JEE Mains · Maths · STD 11 - 8. sequence and series
Let \(A_1\) and \(A_2\) be two arithmetic means and \(G_1, G_2\), \(G _3\) be three geometric means of two distinct positive numbers. The \(G _1^4+ G _2^4+ G _3^4+ G _1^2 G _3^2\) is equal to
- A \(2\left( A _1+ A _2\right) G _1 G _3\)
- B \(\left(A_1+A_2\right)^2 G_1 G_3\)
- C \(\left( A _1+ A _2\right) G _1^2 G _3^2\)
- D \(2\left( A _1+ A _2\right) G _1^2 G _3^2\)
Answer & Solution
Correct Answer
(B) \(\left(A_1+A_2\right)^2 G_1 G_3\)
Step-by-step Solution
Detailed explanation
\(a , A _1, A _2, b\) are in A.P. \(d =\frac{b-a}{3} ; A_1=a+\frac{b-a}{3}=\frac{2 a+b}{3}\) \(A_2=\frac{a+2 b}{3}\) \(A_1+A_2=a+b\) \(a, G_1, G_2, G_3, b \text { are in G.P. }\) \(r=\left(\frac{b}{a}\right)^{\frac{1}{4}}\) \(G_1=\left(a^3 b\right)^{\frac{1}{4}}\)…
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 a die be rolled \(n\) times. Let the probability of getting odd numbers seven times be equal to the probability of getting odd numbers nine times. If the probability of getting even numbers twice is \(\frac{ k }{2^{15}}\), then \(k\) is equal to:JEE Mains 2023 Hard
- If the area enclosed between the curves \(y = kx^2\) and \(x = ky^2, (k > 0)\), is \(1\) square unit. Then \(k\) isJEE Mains 2019 Hard
- Let \(\mathrm{P}(\mathrm{x}, \mathrm{y}, \mathrm{z})\) be a point in the first octant, whose projection in the xy-plane is the point \(\mathrm{Q}\). Let \(\mathrm{OP}=\gamma\); the angle between \(OQ\) and the positive \(\mathrm{x}\)-axis be \(\theta\); and the angle between \(\mathrm{OP}\) and the positive \(\mathrm{z}\)-axis be \(\phi\), where \(\mathrm{O}\) is the origin. Then the distance of \(\mathrm{P}\) from the \(\mathrm{x}\)-axis is :JEE Mains 2024 Medium
- If the mean and the variance of \(6,4, a, 8, b, 12,10\), 13 are 9 and 9.25 respectively, then \(a+b+a b\) is equal to :JEE Mains 2025 Medium
- If the point \((1, 4)\) lies inside the circle \(x^2 + y^2-6x - 10y + p = 0\) and the circle does not touch or intersect the coordinate axes, then the set of all possible values of \(p\) is the intervalJEE Mains 2014 Hard
- Let \(S=\left\{x \in R: 0 < x < 1\right.\) and \(\left.2 \tan ^{-1}\left(\frac{1-x}{1+x}\right)=\cos ^{-1}\left(\frac{1-x^2}{1+x^2}\right)\right\}\). If \(n ( S )\) denotes the number of elements in \(S\) then:JEE Mains 2023 Hard
More PYQs from JEE Mains
- If the system of equations
\(\begin{aligned}
& 2 x-y+z=4 \\
& 5 x+\lambda y+3 z=12 \\
& 100 x-47 y+\mu z=212
\end{aligned}\)
has infinitely many solutions, then \(\mu-2 \lambda\) is equal toJEE Mains 2025 Easy - Let \(A\) be the event that the absolute difference between two randomly choosen real numbers in the sample space \([0,60]\) is less than or equal to \(a\). If \(P(A)=\frac{11}{36}\), then \(a\) is equal to \(...............\).JEE Mains 2023 Hard
- Let \(\alpha=\dfrac{1}{4}+\dfrac{1}{8}+\dfrac{1}{16}+\ldots\infty\) and \(\beta=\dfrac{1}{3}+\dfrac{1}{9}+\dfrac{1}{27}+\ldots\infty\). Then the value of \((0.2)^{\log_{\sqrt{5}}(\alpha)}+(0.04)^{\log_5(\beta)}\) is equal to:JEE Mains 2026 Medium
- If the set of all values of a, for which the equation \(5 x^3-15 x-a=0\) has three distinct real roots, is the interval \((\alpha, \beta)\), then \(\beta-2 \alpha\) is equal to ______JEE Mains 2025 Medium
- For \({x^2} \ne n\pi + 1,\,n \in N\) (the set of natural numbers), the integral \(\int {x\sqrt {\frac{{2\,\sin \,\left( {{x^2} - 1} \right) - \sin \,2\,\left( {{x^2} - 1} \right)}}{{2\,\sin \,\left( {{x^2} - 1} \right) + \sin \,2\,\left( {{x^2} - 1} \right)}}} } \,dx\) isJEE Mains 2019 Hard
- If \(f(x)\, = {x^2} - x + 5,\,\,x > \frac{1}{2},\) and \(g(x)\) is its inverse function, then \(g'(7)\) equalsJEE Mains 2014 Hard