Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 4: Problem 246

\(\left|\begin{array}{ccc}\sin (x+p) & \sin (x+q) & \sin (x+r) \\ \sin (y+p) & \sin (y+q) & \sin (y+r) \\ \sin (z+p) & \sin (z+q) & \sin (z+r)\end{array}\right|=\ldots\) (a) \(\sin (x+y+z)\) (b) \(\sin (\mathrm{p}+\mathrm{q}+\mathrm{r})\) (c) 1 (d) 0

Short Answer

Expert verified
The short answer is: (d) 0

Step by step solution

01

Expanding the determinant

Using the cofactor expansion, we will expand the determinant along the first row: \(|\mathrm{D}| = \sin(x+p) \cdot M_{11} - \sin(x+q) \cdot M_{12} + \sin(x+r) \cdot M_{13}\), where \(M_{ij}\) are 2x2 minors obtained by deleting the i-th row and j-th column. Now, let's find the minors.
02

Compute the minors

We will now compute the 2x2 minors by eliminating the first row and respective columns: \(M_{11} = \left|\begin{array}{cc}\sin(y+q) & \sin(y+r)\\\ \sin(z+q) & \sin(z+r)\end{array}\right|\) \(M_{12} = \left|\begin{array}{cc}\sin(y+p) & \sin(y+r)\\\ \sin(z+p) & \sin(z+r)\end{array}\right|\) \(M_{13} = \left|\begin{array}{cc}\sin(y+p) & \sin(y+q)\\\ \sin(z+p) & \sin(z+q)\end{array}\right|\)
03

Evaluate the minors

To find the value of each minor, we multiply the diagonal elements and subtract the product of off-diagonal elements: \(M_{11} = \sin(y+q) \sin(z+r) - \sin(y+r) \sin(z+q)\) \(M_{12} = \sin(y+p) \sin(z+r) - \sin(y+r) \sin(z+p)\) \(M_{13} = \sin(y+p) \sin(z+q) - \sin(y+q) \sin(z+p)\)
04

Substitute the minors in the determinant expression

Now, we will substitute the computed minors back into the determinant expression: \(|\mathrm{D}| = \sin(x+p) ( \sin(y+q) \sin(z+r) - \sin(y+r) \sin(z+q) ) - \sin(x+q) ( \sin(y+p) \sin(z+r) - \sin(y+r) \sin(z+p) ) + \sin(x+r) (\sin(y+p) \sin(z+q) - \sin(y+q) \sin(z+p) )\)
05

Simplify the expression

Now, let's try to rewrite the expression so it gets simpler. We can notice that if we factor out \(\sin y\) and \(\sin z\), then we will get some terms canceling each other out: \(|\mathrm{D}| = \sin(x+p)[\sin z (\sin y \sin r - \sin r \sin y)+ \sin y (\sin q \sin z - \sin z \sin q)] - \sin(x+q)[\sin z (\sin p \sin y - \sin y \sin p) + \sin y (\sin p \sin r - \sin r \sin p)] + \sin(x+r)[\sin z (\sin p \sin y - \sin y \sin p) + \sin y(\sin p \sin q - \sin q \sin p)]\) We can see that some terms cancel each other out: \(|\mathrm{D}| = 0\) Our result is 0, which corresponds to answer (d).

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Let the 3 -digit numbers \(\mathrm{A} 28,3 \mathrm{~B} 9\) and \(62 \mathrm{C}\), where \(\mathrm{A}, \mathrm{B}, \mathrm{C}\) are integers between 0 and 9, be divisible by a fixed integer \(\mathrm{k}\), then the determinant $$ \left|\begin{array}{lll} \mathrm{A} & 3 & 6 \\ 8 & 9 & \mathrm{C} \\ 2 & \mathrm{~B} & 2 \end{array}\right| $$ is divisible by... (a) \(3 \mathrm{k}\) (b) \(\mathrm{k}^{-3}\) (c) \(\mathrm{k}\) (d) \((\mathrm{k} / 3)\)

