Let the numbers 2, b, c be in an A.P. and A = [ cases 1 & 1 & 1 2 & b & c 4 & b ^ 2 & c ^ 2 cases ] . If det ( A ) ∈ [ 2,16 ] , then c lies in the interval:

Correct answer: B. [4,6] | cases 1 & 1 & 1 2 & b & c 4 & b ^ 2 & c ^ 2 cases | C _ 2 → C _ 2 - C _ 1 , C _ 3 → C _ 3 - C _ 1 → | cases 1 & 0 & 0 2 & b - 2 & c - 2 4 & b ^ 2 - 4 & c ^ 2 - 4 cases | = (b - 2)(c - 2)(c - b) Let common difference of AP = d ∴ b = 2 + d, c = 2 + 2d |A| = d × 2d × d = 2 d ^ 3 ∈ [2, 16] ⇒ d ^ 3 ∈ [1, 8] ⇒ d ∈ [1, 2] ⇒ 2d ∈ [2, 4] ⇒ 2 + 2d ∈ [4, 6] ∴ c ∈ [4, 6]

JEE Main 20198 AprilEvening Shift

Exam: JEE Main
Year: 2019
Date: 8 April
Shift: Evening Shift
Subject: Mathematics
Chapter: Determinants

JEE Main 2019 · 8 April · Evening Shift · Mathematics · Determinants

Let the numbers 2, b, c be in an A.P. and A = [ cases 1 & 1 & 1 2 &...

Let the numbers

2, b, c

be in an A.P. and

A =

\left[

\begin{cases}1 & 1 & 1 \\ 2 & b & c \\ 4 & {b}^{2} & {c}^{2}\end{cases}

\right]

. If

det ( A ) \in [ 2,16 ] , then

c

lies in the interval:

Options

A $\left[2,3\right)$
B $\left[4,6\right]$
C $\left[3,2 + {2}^{\dfrac{3}{4}}\right]$
D $\left(2 + {2}^{\dfrac{3}{4}}, 4\right)$

Correct answer

B. $\left[4,6\right]$

Step-by-step solution

\left|

\begin{cases}1 & 1 & 1 \\ 2 & b & c \\ 4 & {b}^{2} & {c}^{2}\end{cases}

\right|\overset{{C}_{2} \to {C}_{2} - {C}_{1}, {C}_{3} \to {C}_{3} - {C}_{1}}{\to}\left|

\begin{cases}1 & 0 & 0 \\ 2 & b - 2 & c - 2 \\ 4 & {b}^{2} - 4 & {c}^{2} - 4\end{cases}

\right| = \left(b - 2\right)\left(c - 2\right)\left(c - b\right)

<br />Let common difference of

AP = d

<br />

∴ b = 2 + d, c = 2 + 2d

<br />\left|A\right| = d \times 2d \times d = 2{d}^{3} \in \left[2, 16\right]

⇒ {d}^{3} \in \left[1, 8\right] ⇒ d \in \left[1, 2\right] ⇒ 2d \in \left[2, 4\right]

⇒ 2 + 2d \in \left[4, 6\right]

∴ c \in \left[4, 6\right]

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