Question

Aarti and Anu work for X days , Anku and Sham work for next (X+2⅔) days. If remaining 12½ % of total work is completed by Prabhat in 6 more days, then Prabhat will do what portion of total work in (x+12) days ?
(a) ⅔. (b) ¼. (c) ¾. (d) ⅖​

Answer 1

Answer:

\large\underline{\sf{Solution-}}

Solution−

Let assume that

\begin{gathered}\rm \: \alpha ,\gamma \: be \: the \: roots \: of {3x}^{2} - 2x - 5 = 0 \\ \end{gathered}

α,γbetherootsof3x

2

−2x−5=0

Consider,

\begin{gathered}\rm \: {3x}^{2} - 2x - 5 = 0 \\ \end{gathered}

3x

2

−2x−5=0

\begin{gathered}\rm \: {3x}^{2} - 5x + 3x - 5 = 0 \\ \end{gathered}

3x

2

−5x+3x−5=0

\begin{gathered}\rm \: x(3x - 5) + 1(3x - 5) = 0 \\ \end{gathered}

x(3x−5)+1(3x−5)=0

\begin{gathered}\rm \: (x + 1)(3x - 5) = 0 \\ \end{gathered}

(x+1)(3x−5)=0

\begin{gathered}\rm\implies \:x = - 1 \: \: or \: \: x = \dfrac{5}{3} \\ \end{gathered}

⟹x=−1orx=

3

5

\begin{gathered}\rm\implies \: \gamma = - 1 \: \: or \: \: \alpha = \dfrac{5}{3} \\ \rm \: or \\ \rm\implies \: \alpha = - 1 \: \: or \: \: \gamma = \dfrac{5}{3} \\\end{gathered}

⟹γ=−1orα=

3

5

or

⟹α=−1orγ=

3

5

Now,

Let assume that,

\begin{gathered}\rm \: \alpha, \beta \: be \: the \: roots \: of \: {2x}^{2} + px - 1 = 0 \\ \end{gathered}

α,βbetherootsof2x

2

+px−1=0

We know,

\begin{gathered}\boxed{\red{\sf Sum\ of\ the\ roots=\frac{-coefficient\ of\ x}{coefficient\ of\ x^{2}}}} \\ \end{gathered}

Sum of the roots=

coefficient of x

2

−coefficient of x

\begin{gathered}\rm\implies \: \alpha + \beta = - \dfrac{p}{2} \\ \end{gathered}

⟹α+β=−

2

p

Also,

\begin{gathered}\boxed{\red{\sf Product\ of\ the\ roots=\frac{Constant}{coefficient\ of\ x^{2}}}} \\ \end{gathered}

Product of the roots=

coefficient of x

2

Constant

\begin{gathered}\rm\implies \: \alpha \beta = - \dfrac{1}{2} \\ \end{gathered}

⟹αβ=−

2

1

It means, we have

\begin{gathered}\rm\implies \: \alpha + \beta = - \dfrac{p}{2} \: \: and \: \: \alpha \beta = - \dfrac{1}{2} \\ \end{gathered}

⟹α+β=−

2

p

andαβ=−

2

1

Case :- 1

\begin{gathered}\rm \: \alpha = - 1 \\\end{gathered}

α=−1

As, we have

\begin{gathered}\rm \: \alpha \beta = - \dfrac{1}{2} \\ \end{gathered}

αβ=−

2

1

\begin{gathered}\rm \: ( - 1) \beta = - \dfrac{1}{2} \\ \end{gathered}

(−1)β=−

2

1

\begin{gathered}\rm \:\beta = \dfrac{1}{2} \\ \end{gathered}

β=

2

1

Now,

\begin{gathered}\rm \: \alpha + \beta = - \dfrac{p}{2} \\ \end{gathered}

α+β=−

2

p

\begin{gathered}\rm \: - 1 + \dfrac{1}{2} = - \dfrac{p}{2} \\ \end{gathered}

−1+

2

1

=−

2

p

\begin{gathered}\rm \: - \dfrac{1}{2} = - \dfrac{p}{2} \\ \end{gathered}

−

2

1

=−

2

p

\begin{gathered}\rm\implies \:p \: = \: 1 \\ \end{gathered}

⟹p=1

Now, Consider Case :- 2

\begin{gathered}\rm \: \alpha = \dfrac{5}{3} \\\end{gathered}

α=

3

5

As, we have

\begin{gathered}\rm \: \alpha \beta = - \dfrac{1}{2} \\ \end{gathered}

αβ=−

2

1

\begin{gathered}\rm \: \frac{5}{3} \times \beta = - \dfrac{1}{2} \\ \end{gathered}

3

5

×β=−

2

1

\begin{gathered}\rm \: \beta = - \dfrac{3}{10} \\ \end{gathered}

β=−

10

3

Now,

\begin{gathered}\rm \: \alpha + \beta = - \dfrac{p}{2} \\ \end{gathered}

α+β=−

2

p

\begin{gathered}\rm \: \dfrac{5}{3} - \dfrac{3}{10} = - \dfrac{p}{2} \\ \end{gathered}

3

5

−

10

3

=−

2

p

\begin{gathered}\rm \: \dfrac{50 - 9}{30} = - \dfrac{p}{2} \\ \end{gathered}

30

50−9

=−

2

p

\begin{gathered}\rm \: \dfrac{41}{30} = - \dfrac{p}{2} \\ \end{gathered}

30

41

=−

2

p

\begin{gathered}\rm\implies \:p \: = \: - \: \dfrac{41}{15}\\ \end{gathered}

⟹p=−

15

41

So,

\begin{gathered}\begin{gathered}\begin{gathered}\bf\: \rm\implies \:\begin{cases} &\bf{p \: = \: 1} \\ \\ &\sf{or}\\ \\ &\bf{p \: = \: - \: \dfrac{41}{15} } \end{cases}\end{gathered}\end{gathered}\end{gathered}

⟹

⎩

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎨

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎧

p=1

or

p=−

15

41

\rule{190pt}{2pt}

Additional Information :-

Nature of roots :-

Let us consider a quadratic equation ax² + bx + c = 0, then nature of roots of quadratic equation depends upon Discriminant (D) of the quadratic equation.

If Discriminant, D > 0, then roots of the equation are real and unequal.

If Discriminant, D = 0, then roots of the equation are real and equal.

If Discriminant, D < 0, then roots of the equation are unreal or complex or imaginary.

Where,

Discriminant, D = b² - 4ac