OLIMPIADE SAINS TERAPAN NASIONAL SEKOLAH MENENGAH KEJURUAN TINGKAT PROPINSI JAWA TENGAH 2010 BIDANG MATEMATIKA TEKNOLOGI SESI III (ISIAN SINGKAT DAN ESSAY) WAKTU : 180 MENIT ============================================================ I.
Soal Isian Singkat, ada 10 soal dalam tes ini Petunjuk Menjawab Soal a. Tulis jawaban akhir (hasilnya) saja pada kotak di sebelah kanan setiap soal. b. Jika Anda akan mengganti jawaban, maka coret saja pada jawaban yang salah. c. Setiap soal yang dijawab benar diberi nilai + 5, jika jawaban salah diberi nilai – 2 dan jika kosong (tidak dijawab) bernilai 0.
============================================================= 1. Jika 2010 dapat dinyatakan sebagai hasil kali dua bilangan bulat positif yang selisihnya sekecil mungkin, maka selisih tersebut adalah ... 2. Tentukan jumlahan dari : 1 4 9 16 25 36 ... 10000
3. Jika A dan B adalah matriks persegi berukuran sama, dengan A B dan memenuhi A2 B B 2 A dan A3 B 3 , maka nilai det A 2 B 2 4. Jika diketahui A 2, 6, 3 , B 4, 9,1 dan C 6, 3, 2 adalah titik-titik sudut suatu segitiga, maka luas segitiga ABC tersebut adalah ... 5. Sebuah kelas terdiri dari 8 pria dan 12 wanita. Jika dipilih 4 orang untuk menjadi pengurus kelas, berapakah peluang bahwa keempat orang tersebut terdiri dari tepat dua pria dan dua wanita ?
6. Jika a dan b adalah bilangan real tak nol sedemikian hingga akar-akar persamaan kuadrat x 2 ax b 0 adalah a dan b , maka nilai a dan b yang mungkin adalah ... 7. Bola batu beton disusun dalam bentuk piramida dengan alas berupa persegi seperti gambar berikut ini :
Jika panjang sisi bola beton paling bawah ada 20 buah, maka berapakah banyaknya bola beton seluruhnya? 8.
Diberikan sin Z
segitiga
XYZ
dengan
1 Y 5 , maka nilai sin 5 2
sin X
1 2 2
dan
9. Pagar tembok tingginya 3 meter, berjarak 3 meter dari sebuah gedung yang tinggi. Tentukan panjang tangga terpendek yang dapat mencapai gedung jika salah satu ujungnya bertumpu pada tanah di luar tembok ! 10. Perhatikan gambar berikut ini :
Tentukan luas bagian yang diarsir.
II.
Soal Essay, ada 10 soal dalam tes ini Petunjuk Menjawab Soal a. Selesaikan soal-soal di bawah ini dengan memberikan uraian lengkap, jelas langkah-langkahnya dan sebutkan rumus atau dalil yang Anda pakai. Tuliskan jawaban tepat dibawah soal. Tulis yang rapi dan urut serta usahakan jangan banyak coretan. b. Setiap soal bernilai 0 – 10.
============================================================= 1. Misalkan J n menyatakan jumlahan angka ganjil pada bilangan asli n , sebagai contoh J 9168 9 1 10 , maka nilai J 1 J 2 J 3 J 100 Penyelesaian : ................................................................................................................................... ................................................................................................................................... ................................................................................................................................... ................................................................................................................................... ................................................................................................................................... ................................................................................................................................... ................................................................................................................................... ................................................................................................................................... ................................................................................................................................... ................................................................................................................................... ................................................................................................................................... ................................................................................................................................... ................................................................................................................................... ................................................................................................................................... ................................................................................................................................... ................................................................................................................................... ...................................................................................................................................
2. Misalkan ABC dan DAC adalah dua buah segitiga sama kaki dengan AB = AC dan AD = DC. Pada ABC besar BAC 200 , sedangkan pada ADC berlaku ADC 1000 , seperti diberikan oleh gambar di bawah ini.
Buktikan bahwa AB = BC + CD. Penyelesaian : ................................................................................................................................... ................................................................................................................................... ................................................................................................................................... ................................................................................................................................... ................................................................................................................................... ................................................................................................................................... ................................................................................................................................... ................................................................................................................................... ................................................................................................................................... ................................................................................................................................... ................................................................................................................................... ................................................................................................................................... ................................................................................................................................... ...................................................................................................................................
3. Suatu partikel bergerak pada lingkaran x 2 y 2 169 . Tentukan titik-titik di mana partikel bergerak tegak lurus terhadap garis 3x 2 y 100 . Penyelesaian : ................................................................................................................................... ................................................................................................................................... ................................................................................................................................... ................................................................................................................................... ................................................................................................................................... ................................................................................................................................... ................................................................................................................................... ................................................................................................................................... ................................................................................................................................... ................................................................................................................................... ................................................................................................................................... ................................................................................................................................... ................................................................................................................................... ................................................................................................................................... ................................................................................................................................... ................................................................................................................................... ................................................................................................................................... ................................................................................................................................... ................................................................................................................................... ................................................................................................................................... ................................................................................................................................... ...................................................................................................................................
