Question

If the expression $\begin{vmatrix}a&c\\d&b\end{vmatrix}$ has the value $ab-cd$ for all values of $a,$ $b,$ $c,$ and $d,$ then the equation $\begin{vmatrix}2x&1\\x&x\end{vmatrix}=3$

Is satisfied for only $1$ value of $x$

Is satisfied for $2$ values of $x$

Is satisfied for no values of $x$

Is satisfied for an infinite number of values of $x$

None of these

Answer: B

Concepts:determinant quadratic

Difficulty rating: 1600

Small Hint:

Apply the given rule to turn the determinant into a quadratic equation

Big Hint:

The equation is $2x^2-x=3$

Solution:

The given rule turns the equation into $(2x)(x)-(1)(x)=3,$ or $2x^2-x-3=0.$ Factoring gives $(2x-3)(x+1)=0,$ whose two distinct solutions are $x=\tfrac32$ and $x=-1.$

Thus, the correct answer is B.

1950 AMC 12 Problem 38

Answer: B

Difficulty rating: 1600

Small Hint:

Big Hint:

Solution:

Problem 38 in Other Years