XAT Inequalities questions

Given $$A = |x + 3| + | x - 2 | - | 2x -8|$$. The maximum value of $$|A|$$ is:

Consider the real-valued function $$f(x)=\frac{\log{(3x-7)}}{\sqrt{2x^{2}-7x+6}}$$ Find the domain of f(x).

Find z, if it is known that:
a: $$-y^2 + x^2 = 20$$
b: $$y^3 - 2x^2 - 4z \geq -12$$ and
c: x, y and z are all positive integers

Consider the four variables A, B, C and D and a function Z of these variables, $$Z = 15A^2 - 3B^4 + C + 0.5D$$ It is given that A, B, C and D must be non-negative integers and thatall of the following relationships must hold:
i) $$2A + B \leq 2$$
ii) $$4A + 2B + C \leq 12$$
iii) $$3A + 4B + D \leq 15$$
If Z needs to be maximised, then what value must D take?

Consider the function f(x) = (x + 4)(x + 6)(x + 8) ⋯ (x + 98). The number of integers x for which f(x) < 0 is:

If 2 ≤ |x - 1| × |y + 3| ≤ 5 and both x and y are negative integers, find the number of possible combinations of x and y.

If $$x$$ and $$y$$ are real numbers, the least possible value of the expression $$4(x - 2)^{2} + 4(y - 3)^{2} - 2(x - 3)^{2}$$ is :

a, b, c are integers, |a| ≠ |b| ≠|c| and -10 ≤ a, b, c ≤ 10. What will be the maximum possible value of [abc - (a + b + c)]?

Consider the expression $$\frac{(a^2+a+1)(b^2+b+1)(c^2+c+1)(d^2+d+1)(e^2+e+1)}{abcde}$$, where a,b,c,d and e are positive numbers. The minimum value of the expression is

p, q and r are three non-negative integers such that p + q + r = 10. The maximum value of pq + qr + pr + pqr is

If $$x=(9+4\sqrt{5})^{48} = [x] +f$$, where [x] is defined as integral part of x and f is a fraction, then x (1 - f) equals