AP EAMCET · Maths · Quadratic Equation
If \(\alpha, \beta, \gamma\) are the roots of the equation \(x^3-12 x^2+k x-18=0\) and one of them is thrice the sum of the other two roots, then \(\alpha^2+\beta^2+\gamma^2-k=\)
- A \(115\)
- B \(41\)
- C \(56\)
- D \(57\)
Answer & Solution
Correct Answer
(D) \(57\)
Step-by-step Solution
Detailed explanation
\(\alpha+\beta+\gamma=12\) Given \(\alpha = 3(\beta+\gamma)\). So \(4(\beta+\gamma)=12 \implies \beta+\gamma=3 \implies \alpha=9\). Since \(\alpha=9\) is a root: \(9^3-12(9^2)+9k-18=0 \implies 729-972+9k-18=0 \implies 9k=261 \implies k=29\).…
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 Maths
- The set of all real values of c so that the angle between the vectors \(\overline{\mathrm{a}}=\mathrm{cx} \overline{\mathrm{i}}-6 \overline{\mathrm{j}}+3 \overline{\mathrm{k}}\) and \(\overline{\mathrm{b}}=x \overline{\mathrm{i}}+2 \overline{\mathrm{j}}+2 \mathrm{cx} \overline{\mathrm{k}}\) is an obtuse angle for all real \(x\) isAP EAMCET 2025 Medium
- \(\int_{\frac{1}{25}}^3 \frac{e^{\frac{3}{x}}}{x^2} d x=\)AP EAMCET 2023 Easy
- Let \(\overrightarrow{\mathbf{a}}, \overrightarrow{\mathbf{b}}, \overrightarrow{\mathbf{c}}\) be the position vectors of the vertices \(A, B, C\) respectively of \(\triangle A B C\). The vector area of \(\triangle A B C\) is :AP EAMCET 2003 Hard
- The general solution of the differential equation \(\frac{d y}{d x}=\cos ^2(3 x+y)\) is \(\tan ^{-1}\left(\frac{\sqrt{3}}{2} \tan (3 x+y)\right)=f(x)\). Then, \(f(x)=\)AP EAMCET 2022 Medium
- \(3 x+4 y-43=0\) is a tangent to the circle \(S \equiv x^2+y^2-6 x+8 y+k=0\) at a point P. If C is the centre of the circle and Q is a point which divides CP in the ratio \(-1: 2\), then the power of the point Q with respect to the circle \(\mathrm{S}=0\) isAP EAMCET 2025 Medium
- If \(\frac{3-2 i \sin \theta}{1+2 i \sin \theta}\) is purely imaginary number, then \(\theta=\)AP EAMCET 2024 Easy
More PYQs from AP EAMCET
- If \(\frac{2 x^2+5 x+6}{(x+2)^3}=\frac{a}{x+2}+\frac{b}{(x+2)^2}+\frac{c}{(x+2)^3}\) then \(\mathrm{a} . \mathrm{b}+\mathrm{b} . \mathrm{c}+\mathrm{c} . \mathrm{a}=\)AP EAMCET 2023 Easy
- Identify the pair in which both are not extensive propertiesAP EAMCET 2023 Easy
- If \(A=\left[\begin{array}{lll}1 & 0 & 2 \\ 2 & 1 & 3 \\ 3 & 2 & 4\end{array}\right]\), then \(A^2-5 A+6 I=\)AP EAMCET 2024 Easy
- The equation of the circle which passes through the point \((3,2)\) bisects the circumference of the circle \(x^2+y^2=15\) and cuts the circle \(x^2+y^2+4 x+6 y+3=0\) orthogonally isAP EAMCET 2018 Medium
- The roots
\(\begin{aligned}
& (x-a)(x-a-1)+(x-a-1)(x-a-2) \\
& +(x-a)(x-a-2)=0, a \in R \text { are always }
\end{aligned}\)AP EAMCET 2009 Easy - A body falls freely on to a hard horizontal surface. If the coefficient of restitution between the surface and the body is 0.8, then the ratio of the maximum height to which the body rises after second impact and the initial height of the body isAP EAMCET 2025 Medium