SATHEE: Theory of Equations 1 Question 33

Theory of Equations 1 Question 33

34. For real $x$ , the function $\frac{(x - a) (x - b)}{(x - c)}$ will assume all real values provided

(1984, 3M)

(a) $a > b > c$

(b) $a < b < c$

(d) $a \leq c \leq b$

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Answer:

Correct Answer: 34. (d)

Solution:

Let $y = \frac{x^{2} - (a + b) x + a b}{x - c}$

$\Rightarrow y x - c y = x^{2} - (a + b) x + a b$

$\Rightarrow x^{2} - (a + b + y) x + (a b + c y) = 0$

For real roots, $D \geq 0$

$\begin{array}{lrl} \Rightarrow & (a + b + y)^{2} - 4 (a b + c y) & \geq 0 \Rightarrow & (a + b)^{2} + y^{2} + 2 (a + b) y - 4 a b - 4 c y \geq 0 \Rightarrow & y^{2} + 2 (a + b - 2 c) y + (a - b)^{2} \geq 0 \end{array}$

which is true for all real values of $y$ .

$∴ D \leq 0$

$4 (a + b - 2 c)^{2} - 4 (a - b)^{2} \leq 0$

$\Rightarrow 4 (a + b - 2 c + a - b) (a + b - 2 c - a + b) \leq 0$

$\Rightarrow (2 a - 2 c) (2 b - 2 c) \leq 0$

$\Rightarrow (a - c) (b - c) \leq 0$

$\Rightarrow (c - a) (c - b) \leq 0$

$\Rightarrow c$ must lie between $a$ and $b$

i.e. $a \leq c \leq b$ or $b \leq c \leq a$

Theory of Equations 1 Question 33

34. For real $x$ , the function $\frac{(x - a) (x - b)}{(x - c)}$ will assume all real values provided

Answer:

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Theory of Equations 1 Question 33

34. For real x, the function (x−a)(x−b)(x−c) will assume all real values provided

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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34. For real $x$ , the function $\frac{(x - a) (x - b)}{(x - c)}$ will assume all real values provided