1. Given that ‘x’ and ‘y’ are two positive integers. If the value of x^{2} – y^{2} = 11, then what is the value of ‘xy’?

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**Correct answer is option - 4**

**Explanation: **Given x^{2} – y^{2} = 11, and it can be expanded as (x + y) (x – y) = 11.

Now since 11 is a prime number, its only factors are ‘1’ and ‘11’ itself as 1 * 11 = 11 (they cannot be -1 and -11 since its mentioned in the question that ‘x’ and ‘y’ are positive integers).

Hence we can say that either (x + y) or (x – y) should equal to either 1 or 11 accordingly.

In any case, sum of the two factors gives: (x + y) + (x – y) = 1 + 11 = 12.

Here ‘y’ cancels and we get: 2x = 12 and this gives x = 6 which gives y = 5.

Therefore, xy = 6 * 5 = 30.

2. The operation r & s is defined by r & s = (r + s)/ 3s. If (k& 2 = 5), then which of the following could be a possible value of ‘k’?

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**Correct answer is option - 5**

**Explanation: **Given r & s = (r + s)/ 3s. Then applying the given formula we should find the value of k & 2.

Given k & 2 = 5 which implies (k + 2)/ (3 * 2) = 5 ==> (k + 2)/ 6 = 5.

Cross-multiplying gives: k + 2 = 5 * 6 ==> k + 2 = 30 ==> k = 30 – 2.

Hence the value of ‘k’ is 28.

3. If Carl can paint a room in 2 hours and Andy can paint the same room in 3 hours, then how many hours will it take if both Carl and Andy work together?

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**Correct answer is option - 3**

**Explanation: **If Carl can paint the room in 2 hours, which means in 1hr, he can paint 1/2 of the room.

Similarly, if Andy can paint the room in 3hrs, which means in 1hr, he can paint 1/3 of the room.

So if both work together, then it can be written as: 1/2 + 1/3 = 1/x.

This gives: (3 + 2)/ (2 * 3) = 1/x ==> 5/6 = 1/x

Now flipping the numbers we get ==> x = 6/5 ==> x = 1.2 hours.

4. Given the function, f(x) = 3x – p. If f(2) = -4, then what is the value of f(3p)?

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**Correct answer is option - 5**

**Explanation: **Given f(x) = 3x – p and if f(2) = -4, then ==> (3*2 – p) = -4 ==> 6 – p = -4.

This gives p = 6 + 4 ==> p = 10.

The value of f(3p) = f(3*10) ==> f(30 ) = 30 – p = 30 – 10 ==> f(30) = 20.

5. If |4x - 5| ≤ 3, then which of the following exactly describes the interval of ‘x’?

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**Correct answer is option - 5**

**Explanation: **|4x – 5| ≤ 3 is split in this way: -3 ≤ 4x – 5 ≤ 3.

Now to get the interval for ‘x’, add 5 on all the sides.

This gives: (-3 + 5) ≤ (4x – 5 + 5) ≤ 3 + 5.

This gives: 2 ≤ 4x ≤ 8. Now divide the inequality by ‘4’ on all the sides.

This implies: 1/2 ≤ x ≤ 2.

6. If the graph of two lines x – 2y + 3 = 0 and 2x + y – 4 = 0 intersect each other, what is their point of intersection on the X-Y coordinate plane?

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**Correct answer is option - 1**

**Explanation: **The given lines can be rearranged as: x – 2y = -3 and 2x + y = 4.

To find their point of intersection, we can solve these equations either substitution elimination method.

Using Substitution method gives the first equation as: x = 2y – 3.

We can now plug-in the above expression of ‘x’ in the second equation.

This gives: 2(2y -3) + y = 4 ==> 4y – 6 + y = 4 ==> 5y – 6 = 4 ==> 5y = 10 ==> y = 2.

Now solving for ‘x’ gives: x = 2y – 3 ==> x = 2(2) – 3 ==> x = 4 – 3 ==> x = 1.

Therefore the point of intersection is (1, 2).

7. In a triangle ABC, the measure of the angles are (x + 4)°, (2x + 9)° and (3x -7)°. The value of ‘x’ is?

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**Correct answer is option - 5**

**Explanation: **Sum of the angles in a triangle = 180°

Therefore, (x + 4) + (2x + 9) + (3x – 7) = 180°.

This gives: 6x + 6 = 180° ==> 6x = 180° - 6 ==> 6x = 174.

This gives: x = 174/6 ==> x = 29.