Bui Quang Tuyen giai chi tiet mot so cau de toan quoc gia 2017

Cau 37 115 Mot chuyen dong trong 3 gio voi van toc v km / h phu thuoc thoi gian th co do thi van toc nhu hinh ve .trong khoang thoi gian 1 gio ke tu khi bat dau chuyen dong , do thi do l[r]

Giai chi tiet cac cau kho de thi quoc gia 2017

Cau 47 (115): co ? so phuc Z thoa Z 3i 5va Z

Z 4 la so thuan sao

gia su Z = a + bi

va

2

3

Vay co duy nhat so phuc Z thoa dk nen chon dap an C

Cau 49 (115) trong he Oxyz cho (S): x2 y2 z2 9

diem M(1; 1; 2) va (P) : x + y + z – 4 = 0 Goi d la

dg thg di qua M , thuoc (P) cat (S) tai hai diem A, B

sao cho AB nho nhat Biet d co mot vtcp u (1;a;b)

Tinh T = a - b

Nhan xet : OM 6 3 R; toa do M nghiem dung pt (P) nen M nam trong mat cau va nam tren (P) suy ra d qua M nam trong (P) cat (S) tai 2 diem A,B sao cho AB ngan nhat khi va chi khi d HM (H la hinh chieu cua O tren (P)) Goi U la vtcp cua d thi d Ud [MH,n ] p

tu gt thi OH co pt

x t

z t

p d

U [MH,n ] (3; -3;0) vtcp cua d la (1; - 1; 0) = (1; a; b) a b 1 0 1 suy ra

dap an chon la B

Cau 48 (115): Cho ham so y = f(x) Do thi cua ham so y = f’(x) nhu hinh ve

Dat h(x) = 2f (x) x Menh de nao duoi day dung 2

A h(4) h( 2) h(2) B h(2) h(4) h( 2)

C h(2) h( 2) h(4) D h(4) h( 2) h(2)

Nhan xet : Cac diem A(-2;2), B(2;2), C(4; 4) cung nam tren dg thg y = x

Tu gt ta co h’(x) = 2(f’(x) - x) h(x) h '(x)dx [2f '(x) 2x]dx

Goi S la dien tich hinh gioi han :1 y f '(x); y x&S la dien tich hinh gioi han :2 y f '(x); y x

O

2 2 2

2

S [f '(x) x]dx 2S 2 [f '(x) x]dx h'(x)dx g(x) h(2) h( 2) 0

h(2) h( 2) (*)

4

S x f '(x) dx 2S 2 [f '(x) x]dx h'(x)dx h(x)

h(2) h(4) 0 h(2) h(4) (**)

h(4) h(2) dap an D bi loai tiep theo ta so sanh h(4) voi h(- 2) ta co

h(4) h( 2) h '(x)dx 2 [f '(x) x]dx 2 [f '(x) x]dx [f '(x) x]dx

2S 2S 2(S S ) 0 h(4) h( 2) h(4) h( 2)(***)

Tu (*), (**) va (***) suy ra ta co :

2

S la dien tich hinh phang gioi han boi doan CA voi do thi cua f’(x) , C(-2; -2)

suy ra ta co h(2) h(4) > h(-2) Vay dap an chon la dap an B

Cau 46 (115)

Cho hinh tu dien deu canh a M, N lan luot la trung diem AB, BC, E la diem doi xung voi B qua D khi do mp(MNE) chia tu dien thanh hai phan Tinh the tich V cua khoi da dien chua dinh A

Giai tom tat

ACMNPQ E.ACMN E.ACPQ

V d(E,(ACMN)).dt d(E,(ABC)).dt

Q

P N

M

E B

C

D

A

Q

P E

N

M

A

B

C D

E.ACPQ ACPQ ACPQ

AMN

4 4 Tu gt thiet thi P, Q lan luot la trong tam cac tam giac

DAC

dt

d(E,(ABC)).dt 2d(A,(BCD)) dt V

d(E,(ACD)).dt d(A,(BCD)) dt V

ACMNPQ E.ACMN E.ACPQ

8 V

9

2 ABCD

CACH 2

1

3 2 0

BMN

2 2

d(Q,(BMN) d(D,(ABC)) a

Q

D

P N

B

M

P Q

N

M

A

B

C

D

N

C

P

1 1 1 a 6 a 6 d(Q,(BNPD) d(A,(BCD)) d(D,(ABC))

