Question

please solve this questions

Answer 1

$\displaystyle \text{If a trinomial ax^2+bx+c=0 is a perfect square, then the discriminant}\\ \text{will be 0, because the trinomial has equal roots.} \\ \\ \\ \\ b^2-4ac=0 \\ \\ \\ b^2=4ac \\ \\ \\ \large c=\frac{b^2}{4a}$

$\text{We are going to find the value of x according to this concept.}$

$\displaystyle \begin{aligned}(11)\ \ &\text{The trinomial is x^2+6x+\frac{m-1}{2} .}\\ \\ &a=1\ \ \ ; \ \ \ b=6\ \ \ ; \ \ \ c=\frac{m-1}{2}\\ \\ \\ &c=\frac{b^2}{4a}\\ \\ &\frac{m-1}{2}=\frac{6^2}{4 \times 1} \\ \\ &\frac{m-1}{2}=\frac{36}{4} \\ \\ &\frac{m-1}{2}=9\\ \\ &m-1=2 \times 9\\ \\ &m-1=18 \\ \\ & m=18+1 \\ \\ &m =\bold{19}\\ \\ \\ &\text{Thus the answer is option C.}\end{aligned}$

$\displaystyle \begin{aligned}(12)\ \ &\text{The trinomial is x^2-10x+\frac{m+1}{2} .}\\ \\ &a=1\ \ \ ; \ \ \ b=-10\ \ \ ; \ \ \ c=\frac{m+1}{2}\\ \\ \\ &c=\frac{b^2}{4a}\\ \\ &\frac{m+1}{2}=\frac{(-10)^2}{4 \times 1} \\ \\ &\frac{m+1}{2}=\frac{100}{4} \\ \\ &\frac{m+1}{2}=25\\ \\ &m+1=2 \times 25\\ \\ &m+1=50 \\ \\ & m=50-1 \\ \\ &m =\bold{49}\\ \\ \\ &\text{Thus the answer is also option C.}\end{aligned}$