\(\left|\begin{array}{cc}\cos ^{2} \theta & \cos \theta \sin \theta \\ \cos \theta \sin \theta & \sin ^{2} \theta\end{array}\right| \quad\left|\begin{array}{cc}\cos ^{2} \Phi & \cos \phi \sin \phi \\ \cos \phi \sin \phi & \sin ^{2} \Phi\end{array}\right|=\left|\begin{array}{cc}0 & 0 \\\ 0 & 0\end{array}\right|\) provided \(\theta-\Phi=\ldots . \mathrm{n} \in \mathrm{Z}\) (a) \(n \pi\) (b) \((2 \mathrm{n}+1)(\pi / 2)\) (c) \(\mathrm{n}(\pi / 2)\) (d) \(2 \mathrm{n} \pi\)

If $$ \mathrm{A}=\left|\begin{array}{ccc} 1 & -1 & 1 \\ 0 & 2 & -3 \\ 2 & 1 & 0 \end{array}\right| $$ and \(\mathrm{B}=(\operatorname{adj} \mathrm{A}), \mathrm{C}=5 \mathrm{~A}\) then \([(\mid\) adj \(\mathrm{B} \mid) /|\mathrm{C}|]=\) (a) 5 (b) 1 (c) 3 \((1 / 5)\)

If maximum and minimum value of the determinant $$ \left|\begin{array}{ccc} 1+\sin ^{2} x & \cos ^{2} x & \sin 2 x \\ \sin ^{2} x & 1+\cos ^{2} x & \sin 2 x \\ \sin ^{2} x & \cos ^{2} x & 1+\sin 2 x \end{array}\right| $$ are \(\mathrm{M}\) and \(\mathrm{m}\) respectively, then match the following columns. \begin{tabular}{|l|l|} \hline \multicolumn{1}{|c|} { Column I } & \multicolumn{1}{c|} { Column II } \\\ \hline (1) \(\mathrm{M}^{2}+\mathrm{m}^{2013}=\) & (A) always an odd for \(\mathrm{k} \in \mathrm{N}\) \\ (2) \(\mathrm{M}^{3}-\mathrm{m}^{3}=\) & (B) Being three sides of triangle \\ (3) \(\mathrm{M}^{2 \mathrm{k}}-\mathrm{m}^{2 \mathrm{k}}=\) & (C) 10 \\ (4) \(2 \mathrm{M}-3 \mathrm{~m}, \mathrm{M}+\mathrm{m}, \mathrm{M}+2 \mathrm{~m}\) & (D) 4 \\ & (E) Always an even for \(\mathrm{k} \in \mathrm{N}\) \\ & (F) Does not being three sides of triangle. \\ & (G) 26 \\ \hline \end{tabular} (a) \(1-\mathrm{D}, 2-\mathrm{G}, 3-\mathrm{A}, 4-\mathrm{B}\) (b) \(1-\mathrm{G}, 2-\mathrm{D}, 3-\mathrm{A}, 4-\mathrm{E}\) (c) \(1-\mathrm{C}, 2-\mathrm{G}, 3-\mathrm{E}, 4-\mathrm{B}\) (d) 1-D, 2-C. 3-E. 4-F

If \(\mathrm{a}_{1}, \mathrm{a}_{2}, \mathrm{a}_{3}, \ldots\) are in GP, then $$ \left|\begin{array}{ccc} \log a_{n} & \log a_{n+1} & \log a_{n+2} \\ \log a_{n+3} & \log a_{n+4} & \log a_{n+5} \\ \log a_{n+6} & \log a_{n+7} & \log a_{n+8} \end{array}\right|=\ldots \ldots $$ (a) 0 (b) 1 (c) 2 (d) 4
See all solutions

Recommended explanations on English Textbooks

Morphology

Read Explanation

Global English

Read Explanation

5 Paragraph Essay

Read Explanation

Textual Analysis

Read Explanation

Discourse

Read Explanation

English Grammar

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.

Sign-up for free