51 䉶 ñfŠªð‡ mù£‚èœ 1. èíƒèÀ‹ ꣘¹èÀ‹ 1. ªõ¡ðìƒè¬÷Š...
TRANSCRIPT
5 1
䉶 ñFŠªð‡ Mù£‚èœä‰¶ ñFŠªð‡ Mù£‚èœä‰¶ ñFŠªð‡ Mù£‚èœä‰¶ ñFŠªð‡ Mù£‚èœä‰¶ ñFŠªð‡ Mù£‚èœ1. èíƒèÀ‹ ꣘¹èíƒèÀ‹ ꣘¹èíƒèÀ‹ ꣘¹èíƒèÀ‹ ꣘¹èíƒèÀ‹ ꣘¹èèèèèÀ‹À‹À‹À‹À‹
1. ªõ¡ðìƒè¬÷Š ðò¡ð´ˆF H¡õ¼ùõŸ¬ø„ êKò£ âù «ê£Fˆ¶Š 𣘂è. A (B C) = (A B) (A C).
b˜¾ :
LHS: RHS
B C A B
A A C
A (B C) (A B) (A C)
------ (I) -------- (II)
I = II LHS = RHS
A (B C) = (A B) (A C).
2. A (B C) = (A B) (A C).
b˜¾:LHS RHS
B C A B
BA
C
BA
C
BA
C
BA
C
BA
C
BA
C
BA
C
A
C
B
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5 2
A A C
A (B C) (A B) (A C)
------ (I) -------- (II)
I = II LHS = RHS
A (B C) = (A B) (A C).
3. A\(B C) = (A\B) (A\C).b˜¾:
LHS RHS
B C A\B
A A\C
A\(B C) (A\B) (A\C)
------ (I) -------- (II)
I = IILHS = RHS
A\ (B C) = (A\B) (A\ C)
BA
C
BA
C
BA
C
BA
C
BA
C
BA
C
BA
C
A
C
B
A
C
B A
C
B
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5 3
4. A\(B C) = (A\B) (A\C)
b˜¾:
LHS RHS
(B C) (A\B)
A (A\C)
A\(B C) (A\B) (A\C)
---- (I) ------ (II)
I = IILHS = RHS
A\(B C) = (A\B) (A\C)
5. BA)BA( ′′=′ b˜¾:
LHS RHS
BA A′ = U - A
)BA( ′ = U - BA B′ = U - B
--- I
A
C
B BA
C
BA
C
A
C
B
BA
C
BA
C
A B U
A B U A B U
A B U
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5 4
BA ′′
-------- (II)
I = IILHS = RHS
BA)BA( ′′=′ 6. BA)BA( ′′=′
b˜¾:LHS RHS
BA A′ = U - A
)BA( ′ = U - BA B′ = U - B
---- (I)
BA ′′
-------- (II)
I = IILHS = RHS
BA)BA( ′′=′ 7. U = {-2, -1, 0, 1, 2, 3, .... 10}, A = {-2, 2, 3, 4, 5} B = {1, 3, 5, 8, 9} ̄ ñ£˜è¡ èí GóŠH MFè¬÷„ êKð£˜.
b˜¾ ¯ñ£˜è¡ èí GóŠ¹ i) BA)BA( ′′=′ ii) BA)BA( ′′=′
i) L.H.S. = )BA( ′A B = {-2, 2, 3, 4, 5} {1, 3, 5, 8, 9}
= {-2, 1, 2, 3, 4, 5, 8, 9}
)BA( ′ = U \ (A B)
= {-2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10} \ {-2, 1, 2, 3, 5, 8, 9}= {-1, 0, 6, 7, 10} ------- (I)
A B U
B UA A B U
A B
A B
A B U
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5 5
R.H.S. = BA ′′
A′ = U\A
= {-2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10} \ {-2,2,3, 4, 5}
= {-1, 0, 1, 6, 7, 8, 9, 10}
B′ = U \ B
= {-2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10} \ {1, 3, 5, 8, 9}
= {-2, -1, 0, 2, 4, 6, 7, 10}
BA ′′ = {-1, 0, 1, 6, 7, 8, 9, 10} {-2, -1, 0, 2, 4, 6, 7, 10}
= {-1, 0, 6, 7, 10} ----- (II) I = II
)BA( ′ BA ′′
ii) BA)BA( ′′=′
L.H.S. = )BA( ′
)BA( = {-2, 2, 3, 4, 5} {1, 3, 5, 8, 9}
= {3, 5}
)BA( ′ = U \ (A B)
= {-2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10} \ {3, 5}= {-2, -1, 0, 1, 2, 4, 6, 7, 8, 9, 10} ----- (I)
R.H.S. = BA ′′ A′ = U \ A
A′ = {-2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10} \ {-2, 2, 3, 4, 5}
= {-1, 0, 1, 6, 7, 8, 9, 10}
B′ = U \ B
B′ = {-2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10} \ {1, 3, 5, 8, 9}
= {-2, -1, 0, 2, 4, 6, 7, 10}
BA ′′ = {-1, 0, 1, 6, 7, 8, 9, 10} {-2, -1, 0, 2, 4, 6, 7, 10}
= {-2, -1, 0, 1, 2, 4, 6, 7, 8, 9, 10} ----- (II) I = II
BA)BA( ′′=′
8. A = {a, b, c, d, e, f, g, x, y, z}, B = {1, 2, c, d, e} , C = {d, e, f, g, z, y} A\(BC) = (A\B) (A\C)
b˜¾:BC = {1, 2, c, d, e} {d, e, f, g, z, y}
= {1, 2, c, d, e, f, g, y}
A\BC = {a, b, c, d, e, f, g, x, y, z} \ {1, 2, c, d, e, f, g, y}= {a, b, x, z} ---- (I)
A\B = {a, b, c, d, e, f, g, x, y, z}\ {1, 2, c, d, e}= {a, b, f, g, x, y, z}
A\C = {a, b, c, d, e, f, g, x, y, z} \ {d, e, f, g, z, y}= {a, b, c, x, z}
(A\B) (A\C) = {a, b, f, g, x, y, z} {a, b, c, x, z}
= {a, b, x, z} ---- (I) I = II
A\(BC) = (A\B) (A\C)
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5 6
9. å¼ ïèóˆF™ 85% ñ‚èœ îI¿‹, 40% «ð˜ ݃Aôº‹ ñŸÁ‹ 20% «ð˜ Þ‰F»‹ «ð²Aø£˜. 32% «ð˜ÝƒAôº‹ îI¿‹, 13% îI¿‹, Þ‰F»‹, 10% ݃Aôº‹ Þ‰F»‹ «ð²Aø£˜èœ. Í¡Á ªñ£N»‹«ðꈪîK‰îõ˜ êîiî‹ è£‡è.
