Sample 1

Detailed answer to GMAT problems in sample 1.
## Solution to Question 1The length of the third side of a triangle is always less than the sum and greater than the difference of the lengths of other two sides.The sum S and the difference D of the given sides is. S = 3 + 5 = 8 D = 5 - 3 = 2 None of the values suggested in (I), (II) or (III) satisfy the above condition. Answer E ## Solution to Question 2Square both sides of the given equation(√ x) ^{2} = 3^{2}
Simplify to find x x = 9 Elevate both sides to the power 4 x ^{4} = 9^{4}
Simplify x ^{4} = 6561
## Solution to Question 3Since n is odd integer, it can be written asn = 2 k + 1 , where k is an integer Let us express n ^{2} in terms of k as follows
n ^{2} = (2 k + 1)^{2} = 4 k^{2} + 4 k + 1
Let rewrite n ^{2} as follows
n ^{2} = 2(2 k^{2} + 2 k) + 1
Hence n ^{2} is odd.
We now express n ^{2} + 1 in terms of k.
n ^{2} + 1 = 2(2 k^{2} + 2 k) + 1 + 1 = 2(2 k^{2} + 2 k + 1)
Hence n ^{2} + 1 is even.
We now express 3 n ^{2} - 1 in terms of k.
3 n ^{2} - 1 = 3 [2(2 k^{2} + 2 k) + 1
] - 1
= 3 [2(2 k ^{2} + 2 k) ] + 3 - 1
= 6(2 k ^{2} + 2 k) + 2
Hence 3 n ^{2} - 1 is even.
Answer E. ## Solution to Question 4Use the fact that 8 = 2^{3} and 4 = 2^{2}
to rewrite the given expression as follows
8×2 ^{100} + 4×2^{101} = 2^{3}×2^{100} + 2^{2}
×2^{101}
Use rules of exponent to simplify = 2 ^{103} + 2^{103}
= 2×2 ^{103} = 2^{104}
## Solution to Question 5Add the left and right hand sides of the given equations to obtain a new equation(3x + 5y) + (x + 3y) = (5) + (20) Simplify 4x + 8y = 25 Divide all terms of the above equation by 2 2x + 4y = 25 / 2 ## Solution to Question 6n is positive and less than 1 is translated as follows0 < n < 1 Multiply all terms of the above inequality by n to obtain 0 < n ^{2} < n
which also gives n ^{2} - n < 0
Hence statement (I) is true Multiply all terms of the above inequality by n to obtain 0 < n ^{3} < n^{2}
Since n ^{2} < n, we have
0 < n ^{3} < n
Hence statement (II) is true For n = 0.75, statement (III) is not true Hence statement (I) and (II) only are true. ## Solution to Question 7Use decimal numbers to rewrite the above expressionsA) 250% = 250/100 = 2.5 B) 2 + 1/2 = 2 + 0.5 = 2.5 C) 5 × 0.5 = 2.5 D) 1 / 0.1 = 10 E) 4 = 4 The expression 1 / 0.1 has the greatest value. ## Solution to Question 8Factor numerator as follows(4x ^{2} - 4) = 4(x^{2} - 1) = 4(x - 1)(x + 1)
Factor denominator as follows (- 3x + 3) = - 3(x - 1) Substitute in the given expression and simplify (4x ^{2} - 4) / (- 3x + 3) = [ 4(x - 1)(x + 1)
] / [ - 3(x - 1)]
= (- 4/3)(x + 1) ## Solution to Question 9Since x = 2 is a solution of x^{2} + bx = c, then
(2) ^{2} + b(2) = c or 4 + 2b = c
Since x = -3 is a solution of x ^{2} + bx = c, then
(-3) ^{2} + b(-3) = c or 9 - 3b = c
We now have a system of simultaneous equations in b and c to solve. Combining the above equation, we obtain 4 + 2b = 9 - 3b Solve for b 5b = 5 , b = 1 Substitute b by 1 in the equations 4 + 2b = c and solve for c 4 + 2(1) = c or c = 6 Values for b and c b = 1 , c = 6 ## Solution to Question 10Rewrite as(√12 - √3)(-√12 + √3) = - (√12 - √3) (√12 - √3) and simplify = - (12 - 3) = -9
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