Question

Let
$f(x) = \begin{vmatrix} a & -1 & 0\\\ ax & a & -1\\\ ax^2 & ax & a \end{vmatrix}$
a ∈ R. Then the sum of the square of all the values of a, for which 2f′(10) –f′(5) + 100 = 0, is

• A. 117
• B. 106
• C. 125
• D. 136

Answer 1

Answer: (C)

125

Answer 2

The correct answer is (C) : 125
$f(x) = \begin{vmatrix} a & -1 & 0\\\ ax & a & -1\\\ ax^2 & ax & a \end{vmatrix},$ $a∈R$
f(x) = a(a2 + ax) + 1(a2x + ax2) = a(x + a)2
f′(x) = 2a(x + a)
Now, 2f′(10) – f′(5) + 100 = 0
⇒ 2·2a(10 + a) – 2a(5 + a) + 100 = 0
⇒ 2a(a + 15) + 100 = 0
⇒ a2 + 15a + 50 = 0
⇒ a = –10, –5
Therefore,
Sum of squares of values of a is 125.