Q. What is the relationship between the diagonals of a rectangle? (2023)
A.
They are equal
B.
They are perpendicular
C.
They bisect each other at right angles
D.
They are unequal
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Solution
In a rectangle, the diagonals are equal in length.
Correct Answer:
A
— They are equal
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Q. What is the relationship between the exterior angle and the two opposite interior angles of a triangle? (2019)
A.
They are equal
B.
The exterior angle is greater
C.
The exterior angle is less
D.
They are complementary
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Solution
The exterior angle of a triangle is equal to the sum of the two opposite interior angles.
Correct Answer:
B
— The exterior angle is greater
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Q. What is the relationship between the sum of the interior angles and the exterior angles of a polygon?
A.
Interior angles sum to 180 degrees
B.
Exterior angles sum to 360 degrees
C.
Both sum to 360 degrees
D.
Interior angles sum to 360 degrees
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Solution
The sum of the exterior angles of any polygon is always 360 degrees, regardless of the number of sides.
Correct Answer:
B
— Exterior angles sum to 360 degrees
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Q. What is the relationship between the sum of the interior angles of a triangle and the angles formed by a transversal cutting through two parallel lines? (2022)
A.
They are equal
B.
They are supplementary
C.
They are complementary
D.
They are unrelated
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Solution
The sum of the interior angles of a triangle is 180 degrees, and the angles formed by a transversal cutting through two parallel lines are supplementary, meaning they add up to 180 degrees.
Correct Answer:
B
— They are supplementary
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Q. What is the result of (2x + 3)(2x - 3)?
A.
4x² - 9
B.
4x² + 9
C.
4x² - 6
D.
6x² - 9
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Solution
(2x + 3)(2x - 3) = 4x² - 9 by the difference of squares.
Correct Answer:
A
— 4x² - 9
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Q. What is the result of (a - b)(a + b)?
A.
a² + b²
B.
a² - b²
C.
2ab
D.
ab
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Solution
(a - b)(a + b) = a² - b², which is the difference of squares.
Correct Answer:
B
— a² - b²
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Q. What is the result of (x + 1)² - (x - 1)²?
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Solution
(x + 1)² - (x - 1)² = [(x² + 2x + 1) - (x² - 2x + 1)] = 4x.
Correct Answer:
B
— 4
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Q. What is the result of (x + y + z)²?
A.
x² + y² + z² + 2(xy + xz + yz)
B.
x² + y² + z²
C.
x² + y² + z² + 3xyz
D.
x² + y² + z² - 2(xy + xz + yz)
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Solution
(x + y + z)² = x² + y² + z² + 2(xy + xz + yz) is the correct expansion.
Correct Answer:
A
— x² + y² + z² + 2(xy + xz + yz)
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Q. What is the result of (x + y)(x - y)?
A.
x² + y²
B.
x² - y²
C.
2xy
D.
xy
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Solution
(x + y)(x - y) = x² - y², which is the difference of squares.
Correct Answer:
B
— x² - y²
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Q. What is the scalar product of A = 1i + 2j + 3k and B = 4i + 5j + 6k?
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Solution
A · B = (1)(4) + (2)(5) + (3)(6) = 4 + 10 + 18 = 32.
Correct Answer:
A
— 32
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Q. What is the scalar product of the vectors A = 1i + 2j + 3k and B = 4i + 5j + 6k?
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Solution
A · B = (1)(4) + (2)(5) + (3)(6) = 4 + 10 + 18 = 32.
Correct Answer:
A
— 32
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Q. What is the scalar projection of vector H = 6i + 8j onto vector I = 3i + 4j?
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Solution
Scalar projection = (H · I) / |I| = (6*3 + 8*4) / √(3^2 + 4^2) = (18 + 32) / 5 = 50 / 5 = 10.
Correct Answer:
C
— 10
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Q. What is the scalar triple product of vectors A = i + j + k, B = 2i + 3j + k, and C = 3i + j + 2k?
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Solution
Scalar triple product = A · (B × C) = |i j k| |1 1 1| |2 3 1| |3 1 2| = 0.
Correct Answer:
D
— 0
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Q. What is the second derivative of f(x) = 2x^4 - 3x^3 + x?
A.
24x^2 - 18x + 1
B.
24x^2 - 9
C.
12x^2 - 9
D.
8x^3 - 9
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Solution
First derivative f'(x) = 8x^3 - 9x^2 + 1. Second derivative f''(x) = 24x^2 - 18x.
Correct Answer:
A
— 24x^2 - 18x + 1
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Q. What is the second derivative of f(x) = 2x^4 - 4x^3 + 3?
