Q. What is the value of log_5(125)?
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Solution
log_5(125) = log_5(5^3) = 3.
Correct Answer: B — 3
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Q. What is the value of log_5(25) - log_5(5)?
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Solution
log_5(25) = 2 and log_5(5) = 1. Therefore, 2 - 1 = 1.
Correct Answer: A — 1
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Q. What is the value of sec(sin^(-1)(1/2))?
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Solution
sec(sin^(-1)(1/2)) = 1/cos(π/6) = 2.
Correct Answer: B — 2
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Q. What is the value of sec(sin^(-1)(3/5))?
A.
5/3
B.
√(34)/3
C.
√(34)/5
D.
3/5
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Solution
sec(sin^(-1)(3/5)) = √(34)/3
Correct Answer: B — √(34)/3
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Q. What is the value of sec(tan^(-1)(1/√3))?
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Solution
Using the triangle with opposite = 1 and adjacent = √3, hypotenuse = 2. Thus, sec(tan^(-1)(1/√3)) = 2.
Correct Answer: A — 2
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Q. What is the value of sin(tan^(-1)(x))?
A.
x/√(1+x^2)
B.
√(1+x^2)/x
C.
1/x
D.
x
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Solution
sin(tan^(-1)(x)) = x/√(1+x^2)
Correct Answer: A — x/√(1+x^2)
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Q. What is the value of sin^(-1)(1/2) + cos^(-1)(1/2)?
A.
π/3
B.
π/2
C.
π/6
D.
π/4
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Solution
sin^(-1)(1/2) = π/6 and cos^(-1)(1/2) = π/3. Therefore, sin^(-1)(1/2) + cos^(-1)(1/2) = π/6 + π/3 = π/2.
Correct Answer: B — π/2
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Q. What is the value of sin^(-1)(1/2) + sin^(-1)(√3/2)?
A.
π/3
B.
π/2
C.
2π/3
D.
π
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Solution
sin^(-1)(1/2) = π/6 and sin^(-1)(√3/2) = π/3. Therefore, π/6 + π/3 = π/2.
Correct Answer: B — π/2
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Q. What is the value of sin^(-1)(1/2)?
A.
π/6
B.
π/4
C.
π/3
D.
π/2
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Solution
sin^(-1)(1/2) = π/6, since sin(π/6) = 1/2.
Correct Answer: A — π/6
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Q. What is the value of tan^(-1)(1) + tan^(-1)(1)?
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Solution
tan^(-1)(1) = π/4, thus tan^(-1)(1) + tan^(-1)(1) = π/4 + π/4 = π/2.
Correct Answer: A — π/2
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Q. What is the value of tan^(-1)(1) + tan^(-1)(2)?
A.
π/4
B.
π/3
C.
π/2
D.
π/6
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Solution
Using the formula tan^(-1)(a) + tan^(-1)(b) = tan^(-1)((a+b)/(1-ab)), we have tan^(-1)(1) + tan^(-1)(2) = tan^(-1)((1+2)/(1-1*2)) = tan^(-1)(3/-1) = π - tan^(-1)(3) = π/4.
Correct Answer: A — π/4
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Q. What is the value of tan^(-1)(1)?
A.
π/4
B.
π/3
C.
π/2
D.
0
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Solution
tan^(-1)(1) = π/4, since tan(π/4) = 1.
Correct Answer: A — π/4
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Q. What is the value of the 5th term in the expansion of (x + 2)^6?
A.
80
B.
120
C.
160
D.
240
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Solution
The 5th term is given by C(6,4) * (x)^4 * (2)^2 = 15 * x^4 * 4 = 240.
Correct Answer: B — 120
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Q. What is the value of the coefficient of x^3 in the expansion of (x - 1)^6?
A.
-20
B.
-30
C.
-40
D.
-10
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Solution
The coefficient of x^3 is C(6,3) * (-1)^3 = 20 * (-1) = -20.
Correct Answer: B — -30
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Q. What is the value of the coefficient of x^5 in the expansion of (x + 3)^7?
A.
21
B.
63
C.
126
D.
189
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Solution
The coefficient of x^5 is C(7,5) * (3)^2 = 21 * 9 = 189.
Correct Answer: C — 126
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Q. What is the value of the coefficient of x^5 in the expansion of (x + 3)^8?
A.
1680
B.
168
C.
840
D.
280
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Solution
The coefficient of x^5 is C(8, 5) * (3)^3 = 56 * 27 = 1512.
Correct Answer: A — 1680
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Q. What is the value of the determinant of the matrix \( A = \begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{pmatrix} \)?
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Solution
The determinant of matrix A is 0 because the rows are linearly dependent.
Correct Answer: A — 0
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Q. What is the value of the determinant of the matrix \( \begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{pmatrix} \)?
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Solution
The determinant of the matrix is 0 because the rows are linearly dependent.
Correct Answer: A — 0
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Q. What is the value of the determinant \( \begin{vmatrix} 1 & 1 & 1 \\ 1 & 2 & 3 \\ 1 & 3 & 6 \end{vmatrix} \)?
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Solution
The determinant is 0 because the first column is repeated.
Correct Answer: A — 0
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Q. What is the value of the determinant \( \begin{vmatrix} a & b \\ c & d \end{vmatrix} \) when \( a = 1, b = 2, c = 3, d = 4 \)?
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Solution
The determinant is \( 1*4 - 2*3 = 4 - 6 = -2 \).
Correct Answer: A — -2
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Q. What is the value of the determinant | 1 2 3 | | 0 1 4 | | 5 6 0 |?
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Solution
Calculating the determinant gives -12.
Correct Answer: A — -12
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Q. What is the value of the determinant | a b | | c d | when a = 2, b = 3, c = 4, d = 5?
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Solution
det = (2)(5) - (3)(4) = 10 - 12 = -2.
Correct Answer: C — 2
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Q. What is the value of x if 2x + 3 = 11?
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Solution
Subtract 3 from both sides: 2x = 8. Then divide by 2: x = 4.
Correct Answer: C — 4
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Q. What is the value of x if 3x - 5 = 16?
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Solution
Solving for x: 3x = 21 => x = 7.
Correct Answer: A — 7
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Q. What is the value of x in the equation 2x^2 - 8x + 6 = 0?
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Solution
Using the quadratic formula, x = [8 ± sqrt(64 - 48)] / 4 = [8 ± 4] / 4, giving x = 3 or x = 1.
Correct Answer: C — 3
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Q. What is the value of x in the equation 3x - 5 = 7?
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Solution
Solving for x: 3x = 12, thus x = 4.
Correct Answer: A — 4
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Q. What is the value of x in the equation 5(x - 2) = 3x + 4?
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Solution
Expanding gives 5x - 10 = 3x + 4. Rearranging gives 2x = 14, thus x = 7.
Correct Answer: A — -1
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Q. What is the value of x in the equation 5x - 3 = 2x + 12?
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Solution
Rearranging gives 5x - 2x = 12 + 3 => 3x = 15 => x = 5.
Correct Answer: B — 4
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Q. What is the value of x in the equation 5x - 3 = 2x + 6?
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Solution
Rearranging gives 5x - 2x = 6 + 3 => 3x = 9 => x = 3.
Correct Answer: B — 2
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Q. What is the value of z if z^2 = -1?
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Solution
The solutions to z^2 = -1 are z = i and z = -i.
Correct Answer: A — i
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