The partial fractions of x^2 (x - 1) ^3 (x - 2) are - Vidyadip

MCQ

The partial fractions of ${{{x^2}} \over {{{(x - 1)}^3}(x - 2)}}$ are

A
${{ - 1} \over {{{(x - 1)}^3}}} + {3 \over {{{(x - 1)}^2}}} - {4 \over {(x - 1)}} + {4 \over {(x - 2)}}$
B
${{ - 1} \over {{{(x - 1)}^3}}} - {3 \over {{{(x - 1)}^2}}} + {4 \over {(x - 1)}} + {4 \over {(x - 2)}}$
✓
${{ - 1} \over {{{(x - 1)}^3}}} + {{ - 3} \over {{{(x - 1)}^2}}} + {{ - \,4} \over {(x - 1)}} + {4 \over {(x - 2)}}$
D
None of these

✓

Answer

Correct option: C.

${{ - 1} \over {{{(x - 1)}^3}}} + {{ - 3} \over {{{(x - 1)}^2}}} + {{ - \,4} \over {(x - 1)}} + {4 \over {(x - 2)}}$

c
(c) Put the repeated factor

$(x - 1) = y \Rightarrow x = y + 1$

$\therefore {{{x^2}} \over {{{(x - 1)}^3}(x - 2)}} = {{{{(1 + y)}^2}} \over {{y^3}(y - 1)}} = {{1 + 2y + {y^2}} \over {{y^3}( - 1 + y)}}$

Dividing the numerator,

$(1 + 2y + {y^2})$ by $( - 1 + y)$ till ${y^3}$ appears as factor,

we get ${{1 + 2y + {y^2}} \over { - 1 + y}} = ( - 1 - 3y - 4{y^2}) + {{4{y^3}} \over { - 1 + y}}$

Given expression = ${{ - 1} \over {{y^3}}} - {3 \over {{y^2}}} - {4 \over y} + {4 \over { - 1 + y}}$

= ${{ - 1} \over {{{(x - 1)}^3}}} + {{ - 3} \over {{{(x - 1)}^2}}} + {{ - 4} \over {(x - 1)}} + {4 \over {(x - 2)}}$.

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "MCQ" from basic of algoritham→basic of algoritham — all question types→MATHS STD 11 Science question papers→Gujarat Board STD 11 Science question papers→Gujarat Board question paper generator→

Similar questions

The sum of the series ${1^2}.2 + {2^2}.3 + {3^2}.4 + ........$ to $n$ terms is

If the ratio of the coefficient of third and fourth term in the expansion of ${\left( {x - \frac{1}{{2x}}} \right)^n}$ is $1 : 2$, then the value of $ n$ will be

If the point $\left(\alpha, \frac{7 \sqrt{3}}{3}\right)$ lies on the curve traced by the mid-points of the line segments of the lines $x$ $\cos \theta+ y \sin \theta=7, \theta \in\left(0, \frac{\pi}{2}\right)$ between the coordinates axes, then $\alpha$ is equal to

In any case, the difference of the least and greatest term is:

Let $n$ be an odd integer. If $\sin n\theta = \sum\limits_{r = 0}^n {{b_r}{{\sin }^r}\theta } $ for every value of $\theta $, then

If distance between the directrices be thrice the distance between the foci, then eccentricity of ellipse is

Two persons $A$ and $B$ throw a (fair)die (six-faced cube with faces numbered from $1$ to $6$ ) alternately, starting with $A$. The first person to get an outcome different from the previous one by the opponent wins. The probability that $B$ wins is

Tangent to the parabola $y = {x^2} + 6$ at $(1, 7)$ touches the circle ${x^2} + {y^2} + 16x + 12y + c = 0$ at the point

Let $P$ be an interior point of a $\triangle A B C$. Let $Q$ and $R$ be the reflections of $P$ in $A B$ and $A C$, respectively. If $Q, A, R$ are collinear, then $\angle A$ equals

Let a circle $C$ of radius $5$ lie below the $x$-axis. The line $L_{1}=4 x+3 y-2$ passes through the centre $P$ of the circle $C$ and intersects the line $L _{2}: 3 x -4 y -11=0$ at $Q$. The line $L _{2}$ touches $C$ at the point $Q$. Then the distance of $P$ from the line $5 x-12 y+51=0$ is