In a \(\triangle A B C\) (shown in the figure below). State whether the following are true or false
(i) \(\mathbf{A B}+\mathbf{B C}+\mathbf{C A}=\mathbf{0}\) (ii) \(\mathbf{A B}+\mathbf{B C}-\mathbf{A C}=\mathbf{0}\) (iii) \(\mathbf{A B}-\mathbf{C B}+\mathbf{C A}=\mathbf{0}\) (iv) \(\mathbf{A B}+\mathbf{B C}-\mathbf{C A}=\mathbf{0}\)

• Correct answer
  A

  (i) True (ii) True (iii) True (iv) False
• B

  (i) True (ii) False (iii) True, (iv) False
• C

  (i) False (ii) False (iii) True (iv) True
• D

  (i) False (ii) True (iii) False (iv) False

Solution

Following triangle came, \(\begin{array}{llll} & \mathbf{A B}+\mathbf{B C} =\mathbf{A C} \\ \Rightarrow & \mathbf{A B}+\mathbf{B C}+\mathbf{C A} =0 \quad \ldots (i) \\ \text {and } & \mathbf{A B}+\mathbf{B C}-\mathbf{A C} =0 \quad \ldots (ii) \\ \text {Also, } & \mathbf{C A}+\mathbf{A B} =\mathbf{C B} \\ \Rightarrow & \mathbf{C A}+\mathbf{A B}-\mathbf{C B} =0 \\ \text {or } & \mathbf{A B}-\mathbf{C B}+\mathbf{C A} =0 \quad \ldots (iii) \end{array}\)

Hence, the correct answer is option: 'A'.