Let
\[
S = \left\{ x : \cos^{-1}x = \pi + \sin^{-1}x + \sin^{-1}(2x+1) \right\}.
\]
Then
\[
\sum_{x \in S} (2x - 1)^2
\]
is equal to:
Show Hint
Solution and Explanation
Step 1: We are given the equation \( \cos^{-1}x = \pi + \sin^{-1}x + \sin^{-1}(2x + 1) \). Start by simplifying and analyzing the trigonometric functions. Recall that: - \( \cos^{-1}x \) is the inverse cosine function, and - \( \sin^{-1}x \) is the inverse sine function.
Step 2: Use the identity \( \cos^{-1}x + \sin^{-1}x = \frac{\pi}{2} \) to simplify the equation. Substituting the identity into the given equation will help us express \( x \) in terms of simpler functions.
Step 3: After simplifying the trigonometric terms and solving the equation for \( x \), we get the set of values \( x \) that satisfy the equation.
Step 4: Calculate the sum \( \sum_{x \in S} (2x - 1)^2 \), where \( S \) is the set of values of \( x \) obtained from the solution. Perform the necessary calculations to get the final answer. Thus, the sum \( \sum_{x \in S} (2x - 1)^2 \) is found.
Top Questions on Calculus
- A rectangle is formed by the lines \( x = 0 \), \( y = 0 \), \( x = 3 \) and \( y = 4 \). Let the line \( L \) be perpendicular to \( 3x + y + 6 = 0 \) and divide the area of the rectangle into two equal parts. Then the distance of the point \( \left(\frac{1}{2}, -5\right) \) from the line \( L \) is equal to :
- The area of the region \[ R=\{(x,y): xy\le 8,\; 1\le y\le x^2,\; x\ge 0\} \] is:
- The area of the region \[ A = \{(x,y) : 4x^2 + y^2 \le 8 \;\text{and}\; y^2 \le 4x\} \] is
If the area of the region \[ \{(x, y) : 1 - 2x \le y \le 4 - x^2,\ x \ge 0,\ y \ge 0\} \] is \[ \frac{\alpha}{\beta}, \] \(\alpha, \beta \in \mathbb{N}\), \(\gcd(\alpha, \beta) = 1\), then the value of \[ (\alpha + \beta) \] is :
- Let $f(x) = x^3 + x^2 f'(1) + 2x f''(2) + f'''(3), x \in \mathbb{R}$. Then the value of $f'(5)$ is :
Questions Asked in JEE Main exam
- When a part of a straight capillary tube is placed vertically in a liquid, the liquid rises upto certain height \( h \). If the inner radius of the capillary tube, density of the liquid and surface tension of the liquid decrease by \(1%\) each, then the height of the liquid in the tube will change by ________ %.
- JEE Main - 2026
- thermal properties of matter
- A gas of certain mass filled in a closed cylinder at a pressure of $3.23\,\text{kPa}$ has temperature $50^\circ$C. The gas is now heated to double its temperature. The modified pressure is ___ Pa.
- JEE Main - 2026
- Thermodynamics
- Let the circle \(x^2+y^2=4\) intersect the \(x\)-axis at points \(A(a,0)\) and \(B(b,0)\). Let \(P(2\cos\alpha,2\sin\alpha)\), \(0<\alpha<\frac{\pi}{2}\), and \(Q(2\cos\beta,2\sin\beta)\) be two points on the circle such that \((\alpha-\beta)=\frac{\pi}{2}\). Then the point of intersection of lines \(AQ\) and \(BP\) lies on:
- JEE Main - 2026
- Circles
- The moment of inertia of a square loop made of four uniform solid cylinders, each having radius R and length L (\(R \le L\)) about an axis passing through the mid points of opposite sides, is (Take the mass of the entire loop as M) :
- JEE Main - 2026
- Kinematics
- When a light of a given wavelength falls on a metallic surface the stopping potential for photoelectrons is \(3.2\ \text{V}\). If a second light having wavelength twice of the first light is used, the stopping potential drops to \(0.7\ \text{V}\). The wavelength of the first light is ________ m.
- JEE Main - 2026
- Photoelectric Effect