Q. What is the determinant of the matrix \( \begin{pmatrix} 1 & 2 \\ 3 & 5 \end{pmatrix} \)?
Solution
The determinant is calculated as \( 1*5 - 2*3 = 5 - 6 = -1 \).
Correct Answer: B — 1
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Q. What is the determinant of the matrix \( \begin{pmatrix} 3 & 2 \\ 1 & 4 \end{pmatrix} \)?
Solution
The determinant is calculated as (3*4) - (2*1) = 12 - 2 = 10.
Correct Answer: A — 10
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Q. What is the determinant of the matrix \( \begin{pmatrix} 5 & 6 \\ 7 & 8 \end{pmatrix} \)?
Solution
The determinant is calculated as \( 5*8 - 6*7 = 40 - 42 = -2 \).
Correct Answer: A — -2
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Q. What is the determinant of the matrix: | 1 2 3 | | 4 5 6 | | 7 8 10 |?
Solution
Calculating gives a determinant of -3.
Correct Answer: A — -3
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Q. What is the inverse of the matrix A = [[0, 1], [1, 0]]?
-
A.
[[0, 1], [1, 0]]
-
B.
[[1, 0], [0, 1]]
-
C.
[[0, 0], [0, 0]]
-
D.
[[1, 1], [1, 1]]
Solution
The inverse of A is A itself since A is an involutory matrix.
Correct Answer: A — [[0, 1], [1, 0]]
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Q. What is the inverse of the matrix A = [[1, 2], [3, 4]]?
-
A.
[[4, -2], [-3, 1]]
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B.
[[-2, 1], [1.5, -0.5]]
-
C.
[[-2, 1], [1.5, -0.5]]
-
D.
[[2, -1], [-1.5, 0.5]]
Solution
The inverse of A is (1/det(A)) * adj(A) = (1/-2) * [[4, -2], [-3, 1]] = [[-2, 1], [1.5, -0.5]].
Correct Answer: A — [[4, -2], [-3, 1]]
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Q. What is the maximum value of the quadratic function f(x) = -x^2 + 4x + 1?
Solution
The vertex form gives the maximum value at x = 2, f(2) = 5.
Correct Answer: A — 5
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Q. What is the maximum value of the quadratic function f(x) = -x^2 + 6x - 8?
Solution
The vertex form gives the maximum value at x = 3, f(3) = -3^2 + 6*3 - 8 = 4.
Correct Answer: C — 4
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Q. What is the middle term in the expansion of (x + 2)^8?
-
A.
128
-
B.
112
-
C.
256
-
D.
64
Solution
The middle term is T(5) = C(8,4) * (2)^4 = 70 * 16 = 1120.
Correct Answer: A — 128
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Q. What is the modulus of the complex number z = -3 + 4i?
Solution
The modulus |z| = √((-3)^2 + (4)^2) = √(9 + 16) = √25 = 5.
Correct Answer: A — 5
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Q. What is the product of (1 + i) and (1 - i)?
Solution
(1 + i)(1 - i) = 1 - i^2 = 1 - (-1) = 2.
Correct Answer: A — 2
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Q. What is the product of the complex numbers (1 + i) and (1 - i)?
Solution
(1 + i)(1 - i) = 1^2 - i^2 = 1 - (-1) = 2.
Correct Answer: A — 2
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Q. What is the product of the complex numbers z1 = 1 + i and z2 = 1 - i?
Solution
z1 * z2 = (1 + i)(1 - i) = 1 - i^2 = 1 + 1 = 2.
Correct Answer: A — 2
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Q. What is the product of the roots of the equation 2x^2 + 3x - 5 = 0?
-
A.
-5/2
-
B.
5/2
-
C.
2/5
-
D.
-2/5
Solution
The product of the roots is c/a = -5/2.
Correct Answer: A — -5/2
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Q. What is the product of the roots of the equation 2x^2 - 3x + 1 = 0?
Solution
Using Vieta's formulas, the product of the roots is 1/2 (c/a = 1/2).
Correct Answer: A — 1/2
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Q. What is the product of the roots of the equation 3x^2 + 6x + 2 = 0?
-
A.
2/3
-
B.
1/3
-
C.
1/2
-
D.
1/6
Solution
The product of the roots is given by c/a = 2/3.
Correct Answer: B — 1/3
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Q. What is the product of the roots of the equation 3x^2 - 12x + 9 = 0?
Solution
The product of the roots is given by c/a = 9/3 = 3.
Correct Answer: C — 4
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Q. What is the product of the roots of the equation x^2 - 6x + 8 = 0?
Solution
The product of the roots is given by c/a = 8/1 = 8.
Correct Answer: B — 8
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Q. What is the product of the roots of the equation x^2 - 8x + 15 = 0?
Solution
The product of the roots is given by c/a = 15/1 = 15.
Correct Answer: A — 15
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Q. What is the range of sin^(-1)(x)?
-
A.
[-π/2, π/2]
-
B.
[-1, 1]
-
C.
[0, π]
-
D.
[-π, π]
Solution
The range of sin^(-1)(x) is [-π/2, π/2].
Correct Answer: A — [-π/2, π/2]
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Q. What is the range of the function y = cos^(-1)(x)?
-
A.
[0, π]
-
B.
[0, 2π]
-
C.
[−π/2, π/2]
-
D.
[−1, 1]
Solution
The range of y = cos^(-1)(x) is [0, π].
Correct Answer: A — [0, π]
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Q. What is the range of the function y = sin^(-1)(x)?
-
A.
[-π/2, π/2]
-
B.
[0, π]
-
C.
[-1, 1]
-
D.
[0, 1]
Solution
The range of y = sin^(-1)(x) is [-π/2, π/2].
Correct Answer: A — [-π/2, π/2]
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Q. What is the range of the function y = tan^(-1)(x)?
-
A.
(-π/2, π/2)
-
B.
(0, π)
-
C.
(0, 1)
-
D.
(-1, 1)
Solution
The range of the function y = tan^(-1)(x) is (-π/2, π/2).
Correct Answer: A — (-π/2, π/2)
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Q. What is the rank of the matrix A = [[1, 2, 3], [4, 5, 6], [7, 8, 9]]?
Solution
The rows of A are linearly dependent, hence the rank is 2.
Correct Answer: B — 2
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Q. What is the rank of the matrix [[1, 2, 3], [4, 5, 6], [7, 8, 9]]?
Solution
The rank of the matrix is 2, as the rows are linearly dependent.
Correct Answer: B — 2
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Q. What is the real part of the complex number 5 - 7i?
Solution
The real part of 5 - 7i is 5.
Correct Answer: A — 5
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Q. What is the real part of the complex number z = 2e^(iπ/3)?
Solution
The real part is 2 * cos(π/3) = 2 * 1/2 = 1.
Correct Answer: B — 2
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Q. What is the real part of the complex number z = 4e^(iπ/3)?
Solution
The real part is Re(z) = 4 * cos(π/3) = 4 * 1/2 = 2.
Correct Answer: C — 2√3
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Q. What is the real part of the complex number z = 5 - 2i?
Solution
The real part of z = 5 - 2i is 5.
Correct Answer: A — 5
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Q. What is the real part of the complex number z = 5 - 4i?
Solution
The real part of z = 5 - 4i is 5.
Correct Answer: A — 5
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