4. Aditya mencocokkan arlojinya di tengah hari. Pada jam yang seharusnya pukul 13.00, arlojinya menunjukkan waktu 12.58.45. Dengan asumsi arloji Aditya berjalan lebih lambat dengan kecepatan tetap, pukul berapakah waktu sesungguhnya saat arlojinya menunjukkan pukul 23.45 ? Penyelesaian : ................................................................................................................................... ................................................................................................................................... ................................................................................................................................... ................................................................................................................................... ................................................................................................................................... ................................................................................................................................... ................................................................................................................................... ................................................................................................................................... ................................................................................................................................... ................................................................................................................................... ................................................................................................................................... ................................................................................................................................... ................................................................................................................................... ................................................................................................................................... ................................................................................................................................... ................................................................................................................................... ................................................................................................................................... ................................................................................................................................... ................................................................................................................................... ................................................................................................................................... ...................................................................................................................................
5. Sebuah lubang angin dari suatu bangunan terbuat dari besi yang dibentuk dari rangkaian segitiga sama sisi dengan panjang sisi dari segitiga yang paling kecil adalah 10 cm, seperti tampak pada gambar di bawah. Jika panjang sisi segitiga sama sisi yang paling besar adalah 150 cm. Tentukan berapa cm panjang besi yang diperlukan !
Penyelesaian : ................................................................................................................................... ................................................................................................................................... ................................................................................................................................... ................................................................................................................................... ................................................................................................................................... ................................................................................................................................... ................................................................................................................................... ................................................................................................................................... ................................................................................................................................... ................................................................................................................................... ................................................................................................................................... ................................................................................................................................... ................................................................................................................................... ................................................................................................................................... ...................................................................................................................................
1 dan x garis singgung di P memotong sumbu-X di titik A. Tentukan luas segitiga AOP dan bentuk segitiga AOP tersebut !
6. Diketahui pada kuadran pertama titik P a, b terletak pada kurva y
Penyelesaian : ................................................................................................................................... ................................................................................................................................... ................................................................................................................................... ................................................................................................................................... ................................................................................................................................... ................................................................................................................................... ................................................................................................................................... ................................................................................................................................... ................................................................................................................................... ................................................................................................................................... ................................................................................................................................... ................................................................................................................................... ................................................................................................................................... ................................................................................................................................... ................................................................................................................................... ................................................................................................................................... ................................................................................................................................... ................................................................................................................................... ................................................................................................................................... ................................................................................................................................... ...................................................................................................................................
7. Jika x menyatakan bilangan bulat terbesar yang kurang dari atau sama dengan x dan x menyatakan pecahan desimal dari x , maka penyelesaian sistem persamaan
x x x
y
z y z y z
1,1 3,3 adalah ... 4,4
Penyelesaian : ................................................................................................................................... ................................................................................................................................... ................................................................................................................................... ................................................................................................................................... ................................................................................................................................... ................................................................................................................................... ................................................................................................................................... ................................................................................................................................... ................................................................................................................................... ................................................................................................................................... ................................................................................................................................... ................................................................................................................................... ................................................................................................................................... ................................................................................................................................... ................................................................................................................................... ................................................................................................................................... ................................................................................................................................... ……………………………………………………………………………………... ……………………………………………………………………………………... ……………………………………………………………………………………...
8. Tentukan panjang garis terpendek yang menyinggung ellips b 2 x 2 dan memotong sumbu-sumbu koordinat !
a2 y2
a 2b 2
Penyelesaian : ................................................................................................................................... ................................................................................................................................... ................................................................................................................................... ................................................................................................................................... ................................................................................................................................... ................................................................................................................................... ................................................................................................................................... ................................................................................................................................... ................................................................................................................................... ................................................................................................................................... ................................................................................................................................... ................................................................................................................................... ................................................................................................................................... ................................................................................................................................... ................................................................................................................................... ................................................................................................................................... ................................................................................................................................... ................................................................................................................................... ................................................................................................................................... ................................................................................................................................... ................................................................................................................................... ...................................................................................................................................
9. Diberikan sebuah persegi dan lingkaran seperti tampak pada gambar di bawah ini.
Diketahui panjang sisi persegi a cm dan salah sisinya merupakan garis singgung lingkaran. Tentukan perbandingan luas kedua bangun tersebut ! Penyelesaian : ................................................................................................................................... ................................................................................................................................... ................................................................................................................................... ................................................................................................................................... ................................................................................................................................... ................................................................................................................................... ................................................................................................................................... ................................................................................................................................... ................................................................................................................................... ................................................................................................................................... ................................................................................................................................... ................................................................................................................................... ................................................................................................................................... ................................................................................................................................... ................................................................................................................................... ...................................................................................................................................
10. Jika f adalah fungsi riil yang ditentukan oleh f x maka nilai maksimum f adalah ...
6x x 2
10x x 2
24 ,
Penyelesaian : ................................................................................................................................... ................................................................................................................................... ................................................................................................................................... ................................................................................................................................... ................................................................................................................................... ................................................................................................................................... ................................................................................................................................... ................................................................................................................................... ................................................................................................................................... ................................................................................................................................... ................................................................................................................................... ................................................................................................................................... ................................................................................................................................... ................................................................................................................................... ................................................................................................................................... ................................................................................................................................... ................................................................................................................................... ................................................................................................................................... ................................................................................................................................... ................................................................................................................................... ................................................................................................................................... ...................................................................................................................................
