If \(\frac{3x+2}{(x+1)(2x^2+3)} = \frac{A}{x+1}+ \frac{Bx+C}{2x^2+3}\), then A - B + C=
If \(\frac{3x+2}{(x+1)(2x^2+3)} = \frac{A}{x+1}+ \frac{Bx+C}{2x^2+3}\), then A - B + C=
2
1
3
6
The Correct Option is A
Solution and Explanation
To solve the given equation and find \( A - B + C \), let's perform partial fraction decomposition. The given equation is:
\(\frac{3x + 2}{(x+1)(2x^2+3)} = \frac{A}{x+1} + \frac{Bx+C}{2x^2+3}\)
To decompose the fraction, multiply both sides by \((x+1)(2x^2+3)\) to get rid of the denominators:
\(3x + 2 = A(2x^2 + 3) + (Bx + C)(x + 1)\)
Expand the right side of the equation:
- \(A(2x^2 + 3) = 2Ax^2 + 3A\)
- \((Bx + C)(x + 1) = Bx^2 + Bx + Cx + C = Bx^2 + (B + C)x + C\)
Combine the terms:
\(3x + 2 = 2Ax^2 + 3A + Bx^2 + (B + C)x + C\)
Coefficient matching for the powers of \( x \):
- For \(x^2\), \(2A + B = 0\)
- For \(x^1\), \(B + C = 3\)
- For \(x^0\), \(3A + C = 2\)
Now, solve this system of equations:
From equation 1:
\(B = -2A\)
Substitute \(B = -2A\) in equation 2:
\( -2A + C = 3\)
\(C = 3 + 2A\)
Substitute \(C = 3 + 2A\) in equation 3:
\(3A + (3 + 2A) = 2\)
\(5A + 3 = 2\)
\(5A = 2 - 3\)
\(5A = -1\)
\(A = -\frac{1}{5}\)
Find values of \(B\) and \(C\):
\(B = -2A = 2 \times \frac{1}{5} = \frac{2}{5}\)
\(C = 3 + 2A = 3 + 2 \times -\frac{1}{5} = 3 - \frac{2}{5} = \frac{15}{5} - \frac{2}{5} = \frac{13}{5}\)
Calculate \(A - B + C\):
\(-\frac{1}{5} - \frac{2}{5} + \frac{13}{5} = \frac{-1 - 2 + 13}{5} = \frac{10}{5} = 2\)
Therefore, the value of \(A - B + C\) is 2.
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 Algebra of Complex Numbers Questions
If f(x) is a function such that f(x+y) = f(x)+ f(y) and f(1) = 7 then \( \sum_{r=1}^{n}\) f(r) =
If A is a square matrix of order 3, then |Adj(Adj A2)| =
If (-c, c) is the set of all values of x for which the expansion is (7 - 5x)-2/3 is valid, then 5c + 7 =
If a line ax + 2y = k forms a triangle of area 3 sq.units with the coordinate axis and is perpendicular to the line 2x - 3y + 7 = 0, then the product of all the possible values of k is
The quadratic equation whose roots are
\(l = \lim_{\theta\to0} \frac{3sin\theta - 4sin^3\theta}{\theta}\)
m = \(\lim_{\theta\to0} \frac{2tan\theta}{\theta(1-tan^2\theta)}\) is
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:
Algebra of Complex Numbers
Algebra of complex numbers
1. Addition of two complex numbers:
Consider z1 and z2 are two complex numbers.
For example, z1 = 3+4i and z2 = 4+3i
Here a=3, b=4, c=4, d=3
∴z1+ z2 = (a+c)+(b+d)i
⇒z1 + z2 = (3+4)+(4+3)i
⇒z1 + z2 = 7+7i
Properties of addition of complex numbers
- Closure law: While adding two complex numbers the resulting number is also a complex number.
- Commutative law: For the complex numbers z1 and z2 , the commutation can be z1+ z2 = z2+z1
- Associative law: While considering three complex numbers, (z1+ z2) + z?3 = z1 + (z2 + z3)
- Additive identity: An Additive identity is nothing but zero complex numbers that go as 0+i0. For every complex number z, z+0 = z.
- Additive inverse: Every complex number has an additive inverse denoted as -z.
2. Difference between two complex numbers
It is similar to the addition of complex numbers, such that, z1 - z2 = z1 + ( -z2)
For example: (5+3i) - (2+1i) = (5-2) + (-2-1i) = 3 - 3i
3. Multiplication of complex numbers
Considering the same value of z1 and z2 , the product of the complex numbers are
z1 * z2 = (ac-bd) + (ad+bc) i
For example: (5+6i) (2+3i) = (5×2) + (6×3)i = 10+18i
Properties of Multiplication of complex numbers
Note: The properties of multiplication of complex numbers are similar to the properties we discussed in addition to complex numbers.
- Closure law: When two complex numbers are multiplied the result is also a complex number.
- Commutative law: z1* z2 = z2 * z1
Associative law: Considering three complex numbers, (z1 z2) z3 = z1 (z2 z3)
- Multiplicative identity: 1+0i is always denoted as 1. This is multiplicative identity. This means that z.1 = z for every complex number z.
- Distributive law: Considering three complex numbers, z1 (z2 + z3) =z1 z2 + z1 z3 and (z1+ z2) z3 = z1 z2 + z2 z3.
Read More: Complex Numbers and Quadratic Equations
4. Division of complex numbers
If z1 / z2 of a complex number is asked, simplify it as z1 (1/z2 )
For example: z1 = 4+2i and z2 = 2 - i
z1 / z2 =(4+2i)×1/(2 - i) = (4+i2)(2/(2²+(-1)² ) + i (-1)/(2²+(-1)² ))
=(4+i2) ((2+i)/5) = 1/5 [8+4i + 2(-1)+1] = 1/5 [8-2+1+41] = 1/5 [7+4i]