Q. Which of the following matrices is an orthogonal matrix? (2021)
A.
A matrix whose transpose is equal to its inverse
B.
A matrix with all elements equal
C.
A matrix with only one row
D.
A matrix with all diagonal elements equal
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Solution
An orthogonal matrix is defined as a matrix whose transpose is equal to its inverse.
Correct Answer: A — A matrix whose transpose is equal to its inverse
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Q. Which of the following matrices is not invertible? (2019)
A.
[[1, 2], [3, 4]]
B.
[[0, 1], [0, 0]]
C.
[[5, 6], [7, 8]]
D.
[[9, 10], [11, 12]]
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Solution
A matrix is not invertible if its determinant is zero. The matrix [[0, 1], [0, 0]] has a determinant of 0.
Correct Answer: B — [[0, 1], [0, 0]]
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Q. Which of the following matrices is symmetric? (2023)
A.
A = [[1, 2], [3, 4]]
B.
B = [[1, 2], [2, 1]]
C.
C = [[1, 0], [0, 1]]
D.
D = [[1, 2, 3], [4, 5, 6]]
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Solution
A symmetric matrix is one that is equal to its transpose. Matrix B is symmetric because B = B^T.
Correct Answer: B — B = [[1, 2], [2, 1]]
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Q. Which of the following numbers is a perfect square?
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Solution
25 is a perfect square (5^2).
Correct Answer: C — 25
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Q. Which of the following points lies on the parabola y = x^2 - 4?
A.
(2, 0)
B.
(0, -4)
C.
(1, -3)
D.
(3, 5)
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Solution
Substituting x = 1 into the equation gives y = 1^2 - 4 = -3, so the point (1, -3) lies on the parabola.
Correct Answer: C — (1, -3)
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Q. Which of the following points lies on the parabola y^2 = 8x?
A.
(2, 4)
B.
(1, 2)
C.
(4, 4)
D.
(2, 2)
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Solution
To check if a point lies on the parabola, substitute the x-coordinate into the equation. For (2, 4), 4^2 = 16 and 8*2 = 16, so it lies on the parabola.
Correct Answer: A — (2, 4)
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Q. Which of the following represents the identity for (a + b + c)²?
A.
a² + b² + c² + 2ab + 2bc + 2ca
B.
a² + b² + c² + 3abc
C.
a² + b² + c²
D.
a² + b² + c² + ab + ac + bc
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Solution
(a + b + c)² = a² + b² + c² + 2ab + 2bc + 2ca.
Correct Answer: A — a² + b² + c² + 2ab + 2bc + 2ca
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Q. Which of the following statements is true about the function f(x) = 1/(x-1)? (2022)
A.
Continuous at x = 1
B.
Continuous everywhere
C.
Not continuous at x = 1
D.
Continuous at x = 0
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Solution
The function f(x) = 1/(x-1) is not continuous at x = 1 because it is undefined there.
Correct Answer: C — Not continuous at x = 1
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Q. Which of the following statements is true about the function f(x) = 1/(x-3)?
A.
Continuous at x = 3
B.
Continuous everywhere
C.
Not continuous at x = 3
D.
Continuous at x = 0
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Solution
The function f(x) = 1/(x-3) is not defined at x = 3, hence it is not continuous at that point.
Correct Answer: C — Not continuous at x = 3
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Q. Which of the following statements is true about the function f(x) = |x|?
A.
Continuous everywhere
B.
Discontinuous at x = 0
C.
Continuous only at x = 1
D.
Discontinuous everywhere
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Solution
The function f(x) = |x| is continuous everywhere, including at x = 0.
Correct Answer: A — Continuous everywhere
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Q. Which of the following statements is true regarding the function f(x) = 1/(x-3)?
A.
Continuous at x = 3
B.
Discontinuous at x = 3
C.
Continuous everywhere
D.
Discontinuous everywhere
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Solution
The function f(x) = 1/(x-3) is discontinuous at x = 3 because it is undefined at that point.
Correct Answer: B — Discontinuous at x = 3
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Q. Which of the following statements is true regarding the function f(x) = |x|?
A.
Continuous everywhere
B.
Discontinuous at x = 0
C.
Continuous only for x > 0
D.
Discontinuous for x < 0
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Solution
The function f(x) = |x| is continuous everywhere, including at x = 0.
Correct Answer: A — Continuous everywhere
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Q. Which of the following vectors is orthogonal to the vector A = 2i + 3j?
A.
3i - 2j
B.
-3i + 2j
C.
2i + 3j
D.
i + j
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Solution
A vector is orthogonal if A · B = 0. For B = 3i - 2j, A · B = 6 - 6 = 0.
Correct Answer: A — 3i - 2j
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