JEE Mains · Maths · STD 11 - 4.2 Quadratic equations and inequations
If the equation \(\mathrm{a}(\mathrm{b}-\mathrm{c}) \mathrm{x}^2+\mathrm{b}(\mathrm{c}-\mathrm{a}) \mathrm{x}+\mathrm{c}(\mathrm{a}-\mathrm{b})=0\) has equal roots, where \(\mathrm{a}+\mathrm{c}=15\) and \(\mathrm{b}=\frac{36}{5}\), then \(a^2+c^2\) is equal to
- A 115
- B 117
- C 119
- D 121
Answer & Solution
Correct Answer
(B) 117
Step-by-step Solution
Detailed explanation
\begin{aligned} & a(b-c) x^2+b(c-a) x+c(a-b)=0 \\ & x=1 \text { is root } \therefore \text { other root is } 1 \\ & \alpha+\beta=-\frac{b(c-a)}{a(b-c)}=2 \\ & \Rightarrow-\mathrm{bc}+\mathrm{ab}=2 \mathrm{ab}-2 \mathrm{ac} \\ & \Rightarrow 2 \mathrm{ac}=\mathrm{ab}+\mathrm{bc}…
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 random valuable \(X\) follows binomial distribution \(B (n, p)\) for which the difference of the mean and the variance is \(1\). If \(2 P(X=2)=3 P(X=1)\), then \(n^2 P(X > 1)\) is equal to \(......\).JEE Mains 2023 Hard
- The locus of the midpoints of the chord of the circle, \(x^{2}+y^{2}=25\) which is tangent to the hyperbola \(, \frac{ x ^{2}}{9}-\frac{ y ^{2}}{16}=1\) isJEE Mains 2021 Hard
- Let \( |A|=6 \) where A is a \( 3 \times 3 \) matrix. If \( |adj(3adj(A^{2} \cdot adj(2A)))|=2^{m} \cdot 3^{n} \), \( m, n \in N \), then \( m+n \) is equal to:JEE Mains 2026 Medium
- If the sum and product of the first three term in an \(A.P\). are \(33\) and \(1155\), respectively, then a value of its \(11^{th}\) tern isJEE Mains 2019 Hard
- Let \(\mathrm{P}\) be the plane passing through the point \((1,2,3)\) and the line of intersection of the planes \(\vec{r} \cdot(\hat{i}+\hat{j}+4 \hat{k})=16\) and \(\vec{r} \cdot(-\hat{i}+\hat{j}+\hat{k})=6\). Then which of the following points does NOT lie on \(\mathrm{P}\) ?JEE Mains 2021 Medium
- Let the mean of the data
be \(5.\) If \(m\) and \(\sigma^2\) are respectively the mean deviation about the mean and the variance of the data, then \(\frac{3 \alpha}{m+\sigma^2}\) is equal to \(..........\).\(X\) \(1\) \(3\) \(5\) \(7\) \(9\) \((f)\) \(4\) \(24\) \(28\) \(\alpha\) \(8\) JEE Mains 2023 Hard
More PYQs from JEE Mains
- Let \(f(x)=\left\{\begin{array}{cl}x^2 \sin \left(\frac{1}{x}\right) & , x \neq 0 \\ 0 & , x=0\end{array} ;\right.\) Then at \(x=0\)JEE Mains 2023 Hard
- The number of seven digit positive integers formed using the digits \(1,2,3\) and \(4\) only and sum of the digits equal to \(12\) is \(...........\).JEE Mains 2023 Hard
- Let \(\vec{a}=a_{1} \hat{i}+a_{2} \hat{j}+a_{3} \hat{k} \quad a_{i}>0, i=1,2,3\) be \(a\) vector which makes equal angles with the coordinates axes OX, OY and OZ. Also, let the projection of \(\vec{a}\) on the vector \(3 \hat{i}+4 \hat{j}\) be \(7\) . Let \(\vec{b}\) be a vector obtained by rotating \(\vec{a}\) with \(90^{\circ}\). If \(\vec{a}, \vec{b}\) and \(x\)-axis are coplanar, then projection of a vector \(\vec{b}\) on \(3 \hat{i}+4 \hat{j}\) is equal toJEE Mains 2022 Hard
- Let the domains of the functions
\(\mathrm{f}(\mathrm{x})=\log _4 \log _3 \log _7\left(8-\log _2\left(\mathrm{x}^2+4 \mathrm{x}+5\right)\right)\) and \(g(x)=\sin ^{-1}\left(\frac{7 x+10}{x-2}\right)\) be \((\alpha, \beta)\) and \([\gamma, \delta]\), respectively. Then \(\alpha^2+\beta^2+\gamma^2+\delta^2\) is equal to :-JEE Mains 2025 Medium - A line with direction ratios \(1, -1, 2\) intersects the lines \(\dfrac{x}{2} = \dfrac{y}{3} = \dfrac{z+1}{3}\) and \(\dfrac{x+1}{-1} = \dfrac{y-2}{1} = \dfrac{z}{4}\) at the points \(P\) and \(Q\), respectively. If the length of the line segment \(PQ\) is \(\alpha\), then \(225\alpha^2\) is equal to:JEE Mains 2026 Hard
- Let \(\mathrm{f}(\mathrm{x})=\cos \left(2 \tan ^{-1} \sin \left(\cot ^{-1} \sqrt{\frac{1-\mathrm{x}}{\mathrm{x}}}\right)\right)\) \(0<\mathrm{x}<1\). Then :JEE Mains 2021 Hard