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DE THI ~m N G ~ H N& 2013 Men: TO& (De chinh thdc) Thai gian lam bai: 180 phlit.

C l u 1. Cho ma tran d6i a g t h ~ c A =

1 -1 m

(a) Tim tat cA cac gia tri thqc c6a m de" ,4 co ba tri riZng dudng phan biet.

(b) Vdi m = 4, cheo hog &qc giao ma tran A.

(c) Trong he truc toa dg Descartes Oxyz, cho mat bac hai

Nh$n dang m$t cong bat hai nay.

C l u 2. Cho anh xa tuyen tinh f : ItB" R3, biet ma tran cfia f trong cd s6 E =

( 1 1, l ) T , l , 1 2 ) ( 1 2 1 1 A =

(a] Tim tat c5 cac gia tri thgc cfia m de" vectd x = ( 2 . 4 , ,m)T la vectd riZng cfia f .

(b) Tim ma tran c6a anh xa tuyen tinh f trong cd sd El = ( (1 .0 . l j T , ( 1 , 1 , 0 ) ~ > ( 1 , 1 , I ) ~ } .

(c) Tim mQt cd s3 R cGa R3 sao cho ma tran cfia f trong B la ma tran cheo D. Tim D.

C l u 3. Biet r5ng mot ngudi c6 nhom mau AB co the" nhan rnau cfia bat kjr nhom rnau

nao. Neu ngudi 66 co nh6m rnau con lai (A, B hoac 0) thi chi co the" nhan rnau cfia

ngudi cung nhom vdi minh hoac ngudi co nhom mau 0. Cho biet tf le ng~ddi c6 nh6m

rnau 0, A. B va A B tUdng dng la 33,7%, 37 ,5%, 20.9% va 7,9%.

(a) Chon ng5u nhiZn mot ngudi can tiep rnau va met ngudi cho mau. Tink xac sua.t de"

sg truyen rnau thgc hien dudc.

(b) Chon ng5u nhiZn mQt ngudi can tiep rnau va n5rn ngudi cho mau. Tinh xac suat

de" sg truyen mau thgc hien dudc.

Ciu 4. Cho h8m f ( x , a ) = " f O,vdih>o, ~ > 0 . x = o

J f (a) Chung minh rang f (L, a) va -(z, a) lien tuc trong mien a 2 E > 0 , .L 2 0 .

d a

+a3 J'

[bl Chung rninh rang tich ph2n [ f (z, a ) d x hQi tu vdi a 2 E 7 0, z 2 0. 0 2

+*; c? f (c) Chung rninh rang tich ph2n 1 - ( x , a ) d x hQi tu deu vdi a 2 E > 0. x 2 0. " r7a

-+ r e-nz2 - e-/3z'

(dl 'I33 do tinh tich ph2n I ( a ) = 1 d z , vdi a > 0 , p > 0. 0 x

Ciu 5. GiA s3 h8m f (z) khA vi tren doan [ O , :I.] va f l ( 0 ) . f l ( l ) < 0. Chung minh rang trong

khoAng ( 0 , l ) tim dude di6m c sao cho f l ( c ) = 0.

Ciu 6. Cho chudi ham f,, trong do f n : R: - R, f n ( x ) = 2e - (n. 1)'s

n Z O

(a) KhAo sat su hQi tu chugn tgc, ddn, tuyet dbi va deu cGa chudi ham tren. Ky hieu $

18 t6ng cfia n6.

(b) Chur;g minh rang S thuoc ldp C'U tren R*_.

(c) V6,cJudng cong bi6u d i h cGa S.

CHu 7. (a) Tim nghiem cGa phudng trinh vi phan xy" - (J: + 2)y1 + 2y = 0 (*) khai tri6n

dude th8nh chudi lGy thda quanh 0 . Rut gon nghiem nay.

[b] GiAi (*I tren khoAng I c R bat kjr (co xet van d i ghep nbi nghiem].