îI› - T
݃Aô‹ - E
Þ‰F - H
{îI› ªñ£N «ð²ðõ˜} n(T) = 85%
{݃Aô ªñ£N «ð²ðõ˜} n(E) = 40%
{Þ‰F ªñ£N «ð²ðõ˜}n(H) = 20%
{îI¿‹, ݃Aôº‹ «ð²ðõ˜} n(T E) = 32%
{îI¿‹, Þ‰F»‹ «ð²ðõ˜} n(TH) = 13%
{݃Aôº‹, Þ‰F»‹ «ð²ðõ˜} n(E H) = 10%
{Í¡Á ªñ£N»‹ «ð²ðõ˜} n (T E H) = x
Í¡Á ªñ£N»‹ «ðêˆ ªîK‰îõ˜ êîiî‹40 + x + 32 - x + 13 - x + x - 2 + x - 3 + x + 10 - x = 100
95 - 5 + x = 10090+x = 100 x = 100 - 90 x = 10%
Í¡Á ªñ£N»‹ «ðêˆ ªîK‰îõ˜ êîiî‹ = 10%10. 170 õ£®‚¬èò£÷˜èO™ 115 «ð˜ ªî£¬ô‚裆C, 110 «ð˜ õ£ªù£L, 130 «ð˜ ðˆFK‚¬èè¬÷»‹
ðò¡ð´ˆ¶Aø£˜èœ â¡ð¬î å¼ M÷‹ðó GÁõù‹ è‡ìP‰î¶. 85 «ð˜ ªî£¬ô‚裆C ñŸÁ‹ðˆFK‚¬è, 75 «ð˜ ªî£¬ô‚裆C ñŸÁ‹ õ£ªù£L, 95 «ð˜ õ£ªù£L ñŸÁ‹ ðˆFK¬‚¬ò»‹ 70 «ð˜Í¡P¬ù»‹ ðò¡ð´ˆ¶Aø£˜èœ.(i) õ£ªù£L ñ†´‹ (ii) ªî£¬ô‚裆C ñ†´‹(iii) ªî£¬ô‚裆C ñŸÁ‹ ðˆFK‚¬è¬ò ðò¡ð´ˆF õ£ªù£L ðò¡ð´ˆî£îõ˜èœ 裇è.
ªî£¬ô‚裆C = Tõ£ªù£L = RðˆFK‚¬è = M
n(T) = 115n(R) = 110n(M) = 130n(TM) = 85n (T R) = 75n(RM) = 95n(T RM) = 70
i) õ£ªù£L ñ†´‹ = 10ii) ªî£¬ô‚裆C ñ†´‹ = 25iii) ªî£¬ô‚裆C ñŸÁ‹ ðˆFK‚¬èŠ ðò¡ð´ˆF õ£ªù£L ðò¡ð´ˆî£îõ˜ = 15
85 -
(32
-x+x
+13-
x)
85-4
5+x
40+x
32-x
40 - (32-x+x+10-x)
40-42+x
-2+x
x
10-x13-x
20-(13-x+x+10-x)20-23+x
-3+x
T îI›E݃Aô‹
H Þ‰F
115-
(70+
5+15
)11
5-90
25
75-705
110 - (5+70+25)
110 - 100
10
7095-70
2585-70
15
130-(15+70+25)130-110
20
TR
M
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5 7
11. f(x) =
<<−≤≤−<≤−−
74:32
42:23
23:14 2
xx
xx
xx
H¡õ¼õùõŸ¬ø‚ 裇è, (i) f(5) + f(6)
ii) f(1) - f(-3) (iii) f(-2) - f(4) (iv) )1(f)6(f2)1(f)3(f
−−+
b˜¾:
f(x) = )6,5(
)4,3,2(
)1,0,1,2,3(
74:32
42:23
23:14 2 −−−
<<−≤≤−<≤−−
xx
xx
xx
i) f(5) + f(6) =?
f(x) = 2x - 3
f(5) = 2 x 5 - 3
= 10 - 3
f(5) = 7
f(6) = 2× 6 - 3
= 12 - 3
f(6) = 9
f(5) + f(6) = 7 + 9
f(5) + f(6) = 16
ii) f(1) - f(-3) = ?
f(x) = 4x2 - 1
f(1) = 4 x 12 - 1
= 4 - 1
f(1) = 3
f(-3) = 4 x (- 3)2 - 1
= 4 x 9 - 1
= 36 - 1
f(-3) = 35
f(1) - f(-3) = 3 - 35
f(1) - f(-3) = -32
iii) f(-2) - f(4)
f(x) = 4x2 - 1
f(-2) = 4x (- 2)2 - 1
= 4 x 4 - 1
= 16 - 1
f(-2) = 15
f(x) = 3x - 2
f(4) = 3 x 4 - 2
= 12 - 2
f(4) = 10
f(-2) - f(4) = 15 - 10
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5 8
f(-2) - f(4) = 5
iv) )1(f)6(f2)1(f)3(f
−−+
= ?
f(3) + f(-1)
f(x) = 3x - 2
f(3) = 3 x 3 - 2
= 9 - 2
f(3) = 7
f(x) = 4x (-1)2 - 1
= 4 - 1
f(-1) = 3
f(3) + f(-1) = 7 + 3
f(3) + f(-1) = 10
2f(6) - f(-1) = 10
2f(6) - f(1) = ?
