix of the parabola x^2 = 4py is given by y = -p. Here, p = 3, so the directrix is y = -3.
Correct Answer:
B
— y = -3
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Q. What is the equation of the line parallel to y = 3x + 2 that passes through the point (4, 1)? (2020)
-
A.
y = 3x - 11
-
B.
y = 3x + 1
-
C.
y = 3x + 2
-
D.
y = 3x - 2
Solution
Since parallel lines have the same slope, the equation is y - 1 = 3(x - 4) which simplifies to y = 3x - 11.
Correct Answer:
A
— y = 3x - 11
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Q. What is the equation of the line parallel to y = 3x + 4 that passes through the point (1, 2)? (2020)
-
A.
y = 3x - 1
-
B.
y = 3x + 1
-
C.
y = 3x + 2
-
D.
y = 3x - 2
Solution
Parallel lines have the same slope. Using point-slope form: y - 2 = 3(x - 1) gives y = 3x - 1.
Correct Answer:
A
— y = 3x - 1
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Q. What is the equation of the line parallel to y = 3x - 4 that passes through the point (2, 1)? (2020)
-
A.
y = 3x - 5
-
B.
y = 3x + 1
-
C.
y = 3x - 1
-
D.
y = 3x + 4
Solution
Since parallel lines have the same slope, the equation is y - 1 = 3(x - 2) which simplifies to y = 3x - 5.
Correct Answer:
C
— y = 3x - 1
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Q. What is the equation of the line that is perpendicular to y = 3x + 2 and passes through the point (1, 1)? (2022)
-
A.
y = -1/3x + 4/3
-
B.
y = 3x - 2
-
C.
y = -3x + 4
-
D.
y = 1/3x + 2/3
Solution
The slope of the perpendicular line is -1/3. Using point-slope form: y - 1 = -1/3(x - 1) gives y = -1/3x + 4/3.
Correct Answer:
A
— y = -1/3x + 4/3
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Q. What is the equation of the line that is perpendicular to y = 3x + 4 and passes through the point (1, 1)? (2022)
-
A.
y - 1 = -1/3(x - 1)
-
B.
y - 1 = 3(x - 1)
-
C.
y - 1 = 3/1(x - 1)
-
D.
y - 1 = -3(x - 1)
Solution
The slope of the perpendicular line is -1/3. Using point-slope form: y - 1 = -1/3(x - 1).
Correct Answer:
A
— y - 1 = -1/3(x - 1)
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Q. What is the equation of the line that passes through the origin and has a slope of -3? (2022)
-
A.
y = -3x
-
B.
y = 3x
-
C.
y = -x/3
-
D.
y = 1/3x
Solution
The equation of a line through the origin with slope m is y = mx. Thus, y = -3x.
Correct Answer:
A
— y = -3x
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Q. What is the equation of the line that passes through the origin and has a slope of -4? (2023)
-
A.
y = -4x
-
B.
y = 4x
-
C.
y = -x/4
-
D.
y = 1/4x
Solution
Using the slope-intercept form y = mx + b, with m = -4 and b = 0, the equation is y = -4x.
Correct Answer:
A
— y = -4x
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Q. What is the expansion of (2x - 3)²? (2022)
-
A.
4x² - 9
-
B.
4x² - 12x + 9
-
C.
4x² + 12x + 9
-
D.
4x² - 6x
Solution
(2x - 3)² = 4x² - 12x + 9 by the square of a binomial.
Correct Answer:
B
— 4x² - 12x + 9
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Q. What is the expansion of (2x - 3y)²?
-
A.
4x² - 9y²
-
B.
4x² - 12xy + 9y²
-
C.
4x² + 9y²
-
D.
4x² + 12xy + 9y²
Solution
(2x - 3y)² = 4x² - 12xy + 9y² by applying the square of a binomial.
Correct Answer:
B
— 4x² - 12xy + 9y²
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Q. What is the expansion of (x + 2)(x - 2)?
-
A.
x² - 4
-
B.
x² + 4
-
C.
2x² - 4
-
D.
x² - 2
Solution
(x + 2)(x - 2) = x² - 2² = x² - 4 by the difference of squares.
