Q. If log_5(25) = x, then what is the value of log_5(125) in terms of x?
A.
x + 1
B.
2x
C.
3x
D.
x - 1
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Solution
log_5(125) = log_5(5^3) = 3. Since log_5(25) = 2, we have x = 2, thus log_5(125) = 3.
Correct Answer: C — 3x
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Q. If log_5(25) = x, what is the value of log_5(5^x)?
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Q. If log_5(25) = x, what is the value of x?
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Solution
log_5(25) = log_5(5^2) = 2.
Correct Answer: B — 2
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Q. If log_5(x) = 1/2, what is the value of x?
A.
5
B.
25
C.
sqrt(5)
D.
1/5
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Solution
log_5(x) = 1/2 implies x = 5^(1/2) = sqrt(5).
Correct Answer: C — sqrt(5)
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Q. If log_5(x) = 2, what is the value of x?
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Solution
log_5(x) = 2 implies x = 5^2 = 25.
Correct Answer: C — 25
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Q. If log_7(49) = x, what is the value of x?
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Solution
Since 49 = 7^2, log_7(49) = 2, thus x = 2.
Correct Answer: B — 2
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Q. If log_a(2) = x and log_a(3) = y, then log_a(6) is equal to?
A.
x + y
B.
xy
C.
x - y
D.
x/y
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Solution
log_a(6) = log_a(2 * 3) = log_a(2) + log_a(3) = x + y.
Correct Answer: A — x + y
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Q. If log_a(2) = x and log_a(3) = y, what is log_a(6)?
A.
x + y
B.
xy
C.
x - y
D.
x/y
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Solution
log_a(6) = log_a(2 * 3) = log_a(2) + log_a(3) = x + y.
Correct Answer: A — x + y
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Q. If log_a(4) = 2 and log_a(16) = x, what is the value of x?
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Solution
log_a(16) = log_a(4^2) = 2 * log_a(4) = 2 * 2 = 4.
Correct Answer: B — 4
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Q. If log_a(4) = 2, what is the value of a?
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Solution
log_a(4) = 2 implies a^2 = 4 => a = 2.
Correct Answer: B — 4
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Q. If log_a(5) = p and log_a(25) = q, then what is the relationship between p and q?
A.
q = 2p
B.
q = p/2
C.
q = p^2
D.
q = p + 1
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Solution
log_a(25) = log_a(5^2) = 2 log_a(5) = 2p, hence q = 2p.
Correct Answer: A — q = 2p
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Q. If log_a(5) = p and log_a(25) = q, what is the relationship between p and q?
A.
q = 2p
B.
q = p/2
C.
q = p^2
D.
q = p + 1
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Solution
log_a(25) = log_a(5^2) = 2 log_a(5) = 2p, hence q = 2p.
Correct Answer: A — q = 2p
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Q. If log_a(b) = p and log_a(c) = q, then log_a(bc) is equal to?
A.
p + q
B.
pq
C.
p - q
D.
p/q
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Solution
log_a(bc) = log_a(b) + log_a(c) = p + q.
Correct Answer: A — p + q
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Q. If log_a(b) = p and log_a(c) = q, what is log_a(bc)?
A.
p + q
B.
pq
C.
p - q
D.
p/q
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Solution
log_a(bc) = log_a(b) + log_a(c) = p + q.
Correct Answer: A — p + q
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Q. If log_b(27) = 3, what is the value of b?
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Solution
log_b(27) = 3 implies b^3 = 27 => b = 3.
Correct Answer: A — 3
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Q. If log_x(16) = 4, what is the value of x?
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Solution
log_x(16) = 4 implies x^4 = 16 => x^4 = 2^4 => x = 2.
Correct Answer: B — 4
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Q. If log_x(27) = 3, find x.
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Solution
log_x(27) = 3 implies x^3 = 27 => x = 3.
Correct Answer: B — 9
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Q. If log_x(4) = 2, what is the value of x?
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Solution
log_x(4) = 2 implies x^2 = 4 => x = 2.
Correct Answer: C — 8
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Q. If log_x(81) = 4, find x.
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Solution
log_x(81) = 4 implies x^4 = 81 => x^4 = 3^4 => x = 3.
Correct Answer: A — 3
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Q. If log_x(81) = 4, what is the value of x?
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Solution
log_x(81) = 4 implies x^4 = 81 => x = 3.
Correct Answer: A — 3
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Q. If one root of the equation x^2 - 3x + p = 0 is 2, what is the value of p?
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Solution
Substituting x = 2 into the equation gives 2^2 - 3*2 + p = 0 => 4 - 6 + p = 0 => p = 2.
Correct Answer: D — 4
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Q. If one root of the equation x^2 - 6x + k = 0 is 2, what is the value of k?
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Solution
Using the root, we substitute: 2^2 - 6*2 + k = 0 => 4 - 12 + k = 0 => k = 8.
Correct Answer: A — 4
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Q. If one root of the equation x^2 - 7x + k = 0 is 3, what is the value of k?
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Solution
Using Vieta's formulas, if one root is 3, the other root is 7 - 3 = 4. Thus, k = 3 * 4 = 12.
Correct Answer: B — 9
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Q. If R is a relation on set A = {1, 2, 3} defined by R = {(1, 1), (2, 2), (3, 3)}, is R symmetric?
A.
Yes
B.
No
C.
Depends on A
D.
None of the above
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Solution
A relation is symmetric if for every (a, b) in R, (b, a) is also in R. Since R only contains pairs of the form (a, a), it is symmetric.
Correct Answer: A — Yes
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Q. If sin^(-1)(x) + cos^(-1)(x) = π/2, then the value of x is:
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Solution
The equation sin^(-1)(x) + cos^(-1)(x) = π/2 holds for all x in the domain of the functions, which is [-1, 1]. Therefore, x can be any value in this range.
Correct Answer: A — 0
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Q. If sin^(-1)(x) + cos^(-1)(x) = π/2, then what is the value of x?
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Solution
Using the identity sin^(-1)(x) + cos^(-1)(x) = π/2, we can conclude that x can take any value in the range [-1, 1].
Correct Answer: A — 0
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Q. If sin^(-1)(x) = π/4, what is the value of x?
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Solution
If sin^(-1)(x) = π/4, then x = sin(π/4) = √2/2.
Correct Answer: B — √2/2
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Q. If tan^(-1)(x) = π/4, then the value of x is:
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Solution
tan^(-1)(x) = π/4 implies that x = tan(π/4) = 1.
Correct Answer: B — 1
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Q. If the determinant | x 2 | | 3 4 | = 0, find x.
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Solution
Setting the determinant to zero gives x*4 - 2*3 = 0, thus x = 1.5.
Correct Answer: B — 2
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Q. If the equation 2x^2 + 3x + k = 0 has one root equal to 1, what is the value of k?
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Solution
Substituting x = 1 gives 2(1)^2 + 3(1) + k = 0 => 2 + 3 + k = 0 => k = -5.
Correct Answer: A — -1
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