If \( n_1 \) and \( n_2 \) are the number of real valued solutions of \( x = |\sin^{-1} x| \) \(\text{and}\) \( x = \sin(x) \text{ respectively, then the value of} \, n_2 - n_1 \text{ is:}\)
We are given two equations: 1. \( x = |\sin^{-1} x| \) 2. \( x = \sin(x) \)
Step 1: Analyze the first equation \( x = |\sin^{-1} x| \) We know that \( \sin^{-1} x \) (the inverse sine function) gives values in the range \( [-\frac{\pi}{2}, \frac{\pi}{2}] \) for \( x \in [-1, 1] \). For the equation \( x = |\sin^{-1} x| \), we examine the two cases:
- Case 1: When \( x \geq 0 \), \( |\sin^{-1} x| = \sin^{-1} x \). Thus, \( x = \sin^{-1} x \).
- Case 2: When \( x < 0 \), \( |\sin^{-1} x| = -\sin^{-1} x \). Thus, \( x = -\sin^{-1} x \).
These two cases give us real solutions in the interval \( [-1, 1] \). Through solving, we find that there are 2 real solutions for \( x = |\sin^{-1} x| \). Hence, \( n_1 = 2 \).
Step 2: Analyze the second equation \( x = \sin(x) \) Now, we solve \( x = \sin(x) \). This is a transcendental equation, and we solve it by plotting or numerically approximating. The solutions occur within the interval \( [-1, 1] \).
We find 3 real solutions for this equation. Thus, \( n_2 = 3 \).
Step 3: Calculate \( n_2 - n_1 \) Now, we calculate the difference: \[ n_2 - n_1 = 3 - 2 = 1 \] Thus, the correct answer is \( \boxed{(a) \, 1} \).
An observer at a distance of 10 m from tree looks at the top of the tree, the angle of elevation is 60\(^\circ\). To find the height of tree complete the activity. (\(\sqrt{3} = 1.73\))
Activity :
In the figure given above, AB = h = height of tree, BC = 10 m, distance of the observer from the tree.
Angle of elevation (\(\theta\)) = \(\angle\)BCA = 60\(^\circ\)
tan \(\theta\) = \(\frac{\boxed{\phantom{AB}}}{BC}\) \(\dots\) (I)
tan 60\(^\circ\) = \(\boxed{\phantom{\sqrt{3}}}\) \(\dots\) (II)
\(\frac{AB}{BC} = \sqrt{3}\) \(\dots\) (From (I) and (II))
AB = BC \(\times\) \(\sqrt{3}\) = 10\(\sqrt{3}\)
AB = 10 \(\times\) 1.73 = \(\boxed{\phantom{17.3}}\)
\(\therefore\) height of the tree is \(\boxed{\phantom{17.3}}\) m.
In the figure given below, find RS and PS using the information given in \(\triangle\)PSR.
A remote island has a unique social structure. Individuals are either "Truth-tellers" (who always speak the truth) or "Tricksters" (who always lie). You encounter three inhabitants: X, Y, and Z.
X says: "Y is a Trickster"
Y says: "Exactly one of us is a Truth-teller."
What can you definitively conclude about Z?
Consider the following statements followed by two conclusions.
Statements: 1. Some men are great. 2. Some men are wise.
Conclusions: 1. Men are either great or wise. 2. Some men are neither great nor wise. Choose the correct option: