JEE Mains · Maths · STD 11 - 4.2 Quadratic equations and inequations
Let \(\mathrm{S}=\left\{x \in R:(\sqrt{3}+\sqrt{2})^x+(\sqrt{3}-\sqrt{2})^x=10\right\}\). Then the number of elements in \(\mathrm{S}\) is :
- A \(4\)
- B \(0\)
- C \(2\)
- D \(1\)
Answer & Solution
Correct Answer
(C) \(2\)
Step-by-step Solution
Detailed explanation
\((\sqrt{3}+\sqrt{2})^{\mathrm{x}}+(\sqrt{3}-\sqrt{2})^{\mathrm{x}}=10\) \(\text { Let }(\sqrt{3}+\sqrt{2})^{\mathrm{x}}=\mathrm{t}\) \(\mathrm{t}+\frac{1}{\mathrm{t}}=10\) \(\mathrm{t}^2-10 \mathrm{t}+1=0\) \(\mathrm{t}=\frac{10 \pm \sqrt{100-4}}{2}=5 \pm 2 \sqrt{6}\)…
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 number of real solutions of the equation \(x\left(x^2+3|x|+5|x-1|+6|x-2|\right)=0\) isJEE Mains 2024 Hard
- The mean and variance of \(n\) observations are \(8\) and \(16\), respectively. If the sum of the first \((n-1)\) observations is \(48\) and the sum of squares of the first \((n-1)\) observations is \(496\), then the value of \(n\) is:JEE Mains 2026 Medium
- Let
\(\mathrm{f}(x)=\left\{\begin{array}{lc}3 x, & x \lt 0 \\ \min \{1+x+[x], x+2[x]\}, & 0 \leq x \leq 2 \\ 5, & x\gt2,\end{array}\right.\)
where [.] denotes greatest integer function. If \(\alpha\) and \(\beta\) are the number of points, where f is not continuous and is not differentiable, respectively, then \(\alpha+\beta\) equals __________JEE Mains 2025 Hard - The number of critical points of the function \(f(x) = \begin{cases} \left|\dfrac{\sin x}{x}\right|, & x \neq 0 \\ 1, & x = 0 \end{cases}\) in the interval \((-2\pi, 2\pi)\) is equal to :JEE Mains 2026 Hard
- If \(x,y,z\) are in \(A.P.\) and \({\tan ^{ - 1}}x,{\tan ^{ - 1}}y\) and \({\tan ^{ - 1}}z\) are also in other \(A.P.\) then . . .JEE Mains 2013 Medium
- Let the foot of perpendicular from the point \((\lambda, 2, 3)\) on the line \(\dfrac{x-4}{1} = \dfrac{y-9}{2} = \dfrac{z-5}{1}\) be the point \((1, \mu, 2)\). Then the distance between the lines \(\dfrac{x-1}{2} = \dfrac{y-2}{3} = \dfrac{z+4}{6}\) and \(\dfrac{x-\lambda}{2} = \dfrac{y-\mu}{3} = \dfrac{z+5}{6}\) is equal to:JEE Mains 2026 Hard
More PYQs from JEE Mains
- \( 6\int_{0}^{\pi}|(\sin 3x+\sin 2x+\sin x)| dx \) is equal to ....JEE Mains 2026 Easy
- Let \(\mathrm{A}\) be a fixed point \((0,6)\) and \(\mathrm{B}\) be a moving point \((2 \mathrm{t}, 0)\). Let \(\mathrm{M}\) be the mid-point of \(\mathrm{AB}\) and the perpendicular bisector of \(\mathrm{AB}\) meets the \(\mathrm{y}\)-axis at \(\mathrm{C}\). The locus of the mid-point \(\mathrm{P}\) of \(\mathrm{MC}\) is :JEE Mains 2021 Hard
- The number of arrangements of the letter of the word "\(INDEPENDENCE\)" in which all the vowels always occur together isJEE Mains 2023 Medium
- A bag contains \(6\) white and \(4\) black balls. A die is rolled once and the number of balls equal to the number obtained on the die are drawn from the bag at random. The probability that all the balls drawn are white is:JEE Mains 2023 Medium
- If \(\int_{0}^{\pi}\left(\sin ^{3} x\right) e^{-\sin ^{2} x} d x=\alpha-\frac{\beta}{e} \int_{0}^{1} \sqrt{t} e^{t} d t\), then \(\alpha+\beta\) is equal to \(....\)JEE Mains 2021 Hard
- The distance of the point \((7,-3,-4)\) from the plane passing through the points \((2,-3,1),(-1,1,-2)\) and \((3,-4,2)\) is:JEE Mains 2023 Easy