JEE Mains · Maths · STD 11 - 8. sequence and series
Let \( \alpha_{1}, \alpha_{2}, \alpha_{3}, \alpha_{4} \) be an A.P. of four terms such that each term of the A.P. and its common difference \( l \) are integers. If \( \alpha_{1}+\alpha_{2}+\alpha_{3}+\alpha_{4}=48 \) and \( \alpha_{1},\alpha_{2},\alpha_{3},\alpha_{4}+l^{4}=361 \) then the largest term of the A.P. is equal to
- A 27
- B 24
- C 21
- D 23
Answer & Solution
Correct Answer
(A) 27
Step-by-step Solution
Detailed explanation
\(a_1, a_2, a_3, a_4\) as \(a-3 d, a-d, a+d, a+3 d\) where \(d =\frac{\ell}{2}\) \(\because a_1+a_2+a_3+a_4=48 \Rightarrow 4 a=48 \Rightarrow a=12\) & \(a_1 a_2 a_3 a_4+\ell^4=361 \Rightarrow\left(a^2-9 d^2\right)\left(a^2-d^2\right)+16 d^4\) = 361…
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 integral \(\int \frac{ e ^{3 \log _{e} 2 x }+5 e ^{2 \log _{ e } 2 x }}{ e ^{4 \log _{e} x }+5 e ^{3 \log _{e} x }-7 e ^{2 \log _{e} x }} dx , x > 0\), is equal to ....... . (where \(c\) is a constant of integration)JEE Mains 2021 Hard
- If three distinct number \(a, b, c\) are in \(G.P.\) and the equations \(ax^2 + 2bc + c = 0\) and \(dx^2 + 2ex + f = 0\) have a common root, then which one of the following statements is correct?JEE Mains 2019 Hard
- A bag contains \(30\) white balls and \(10\) red balls. \(16\) balls are drawn one by one randomly from the bag with replacement. If \(X\) be the number of white balls drawn, then \(\left( {\frac{{{\rm{mean\, of\, X}}}}{{{\rm{standard\, deviation\, of\, X}}}}} \right)\) is equal toJEE Mains 2019 Hard
- For \(\mathrm{n} \geq 2\), let \(S_n\) denote the set of all subsets of \(\{1,2 \ldots . . ., n\}\) with no two consecutive numbers. For example \(\{1,3,5\} \in \mathrm{S}_6\), but \(\{1,2,4\} \notin \mathrm{S}_6\). Then \(n\left(\mathrm{~S}_5\right)\) is equal to ________JEE Mains 2025 Easy
- If \(f(x)=\left\{\begin{array}{ccc}\frac{1}{|x|} & ; & |x| \geq 1 \\ a x^{2}+b & ; & |x|<1\end{array}\right.\) is differentiable at every point of the domain, then the values of \(a\) and \(b\) are respectivelyJEE Mains 2021 Hard
- Let \(x=x(y)\) be the solution of the differential equation \(y=\left(x-y \frac{\mathrm{~d} x}{\mathrm{~d} y}\right) \sin \left(\frac{x}{y}\right), y\gt0\) and \(x(1)=\frac{\pi}{2}\). Then \(\cos (x(2))\) is equal to :JEE Mains 2025 Hard
More PYQs from JEE Mains
- Let \(\vec{a}=\hat{i}+4 \hat{j}+2 \hat{k}, \vec{b}=3 \hat{i}-2 \hat{j}+7 \hat{k}\) and \(\overrightarrow{ c }=2 \hat{ i }-\hat{ j }+4 \hat{ k }\). If a vector \(\overrightarrow{ d }\) satisfies \(\overrightarrow{ d } \times \overrightarrow{ b }=\overrightarrow{ c } \times \overrightarrow{ b }\) and \(\overrightarrow{ d } \cdot \overrightarrow{ a }=24\), then \(|\overrightarrow{ d }|^2\) is equal to \(.........\).JEE Mains 2023 Hard
- If \(\alpha \neq \mathrm{a}, \beta \neq \mathrm{b}, \gamma \neq \mathrm{c}\) and \(\left|\begin{array}{lll}\alpha & \mathrm{b} & \mathrm{c} \\ \mathrm{a} & \beta & \mathrm{c} \\ \mathrm{a} & \mathrm{b} & \gamma\end{array}\right|=0\),then \(\frac{a}{\alpha-a}+\frac{b}{\beta-b}+\frac{\gamma}{\gamma-c}\) is equal to :JEE Mains 2024 Hard
- \(\max _{0 \leq x \leq \pi}\left\{x-2 \sin x \cos x+\frac{1}{3} \sin 3 x\right\}=\)JEE Mains 2023 Hard
- Let the function \(f :[0,2] \rightarrow R\) be defined as \(f(x)=\left\{\begin{array}{cc}e^{\min \left[x^2, x-[x]\right\}}, & x \in[0,1) \\e^{\left[x-\log _e x\right]}, & x \in[1,2]\end{array}\right.\) where [t] denotes the greatest integer less than or equal to \(t\). Then the value of the integral \(\int \limits_0^2 x f(x) d x\) isJEE Mains 2023 Hard
- If the solution curve of the differential equation \(\left(y-2 \log _e x\right) d x+\left(x \log _e x^2\right) d y=0, x > 1\) passes through the points \(\left(e, \frac{4}{3}\right)\) and \(\left(e^4, \alpha\right)\), then \(\alpha\) is equal to \(................\).JEE Mains 2023 Hard
- The plane \(2 x-y+z=4\) intersects the line segment joining the points \(A ( a ,-2\), 4) and \(B (2, b ,-3)\) at the point \(C\) in the ratio \(2: 1\) and the distance of the point \(C\) from the origin is \(\sqrt{5}\). If \(ab <0\) and \(P\) is the point \(( a - b , b , 2 b - a )\) then \(CP ^2\) is equal to :JEE Mains 2023 Hard