JEE Mains · Maths · STD 11 - 4.2 Quadratic equations and inequations
The number of real solutions of the equation \(e ^{4 x }+4 e ^{3 x }-58 e ^{2 x }+4 e ^{ x }+1=0\) is..........
- A \(6\)
- B \(9\)
- C \(20\)
- D \(2\)
Answer & Solution
Correct Answer
(D) \(2\)
Step-by-step Solution
Detailed explanation
\(e^{4 x}+4 e^{3 x}-58 e^{2 x}+4 e^{x}+1=0\) Let \(f(x)=e^{2 x}\left(e^{2 x}+\frac{1}{e^{2 x}}+4\left(e^{x}+\frac{1}{e^{x}}\right)-58\right)\) \(e^{x}+\frac{1}{e^{x}}\) Let \(h(t)=t^{2}+4 t-58=0\) \(t =\frac{-4 \pm \sqrt{16+4.58}}{2}\) \(\frac{-4 \pm 2 \sqrt{62}}{2}\)…
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
- Let \(f: R \rightarrow R\) be a function defined \(f(x)=\frac{2 e^{2 x}}{e^{2 x}+\varepsilon}\). Then \(f\left(\frac{1}{100}\right)+f\left(\frac{2}{100}\right)+f\left(\frac{3}{100}\right)+\ldots .+f\left(\frac{99}{100}\right)\) is equal toJEE Mains 2022 Hard
- If the tangent to the curve \(y=x+\sin y\) at a point \((a, b)\) is parallel to the line joining \(\left(0, \frac{3}{2}\right)\) and \(\left(\frac{1}{2}, 2\right),\) thenJEE Mains 2020 Medium
- If \(8=3+\frac{1}{4}(3+p)+\frac{1}{4^2}(3+2 p)+\frac{1}{4^3}(3+3 p)+\ldots \infty,\) then the value of \(p\) isJEE Mains 2024 Medium
- Let \(E _{1}, E _{2}, E _{3}\) be three mutually exclusive events such that \(P \left( E _{1}\right)=\frac{2+3 p }{6}, P \left( E _{2}\right)=\frac{2- p }{8}\) and \(P \left( E _{3}\right)\) \(=\frac{1- p }{2}\). If the maximum and minimum values of \(p\) are \(p _{1}\) and \(p _{2}\), then \(\left( p _{1}+ p _{2}\right)\) is equal to.JEE Mains 2022 Hard
- Let the system of linear equations \(x +2 y + z =2\), \(\alpha x +3 y - z =\alpha,-\alpha x + y +2 z =-\alpha\) be inconsistent. Then \(\alpha\) is equal toJEE Mains 2022 Medium
- If \(\alpha+\beta+\gamma=2 \pi\), then the system of equations \(x+(\cos \gamma) y+(\cos \beta) z=0\) \((\cos \gamma) x+y+(\cos \alpha) z=0\) \((\cos \beta) x+(\cos \alpha) y+z=0\) has :JEE Mains 2021 Hard
More PYQs from JEE Mains
- Let \(f\) be an odd function defined on the set of real numbers such that for \(x \geq 0\) , \(f(x)\, =3\, sin\, x + 4\, cos\, x\). Then \(f(x)\) at \(x = - \frac{{11\pi }}{6}\) is equal toJEE Mains 2014 Hard
- If the lines \(x\,=\,ay\,+\,b,\,\,z\,=\,cy\,+\,d\) and \(x\, = \,a\,'z + \,b\,',\,\,y = \,c\,'z\, + \,d\,'\) are perpendicular, thenJEE Mains 2019 Easy
- For \(a , b \in Z\) and \(| a - b | \leq 10\), let the angle between the plane \(P : ax + y - z = b\) and the line \(l: x -1= a\) \(-y=z+1\) be \(\cos ^{-1}\left(\frac{1}{3}\right)\). If the distance of the point \((6,-6,4)\) from the plane \(P\) is \(3 \sqrt{6}\), then \(a^4+b^2\) is equal toJEE Mains 2023 Hard
- Let \(\mathrm{C}\) be the set of all complex numbers. Let \(\mathrm{S}_{1} =\left\{\mathrm{z} \in \mathrm{C}|| \mathrm{z}-3-\left.2 \mathrm{i}\right|^{2}=8\right\}\) \(\mathrm{S}_{2} =\{\mathrm{z} \in \mathrm{C} \mid \operatorname{Re}(\mathrm{z}) \geq 5\} \text { and }\) \(\mathrm{S}_{3} =\{\mathrm{z} \in \mathrm{C} \| \mathrm{z}-\bar{z} \mid \geq 8\}\) Then the number of elements in \(\mathrm{S}_{1} \cap \mathrm{S}_{2} \cap \mathrm{S}_{3}\) is equal to:JEE Mains 2021 Hard
- Let the latus rectum of the hyperbola \(\frac{x^2}{9}-\frac{y^2}{b^2}=1\) subtend an angle of \(\frac{\pi}{3}\) at the centre of the hyperbola. If \(\mathrm{b}^2\) is equal to \(\frac{l}{\mathrm{~m}}(1+\sqrt{\mathrm{n}})\), where \(l\) and \(\mathrm{m}\) are co-prime numbers, then \(l^2+\mathrm{m}^2+\mathrm{n}^2\) is equal to ...........JEE Mains 2024 Hard
- Let \([t]\) be the greatest integer less than or equal to \(t\). Let \(A\) be the set of al prime factors of \(2310\) and \(f: A \rightarrow \mathbb{Z}\) be the function \(f(x)=\left[\log _2\left(x^2+\left[\frac{x^3}{5}\right]\right)\right]\). The number of one-to-one functions from \(A\) to the range of \(f\) is :JEE Mains 2024 Hard