2(x) = 2x - 3
f(6) = 2 x 6 - 3
= 12 - 3
f(6) = 9
2f(6) = 18
f(1) = 4 x 12 - 1
= 4 - 1
f(1) = 3
2f(6) - f(1) = 18 - 3
= 15
)1(f)6(f2)1(f)3(f
−−+
= 1510
Ans: 32
12. A= { 0, 1, 2, 3 } , B = { 1, 3, 5, 7, 9 } Þ¼ èíƒèœ f : A → B â¡Â‹ ꣘¹ f(x)=2x+1 âù‚
ªè£´‚èŠð†´œ÷¶. ބ꣘H¬ù (i) õK¬ê «ê£®èO¡ èí‹ (ii) ܆ìõ¬í (iii) Ü‹¹‚°PŠðì‹
(iv) õ¬óðì‹ ÝAòõŸø£™ °PŠH´è,
b˜¾
f(x) = 2x + 1
f (0) = 2 x 0 + 1 = 0+1 = 1
f (1) = 2 x 1 +1 = 2 + 1 = 3
f (2) = 2 x 2 + 1 = 4 + 1 = 5
f (3) = 2 x 3 + 1 = 6 + 1 = 7
(i) õK¬ê «ê£®èO¡ èí‹
{ (0, 1), (1, 3), (2, 5), (3, 7) }
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5 9
(ii) ܆ìõ¬í
x 0 1 2 3
f (x) 1 3 5 7
(iii) Ü‹¹‚°P ðì‹
A B
0 1
1 3
2 5
3 7
(iv) õ¬óðì‹
13. f(x) =
<≤−
<≤−<≤+
64103
4212
211
2 xx
xx
xx
(1, 6) = {x∈R; 1 ≤ 6)
(i) f(5) (ii) f (3) (iii) f (1) (iv) f(2)- f(4) (v) 2 f(5) - 3f (1) ñFŠ¹è¬÷‚ 裇è.
b˜¾:
f(x) = )5,4(
)3,2(
)1(
64103
4212
211
2
<≤−
<≤−<≤+
xx
xx
xx
i) f(x) = 3x2 - 10f(5) = 3 x 52 - 10
= 3 x 25 - 10= 75 - 10
f(5) = 65
ii) f(x) = 2x - 1
f(3) = 2 x 3 - 1
= 6 - 1
f(3) = 5
iii) f(x) = 1 + x
f(1) = 1 + 1
f(1) = 2
iv) f(2) - f(4)
f
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6 0
f(x) 2x - 1
f(2) = 2 x 2 - 1
= 4 - 1
f(2) = 3
f(x) 3x2 - 10
f(4) = 3 x 42 - 10
= 3 x 16 - 10
= 48 - 10
f(4) = 38
f(2) - f(4) = 3 - 38
f(2) - f(4) = -35
v) 2f(5) - 3f(1)
2f(5) = 2 x 65
= 130
3f(1) = 3 x 2
= 6
2f(5) - 3f(1) = 130 - 6
2f(5) - 3f(1) = 124
14. A = {4, 6, 8, 10} B = {3, 4, 5, 6, 7} â¡è. f : A → B f(x) = 21
x + 1 õ¬óòÁ‚èŠð†´œ÷¶. (i) Ü‹¹‚°PŠðì‹
(ii) õK¬ê„ «ê£®èO¡ èí‹ (iii) ܆ìõ¬í (iv) õ¬óðì‹ ÝAòõŸP¡ Íô‹ °P‚è.
b˜¾:
f(x) = 21
x + 1
f(4) = 21
x 4 + 1 = 2 + 1 = 3
f(6) = 21
x 6 + 1 = 3 + 1 = 4
f(8) = 21
x 8 + 1 = 4 + 1 = 5
f(10) = 21
x 10 + 1 = 5 + 1 = 6
i) Ü‹¹‚°PŠðì‹
A B
4 3
6 4
8 5
10 6
ii) õK¬ê «ê£®èO¡ èí‹
f = {(4, 3) (6, 4) (8, 5) (10, 6)}
iii) ܆ìõ¬í
x 4 6 8 10f(x) 3 4 5 6
f
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6 1
iv) õ¬óðì‹
15. f(x) =
<<≤≤−
−<≤−
−+
++
62
25
57
1
5
122
x
x
x
x
x
xx
H¡õ¼õùõŸ¬ø‚ 裇è.
(i) 2f(-4) + 3f(2) ii) f(-7) -f(-3) iii) )1(f3)6(f)4(f2)3(f4
−−+−
f(x) = )5,4,3(
)2,1,0,1,2,3,4,5(
)6,7(
62
25
57
1
5
122
−−−−−−−
<<≤≤−
−<≤−
−+
++
x
x
x
x
x
xx
i) 2f(-4) + 3f(2)
f(x) = x + 5
f(-4) = - 4 + 5 = 1
2xf(-4)= 1 x 2
2f(-4) = 2
f(2) = 2 + 5 = 7
3 x f(2) = 7 x 3
3f(2) = 21
2f(-4) + 3f(2) = 2 + 21
2f(-4) + 3f(2) = 23
ii) f(-7) - f(-3)
f(x) = x2 + 2x + 1
f(-7) = (-7)2 + 2x (- 7) + 1
= 49 - 14 + 1
= 50 - 14
f(-7) = 36
f(x) x + 5
f(-3) = - 3 + 5
= 2
f(-3) = 2
f(-7) - f(-3) = 36 - 2
f(-7) - f (-3) = 34
654
32
1
1 2 3 4 5 6 7 8 9 10
(4,3)
(6,4)
(8,5)
(10,6)
x
y
O
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6 2
iii) )1(f3)6(f)4(f2)3(f4
−−+−
f(x) = x + 5
f(-3) = - 3 + 5 = 2
4f(-3) = 2 x 4
4f(-3) = 8
f(x) = x - 1
f(4) = 4 - 1 = 3
2f(4) = 3 x 2
2f(4) = 6
4f(-3) + 2f(4) = 8 + 6
4f (-3) + 2f(4) = 14
f(x) = x2 + 2x + 1
f(-6) = (-6)2 + 2 x (- 6) + 1
= 36 - 12 + 1
= 37 - 12
f(-6) = 25
f(x) = x + 5
3(1) = 1 + 5 = 6
3f(1) = 6 x 3
3f(1) = 18
f(-6) - 3f(1) = 25 - 18
f(-6) - 3f(1) = 7
)1(f3)6(f)4(f2)3(f4
−−+−
= 7
14
M“: 2
16. A = { 6, 9, 15, 18, 21 }; B = { 1, 2, 4, 5, 6 } f : A → B â¡ð¶ f(x) =3
3−x âù õ¬óòÁ‚èŠð†´œ÷¶.
