1. The expression (x^{-1} + y^{-1})/ y^{-1} is equivalent to:
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Correct answer is option - 3
Explanation: The numerator can be simplified as (x^{-1} + y^{-1}) = (1/x + 1/y) = (x + y)/ xy.
Now writing the whole expression together gives: [(x + y)/xy]/ (1/y).
This gives: [(x + y)] * y / xy. Now we can cancel the ‘y’.
This gives: (x + y)/ x.
2. If G = (x + y)/ 2 and H = 2xy/ (x + y) and G = H, then which of the following expressions is true?
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Correct answer is option - 5
Explanation: Given, (x + y)/ 2 = 2xy/ (x + y) ==> Here we can cross-multiply the numbers to the other sides.
(x + y)^{2} = 2xy * 2 ==> (x + y)^{2} = 4xy ==> x^{2} + 2xy + y^{2} = 4xy.
This gives: x^{2} + 2xy – 4xy + y^{2} = 0. This gives, x^{2} – 2xy + y^{2} = 0.
This implies: (x – y)^{2} = 0. Taking the square root on both sides gives: (x – y) = 0.
This gives: x = y.
3. If ‘x’ is one-fifth of ‘y’ and ‘y’ is one-third of ‘z’, then what is ‘z’ in terms of ‘x’?
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Correct answer is option - 4
Explanation: Given: x = 1/5 * y and y = 1/3 * z.
This gives: x = (1/5) * (1/3) * z = 1/15 * z
Now since x = 1/15 * z, this implies z = 15x
4. If the average of ‘r’ and ‘s’ is equal to 80% of 3 times ‘r’, then what is the value of r/s?
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Correct answer is option - 3
Explanation:
The average of ‘r’ and ‘s’ can be written as (r + s)/2.
80% of 3 times ‘r’ is written as: (80/100) * 3r ==> (4/5) * 3r = 12r/ 5.
Now both the above expressions are equal to each other ==> (r + s)/ 2 = 12r /5
When we cross-multiply we get ==> 5 * (r + s) = 12r * 2.
This gives: 5r + 5s = 24r ==> 5s = 24r – 5r ==> 5s = 19r.
This implies ==> r/s = 5/19.
5. The lines ‘m’ and ‘n’ are perpendicular to each other. If the equation of the line ‘m’ is x – 2y + 9 = 0, then the equation of the line ‘n’ passing through (2, 3) is?
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Correct answer is option - 1
Explanation: The line ‘m’ can rearranged as 2y = (x + 9) ==> y = (1/2)x + 9/2.
This is in the form of y= mx + b. The slope of this line is 1/2.
Since the lines ‘m’ and ‘n’ are perpendicular to each other ==> hence their product of their slopes is -1.
Therefore slope of the line ‘n’ is -2 and the point is (2, 3).
The equation of this line can be written in point –slope form as: y – 3 = -2 (x – 2).
This gives: y – 3 = -2x + 4 ==> 2x + y – 7 = 0.
6. The sum of 3 consecutive integers is 15. What is the product of these integers?
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Correct answer is option - 4
Explanation:
Let the three consecutive integers be x, (x + 1), (x + 2).
Sum of these three integers is 15 ==> x + x + 1 + x + 2 = 15.
This gives: 3x + 3 = 15. Now we should solve for ‘x’.
3x = 15 – 3 ==> 3x = 12 ==> x = 12/3 ==> x = 4.
Therefore the integers are 4, 5, and 6.
Product of these integers is 4 * 5 * 6 = 120.
7. In triangle ABC, the angle ABC is 60° and side AD is perpendicular to side BCas shown in the figure below. If the length of the sideAB is12m andside BC is 20m, then what is the length of CD?
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Correct answer is option - 3
Explanation: Given ABD is a right angled triangle; hence we can use the appropriate trigonometric function to get the length of the other sides.
cos(60°) = (adjacent side)/ (hypotenuse) ==>cos(60°) = BD/ AB ==> BD = 12 * cos(60°)
This gives: BD = 12 * 1/2 = 6m.
The total side length BC is 20m, then CD = 20m – 6m ==> CD = 14m.