JEE Mains · Maths · STD 12 - 1. relation and function
The number of points, where the curve \(f(x)=e^{8 x}-e^{6 x}-3 e^{4 x}-e^{2 x}+1, x \in R\) cuts \(x\)-axis, is equal to
- A \(2\)
- B \(4\)
- C \(6\)
- D \(8\)
Answer & Solution
Correct Answer
(A) \(2\)
Step-by-step Solution
Detailed explanation
Let \(e ^{2 x }= t\) \(\Rightarrow t^4-t^3-3 t^2-t+1=0\) \(\Rightarrow t^2+\frac{1}{t^2}-\left(t+\frac{1}{t}\right)-3=0\) \(\Rightarrow\left(t+\frac{1}{t}\right)^2-\left(t+\frac{1}{t}\right)-5=0\) \(\Rightarrow t+\frac{1}{t}=\frac{1+\sqrt{21}}{2}\) Two real values of \(t\).
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
- Three points \(\mathrm{O}(0,0), \mathrm{P}\left(\mathrm{a}, \mathrm{a}^2\right), \mathrm{Q}\left(-\mathrm{b}, \mathrm{b}^2\right), \mathrm{a}>0, \mathrm{~b}>0\), are on the parabola \(y=x^2\). Let \(S_1\) be the area of the region bounded by the line \(P Q\) and the parabola, and \(S_2\) be the area of the triangle \(O P Q\). If the minimum value of \(\frac{\mathrm{S}_1}{\mathrm{~S}_2}\) is \(\frac{\mathrm{m}}{\mathrm{n}}, \operatorname{gcd}(\mathrm{m}, \mathrm{n})=1\), then \(\mathrm{m}+\mathrm{n}\) is equal to :JEE Mains 2024 Hard
- If the tangent to the curve, \(y =f( x )= x \log _{ e } x\) \((x>0)\) at a point \((c, f(c))\) is parallel to the line segement joining the points \((1,0)\) and \(( e , e ),\) then \(c\) is equal toJEE Mains 2020 Medium
- If the normal to the curve \(y(x)=\int_{0}^{x}\left(2 t^{2}-15 t+10\right) d t\) at a point \((a, b)\) is parallel to the line \(x+3 y=-5, a>1,\) then the value of \(|a +6 b|\) is equal to..........JEE Mains 2021 Hard
- If \(^{20}{C_1} + \left( {{2^2}} \right){\,^{20}}{C_3} + \left( {{3^2}} \right){\,^{20}}{C_3} + \left( {{2^2}} \right) + ..... + \left( {{{20}^2}} \right){\,^{20}}{C_{20}} = A\left( {{2^\beta }} \right)\), then the ordered pair \((A, \beta )\) is equal toJEE Mains 2019 Hard
- A function \(f\) is defined on \([-3,3]\) as \(f(x)=\left\{\begin{array}{cc}\min \left\{|x|, 2-x^{2}\right\} & , \quad-2 \leq x \leq 2 \\ {[|x|]} & , \quad 2<|x| \leq 3\end{array}\right.\) where \([x]\) denotes the greatest integer \(\leq x .\) The number of points, where \(f\) is not differentiable in \((-3,3)\) isJEE Mains 2021 Hard
- The value of \(1^3 - 2^3 + 3^3 - \ldots + 15^3\) is:JEE Mains 2026 Medium
More PYQs from JEE Mains
- Let \(a, b\) and \(c\) be in \(G.P\) with common ratio \(r,\) where \(a \ne 0\) and \(0\, < \,r\, \le \,\frac{1}{2}\). If \(3a, 7b\) and \(15c\) are the first three terms of an \(A.P.,\) then the \(4^{th}\) term of this \(A.P\) isJEE Mains 2019 Hard
- Consider the hyperbola \(\frac{x^2}{a^2}-\frac{y^2}{b^2}=1\) having one of its focus at \(\mathrm{P}(-3,0)\). If the latus ractum through its other focus subtends a right angle at P and \(a^2 b^2=\alpha \sqrt{2}-\beta, \alpha, \beta \in \mathbb{N}\).JEE Mains 2025 Medium
- The point of intersection of the normals to the parabola \(y^ 2\, = 4x\) at the ends of its latus rectum isJEE Mains 2013 Hard
- Let \(a_1, a_2, a_3, \ldots\) be in an arithmetic progression of positive terms. Let \(\mathrm{A}_{\mathrm{k}}=\mathrm{a}_1{ }^2-\mathrm{a}_2{ }^2+\mathrm{a}_3{ }^2-\mathrm{a}_4{ }^2+\ldots+\mathrm{a}_{2 \mathrm{k}-1}{ }^2-\mathrm{a}_{2 \mathrm{k}}{ }^2\). If \(\mathrm{A}_3=-153, \mathrm{~A}_5=-435\) and \(\mathrm{a}_1{ }^2+\mathrm{a}_2{ }^2+\mathrm{a}_3{ }^2=66\), then \(\mathrm{a}_{17}-\mathrm{A}_7\) is equal to ....................JEE Mains 2024 Hard
- Given three points \(P, Q, R\) with \(P(5, 3)\) and \(R\) lies on the \(x-\) axis. If equation of \(RQ\) is \(x - 2y = 2\) and \(PQ\) is parallel to the \(x-\) axis, then the centroid of \(\Delta PQR\) lies on the lineJEE Mains 2014 Hard
- Let a common tangent to the curves \(y^2=4 x\) and \((x-4)^2+y^2=16\) touch the curves at the points \(P\) and \(Q\). Then \(( PQ )^2\) is equal to \(..........\).JEE Mains 2023 Hard