Q. What is the value of k if the roots of the equation x^2 + kx + 9 = 0 are imaginary?
-
A.
k < 0
-
B.
k > 0
-
C.
k = 0
-
D.
k ≤ 0
Solution
The discriminant must be less than zero: k^2 - 4*1*9 < 0 leads to k^2 < 36, hence k < 0 or k > 0.
Correct Answer: A — k < 0
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Q. What is the value of k if the roots of the equation x^2 - 5x + k = 0 are equal? (2020)
Solution
For the roots to be equal, the discriminant must be zero. Thus, (-5)^2 - 4(1)(k) = 0. Solving gives k = 6.25.
Correct Answer: A — 6.25
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Q. What is the value of k if the roots of the equation x^2 - kx + 16 = 0 are real and distinct?
Solution
For the roots to be real and distinct, the discriminant must be positive: k^2 - 64 > 0, leading to k > 8 or k < -8.
Correct Answer: C — 12
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Q. What is the value of k if the roots of the equation x^2 - kx + 8 = 0 are 2 and 4? (2023)
Solution
The sum of the roots is 2 + 4 = 6, so k = 6.
Correct Answer: A — 6
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Q. What is the value of k if the roots of the equation x^2 - kx + 9 = 0 are 3 and 3?
Solution
The sum of the roots is 3 + 3 = 6, so k = 6.
Correct Answer: A — 6
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Q. What is the value of log_10(0.01)?
Solution
log_10(0.01) = log_10(10^-2) = -2.
Correct Answer: B — -2
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Q. What is the value of log_2(1/8)? (2023)
Solution
log_2(1/8) = log_2(2^-3) = -3.
Correct Answer: A — -3
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Q. What is the value of log_4(16)?
Solution
log_4(16) = log_4(4^2) = 2.
Correct Answer: B — 2
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Q. What is the value of log_5(1)?
-
A.
0
-
B.
1
-
C.
5
-
D.
undefined
Solution
log_5(1) = 0 because 5^0 = 1.
Correct Answer: A — 0
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Q. What is the value of the 5th term in the expansion of (x + 2)^7?
-
A.
672
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B.
672x^4
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C.
672x^3
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D.
672x^2
Solution
The 5th term is C(7,4) * (2)^4 * x^3 = 35 * 16 * x^3 = 560x^3.
Correct Answer: C — 672x^3
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Q. What is the value of the coefficient of x^4 in the expansion of (x + 5)^6?
-
A.
150
-
B.
300
-
C.
600
-
D.
750
Solution
The coefficient of x^4 in (x + 5)^6 is given by 6C4 * 5^2 = 15 * 25 = 375.
Correct Answer: B — 300
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Q. What is the value of the discriminant for the equation 3x^2 + 12x + 9 = 0?
Solution
The discriminant is b^2 - 4ac = 12^2 - 4*3*9 = 144 - 108 = 36.
Correct Answer: A — 0
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Q. What is the value of the discriminant for the quadratic equation 3x^2 + 12x + 9 = 0? (2019)
Solution
The discriminant D = b^2 - 4ac = 12^2 - 4*3*9 = 0, indicating equal roots.
Correct Answer: A — 0
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Q. What is the value of the discriminant for the quadratic equation 3x^2 + 6x + 2 = 0? (2023)
Solution
The discriminant is b^2 - 4ac = 6^2 - 4(3)(2) = 36 - 24 = 12.
Correct Answer: B — 4
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Q. What is the value of x in the equation 3x - 7 = 2x + 5? (2023)
Solution
Rearranging gives 3x - 2x = 5 + 7, thus x = 12.
Correct Answer: B — 6
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Q. What is the value of x in the equation 4x + 3 = 19? (2023)
Solution
Solving for x gives 4x = 16, thus x = 4.
Correct Answer: A — 4
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Q. What is the value of √(144) + √(64)?
Solution
√(144) = 12 and √(64) = 8, so 12 + 8 = 20.
Correct Answer: C — 18
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Q. What is the value of √(16) + √(25)? (2021)
Solution
√(16) = 4 and √(25) = 5, so 4 + 5 = 9.
Correct Answer: C — 11
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Q. What is the vertex of the quadratic function f(x) = 2x^2 - 8x + 5? (2021)
-
A.
(2, -3)
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B.
(2, -7)
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C.
(4, -3)
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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. 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
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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 smallest positive integer? (2023)
Solution
The smallest positive integer is 1.
Correct Answer: B — 1
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Q. Which of the following is true for log_10(0.1)?
Solution
log_10(0.1) = log_10(10^-1) = -1.
Correct Answer: A — -1
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Q. Which of the following is true for log_a(b) + log_a(c)?
-
A.
log_a(bc)
-
B.
log_a(b/c)
-
C.
log_a(b-c)
-
D.
log_a(b+c)
Solution
log_a(b) + log_a(c) = log_a(bc) by the product property of logarithms.
Correct Answer: A — log_a(bc)
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