SATHEE: Theory of Equations 1 Question 15

Theory of Equations 1 Question 15

16. The sum of all real values of $x$ satisfying the equation ${(x^{2} - 5 x + 5)}^{x^{2} + 4 x - 60} = 1$ is

(2016 Main)

(a) 3

(b) -4

(d) 5

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Answer:

Correct Answer: 16. (a)

Solution:

Given, ${(x^{2} - 5 x + 5)}^{x^{2} + 4 x - 60} = 1$

Clearly, this is possible when I. $x^{2} + 4 x - 60 = 0$ and $x^{2} - 5 x + 5 \neq 0$ or II. $x^{2} - 5 x + 5 = 1$ or

III. $x^{2} - 5 x + 5 = - 1$ and $x^{2} + 4 x - 60 =$ Even integer.

$\begin{array}{lrl} Case I & When x^{2} + 4 x - 60 & = 0 \\ \Rightarrow & x^{2} + 10 x - 6 x - 60 & = 0 \\ \Rightarrow & x (x + 10) - 6 (x + 10) & = 0 \\ \Rightarrow & (x + 10) (x - 6) & = 0 \\ \Rightarrow & x = - 10 or x & = 6 \end{array}$

Note that, for these two values of $x, x^{2} - 5 x + 5 \neq 0$

$\begin{array}{lr} Case II When & x^{2} - 5 x + 5 = 1 \\ \Rightarrow & x^{2} - 5 x + 4 = 0 \\ \Rightarrow & x^{2} - 4 x - x + 4 = 0 \\ \Rightarrow & x (x - 4) - 1 (x - 4) = 0 \\ \Rightarrow & (x - 4) (x - 1) = 0 \Rightarrow x = 4 or x = 1 \\ Case III When & x^{2} - 5 x + 5 = - 1 \\ \Rightarrow & x^{2} - 5 x + 6 = 0 \\ \Rightarrow & x^{2} - 2 x - 3 x + 6 = 0 \\ \Rightarrow & x (x - 2) - 3 (x - 2) = 0 \\ \Rightarrow & (x - 2) (x - 3) = 0 \\ \Rightarrow & x = 2 or x = 3 \end{array}$

Now, when $x = 2, x^{2} + 4 x - 60 = 4 + 8 - 60 = - 48$ , which is an even integer.

When $x = 3, x^{2} + 4 x - 60 = 9 + 12 - 60 = - 39$ , which is not an even integer.

Thus, in this case, we get $x = 2$ .

Hence, the sum of all real values of

$x = - 10 + 6 + 4 + 1 + 2 = 3$

Theory of Equations 1 Question 15

16. The sum of all real values of $x$ satisfying the equation ${(x^{2} - 5 x + 5)}^{x^{2} + 4 x - 60} = 1$ is

Answer:

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Theory of Equations 1 Question 15

16. The sum of all real values of x satisfying the equation (x2−5x+5)x2+4x−60=1 is

Answer:

Engineering Preparation

Medical Preparation

NCERT Books & Solutions

Important Resources

Other Resources

Syllabus

Test Platform

Sample Mock Test

Mentorship

NCERT Exercise Video Solution

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16. The sum of all real values of $x$ satisfying the equation ${(x^{2} - 5 x + 5)}^{x^{2} + 4 x - 60} = 1$ is