Q. If a matrix is said to be orthogonal, what property does it have?
A.
All elements are zero
B.
Transpose is equal to its inverse
C.
All diagonal elements are equal
D.
It is a square matrix
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Solution
An orthogonal matrix is defined as a square matrix whose transpose is equal to its inverse.
Correct Answer:
B
— Transpose is equal to its inverse
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Q. If a matrix is said to be skew-symmetric, what must be true about its elements? (2023)
A.
All elements are zero
B.
a_ij = -a_ji
C.
a_ij = a_ji
D.
All diagonal elements are zero
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Solution
A skew-symmetric matrix satisfies the condition a_ij = -a_ji for all i and j.
Correct Answer:
B
— a_ij = -a_ji
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Q. If a matrix is symmetric, what can be said about its elements? (2021)
A.
Aij = Aji
B.
Aij = -Aji
C.
Aij = 0
D.
Aij = Aii
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Solution
A symmetric matrix satisfies the condition Aij = Aji for all i and j.
Correct Answer:
A
— Aij = Aji
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Q. If a matrix is symmetric, what property does it have?
A.
A = A^T
B.
A = -A
C.
A^2 = I
D.
A = 0
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Solution
A symmetric matrix is defined by the property A = A^T, meaning it is equal to its transpose.
Correct Answer:
A
— A = A^T
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Q. If a matrix is symmetric, which of the following must be true? (2021)
A.
A = A^T
B.
A = -A^T
C.
A^2 = I
D.
A^T = 0
Show solution
Solution
A matrix A is symmetric if it is equal to its transpose, i.e., A = A^T.
Correct Answer:
A
— A = A^T
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Q. If a parabola has its vertex at (3, -2) and opens downwards, what is the general form of its equation?
A.
y + 2 = a(x - 3)^2
B.
y + 2 = -a(x - 3)^2
C.
y - 2 = a(x + 3)^2
D.
y - 2 = -a(x + 3)^2
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Solution
For a downward-opening parabola with vertex (h, k), the equation is y - k = -a(x - h)^2. Here, h = 3 and k = -2.
Correct Answer:
B
— y + 2 = -a(x - 3)^2
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Q. If a parabola opens to the left, which of the following is its standard form?
A.
y^2 = -4px
B.
x^2 = -4py
C.
y^2 = 4px
D.
x^2 = 4py
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Solution
The standard form of a parabola that opens to the left is y^2 = -4px.
Correct Answer:
A
— y^2 = -4px
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Q. If a parabola opens to the right and has its vertex at the origin, what is the general form of its equation?
A.
y^2 = 4px
B.
x^2 = 4py
C.
y = mx + c
D.
x = ay^2
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Solution
The general form of a parabola that opens to the right is y^2 = 4px.
Correct Answer:
A
— y^2 = 4px
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Q. If a person standing 30 m away from a building observes the top of the building at an angle of elevation of 60 degrees, what is the height of the building? (2023)
A.
15 m
B.
25 m
C.
30 m
D.
51.96 m
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Solution
Height = distance * tan(60) = 30 * √3 ≈ 30 * 1.732 = 51.96 m.
Correct Answer:
D
— 51.96 m
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Q. If a square has a perimeter of 32 cm, what is the length of one side? (2023)
A.
8 cm
B.
10 cm
C.
12 cm
D.
6 cm
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Solution
The perimeter P of a square is given by P = 4 * side. If P = 32 cm, then side = 32/4 = 8 cm.
Correct Answer:
A
— 8 cm
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Q. If a student is selected at random from a group of 15 students, where 9 are girls and 6 are boys, what is the probability that the selected student is a boy? (2021)
A.
1/3
B.
2/5
C.
1/2
D.
3/5
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Solution
Probability of selecting a boy = Number of boys / Total students = 6/15 = 2/5.
Correct Answer:
B
— 2/5
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Q. If a student is selected at random from a group of 15 students, where 9 are girls and 6 are boys, what is the probability that the student is a boy? (2023)
A.
2/5
B.
1/3
C.
1/2
D.
3/5
Show solution
Solution
Probability of selecting a boy = Number of boys / Total students = 6/15 = 2/5.
Correct Answer:
B
— 1/3
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Q. If a student is selected at random from a group of 40 students, where 25 are girls and 15 are boys, what is the probability that the selected student is a boy?
A.
3/8
B.
1/2
C.
3/5
D.
1/3
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Solution
Probability of selecting a boy = Number of boys / Total students = 15/40 = 3/8.
Correct Answer:
A
— 3/8
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Q. If a transversal intersects two parallel lines, how many pairs of corresponding angles are formed? (2023)
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Solution
A transversal intersecting two parallel lines forms 4 pairs of corresponding angles.
Correct Answer:
D
— 4
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Q. If a transversal intersects two parallel lines, what can be said about the same-side interior angles? (2023)
A.
