Contoh Soal 6 Fungsi Kuadrat (Kelas X/Fase E Semester Genap) Tahun 2024

 $\begin{array}{ll}\\ 26.&\textrm{Misalkan}\: \: F=\displaystyle \frac{6x^{2}+16x+3m}{6}\: \: \textrm{merupakan kuadrat}\\ &\textrm{dari bentuk linear terhadap}\: \: x(ax+b).\: \: \textrm{Nilai}\\ &m\: \: \textrm{yang memungkinkan kondisi tersebut terjadi}\\ &\textrm{terletak di antara}\: ....\\ &\begin{array}{lllllllll}\textrm{A}.&\displaystyle -6\: \: \textrm{dan}\: \: -5&&&&&\textrm{D}.&\displaystyle \textrm{4 dan 5}\\ \textrm{B}.&\displaystyle -4\: \: \textrm{dan}\: \: -3&&&\displaystyle &&\textrm{E}.&\displaystyle \textrm{5 dan 6}\\ \textrm{C}.&\color{red}\displaystyle 3\: \: \textrm{dan}\: \: 4 \end{array}\\\\ &\textbf{Jawab}:\textbf{C}\\ &\begin{aligned}&\textrm{Diketahui FK}:\displaystyle \frac{6x^{2}+16x+3m}{6}\\ &\textrm{merupakan bentuk kuadrat, maka}\\ &f(x)=\displaystyle \frac{6x^{2}+16x+3m}{6}=x^{2}+\displaystyle \frac{8}{3}x+\frac{1}{2}m\\ &\textrm{dengan}\: \: a=1,\: b=\displaystyle \frac{8}{3},\: \: \textrm{dan}\: \: \: c=\displaystyle \frac{1}{2}m\\ &\textrm{dan istilah bentuk linier mengarah kepada}\\ &\textrm{jenis akarnya real, maka}\: \: D=b^{2}-4ac>0\\ &\left ( \displaystyle \frac{8}{3} \right )^{2}-4.1.\left ( \displaystyle \frac{1}{2}m \right )>0\\ &\Leftrightarrow \displaystyle \frac{64}{9}-2m>0\\ &n \to \Leftrightarrow \displaystyle \frac{32}{9}-m>0\: \: \: (\textrm{dikali})-1\\ &m-\displaystyle \frac{32}{9}>0\\ &\Leftrightarrow m>\displaystyle \frac{32}{9}\Leftrightarrow m>\color{red}3\displaystyle \frac{5}{9} \end{aligned} \end{array}$.

$\begin{array}{ll}\\ 27.&\textrm{Ekspresi}\: \: 21x^{2}+ax+21\: \: \textrm{dapat dituliskan}\\ &\textrm{sebagai perkalian dua bentuk linear dengan}\\ &\textrm{koefisien bilangan bulat. Hal tersebut hanya}\\ &\textrm{dapat dilakukan jika}\: \: a\: \: \: \textrm{adalah}\: ....\\ &\begin{array}{lllllllll}\textrm{A}.&\displaystyle \textrm{bilangan 0}&&&&&\\ \textrm{B}.&\displaystyle \textrm{beberapa bilangan ganjil}&&\\ \textrm{C}.&\color{red}\textrm{beberapa bilangan genap}\\ \textrm{D}.&\textrm{sembarang bilangan ganjil}\\ \textrm{E}.&\textrm{sembarang bilangan genap} \end{array}\\\\ &\textbf{Jawab}:\textbf{C}\\ &\textrm{Ekspresi dapat dituliskan dalam perkalian dua}\\ &\textrm{bentuk linear, maka}\: D=b^{2}-4ac\geq 0\\ &\textrm{Dari}\: \: 21x^{2}+ax+21\\&a=21,\: b=a,\: c=21,\: \: \textrm{maka}\\ &a^{2}-4.21.21\geq 0\\ &\Leftrightarrow a^{2}-42^{2}\geq 0\\ &\Leftrightarrow (a+42)(a-42)\geq 0\\ &\Leftrightarrow \color{red}a\leq -42\: \: \color{black}atau\: \: \color{red}a\geq 42 \end{array}$.