BMNPDQ

1 a 3 2a 6 a 3 a 6 1a 3 2 1 1 7 2a

Vay dap an chon la C co the ta lam truc tiep nhu sau :

ACMNPQ A.MNPQ A.CNP A.CNP N.APQ N.AMQ

1

dt d(A,(CNP)) dt d(N,(APQ)) dt d(N,(AMQ)) 3

2

1 a 2a 3 a 3

2 2 3 2 12 ;

2

2

ACMNPQ

1 a 3 a 6 a 3 a 6 a 3 a 6 1 11a 18 11a 2

Vay Chon dap an C

Cau 50 (115) Xet cac so thuc duong thoa log3 1 xy 3xy x 2y 4

x 2y Tim gia tri nho nhat cua P = x + y

Giai tom tat

1 xy

log 3xy x 2y 4 log (1 xy) 3(1 xy) 1 log (x 2y) x 2 y

x 2y

log 3(1 xy) 3(1 xy) log (x 2y) (x 2 y) (*)

D

C

Q

P N

M A

P

Q

E

N

M A

C

Dat a = 3(1 – 3xy) >0 va b = (x + 2y) > 0 (*) log a3 a log b3 b

Xet ham so f(t) = log t3 t f '(t) 1 1 0 t 0

t ln 3 (*) xay ra khi va chi khi 3(1 – xy ) = x + 2y y(2 3x) 3 x y x 3(do x 0)

2

2

3

3

suy ra ta co bbt cua P la

SUY RA DAP AN CHON LA C

Cau 33 (115) Mot nguoi gui 50Trieu voi lai suat 6% neu khong rut thi sau moi nam so lai nhap

vao goc de tinh lai cho nam tiep theo Hoi sau it nhat bao nhieu nam nguoi do nhan duoc so tien hon 100Trieu

A 12 nam B 14 nam C 13 nam D 11 nam

Ta da biet neu mot nguoi co so goc von ban dau la G gui vao ngan hang voi lai suat la r thi sau n nam nguoi do duoc lanh ca goc va lai la Tn G(1 r) n

Goi n la so nam gui vao duoc lanh ra lon hon 100 trieu khi va chi khi

100 = 50(1 0,06)n (1 0,06)n 100 2 n log(1,06)2 log 2 11,9

vay yeu cau bai toan thoa so nam it nhat la 12 nen dap an chon la A

Cau 37 (115) Mot chuyen dong trong 3 gio voi van toc v (km / h)

phu thuoc thoi gian t(h) co do thi van toc nhu hinh ve trong

khoang thoi gian 1 gio ke tu khi bat dau chuyen dong , do thi

do la mot phan cua parabon co ding I(2; 9) khoang thoi gian

con lai do thi chuyen dong la mot doan thg song song voi

truc Ox Tinh quang dg cua chuyen dong trong 3 gio

A.s 13,83(km) B s 23, 25(km)

C.s 21,59(km) D s 15,50(km)

Trong mot gio dau vat chuyen dong co do thi van toc la parbol

co pt la y 5x2 5x 4

4 tai x = 1 ta co y(1) = 7,75 =

31

4 suy ra quang duong ma vat di duoc

trong 3 gio la

2

s ( x 5x 4)dx dx 6,08(3) 15.5 21,58

+ 

0

7,75 9

3

4

1

hay

1

2

0

4 4 tuc hai gio sau chuyen dong deu voi van toc la 31(km/ h)

4 Vay chon dap an C

Cau 42 (115) Cho F(x) = x2 la mot nguyen ham cua ham so f (x)e Tim nguyen ham cua 2x ham so f '(x)e 2x

Cach 1:

2x

4x

x f (x)e dx 2x f (x)e f (x) f '(x)

2e (1 2x)

e

dap an A

Cach 2: dat

2x 2x

2x

du 2e dx

2x

dv f '(x)dx v f (x)

e

Cau 40 (115)Cho ham so y = x m

x 1 thoa man [2;4]

y 3

min Menh de nao duoi day dung

A m > 4 B 3 < m 4 C m < - 1 D 1 m < 3

[2;4]

