Algebra
Q. Find the value of the coefficient of x^2 in the expansion of (3x - 4)^4.
-
A.
-144
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B.
-216
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C.
216
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D.
144
Solution
The coefficient of x^2 is C(4,2) * (3)^2 * (-4)^2 = 6 * 9 * 16 = 864.
Correct Answer: A — -144
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Q. Find the value of the determinant \( \begin{pmatrix} a & b \\ c & d \end{pmatrix} \) when \( a=1, b=2, c=3, d=4 \).
Solution
The determinant is \( 1*4 - 2*3 = 4 - 6 = -2 \).
Correct Answer: A — -2
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Q. Find the value of the determinant \( |D| \) where \( D = \begin{pmatrix} 1 & 2 & 3 \\ 0 & 1 & 4 \\ 5 & 6 & 0 \end{pmatrix} \).
-
A.
-12
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B.
-10
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C.
-8
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D.
-6
Solution
The determinant is calculated as 1(1*0 - 4*6) - 2(0*0 - 4*5) + 3(0*6 - 1*5) = 1(0 - 24) - 2(0 - 20) + 3(0 - 5) = -24 + 40 - 15 = 1.
Correct Answer: A — -12
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Q. Find the value of the determinant: | 2 3 1 | | 1 0 4 | | 0 5 2 |
Solution
Using the determinant formula, we calculate it to be 1.
Correct Answer: B — 1
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Q. Find the value of the determinant: | 2 3 1 | | 1 0 4 | | 5 6 2 |
-
A.
-20
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B.
-10
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C.
10
-
D.
20
Solution
Using the determinant formula, we calculate it to be -10.
Correct Answer: B — -10
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Q. Find the value of the determinant: | 2 3 1 | | 1 0 4 | | 5 6 7 |
-
A.
-30
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B.
-20
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C.
20
-
D.
30
Solution
Using the determinant formula, we calculate it to be -20.
Correct Answer: B — -20
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Q. Find the value of the determinant: | x 1 2 | | 3 x 4 | | 5 6 x | when x = 1.
Solution
Substituting x = 1 gives the determinant | 1 1 2 | | 3 1 4 | | 5 6 1 | = 6.
Correct Answer: C — 6
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Q. Find the value of x if 3x + 5 = 20.
Solution
Subtracting 5 from both sides gives 3x = 15, thus x = 15/3 = 5.
Correct Answer: A — 5
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Q. Find the value of z if z^2 + 4z + 8 = 0.
-
A.
-2 + 2i
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B.
-2 - 2i
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C.
-4 + 0i
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D.
-4 - 0i
Solution
Using the quadratic formula, z = [-4 ± √(16 - 32)]/2 = -2 ± 2i.
Correct Answer: A — -2 + 2i
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Q. Find the value of z if z^2 = -16.
Solution
Taking square root, z = ±√(-16) = ±4i.
Correct Answer: A — 4i
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Q. Find the value of z^2 if z = 1 + i.
Solution
z^2 = (1 + i)^2 = 1 + 2i + i^2 = 1 + 2i - 1 = 2i.
Correct Answer: B — 2
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Q. Find the value of \( k \) for which the determinant \( \begin{vmatrix} 1 & 2 & 3 \\ 4 & k & 6 \\ 7 & 8 & 9 \end{vmatrix} = 0 \)
Solution
Setting the determinant to zero gives \( k = 6 \).
Correct Answer: B — 6
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Q. Find the value of \( \det \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix} \).
Solution
The determinant of the identity matrix is always 1.
Correct Answer: B — 1
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Q. For the matrix \( B = \begin{pmatrix} 1 & 2 \\ 2 & 4 \end{pmatrix} \), what is the determinant \( |B| \)?
Solution
The determinant is 0 because the rows are linearly dependent.
Correct Answer: A — 0
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Q. For the quadratic equation 2x^2 - 4x + k = 0 to have real roots, what is the condition on k?
-
A.
k >= 0
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B.
k <= 0
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C.
k >= 2
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D.
k <= 2
Solution
The discriminant must be non-negative: (-4)^2 - 4*2*k >= 0, which simplifies to k <= 2.
Correct Answer: C — k >= 2
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Q. For the quadratic equation ax^2 + bx + c = 0, if a = 1, b = -3, and c = 2, what are the roots?
