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Quadratic Equation and Inequalities question

2006 · Shift 0 · Q70
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Quadratic Equation and Inequalities question

2006 · Shift 0 · Q70

JEE MainMathematicsQuadratic Equation and InequalitiesMCQ+4 / −1
All the values of mmm for which both roots of the equation x2−2mx+m2−1=0{x^2} - 2mx + {m^2} - 1 = 0x2−2mx+m2−1=0 are greater than −2- 2−2 but less then 4, lie in the interval
  1. A
    −2<m<0- 2 \lt m \lt 0−2<m<0
  2. B
    m>3m \gt 3m>3
  3. C
    −1<m<3- 1 \lt m \lt 3−1<m<3
  4. D
    1<m<41 \lt m \lt 41<m<4
View written solutionFree

Correct answer: C

  1. Write the quadratic and identify its roots

Given x2−2mx+(m2−1)=0x^2-2mx+(m^2-1)=0x2−2mx+(m2−1)=0

We can factor it as x2−2mx+m2−1=(x−m)2−1=0x^2-2mx+m^2-1=(x-m)^2-1=0x2−2mx+m2−1=(x−m)2−1=0

So, ((x−m)−1)((x−m)+1)=0((x-m)-1)((x-m)+1)=0((x−m)−1)((x−m)+1)=0

Hence the roots are x=m+1andx=m−1x=m+1 \quad \text{and} \quad x=m-1x=m+1andx=m−1

  1. Apply the condition on both roots

Both roots must satisfy −2<x<4-2<x<4−2<x<4

So we need:

  • For the smaller root m−1m-1m−1: −2<m−1<4-2<m-1<4−2<m−1<4
  • For the larger root m+1m+1m+1: −2<m+1<4-2<m+1<4−2<m+1<4

But it is enough to impose:

  • the smaller root is greater than −2-2−2: m−1>−2⇒m>−1m-1>-2 \Rightarrow m>-1m−1>−2⇒m>−1
  • the larger root is less than 444: m+1<4⇒m<3m+1<4 \Rightarrow m<3m+1<4⇒m<3

Thus, −1<m<3-1<m<3−1<m<3

  1. Match with the options

This corresponds to:

C: −1<m<3\boxed{\text{C: } -1<m<3}C: −1<m<3​

  1. Compare with stored correct answer

Stored correct answer is C, which matches our result.

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