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QUESTION: 1

The area of a triangle is equal to the area of the rectangle. Find the perimeter of the rectangle.

1. The perimeter of the square is 24 inches.

2. The sum of the length and the width is 13 inches.

Solution:

Area of triangle = Area of rectangle

In statement 1, perimeter of the square is 24 = 4s, where s represents the side, s = 6. Therefore, area = 6^{2} = 36 square inches.

Let l and w be the length and width of the rectangle respectively, then lw =36.

Thus, we cannot determine w or l hence the statement is insufficient.

In statement 2, the sum of the length (l) and width(w) is 13.

Hence l + w = 13 hence w = 13 - l. This is an equation in two unknowns hence, we cannot determine the value of w or l. Therefore, the statement is not sufficient.

Combining the two statements, we have l = 13 - w and lw =36, substituting for l, we have (13 - w)w = 36

w^{2} - 13w + 36 = 0; w^{2} - 9w - 4w + 36 = 0

(w - 9)(w - 4) = 0, w =9, w = 4.

Since let w = 9, then l = 4 so that 9 + 4 =13, thus, w must be 4.

w = 4.

Perimeter = 2(4 + 9) = 26 inches. Therefore, Correct answer: BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

QUESTION: 2

A particle moving in air increases its speed within 30 minutes. Find its acceleration.

1. Its initial velocity is 20miles per hour and its final velocity is 25 miles per hour.

2. The particle increases its speed by 5 miles per hour.

Solution:

Time taken = change in time = 30 minutes = 1/2 hours

In statement 1, Initial velocity (v_{1}) = 20 miles per hour and change in time = 0.5 hours (given)

Final velocity (v_{2}) = 25 miles per hour.

Acceleration = (v_{2} - v_{1})/ change in time = (25 - 20)/ 0.5 = 10 miles per hour per hour. Therefore, the statement is sufficient.

In statement 2, Change in velocity = 5 miles per hour and change in time = 0.5 hours (given).

Acceleration = Change in velocity/change in time = 5/0.5 =10 miles per hour per hour. Therefore, the statement is sufficient.

Therefore, EACH statement ALONE is sufficient.

QUESTION: 3

Are the two lines L1 and L2 parallel?

1. Both lines lie in the first, second and fourth quadrants.

2. The y intercepts of the lines L1 and L2 are 8 and 4 respectively.

Solution:

Parallel lines have equal slopes.

In statement 1, if both lines are in the first, second and fourth quadrant then they have a negative slope. This alone is not enough to prove that they are parallel or not, hence, the statement is insufficient.

In statement 2, the y intercepts are 8 and 4 to imply that the equation are of the form y = mx + 8 and y = bx + 4. But this does not enough to determine if the lines area parallel since, notheing is said about the value of m and b, hence, the statement too is insufficient.

Combining the two statements, we have equations having negative gradients, thus, y =-mx + 8 and y = -bx + 4. Since we are not sure of -m = -b, we cannot say that they are parallel on not. Therefore, Statements (1) and (2) TOGETHER are NOT sufficient.

QUESTION: 4

s,p and q are interior angles of an Isosceles triangle. Find the value of q.

1. s = 72°.

2. p and q are base angles of the triangle.

Solution:

Since s, p and q are interior angles of Isosceles triangle, s + p + q = 180°.

In statement 1, If s = 72°, then p + q + 72 = 180° and

p + q = 180°.

Since we have two unknowns in one equation and we are not sure which angles are base angles, we cannot determine the value of q, hence the statement is not sufficient.

In statement 2, p and q are base angles of the triangle, hence from s + p + q = 180°, we have s + 2q = 180°, but p = q.

But the statement is not sufficient since it does not have any information about s.

Combining the two, we have s + 2q = 180° and s = 72°, we have

72° + 2q = 180°; 2q = 180 - 72 = 108°. q = 54°.

Therefore, BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

QUESTION: 5

Is A an obtuse angle?

1. A is more than 90°.

2. A is a supplement of an angle B, an acute triangle.

Solution:

An obtuse angle is an angle more than 90° and less than 180°.

