Q. Which identity represents the expansion of (a + b + c)²?
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A.
a² + b² + c² + 2ab + 2bc + 2ca
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B.
a² + b² + c² + 3abc
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C.
a² + b² + c²
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D.
2a² + 2b² + 2c²
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 functions is continuous at all points?
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A.
f(x) = 1/x
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B.
f(x) = x^3
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C.
f(x) = sqrt(x)
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D.
f(x) = tan(x)
Solution
f(x) = x^3 is a polynomial function, which is continuous everywhere.
Correct Answer: B — f(x) = x^3
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Q. Which of the following functions is continuous at x = 0?
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A.
f(x) = 1/x
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B.
f(x) = e^x
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C.
f(x) = tan(x)
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D.
f(x) = 1/(x^2 + 1)
Solution
The function f(x) = e^x is continuous everywhere, including at x = 0.
Correct Answer: B — f(x) = e^x
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Q. Which of the following functions is continuous on the interval [0, 1]?
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A.
f(x) = 1/x
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B.
f(x) = x^3
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C.
f(x) = sqrt(x)
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D.
f(x) = 1/(x-1)
Solution
f(x) = x^3 is a polynomial function and is continuous on the interval [0, 1].
Correct Answer: B — f(x) = x^3
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Q. Which of the following is a characteristic of a diagonal matrix? (2022)
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A.
All elements are zero
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B.
Only diagonal elements are non-zero
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C.
All elements are equal
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D.
It is always square
Solution
A diagonal matrix has non-zero elements only on its diagonal, while all off-diagonal elements are zero.
Correct Answer: B — Only diagonal elements are non-zero
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Q. Which of the following is a singular matrix? (2019)
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A.
[[1, 2], [3, 4]]
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B.
[[2, 3], [4, 6]]
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C.
[[5, 0], [0, 5]]
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D.
[[0, 1], [1, 0]]
Solution
A singular matrix is one that does not have an inverse, which occurs when its determinant is zero. The matrix [[2, 3], [4, 6]] is singular.
Correct Answer: B — [[2, 3], [4, 6]]
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Q. Which of the following is a square matrix? (2020)
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A.
2x3
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B.
3x3
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C.
4x2
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D.
5x1
Solution
A square matrix is one where the number of rows is equal to the number of columns. Among the options, only 3x3 has equal rows and columns.
Correct Answer: B — 3x3
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Q. Which of the following is a symmetric matrix? (2020)
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A.
[[1, 2], [3, 4]]
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B.
[[1, 2], [2, 1]]
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C.
[[1, 0], [0, 1]]
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D.
[[0, 1], [1, 0]]
Solution
A symmetric matrix is one that is equal to its transpose. The matrix [[1, 2], [2, 1]] is symmetric.
Correct Answer: B — [[1, 2], [2, 1]]
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Q. Which of the following is an identity matrix? (2019)
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A.
[[1, 0], [0, 1]]
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B.
[[1, 1], [1, 1]]
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C.
[[0, 1], [1, 0]]
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D.
[[2, 0], [0, 2]]
Solution
An identity matrix is a square matrix with ones on the diagonal and zeros 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 is an irrational number?
Solution
√4 = 2, 0.25 = 1/4, and 1 is rational. √2 is irrational.
Correct Answer: B — √2
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Q. Which of the following is equal to 0.125?
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A.
1/8
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B.
1/4
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C.
1/2
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D.
1/16
Solution
0.125 is equal to 1/8.
Correct Answer: A — 1/8
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Q. Which of the following is equal to log_10(0.01)? (2019)
Solution
log_10(0.01) = log_10(10^-2) = -2.
Correct Answer: B — -2
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Q. Which of the following is equivalent to log_2(32)?
Solution
log_2(32) = log_2(2^5) = 5.
Correct Answer: B — 5
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Q. Which of the following is not a rational number? (2023)
Solution
√3 is an irrational number, while 1/2, 0, and 3 are rational.
Correct Answer: C — √3
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Q. Which of the following is the conjugate of the complex number z = 5 - 2i? (2021)
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A.
5 + 2i
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B.
5 - 2i
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C.
-5 + 2i
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D.
-5 - 2i
Solution
The conjugate of z = 5 - 2i is z* = 5 + 2i.
Correct Answer: A — 5 + 2i
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Q. Which of the following is the conjugate of the complex number z = 5 - 3i?
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A.
5 + 3i
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B.
5 - 3i
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C.
3 - 5i
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D.
3 + 5i
Solution
The conjugate of a complex number z = a + bi is given by z* = a - bi. Here, the conjugate of 5 - 3i is 5 + 3i.
Correct Answer: A — 5 + 3i
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Q. Which of the following is the correct expansion of (x - y)³?
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A.
x³ - 3x²y + 3xy² - y³
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B.
x³ - y³
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C.
x³ - 3xy² + 3x²y - y³
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D.
3x²y - 3xy²
Solution
(x - y)³ = x³ - 3x²y + 3xy² - y³ by the binomial theorem.
Correct Answer: A — x³ - 3x²y + 3xy² - y³
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Q. Which of the following is the correct factorization of a² - b²?
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A.
(a + b)(a - b)
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B.
(a - b)(a - b)
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C.
(a + b)(a + b)
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D.
a² - 2ab + b²
Solution
a² - b² = (a + b)(a - b) by the difference of squares identity.
Correct Answer: A — (a + b)(a - b)
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Q. Which of the following is the correct identity for (a - b)²?
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A.
a² - 2ab + b²
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B.
a² + 2ab + b²
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C.
a² - b²
-
D.
2a² - 2b²
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?
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A.
x³ + y³ + z³ + 3(x + y)(y + z)(z + x)
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B.
x³ + y³ + z³ + 3xyz
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C.
x³ + y³ + z³ + 3xy + 3yz + 3zx
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D.
x³ + y³ + z³ - 3xyz
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?
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A.
x³ + y³ + 3xy(x + y)
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B.
x³ + y³ - 3xy(x + y)
-
C.
x³ + y³ + 3xy²
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D.
x³ - y³
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?
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A.
x² + 4
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B.
x² + 4x + 4
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C.
x² + 2x + 2
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D.
x² + 2
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)
Solution
The smallest positive integer is 1.
Correct Answer: B — 1
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Q. Which of the following is true for log_a(b) + log_a(c)?
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A.
log_a(bc)
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B.
log_a(b/c)
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C.
log_a(b-c)
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D.
log_a(b+c)
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)?
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A.
log_a(b) + log_a(c)
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B.
log_a(b) - log_a(c)
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C.
log_a(bc) = log_a(b) * log_a(c)
-
D.
None of the above
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?
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A.
a³ + b³
-
B.
a³ + 3a²b + 3ab² + b³
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C.
a² + b²
-
D.
3ab
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
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
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B.
4x + 3y - 7 = 0
-
C.
3x + 4y - 5 = 0
-
D.
2x - 3y + 1 = 0
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 zero matrix? (2022)
-
A.
[[0, 0], [0, 0]]
-
B.
[[1, 0], [0, 1]]
-
C.
[[1, 2], [3, 4]]
-
D.
[[0, 1], [1, 0]]
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]]
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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