Let a and b are roots of x2 – 7x – 1 = 0. The value of \(\frac{a^{21} + b^{21} + a^{17} + b^{17}}{a^{19} + b^{19}}\) is?
29
49
53
51
The Correct Option is D
Solution and Explanation
‘a’ and ‘b’ are roots of \(x^2 -7x -2 =0\) to find \(\frac{a^{17}( a^4+1) + b^{17}(b^4 + 1) }{a^{19} + b^{19}}\)
Considering one of the root ‘\(\alpha\)’ for the equation;
\(\alpha ^2 - 1 = 7\alpha\)
⇒ \(\alpha ^4 + 1 = 51\alpha ^2\)
∴\(\frac{51a^{19} + 51b^{29}}{a^{19}+ b^{19}}\) [Here, consider as \(\large\alpha^2\)\(\large<^{\large{a}}_{\large{b}}\) ]
\(=51(\frac{a^{19} + b^{29}}{a^{19}+ b^{19}})\)
\(=51\)
Hence, The correct answer is the option (D) 51.
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Concepts Used:
Quadratic Equations
A polynomial that has two roots or is of degree 2 is called a quadratic equation. The general form of a quadratic equation is y=ax²+bx+c. Here a≠0, b, and c are the real numbers.
Consider the following equation ax²+bx+c=0, where a≠0 and a, b, and c are real coefficients.
The solution of a quadratic equation can be found using the formula, x=((-b±√(b²-4ac))/2a)
Two important points to keep in mind are:
- A polynomial equation has at least one root.
- A polynomial equation of degree ‘n’ has ‘n’ roots.
Read More: Nature of Roots of Quadratic Equation
There are basically four methods of solving quadratic equations. They are:
- Factoring
- Completing the square
- Using Quadratic Formula
- Taking the square root