**Question 1 :**

What is the least possible value of (x^{2} - 1)/x, if x ≥ 1

(A) -1 (B) 0 (C) 3 (D) 3/4 (E) 2/3

**Solution :**

To find the least possible value of (x^{2} - 1)/x, we have to simplify

(x^{2} - 1)/x = (x + 1)(x - 1)/x

Since the value of x is greater than or equal to 1, we may apply x = 1.

= (1 + 1) (1 - 1)/2

= 0/2

= 0

Hence the least possible value for x is 0.

**Question 2 :**

What is the maximum number of points in which a circle and triangle can intersect ?

(A) 3 (B) 5 (C) 6 (D) 8 (E) ∞

**Solution :**

Let us draw a circle and triangle to intersect each other at maximum points.

From the picture, there are 6 points can intersect both circle and triangle.

**Question 3 :**

What is the area of the shaded region ?

(A) 48 - 10π (B) 64 - 22π (C) 10π

(D) 48 - 6π (E) 16π

**Solution :**

In order to find the area of shaded region, we have to subtract the areas of two circles from the area of rectangle.

Area of circle = πr^{2}

Radius of larger circle = 3, radius of small circle = 1

Area of shaded region

= Area of rectangle - Areas of circles

= length ⋅ width - [π(3)^{2} + π(1)^{2}]

= 8 ⋅ 6 - π [9 + 1]

= 48 - 10π

**Question 4 :**

There are 45 plastic ducks in a bag. If there are black, green, blue and purple plastic ducks and 1/3 of the plastic ducks are black, 1/5 of the plastic ducks are blue, one third of the number of black plastic ducks are green, then how many purple plastic ducks are in the bag ?

(A) 6 (B) 15 (C) 16 (D) 21 (E) 39

**Solution :**

There are 4 colors of plastic ducks, black, green, blue and purple.

Number of black ducks = (1/3) of 45 = 15

Number of blue ducks = (1/5) of 45 = 9

Number of green ducks = (1/3) of 15 = 5

Number of purple ducks = 45 - (15 + 9 + 5)

= 45 - 29

= 16

Hence the number of purple ducks is 16.

**Question 5 :**

In the figure above, point B is on the line segment DC. If AB = BC, what is the measure of the angle ABE ?

(A) 20° (B) 40° (C) 80° (D) 90° (E) 100°

**Solution :**

Since AB and BC are equal, it forms same angle.

<ABC = 180 - (<BAC + <BCA)

= 180 - (30 + 30)

= 180 - 60

= 120

<DBE + <EBA + <ABC = 180

4x + 2x + 120 = 180

6x = 180 - 120

6x = 60 ==> x = 10

2x = 2(10) = 20°

**Question 6 :**

According to the figure above, what is the value of y ?

(A) 3 (B) 6 (C) 12 (D) 15 (E) 18

**Solution :**

Since it is right triangle, we may apply the concept Pythagorean theorem.

30^{2} = 24^{2} + (12+y)^{2}

900 = 576 + (144 + 24y + y^{2})

900 = 720 + 24y + y^{2}

y^{2} + 24y + 720 - 900 = 0

y^{2} + 24y - 180 = 0

y^{2} + 30y - 6y - 180 = 0

y(y + 30) - 6(y + 30) = 0

(y - 6) (y + 30) = 0

y = 6 (or) y = -30

The negative values is not possible. Hence the value of y is 6.

**Question 7 :**

A painter needs 4 gallons of paint to paint each room. If a house has 8 rooms in total, how many quarts of paint is he going to need ?

(A) 12 (B) 32 (C) 128 (D) 200 (E) 512

**Solution :**

A painter needs 4 gallons of paint to paint each room.

Number of gallons of points needed to paint 8 rooms

= 8(4)

= 32 gallons

1 gallon = 4 quats

Number of quats of paint needed to paint = 32 (4) = 128 quats.

**Question 8 :**

A rectangle is inscribed in a circle. The rectangle is tangent at the points A, B, C and D. If the diagonal of the rectangle is 20 inches long. What is the area of the circle ?

(A) 10π (B) 15π (C) 20π (D) 100π (E) 400π

**Solution :**

AC is the diameter of the circle, then radius of the circle

= 10 inches

Area of circle = πr^{2}

= π(10)^{2}

= 100π

**Question 9 :**

The prices in the table given below show the different types of gas offered at a gas station and the prices of gas per gallon. If Emily has $50, what is the least amount of gas, in gallons, she can purchase subtracted from the greatest amount of gas, in gallons, she can purchase ?

(A) 2 gallons (B) 3 gallons (C) 5 gallons

(D) 20 gallons (E) 25 gallons

**Solution :**

Number of gallons of gas filled by choosing unloaded gas

= 50/2.50

= 20 gallons

Number of gallons of gas filled by choosing regular gas

= 50/2

= 25 gallons

= 25 - 20

Required number of gallons = 5 gallons

**Question 10 :**

3[9 ÷ (-3)] + [-3 - (-8)] =

(A) -20 (B) -4 (C) -2 (D) 2 (E) 14

**Solution :**

= 3[9 ÷ (-3)] + [-3 - (-8)]

= 3(-3) + (-3 + 8)

= -9 + 5

= -4

Hence the answer is -4.

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