Consider the following two sets: $$A={x$$:|x−$$1$$|<$$3}$$;$$B=x$$:$$x2$$−$$2ax+a2$$−$$4$$≤$$0$$ If A∩$$B={x$$:−$$2$$

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Consider the following two sets: $A = {x : | x - 1 | < 3}$ ; $B = \{x : x^{2} - 2 a x + a^{2} - 4 \leq 0\}$
$If A \cap B = {x : - 2 < x \leq 1}, then the number of integers in the range of a, is$

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Solution

$\begin{array}{l} | x - 1 | < 3 \\ \Rightarrow - 3 < x - 1 < 3 \\ \Rightarrow - 2 < x < 4 - - - - - (1) \\ Also, x^{2} - 2 a x + a^{2} - 4 \leq 0 \\ \Rightarrow (x - a)^{2} \leq 4 \\ \Rightarrow | x - a | \leq 2 \\ \Rightarrow - 2 \leq x - a \leq 2 \\ \Rightarrow - 2 + a \leq x \leq 2 + a - - - - - (2) \\ Now, for A \cap B to be (- 2, 1], \\ a + 2 = 1 \\ \Rightarrow a = - 1 \end{array}$

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Introduction to linear inequalities

Remember concepts with our Masterclasses.

Consider the following two sets: A={x:|x−1|<3};B=x:x2−2ax+a2−4≤0 If A∩B={x:−2<x≤1}, then the number of integers in the range of a , is

|x−1|<3⇒ −3<x−1<3⇒ −2<x<4-----(1) Also, x2−2ax+a2−4≤0⇒ (x−a)2≤4⇒ |x−a|≤2⇒ −2≤x−a≤2⇒ −2+a≤x≤2+a-----(2) Now, for A∩B to be (−2,1] , a+2=1⇒ a=−1

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free study material

Consider the following two sets: $A = {x : | x - 1 | < 3}$ ; $B = \{x : x^{2} - 2 a x + a^{2} - 4 \leq 0\}$
$If A \cap B = {x : - 2 < x \leq 1}, then the number of integers in the range of a, is$