(i) Ü‹¹‚°Pðì‹ (ii) õK¬ê «ê£®èO¡ èí‹ (iii) ܆ìõ¬í (iv) õ¬óðì‹ °P‚辋.b˜¾:
f(x) = 3
3−x
f(6) = 3
36 − =
33
= 1
f(9) = 3
39 − =
36
= 2
f(15) = 3
315− =
312
= 4
f(18) = 3
318− =
315
= 5
f(15) = 3
321− =
318
= 6 www.mathstimes.com
6 3
i) Ü‹¹‚°PŠðì‹
A B
6 1
9 2
15 4
18 5
21 6
ii) õK¬ê «ê£®èO¡ èí‹
f = {(6, 1), (9, 2), (15, 4), (18, 5), (21, 6)}
iii) ܆ìõ¬í
x 6 9 15 18 21
f(x) 1 2 4 5 6
iv) õ¬óðì‹
17. A = {5, 6, 7, 8}, B = {-11, - 4, 7, -1, -7, -9, -13} â¡è. f = {(x, y); y = 3-2x, x∈A, y∈B}
i) f ¡ àÁŠ¹ ii) Üî¡ ¶¬í ñFŠðè‹ iii) i„êè‹
iv) âšõ¬è ꣘¹
b˜¾:
y = 3-2x
x = 5, y = 3-2 x 5 = 3 - 10 = -7
x = 6, y = 3 - 2 x 6 = 3 - 12 = -9
x = 7, y = 3 - 2 x 7 = 3 - 14 = -11
x = 8, y = 3 - 2 x 8 = 3 - 16 = -13
i) f ¡ àÁŠ¹èœ
f = {(5, -7), (6, -9), (7, -11), (8, -13)}
ii) ¶¬í ñFŠðè‹ = {-11, 4, 7, -10, -7, -9, -13}
iii) i„êè‹ = {-7, -9, -11, -13}
iv) ꣘H¡ õ¬è
å¡Á‚° å¡ø£ù ꣘¹
f
7654
32
1
3 6 9 12 15 18 21 24
(6,1)(9,2)
(15,4)
(18,5)
x
y
(21,6)
O
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6 4
2. ªñŒªò‡èO¡ ªî£ì˜õK¬èèÀ‹ ªî£ì˜èÀ‹ªñŒªò‡èO¡ ªî£ì˜õK¬èèÀ‹ ªî£ì˜èÀ‹ªñŒªò‡èO¡ ªî£ì˜õK¬èèÀ‹ ªî£ì˜èÀ‹ªñŒªò‡èO¡ ªî£ì˜õK¬èèÀ‹ ªî£ì˜èÀ‹ªñŒªò‡èO¡ ªî£ì˜õK¬èèÀ‹ ªî£ì˜èÀ‹
1. å¼ Ã†´ˆ ªî£ì˜ õK¬êJ™ 10 ñŸÁ‹ 18õ¶ àÁŠ¹èœ º¬ø«ò 41 ñŸÁ‹ 73 âQ™ 27õ¶ àÁŠ¬ð‚裇è.b˜¾:
ªè£´‚èŠð†ì¶ t10 = 41 a + 9d = 41 ---- (1) t18 = 73 a + 17d = 73 ---- (2)(1) - (2) - 8d = 32
d = 832
−−
d = 4 ä (1) ™ HóFJì a + 9d = 41a + 9 x 4 = 41
a + 36 = 41 a = 5
a = 41 - 36a = 5
t27 = a + 26 d a = 5 , d = 4 âQ™ t27 = 5 + 26 (4)
= 5 + 104 t27= 109
2. a, b, c ÝAòù Æ´ˆ ªî£ì˜ õK¬êJ™ Þ¼ŠH¡ bc1
, ca1
, ab1
ÝAòù å¼ Ã†´ˆ ªî£ì˜õK¬êJ™
Þ¼‚°‹ âù GÁ¾è.b˜¾:
a, b, c å¼ A.P.÷ abc.
abca
, abcb
, abcc
»‹ å¼ A.P.
bc1
, ca1
, ab1
»‹ å¼ A.P.
3. å¼ ªð¼‚°ˆ ªî£ì˜ õK¬êJ¡ 裋 àÁŠ¹ 32
ñŸÁ‹ Üî¡ ãö£õ¶ àÁŠ¹8116
âQ™ ÜŠªð¼‚°ˆ
ªî£ì˜ õK¬ê¬ò 裇è.
t4 = 32
ar3 = 32
--- (1)
t7 = 8116
ar6 = 8116
--- (2)
(2) ÷ (1) 3
6
ar
ar = 32
8116
r6-3 = 8116
x 23
r3 = 278
r3 =
3
32
www.mathstimes.com
6 5
r = 32 ä (1) ™ HóFJì
ar3 = 32
a x
3
32
= 2/3
a = 32
x 23
x 23
x 23
a = 49
ªð¼‚°ˆ ªî£ì˜ õK¬ê = a, ar, ar2 .....
49
,
49
32
,
49 2
32
.....
4. å¼ ªð¼‚°ˆ ªî£ì˜õK¬êJ™ ºî™ ñŸÁ‹ Ýø£õ¶ àÁŠ¹èœ º¬ø«ò 31
, 729
1 âQ™ ÜŠªð¼‚°ˆ
ªî£ì˜õK¬ê¬ò‚ 裇è.
a = 31
t6 = 729
1 ar5 =
7291
31
r5 = 729
1
r5 = 729
1 x 3
= 243
1
r5 = 53
1
r5 =
5
31
r = 31
ªð¼‚°ˆ ªî£ì˜ õK¬ê,= a, ar, ar2 .....
= 31
,
31
31
, 31 2
31
......
= 31
, 91
, 271
.....
www.mathstimes.com
6 6
5. å¼ ªð¼‚°ˆ ªî£ì˜ õK¬êJ™ 4õ¶ ñŸÁ‹ 7õ¶ àÁŠ¹èœ º¬ø«ò 54 ñŸÁ‹ 1458 âQ™,܈ªî£ì˜õK¬ê¬ò‚ 裇è.
t4 = 54 ar3 = 54 ---- (1)t7 = 1458 ar6 = 1458 ---- (2)
(2) ÷ (1) 3
6
ar
ar =
541458
r3 = 27r3 = (3)3
r = 3 ä (1) ™ HóFJìar3 = 54a(3)3 = 54
a = 33354
××a = 2
ªð¼‚°ˆ ªî£ì˜ õK¬ê a, ar, ar2
= 2, (2) (3), (2) (3)2 ....= 2, 6, 18 .....
