Question:
If \( \frac{e^x + 2}{(e^x - 1)(2e^x - 3)} = -\frac{3}{e^x - 1} + \frac{B}{2e^x - 3} \), then \( B \) is equal to
If \( \frac{e^x + 2}{(e^x - 1)(2e^x - 3)} = -\frac{3}{e^x - 1} + \frac{B}{2e^x - 3} \), then \( B \) is equal to
Show Hint
When solving partial fractions, expand both sides and compare the coefficients of like terms to solve for unknowns.
Updated On: Apr 22, 2026
- 1
- \( -1 \)
- 4
- 7
Show Solution
Verified By Collegedunia
The Correct Option is D
Solution and Explanation
Step 1: Set up the partial fractions.
We are given the equation: \[ \frac{e^x + 2}{(e^x - 1)(2e^x - 3)} = -\frac{3}{e^x - 1} + \frac{B}{2e^x - 3} \] To combine the fractions on the right-hand side, get a common denominator: \[ -\frac{3}{e^x - 1} + \frac{B}{2e^x - 3} = \frac{-3(2e^x - 3) + B(e^x - 1)}{(e^x - 1)(2e^x - 3)} \]
Step 2: Set the numerators equal.
Since the denominators are the same on both sides, we can set the numerators equal to each other: \[ e^x + 2 = -3(2e^x - 3) + B(e^x - 1) \]
Step 3: Expand both sides.
Expand the right-hand side: \[ e^x + 2 = -6e^x + 9 + B(e^x - 1) \] Now distribute \( B \): \[ e^x + 2 = -6e^x + 9 + Be^x - B \]
Step 4: Combine like terms.
Combine the \( e^x \) terms and constant terms: \[ e^x + 2 = (-6 + B)e^x + (9 - B) \]
Step 5: Compare coefficients.
Equate the coefficients of \( e^x \) and the constant terms on both sides:
- From the \( e^x \)-terms: \( 1 = -6 + B \quad \implies \quad B = 7 \)
- From the constant terms: \( 2 = 9 - B \quad \implies \quad B = 7 \)
Step 6: Conclusion.
Thus, \( B = 7 \), corresponding to option (D).
We are given the equation: \[ \frac{e^x + 2}{(e^x - 1)(2e^x - 3)} = -\frac{3}{e^x - 1} + \frac{B}{2e^x - 3} \] To combine the fractions on the right-hand side, get a common denominator: \[ -\frac{3}{e^x - 1} + \frac{B}{2e^x - 3} = \frac{-3(2e^x - 3) + B(e^x - 1)}{(e^x - 1)(2e^x - 3)} \]
Step 2: Set the numerators equal.
Since the denominators are the same on both sides, we can set the numerators equal to each other: \[ e^x + 2 = -3(2e^x - 3) + B(e^x - 1) \]
Step 3: Expand both sides.
Expand the right-hand side: \[ e^x + 2 = -6e^x + 9 + B(e^x - 1) \] Now distribute \( B \): \[ e^x + 2 = -6e^x + 9 + Be^x - B \]
Step 4: Combine like terms.
Combine the \( e^x \) terms and constant terms: \[ e^x + 2 = (-6 + B)e^x + (9 - B) \]
Step 5: Compare coefficients.
Equate the coefficients of \( e^x \) and the constant terms on both sides:
- From the \( e^x \)-terms: \( 1 = -6 + B \quad \implies \quad B = 7 \)
- From the constant terms: \( 2 = 9 - B \quad \implies \quad B = 7 \)
Step 6: Conclusion.
Thus, \( B = 7 \), corresponding to option (D).
Was this answer helpful?
0
0
Top MET Mathematics Questions
- $\int\limits_{0}^{8}| x -5| d x$ is equal to
- The equation $3^{3x + 4} = 9^{2x - 2},\ x>0$ has the solution
- A parallelogram is constructed on the vectors $\mathbf{a} = 3\alpha - \beta$, $\mathbf{b} = \alpha + 3\beta$. If $|\alpha| = |\beta| = 2$ and the angle between $\alpha$ and $\beta$ is $\dfrac{\pi}{3}$, then the length of a diagonal of the parallelogram is
- Every group of order 7 is
- Let $U_{n} = 2 + 2^{3} + 2^{5} + \cdots + 2^{2n+1}$ and $V_{n} = 1 + 4 + 4^{2} + \cdots + 4^{n-1}$. Then $\displaystyle\lim_{n \to \infty} \dfrac{U_n}{V_n}$ is equal to
View More Questions
Top MET Integral Calculus Questions
- If \(\int f(x) dx = F(x)\), then \(\int x^3 f(x^2) dx\) is equal to
- The approximate value of \(\int_1^5 x^2 dx\) using trapezoidal rule with \(n = 4\) is
- If \(I_n = \int \sin^n x dx\), then \(nI_n - (n - 1)I_{n-2}\) equals
- If \[ \int \sin^2 t \tan^{-1}\sqrt{\frac{1-x}{1+x}}\,dx = A\sin^{-1}x + B\sqrt{1-x^2} + C, \] then \(A+B\) is equal to
- By Simpson’s \(1/3\)rd rule, the approximate value of the integral \(\int_1^2 e^{-x/2}dx\) using four intervals, is
View More Questions
Top MET Questions
- A coil of 100 turns and area $2 \times 10^{-2}m^2 $ is pivoted about a vertical diameter in a uniform magnetic field and carries a current of 5A. When the coil is held with its plane in north-south direction, it experiences a couple of 0.33 Nm. When the plane is east-west, the corresponding couple is 0.4 Nm, the value of magnetic induction is [Neglect earth's magnetic field]
- $\int\limits_{0}^{8}| x -5| d x$ is equal to
- Radiation, with wavelength 6561 $?$ falls on a metal surface to produce photoelectrons. The electrons are made to enter a uniform magnetic field of $3 \times 10^{-4} T$. If the radius of the largest circular path followed by the electrons is 10 mm, the work function of the metal is close to :
- In intrinsic semiconductor at room temperature number of electrons and holes are
- Which of the following product is formed in the reaction $ CH_3MgBr {->[(i)CO_2][(ii)H_2O]} ? $
View More Questions