-
A.
1 and 2
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B.
2 and 1
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C.
3 and 0
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D.
0 and 3
Solution
The roots can be found using the quadratic formula: x = (3 ± √(9-8))/2 = 1 and 2.
Correct Answer: A — 1 and 2
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Q. For the quadratic equation x^2 + 2x + 1 = 0, what is the nature of the roots?
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A.
Real and distinct
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B.
Real and equal
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C.
Complex
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D.
None of the above
Solution
The discriminant is 0, indicating that the roots are real and equal.
Correct Answer: B — Real and equal
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Q. For the quadratic equation x^2 + 2x + 1 = 0, what is the vertex of the parabola?
-
A.
(-1, 0)
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B.
(-1, 1)
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C.
(0, 1)
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D.
(1, 1)
Solution
The vertex can be found using the formula (-b/2a, f(-b/2a)). Here, vertex is (-1, 0).
Correct Answer: A — (-1, 0)
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Q. For the quadratic equation x^2 + 2x + k = 0 to have no real roots, k must be:
-
A.
< 0
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B.
≥ 0
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C.
≤ 0
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D.
> 0
Solution
The discriminant must be negative: 2^2 - 4*1*k < 0 => 4 < 4k => k > 1.
Correct Answer: A — < 0
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Q. For the quadratic equation x^2 + 4x + 4 = 0, what is the nature of the roots?
-
A.
Real and distinct
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B.
Real and equal
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C.
Complex
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D.
None of the above
Solution
The discriminant is 0 (b^2 - 4ac = 16 - 16 = 0), indicating real and equal roots.
Correct Answer: B — Real and equal
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Q. For the quadratic equation x^2 + 4x + k = 0 to have no real roots, k must be:
Solution
The discriminant must be negative: 4^2 - 4*1*k < 0 => 16 < 4k => k > 4.
Correct Answer: A — 0
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Q. For the quadratic equation x^2 + 4x + k = 0 to have real roots, what is the condition on k?
-
A.
k >= 4
-
B.
k <= 4
-
C.
k > 0
-
D.
k < 0
Solution
The discriminant must be non-negative: 4^2 - 4*1*k >= 0, which simplifies to k <= 4.
Correct Answer: A — k >= 4
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Q. For the quadratic equation x^2 + 6x + 8 = 0, what are the roots?
-
A.
-2 and -4
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B.
-4 and -2
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C.
2 and 4
-
D.
0 and 8
Solution
Factoring gives (x+2)(x+4) = 0, hence the roots are -2 and -4.
Correct Answer: B — -4 and -2
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Q. For the quadratic equation x^2 + 6x + 9 = 0, what is the nature of the roots?
-
A.
Two distinct real roots
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B.
One real root
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C.
No real roots
-
D.
Complex roots
Solution
The discriminant is 0, indicating one real root (a repeated root).
Correct Answer: B — One real root
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Q. For the quadratic equation x^2 + mx + n = 0, if the roots are 2 and 3, what is the value of n?
Solution
The product of the roots is n = 2 * 3 = 6.
Correct Answer: B — 6
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Q. For the quadratic equation x^2 + px + q = 0, if the roots are 1 and -3, what is the value of p?
Solution
The sum of the roots is 1 + (-3) = -2, hence p = -2.
Correct Answer: A — 2
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Q. For the quadratic equation x^2 - 10x + 25 = 0, what is the double root?
Solution
The equation can be factored as (x-5)^2 = 0, hence the double root is x = 5.
Correct Answer: A — 5
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Q. For the quadratic equation x^2 - 6x + k = 0 to have equal roots, what must be the value of k?
Solution
Setting the discriminant to zero: (-6)^2 - 4*1*k = 0 gives k = 9.
Correct Answer: B — 9
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Q. For which value of k does the equation x^2 + kx + 16 = 0 have real and distinct roots?
Solution
The discriminant must be positive: k^2 - 4*1*16 > 0 => k^2 > 64 => k > 8 or k < -8. Thus, k = -4 is valid.
Correct Answer: B — -4
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Q. For which value of k does the equation x^2 + kx + 4 = 0 have one root equal to 2?
Solution
Substituting x = 2 into the equation gives 2^2 + 2k + 4 = 0, leading to k = -4.
Correct Answer: B — -2
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