IIT-JEE 2007 · Paper 2 · Mathematics Q64

MathematicsMatrix match+6Question 64 of 66

Let $f(x)=\dfrac{x^2-6x+5}{x^2-5x+6}$ .

Match the expressions/statements in Column I with expressions/statements in Column II and indicate your answer by darkening the appropriate bubbles in the 4 × 4 matrix given in the ORS.

Column I

A
If $-1<x<1$ , then $f(x)$ satisfies
B
If $1<x<2$ , then $f(x)$ satisfies
C
If $3<x<5$ , then $f(x)$ satisfies
D
If $x>5$ , then $f(x)$ satisfies

Column II

P
$0<f(x)<1$
Q
$f(x)<0$
R
$f(x)>0$
S
$f(x)<1$

Answer and worked solution

Official answer: A → P, R, S; B → Q, S; C → Q, S; D → P, R, S

Factor. $f(x)=\dfrac{(x-1)(x-5)}{(x-2)(x-3)}$ and
$f(x)-1=\frac{-(x+1)}{(x-2)(x-3)}.$

A, $-1<x<1$ : numerator and denominator of $f$ both positive, so $f>0$ ; $f-1<0$ since $x+1>0$ and the denominator is positive. → P, R, S.
B, $1<x<2$ : $(x-1)(x-5)<0$ , denominator $>0$ : $f<0$ (so also $f<1$ ). → Q, S.
C, $3<x<5$ : same signs: $f<0$ . → Q, S.
D, $x>5$ : $f>0$ and $f-1<0$ . → P, R, S.

Answer: A → P, R, S; B → Q, S; C → Q, S; D → P, R, S

Topics:Inequalities and modulus Graphs of functions

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IIT-JEE 2007 · Paper 2, Mathematics question 64