Question

If 2 is a root of $k{x^4} - 11{x^3} + k{x^2} + 13x + 2$ , what is the value of k?
A). 1
B). 2
C). 3
D). 4

Answer 1

Hint: Roots of any equation satisfy its equation. Means if we replace the variable in an equation by any of its roots then that equation will become zero. In this question, we are going to use this approach to solve.

Complete step-by-step solution:
Given equation: $k{x^4} - 11{x^3} + k{x^2} + 13x + 2$ ………….. (1)
Since it is also given that 2 is a root of $k{x^4} - 11{x^3} + k{x^2} + 13x + 2$ ,So 2 must satisfy this equation.
Putting $x = 2$ in equation (1) , we get
$k{\left( 2 \right)^4} - 11{\left( 2 \right)^3} + k{\left( 2 \right)^2} + 13\left( 2 \right) + 2 = 0$
Solving above equation, we get
$ \Rightarrow 16k - 88 + 4k + 26 + 2 = 0$
$ \Rightarrow 20k - 60 = 0$
$\therefore k = 3$
The value of k is 3. Therefore, option (c) is the correct option.

Note: In these types of questions if the root of the given equation is given and some other value has been asked then put the variable is equal to root in the equation and equate it to 0 to solve the problem.