JEE Mains · Maths · STD 11 - 4.2 Quadratic equations and inequations
Consider the quadratic equation \((n^2 - 2n + 2)x^2 - 3x + (n^2 - 2n + 2)^2 = 0\), \(n \in \mathbb{R}\). Let \(\alpha\) be the minimum value of the product of its roots and \(\beta\) be the maximum value of the sum of its roots. Then the sum of the first six terms of the G.P., whose first term is \(\alpha\) and the common ratio is \(\dfrac{\alpha}{\beta}\), is :
- A \(\dfrac{61}{37}\)
- B \(\dfrac{121}{81}\)
- C \(\dfrac{364}{243}\)
- D \(\dfrac{1093}{729}\)
Answer & Solution
Correct Answer
(C) \(\dfrac{364}{243}\)
Step-by-step Solution
Detailed explanation
The given quadratic equation is \((n^2 - 2n + 2)x^2 - 3x + (n^2 - 2n + 2)^2 = 0\). Let \(k = n^2 - 2n + 2 = (n-1)^2 + 1\). Since \((n-1)^2 \ge 0\) for all \(n \in \mathbb{R}\), the minimum value of \(k\) is \(1\) at \(n=1\). The product of the roots is given by…
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
- If the distance between the plane, \(23 \mathrm{x}-10 \mathrm{y}-2 \mathrm{z}+48=0\) and the plane containing the lines \(\frac{x+1}{2}=\frac{y-3}{4}=\frac{z+1}{3}\) and \(\frac{x+3}{2}=\frac{y+2}{6}=\frac{z-1}{\lambda}(\lambda \in R)\) is equal to \(\frac{\mathrm{k}}{\sqrt{633}},\) then \(\mathrm{k}\) is equal toJEE Mains 2020 Hard
- Consider the triangles with vertices \(A (2,1) B (0,0)\) and \(C ( t , 4), t \in[0,4]\). It the maximum and the minimum perimeters of such triangles are obtained at \(t=\alpha\) and \(t=\beta\) respectively, then \(6 \alpha+21 \beta\) is equal to \(.........\).JEE Mains 2023 Hard
- If \(z\) and \(w\) are two complex numbers such that \(|zw| = 1\) and \(arg(z) -arg(w) =\frac {\pi }{2},\) thenJEE Mains 2019 Hard
- Let \(f\left( x \right) = \left\{ \begin{array}{l}
\max \left\{ {\left| x \right|,{x^2}} \right\},\,\,\,\,\left| x \right| \le 2\\
8 - 2\left| x \right|,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,2 < \left| x \right| \le 4\,\,\,\,
\end{array} \right.\). Let \(S\) be the set of points in the interval \((-4, 4)\) at which \(f\) is not differentiable. Then \(S\)JEE Mains 2019 Hard - Let \(y = y(x)\) be the solution of the differential equation \(x\sin\left(\dfrac{y}{x}\right)dy = \left(y\sin\left(\dfrac{y}{x}\right) - x\right)dx\), \(y(1) = \dfrac{\pi}{2}\) and let \(\alpha = \cos\left(\dfrac{y(e^{12})}{e^{12}}\right)\). Then the number of integral values of \(p\), for which the equation \(x^2 + y^2 - 2px + 2py + \alpha + 2 = 0\) represents a circle of radius \(r \leq 6\), is __________.JEE Mains 2026 Hard
- Let \(\vec{a}=4 \hat{i}+3 \hat{j}\) and \(\vec{b}=3 \hat{i}-4 \hat{j}+5 \hat{k}\) and \(\vec{c}\) is a vector such that \(\overrightarrow{ c } \cdot(\overrightarrow{ a } \times \overrightarrow{ b })+25=0, \overrightarrow{ c } \cdot(\hat{ i }+\hat{ j }+\hat{ k })=4\) and projection of \(\overrightarrow{ c }\) on \(\overrightarrow{ a }\) is \(1,\) then the projection of \(\overrightarrow{ c }\) on \(\overrightarrow{ b }\) equals:JEE Mains 2023 Hard
More PYQs from JEE Mains
- Let \( f(\alpha) \) denote the area of the region in the first quadrant bounded by \( x=0, x=1, y^{2}=x \) and \( y=|\alpha x-5|-|1-\alpha x|+\alpha x^{2}. \) Then \( (f(0)+f(1)) \) is equal toJEE Mains 2026 Hard
- Let \(a, b\) and \(c\) be distinct positive numbers. If the vectors \(a \hat{i}+a \hat{j}+c \hat{k}, \hat{i}+\hat{k}\) and \(c \hat{i}+c \hat{j}+b \hat{k}\) are co-planar, then \(\mathrm{c}\) is equal to:JEE Mains 2021 Easy
- Let \(q\) be the maximum integral value of \(p\) in \([0,10]\) for which the roots of the equation \(x ^2- px +\frac{5}{4} p =0\) are rational. Then the area of the region \(\{(x, y): 0 \leq y\) \(\left.\leq(x-q)^2, 0 \leq x \leq q\right\}\) isJEE Mains 2023 Hard
- Let the centre of the circle \(x^2 + y^2 + 2gx + 2fy + 25 = 0\) be in the first quadrant and lie on the line \(2x - y = 4\). Let the area of an equilateral triangle inscribed in the circle be \(27\sqrt{3}\). Then the square of the length of the chord of the circle on the line \(x = 1\) is _______.JEE Mains 2026 Hard
- If \(A\) and \(B\) are two events such that \(P ( A )=\frac{1}{3}, P ( B )=\frac{1}{5} \) and \(P ( A \cup B )=\frac{1}{2}\), then \(P \left( A \mid B ^{\prime}\right)+ P \left( B \mid A ^{\prime}\right)\) is equal toJEE Mains 2022 Hard
- Consider the relations \(R_1\) and \(R_2\) defined as \(a R_1 b\) \(\Leftrightarrow a^2+b^2=1\) for all \(a, b, \in R\) and \((a, b) R_2(c, d)\) \(\Leftrightarrow a+d=b+c\) for all \((a, b),(c, d) \in N \times N\). ThenJEE Mains 2024 Medium