Q. What is the product of the roots of the quadratic equation x^2 - 7x + 10 = 0? (2023)
Solution
Using Vieta's formulas, the product of the roots is c/a = 10/1 = 10.
Correct Answer: A — 10
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Q. What is the real part of the complex number z = 4 + 5i? (2022)
Solution
The real part of z = 4 + 5i is 4.
Correct Answer: A — 4
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Q. What is the sum of the coefficients in the expansion of (x + 1)^8?
-
A.
256
-
B.
512
-
C.
128
-
D.
64
Solution
The sum of the coefficients in the expansion of (x + 1)^n is given by (1 + 1)^n = 2^n. Here, n=8, so the sum is 2^8 = 256.
Correct Answer: B — 512
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Q. What is the sum of the roots of the equation x² - 5x + 6 = 0? (2022)
Solution
The sum of the roots is given by -b/a = 5/1 = 5.
Correct Answer: A — 5
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Q. What is the value of (−1) × (−1) × (−1)? (2021)
Solution
Multiplying three negative ones gives −1.
Correct Answer: C — −1
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Q. What is the value of (−1)^3?
Q. What is the value of 3 + 4 × 2?
Solution
According to BODMAS, 4 × 2 = 8, so 3 + 8 = 11.
Correct Answer: B — 11
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Q. What is the value of k for which the equation x^2 + kx + 16 = 0 has no real roots? (2021)
Solution
The discriminant must be less than zero. Thus, k^2 - 4*1*16 < 0 leads to k^2 < 64, giving k < 8 and k > -8.
Correct Answer: A — -8
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Q. What is the value of k if the equation x^2 + kx + 4 = 0 has equal roots? (2022)
Solution
For equal roots, the discriminant must be zero. Thus, k^2 - 4*1*4 = 0, which gives k^2 = 16, so k = ±4.
Correct Answer: A — 4
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Q. What is the value of k if the quadratic equation x^2 + kx + 16 = 0 has roots that are real and distinct? (2019)
Solution
For real and distinct roots, the discriminant must be positive: k^2 - 4(1)(16) > 0. Thus, k^2 > 64, leading to k < -8 or k > 8.
Correct Answer: B — -4
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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 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_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 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. 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 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 true for log_a(bc)?
-
A.
log_a(b) + log_a(c)
-
B.
log_a(b) - log_a(c)
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C.
log_a(bc) = log_a(b) * log_a(c)
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D.
None of the above
Solution
log_a(bc) = log_a(b) + log_a(c) by the product rule of logarithms.
Correct Answer: A — log_a(b) + log_a(c)
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Q. Which of the following numbers is a perfect square?
Solution
25 is a perfect square (5^2).
Correct Answer: C — 25
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