j 2013.
1
1. :6
1 + x2
x+
1 + x2 x =
1
2(
3
2
2
3) j:
() 6 () 5 ()2
3()
3
2() 4.
: x =1
2 3 2
6=
1
2 1
6
66
=
6
12
:6
1 + (6
12)2
6
12+
1 + (6
12)2
=6
150144
6
12+
150144
=306
126
12+ 5
6
12
=
30
6
6
6= 5
2. a b j a2 6= b2, j : aa2 + b2
b(a b)2
a4 b4
j:
()1
a b()
a
a+ b()
1
a+ b()
b
a+ b()
a ba+ b
.
: :
a
a2 + b2 b(a b)
2
(a b)(a+ b)(a2 + b2)=a(a+ b) b(a b)
(a2 + b2)(a+ b)=
=a2 + ab ab+ b2
(a2 + b2)(a+ b)=
1
a+ b
3. j j a. j j j j:
() a
5
2() a
2
3() a
3
2() a
2
2() a
6
4.
:
1
AB ABC a, DE DEC y, ABC h DEC x. y. :, ABC DEC , j
: a : y = h : x, ax = yh, x =yh
a.
, ABC j DEC.
j:ah
2= 2
xy
2, x =
ah
2y.
j:yh
a=ah
2y,y
a=
a
2y, 2y2 = a2, y2 =
a2
2, y =
a2.
y a
2
2.
4. j 44 j?
() 11 () 10 () 13 () 9 () 8.
: j j j: Dn =n(n 3)
2. j:
44 =n(n 3)
2, 88 = n(n 3), 8 11 = n(n 3), j j : n = 11.
5. j x2 2|x| 3 = 0 j j:
() 1 () 0 () 3 () 2 () 1.
: j j: j x 0, j :
x2 2x 3 = 0. j: x1,2 =2
4 + 12
2, x1 = 3 x2 = 1.
j j x 0, x1. j, j x < 0, j :
x2 + 2x 3 = 0. j: x3,4 =2
4 + 12
2, x3 = 1 x4 = 3.
x4. j: x1 + x4 = 3 + (3) = 0.
6. j log2(x2 + 1) < 1 j:
() (0, 2) () (2, 0) () (1, 1) () (1,+) () (1, 0).
: j j f(t) = log2(t) , : log2(x2 + 1) < log22,
j x2 + 1 < 2, x2 < 1. j: x (1, 1).
7. m j j j x2 + (m4)xm 4 = 0 j :
() (0, 2) () (4, 5) () (5,+) () (1, 4) () (4, 6).
2
: x1 x2 j :
x21 + x22 = (x1 + x2)
2 2x1x2 = (B
A)2 2C
A= ((m 4))2 2(m 4) =
= m2 8m+ 16 + 2m+ 8 = m2 6m+ 24.j :
mmin = b
2a= 6
2= 3 (1, 4).
8. j j j 19 j j. j 1, j j. j :
() 162 () 316 () 126 () 180 () 216.
:
j : a1, a2 = a1q, a3 = a1q2.
j 19, j: a1 + a1q + a1q2 = 19. j
, :a1q
2 1 a1q = a1q a1. :{a1 + a1q + a1q
2 = 19
a1q2 1 a1q = a1q a1
{a1(1 + q + q
2) = 19
a1(q2 q q + 1) = 1
19
1 + q + q2=
1
1 2q + q2, 19 38q + 19q2 = 1 + q + q2
18q2 39q + 18 = 0, q1,2 =39
1521 1296
36,
q1 =19
36, q2 =
3
2.
j , q2.a1 =
1(1q)2 =
1( 1
2)2
= 4, a2 = 6, a3 = 9. j: a1 a2 a3 = 216.
9. j cosx+
3 sinx = 1 (0, )?() 1 () 0 () 2 () 3 () 4.
:
j 2 j:1
2cosx+
3
2sinx =
1
2
sin
6cosx+ cos
6sinx =
1
2, : sin
6+ x =
1
2. j :
6+ x1 =
6+ 2k, k Z
6+ x2 =
5
6+ 2l, l Z
, :
x1 = 2k, k Zx2 = 23
+ 2l, l Z.
J j x =2
3 (0, ).
10. j j j 3,
3 1 j j:
3
() 3(2 +
3) () 2(3 +
3) () 3(1 +
3) () 2(3 + 5
3) () 4
3.
:
j (30, 60, 90) c = 2b, a =c
3
2=
b
3. : P =ab
2 P = rs,
j:
s =a+ b+ c
2=b
3 + b+ 2b
2=b(
3 + 3)
2.
j:ab
2=r(a+ b+ c)
2, :
b2
3
2= (
3 1)(
3 + 3)b
2,
b
3 = 3 +
3
3 3, b = 2, a = 2
3, c = 4. j: a+ b+ c = 2
3 + 2 + 4 = 2(
3 + 3).
