A tetrahedron vessel has vertices ABCD such that AB→=i^,AC→=i^+j^,AD→=λi^+j^+k^ and it is completely filled with water List-I List-II(I)If the water evaporates from the top surface at the rate proportional to height of water level from the bottom (vertex A)(P)1927 of tetrahedron when λ=3(II)If the water evaporates from the top surface at positive rate proportional to area of water exposed to air(Q)time to empty of 78 total volume is 34 th of the time to empty whole of the vessel(III)If the tetrahedron is cut through a parallel plane such that area of ΔB'C'D' is 49 of area of ΔBCD then volume of figure BDCB'D'C' is(R)time to empty 78 of the total volume is 12 of the time to empty whole of the vessel(IV)For λ=2 the volume of the tetrahedron is(S)278 of tetrahedron when λ=3 (T)same as of the tetrahedron having λ=1The correct option is:

Let AB'→=αiˆ,AC→=αiˆ+αjˆ,AD'→=αλiˆ+αjˆ+αkˆ for some α∈(0,1)Volume of tetrahedron AB'C'D'=α36Area of ΔB'C'D'=α22λ2−2λ+2⇒ Height of tetrahedron from vertex A=αλ2−2λ+2

A tetrahedron vessel has vertices ABCD such that AB→=i^,AC→=i^+j^,AD→=λi^+j^+k^ and it is completely filled with water List-I List-II(I)If the water evaporates from the top surface at the rate proportional to height of water level from the bottom (vertex A)(P)1927 of tetrahedron when λ=3(II)If the water evaporates from the top surface at positive rate proportional to area of water exposed to air(Q)time to empty of 78 total volume is 34 th of the time to empty whole of the vessel(III)If the tetrahedron is cut through a parallel plane such that area of ΔB'C'D' is 49 of area of ΔBCD then volume of figure BDCB'D'C' is(R)time to empty 78 of the total volume is 12 of the time to empty whole of the vessel(IV)For λ=2 the volume of the tetrahedron is(S)278 of tetrahedron when λ=3 (T)same as of the tetrahedron having λ=1The correct option is:

Let AB'→=αiˆ,AC→=αiˆ+αjˆ,AD'→=αλiˆ+αjˆ+αkˆ for some α∈(0,1)Volume of tetrahedron AB'C'D'=α36Area of ΔB'C'D'=α22λ2−2λ+2⇒ Height of tetrahedron from vertex A=αλ2−2λ+2

AI Mentor

Check Your IQ

Free Expert Demo

Try Test

Courses

Dropper NEET Course Dropper JEE Course Class - 12 NEET Course Class - 12 JEE Course Class - 11 NEET Course Class - 11 JEE Course Class - 10 Foundation NEET Course Class - 10 Foundation JEE Course Class - 10 CBSE Course Class - 9 Foundation NEET Course Class - 9 Foundation JEE Course Class -9 CBSE Course Class - 8 CBSE Course Class - 7 CBSE Course Class - 6 CBSE Course

Offline Centres

A tetrahedron vessel has vertices $A B C D$ such that $\vec{A B} = \hat{i}, \vec{A C} = \hat{i} + \hat{j}, \vec{A D} = λ \hat{i} + \hat{j} + \hat{k}$ and it is completely filled with water
List-I List-II
(I) If the water evaporates from the top surface at the rate proportional to height of water level from the bottom (vertex A) (P) $\frac{19}{27}$ of tetrahedron when $λ = 3$
(II) If the water evaporates from the top surface at positive rate proportional to area of water exposed to air (Q) time to empty of $\frac{7}{8}$ total volume is $\frac{3}{4}$ th of the time to empty whole of the vessel
(III) If the tetrahedron is cut through a parallel plane such that area of $Δ B^{'} C^{'} D^{'}$ is $\frac{4}{9}$ of area of $Δ B C D$ then volume of figure $B D C B^{'} D^{'} C^{'}$ is (R) time to empty $\frac{7}{8}$ of the total volume is $\frac{1}{2}$ of the time to empty whole of the vessel
(IV) For $λ = 2$ the volume of the tetrahedron is (S) $\frac{27}{8}$ of tetrahedron when $λ = 3$
(T) same as of the tetrahedron having $λ = 1$
The correct option is:

see full answer

Your Exam Success, Personally Taken Care Of

1:1 expert mentors customize learning to your strength and weaknesses – so you score higher in school , IIT JEE and NEET entrance exams.

An Intiative by Sri Chaitanya

(I)-(R) (II)-(S) (III)-(T) (IV)-(P)

(I)-(Q) (II)-(R) (III)-(P) (IV)-(T)

(I)-(R) (II)-(Q) (III)-(P) (IV)-(S)

(I)-(Q) (II)-(R) (III)-(S) (IV)-(T)

answer is B.

(Unlock A.I Detailed Solution for FREE)

Best Courses for You

JEE

NEET

Foundation JEE

Foundation NEET

CBSE

Detailed Solution

Let $\vec{{AB}^{'}} = α \hat{i}, \vec{AC} = α \hat{i} + α \hat{j}, \vec{{AD}^{'}} = α λ \hat{i} + α \hat{j} + α \hat{k}$ for some $α \in (0, 1)$
Volume of tetrahedron $A B^{'} C^{'} D^{'} = \frac{α^{3}}{6}$
Area of $Δ B^{'} C^{'} D^{'} = \frac{α^{2}}{2} \sqrt{λ^{2} - 2 λ + 2}$
$\Rightarrow$ Height of tetrahedron from vertex $A = \frac{α}{\sqrt{λ^{2} - 2 λ + 2}}$

Watch 3-min video & get full concept clarity

courses

No courses found

Result Producing online coaching for JEE/NEET of south India

Largest All India Test Series for JEE/NEET

Best Online Course for IIT JEE

Best Online Course for NEET

Best Online Course for CBSE

Ready to Test Your Skills?

Check your Performance Today with our Free Mock Test used by Toppers!

Take Free Test

Get Expert Academic Guidance – Connect with a Counselor Today!