Question

Let $a,b,c,d$ be numbers in the set {1,2,3,4,5,6} such that the curve $y = 2{x^3} + ax + b$ and $y = 2{x^3} + cx + d$ have no point in common. The maximum possible value of ${\left( {a - c} \right)^2} + b - d$ is
(a). 0
(b). 5
(c). 30
(d). 36

Answer 1

Hint: For the two curves in a variable to have no point in common, there should be no valid value of the variable obtained when both the curves are equated. Use this condition to simplify the given expression and then use hit and trial method to find the maximum possible value.

Complete step-by-step solution -

Given the problem, two curves:
$y = 2{x^3} + ax + b{\text{ (1)}}$
$y = 2{x^3} + cx + d{\text{ (2)}}$
It is given that the above two curves have no pint in common.
Also, $a,b,c,d$ be numbers in the set {1,2,3,4,5,6}.
We need to find the maximum possible value of expression ${\left( {a - c} \right)^2} + b - d$.
For the curves to intersect or to have common points equation $(1)$ and equation $(2)$ should be equal.
Hence for the curves to have no point in common, equation $(1)$ and equation $(2)$ should not be equal.
$
   \Rightarrow 2{x^3} + ax + b \ne 2{x^3} + cx + d \\
   \Rightarrow ax + b \ne cx + d \\
   \Rightarrow \left( {a - c} \right)x \ne d - b \\
   \Rightarrow x \ne \dfrac{{d - b}}{{a - c}}{\text{ (3)}} \\
$
Hence for curve $(1)$ and curve $(2)$ to have no point in common, equation $(3)$ should be valid.
If $a = c$, the quantity $\left( {\dfrac{{d - b}}{{a - c}}} \right)$ will be not defined.
Since the above curves are made of finite points lying on the real plane, equation $(3)$ will be valid for the above condition.
Therefore, for curve $(1)$ and curve $(2)$ to have no point in common, $a = c$ is true.
We need to find the maximum possible value of the expression
${\left( {a - c} \right)^2} + b - d$
Using the above obtained result $a = c$ in the above expression
$
   \Rightarrow {\left( 0 \right)^2} + b - d \\
   \Rightarrow b - d{\text{ (4)}} \\
$
It is given in the problem that numbers $a,b,c,d$ lie in the set {1,2,3,4,5,6}.
Thus, to have maximum value of the expression $(4)$, we choose $b = 6$ and $d = 1$
$ \Rightarrow b - d = 6 - 1 = 5$
Hence expression ${\left( {a - c} \right)^2} + b - d$ has the maximum possible value of 5.
Therefore, option (B). 5 is the correct answer.

Note: In geometry, an intersection is a point, line, or curve common to two or more objects. A point or an ordered pair with either of the coordinates having denominator zero is not defined on the real plane. In the problems related to intersection of curves like above, the equations of the curves are always needed to be equated in order to get conditions regarding the points of intersection and to verify that the same exist or not.