GRE
Practice Test 4
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If the radius of the circle with centre O is 7 and the measure of angle AOB is 100, what is the best approximation to the length of arc AB ?

  • Solution

    If the radius is 7, the circumference = 14π
    The length of the arc is 100/360 of the circumference
    Taking π as 22/7 we get
    (100 x 14 x 22) / (360 x 7) which reduces to 440/ 36 = 12.22 (i.e. approx. 12)

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All the dots in the array are 2 units apart vertically and horizontally. What is the length of the longest line segment that can be drawn joining any two points in the array without passing through any other point ?

 

  • Solution

    The longest line segment that can be drawn without passing through any dots other than those at the beginning and end of the segment, such a line could go from the middle dot in the top row to either the bottom left or right dot. In any case the segment will be the hypotenuse of a right triangle with sides 2 and 4. Using Pythagoras theorem the hypotenuse will be √(2 ² + 4 ² ) = √20

A dress on sale in a shop is marked at $D. During the discount sale its price is reduced by 15%. Staff are allowed a further 10% reduction on the discounted price. If a staff member buys the dress what will she have to pay in terms of D ?

 

  • Solution

    If the price is reduced by 15 %, then the new price will be 0.85D
    If this new price is further reduced by 10%, the discounted price will be 0.9 x 0.85D = 0.765D

n is an integer chosen at random from the set

{5, 7, 9, 11 }

p is chosen at random from the set
{2, 6, 10, 14, 18}

What is the probability that n + p = 23 ?

 

  • Solution

    Each of the integers in the first set could be combined with any from the second set, giving a total of 4 x 5 = 20 possible pairs.
    Of these the combinations that could give a sum of 23 are (5 + 18), and (9 + 14)
    This means that the probability of getting a sum of 23 is 2/20 = 1/10

Half the people on a bus get off at each stop after the first, and no one gets on after the first stop. If only one person gets off at stop number 7, how many people got on at the first stop?

 

  • Solution

    Work backwards. If one person got off at stop 7, as the bus approached the stop there must have been two people on board. As it approached the previous stop (no. 6) there were four on board. Approaching stop 5, there were 8. Approaching stop 4, there were 16 on board. Approaching stop 3, there were 32. Approaching stop 2, there were 64. As stop 2 was the first stop at which people got off there must have been a total of 64 people who boarded at the first stop. (We are assuming that no people could have been on the bus before the first official stop.)

A perfect cube is an integer whose cube root is an integer. For example, 27, 64 and 125 are perfect cubes. If p and q are perfect cubes, which of the following will not necessarily be a perfect cube?

 

  • Solution

    A perfect cube will have prime factors that are in groups of 3; for example 125 has the prime factors 5 x 5 x 5 , and 64 x 125 will also be a cube because its factors will be 4 x 4 x 4 x 5 x 5 x 5
    Consider the answer choices in turn.
    8 is the cube of 2, and p is a cube, and so the product will also be a cube.
    pq will also be a cube as shown above.
    pq is a cube and so is 27, but their sum need not be a cube. Consider the case where p =1 and q = 8, the sum of pq and 27 will be 35 which has factors 5 x 7 and is not a cube.
    -p will be a cube.
    Since the difference between p and q is raised to the power of 6, this expression will be a cube (with cube root = difference squared).

certain animal in the zoo has consumed 39 pounds of food in six days. If it continues to eat at the same rate, in how many more days will its total consumption be 91 pounds?

  • Solution

    Food consumed per day = 39/6.
    In the remaining days it will consume 91 - 39 pounds = 52 pounds
    Now divide the food by the daily consumption to find the number of days
    52 / (39/6) = 52 x (6 / 39) = 8

One side of a triangle has length 8 and a second side has length 5. Which of the following could be the area of the triangle?

Select ALL that apply.

 

 

  • Solution

    Correct Answer: B,C

    Explanation:

    The maximum area of the triangle will come when the given sides are placed at right angles. If we take 8 as the base and 5 as the height the area = ½ x 8 x 5 = 20
    We can alter the angle between the sides to make it less or more than 90, but this will only reduce the area. (Draw it out for yourself). Hence the area can be anything less than or equal to 20.

If n is even, which of the following cannot be odd?

Select all that apply.

 

  • Solution

    Correct Answer: B,D

    Explanation:

    In case A , even plus odd will give odd
    In case B, odd times even will give even
    In case C, even squared is even, and even minus odd is odd.
    In case D, two times an odd number is even
    (You can check this by using an easy even number like 2 in place of n)
    Only B and D cannot be odd.

If a² = 12, then a4 =

  • Solution

    a4 = a2 x a2 = 12 x 12 = 144

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