If x = log (y +√y2 + 1 ) then y =
If x = log (y +√y2 + 1 ) then y =
tanh x
coth x
sinh x
cosh x
The Correct Option is C
Solution and Explanation
To determine the value of \( y \) in terms of \( x \), given \( x = \log\left(y + \sqrt{y^2 + 1}\right) \), we proceed as follows:
- Convert to exponential form: \[ e^x = y + \sqrt{y^2 + 1} \]
- Isolate the square root: \[ e^x - y = \sqrt{y^2 + 1} \]
- Square both sides: \[ (e^x - y)^2 = y^2 + 1 \]
- Expand: \[ e^{2x} - 2ye^x + y^2 = y^2 + 1 \]
- Cancel \( y^2 \): \[ e^{2x} - 2ye^x = 1 \]
- Solve for \( y \): \[ 2ye^x = e^{2x} - 1 \quad \Rightarrow \quad y = \frac{e^{2x} - 1}{2e^x} \]
- Recognize the identity: \[ y = \frac{e^x - e^{-x}}{2} \]
Thus, the value of \( y \) is:
\( \sinh x \)
Top TS EAMCET Mathematics Questions
- A bag contains $5$ red balls, $3$ black balls and $4$ white balls are drawn at random. The probability that they are not of same colour is
- If $cosec\, \theta - \cot \theta = 2017$, then quadrant in which $\theta $ lies is
- If $a$ is a unit vector, then $| a \times \hat{ i }|^{2}+| a \times \hat{ j }|^{2}+| a \times \hat{ k }|^{2}=$
- $A$ straight line makes an intercept on the $Y$-axis twice as long as that on $X$-axis and is at a unit distance from the origin. Then the line is represented by the equations
- Let $S$ and $S'$ be the foci of an ellipse and B be one end of its minor axis. If $SBS'$ is an isosceles right angled triangle then the eccentricity of the ellipse is
Top TS EAMCET Quadratic Equations Questions
If A = \(\begin{bmatrix} 0 & 3\\ 0 & 0 \end{bmatrix}\)and f(x) = x+x2+x3+.....+x2023, then f(A)+I =
If x2 + 2px - 2p + 8 > 0 for all real values of x, then the set of all possible values of p is
The quadratic equation whose roots are sin218° and cos2 36° is
The roots of the equation x4 + x3 - 4x2 + x + 1 = 0 are diminished by h so that the transformed equation does not contain x2 term. If the values of such h are α and β, then 12(α - β)2 =
- If are the roots of the equation \[ 2x^3 \;-\; 5x^2 \;+\; 4x \;-\; 3 \;=\; 0, \] then \[ \sum \alpha\beta(\alpha+\beta) \;=\;? \]
Top TS EAMCET Questions
- A bag contains $5$ red balls, $3$ black balls and $4$ white balls are drawn at random. The probability that they are not of same colour is
- If $cosec\, \theta - \cot \theta = 2017$, then quadrant in which $\theta $ lies is
- If $a$ is a unit vector, then $| a \times \hat{ i }|^{2}+| a \times \hat{ j }|^{2}+| a \times \hat{ k }|^{2}=$
- $A$ straight line makes an intercept on the $Y$-axis twice as long as that on $X$-axis and is at a unit distance from the origin. Then the line is represented by the equations
- Let $S$ and $S'$ be the foci of an ellipse and B be one end of its minor axis. If $SBS'$ is an isosceles right angled triangle then the eccentricity of the ellipse is
Concepts Used:
Quadratic Equations
A polynomial that has two roots or is of degree 2 is called a quadratic equation. The general form of a quadratic equation is y=ax²+bx+c. Here a≠0, b, and c are the real numbers.
Consider the following equation ax²+bx+c=0, where a≠0 and a, b, and c are real coefficients.
The solution of a quadratic equation can be found using the formula, x=((-b±√(b²-4ac))/2a)
Two important points to keep in mind are:
- A polynomial equation has at least one root.
- A polynomial equation of degree ‘n’ has ‘n’ roots.
Read More: Nature of Roots of Quadratic Equation
There are basically four methods of solving quadratic equations. They are:
- Factoring
- Completing the square
- Using Quadratic Formula
- Taking the square root