Let α, β(α ≠ β) be the values of m , for which the equations x+y+z=1 ; x+2 y+4 z=m and x+4 y+10 z=m^2 have infinitely many solutions. Then the value of ∑_ n=1 ^ 10 (n^α+n^β) is equal to :

Correct answer: D. 440 aligned Δ & =| array llc 1 & 1 & 1 1 & 2 & 4 1 & 4 & 10 array |=1(20-16)-1(10-4)+1(4-2) & =4-6+2=0 aligned For infinite solutions aligned & Δ_x=Δ_y=Δ_z=0 & m^2-3 x+2=0 & m=1,2 & α=1, β=2 & ∑_ n=1 ^ 10 (n^α+n^β)=∑_ n=1 ^ 10 n^1+∑_ n=1 ^ 10 n^2 &= (10(11))/(2)+(10(11)(21))/(6) &= 55+385 &= 440 aligned

JEE Main 202529 JanuaryEvening Shift

Exam: JEE Main
Year: 2025
Date: 29 January
Shift: Evening Shift
Subject: Mathematics
Chapter: Determinants

JEE Main 2025 · 29 January · Evening Shift · Mathematics · Determinants

Let α, β(α ≠ β) be the values of m , for which the equations...

Let

\alpha, \beta(\alpha \neq \beta)

be the values of m , for which the equations

x+y+z=1 ; x+2 y+4 z=\mathrm{m} and

x+4 y+10 z=m^2

have infinitely many solutions. Then the value of

\sum_{n=1}^{10}

\left(n^\alpha+n^\beta\right)

is equal to :

Options

A3080
B560
C3410
D440

Correct answer

D. 440

Step-by-step solution

\begin{aligned} \Delta & =\left|\begin{array}{llc} 1 & 1 & 1 \\ 1 & 2 & 4 \\ 1 & 4 & 10 \end{array}\right|=1(20-16)-1(10-4)+1(4-2) \\ & =4-6+2=0 \end{aligned}

For infinite solutions

\begin{aligned} & \Delta_x=\Delta_y=\Delta_z=0 \\ & m^2-3 x+2=0 \\ & m=1,2 \\ & \alpha=1, \beta=2 \\ & \therefore \sum_{n=1}^{10}\left(n^\alpha+n^\beta\right)=\sum_{n=1}^{10} n^1+\sum_{n=1}^{10} n^2 \\ &= \frac{10(11)}{2}+\frac{10(11)(21)}{6} \\ &= 55+385 \\ &= 440 \end{aligned}

Practice Determinants on Quantrex Academy →

More from Determinants

If the system of linear equations: x+y+z=6 , x+2y+5z=10 , 2x+3y+λ z=μ has infinitely many… JEE Main 2026 The sum of all possible values of θ ∈ [0, 2π] , for which the system of equations : x 3θ… JEE Main 2026 If f: N arrow Z is defined by f(n) = vmatrix n & -1 & -5 -2n^2 & 3(2k+1) & 2k+1 -3n^3 &… JEE Main 2026 Consider the system of linear equations in x, y, z : x + 2y + tz = 0 , 6x + y + 5tz = 0… JEE Main 2026 If the system of equations: x+y+z=5 x+2y+3z=9 x+3y+λ z=μ has infinitely many solutions… JEE Main 2026 If the system of equations x + 5y + 6z = 4 , 2x + 3y + 4z = 7 , x + 6y + az = b has… JEE Main 2026 Let α, β ∈ ℝ be such that the system of linear equations x + 2y + z = 5 2x + y + α z = 5… JEE Main 2026 The system of linear equations x+y+z=6 2 x+5 y+a z=36 x+2 y+3 z=b has JEE Main 2026 Among the statements : I: If | array ccc 1 & α & β α & 1 & γ β & γ & 1 array |=| array… JEE Main 2026 Let n be the number obtained on rolling a fair die. If the probability that the system… JEE Main 2026 If the system of equations 3 x+y+4 z=3 2 x+α y-z=-3 x+2 y+z=4 has no solution, then the… JEE Main 2026 Let the system of equations aligned & x+5 y-z=1 & 4 x+3 y-3 z=7 & 24 x+y+λ z=μ aligned λ… JEE Main 2025 Let the system of equations : aligned & 2 x+3 y+5 z=9 & 7 x+3 y-2 z=8 & 12 x+3 y-(4+λ)… JEE Main 2025 If y(x)=| array ccc x & x & x+ x+1 27 & 28 & 27 1 & 1 & 1 array |, x ∈ ℝ , then (d^2… JEE Main 2025 If the system of equation aligned & 2 x+λ y+3 z=5 & 3 x+2 y-z=7 & 4 x+5 y+μ z=9 aligned… JEE Main 2025 If the system of linear equations aligned & 3 x+y+β z=3 & 2 x+α y-z=-3 & x+2 y+z=4… JEE Main 2025 Full Determinants list Determinants JEE Maths PYQs JEE Mathematics PYQs All JEE Main PYQs