In statement 1, A is more than 90°. This implies that A can be as well take a value more than 180° to be a reflex angle or between 90° and 180° to be an obtuse angle. Therefore, the statement is not sufficient.

In statement 2, if B is an acute angle, let B = 45, then A will be its supplement, that is, A = 180° - 45 = 135° an obtuse angle. This is true for all values of B taken. Hence statement B is sufficient.

QUESTION: 6

Determine the value of angle k.

1. Angle k and m lies on a straight line.

2. Angle m = 39° .

Solution:

In statement1, k and m lie on a straight line hence k + m =180°. Since m is unknown, the statement is insufficient.

In statement 2, m = 39°; since there is no relationship between m and k, we cannot find the value of k. Thus the statement is not sufficient.

Combining the two statements, we have k + m = 180°, and m = 39°.

Substituting for m, we have k + 39 = 180, k = 141°. Thus, BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

QUESTION: 7

A straight line L passes through (2,8) and the origin. Find the equation of a line perpendicular to L.

1. The line passes through the origin.

2. The line passes through (2,-0.5).

Solution:

The line L passes through (2,8) and (0,0) hence its slope is

slope = (8 - 0)/(2 - 0) = 4.

Since L is perpendicular to the line in question, the product of their slope is -1. Therefore, the slope of the line in question is -1/4.

In statement 1, the line passes the origin, (0,0) and its slope is -1/4 hence its equation is y = -x/4. The statement is sufficient.

In statement 2, line passes through the (2,0.5) and its slope is -1/4. Thus we have (y + 0.5)/(x - 2) = -1/4

y + 1/2 = -x/4 + 1/2

thus y = -x/4. The statement is sufficient too.

Thus, EACH statement ALONE is sufficient.

QUESTION: 8

Two pipes supply waters to a cistern whose capacity of 15 cubic feet. How long does it take the two pipes to fill the cistern?

1. The first pipe supplies water at a rate (per minute) that is thrice faster than the second pipe.

2. The pipes fill 8 cubic feet of the tank in ten minute.

Solution:

The capacity of cistern is 15 cubic feet.

In statement 1, let the second pipe supply water a rate of x feet per minute. The rate of the first pipe is 3x feet per minute. Thus, the statement is insufficient.

In statement 2, the pipes fill 8 cubic feet in 10 minute. Therefore, 15 cubic feet will be filled in (15 × 10)/8 = 18.75 minutes. Hence, the statement is sufficient.

QUESTION: 9

Is 2x + 1 > 0.

1. x is an integer

2. |x| < 1.5

Solution:

2x + 1 > 0

In statement 1, a is an integer, when x = -2, 2x + 1 = -3 < 0. When x = 2, 2x + 1 = 5 > 0, hence the statement is not sufficient.

In statement 2, |x| < 1.5 implies that -1.5 < x < 1.5.

When x = 1.4, 2x + 1 = 3.8 > 0. When x = -1.4, 2x + 1 = -1.8 < 0.

Hence, the statement is not sufficient.

Combining the two statements, we have, x, an integer and -1.5 < x < 1.5, considering the more strict condition, -1.5 < x < 1.5, we find that 2x + 1 < 0, when x = -1.4 and 2x + 1 > 0 when x = 1.4.

Therefore, Statements (1) and (2) TOGETHER are NOT sufficient.

QUESTION: 10

Two numbers 12 and t are two positive numbers with some similar properties. What is the value of t.

1. The Least Common Multiple of the two numbers 48.

2. The Greatest Common multiple of the two numbers is 4.

Solution:

We consider 12 and t.

In statement 1, the Least Common multiple of 12 and t is 48 to mean that t is a factor of 48. This mean that, t can take 16, 24 and 48. Hence the statement is not sufficient.

In statement 2, the Greatest Common Multiple is 4. Therefore, t can take 4, 8, 16, 28, 32, 40 among others. Thus, the statement is not sufficient.

Combining statements 1and 2, we have 16 featuring in both lists, hence it satisfies the two conditions. Therefore, t =16. Thus BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

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