1. If (4 – x/7) = (5 – x/6), then the value of x is?

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**Correct answer is option - 4**

**Explanation:**

To find the value of ‘x’, we combine the like terms together.

Hence we get: (x/6 – x/7) = (5 – 4).

This gives: (7x – 6x)/ 42 = 1 ==> x/42 = 1.

This gives the value of ‘x’, x = 42.

2. Find the LCM of the expressions (3x^{3} + 9x^{2}) and (6x + 12x^{2}).

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**Correct answer is option - 4**

**Explanation:**

To find the LCM we can first split them into their factors. This gives:

(3x^{3} + 9x^{2}) = 3 * x^{2} * (x + 3)

(6x^{2} + 12x) = 2 * 3 * x * (x + 2)

Now the LCM of these above two expressions will be 6x^{2}(x + 3) (x + 2) since these are the factors appearing at least once in both the given expressions.

3. Two quantities ‘x’ and ‘y’ are directly proportional to each other. If the ‘y’ value is 36 when ‘x’ is 12, then what is the value of ‘x’ when y is 24?

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**Correct answer is option - 5**

**Explanation:**

Since the quantities ‘x’ and ‘y’ are directly proportional, hence they are related in this way: x = ky where ‘k’ is the constant of proportionality ==> x/y = k = constant

Hence if x_{1}, y_{1} and x_{2}, y_{2}are the quantities in direct proportion, then they are written as: x_{1} / y_{1} = x_{2} / y_{2}.

This gives: 12/ 36 = x_{2}/ 24 ==> x_{2} = (12 * 24)/ 36.

This implies: x_{2} = 8

4. Which of the following represents the exact solution set for the linear inequality, 5x – 16 ≥ 9?

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**Correct answer is option - 3**

**Explanation: **First we should solve the given linear inequality: 5x – 16 ≥ 9.

This implies: 5x ≥ 9 + 16 ==> 5x ≥ 25.

This gives: 5x/5 ≥ 25/5.

So we get, x ≥ 5.

The interval notation for this solution is [5, ∞) since ‘x’ can take the value of either ‘5’ or any other number greater than ‘5’ until infinity.

5. A circle with center ‘O’ is exactly inscribed in a square of side length 8cm. The area of the inscribed circle in cm^{2} is?

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**Correct answer is option - 5**

**Explanation: **If the circle is exactly inscribed in a square, then the side length of the square will serve as the diameter of the circle.

Radius of the circle = 1/2 * (Diameter of the circle)

Therefore, radius, r = 1/2 * (8cm) = 4cm.

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

6. If given, log_{2}(x^{2} – 20) = log_{2}(x), then the value of ‘x’ is?

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**Correct answer is option - 1**

**Explanation: **For the given logarithmic equation, we can equate the ‘insides’ of the log function equal to each other since the bases are the same.

This gives: x^{2} – 20 = x which implies: x^{2} – x – 20 = 0.

We can solve this quadratic equation by taking the factors (x – 5) (x – 4) = 0.

This gives x – 5 = 0 and x – 4 = 0.

This gives x = 5 and x = - 4, however ‘x’ cannot take negative values since there cannot be a negative number inside the logarithm.

Hence we get, x = 5 as the only answer.

7. What is the discriminant of the quadratic equation 3x^{2} – 5x - 9 = 0?

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**Correct answer is option - 2**

**Explanation: **When a quadratic equation is written in the form of ax^{2} + bx + c = 0, then the discriminant of the equation is b^{2} – 4ac.

In the given quadratic equation, 3x^{2} – 5x - 9 = 0, the value of a = 3, b = -5 and c = -9.

Therefore, the discriminant = (-5)^{2} – (4 * 3 * -9) = 25 + 108 = 133.