1. 3^{p} = x, then 3^{p + 3} is:
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Correct answer is option - 3
Explanation:
Applying the laws of exponents, it gives: 3^{p + 3} = 3^{p} * 3^{3} = x * 27 = 27x.
2. The expression 1/i is equivalent to:
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Correct answer is option - 4
Explanation:
The expression 1/i can be multiplied in the numerator and the denominator by i.
This gives: (1 * i)/ (i * i) = i/ i^{2}. In complex numbers, i^{2} = -1.
Hence we get, i/-1 = -i.
3. If 2x^{3} = 8, then what is the value of (6x^{3})^{2}?
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Correct answer is option - 5
Explanation:
The given expression, (6x^{3})^{2} can be written in terms of 2x^{3} as: (3 * 2x^{3})^{2} = (3)^{2} * (2x^{3})^{2}.
This gives: 9 * (8)^{2} = 9 * 64 = 576.
4. The expression √(5/3) can also be written as:
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Correct answer is option - 3
Explanation: Multiplying the numerator and denominator of the expression by √3, we get:
(√5 * √3)/ (√3 * √3) = √(5 * 3)/ √(3 * 3) = √15/√9 = (√15)/3.
Therefore the expression √(5/3) can also be written as (√15)/3.
5. The daily income of Laura has increased from $120 to $150. The percentage increase in her income is:
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Correct answer is option - 3
Explanation: Laura’s original income = $120 and new income = $150.
Increase in Laura’s income = $150 - $120 = $30.
In order to find the percentage increase in her income, we can use the formula:
% percentage increase = (increase in income * 100%) / (original income)
This gives: % percentage increase = ($30 * 100%)/ $120 = 25%.
6. If |x – 4| < 2, then which of the following is true?
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Correct answer is option - 1
Explanation:
If | f(x) | = a, then according to the absolute value property: f(x) = a, or f(x) = -a.
Similarly the absolute value expression |x – 4| < 2 can be written as: -2 < x – 4 < 2.
Here 4 can be added to each side and this gives: (-2 + 4) < (x – 4 + 4) < (2 + 4)
This gives: 2 < x < 6.
7. A circle is inscribed in a square ABCD as shown in the figure. If the length of the line segment PQ is 6, then the area of the circle is:
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Correct answer is option - 3
Explanation:
From the given figure, it can be observed that PCQ is a right angled triangle with PQ as the hypotenuse.
The length PC = CQ as tangents drawn to a circle from the same external point are equal in length.
Let the length PC = CQ = x, then by the Pythagorean Theorem we get, 2x^{2} = 6^{2} = 36.
This gives: x^{2} = 18. From the figure, the length PC = radius of the circle.
Hence area of the circle = π * (radius)^{2} = π * 18 = 18π.