A.
24x^2 - 24x
B.
12x^2 - 12
C.
24x^3 - 12x^2
D.
8x^2 - 12
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Solution
First derivative f'(x) = 8x^3 - 12x^2. Second derivative f''(x) = 24x^2 - 24x.
Correct Answer:
A
— 24x^2 - 24x
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Q. What is the second derivative of the function f(x) = 2x^4 - 4x^3 + 3?
A.
24x^2 - 24x
B.
24x^2 - 12
C.
12x^2 - 12
D.
8x^2 - 12
Show solution
Solution
First derivative f'(x) = 8x^3 - 12x^2. Second derivative f''(x) = 24x^2 - 24x.
Correct Answer:
A
— 24x^2 - 24x
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Q. What is the slope of the line perpendicular to the line 4x - 5y + 10 = 0? (2022)
A.
5/4
B.
-4/5
C.
4/5
D.
-5/4
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Solution
The slope of the line is 4/5, so the slope of the perpendicular line is -5/4.
Correct Answer:
B
— -4/5
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Q. What is the slope of the line perpendicular to the line 4x - 5y = 10? (2022)
A.
5/4
B.
-4/5
C.
4/5
D.
-5/4
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Solution
The slope of the line is 4/5, so the slope of the perpendicular line is -5/4.
Correct Answer:
B
— -4/5
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Q. What is the slope of the line represented by the equation 4x - 2y + 8 = 0? (2021)
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Solution
Rearranging gives y = 2x + 4, so slope = 2.
Correct Answer:
C
— -2
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Q. What is the slope of the line represented by the equation y = 3x + 4? (2022)
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Solution
The slope of the line in the equation y = mx + b is m. Here, m = 3.
Correct Answer:
A
— 3
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Q. What is the slope of the tangent line to the curve y = x^2 + 2x at x = 1?
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Solution
The derivative y' = 2x + 2. At x = 1, y' = 2(1) + 2 = 4.
Correct Answer:
A
— 3
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Q. What is the slope of the tangent line to the curve y = x^2 at the point (1,1)? (2023)
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Solution
The derivative y' = 2x; At x = 1, slope = 2(1) = 2.
Correct Answer:
B
— 2
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Q. What is the slope of the tangent line to the curve y = x^2 at the point (2, 4)?
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Solution
The derivative y' = 2x. At x = 2, the slope is y'(2) = 2(2) = 4.
Correct Answer:
A
— 2
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Q. What is the slope of the tangent line to the curve y = x^2 at the point (3, 9)? (2020)
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Solution
The derivative y' = 2x. At x = 3, y' = 2(3) = 6.
Correct Answer:
B
— 6
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Q. What is the slope of the tangent line to the curve y = x^3 at x = 1? (2019)
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Solution
The derivative y' = 3x^2. At x = 1, y' = 3(1^2) = 3.
Correct Answer:
C
— 3
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Q. What is the solution of the differential equation dy/dx = y/x?
A.
y = Cx
B.
y = Cx^2
C.
y = C/x
D.
y = C ln(x)
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Solution
This is separable. Integrating gives y = Cx.
Correct Answer:
A
— y = Cx
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Q. What is the solution to the differential equation dy/dx = 3x^2?
A.
x^3 + C
B.
3x^3 + C
C.
x^2 + C
D.
3x^2 + C
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Solution
Integrating both sides gives y = x^3 + C, where C is the constant of integration.
Correct Answer:
A
— x^3 + C
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Q. What is the solution to the differential equation dy/dx = 6x?
A.
y = 3x^2 + C
B.
y = 6x^2 + C
C.
y = 2x^2 + C
D.
y = 3x + C
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Solution
Integrating gives y = (6/2)x^2 + C = 3x^2 + C.
Correct Answer:
A
— y = 3x^2 + C
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Q. What is the solution to the differential equation dy/dx = xy?
A.
y = Ce^(x^2/2)
B.
y = Ce^(-x^2/2)
C.
y = Cx^2
D.
y = C/x
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Solution
This is separable. Separating and integrating gives y = Ce^(x^2/2).
Correct Answer:
A
— y = Ce^(x^2/2)
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Q. What is the solution to the initial value problem dy/dx = 4y, y(1) = 2?
A.
y = 2e^(4x)
B.
y = 2e^(4x-4)
C.
y = e^(4x)
D.
y = 4e^(x)
Show solution
Solution
The general solution is y = Ce^(4x). Using y(1) = 2, we find C = 2e^(-4).
Correct Answer:
B
— y = 2e^(4x-4)
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