# 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)
Let x and y be real numbers and let
f(x,y) = |x + y|, F(f(x,y)) = -f(x,y) and G(f(x,y)) = -F(f(x,y))
Then which of the following statements is true?
a. F(f(x,y)).G(f(x,y)) = -F(f(x,y)).G(f(x,y)
b. F(f(x,y)).G(f(x,y)) > -F(f(x,y)).G(f(x,y)
c. F(f(x,y)).G(f(x,y)) # G(f(x,y).F(f(x,y))
d. F(f(x,y)) + G(f(x,y)) + f(x,y) = f(-x, -y)

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

Q6. (CAT 1993)
A function f(x) is said to be even if f(-x) = f(x), and odd if f(-x) = -f(x). Thus, for example, the function given by f(x) = x^2 is even, while the function given by f(x) = x^3 is odd.
The function given by f(x) = (|x|)^3 is
a) Even
b) Odd
c) Neither
d) Both

Q7. (CAT 1994)
If f(x) = 2x + 3 and g(x) = (x - 3)/2, What is the value of (gofofogogof)(x)(fogofog)(x)
a) x
b) x^2
c) (5x + 3)/(4x - 1)
d) (x + 3)(5x + 3)/(4x - 5)(4x - 1)

Q8. (CAT 1994)
In the above question, what is the value of fo(fog)o(gof)(x)
a) x
b) x^2
c) 2x + 3
d) (x + 3)/(4x - 5)

Q9. (CAT 1993)
The maximum possible value of y = min ( 1/2 - 3x^2/4, 5x^2/4) for the range 0 < x < 1 is
(a) 1/3
(b) 1/2
(c) 5/27
(d) 5/16

Q10. (CAT 1999)
Let x and y be real numbers and let
f(x,y) = |x + y|, F(f(x,y)) = -f(x,y) and G(f(x,y)) = -F(f(x,y))
What is the value of f( G (f (1,0)), f (F (f (1,2)), G (f (1, 2)))) ?
a) 3
b) 2
c) 1
d) 0

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