(x 1)

3

Min Min

suy ra chon dap an A

DE 102

Cau 35 Cho ham so y = x m

x 1 thoa man [1;2] [1;2

16

3

max min Menh de nao duoi day dung

A m 0 B m > 4 C 0 < m 2 D 2< m 4

2

[1;2] [1;2]

[1;2] [1;2]

Max Min

Dap an chon la B

Cau 37 Cho x,y la cac so thuc duong thoa x2 9y2 6xy Tinh 12 12

12

1 log x log y M

2log (x 3y)

A M 1

1 M

1 M 3

Tu gt x2 6xy 9y2 12xy (x 3y)2 12xy

2

1 log x log y log 12xy log 12xy

2log (x 3y) log (x 3y) log 12xy

Chon dap an B

Cau 38 : Mot chuyen dong trong 3 gio voi van toc v (km / h)

phu thuoc thoi gian t(h) co do thi do la mot phan cua parabon

co ding I(2; 9) Tinh quang dg cua chuyen dong trong 3 gio

A.s 24, 25(km) B s 26,75(km)

C.s 24,75(km) D s 25, 25(km)

tu gt thi pt van toc cua chuyen dong duoc cho boi pt :

y = 3x2 3x 6

4 suy ra quang duong vat di duoc trong 3 gio

la s =

3

2 0

Chon dap an C

Cau 39 Cho so phuc Z = a + bi thoa Z + 2 + i = Z Tinh S = 4a + b

2

4

Chon dap an D

Cau 40 Cho F(x) = (x 1)e la mot nguyen ham cua ham so x f (x)e Tim nguyen ham cua 2x ham so f '(x)e x

2 x

A f '(x)e dx (4 2 x)e c B f '(x)e dx e c

2

C f '(x)e dx (2 x)e c D f '(x)e dx (x 2)e c

tu gt (x 1)ex f(x)e dx2x ex (x 1)ex xex e f (x)2x f(x) xx

e dat

x x

x

x

e

v ' f '(x) v f(x)

e

9

3

6

1

Dap an chon C

Cau 41 : Ong A lap cong ty tong so tien cong A phai tra cho cong nhan trong mot nam la 1 ty

dong Biet so tien phai tra sau moi nam tang them 15 % so voi nam truoc do Hoi neu A thanh lap cong ty tu nam 2016 thi nam dau tien nao A dung de tra luong cho cong nhan trong ca nam lon hon 2 ty dong

A Nam 2023 B Nam 2022 C Nam 2021 D Nam 2020

so tien phai tr sau nam thu 1 la N1 1 1.0,15 tiep theo sau nam thu 2 la

2

N N N 0,15 1 1.0,15 (1 1.0,15)0,15 (1 1.0,15)[1 0,15] (1 1.0,15) … nam thu n so tien phai tra la Nn (1 0,15) cho n lay gia tri tu1,2,3… n ta co n

N 1,15ty; N 1,3225ty; N 1,520875ty; N 1,74900625ty, N 2,011357188ty

hay 2ty = (1 0,15)n so nam n = log 2 4.95948 5

Suy ra chon dap an C

Cau 42 (112) Cho ham so y= f(x) co bang bien thien nhu hinh ve

Do thi ham so y = f (x) co bao nhieu diem cuc tri

hinh ve ben la do thi cua ham so y = f (x) nen so diem cuc

tri cua do thi la 3

nen dap an chon C

Cau 44 : cobao nhieu so phuc Z thoa z 2 i 2 2 va (z 1) la so thuan ao 2

(z 1) [(a 1) bi] (a 1) 2(a 1) bi b la so thuan ao

2(a 1) b 0

a 1& b 0

(a 1) b

Neu a – 1 = b thi ta co : a2 4a (a 1)2 2(a 1) 3 2a2 3 3 a 0;b 1

Neu a – 1 = - b thi ta co : a2 4a (a 1)2 2(1 a) 3 2a2 4a 1 3 a2 2a 2 0

1 5

+ 0

- 

0

3

0

3 -1

1 5

a 1 3 b 2 3

vay co ba so phuc thoa dk nen chon dap C

Cau 45 Tim cac gia tri thuc m de dg thg y = - mx cat do thi ham so y = x3 3x2 m 2 tai 3 diem phan biet A, B, C sao cho AB = BC