6. 8 Ý™ õ°ð´‹ ܬùˆ¶ Í¡Pô‚è Þò™â‡èO¡ Ã´î™ è£‡è.Í¡Pô‚è Þò™ â‡èœ 100, 101, ..... 999.a = 104, d = 8, l = 992
ð®1 :
n =
−
da
+ 1
=
−
8104992
+1
=
8888
+ 1
= 111 + 1n = 112
ð® 2:
Sn = 2n
[a + l]
S112 = 2
112[104 + 992]
= 56 x 1096S112 = 61376
7. 9Ý™ õ°ð´‹ ܬùˆ¶ Í¡Pô‚è Þò™ â‡èO¡ Ã´î™ è£‡.Í¡Pô‚è â‡èœ 100, 101, ... 999.a = 108, d = 9, l = 999
ð® 1 :
n =
−
da
+ 1
=
−
9108999
+1
12
8 100 + 4 = 104
8
20
16
4+4
125
8 999 -7 = 992
8
19
16
39
32
7
11
8 100 + 8
9
10
9
1+8
111
8 999 - 0
9
9
9
9
9
0www.mathstimes.com
6 7
=
9891
+ 1
= 99 + 1n = 100
ð® 2:
Sn = 2n
[a + l ]
S100 = 2
100[108 + 999]
= 56 x 1107
S100 = 55350
8. 300‚°‹ 500‚°‹ Þ¬ì«ò»œ÷ 11 Ý™ õ°ð´‹ ܬùˆ¶ Þò™ â‡èO¡ ÆìŸðô¬ù‚ 裇è.
a = 308, d = 11, l = 495
ð® 1 :
n =
−
da
+ 1
=
−
11308495
+1
=
11187
+ 1
= 17 + 1n = 18
ð® 2:
Sn = 2n
[a + l ]
S18 = 2
18[308 + 495]
= 9 x 803S18 = 7227
9. 100‚°‹ 200‚°‹ Þ¬ì«ò»œ÷ 5Ý™ õ°ðì£î ܬùˆ¶ Þò™ â‡èO¡ ÆìŸðô¬ù‚ 裇è.5Ý™ õ°ð´‹ Þò™ â‡èO¡ Ã´î™ 105 + 110 + ..... +195, a = 105, d = 5, l = 195ð® 1 :
n =
−
da
+ 1
=
−
5105195
+1
=
590
+ 1
= 18 + 1n = 19
Sn = 2n
[a + l ]
S19 = 2
19[105 + 195]
27
11 300 + 8 =308
22
80
77
3+8
45
11 499 - 4 = 495
44
59
55
4
www.mathstimes.com
6 8
= 19 x 150S19 = 2850
ð® 2 :Þò™ â‡èO¡ Ã´î™ 101 + 102 + ... 199
2)1n(n
n+=
101 + 102 + .... + 199= (1 + 2 + .... + 199) - (1 + 2 + .... + 100)
= 2
200199× -
2101100×
= 19900 - 5050= 14850
ð® 3:5Ý™ õ°ðì£î â‡èO¡ Ã´î™ = 14850 - 2850
= 1200010. 6 + 66+ 666 +... â‹ ªî£ìK™ ºî™ n àÁŠ¹èO¡ Ã´î™ è£‡.
Sn = 6 + 66 + 666 + .... n àÁŠ¹èœ õ¬ó= 6 (1 + 11 + 111 + ....n àÁŠ¹èœ õ¬ó)
= 96
(9 + 99 + 999 + ..... n àÁŠ¹èœ õ¬ó)
= 32
[(10 - 1) + (100 - 1) + (1000 - 1) ..... n àÁŠ¹èœ õ¬ó]
= 32
[(10 + 100 + 1000 + .... n àÁŠ¹èœ õ¬ó) - n] a = 10, r = 10>1
= 32
−
−−
n110
)110(10 n
Sn = 1r
)1r(a n
−−
= 110
)110(10 n
−−
Sn = 32
−−
n9
)110(10 n
11. 7 + 77 + 777 +... â‹ ªî£ìK™ ºî™ n àÁŠ¹èO¡ Ã´î™ è£‡. Sn = 7 + 77 + 777 + ....n àÁŠ¹èœ õ¬ó
= 7 (1 + 11 + 111 + .... n àÁŠ¹èœ õ¬ó)
= 97
(9 + 99 + 999 + ..... n àÁŠ¹èœ õ¬ó)
= 97
[(10 - 1) + (100 - 1) + (1000 - 1) ..... n àÁŠ¹èœ õ¬ó]
= 97
[(10 + 100 + 1000 + .... n àÁŠ¹èœ õ¬ó) - n] a = 10, r = 10>1
= 97
−
−−
n110
)110(10 n
Sn = 1r
)1r(a n
−−
= 110
)110(10 n
−−
Sn = 97
−−
n9
)110(10 n
12. 1 + 11 + 111 +... 20 àÁŠ¹èœ õ¬ó â‹ ªî£ìK¡ Ã´î™ è£‡. Sn = 1 + 11 + 111 + .... n àÁŠ¹èœ õ¬ó
= 91
(9 + 99 + 999 + ..... n àÁŠ¹èœ õ¬ó)
= 91
[(10 - 1) + (100 - 1) + (1000 - 1) ..... n àÁŠ¹èœ õ¬ó]www.mathstimes.com
6 9
= 91 [(10 + 100 + 1000 + .... n àÁŠ¹èœ õ¬ó) - n]
Sn =91
−
−−
n110
)110(10 n
a = 10, r = 10>1
S20 = 91
−
−−
20110
)110(10 20
Sn = 1r
)1r(a n
−−
= 91
−−
209
)110(10 20
S20 =
−−
920
)110(8110 20
13. Ã´î™ è£‡ 162 + 172 + 182 .... + 252
6)1n2()1n(n
n2 ++=
162 + 172 + 182 .... + 252 = (12 + 22 + ... + 252 ) - (12 + 22 .... + 152)
= 6
3116156
512625 ××−××
= 25 x 13 x 17 - 5 x 8 x 31= 5525 - 1240= 4285
14. Ã´î™ è£‡ 162 + 172 + ... 352
6)1n2()1n(n
n2 ++=
162 + 172 + .... + 352 = (12 + 22 + ... + 352 ) - (12 + 22 .... + 152)
= 6
3116156
713635 ××−××
= 35 x 6 x 17 - 5 x 8 x 31= 14910 - 1240= 13670
15. 11 ªê.e, 12 ªê.e, 13 ªê.e .... 24 ªê.e ÝAòõŸ¬ø º¬ø«ò ð‚è Ü÷¾è÷£è‚ ªè£‡ì 14ê¶óƒèO¡ ªñ£ˆîŠðóŠ¹ è£‡è.