Correct Answer:
A
— x² - 4
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Q. What is the first derivative of f(x) = ln(x)? (2019)
-
A.
1/x
-
B.
x
-
C.
ln(x)
-
D.
e^x
Solution
The derivative f'(x) = 1/x.
Correct Answer:
A
— 1/x
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Q. What is the first derivative of f(x) = tan(x)? (2021)
-
A.
sec^2(x)
-
B.
cos^2(x)
-
C.
sin^2(x)
-
D.
csc^2(x)
Solution
The derivative of f(x) = tan(x) is f'(x) = sec^2(x).
Correct Answer:
A
— sec^2(x)
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Q. What is the focus of the parabola defined by the equation y^2 = 20x?
-
A.
(5, 0)
-
B.
(0, 5)
-
C.
(0, 10)
-
D.
(10, 0)
Solution
In the equation y^2 = 4px, we have 4p = 20, thus p = 5. The focus is at (5, 0).
Correct Answer:
A
— (5, 0)
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Q. What is the focus of the parabola given by the equation y^2 = 20x?
-
A.
(5, 0)
-
B.
(0, 5)
-
C.
(0, -5)
-
D.
(10, 0)
Solution
For the parabola y^2 = 4px, here 4p = 20, so p = 5. The focus is at (5, 0).
Correct Answer:
A
— (5, 0)
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Q. What is the general solution of the differential equation dy/dx = 3x^2?
-
A.
y = x^3 + C
-
B.
y = 3x^3 + C
-
C.
y = x^2 + C
-
D.
y = 3x^2 + C
Solution
Integrating both sides gives y = ∫3x^2 dx = x^3 + C.
Correct Answer:
A
— y = x^3 + C
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Q. What is the general solution of the differential equation dy/dx = 4y? (2019)
-
A.
y = Ce^(4x)
-
B.
y = Ce^(x/4)
-
C.
y = 4Ce^x
-
D.
y = 4Ce^(4x)
Solution
The differential equation dy/dx = 4y can be solved using separation of variables, leading to y = Ce^(4x).
Correct Answer:
A
— y = Ce^(4x)
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Q. What is the indefinite integral of e^x? (2020)
-
A.
e^x + C
-
B.
e^x
-
C.
x e^x + C
-
D.
x^2 e^x + C
Solution
The indefinite integral of e^x is e^x + C.
Correct Answer:
A
— e^x + C
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Q. What is the integral of cos(3x) dx?
-
A.
(1/3)sin(3x) + C
-
B.
3sin(3x) + C
-
C.
(1/3)cos(3x) + C
-
D.
sin(3x) + C
Solution
The integral of cos(3x) is (1/3)sin(3x) + C, where C is the constant of integration.
Correct Answer:
A
— (1/3)sin(3x) + C
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Q. What is the integral of e^(2x) dx?
-
A.
(1/2)e^(2x) + C
-
B.
2e^(2x) + C
-
C.
e^(2x) + C
-
D.
(1/2)e^(x) + C
Solution
The integral of e^(2x) is (1/2)e^(2x) + C, where C is the constant of integration.
Correct Answer:
A
— (1/2)e^(2x) + C
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Q. What is the integral of e^(2x)? (2023)
-
A.
(1/2)e^(2x) + C
-
B.
2e^(2x) + C
-
C.
e^(2x) + C
-
D.
(1/2)e^(x) + C
Solution
The integral of e^(2x) is (1/2)e^(2x) + C.
Correct Answer:
A
— (1/2)e^(2x) + C
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Q. What is the integral of e^x dx?
-
A.
e^x + C
-
B.
e^x
-
C.
x e^x + C
-
D.
x e^x
Solution
The integral of e^x is e^x + C, where C is the constant of integration.
Correct Answer:
A
— e^x + C
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Q. What is the integral of f(x) = 1/x? (2023)
-
A.
ln
-
B.
x
-
C.
+ C
-
D.
1/x + C
-
.
x + C
-
.
e^x + C
Solution
The integral of 1/x is ln|x| + C.
Correct Answer:
A
— ln
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