Q. What is the value of (x + y + z)²?
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A.
x² + y² + z²
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B.
x² + y² + z² + 2xy + 2yz + 2zx
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C.
x² + y² + z² + 3xyz
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D.
x² + y² + z² - 2xy - 2yz - 2zx
Solution
(x + y + z)² = x² + y² + z² + 2xy + 2yz + 2zx by the expansion of the square of a trinomial.
Correct Answer: B — x² + y² + z² + 2xy + 2yz + 2zx
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Q. What is the value of (x + y)(x - y)?
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A.
x² + y²
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B.
x² - y²
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C.
2xy
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D.
xy
Solution
(x + y)(x - y) = x² - y² by the difference of squares identity.
Correct Answer: B — x² - y²
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Q. What is the value of (x - 4)(x + 4)?
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A.
x² - 16
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B.
x² + 16
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C.
x² - 8
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D.
x² + 8
Solution
(x - 4)(x + 4) = x² - 16.
Correct Answer: A — x² - 16
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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 (−1)² + (−2)²? (2019)
Solution
(−1)² = 1 and (−2)² = 4, so 1 + 4 = 5.
Correct Answer: C — 5
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Q. What is the value of (−2)²? (2022)
Q. What is the value of (−5)²? (2021)
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A.
25
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B.
−25
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C.
10
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D.
−10
Q. What is the value of (−7) + (−3)? (2019)
Solution
−7 + (−3) = −10.
Correct Answer: A — −10
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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 3.5 + 2.8? (2021)
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A.
5.3
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B.
6.3
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C.
7.3
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D.
8.3
Solution
3.5 + 2.8 = 6.3.
Correct Answer: B — 6.3
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Q. What is the value of 3² + 4²? (2020)
Solution
3² = 9 and 4² = 16. So, 9 + 16 = 25.
Correct Answer: C — 25
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Q. What is the value of 3√(27)? (2023)
Solution
3√(27) = 3 × 3 = 9.
Correct Answer: A — 9
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Q. What is the value of 7^2? (2020)
Solution
7^2 = 7 × 7 = 49.
Correct Answer: A — 49
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Q. What is the value of cot 60°?
Solution
Cotangent is the reciprocal of tangent. Since tan 60° = √3, cot 60° = 1/√3.
Correct Answer: A — 1/√3
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Q. What is the value of csc 30°?
Solution
Cosecant is the reciprocal of sine. Since sin 30° = 1/2, csc 30° = 1 / (1/2) = 2.
Correct Answer: A — 2
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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 equation x^2 - kx + 9 = 0 has roots that are both positive?
Solution
For both roots to be positive, k must be greater than 6 (sum of roots) and k^2 - 36 > 0 (discriminant). Thus, k > 6 and k < 12.
Correct Answer: C — 10
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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 quadratic equation x^2 + kx + 16 = 0 has roots that are both negative? (2019)
Solution
For both roots to be negative, k must be negative and |k| > 8. Thus, k = -8.
Correct Answer: A — -8
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Q. What is the value of k if the quadratic equation x^2 + kx + 9 = 0 has roots that are both positive? (2023)
Solution
For both roots to be positive, k must be negative and k^2 > 36. Thus, k < -6.
Correct Answer: A — -6
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Q. What is the value of k if the quadratic equation x^2 + kx + 9 = 0 has roots that are both negative? (2023)
Solution
For both roots to be negative, k must be positive and k^2 > 4(1)(9). Thus, 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 + 4 = 0 are -2 and -2?
Solution
The sum of the roots is -2 + -2 = -4, so k = 4.
Correct Answer: C — 6
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Q. What is the value of k if the roots of the equation x^2 + kx + 9 = 0 are imaginary?
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A.
k < 0
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B.
k > 0
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C.
k = 0
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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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