ðóŠ¹ = 112 +122 + 132 +.... + 242
6)1n2()1n(n
n2 ++=
112 + 122 +132 .... + 242 = (12 + 22 + ... + 242 ) - (12 + 22 .... + 102)
= 6
2111106
492524 ××−××
= 4 x 25 x 49 - 5 x 11 x 7= 4900 - 385= 4515
ðóŠ¹ = 4515 ªê.e2
16. 12 ªê.e, 13 ªê.e...... 23 ªê.e ÝAòùõŸ¬ø º¬ø«ò ð‚è Ü÷¾è÷£è‚ ªè£‡ì 12 ê¶óƒèO¡ªñ£ˆîŠðóŠð÷¾‚ 裇è.b˜¾ :
ðóŠ¹ = 122 + 132 + 142 + .... + 232
6)1n2()1n(n
n2 ++= www.mathstimes.com
7 0
122 + 132 + .... + 232 = (12 + 22 + .... +232) - (12 + 22 + .... + 112)
= 6
2312116
472423 ××−××
= 23 x 4 x 47 - 22 x 23= 4324 - 506= 3818
ðóŠ¹ = 3818 ªê.e2.17. 16 ªê.e, 17 ªê.e, 18 ªê.e..... 30 ªê.e ÝAòùõŸ¬ø º¬ø«ò ð‚è Ü÷¾è÷£è‚ ªè£‡ì 15
èù„ê¶óƒèO¡ èù Ü÷¾èO¡ Ã´î™ è£‡è.b˜¾ :
èùÜ÷¾ = 163 + 173 + 183 + .... + 303
23
2)1n(n
n
+=
163 + 173 + 183 + .... + 303 = (13 + 23 + .... +303) - (13 + 23 + .... + 153)
= 22
21615
23130
×−
×
= (15 x 31)2 - (15 x 8)2
= (465)2 − (120)2
= (465 + 120) (465 - 120)
= 585 x 345
èù Ü÷¾ = 201825 ªê.e3
18. å¼ Ã†´ˆ ªî£ì˜ õK¬êJ™ Ü´ˆî´ˆî Í¡Á àÁŠ¹èO¡ Ã´î™ 6 ñŸÁ‹ ÜõŸP¡ ªð¼‚°ˆ
ªî£¬è &120 âQ™ Ü‹Í¡Á ⇂¬÷‚ 裇è.
a - d, a, a+d â¡ðù Æ´ˆªî£ì˜ õK¬êJ¡ Í¡Á àÁŠ¹èœ â¡è.
Ã´î™ = 6
a - d + a + a + d = 6
3a = 6
a = 6/3
a = 2
ªð¼‚°ˆ ªî£¬è= - 120
(a-d) (a) (a+d) = -120
(a2 - d2) a = -120
a = 2 ä HóFJì
(22 - d2)2 = -120
4 - d2 = 2120−
-d2 = -60 -4d2 = 64
d = 88×
d = ± 8a = 2 ñŸÁ‹ d = 8 âQ™Í¡Á â‡èœ = 2- 8, 2, 2+8 (Ü™ô¶) = -6, 2 10
a = 2 ñŸÁ‹ d = -8 âQ™
Í¡Á â‡èœ = 2 - (-8), 2, 2 - 8 = 10, 2, -6 www.mathstimes.com
7 1
19. 5 + 11 + 17 + .... + 95 â¡ø ªî£ìK¡ Ã´î™ è£‡.
a = 5, d = 11-5 = 6; l = 95
ð® 1:
n =
−
da
+ 1
=
−6
595+1
=
6
90 + 1
= 15 + 1n = 16
ð® 2:
Sn = 2n
[a + l ]
S16 = 2
16[5+95]
= 7 x 100S16 = 800
3. ÞòŸèEî‹ÞòŸèEî‹ÞòŸèEî‹ÞòŸèEî‹ÞòŸèEî‹1. c‚è™ º¬ø¬ò ðò¡ð´ˆFˆ b˜‚è 101x + 99y = 499, 99x + 101y = 501
101x + 99y = 499 --- (1)99x + 101y = 501 --- (2)
(1)+(2) 200x + 200y = 1000
÷ 200x + y = 5 --- (3)
(1)-(2) 2x - 2y = -2
÷ 2x - y = -1 ---- (4)
(3)+(4) 2x = 4x = 4/2 = 2
(1) ™ HóFJì2 + y = 5
y = 5 - 2y = 3
x = 2y = 3
2. è£óEŠð´ˆ¶è : x3 - 2x2 - 5x + 61 1 -2 -5 6
0 1 -1 -63 1 -1 -6 0 (x-1) è£óE
0 3 61 2 0 (x-3) è£óE
(x +2) è£óE(x - 1), (x - 3), (x+2) è£óEèœ Ý°‹. www.mathstimes.com
7 2
3. è£óEŠð´ˆ¶è 4x3 - 7x + 3
1 4 0 -7 3
0 4 4 -3
4 4 -3 0 (x - 1) è£óE
4x2 + 4x - 3 = (2x + 3) (2x -1) è£óEèœ
∴ (x-1), (2x-1), (2x+3) è£óEèœ.
4. è£óEŠð´ˆ¶è: x3 - 7x + 6
1 1 0 -7 6
0 1 1 -6
2 1 1 -6 0 (x-1) è£óE
0 2 6
1 3 0 (x-2) è£óE
(x+3) is a factor
∴ (x-1), (x-2), (x+3) are factors.
5. è£óEŠð´ˆ¶è: x3 - 3x2 - 10x + 24
2 1 -3 -10 24
0 2 -2 -24
-3 1 -1 -12 0 (x - 2) è£óE
0 -3 12
1 -4 0 (x+3) è£óE
(x-4) è£óE
(x-2), (x + 3), (x - 4) è£óEèœ(°PŠ¹ : å¼ è£óE è‡ìH¡, ã¬ùòõŸ¬ø è£óEŠð´ˆî ÞòôM™¬ôªòQ™ ÜŠð®«ò â¿î¾‹
ðJŸC 3.5™ àœ÷ IV, VIII ñŸÁ‹ XI Þ‰î õ¬è¬ò„ ꣼‹)
6. P = y+xx
, Q = yy+x âQ™ 22 QP
Q2QP
1
−−
− 裇è.
