Q. Which of the following is the correct identity for (a - b)²?
A.
a² - 2ab + b²
B.
a² + 2ab + b²
C.
a² - b²
D.
2a² - 2b²
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Solution
(a - b)² = a² - 2ab + b² by the expansion of the square of a binomial.
Correct Answer: A — a² - 2ab + b²
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Q. Which of the following is the correct identity for (x + y + z)³?
A.
x³ + y³ + z³ + 3(x + y)(y + z)(z + x)
B.
x³ + y³ + z³ + 3xyz
C.
x³ + y³ + z³ + 3xy + 3yz + 3zx
D.
x³ + y³ + z³ - 3xyz
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Solution
(x + y + z)³ = x³ + y³ + z³ + 3xyz.
Correct Answer: B — x³ + y³ + z³ + 3xyz
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Q. Which of the following is the correct identity for (x + y)³?
A.
x³ + y³ + 3xy(x + y)
B.
x³ + y³ - 3xy(x + y)
C.
x³ + y³ + 3xy²
D.
x³ - y³
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Solution
(x + y)³ = x³ + y³ + 3xy(x + y) by the binomial expansion.
Correct Answer: A — x³ + y³ + 3xy(x + y)
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Q. Which of the following is the expanded form of (x + 2)²?
A.
x² + 4
B.
x² + 4x + 4
C.
x² + 2x + 2
D.
x² + 2
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Solution
(x + 2)² = x² + 2*2*x + 2² = x² + 4x + 4.
Correct Answer: B — x² + 4x + 4
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Q. Which of the following is the smallest positive integer? (2023)
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Solution
The smallest positive integer is 1.
Correct Answer: B — 1
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Q. Which of the following is true for log_10(0.1)?
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Solution
log_10(0.1) = log_10(10^-1) = -1.
Correct Answer: A — -1
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Q. Which of the following is true for log_a(b) + log_a(c)?
A.
log_a(bc)
B.
log_a(b/c)
C.
log_a(b-c)
D.
log_a(b+c)
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Solution
log_a(b) + log_a(c) = log_a(bc) by the product property of logarithms.
Correct Answer: A — log_a(bc)
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Q. Which of the following is true for log_a(bc)?
A.
log_a(b) + log_a(c)
B.
log_a(b) - log_a(c)
C.
log_a(bc) = log_a(b) * log_a(c)
D.
None of the above
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Solution
log_a(bc) = log_a(b) + log_a(c) by the product rule of logarithms.
Correct Answer: A — log_a(b) + log_a(c)
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Q. Which of the following is true for the identity (a + b)³?
A.
a³ + b³
B.
a³ + 3a²b + 3ab² + b³
C.
a² + b²
D.
3ab
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Solution
(a + b)³ = a³ + 3a²b + 3ab² + b³.
Correct Answer: B — a³ + 3a²b + 3ab² + b³
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Q. Which of the following is true for the identity (x + y)²?
A.
x² + y²
B.
x² + 2xy + y²
C.
x² - 2xy + y²
D.
2xy
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Solution
(x + y)² = x² + 2xy + y².
Correct Answer: B — x² + 2xy + y²
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Q. Which of the following lines is parallel to the line 3x - 4y + 5 = 0?
A.
6x - 8y + 10 = 0
B.
4x + 3y - 7 = 0
C.
3x + 4y - 5 = 0
D.
2x - 3y + 1 = 0
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Solution
Parallel lines have the same slope. The slope of the given line is 3/4, and the line 6x - 8y + 10 = 0 also has the same slope.
Correct Answer: A — 6x - 8y + 10 = 0
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Q. Which of the following matrices is a diagonal matrix? (2023)
A.
[[1, 0], [0, 2]]
B.
[[1, 2], [3, 4]]
C.
[[0, 1], [1, 0]]
D.
[[1, 1], [1, 1]]
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Solution
A diagonal matrix is one where all off-diagonal elements are zero. The matrix [[1, 0], [0, 2]] is a diagonal matrix.
Correct Answer: A — [[1, 0], [0, 2]]
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Q. Which of the following matrices is a zero matrix? (2022)
A.
[[0, 0], [0, 0]]
B.
[[1, 0], [0, 1]]
C.
[[1, 2], [3, 4]]
D.
[[0, 1], [1, 0]]
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Solution
A zero matrix is one where all elements are zero. The matrix [[0, 0], [0, 0]] meets this criterion.
Correct Answer: A — [[0, 0], [0, 0]]
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Q. Which of the following matrices is an identity matrix? (2023)
A.
[[1, 0], [0, 1]]
B.
[[0, 1], [1, 0]]
C.
[[1, 1], [1, 1]]
D.
[[0, 0], [0, 0]]
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Solution
An identity matrix has 1s on the diagonal and 0s elsewhere. The matrix [[1, 0], [0, 1]] fits this definition.
Correct Answer: A — [[1, 0], [0, 1]]
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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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