Solved CAT Questions (Algebra)  Set 5

Q1. (CAT 2000)
Each of the number x1, x2, x3 ... xn (n > 4) is equal to 1 or 1.
Suppose x1x2x3x4 + x2x3x4x5 + x3x4x5x6 + .... + x(n1)xnx1x2 + xnx1x2x3 = 0. then
a) n is even
b) n is odd
c) n is an odd multiple of 3
d) n is primeWe can see that total number of terms in the given expression is n.
Now to get a sum of 0 there should be PAIRS of 1 and 1.
So n should be even.Q2. (CAT 1999)
G(f(x, y)) + F(f(x, y) = 0  (1)
as f (x, y) =  x + y 
f(x, y) = f(x, y)
So d satisfies.Q3. (CAT 1997)
Log_2 [ Log_7 (x^2  x + 37) ] = 1, then what could be the value of x ?
a) 3
b) 4
c) 5
d) None of theseLog_2 [ Log_7 (x^2  x + 37) ] = 1
=> Log_7 (x^2  x + 37) = 2^1
=> x^2  x + 37 = 7^2 = 49
=> x^2  x  12 = 0
x = 4Q4. (CAT 1997)
If roots p and q of the quadratic equation x^2  2x + c = 0 also satisfy the equation 7q  4p = 47, then which of the following is true
a. c = 15
b. p = 5, q = 3
c. p = 4.5, q = 2.5
d. None of thesep + q = 2
7q  4p = 47
solve and we have, p = 3 and q = 5
c = p x q = 15Q5. (CAT 1996)
Which of the following values of x do not satisfy the equation x^2  3x + 2 > 0 at all.
a. 1 ≤ x ≤ 2
b. 1 ≥ x ≥ 2
c. 0 ≤ x ≤ 2
d. 0 ≥ x ≥ 2On solving x^23x+2>0 , we get (x1)(x2)>0
That is the roots are x = 1 or x = 2
The equation assumes a positive value for all values which belong to the interval less than 1 and greater than 2
For x = 1 and x = 2, it becomes zero and for any value of x between 1 and 2 it becomes negative.
So, option a