 # GMAT - Questions on Algebra, Sample Problems  ### 1. Algebra

1. Statement: a + c > b + c

The statement above logically conveys which of the following?

A. ac>bc
B. a/c>b/c
C. a-c > b-c
D. a2 >bc
E.b2 = ac

2. If 42.28 = n(14 + m/50), where k and m are positive integers and m < 50, then which of the following is true?
A. n + m  >n - m
B. nm = 4
C. n + m  >4
D. n + m <4
E. n + m  = 4

3. Given a2 + b2 = 1, c2 + d2 = 1, p2 + q2 = 1, where the numbers considered are all real, then find the maximum value of  ab + cd + pq.
A. 3/2
B. 5/2
C. 3
D. 6
E. None of these

4. A spiral is made up of successive semicircles, with centres alternately at A and B, starting with centre at A, of radii 0.5 cm, 1.0 cm, 1.5 cm, 2.0 cm,.....as shown in figure. What is the total length of such a spiral made up of thirteen consecutive semicircles? A. 143 cm
B. 146 cm
C. 149 cm
D. 151 cm
E. None of these

5. Let N = 0.011010100010100010100010000010100001........

Where the digit in the nth decimal place is 1 when n = 1 or a prime and 0 when ‘n’ is composite.

The statement above logically conveys which of the following?

A. N is a prime number

B. N is a composite number

C. N is a rational number

D. N is an irrational number

E. N is either a prime number or a composite number

6. Statement:2K – 1 is a prime number.

The statement above logically conveys which of the following?

A. k is a prime number

B. k is a composite number

C. k is an even number

D. k is an odd number

E. None of these

7. Natashapractices for a cross-country meet. On Monday, she ran a certain distance. On Tuesday, she ran twice as far as she ran on Monday. On Wednesday, she ran half as far as she ran on Tuesday. On Thursday, she ran half as far as she ran on Wednesday. On Friday, she ran twice as far as she ran on Thursday. If the shortest distance that she ran on any of the ï¬ve days is 10 km, how far did she run in total?
A. 72 km
B. 75 km
C. 85 km
D. 100 km
E. 110 km

8. If x2 + 8x + 7 < 0, then
A. –7 < x < –1
B. 7 < x < –1
C. -7 < x <1
D. -7 ≤ x < –1
E. None of these

9. Statement:  <1 and -1

Which of the following statement is logically correct?

A. a < b < 0
B. b>a
C. b     D. a > b
E. b2 = a2

10. The number of solutions of the equation 2x + 3y = 40, such that both x and y are natural numbers, is

A. 20
B. 13
C. 6
D. 9
E. None of these

## Solutions

1. C
Given, a + c > b + c, =>a > b.
If c < 0, then same is not true for ac > bc.
Similarly, If c < 0, then same is not true for a/c > b/c. (In both cases inequality must reverse).

2. E
Given, 42.28 = n(14 + m/50), => m = 50(42.28/n-14)

But m < 50, which is true only when  (42.28/n-14 )<1, => n>42.28/15.

=> n > 2.8, When n =3 and m = 1, m + n = 4.

3. A
Since (a – b)2 ≥ 0 , a2 + b2> 2ab,
Similarly, c2 + d2 ≥ 2cd and p2 + q2 ≥ 2pq a2 + b2 + c2 + d2 + p2 + q2 ≥ 2 (ab + cd + pq)
=>3/2 ≥ ab + cd + pq,   ab + cd + pq ≤ 3/2.

4. A
First semicircle is drawn with centre A and radius 0.5 cm.

Its semi - circumference = 22/7 X 1/2 = 11/7

Second semicircle is drawn with centre B and radius 1 cm.

Its semi - circumference = 22/7 X 1  = 22/7

Third semicircle is drawn with centre A and radius 1.5 cm.

Its semi - circumference = 22/7 X 1.5  = 33/7 and so on.  l1,l2,l3,...are in A.P. with common difference = 11/7. Total length = S13 = 13/2[2x11/7+(13-1)x11/7] = 143 cm

5. D
i.    We have to observe that ‘N’ is made up of the digits 0 and 1.
ii.    And we have to understand that 1’s are placed when the decimal place is a prime place and 0’s are placed when the decimal place is a Composite place
iii.    We know that primes occur at irregular intervals on the number line. Hence we put 1’s irregularly that is non-periodically . Therefore ‘N’ is non-periodic.
iv.    Primes as well as composites are infinitely many.  Therefore ‘N’ is a nonterminating decimal number.
v.    Now, ‘N’ is a decimal which is non-periodic and nonterminating
’N’ is an Irrational number.

6. C
Since every number except 1 is either a prime or a composite number but cannot be both. 1 is neither a prime number nor a composite number.

7. E

Let distance covered by Natasha on Monday be x km. Distance covered on Tuesday = 2x, distance covered on Wednesday =x km, distance covered on Thursday = 1/2x km and distance covered on Friday = x km. The shortest covered on any of the five days is 5 km, It must be on Thursday, => 1/2x = 10 or x = 20. Total distance covered = 20 km +40 km + 20 km +10 km + 20 km = 110 km

8. A x2 + 8x + 7 < 0
=>(x + 1) (x + 7) < 0
=>–7 < x < –1.

9. C
Working with options :

A. If b = -1 then a/b < 1 but b < a.

B. If a = -0.5 and b = -2 then a/b < 1 but b < a.

C. If b < a then a/b < 1 as -1< a < 0

10. C
2x and 40 are even numbers. Therefore 3y must an even number and a multiple of 6. Since between 0 and 40, there are 6 even multiple of 6, hence there are 6 values of (x, y) which satisfy the given equation.