Answer

Verified

483.3k+ views

Hint- We will use the basic definitions of inverse cosine to solve this question.

As we know, if $y = {\cos^{ - 1}}x$ then \[x\] must be in the range of (-1, 1) and y must be in the range of $(0,\pi ).$

Complete step-by-step solution -

Given equation is ${\cos ^{ - 1}}p + {\cos ^{ - 1}}q + {\cos ^{ - 1}}r = 3\pi $

Let ${y_1} = {\cos ^{ - 1}}p,{y_2} = {\cos ^{ - 1}}q,{y_3} = {\cos ^{ - 1}}r$

Now, we have to evaluate the values of p, q, r.

As we know if $y = {\cos^{ - 1}}x,{\text{ then - 1}} \leqslant {\text{x}} \leqslant {\text{1 and 0}} \leqslant {\text{y}} \leqslant \pi $

Hence, the given equation will hold only when each has the highest value of y.

So, ${y_1} = {y_2} = {y_3} = \pi $

$

\Rightarrow {\cos ^{ - 1}}p = {\cos ^{ - 1}}q = {\cos ^{ - 1}}r = \pi \\

\Rightarrow p = q = r = \cos \pi \\

\Rightarrow p = q = r = - 1 \\

$

Now, we have to find the value of ${p^2} + {q^2} + {r^2} + 2pqr$

By substituting the value $p = q = r = - 1$ we get

$

\Rightarrow {( - 1)^2} + {( - 1)^2} + {( - 1)^2} + 2( - 1)( - 1)( - 1) \\

\Rightarrow 3 - 2 \\

\Rightarrow 1 \\

$

Hence, the correct option is B.

Note- To solve inverse trigonometric equations remember the basic concept of solving the algebraic equations and remember the definitions of inverse trigonometric functions such as about their domain and range. The identities of inverse trigonometric functions must be remembered with their domain and range.

As we know, if $y = {\cos^{ - 1}}x$ then \[x\] must be in the range of (-1, 1) and y must be in the range of $(0,\pi ).$

Complete step-by-step solution -

Given equation is ${\cos ^{ - 1}}p + {\cos ^{ - 1}}q + {\cos ^{ - 1}}r = 3\pi $

Let ${y_1} = {\cos ^{ - 1}}p,{y_2} = {\cos ^{ - 1}}q,{y_3} = {\cos ^{ - 1}}r$

Now, we have to evaluate the values of p, q, r.

As we know if $y = {\cos^{ - 1}}x,{\text{ then - 1}} \leqslant {\text{x}} \leqslant {\text{1 and 0}} \leqslant {\text{y}} \leqslant \pi $

Hence, the given equation will hold only when each has the highest value of y.

So, ${y_1} = {y_2} = {y_3} = \pi $

$

\Rightarrow {\cos ^{ - 1}}p = {\cos ^{ - 1}}q = {\cos ^{ - 1}}r = \pi \\

\Rightarrow p = q = r = \cos \pi \\

\Rightarrow p = q = r = - 1 \\

$

Now, we have to find the value of ${p^2} + {q^2} + {r^2} + 2pqr$

By substituting the value $p = q = r = - 1$ we get

$

\Rightarrow {( - 1)^2} + {( - 1)^2} + {( - 1)^2} + 2( - 1)( - 1)( - 1) \\

\Rightarrow 3 - 2 \\

\Rightarrow 1 \\

$

Hence, the correct option is B.

Note- To solve inverse trigonometric equations remember the basic concept of solving the algebraic equations and remember the definitions of inverse trigonometric functions such as about their domain and range. The identities of inverse trigonometric functions must be remembered with their domain and range.

Recently Updated Pages

what is the correct chronological order of the following class 10 social science CBSE

Which of the following was not the actual cause for class 10 social science CBSE

Which of the following statements is not correct A class 10 social science CBSE

Which of the following leaders was not present in the class 10 social science CBSE

Garampani Sanctuary is located at A Diphu Assam B Gangtok class 10 social science CBSE

Which one of the following places is not covered by class 10 social science CBSE

Trending doubts

Which are the Top 10 Largest Countries of the World?

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

Give 10 examples for herbs , shrubs , climbers , creepers

The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths

How do you graph the function fx 4x class 9 maths CBSE

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

Change the following sentences into negative and interrogative class 10 english CBSE

Why is there a time difference of about 5 hours between class 10 social science CBSE

Differentiate between homogeneous and heterogeneous class 12 chemistry CBSE