question_answer

DIRECTION (Qs. 80): Each of these questions contains two statements: Statement-1 (Assertion) and Statement-2 (Reason). Choose the correct answer (ONLY ONE option is correct) from the following-

Statement-1: If \[|{{z}_{1}}|=30,\,\,|{{z}_{2}}-(12+5i)|=6\], then maximum value of\[|{{z}_{1}}-{{z}_{2}}|\]is\[49\].

Statement-2: If \[{{z}_{1}},\,\,\,{{z}_{2}}\] are two complex numbers, then\[|{{z}_{1}}-{{z}_{2}}|\le |{{z}_{1}}|+|{{z}_{2}}|\]and equality holds when origin, \[{{z}_{1}}\] and \[{{z}_{2}}\] are collinear and \[{{z}_{1}},\,\,{{z}_{2}}\] are on the opposite side of the origin.

A)  Statement-1 is false, Statement-2 is true.

B)  Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1.

C)  Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statement-1.

D)  Statement-1 is true, Statement-2 is false.

Correct Answer: C

Solution :

                \[{{C}_{1}}{{C}_{2}}=13\]                 \[{{r}_{1}}=30,\,\,{{r}_{2}}=6\]                 \[{{C}_{1}}{{C}_{2}}<{{r}_{1}}-{{r}_{2}}\] \[\therefore \]The circle\[|{{z}_{2}}-(12+5i)|\,=6\] lies within the circle\[|{{z}_{1}}|\,=30\] \[\therefore \]\[\max |{{z}_{1}}-{{z}_{2}}|\,\,=30+13+6=49\] \[\therefore \]Statement-1 is true. Statement-2\[|{{z}_{1}}-{{z}_{2}}|\le |{{z}_{1}}|+|{{z}_{2}}|\]is always true. Equality sign holds if \[{{z}_{1}},\,\,{{z}_{2}}\] origin are collinear and \[{{z}_{1}}\,\,\text{and}\]\[{{z}_{2}}\] lies on opposite sides of the origin. \[\therefore \]Statement-2 is true.