maths _21

51. x - 5 > -7 and 2x - 4 <= 6

x > -7 + 5 and 2x <= 6 + 4

x > -2 and x <= 10/2

x > -2 and x <= 5

So, the combined solution is -2 < x <= 5

Therefore, the possible values of x are -1, 0, 1, 2, 3, 4 and 5

So, the number of possible values of x is 7. The answer is D.


52. For x >= 0, it is the region to the right of the y-axis, and the four regions fulfill the condition.

For y >= 0, it is the region above the x-axis, again, the four regions fulfill the condition.

For 2x - y >= 4, it is the region to the right of the line 2x - y = 4, so, it is either region III or region II.

For x + y <= 3, it is to the left of the line x + y = 3

So, it is region III.


50. f(x) = m2x4 + mx3 + m2x2 - 1

As f(x) is divisible by x + m, so by factor theorem,

f(-m) = 0

m2(-m)4 + m(-m)3 + m2(-m)2 - 1 = 0

m6 - 1 = 0

(m3 - 1)(m3 + 1) = 0

m = 1 or -1

Without loss of generosity, take m = 1

So, by remainder theorem, the remainder when f(x) is divided by x - m (x - 1) is

f(1) = 1 + 1 + 1 - 1 = 2

So, the answer is C.

P.S. you will get the same answer if you take m = -1.


47. 4x - 3‧2x - 4 = 0

(2x)2 - 3‧2x - 4 = 0

(2x - 4)(2x + 1) = 0

2x = 4 or -1 (rej.)

2x = 22

x = 2

So, the answer is B.