JEE Advanced · Mathematics · 2. Quadratic Equations
Paragraph:
Let \(p, q\) be integers and let \(\alpha, \beta\) be the roots of the equation, \(x^{2}-x-1=0\), where \(\alpha \neq \beta\). For \(n=0,1,2, \ldots\), let \(a_{n}=p \alpha^{n}+q \beta^{n}\)
FACT: If \(a\) and \(b\) are rational numbers and \(a+b \sqrt{5}=0\), then \(a=0=b\).
Question:
\(a_{12}=\)
- A
- B
- C
- D
Answer & Solution
Correct Answer
(C)
Step-by-step Solution
Detailed explanation
\(\alpha^2=\alpha+1 \Rightarrow \alpha^n=\alpha^{n-1}+\alpha^{ n -2}\)
\(\Rightarrow p \alpha^n+q \beta^n=p\left(\alpha^{n-1}+\alpha^{n-2}\right)+ q\) \(\left(\beta^{n-1}+\beta^{n-2}\right)\)
\(a_n=a_{n-1}+a_{ n -2}\)
\(\Rightarrow a_{12}=a_{11}+a_{10}\)
\(\Rightarrow p \alpha^n+q \beta^n=p\left(\alpha^{n-1}+\alpha^{n-2}\right)+ q\) \(\left(\beta^{n-1}+\beta^{n-2}\right)\)
\(a_n=a_{n-1}+a_{ n -2}\)
\(\Rightarrow a_{12}=a_{11}+a_{10}\)
See the Complete Solution
Get step-by-step explanations for this and 2.5 Lakh+ more JEE, NEET & CET questions.
- Unlock all solutions
- Practice the full chapter
- Track accuracy across PYQs
4.8 rated on Google Play · 14,000+ reviews
More questions from Mathematics
- If \(y(x)\) satisfies the differential equation \(y^{\prime}-y \tan x\) \(=2 x \sec x\) and \(y(0)=0\), thenJEE Advanced 2012 Medium
- A line passing through the origin is perpendicular to the lines
Then, the coordinate(s) of the point(s) on at a distance of from the point of intersection of and is (are)JEE Advanced 2013 Medium - For a real number if the system of linear equations, has infinitely many solutions, thenJEE Advanced 2017 Medium
- Let \(f:\left[0, \frac{\pi}{2}\right] \rightarrow[0,1]\) be the function defined by \(f(x)=\sin ^2 x\) and let \(g:\left[0, \frac{\pi}{2}\right] \rightarrow[0, \infty)\) be the function defined by \(g(x)=\sqrt{\frac{\pi x}{2}-x^2}\).
The value of \(\frac{16}{\pi^3} \int_0^{\frac{\pi}{2}} f(x) g(x) d x\) isJEE Advanced 2024 Medium - The value of \(b\) for which the equations \(x^2+b x-1=0, x^2+x+b=0\) have one root in common isJEE Advanced 2011 Hard
- Let \(\alpha, \beta\) and \(\gamma\) be real numbers such that the system of linear equations
\[
\begin{array}{c}
x+2 y+3 z=\alpha \\
4 x+5 y+6 z=\beta \\
7 x+8 y+9 z=\gamma-1
\end{array}
\]
is consistent. Let \(|M|\) represent the determinant of the matrix
\[
M=\left[\begin{array}{ccc}
\alpha & 2 & \gamma \\
\beta & 1 & 0 \\
-1 & 0 & 1
\end{array}\right]
\]
Let \(P\) be the plane containing all those \((\alpha, \beta, \gamma)\) for which the above system of linear equations is consistent, and \(D\) be the square of the distance of the point \((0,1,0)\) from the plane \(P\).
The value of isJEE Advanced 2021 Medium
More PYQs from JEE Advanced
- The correct option(s) to distinguish nitrate salts of taken separately is(are)JEE Advanced 2018 Easy
- The correct combination of names for isomeric alcohols with molecular formula is/areJEE Advanced 2014 Medium
- One mole of a monatomic ideal gas undergoes the cyclic process \(\mathrm{J} \rightarrow \mathrm{K} \rightarrow \mathrm{L} \rightarrow \mathrm{M} \rightarrow \mathrm{J}\), as shown in the P-T diagram.
Match the quantities mentioned in List-I with their values in List-II and choose the correct option.
[\(\mathcal{R}\) is the gas constant.]List-I List-II (P) Work done in the complete cyclic process (1) \(R T_0-4 R T_0 \ln 2\) (Q) Change in the internal energy of the gas in the process JK (2) \(0\) (R) Heat given to the gas in the process KL (3) \(3 K T_0\) (S) Change in the internal energy of the gas in the process MJ (4) \(-2 R T_0 \ln 2\) \((5)-3 R T_0 \ln 2\) JEE Advanced 2024 Medium - A horizontal circular platform of radius 0.5 m and mass 0.45 kg is free to rotate about its axis. Two massless spring toy-guns, each carrying a steel ball of mass 0.05 kg are attached to the platform at a distance 0.25 m from the centre on its either sides along its diameter (see figure). Each gun simultaneously fires the balls horizontally and perpendicular to the diameter in opposite directions. After leaving the platform, the balls have horizontal speed of 9 with respect to the ground. The rotational speed of the platform in rad after the balls leave the platform is
JEE Advanced 2014 Medium - The total number of lone pairs of electrons in melamine isJEE Advanced 2013 Easy
- The dimensions of a cone are measured using a scale with a least count of \(2 \mathrm{~mm}\). The diameter of the base and the height are both measured to be \(20.0 \mathrm{~cm}\). The maximum percentage error in the determination of the volume is _______ .JEE Advanced 2024 Easy