11. j 3:11:17. j 815cm3, j :
() 386cm2 () 360cm2 () 380cm2 () 364cm2 () 368cm2.
:
j :P = (r21 + r
22 + s(r1 + r2)). : r1 =
11k, r2 = 3k, s = 17k. j k.
V =H
3(r21 + r1r2 + r
22), 815 =
H
3(121k2 + 33k2 + 9k2)
815 =H
3163k2, H =
15
k2.
j: H2 = s2 (r1 r2)2
H2 = 289k2 64k2, H = 15k. j: 15k = 15k2, k3 = 1, k = 1.
: P = (121 + 9 + 17(11 + 3)) = 368cm2.
12. sin3 x+ cos3 x
2 sin 2x j j:
4
()sinx+ cosx
2()
sinx cosx2
()sin2 x cosx
2()
sinx cos2 x2
()sinx+ cos2 x
2.
:
(sinx+ cosx)(sin2 x sinx cosx+ cos2 x)2 2 sinx cosx
=(sinx+ cosx)(1 sinx cosx)
2(1 sinx cosx)=
sinx+ cosx
2.
13. j x4 + 3x3 7x2 27x 18 = 0 j:
() 23 () 31 () 26 () 18 () 21.
:
: x1 + x2 + x3 + x4 = 3
1
x1x2+x1x3+x1x4+x2x3+x2x4+x3x4 =71. j j:
(x1 + x2 + x3 + x4)2 = 9, (x1 + x2)
2 + 2(x1 + x2)(x3 + x4) + (x3 + x4)2 = 9
x21 + x22 + x
23 + x
24 + 2(x1x2 + x1x3 + x1x4 + x2x3 + x2x4 + x3x4) = 9
x21 + x22 + x
23 + x
24 + 2(7) = 9, x21 + x22 + x23 + x24 = 23.
14. j5 + i
1 2ij:
()2
5()
3
5() 2
5() 3
4()
4
5.
: z =5 + i
1 2i 1 + 2i
1 + 2i=
5 + 10i+ i 21 4(1)
=3 + 11i
5, Re(z) =
3
5
15. j
2 tan = 11
12 j sin j:
()2
5()
3
5()
11265
() 34()
4
5.
:sin
cos= 11
12,
j :sin
1 sin2 = 11
12, 12 sin = 11
1 sin2
144 sin2 = 121(1 sin2 )sin2 =
121
265, : sin =
11265
16. q p : y = 14x + 3 M(0, 3) . q
x- N(xN , 0) j xN =?
()2
5()
3
5() 2
5() 3
4()
4
5.
: q j: y = kx + n. o je y- (0,3) j y- n = 3. j k j
5
j p:
k = 11
4
= 4. q : y = 4x+ 3 x- xN = 3
4.
17. j j 5
3, j j:
()125
6
4()
25
6
4() 5
5
6
4()
25
6
2()
3755
6
4.
: V =BH
3, B =
a2
3
4=
25 3
3
4=
75
3
4, j:
h =a
3
2=
15
2, : H2 =
h2 (h3
)2 =225
4 25
4=
200
4, H = 5
2, V =
1
3 75
3
4 5
2 =125
6
4.
18. y- jx2 + y2 + 3x 2y + 3 = 0 j:
()2
5()
3
5() 3
5() 2
5()
3
2.
: J :
x2 + 3x+9
4 9
4+ y2 2y + 1 1 + 3 = 0, j: (x+ 3
2)2 + (y 1)2 = 1
4,
j : r =1
2, a : yC = 1, j
: r + yC =3
2.
19. 3x+ 2y = 26 6x 3y = 3 M(x, y), j x+ y j:
() 11 () 316 () 126 () 180 () 216.
: M jj j:{3x+ 2y = 26
6x 3y = 3. :
=
3 26 3 = 9 12 = 21, : x = 26 23 3
= 78 6 = 84 y =
3 266 3 = 9 156 = 147. j j: x = 8421 = 4 y = 14721 = 7,
j: x+ y = 4 + 7 = 11.
20. j |z|+ z = 2 + i j:
6
()2
5()
3
4() 2
5() 3
5()
4
5.
: j z z = x+ iy, j |z| =
x2 + y2, j :
x2 + y2 + (x+ iy) = 2 + i.
j, j:x2 + y2 + x = 2
i(1y) = 0. j y = 1, j :x2 + 1 = 2x. j :
2 x 0, j: x2 + 1 = 4 4x+ x2, 4x = 3, j j : x =
3
4.
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