kuadrat | Belajar Matematika dan Sains SMP-SMA

Soal Matematika SMA dan Penyelesaian

21 Des 2011 443 Komentar

by Raharja in Matematika SMA XII, Soal Latihan UN Tag:differensial, fungsi, integral, kuadrat, logaritma, matematika sma, trigonometri

Nomor 1

Jika $^{9}\log{18}=m-1$ , maka $^{36}\log{128}=...$ .

Penyelesaian :

Langkah pertama :

$^{9}\log{18}=m-1 \\\Rightarrow ^{9}\log{\(9\times 2)}=m-1\\\Rightarrow^{9}\log{9}+^{9}\log{2}=m-1 \\\Rightarrow 1+^{9}\log{2}=m-1 \\\Rightarrow ^{9}\log{2}=m-2 \\\Rightarrow ^{2}\log{9}=\frac{1}{m-2}$

Langkah kedua :

$^{36}\log{128}\\=\frac{1}{^{128}\log{36}}\\=\frac{1}{^{128}\log{4}+^{128}\log{9}}\\=\frac{1}{^{2^7}\log{2^2}+^{2^7}\log{9^1}}\\=\frac{1}{\frac{2}{7}\(^{2}\log{2})+\frac{1}{7}\(^{2}\log{9})}\\=\frac{1}{\frac{2}{7}\(1)+\frac{1}{7}\(\frac{1}{m-2})}\\=\frac{7}{\frac{2(m-2)+1}{m-2}}\\=\frac{7(m-2)}{2(m-2)+1}\\=\frac{7m-14}{2m-3}$ selesai

Nomor 2:

Diketahui $^{2}\log{5}=a$ dan $^{7}\log{2}=b$ , maka $^{35}\log{98}=...$ .

Penyelesaian :

$^{2}\log{5}=a \Rightarrow ^{5}\log{2}=\frac{1}{a} \\ ^{7}\log{2}=b \Rightarrow ^{2}\log{7}=\frac{1}{b}$

Kemudian :

$^{35}\log{98}\\=^{35}\log{\(2\times 7^2)}\\=^{35}\log{2}+^{35}\log{7^2}\\=^{35}\log{2}+2 \times \(^{35}\log{7})\\=\frac{1}{^{2}\log{35}}+\frac{2}{^{7}\log{35}}\\=\frac{1}{^{2}\log{5}+^{2}\log{7}}+\frac{2}{^{7}\log{7}+^{7}\log{5}}\\=\frac{1}{^{2}\log{5}+^{2}\log{7}}+\frac{2}{^{7}\log{7}+\(^{7}\log{2}) \times \(^{2}\log{5})}\\=\frac{1}{a+\frac{1}{b}}+\frac{2}{1+ba}\\=\frac{1}{\frac{ab}{b}+\frac{1}{b}}+\frac{2}{ab+1}\\=\frac{b}{ab+1}+\frac{2}{ab+1}\\=\frac{b+2}{ab+1}$ selesai

Nomor 3 :

Supaya $y=\: ^x \log\(x^2-5x+6\)$ terdefinisi maka haruslah ….

A. x < 2 atau x > 3

B. 0 < x < 2 atau x > 3

C. 0 < x < 1 atau x > 3

D. 0 < x < 1 atau 1 < x < 2

E. 0 < x < 1 atau 1 < x < 2 atau x > 3

Penyelesaian :

Ingat bahwa $^p \log q$ terdefinisi jika p, q > 0. Sehingga

$x>0\: \: dan\: \: x^2-5x+6>0 \\\\\Rightarrow x>0\: \: dan\: \: (x-2)(x-3)>0 \\\\\Rightarrow x>0\: \: dan\: \:\( x<2 \:\: atau \:\: x>3\) \\\\\Rightarrow \(x>0\: \: dan\: \:x<2\) \:\: atau \(\:\:x>0 \:\: dan \:\: x>3\)\\\\\Rightarrow 0<x<2 \:\: atau \:\: x>3$

Jawab (C)