WBJEE · Maths · Quadratic Equation
If \(a, b, c\) are real, then both the roots of the equation \((x-b)(x-c)+(x-c)(x-a)+(x-a)(x-b)=0\) are always
- A positive
- B negative
- C real
- D imaginary
Answer & Solution
Correct Answer
(C) real
Step-by-step Solution
Detailed explanation
Hints : \(3 x^2-2 x(a+b+c)+a b+b c+c a=0\) \[ \begin{aligned} & \mathrm{D}=4(a+b+c)^2-4.3(a b+b c+c a) \\ & =4\left(a^2+b^2+c^2-a b-b c-c a\right) \\ & =2\left[(a-b)^2+(b-c)^2+(c-a)^2\right] \\ & =\left[(a-b)^2+(b-c)^2+(c-a)^2\right] \\ & \geq 0 \end{aligned} \]
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
- A line passing through the point of intersection of \(x+y=4\) and \(x-y=2\) makes an angle \(\tan ^{-1}\left(\frac{3}{4}\right)\) with the \(X\) -axis. It intersects the parabola \(y^{2}=4(x-3)\) at points \(\left(x_{1}, y_{1}\right)\) and \(\left(x_{2}, y_{2}\right),\) respectively. Then \(\left|x_{1}-x_{2}\right|\) is equal toWBJEE 2016 Hard
- The value of \(\lim _{x \rightarrow 0} \frac{1}{x}\left[\int_{y}^{a} e^{\sin ^{2} t} d t-\int_{x+y}^{a} e^{\sin ^{2} t} d t\right]\) is equal toWBJEE 2019 Easy
- The area of the region lying above \(X\) -axis, and included between the circle \(x^{2}+y^{2}=2 a x\) and the parabola \(y^{2}=a x, a>0\) isWBJEE 2018 Medium
- Let \(X\) be a nonvoid set. If \(\rho_1\) and \(\rho_2\) be the transitive relations on \(X\), thenWBJEE 2023 Easy
- If \(x+\log _{10}\left(1+2^{x}\right)=x \log _{10} 5+\log _{10} 6,\) then
the value of \(x\) isWBJEE 2018 Medium - If \(n\) is even posirive integer, then the condition that the greatest term in the expansion of \((a+x)^{n}\) may also have the greatest coefficient, isWBJEE 2018 Medium
More PYQs from WBJEE
- The line joining \(A(b \cos \alpha, b \sin \alpha)\)
and
\(B(a \cos \beta, a \sin \beta),\) where \(a \neq b,\) is produced to the
point \(M(x, y)\) so that \(A M: M B=b: a\). Then, \(x \cos \frac{\alpha+\beta}{2}+y \sin \frac{\alpha+\beta}{2}\) is equal toWBJEE 2012 Easy - The expression \(a x^{2}+b x+c(a, b\) and \(c\) are real \()\) has the same sign as that of a for all \(x\) ifWBJEE 2020 Easy
- In a GP series consisting of positive terms, each term is equal to the sum of next two terms. Then, the common ratio of this GP series isWBJEE 2017 Easy
- If the four points with position vectors \(-2\hat i+\hat j + \hat k, \hat i+\hat j+\hat k,\hat j- \hat k\) and \(\lambda \hat j + \hat k\) are coplanar, then \(\lambda\) is equal toWBJEE 2015 Easy
- Which one of the following formulae does not represent an organic compound?WBJEE 2009 Medium
- \(\int_0^{2 \pi} \theta \sin ^6 \theta \cos \theta d \theta\) is equal toWBJEE 2023 Medium