
If the roots of the equation $12{{x}^{2}}+mx+5=0$ are in the ratio 3: 2 the value of m is?
( a ) $5\sqrt{10}$
( b ) $3\sqrt{10}$
( c ) $2\sqrt{10}$
( d ) none of these
Answer
164.4k+ views
Hint: In this question, a quadratic equation is given with their roots and we have to find the value of m. as the roots are in ratio so we suppose the variable a with the ratios, and then we apply the conditions for the sum and the product of roots and after solving it, we get the value of m.
Formula used:
Sum of roots = $\dfrac{-b}{a}$
Product of roots = $\dfrac{c}{a}$
Complete step by step Solution:
Given equation is $12{{x}^{2}}+mx+5=0$
Compare the above equation with the standard form of quadratic equation $a{{x}^{2}}+bx+c=0$, we get
a = 12 , b = m and c = 5
And their roots are in the ratio 3 : 2
Let the roots of the equation are 3a and 2a
Now we apply the condition for the sum and the product of roots, and we get
Sum of roots ( 3a + 2a) = $-\dfrac{b}{a}$= $-\dfrac{m}{12}$
That is 5a = $-\dfrac{m}{12}$
And m = - 60a
And the product of roots ( 3a $\times $ 2a) = $\dfrac{c}{a}$= $\dfrac{5}{12}$
That is 6${{a}^{2}}$= $\dfrac{5}{12}$
${{a}^{2}}$= $\dfrac{5}{72}$
That is a = $\pm \dfrac{\sqrt{5}}{6\sqrt{2}}$
Now put the value of a in m = -60 a, and we get
m = -60 $\times $$\pm \dfrac{\sqrt{5}}{6\sqrt{2}}$
By solving the above equation, we get
m = $\pm 5\sqrt{10}$
Therefore, the correct option is (a).
Note:We also find out the sum and the product of roots by simply putting the formula if x and y are the roots of any quadratic equation the value of xy will be equal to $\dfrac{ constant\, term}{coefficient\, of\, x^2}$ and sum of the roots that is x + y is equal to $\dfrac{ -coefficient\, of \,x}{coefficient\, of\, x^2}$ and by solving it we get the desired answer.
Formula used:
Sum of roots = $\dfrac{-b}{a}$
Product of roots = $\dfrac{c}{a}$
Complete step by step Solution:
Given equation is $12{{x}^{2}}+mx+5=0$
Compare the above equation with the standard form of quadratic equation $a{{x}^{2}}+bx+c=0$, we get
a = 12 , b = m and c = 5
And their roots are in the ratio 3 : 2
Let the roots of the equation are 3a and 2a
Now we apply the condition for the sum and the product of roots, and we get
Sum of roots ( 3a + 2a) = $-\dfrac{b}{a}$= $-\dfrac{m}{12}$
That is 5a = $-\dfrac{m}{12}$
And m = - 60a
And the product of roots ( 3a $\times $ 2a) = $\dfrac{c}{a}$= $\dfrac{5}{12}$
That is 6${{a}^{2}}$= $\dfrac{5}{12}$
${{a}^{2}}$= $\dfrac{5}{72}$
That is a = $\pm \dfrac{\sqrt{5}}{6\sqrt{2}}$
Now put the value of a in m = -60 a, and we get
m = -60 $\times $$\pm \dfrac{\sqrt{5}}{6\sqrt{2}}$
By solving the above equation, we get
m = $\pm 5\sqrt{10}$
Therefore, the correct option is (a).
Note:We also find out the sum and the product of roots by simply putting the formula if x and y are the roots of any quadratic equation the value of xy will be equal to $\dfrac{ constant\, term}{coefficient\, of\, x^2}$ and sum of the roots that is x + y is equal to $\dfrac{ -coefficient\, of \,x}{coefficient\, of\, x^2}$ and by solving it we get the desired answer.
Recently Updated Pages
Geometry of Complex Numbers – Topics, Reception, Audience and Related Readings

JEE Main 2021 July 25 Shift 1 Question Paper with Answer Key

JEE Main 2021 July 22 Shift 2 Question Paper with Answer Key

JEE Atomic Structure and Chemical Bonding important Concepts and Tips

JEE Amino Acids and Peptides Important Concepts and Tips for Exam Preparation

JEE Electricity and Magnetism Important Concepts and Tips for Exam Preparation

Trending doubts
JEE Main 2025 Session 2: Application Form (Out), Exam Dates (Released), Eligibility, & More

Atomic Structure - Electrons, Protons, Neutrons and Atomic Models

JEE Main 2025: Derivation of Equation of Trajectory in Physics

Displacement-Time Graph and Velocity-Time Graph for JEE

Electric Field Due to Uniformly Charged Ring for JEE Main 2025 - Formula and Derivation

Degree of Dissociation and Its Formula With Solved Example for JEE

Other Pages
JEE Advanced Marks vs Ranks 2025: Understanding Category-wise Qualifying Marks and Previous Year Cut-offs

JEE Advanced Weightage 2025 Chapter-Wise for Physics, Maths and Chemistry

NCERT Solutions for Class 11 Maths Chapter 4 Complex Numbers and Quadratic Equations

Instantaneous Velocity - Formula based Examples for JEE

NCERT Solutions for Class 11 Maths Chapter 6 Permutations and Combinations

NCERT Solutions for Class 11 Maths Chapter 8 Sequences and Series
