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

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

$\begin{array}{l}\\ 27.& \text{Ekspresi}21x^2+ax+21\text{dapat dituliskan}\\ & \text{sebagai perkalian dua bentuk linear dengan}\\ & \text{koefisien bilangan bulat. Hal tersebut hanya}\\ & \text{dapat dilakukan jika}a\text{adalah}....\\ & \begin{array}{lllllll}\text{A}.& {\displaystyle \text{bilangan 0}}& & & & & \\ \text{B}.& {\displaystyle \text{beberapa bilangan ganjil}}& & \\ \text{C}.& \text{beberapa bilangan genap}\\ \text{D}.& \text{sembarang bilangan ganjil}\\ \text{E}.& \text{sembarang bilangan genap}\end{array}\\ \\ & \mathbf{\text{Jawab}}:\mathbf{\text{C}}\\ & \text{Ekspresi dapat dituliskan dalam perkalian dua}\\ & \text{bentuk linear, maka}D=b^2-4ac\ge 0\\ & \text{Dari}21x^2+ax+21\\ & a=21,b=a,c=21,\text{maka}\\ & a^2-4.21.21\ge 0\\ & \iff a^2-{42}^2\ge 0\\ & \iff (a+42)(a-42)\ge 0\\ & \iff a\le -42ataua\ge 42\end{array}$.

Hubungan Fungsi Kuadrat dan Sebuah garis linear

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

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

$\begin{array}{l}\\ 30.& \text{Grafik}y=2x^2+ax+b\text{berpotongan dengan}\\ & \text{garis}y=3x-1\text{di titik}(x_1,y_1)\text{dan}(x_2,y_2)\\ & \text{Jika}{x}_1+x_2=3\text{dan}{x}_1\times x_2=4,\text{maka}\\ & a+b=....\\ & \begin{array}{llllllll}\text{A}.& 1& & & & & \text{D}.& {\displaystyle 4}\\ \text{B}.& {\displaystyle 2}& & & {\displaystyle }& & \text{E}.& {\displaystyle 5}\\ \text{C}.& {\displaystyle 3}\end{array}\\ \\ & \mathbf{\text{Jawab}}:\mathbf{\text{D}}\\ & \end{array}$