1. If ‘p’ and ‘q’ are two positive integers, and p^{2} – q^{2} = 11, then what is the value of ‘pq’?

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**Correct answer is option - 3**

**Explanation: **Given p^{2} – q^{2} = 11 and (p^{2} – q^{2}) can be expanded as (p + q) (p – q) = 11 * 1

This gives: p + q = 11 and p – q = 1. (We can take it otherwise as well saying (p + q) = 1 and (p – q) = 11, but it is mentioned in the question that ‘p’ and ‘q’ are positive integers and that should be considered).

Hence solving the above two equations we get ==> 2p = 12 ==> p = 6.

Since p + q = 11 ==> q = 11 – 6 ==> q = 5.

This gives: pq = 6 * 5 = 30 as the answer!

2. The function f(x) is defined as f(x) = 2x – k. If the function value, f(3) = -4, then what is the value of f(3k)?

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**Correct answer is option - 5**

**Explanation: **Given f(x) = 2x – k and f(3) = -4 which means f(3) = 2(3) – k = -4 ==> 6 – k = -4.

This gives: k = 6 + 4 ==> k = 10.

f(3k) = f(3 * 10) = f(30) = 2(30) – k. This gives: f(30) = 60 – k and k = 10.

Therefore we get: f(3k) = f(30) = 60 – 10 = 50.

3. In a box there are 5 blue pens and 6 green pens. If Ben needs 2 pens out from the box and hence randomly picks one pen after another without replacement, then what is the probability of Ben selecting two green pens?

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**Correct answer is option - 4**

**Explanation: **Total number of pens in the box = 5 + 6 = 11 pens.

There are 6 green pens and hence probability of Ben first picking a green pen = 6/11.

While picking the second green pen, now the box has total of (11 – 1) = 10 pens and out of them (6 – 1) = 5 green pens.

Therefore the probability of picking another green pen = 5/10.

Since Ben is selecting one pen after another, hence we multiply both the outcomes.

This gives: 6/11 * 5/10 ==> 3/11.

4. If given G = 3(x + y)/ 2 and H = 6xy/ (x + y) and let G = H, then which of the following must be true?

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**Correct answer is option - 1**

**Explanation:**Given G = H, hence 3(x + y)/2 = 6xy/ (x + y). Now we can cross- multiply and then we get: 3(x + y) (x + y) = 12xy ==> 3 (x + y)^{2} = 12xy.Dividing 3 on both sides and expanding (x + y)^{2}, we get: (x^{2} + 2xy + y^{2}) = 4xy.

Subtracting ‘4xy’ on both sides gives: x^{2} + 2xy + y^{2} – 4xy = 0 ==> x^{2} – 2xy + y^{2} = 0.

The above expression is equal to: (x – y)^{2} = 0 ==> x = y.

Option (d) cannot be the answer because if x and y are equal to 0, then ‘H’ becomes undefined.

Option (e) cannot be the answer because x = y = 1 could be the answer but the question clearly states which of them MUST be true and it is not necessary that x and y have to be 1.

5. A circle with center O at (4, -3) has the Y-axis as its tangent. The area of the circle is?

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**Correct answer is option - 3**

**Explanation:**Center of the circle is at (4, -3) and Y-axis is the tangent.

Hence the distance from the center to the point where the tangent touches the circle is = 4 units (counted on the X-axis).

This gives the radius of the circle, r = 4.

Hence the area of the circle, A = π * (radius)^{2} = π * (4)^{2} ==> A = 16π.

6. Which of the following is equivalent to the expression: 5(1 + i)/ (4 – 3i)?

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**Correct answer is option - 2**

**Explanation: **Multiplying the numerator and the denominator by the conjugate of (4 – 3i) we get:

[5 (1 + i) (4 + 3i)]/ [(4 – 3i)(4 + 3i)].

This gives: 5(4 + 7i + 3i^{2})/ (16 – 9i^{2}) and in complex numbers, i^{2} = -1.

This implies: 5(4 + 7i – 3)/ (16 + 9)

Therefore, it gives 5(1 + 7i)/ 25 ==> (1 + 7i)/ 5.

7. 69 is what percent of 1150?

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**Correct answer is option - 2**

**Explanation: **Let the percent value be = x%.

Then we can say that: 69 = (x/ 100) * 1150.

Now solving for ‘x’ gives: (69 * 100)/ 1150 = x.

This implies: x = 6%.