22 QP
Q2QP
1
−−
− = QP
1− − )QP()QP(
Q2−+
= )QP()QP(Q2QP−+
−+
= )QP()QP(QP
−+−
= QP
1+
=
yy
y
1
++
+ xxx
= yy
1
++
xx
= 1 www.mathstimes.com
7 3
7. õ˜‚è Íô‹ 裇è (x2 - 25) (x2 + 8x + 15) (x2 -2x -15)= (x + 5) (x - 5) (x + 3) (x + 5) (x - 5) (x + 3)= (x+5)2 (x - 5)2 (x + 3)2
õ˜‚è Íô‹ = |(x + 5) (x - 5) (x + 3)|8. õ˜‚è Íô‹ 裇è 9x4 + 12x3 + 10x2 + 4x + 1
3 2 13 9 12 10 4 1
96 2 12 10
12 46 4 1 6 4 1
6 4 10
õ˜‚è Íô‹ = |3x2 + 2x + 1|
9. õ˜‚è Íô‹ 裇è x4 - 10x3 + 37x2 - 60x + 36
1 -5 6
1 1 -10 37 -60 36
1
2 -5 -10 37
-10 25
2 -10 6 12 -60 36
12 -60 36
0
õ˜‚è Íô‹ = |x2 - 5x + 6|
10. õ˜‚è Íô‹ 裇è 4+25x2−12x−24x3+16x4
F†ì ܬñŠH™ â¿î
16x4 - 24x3 + 25x2 - 12x + 4
4 − 3 2
4 16 − 24 25 − 12 4
16
8 −3 −24 25
−24 9
8 −6 2 16 − 12 4
16 − 12 40
õ˜‚è Íô‹ = |4x2 − 3x + 2|
11. m - nx + 28x2 + 12x3 + 9x4 Ýù¶ å¼ º¿ õ˜‚è‹ âQ™ m , n 裇è.
Þøƒ° õK¬êJ™ â¿î
9x4 + 12x3 + 28x2 - nx + m
3 2 43 9 12 28 -n m
96 2 12 28
12 46 4 4 24 -n m
24 16 160
º¿ õ˜‚è‹ Ýîô£™, m = 16, n = -16 www.mathstimes.com
7 4
12. ax4 - bx3 + 40x2 + 24x + 36 å¼ º¿ õ˜‚è‹ âQ™ a ñŸÁ‹ b 裇è.
ãÁõK¬êJ™ â¿î
36 + 24x + 40x2 - bx3 + ax4
6 2 3
6 36 24 40 -b a
36
12 2 24 40
24 4
12 4 3 36 -b a
36 12 9
0
º¿ õ˜‚è‹ Ýîô£™ a = 9
b = -12
13. å¼ â‡ Üî¡ î¬ôWN ÝAòõŸP¡ Ã´î™ 551âQ™ ܉î â‡¬í‚ è£‡è.
b˜¾ :܉î ⇠= xî¬ôWN = 1/x
Ã´î™ = 5 51
x
x1+ =
526
x
x 12 +=
526
5(x2 +1) = 26x
5x2 + 5 - 26x = 0
5x2 - 26x + 5 = 0
(5x - 1) (x - 5)= 0
5x - 1 = 0 or x = 5
x = 1/5 or x = 5
x = {1/5, 5}
14. êñ¡ð£´ (1+m2) x2 + 2mcx+c2 - a2 = 0 ¡ Íôƒèœ êñ‹ âQ™ c2 = a2 (1+m2) âù GÁ¾è.
êñ¡ð£´ (1+m2) x2 + 2mcx+c2 - a2 = 0
a = 1 + m2, b = 2mc, c = c2 - a2
Íôƒèœ êñ‹ = b2 - 4AC = 0
(2mc)2 - 4 (1+m2) (c2 - a2) = 0
4m2c2 - 4 (c2 - a2 + m2c2 - m2a2) = 0
4m2 c2 - 4c2 + 4a2 - 4m2c2 + 4m2 a2 = 0
-4c2 = -4a2 - 4m2a2 = 0
÷ −−−−− 4
c2 = a2 + m2 a2
c2 = a2 (1+m2)
∴ c2 = a2 (1+m2) GÏH‚èŠð†ì¶
°PŠ¹: ðJŸC 3.5 ñŸÁ‹ 3.13 º¿õ¶‹ 𣘂辋. www.mathstimes.com
7 5
4. ÜEèœÜEèœÜEèœÜEèœÜEèœ
1.
21
53 ñŸÁ‹
−
−31
52 ÜEŠªð¼‚è¬ôŠ ªð£¼ˆ¶ å¡Á‚ªè£¡Á «ï˜ñ£Á ÜE âù GÁ¾è.
b˜¾:
21
53
−
−31
52=
+−−+−−
6522
151556
=
10
01 = I
−
−31
52
21
53=
+−+−
−−6533
101056
=
10
01 = I
«ï˜ñ£Á ÜEò£°‹.
2. A =
37
25 B =
−
−57
23 å¡Á‚ªè£¡Á ªð¼‚è™ «ï˜ñ£Á ÜE âù GÁ¾è.
b˜¾ :
AB =
37
25
−
−57
23
=
+−−+−−
15142121
10101415 =
10
01= I
BA =
−
−57
23
37
25
=
+−+−
−−15143535
661415=
10
01= I
«ï˜ñ£Á ÜEò£°‹
3. A =
04
23 ñŸÁ‹ B =
23
03 âQ™ AB , BA 裇. ܬõ êññ£è Þ¼‚°ñ£?
b˜¾:
AB =
04
23
23
03
=
++++
00012
4069 =
012
415
BA =
23
03
04
23
=
++++
0689
0609=
BA =
618
69
AB ≠ BA
www.mathstimes.com
7 6
4. A =
−
5
4
2
ñŸÁ‹ B = (1 3 -6) âQ™ (AB)T = BTAT GÁ¾è.
b˜¾:
AB =
−
5
4
2
(1 3 -6)
=
−−
−−
30155
24124
1262
(AB)T =
−−−−
302412
15126
542
----- (1)
BT =
− 6
3
1
AT = ( -2 4 5)
BT AT =
− 6
3
1
( -2 4 5)
=
−−−−
302412
15126
542
--- (2)
(AB)T = BT AT êK𣘂èŠð†ì¶.
5. A =
37
25 B =
−
−11
12 âQ™ (AB)T = BT AT êKð£˜.
b˜¾:
AB =
37
25
−
−11
12
=
+−−+−−
37314
25210
AB =
−−
411
38
(AB)T =
−− 43
118------ (1)
BT =
−
−11
12
www.mathstimes.com
7 7
AT =
32
75
BTAT =
−
−11
12
32
75
=
+−+−−−
3725
314210
BTAT =
−− 43
118----- (2)
(1) = (2)
(AB)T = BT AT êK𣘂èŠð†ì¶.
