**Question 1 :**

(-39)^{2}/13^{3}

**Solution :**

= (-39)^{2}/13^{3}

Since we have even power, so the negative sign will become positive.

= (39 ⋅ 39) / (13 ⋅ 13 ⋅ 13)

= (3 ⋅ 3) / 13

= 9/13

**Question 2 : **

In a scaled diagram, 1 inch represents 20 feet. How many square inches on the diagram represent 1 square foot ?

**Solution :**

1 inch = 20 feet

1 feet = 1/20 inch

1 square foot = (1/20)^{2}

= 1/400

= 0.0025

Hence 0.0025 square inches represents 1 square foot.

**Question 3 :**

What is the greatest common factor of 2,240 and 3,360 ?

**Solution :**

**To find the greatest common factor, we have to decompose the given numbers into the prime factors until we get prime numbers.**

To find the greatest common divisor, take all the common divisors of 2240 and 3360 and multiply them.

Greatest common factor = 5 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ 7

= 1120

**Question 4 :**

If 50% of 2y is 12, what is y^{2} ?

**Solution :**

50% of 2y = 12

(50/100) ⋅ (2y) = 12

(1/2) ⋅ (2y) = 12

y = 12

y^{2} = 12^{2 }==> 144

Hence the value of y^{2 }is 144.

**Question 5 :**

(15) In the figure given below ∢ PRL is 150 degree. What is (a^{2} - b^{2})/(a - b) ?

**Solution :**

To find the value of (a^{2} - b^{2})/(a - b), first let us simplify this using algebraic identities.

(a^{2} - b^{2})/(a - b) = (a + b)(a - b)/(a - b)

(a^{2} - b^{2})/(a - b) = a + b

From the given picture, we have

<PRL = 150

<PRL = <RPQ + <PQR

150 = a + b

Hence the value of a + b is 150.

**Question 6 :**

If y = 3/4 and xy^{2} = 9/16, what is (x - 4)?

**Solution :**

Given that :

y = 3/4 and xy^{2} = 9/16

x(3/4)^{2} = 9/16

x(9/16) = 9/16

x = 1

then x - 4 = 1 - 4 = -3

Hence the value of x - 4 is -3

**Question 7 :**

If P is an odd integer, which of the following must be an even number ?

**Solution :**

By observing the options, D is having an even number in the denominator. So we might have even number in the numerator.

Even number / even number = even number

Hence (2p^{2} - 2P^{3}) / 2P is the even number.

**Question 8 :**

If the diameter of a circle is P, and P^{2}/4 = 2, what is the area of the circle ?

**Solution :**

Diameter of the circle = P

Radius of the circle = P/2

Area of the circle = π r^{2}

= π (P/2)^{2}

= π (P^{2}/4)

= 2π

Hence the required area of the circle is 2π.

**Question 9 :**

If a regular polygon has (N - 10) sides, where, N = (40/10)^{2}, what is the measure of one of its angles?

**Solution :**

N = (40/10)^{2 }==> 4^{2 }= 16

Number of sides in the given polygon

N - 10 ==> 16 - 10 = 6

So, the given polygon has 6 sides, it is hexagon.

Sum of the angles of a polygon = (n - 2) 180

= (6 - 2) 180

= 4 (180)

= 720

Angle measure = 720/6

= 120 degree

**Question 10 :**

If the perimeter of the triangle shown above is the circumference of a circle, then what is the radius of the circle?

**Solution :**

Perimeter of triangle = Sum of lengths of all sides

Let "x" be the length of unknown side.

(5π)^{2} = (4π)^{2} + x^{2}

25π^{2 }- 16π^{2 } = x^{2}

x^{2 } = 9π^{2}

x = 3π

Perimeter of triangle = 3π + 5π + 4π

Circumference of circle = 12 π

2π r = 12π

r = 6

Hence the radius of circle is 6

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