Question

Please solve both the questions ! Please answer quickly Please please !

NOTE :- The both questions may have More than one correct answer
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Answer 1

$\displaystyle\huge\red{\underline{\underline{Solution}}}$

ANSWER TO QUESTION : 7

Here f(x) = ax + b

By the given condition

$\sf{f(1) = 1 \: \: \: \: and \: \: f(0) = - 1 \: \: }$

$\sf{Now \: \: \:f(1) = 1 \: } \: gives$

$\sf{ a + b = 1\: \: } \: \: ......(1)$

$\sf{Again \: f(0) = - 1} \: \: gives$

$\sf{0 + b = - 1 }$

$\implies \: \sf{ b = - 1 } \: \: \: .....(2)$

Solving we get

$\sf{ a = 2 \: \: \: \: and \: \: \: b = - 1\: \: }$

CHECKING FOR OPTION (a)

$\sf{ 2a + 3b = 4 - 3 = 1\: \: }$

This option is CORRECT

CHECKING FOR OPTION (b)

$\sf{ 3a - 2b = 6 + 2 = 8\: \: }$

This option is CORRECT

CHECKING FOR OPTION (c)

$\sf{a + 2b = 2 - 2 = 0 \: \: }$

This option is CORRECT

CHECKING FOR OPTION (d)

$\sf{ ab = 2 \times - 1 = - 2\: \: }$

This option is INCORRECT

ANSWER TO QUESTION : 8

Given

$\sf{f(x) = 6x \: \: \: \: and \: \: \: \: 2b = a + c \: \: }$

Now

$\sf{ f(a - 3) = 6a - 18\: \: }$

$\sf{ f(b - 3) = 6b - 18\: \: }$

$\sf{ f(c - 3) = 6c- 18\: \: }$

CHECKING FOR OPTION (a)

$\sf{ f(a - 3) + f(c- 3) \: \: \: }$

$\sf{ = 6a - 18 + 6c - 18 \: \: }$

$\sf{ = 6(a + c) - 36 \: \: }$

$\sf{ = 12b - 36 \: \: }$

$\sf{ = 2 \times 6(b - 3)\: \: }$

$\sf{ =2 \: f(b - 3) \: \: }$

This option is CORRECT

CHECKING FOR OPTION (b)

$\displaystyle \sf{ \frac{f(a - 3) - f(b - 3) \: }{f(b - 3) - f(c - 3) } \: \: }$

$= \displaystyle \sf{ \frac{6(a - 3) - 6(b - 3) \: }{6(b - 3) - 6(c - 3) } \: \: }$

$= \displaystyle \sf{ \frac{6(a -b ) \: }{6(b - c )} \: \: }$

$= \displaystyle \sf{ \frac{(a -b ) \: }{(b - c )} \: \: }$

$= \displaystyle \sf{ \frac{(a -b ) \: }{(b - 2b + a)} \: \: }$

$= \displaystyle \sf{ \frac{(a -b ) \: }{(a - b)} \: \: }$

$= \displaystyle \sf{1\: }$

This option is CORRECT

CHECKING FOR OPTION (c)

$\displaystyle \sf{ \frac{f(a - 3) - f(c - 3) \: }{f(b - 3) - f(a - 3) } \: \: }$

$= \displaystyle \sf{ \frac{6(a - 3) - 6(c - 3) \: }{6(b - 3) - 6(a - 3) } \: \: }$

$= \displaystyle \sf{ \frac{(a -c ) \: }{(b - a ) } \: \: }$

$= \displaystyle \sf{ \frac{(a + a - 2b ) \: }{(b - a ) } \: \: }$

$= \displaystyle \sf{ \frac{(2a - 2b ) \: }{(b - a ) } \: \: }$

$= \displaystyle \sf{ - 2\: \: }$

This option is INCORRECT

CHECKING FOR OPTION (d)

$\sf{ \:f(a - 3) - f(c - 3) \: }$

$\sf{ \: = 6(a - 3) - 6(c - 3) \: }$

$\sf{ \: = 6a - 18 - 6c + 18 \: }$

$\sf{ \: = 6a - 6c \: }$

$\sf{ \: = 6(a - c ) \: }$

This option is CORRECT

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