Question Bank  100 Algebra Questions From Previous CAT Papers (Solved)

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

Q41. (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

Q42. (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 (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

Q43. (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

Q44. (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)

Q45. (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)

Q46. (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

Q47. (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

Q48. (CAT 1991)
A function can sometimes reflect on itself, i.e. if y = f(x), then x = f(y). Both of them retain the same structure and form. Which of the following functions has this property?
(a) y = ( 2x + 3 ) / ( 3x + 4 )
(b) y = ( 2x + 3 ) / ( 3x  2 )
(c) y = ( 3x + 4 ) / ( 4x  5 )
(d) None of the above

Q49. (CAT 1991)
What is the value of k for which the following system of equations has no solution:
2x – 8y = 3 and kx +4y = 10
(a) –2
(b) 1
(c) –1
(d) 2

Q50) Q3. (CAT 1991)
If y = f(x) and f(x) = (1x) / (1 + x), which of the following is true?
(a) f (2x) = f (x) – 1
(b) x = f (2y)1
(c) f (1/x) = f (x)
(d) x = f (y)

Q51. (CAT 1991)
Let Y = minimum of {(x+2), (3x)}. What is the maximum value of Y for 0 ≤ x ≤ 1
(a) 1.0
(b) 1.5
(c) 3.1
(d) 2.5

Q52. (CAT 1997)

Q53. (CAT 1990)
The roots of the equation ax^2 + 3x + 6 = 0 will be reciprocal to each other if the value of a is
(a) 3
(b) 4
(c) 5
(d) 6

Q54. (CAT 2003 Retest)
The number of roots common between the two equations x^3 + 3x^2 + 4x + 5 = 0 and x^3 + 2x^2 + 7x + 3 = 0 is:
(1) 0
(2) 1
(3) 2
(4) 3

Q55. (CAT 2001)
Ujakar and Keshab attempted to solve a quadratic equation. Ujakar made a mistake in writing down the constant term. He ended up with the roots (4, 3). Keshab made a mistake in writing down the coefficient of x. He got the root as (3, 2). What will be the exact roots of the original quadratic equation?
(1) (6, 1)
(2) (–3, –4)
(3) (4, 3)
(4) (–4, –3)

Q56. (CAT 2003 Retest)
The number of non negative real roots of 2^x  1  1 = 0 is
a) 0
b) 1
c) 2
d) 3

Q57. (CAT 2003 Leaked)
When the curves y = logx (base 10) and y = 1/x are drawn in the XY plane, how many times do they intersect for values of x ≥ 1
a) Never
b) Once
c) Twice
d) More than twice