Question:

Step 1: Set up equations from the given conditions. Since $b$ is a root of $x^2+ax+10=0$: $$b^2+ab+10=0 \quad \cdots (1)$$ Since $a$ is a root of $x^2+bx+10=0$: $$a^2+ab+10=0 \quad \cdots (2)$$ Step 2: Subtract equation (2) from equation (1). $$b^2 - a^2 = 0 \implies (b-a)(b+a)=0$$ Since $a\neq b$ (distinct), we get $a+b=0$. Step 3: Substitute $b = -a$ into equation (1). $$a^2 - a^2 + 10 = 0 \implies 10 = 0$$ This is a contradiction, so no such distinct real $a$ and $b$ exist. Step 4: Identify incorrect options. Option (a): $a-b=0$ means $a=b$, contradicts distinctness — this is incorrect. Option (b): $a+b=0$ — this is what we derived, so it is correct. Option (c): $a+b=2$ — incorrect. Option (d): 'No such $a$ and $b$ exists' — actually true (contradiction found), so this is correct. The incorrect statements are (a) and (c), i.e., options 1 and 3.