pt hoanh do giao diem

2

x 1

x 2 x m 2 0(*)

dk can la pt(*) co hai nghiemphan biet khac 1 ' 3 m 0 m 3 m 3

khi do (*) co hai nghiem x1,2 1 3 m suy ra do thi ham so co ba diem cuc tri

A(1 3 m; m(1 3 m));B(1; m);C(1 3 m; m(1 3 m))

do xA xC yA yC

2 2 nen AB = BC Vay dap an chon la A

thanh cap so cong nen ba giao diem nam tren cung mot dg thg thoa man AB = BC

Cau 46: Xet cac so thuc duong x, y thoa log31 xy 2xy x y 3

x y Tim gia tri nho nhat cua P = x + 2y

Giai tom tat

1 xy

log 2xy x y 3 log (1 xy) 2(1 xy) 1 log (x y) x y

x y

log 2(1 xy) 2(1 xy) log (x y) (x y) (*)

Dat a = 2(1 – 3xy) >0 va b = (x + y) > 0 (*) log a3 a log b3 b

Xet ham so f(t) = log t3 t f '(t) 1 1 0 t 0

t ln 3 (*) xay ra khi va chi khi 2(1 – xy ) = x + y y(1 2 x) 2 x y x 2 (do x 0)

2

2

2

2

suy ra ta co bbt cua P la

+

0

P

SUY RA DAP AN CHON LA A

Cau 47 : Trong he Oxyz chohai diem A(4;6;2), B(2; - 2;0)

mf(P): x + y + z = 0 Xet dg thg d thay doi thuoc (P)

va di qua B, goi H la hinh chieu cua A tren d biet rang

khi thay doi H luon nam tren mot dg tron co dinh

Tinh R cua dg tron do

Nhan xet : tu gt suy ra B (P)do AH d

EH d ,( Ela hinh chieu cua A tren (P)) khi d thay doi H luon nam tren dg tron duong kinh EB Toa do diem E la nghiem cua he

Chon dap an A

Cau 48: Cho ham so y = f(x)co do thi cua y = f’(x) nhu hinh ve Dat g(x) = 2f(x) (x 1) 2 Menh de nao duoi day dung

A g( 3) g(3) g(1) B g(1) g( 3) g(3)

C g(3) g( 3) g(1) D g(1) g(3) g( 3)

Tu hinh ve ta thay cac diem A(-3; -2), B(1;2), C(3;4) nam tren dg thg d: y = x + 1

Tu gt ta co g’(x) = 2f’(x) 2(x + 1)

Goi S la dt hinh gioi han:1 y f '(x), y x

x 3; x 1 ; 2

y f '(x), y x

S la dt hinh gioi han:

1

s f '(x) (x 1) dx 2S 2 f '(x) (x 1) dx g '(x)dx [g(x)]

(g(1) g( 3)) 0 g(1) g( 3) (*)

S [(x 1) f '(x))]dx 2S 2 [f'(x) (x 1)]dx 2 g'(x)dx

2[g(3) g(1)] 0 g(3) g(1) 0 g(3) g(1) (**)

3 3 3 3

T a co : g(3) g( 3) g(x) g '(x)dx

3

3

2 [f '(x) (x 1)]dx

2 f '(x) (x 1) dx f '(x) (x 1) dx 2S1 2S2 2(S1 S )2 0

2[g(3) g( 3)] 0 g(3) g( 3) (***)

P

d

A

E

Tom lai ta co : g(3) g( 3) g(1) Suy ra dap an chon D

Cau 49 : Xet khoi tu dien ABCD co canh AB = x va cac canh con lai deu bang 2 3 Tim x de the tich V cua khoi tu dien nay lon nhat

Ta co: V =

dt d(A,(BCD)) 2V d(C,(ABI)).dt

2

Vlon nhat khi sin(AIB) lon nhat AIB 90

khi do AIB la tam giac vuong can nen AB = x = BI 2 2.2 3 3 3 2

Chon dap an C

Cau 50: Cho mat cau (S) co R = 4, , hinh tru (H) co chieu cao h = 4 va hai dg tron day nam tren

