Q. What is the variance of the numbers: 5, 5, 5, 5? (2023)
Solution
All values are the same, so variance = 0.
Correct Answer: A — 0
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Q. What is the vertex of the parabola given by the equation y = 2x^2 - 4x + 1?
-
A.
(1, -1)
-
B.
(1, 0)
-
C.
(2, 1)
-
D.
(0, 1)
Solution
To find the vertex, use the formula x = -b/(2a). Here, a = 2, b = -4, so x = 1. Substitute x = 1 into the equation to find y = -1.
Correct Answer: A — (1, -1)
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Q. What is the vertex of the quadratic function f(x) = 2x^2 - 8x + 5? (2021)
-
A.
(2, -3)
-
B.
(2, -7)
-
C.
(4, -3)
-
D.
(4, -7)
Solution
The vertex can be found using x = -b/(2a). Here, x = 8/(2*2) = 2. Substituting x = 2 into f(x) gives f(2) = -3, so the vertex is (2, -3).
Correct Answer: A — (2, -3)
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Q. What is the x-intercept of the line given by the equation 4x + 5y - 20 = 0?
Solution
To find the x-intercept, set y = 0. Thus, 4x = 20, giving x = 5.
Correct Answer: A — 4
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Q. What is the y-intercept of the line given by the equation 5x - 2y = 10? (2023)
Solution
Rearranging to slope-intercept form: -2y = -5x + 10, thus y = (5/2)x - 5. The y-intercept is -5.
Correct Answer: D — -2
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Q. What is the y-intercept of the line represented by the equation 4x - y + 8 = 0? (2019)
Solution
Setting x = 0 in the equation gives -y + 8 = 0, thus y = 8.
Correct Answer: A — 8
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Q. What type of matrix has all its diagonal elements as 1 and all other elements as 0? (2023)
-
A.
Identity matrix
-
B.
Zero matrix
-
C.
Diagonal matrix
-
D.
Scalar matrix
Solution
An identity matrix is defined as a square matrix with all diagonal elements equal to 1 and all other elements equal to 0.
Correct Answer: A — Identity matrix
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Q. What type of matrix has all its elements equal to zero? (2022)
-
A.
Identity matrix
-
B.
Zero matrix
-
C.
Diagonal matrix
-
D.
Scalar matrix
Solution
A matrix where all elements are zero is called a zero matrix.
Correct Answer: B — Zero matrix
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Q. What type of matrix has all its elements equal? (2022)
-
A.
Identity matrix
-
B.
Zero matrix
-
C.
Scalar matrix
-
D.
Diagonal matrix
Solution
A scalar matrix is a square matrix in which all the elements are equal to the same scalar value.
Correct Answer: C — Scalar matrix
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Q. What type of matrix has the property A = -A? (2022)
-
A.
Symmetric matrix
-
B.
Skew-symmetric matrix
-
C.
Identity matrix
-
D.
Diagonal matrix
Solution
A skew-symmetric matrix is defined by the property A = -A.
Correct Answer: B — Skew-symmetric matrix
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Q. Which identity represents the difference of squares?
-
A.
(a + b)² = a² + 2ab + b²
-
B.
(a - b)² = a² - 2ab + b²
-
C.
a² - b² = (a + b)(a - b)
-
D.
a² + b² = (a + b)² - 2ab
Solution
The difference of squares is given by a² - b² = (a + b)(a - b).
Correct Answer: C — a² - b² = (a + b)(a - b)
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Q. Which identity represents the expansion of (a + b + c)²?
-
A.
a² + b² + c² + 2ab + 2bc + 2ca
-
B.
a² + b² + c² + 3abc
-
C.
a² + b² + c²
-
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?
-
A.
f(x) = 1/x
-
B.
f(x) = x^3
-
C.
f(x) = sqrt(x)
-
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?
-
A.
f(x) = 1/x
-
B.
f(x) = e^x
-
C.
f(x) = tan(x)
-
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]?
-
A.
f(x) = 1/x
-
B.
f(x) = x^3
-
C.
f(x) = sqrt(x)
-
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)
-
A.
All elements are zero
-
B.
Only diagonal elements are non-zero
-
C.
All elements are equal
-
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)
-
A.
[[1, 2], [3, 4]]
-
B.
[[2, 3], [4, 6]]
-
C.
[[5, 0], [0, 5]]
-
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)
-
A.
2x3
-
B.
3x3
-
C.
4x2
-
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)
-
A.
[[1, 2], [3, 4]]
-
B.
[[1, 2], [2, 1]]
-
C.
[[1, 0], [0, 1]]
-
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)
-
A.
[[1, 0], [0, 1]]
-
B.
[[1, 1], [1, 1]]
-
C.
[[0, 1], [1, 0]]
-
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?
-
A.
1/8
-
B.
1/4
-
C.
1/2
-
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 0.25? (2023)
-
A.
1/2
-
B.
1/4
-
C.
1/3
-
D.
1/5
Solution
0.25 is equal to 1/4.
Correct Answer: B — 1/4
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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)
-
A.
5 + 2i
-
B.
5 - 2i
-
C.
-5 + 2i
-
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?
-
A.
5 + 3i
-
B.
5 - 3i
-
C.
3 - 5i
-
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?
-
A.
x³ - 3x²y + 3xy² - y³
-
B.
x³ - y³
-
C.
x³ - 3xy² + 3x²y - y³
-
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?
-
A.
(a + b)(a - b)
-
B.
(a - b)(a - b)
-
C.
(a + b)(a + b)
-
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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