Hubungan Fungsi Kuadrat dan Sebuah garis linear

$\begin{array}{ll}\\ 28.&\textrm{Garis}\: \: 2x+y=p\: \: \textrm{akan memotong parabola}\\ &4x^{2}-y=0\: \: \textrm{di dua titik apabila nilai}\: \: p=\: ....\\ &\begin{array}{lllllllll}\textrm{A}.&p<-\displaystyle \frac{1}{4}&&&&&\textrm{D}.&\displaystyle p<-\frac{3}{4}\\ \textrm{B}.&\color{red}\displaystyle p>-\displaystyle \frac{1}{4}&&&\displaystyle &&\textrm{E}.&\displaystyle p<-1\\ \textrm{C}.&\displaystyle p<\displaystyle \frac{1}{4}\end{array}\\\\ &\textbf{Jawab}:\textbf{B}\\ &\begin{aligned}&\textrm{Diketahui bahwa garis}\: \: 2x+y=p\Rightarrow y=p-2x\\ &\textrm{memotong parabola}\: \: 4x^{2}-y=0\: \:  \textrm{di dua titik}\\ &\textrm{berbeda, maka}\\ &4x^{2}-y=0\Leftrightarrow 4x^{2}-(p-2x)=0\\ &4x^{2}+2x-p=0,\: \: \textrm{dengan}\: a=4,b=2,\: c=-p\\ &\textrm{Syarat memotong di dua titik berbeda adalah}:\\ &D=b^{2}-4ac>0\\ &\Leftrightarrow 2^{2}-4.4.(-p)>0\\ &\Leftrightarrow 4+16p>0\Leftrightarrow 16p>-4\Leftrightarrow p>-\displaystyle \frac{4}{16}\\ &\Leftrightarrow p>\color{red}-\displaystyle \frac{1}{4} \end{aligned} \end{array}$.

$\begin{array}{ll}\\ 29.&\textrm{Jika}\: \: ax^{2}+bx+c=0\: \: \textrm{tidak mempunyai akar real}\\ &\textrm{maka grafik fungsi}\: \: y=ax^{2}+bx+c\: \: \textrm{akan}\\ &\textrm{menyinggung garis}\: \: y=x\: \: \textrm{apabila}\: ....\\ &\begin{array}{lllllllll}\textrm{A}.&\color{red}b<\displaystyle \frac{1}{2}&&&&&\textrm{D}.&\displaystyle b>2\\ \textrm{B}.&\displaystyle b>\displaystyle \frac{1}{2}&&&\displaystyle &&\textrm{E}.&\displaystyle 1<b<2\\ \textrm{C}.&\displaystyle b>1\end{array}\\\\ &\textbf{Jawab}:\textbf{A}\\ &\begin{aligned}&\textrm{Diketahui sebuah kurva parabola}\\ &y=ax^{2}+bx+c\: \: (\textrm{tidak punya akar real})\\ &\textrm{dan}\: \: \textrm{garis}\: \: y=x\: \: \textrm{saling bersinggungan, }\\&\textrm{sehingga}\: \: y=y\\ &\Leftrightarrow ax^{2}+bx+c=x\\ &\Leftrightarrow ax^{2}+bx-x+c=0\\ &\Leftrightarrow ax^{2}+(b-1)x+c=0\\ &\textrm{dengan}\: \: a=a,\: b=b-1,\: c=c \\&\textrm{Selanjutnya syarat menyinggung adalah}:\\ &D=b^{2}-4ac=0\\ &\Leftrightarrow (b-1)^{2}-4ac=0\Leftrightarrow (b-1)^{2}=4ac.\\ &\textrm{Karena}\: \: ax^{2}+bx+c\: \: (\textrm{tidak punya akar real})\\ &\textrm{maka}\: \: D<0\Leftrightarrow b^{2}-4ac<0\\ &\Leftrightarrow b^{2}-(b-1)^{2}<0\Leftrightarrow b^{2}-(b^{2}-2b+1)<0\\ &\Leftrightarrow 2b-1<0\Leftrightarrow b<\color{red}\displaystyle \frac{1}{2} \end{aligned} \end{array}$.

$\begin{array}{ll}\\ 30.&\textrm{Grafik}\: \: y=2x^{2}+ax+b\: \: \textrm{berpotongan dengan}\\ &\textrm{garis}\: \: y=3x-1\: \: \textrm{di titik}\: \: (x_{1},y_{1})\: \: \textrm{dan}\: \: (x_{2},y_{2})\\ &\textrm{Jika}\: \: x_{1}+x_{2}=3\: \: \textrm{dan}\: \: x_{1}\times x_{2}=4,\: \textrm{maka}\\ &a+b=....\\ &\begin{array}{lllllllll}\textrm{A}.&1&&&&&\textrm{D}.&\color{red}\displaystyle 4\\ \textrm{B}.&\displaystyle 2&&&\displaystyle &&\textrm{E}.&\displaystyle 5\\ \textrm{C}.&\displaystyle 3\end{array}\\\\ &\textbf{Jawab}:\textbf{D}\\ & \end{array}$