(S) Goi V la the tich cua (H) va V’ la the tich cua (S) Tinh ti so V

V '

Goi r la ban kinh day cua (H) thi

2

DE 103

Cau 35 : mot vat chuyen dong trong 4 gio voi van toc v (km / h) phu thuoc thoi gian t (h) co do thi van toc nhu

hinh ve Trong thoi gian 3 gio tu khi bat dau chuyen dong , do thi la mot phan Prabol co dinh I(2; 9) khoang thoi gian con lai do thi la mot dg thg song song voi truc Ox Tinh quang dg vat di duoc trong 4 gio

A s 26,5(km) B s 28,5(km)

C s 27,0km) D s 24,0(km)

Tu gt do thi van toc v la do thi ham so y = 9x2 9x

4

27 y(3)

4

trong khoang thoi gian tu 3h den 4h vat chuyen

dong deu voi van toc v = 27

4 suy ra quang dg vat di duoc trong

4 gio bang :

3

2 0

Vay chon dap an C

I

B

A H

9

4

2 3 O

B

t v

Cau 37: Cho F(x) = 13

3x la mot nguyen ham cua ham so

f (x)

x

Tim nguyen ham cua ham so f’(x)lnx

Tu gt

4

3

dx

1

V x

3x

Cach khac dat :

3

dx du

v f (x)

x

Chon dap an C

Cau 41: Mot vat chuyen dong theo qui luat S 1t3 6t2

2 voi t (giay ) la khoang

thoi gian tinh tu khi vat bat dau chuyen dong va s (met) la quang duong di duoc cua cua vat trong khoang thoi gian do Hoi trong khoang thoi gian 6 giay ke tu khi bat dau chuyen dong van toc lon nhat cua vat dat duoc bang nhieu

A 24 (m/s) B 108 (m/s) C 18 (m/s) D 64 (m/s)

Ta co v toc cua chuyen dong trong khoang thoi gian 6 giay duoc cho boi pt :

2

max

3

v S' t 12t v' 3t 12 0 t 4 van toc lon nhat cua

2 chuyen dong trong khoang thoi gian 6giayla tai thoi diem t 4giay V V(4) 24(m/ s)

Chon dap an A

Cau 44 : Xet khoi chop S.ABC day la tam gic vuong can tai A, SA vuong goc day khoang cach

tu A den mf(SBC) bang 3 Goi la goc giua hai mat phang (SBC) & (ABC), tinh cos sao cho V cua khoi chop do nho nhat

3



S

A

B H

Tu gt thi AI = AH 3 BC 6 &

SA AI tan

sin cos cos ABC

2sin cos 2 (1 cos )cos

V nho nhat khi f(t) = 1 3 nho nhat voi moi t (0;1) va t cos

f’(t) =

2

3 2

0 t (t t ) 3 3 suy ra bbt cua f(t) tu bbt suy ra V nho nhat

khi t cos 3

3 Nen ta chon dap an B

Cau 45: Tim cac gia tri m de do thi ham so y = x4 2mx2 co ba diem cuc tri tao thanh tam giac co dt nho hon 1

A m > 0 B m < 1 C 0 < m < 3 4 D 0 < m < 1

y' 4x 4mx 4x(x m) dk can la y' 0 phaicoba nghiemphan biet m 0

A( m; m );O(0;0);B( m; m )

AOB

Nen ta chon dap an D

Cau 46 : Cho ham so y = f(x) co do thi cua y = f’(x) nhu hinh ve

Dat g(x) = 2f(x) + x thi menh de nao duoi day dung 2

A g(3) g( 3) g(1) B g(1) g(3) g( 3)

C g(1) g( 3) g(3) D g( 3) g(3) g(1)

Nhan xet: A(-3; 3), B(1; - 1), C(3; -3) cung nam tren dg thg y = - x

Tu gt ta co g’(x) = 2f’(x) + 2x

1

1 3 3

g '(x)dx g(x) g(1) g( 3)

Goi

1

y f '(x), y x

S la dt hinh phang gioi han:

x 3; x 1 ; 2

y f '(x), y x

S la dt hinh phang gioi han:

x 1; x 3

1

s ( x) f '(x) dx 2S 2 f '(x) x dx 2 g '(x)dx 2[g(x)]