They are equal
B.
They are supplementary
C.
They are complementary
D.
They are unequal
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Solution
When a transversal intersects two parallel lines, the same-side interior angles are supplementary, meaning they add up to 180 degrees.
Correct Answer:
B
— They are supplementary
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Q. If B = [[1, 0, 2], [0, 1, 3], [0, 0, 1]], what is the rank of matrix B?
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Solution
The rank of matrix B is the number of non-zero rows in its row echelon form, which is 3.
Correct Answer:
C
— 3
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Q. If B = [[2, 3], [5, 7]], what is the value of det(B)? (2020)
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Solution
The determinant of B is calculated as (2*7) - (3*5) = 14 - 15 = -1.
Correct Answer:
A
— -1
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Q. If C = [[1, 0, 0], [0, 1, 0], [0, 0, 0]], what is the determinant of C? (2022)
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Solution
The determinant of C is 0 because it has a row of zeros.
Correct Answer:
B
— 0
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Q. If C = [[1, 0, 2], [-1, 3, 1], [2, 1, 0]], find det(C).
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Solution
Using the determinant formula for 3x3 matrices, det(C) = 1(3*0 - 1*1) - 0 + 2(-1*1 - 3*2) = 0 - 0 - 12 = -12.
Correct Answer:
A
— -9
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Q. If C = [[1, 0, 2], [0, 1, 3], [0, 0, 1]], what is det(C)? (2019)
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Solution
The determinant of an upper triangular matrix is the product of its diagonal elements. Here, det(C) = 1 * 1 * 1 = 1.
Correct Answer:
B
— 1
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Q. If C = [[1, 2], [3, 5]], find C^2.
A.
[[7, 14], [21, 35]]
B.
[[11, 28], [15, 35]]
C.
[[11, 16], [18, 35]]
D.
[[11, 16], [15, 25]]
Show solution
Solution
C^2 = C * C = [[1*1 + 2*3, 1*2 + 2*5], [3*1 + 5*3, 3*2 + 5*5]] = [[11, 16], [18, 35]].
Correct Answer:
C
— [[11, 16], [18, 35]]
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Q. If cos A = 5/13, what is the value of tan A?
A.
12/5
B.
5/12
C.
13/5
D.
5/13
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Solution
Using the identity tan A = sin A / cos A, we find sin A = √(1 - cos²A) = √(1 - (5/13)²) = 12/13. Thus, tan A = (12/13) / (5/13) = 12/5.
Correct Answer:
A
— 12/5
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Q. If cos B = 0.6, what is the value of sin B?
A.
0.8
B.
0.6
C.
0.4
D.
0.2
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Solution
Using the Pythagorean identity, sin B = √(1 - cos²B) = √(1 - 0.6²) = √(1 - 0.36) = √(0.64) = 0.8.
Correct Answer:
A
— 0.8
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Q. If cos B = 1/2, what is the value of angle B in degrees?
A.
30°
B.
60°
C.
90°
D.
120°
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Solution
Cosine of 60° is 1/2, hence angle B = 60°.
Correct Answer:
B
— 60°
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Q. If cos B = 5/13, what is the value of sin B?
A.
12/13
B.
5/13
C.
13/5
D.
3/5
Show solution
Solution
Using the Pythagorean identity, sin B = √(1 - cos²B) = √(1 - (5/13)²) = √(1 - 25/169) = √(144/169) = 12/13.
Correct Answer:
A
— 12/13
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Q. If cos C = 0.5, what is the value of angle C in degrees?
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Solution
The angle C for which cos C = 0.5 is 60°.
Correct Answer:
B
— 60
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Q. If cot A = 1, what is the value of A? (2022)
A.
45°
B.
90°
C.
60°
D.
30°
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Solution
Cotangent is 1 when the angle A is 45°, as cot A = cos A/sin A = 1 when both are equal.
Correct Answer:
A
— 45°
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Q. If cot A = 3/4, what is the value of sin A?
A.
3/5
B.
4/5
C.
5/3
D.
5/4
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Solution
cot A = cos A/sin A = 3/4 implies sin A = 4/5 using the identity sin²A + cos²A = 1.
Correct Answer:
A
— 3/5
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Q. If cot D = 3/4, what is the value of sin D?
A.
3/5
B.
4/5
C.
5/3
D.
5/4
Show solution
Solution
cot D = 3/4 implies tan D = 4/3. Therefore, sin D = 4/5 (using the identity sin²D + cos²D = 1).
Correct Answer:
A
— 3/5
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Q. If cot θ = 1, what is the value of θ?
A.
0°
B.
45°
C.
90°
D.
180°
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Solution
cot θ = 1 implies tan θ = 1, which occurs at θ = 45°.
Correct Answer:
B
— 45°
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