MHT CET · Maths · Inverse Trigonometric Functions
If \(\cos ^{-1} x+\cos ^{-1} y+\cos ^{-1} z=3 \pi\), then the value of \(x^2+y^2+z^2-2 x y z\) is
- A 3
- B 2
- C 1
- D 5
Answer & Solution
Correct Answer
(D) 5
Step-by-step Solution
Detailed explanation
Since, \(0 \leq \cos ^{-1} x \leq \pi\)
\(\therefore \quad \cos ^{-1} x\) cannot be greater than \(\pi\)
\(\therefore \quad \cos ^{-1} x=\cos ^{-1} y=\cos ^{-1} x=\pi\)
Therefore, \(x=y=\mathrm{z}=-1\)
Putting these values in the given expression, we get
\(\begin{aligned}
& x^2+y^2+z^2-2 x y z \\
& =(-1)^2+(-1)^2+(-1)^2-2(-1)(-1)(-1) \\
& =1+1+1-2(-1) \\
& =3+2 \\
& =5
\end{aligned}\)
\(\therefore \quad \cos ^{-1} x\) cannot be greater than \(\pi\)
\(\therefore \quad \cos ^{-1} x=\cos ^{-1} y=\cos ^{-1} x=\pi\)
Therefore, \(x=y=\mathrm{z}=-1\)
Putting these values in the given expression, we get
\(\begin{aligned}
& x^2+y^2+z^2-2 x y z \\
& =(-1)^2+(-1)^2+(-1)^2-2(-1)(-1)(-1) \\
& =1+1+1-2(-1) \\
& =3+2 \\
& =5
\end{aligned}\)
See the Complete Solution
Get step-by-step explanations for this and 2.5 Lakh+ more JEE, NEET & CET questions.
- Unlock all solutions
- Practice the full chapter
- Track accuracy across PYQs
4.8 rated on Google Play · 14,000+ reviews
More questions from Maths
- If \(y=\tan ^{-1}\left(\frac{4 \sin 2 x}{\cos 2 x-6 \sin ^2 x}\right)\), then \(\left(\frac{\mathrm{d} y}{\mathrm{~d} x}\right)\) at \(x=0\) isMHT CET 2023 Hard
- \(\int_{0}^{\frac{\pi}{2}} \frac{\sin ^{\frac{2}{3}} x}{\sin ^{\frac{2}{3}} x+\cos ^{\frac{2}{3}} x} d x=\)MHT CET 2020 Easy
- A plane is parallel to two lines, whose direction ratios are \(1,0,-1\) and \(-1,1,0\) and it contains the point \((1,1,1)\). If it cuts co-ordinate axes (X, Y, Z - axes resp.) at \(\mathrm{A}, \mathrm{B}, \mathrm{C}\), then the volume of the tetrahedron \(\mathrm{OABC}\) is cu. units.MHT CET 2023 Hard
- If \(\int \frac{\mathrm{d} x}{\sqrt[3]{\sin ^{11} x \cos x}}=-\left(\frac{3}{8} \mathrm{f}(x)+\frac{3}{2} \mathrm{~g}(x)\right)+\mathrm{c}\) thenMHT CET 2024 Hard
- If \(x=-1\) and \(\cdot x=2\) are extreme points of \(f(x)=\alpha \log |x|+\beta x^2+x\), thenMHT CET 2024 Easy
- If a fair coin is tossed 8 times, then the probability that it shows heads more than tails isMHT CET 2020 Medium
More PYQs from MHT CET
- Which of the following compounds has maximum covalent character?MHT CET 2025 Medium
- Find the freezing point depression of solution having molality \(0.25 \mathrm{~mol} \mathrm{~kg}^{-1}\). \(\left(\mathrm{K}_{\mathrm{f}}=4.0\right)\)MHT CET 2022 Easy
- The amine 'A', when treated with nitrous acid, gives a yellow oily substance. The amine A isMHT CET 2016 Hard
- Identify the product 'C' in the following reaction.
Aniline \(\xrightarrow[\text { Pyridine }]{\left( CH _3 CO \right)_2 O } A \xrightarrow[ CH _3 COOH ]{ Br _2}\) \(B \xrightarrow{ H ^{+} \text {or } OH ^{-}} C\)MHT CET 2018 Hard - A simple pendulum of length has mass and it oscillates freely with amplitude . At the extreme position, its potential energy is ( = acceleration due to gravity)MHT CET 2017 Medium
- Which compound is used in medicine as barium meal for intestinal x-ray?MHT CET 2024 Medium