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(n-1)xnx1x2 + xnx1x2x3 = 0. then
a) n is even
b) n is odd
c) n is an odd multiple of 3
d) n is prime

We 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 these

Log_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 = 4

Q4. (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 these

p + q = 2
7q - 4p = 47
solve and we have, p = -3 and q = 5
c = p x q = -15

Q5. (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 ≥ 2

On solving x^2-3x+2>0 , we get (x-1)(x-2)>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

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