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Exam
JEE Main
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Mathematics
Chapter
Basics of Mathematics

JEE Main · Mathematics · Basics of Mathematics

The set of values of x for which the inequalities x^2-3 x-10 0 hold...

The set of values of xx for which the inequalities x2−3x−10<0,10x−x2−16>0x^2-3 x-10 < 0, \quad 10 x-x^2-16>0 \quad hold simultaneously, is

Options

  1. A(−2,5)(-2,5)
  2. B(2,8)(2,8)
  3. C(−2,8)(-2,8)
  4. D(2,5)(2,5)

Correct answer

D. (2,5)(2,5)

Step-by-step solution

Given inequalities are x2−3x−10⇒x2−5x+2x−10<0⇒x∈(−2,5)⇒(x+2)(x−5)<0 and 10x−x2−16>0⇒x2−10x+16<0⇒x2−8x−2x+16<0⇒(x−2)(x−8)<0⇒x∈(−2,5)∩(2,8)∴x∈(2,5)\begin{array}{lrl}x^2-3 x-10 & \\ \Rightarrow & x²-5 x+2 x-10 & < 0 \\ \Rightarrow & & x \in(-2,5) \\ \Rightarrow & & (x+2)(x-5) < 0 \\ \text { and }& & 10 x-x²-16>0 \\ \Rightarrow & x²-10 x+16 < 0 \\ \Rightarrow & x²-8 x-2 x+16 < 0 \\ \Rightarrow & & (x-2)(x-8) < 0 \\ \Rightarrow & x \in(-2,5) \cap(2,8) \\ \therefore & x \in(2,5)\end{array}
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