Q. What is the rank of the matrix D = [[1, 2, 3], [4, 5, 6], [7, 8, 9]]? (2019)
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Solution
The rows of D are linearly dependent, hence the rank of D is 2.
Correct Answer: B — 2
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Q. What is the rank of the matrix E = [[1, 2, 3], [0, 0, 0], [4, 5, 6]]?
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Solution
The rank of a matrix is the maximum number of linearly independent row vectors. Here, the first and third rows are independent, while the second row is zero. Thus, the rank is 2.
Correct Answer: B — 2
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Q. What is the rank of the matrix E = [[1, 2, 3], [4, 5, 6], [7, 8, 9]]? (2023)
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Solution
The rank of E is 2 because the rows are linearly dependent (the third row is a linear combination of the first two).
Correct Answer: B — 2
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Q. What is the real part of the complex number z = -2 + 3i? (2020)
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Solution
The real part of z is -2.
Correct Answer: A — -2
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Q. What is the real part of the complex number z = 4 + 5i? (2022)
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Solution
The real part of z = 4 + 5i is 4.
Correct Answer: A — 4
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Q. What is the real part of the complex number z = 4 - 3i? (2023)
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Solution
The real part of z is 4.
Correct Answer: A — 4
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Q. What is the relationship between the angles formed when two lines intersect? (2022)
A.
They are all equal
B.
They are supplementary
C.
They are complementary
D.
They are congruent
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Solution
When two lines intersect, the angles formed are supplementary, meaning they add up to 180 degrees.
Correct Answer: B — They are supplementary
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Q. What is the relationship between the angles of an equilateral triangle? (2023)
A.
All angles are 60 degrees
B.
All angles are 90 degrees
C.
Two angles are equal
D.
No angles are equal
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Solution
In an equilateral triangle, all three angles are equal and measure 60 degrees each.
Correct Answer: A — All angles are 60 degrees
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Q. What is the relationship between the corresponding angles when two parallel lines are cut by a transversal? (2022)
A.
They are equal
B.
They are supplementary
C.
They are complementary
D.
They are unequal
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Solution
When two parallel lines are cut by a transversal, the corresponding angles are equal.
Correct Answer: A — They are equal
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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 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
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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 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 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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