If 2, 3 be the roots of 2x³ + mx² - 13x + n = 0 then the values of m and n are respectively

Correct Answer is : −5,30-5, 30−5,30

Correct Answer is : −5,30-5, 30−5,30

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AP DSC School Assistant Math Model Test 1

Section: Subject Specific

Question : 44 of 160

Marks: +1, -0

If 2, 3 be the roots of 2x^{3} + mx^{2} - 13x +

then the values of m and n are respectively

Solution:

Concept:

For a cubic equation

ax^3+bx^2+cx+d=0

with roots

\alpha,\beta,\gamma

: sum of roots

=-b/a

, sum of pairwise products

=c/a

, product

=-d/a

. These relations are used to find unknown coefficients.

Explanation:

Given:

2x^3+mx^2-13x+n=0

has roots 2, 3, and a third root

\delta

Sum of roots:

2+3+\delta = -\frac{m}{2}

→

5+\delta = -\frac{m}{2}

...(1)

Sum of pairwise products:

(2)(3)+3\delta+2\delta = \frac{-13}{2}

→

6+5\delta = -\frac{13}{2}

→

\delta = -\frac{5}{2}

Product of roots:

2 \times 3 \times \delta = -\frac{n}{2}

→

6 \times (-\frac{5}{2}) = -\frac{n}{2}

→

-15 = -\frac{n}{2}

→

n=30

From (1):

5+(-\frac{5}{2}) = -\frac{m}{2}

→

\frac{5}{2} = -\frac{m}{2}

→

m=-5

Answer:

Option B:

-5, 30

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