Find the area of a right angle triangle whose base is 12 inches.
1. The hypotenuse is 13 inches.
2. The perpendicular height of the triangle is one less than half its base.
Base = 12 inches.
In statement 1, hypotenuse = 13 inches. Using Pythagorean theorem, we have height = √(13²  12²) = 5.
Area = 1/2 × 5 × 12 = 30 square inches. The statement is sufficient.
In statement 2, perpendicular height = 1/2 (base)  1 = 1/2 × 12 1 = 5 inches. Base = 12 inches.
Area =1/2 × 5 × 12 = 30 square inches. The statement is sufficient.
Is the number a prime number?
1. The number is divisible by a prime factor.
2. The number is positive
In statement 1, the number is divisible by a prime factor, hence the number can be 2, 4, 6, 2, 4, 6, 8 , 9 . . . . Hence the statement is insufficient.
In statement 2, the number is positive. But there are infinitely many numbers that are positive but not prime, hence, the statement is not sufficient.
Combining the two statements, we have the numbers being positive, thus, 2,4,6,8,9,10,. . . . The numbers are infinitely many, hence Statements (1) and (2) TOGETHER are NOT sufficient.
Find the direction in which the parabola y = ax^{2 }+ bx  2 is facing.
1. a = b
2. a < 0
The parabola y = ax^{2} + bx  2 faces upwards when a > 0 and downwards when a < 0.
In statement 1, a = b, this does not give the numerical equivalent to a, hence it is not sufficient.
In statement 2, a < 0, hence the parabola faces downwards. Thus the statement is sufficient.
Find the equation of a line.
1. Its x and y intercept is 2 and 2 respectively.
2. The slope of the line is 1.
In statement 1, the x and y intercept is 2 and 2 respectively, hence the line passes through (2,0) and (0,2). The slope is
(2  0)/(2  0) = 1.
The equation is y/(x  2) = 1 hence x = x + 2.
The statement is sufficient.
In statement 2, the slope is 1. Since we are not given any point where the line is passing, the statement is insufficient. Thus, Statement (1) ALONE is sufficient but statement (2) is not sufficient.
Determine the size of an interior angle of the polygon.
1. The ratio of its interior angle to the exterior angle is 2:1.
2. The polygon is a regular hexagon
In statement 1, the ratio interior angle to the exterior angle is 2:1. Since two angles lies on a straight line they add up to 180°. Taking x as the constant of proportionality, we have 2/3x = 180, x = 270.
The interior angle is 2/3x = 2/3 × 270 =180°.
In statement 2, the polygon is a regular hexagon, a 6 sided figure. The sum of interior sides is given by (n  2)180°; since n = 6, we have (6  2) 180 =720.
Interior side = 720/6 =120°. The statement is sufficient. Thus, Statement (2) ALONE is sufficient but statement (1) is not sufficient.
Find out if t < 0.
1. t > t
2. t^{2} > 0
In statement 1, t > t. But is t = t. Hence t can be positive or negative. When t > t, it implies that t is a negative value, hence t < 0. The statement is sufficient.
In statement 2, t^{2} > 0 , but when t is negative or positive, t^{2} > 0 , hence the statement is not sufficient. Thus Statement (1) ALONE is sufficient but statement (2) is not sufficient.
Determine the value of t.
1. 2t + 6s = 8
2. t/2  2 =  3s/4
In statement 1, 2t + 6s = 8 is one equation is two unknowns, hence we cannot determine the value of t. The statement is insufficient.
In statement 2, t/2  2 = 3s/4 can be transformed to t  4 = 6s/2. But this is an equation with two unknowns hence we cannot determine the value of t. The statement is insufficient.
Combining the two statements, we have 2, t/2  2 = 3s/4 which can be transformed to 2t  8 = 3s. Thus, we can determine the value of t and s. Therefore, statements (1) and (2) TOGETHER are sufficient.
Find the percentage change in the volume of cylinder.
1. The diameter is increased by 20%.
2. The height is increased by 21%.
The volume of the cylinder is given buy v = πr²h.
In statement 1, diameter (d) is increased by 20%, hence the radius is increased by 10%. The new radius is given by 1.1r. The new volume = π(1.1r)²h = 1.21πr²h
Percentage change in volume = (1.21πr²h  πr²h )/pr²h × 100% = 21%. The statement is sufficient.
In statement 2, height is increased by 21%, hence the new height is 1.21h. The new volume = 1.21πr²h.
Percentage change in volume = (1.21πr²h  πr²h )/πr²h × 100% = 21%. The statement is sufficient. Therefore, EACH statement ALONE is sufficient.
a < b. Is a positive?
1. b = 0.
2. √a < a
In statement 1, b =0, when a < b and b = 0, it implies that a < 0, hence, a is negative and a is not positive. The statement is sufficient.
In statement 2, √a < a, the square root of a number is either negative or positive, when √a < a, then √a is negative.
√a < a. But this is not enough to determine the sign of a. Hence, the statement is not sufficient. Thus, Statement (1) ALONE is sufficient but statement (2) is not sufficient.
Determine the equation of the circle passing through (4,2).
1. (1,1) lies in the circle.
2. The center of the circle is the origin.
The circle passes through (4,2).
In statement 1, (1,1) lies in the circle; since the statement does not specify if the point is the center, we cannot determine its radius and the center, hence, it is insufficient.
In statement 2, the center is the origin, using (4,2) the radius of the circle is √((4  0)² + (2  0)²) = √(16 + 4) = √20 units.
The equation is (x  0)² + (y  0)² = 20 ; x² + y² = 20. The statement is sufficient. Thus, Statement (2) ALONE is sufficient but statement (1) is not sufficient.
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