Ex.4.3 Q5 Quadratic Equations Solutions - NCERT Maths Class 10

Go back to  'Ex.4.3'

Question

In a class test the sum of Shefali’s marks in Mathematics and English is \(30.\) Had She got \(2\) marks more in Mathematics and \(3\) marks less in English, the product of their marks would have been \(210.\) Find her marks in the two subjects.

 Video Solution
Quadratic Equations
Ex 4.3 | Question 5

Text Solution

What is known?

(i) Sum of Shefali’s marks in Mathematics and English is \(30.\)

(ii) Had she got \(2\) marks more in Mathematics and \(3\) marks less in English, product of marks would have been \(210 .\)

What is Unknown?

Marks of Shefali in two subjects.

Reasoning:

Let the marks Shefali scored in mathematics be \(x​​​​​​​.\)

i) Then, marks scored by her in English \( \Rightarrow 30 – \) Marks scored in Mathematics \(= 30 – x\)

ii) 2 more marks in Mathematics \(= x + 2\)

 \[\begin{align} 3 \text{ marks less in English} &= 30-x-3\\&= 27-x \end{align}\]

\[\begin{align}\text{Product of these two }&= 210\\\left( {x + 2} \right)\left( {27 - x} \right) &= 210 \end{align}\]

Steps:

\[\begin{align}({{x}} + 2)(27 - {{x}}) &= 210\\27{{x}} - {{{x}}^2} + 54 - 2{{x}} &= 210\\ - {{{x}}^2} + 25{{x}} + 54 &= 210\\ - {{{x}}^2} + 25{{x}} + 54 - 210 &= 0\\ - {{{x}}^2} + 25{{x}} - 156 &= 0\end{align}\]

Multiplying both sides by \( -1:\)

\[{{{x}}^2} - 25{{x + }}156 = 0\]

Comparing with \(ax^\text{2}+bx+c=0\)

\[\begin{align}
{a{\rm{ }} = 1,{b}{\rm{ }} = {\rm{ }} - 25,c{\rm{ }} = {\rm{ }}156}\\{{b^2}\rm{ }} - {\rm{ }4ac{\rm{ }} = {\rm{ }}{( - 25)^2}{\rm{ }} - {\rm{ }}4{\rm{ }}(1){\rm{ }}(156)}\\= {\rm{ }}625{\rm{ }}-{\rm{ }}624\\ = {\rm{ }}1\\
{{b^2} - {\rm{ }}4ac{\rm{ }} > {\rm{ }}0}
\end{align}\]

\[\begin{align}{{x}} &= \frac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}\\ &= \frac{{ - ( - 25) \pm \sqrt {{{( - 25)}^2} - 4(1)(156)} }}{{2(1)}}\\ &= \frac{{ - ( - 25) \pm \sqrt 1 }}{{2(1)}}\\
 &= \frac{{25 \pm 1}}{2}\\x &= \frac{{25 + 1}}{2}\quad x = \frac{{25 - 1}}{2}\\x &= \frac{{26}}{2}\qquad \;x = \frac{{24}}{2}\\x &= 13 \qquad\;\; x = 12\end{align}\]

Two possible answers for this given question:

If Shefali scored \(13\) mark in mathematics, then mark in English \( = \left( {30 - 13} \right) = 17.\)

If Shefali scored \(12\) mark in mathematics, then mark in English \( = \left( {30 - 12} \right) = 18.\)