2(g(1) g( 3)) 0 g(1) g( 3) (*)

S [f '(x) ( x)]dx 2S 2 [f '(x) ( x)]dx 2 g'(x)dx

2[g(3) g(1)] 0 g(3) g(1) (**)

f( )

1 + f'(t) t 0

-0 f(t)

3 3 3 3

T a co : [g(3) g( 3)] g(x) g '(x)dx

3

3

2 [f '(x) x]dx

2 f '(x) ( x) dx f '(x) ( x) dx 2S1 2S2 2(S1 S )2 0

g(3) g( 3) 0 g(3) g( 3) (***)

Tom lai ta co : g(1) < g(3) < g(-3) Suy ra dap an chon B

Cau 48 : Co bao nhieu so phuc Z tho z 3i 13 va z

z 2 la so thuan ao

2

la so thuan ao

2

b 0 a 2(loai)

Vay chon dap an D

Cau 49 Trong he toa do Oxyz Cho hai diem A(3; - 2 ; 6), B(0; 1; 0) va mat cau (S):

(x 1) (y 2) (z 3) 25 Mat phang (P): ax + by +cz – 2 = 0 di qua A, B va cat (S) theo giao tuyen la dg tron co ban kinh nho nhat Tinh T = a + b + c

Tu gt suy ra A nam ngoai , B nam trong mat cau nen AB

cat mat cau tai hai diem co dinh M, N Goi H la tam dg

tron giao tuyen thi H nam tren trung truc cua MN va H

la hinh chieu cua I tren (P) Goi K la hinh chieu cua I tren

AB thi HK AB khi do ban kinh r = HK2 KN2

r nho nhat khi HK= 0 xay khi HI cung vuong goc AB do mat

phang (P) nhan IK lam vtpt

BA (3; 3;6) n (1; 1;2) la mot vtpt cua mf(Q) qua I

vuong goac AB co pt la co pt: x – y + 2z – 5 = 0

AB co pt

x t

y 1 t

z 2t

suy ra toa do K la nghiem cua he

I

K

A

H'

H B

K H B

A I

M

N

x t

la mot vtpt cua (P)

nen (P) coa pt la 0x + 2y +z + 2 =0 T a b c 3 Chon dap an A

Cau 50: Xet ham so

t

9

f (t)

9 m (m la so thuc ) Goi S la tap hop tat ca cac gia tri cua m sao cho f(x) + f(y) = 1 voi moi so thuc x, y thoa man ex y e(x y) tim so phan tu cua S

x y

x y

xet hamso f (t) e e.t f '(t) e e 0 t 1

t 1 f '(t) 0

bbt cua y f (t) tu bbt ta thay voi moi 0 < t < 1 va

voi moi 1 < t < 0 thi f (t) et e.t > 0 tuc la

t

e e.t voi moi t 0, t 1 ex y e(x y) chi dung

khi x+ y = 1 khi do ta co m2 31 3 m 3

nen chi co hai gia tri cua m thoa man Vay chon dap an D

DE 104

Cau 34: Mot vat chuyen dong theo qui luat S = 1t3 6t2

3 voi t (giay)la khong thoi gian tu khi vat chuyen dong va S la (m) la quang dg vat di duoc trong khoang thoi gian do hoi trong 9 giay van toc lon nhat cua vat dat duoc bang bao nhieu

A 144(m/s) B 36(m/s) C 243(m/s) D 27(m/s)

S' t 12t (t 6) 36 36 vay van toc lon nhat la 36(m/s)

suy ra dap an B

Cau 35: Mot nguoi chay trong thoi gian t gio van toc (km/h) phu thuoc thoi gian t(h)

co do thi la mot phan cua Pra bol dinh I(( ;8)1

2 )…, nhu hinh ve Tinh quang dg S nguoi do

chay trong khoang thoi gian 45 phut

A S= 4, 0 (km) B S=2,3(km) C S = 4,5 (km) D 5,3 (km)

Vi do thi cua chuyen dong la mot phan cua do thi ham so y = y 32x2 32x

Nen quang duong di trong khoang thoi gian 45 phut la :

1 + 0 f'(t) t 0 - 0 f(t)