6. A =
−32
11âQ™ A2 - 4A + 5I2 = 0 GÁ¾è.
b˜¾
A2 = A x A =
−32
11
−32
11
=
+−+−−−
9262
3121
=
−−78
41
4A = 4
−32
11
=
−128
44
5I2 = 5
10
01
=
50
05
A2 - 4A = 5I2 =
−−78
41 -
−128
44 +
50
05
=
+−+−++−+−−
5127088
044541
=
00
00
A2 - 4A + 5I2 = 0 GÁõŠð†ì¶.
7. A =
− 41
23, B =
−76
52 C =
− 35
11 âQ™ A(B+C) = AB + AC â¡ð¬î êKð£˜.
b˜¾:
B + C =
−76
52 +
− 35
11 =
−101
61
www.mathstimes.com
7 8
A(B+C) =
− 41
23
−101
61
=
+−+++−
40641
201823
=
−345
381----- (1)
AB =
− 41
23
−76
52
=
+−+++−
285242
1415126
=
2326
296
AC =
− 41
23
− 35
11
=
+−−−+−
121201
63103
=
−−
1121
97
AB + AC =
2326
296 +
−−
1121
97
=
−345
381----- (2)
A (B+C) = AB + AC êK𣘂èŠð†ì¶
5. Ýòˆªî£¬ôõ®Mò™Ýòˆªî£¬ôõ®Mò™Ýòˆªî£¬ôõ®Mò™Ýòˆªî£¬ôõ®Mò™Ýòˆªî£¬ôõ®Mò™
1. A(2,-2) ñŸÁ‹ B(-7, 4) â¡ø ¹÷Oè¬÷ ެ킰‹ «è£†´ˆ¶‡¬ì Í¡Á êñð£èƒè÷£èŠ HK‚°‹¹œOè¬÷‚ 裇è.
P , Q â¡ðù «è£†´ˆ¶‡´ AB ä Í¡Á êñð£èƒè÷£è AP = PQ = QB â¡øõ£Á HK‚°‹¹œOèœ â¡è. Ýè«õ P â¡ð¶ AB ä 1 : 2 â¡ø MAîˆF½‹ Q â¡ð¶ AB¬ò 2 : 1 â¡øMAîˆF½‹ à†¹øñ£è HK‚A¡øù, HK¾ ňFóˆF¡ ð®
P =
+
−+×+
×+−×21
)2(2)41(,
21)22()7(1
=
−+−3
44,
347
= (-1, 0)
Ýè«õ ¹œO P â¡ð¶ (-1, 0) Ý°‹.
QPA B
www.mathstimes.com
7 9
«ñ½‹ Q =
+
−×+×+
×+−×12
)2(1)42(,
12)21()7(2
= (-4, 2)
Ýè«õ ¹œO Q â¡ð¶ (-4, 2) Ý°‹.
2. A (-4, 0) ñŸÁ‹ (0,6) â¡ø ¹œOè¬÷ ެ킰‹ «è£†´ˆ¶‡¬ì ° êñð£èƒè÷£èŠ HK‚°‹
¹œOè¬÷‚ 裇è.
P,Q,R â¡ðù AB â¡ø «è£†´ˆ¶‡¬ì ° êñð£èƒè÷£è HK‚°‹ ¹œOèœ â¡è.
Q â¡ð¶ AB ¡ ¬ñòŠ¹œO
âù«õ Q =
++−2
60,
204
= (-2, 3)
Pâ¡ð¶ AQ ¡ ¬ñòŠ¹œO
P =
+−−2
30,
224
P =
−23
,26
=
−23
,3
R â¡ð¶ QB ¡ ¬ñòŠ¹œO
R =
++−2
63,
202
=
−29
,1
Ýè«õ «î¬õò£ù ¹œOèœ P =
−23
,3 Q = (-2, 3) R =
−29
,1 .
3. (-5, 1) ñŸÁ‹ (2, 3) â¡ø ¹œOè¬÷ ެ킰‹ «è£†´ˆ¶‡®¬ù y Ü„² HK‚°‹ MA
ñŸÁ‹ HK‚°‹ ¹œO¬ò»‹ 裇è.
A (-5, 1) ñŸÁ‹ B(2, 3) â¡ðù ªè£´‚èŠð†ì ¹œOèœ
P (0, y) â¡ð¶ AB¬ò l : m â¡ø MAîˆF™ à†¹øñ£è HK‚Aø¶ â¡è.
P(0, y) = P
+
×+×+
−×+×m
)1m)3(,
m)5(m()2(
---- (1)
P(0, y) = P
++
+−
mm3
,mm52
---- (1)
x Ü„²ˆªî£¬ô¾è¬÷ êñŠð´ˆî
mm52
+−
= 0 2l - 5m = 0 m
= 25
âù«õ «î¬õò£ù MAî‹ l:m = 5 : 2 «ñ½‹ I L¼‰¶  ªðÁõ¶
P(0,y) = P
+
×+×25
)12()35(,0
QPA B
R
(2,3)(-5,1)
x
y
P(0,y)
www.mathstimes.com
8 0
= P
7
17,0
Ýè«õ, y Ü„² HK‚°‹ ¹œO
7
17,0 Ý°‹,
l : m = 5 : 2 ñŸÁ‹ P(0, y) = P (0, 17/4)4. å¼ º‚«è£íˆF¡ º¬ùèœ (1, -1) , (0, 4) ñŸÁ‹ (-5, 3) âQ™ Ü‹º‚«è£íˆF¡ ï´‚«è£´èO¡
c÷ƒè¬÷‚ èí‚A쾋.b˜¾:
A (1, -1), B (0, 4) C (-5, 3) â¡ðù º‚«è£íˆF¡ à„Cèœ D, E, F â¡ðù º¬ø«ò BC, AC,AB¡ ¬ñòŠ¹œOèœ â¡è.
âù«õ BC¡ ¬ñòŠ¹œO D =
+−2
34,
250
= D
−27
,25
AC¡ ¬ñòŠ¹œO E =
+−−2
31,
251
= E (-2, 1)
AB¡ ¬ñòŠ¹œO F =
+−+2
41,
201
= F
23
,21
Ýè«õ ï´‚«è£´ AD¡ c÷‹ AD = 22
27
125
1
−−+
+
= 22
29
27
−+
=
481
449 + =
4130
ï´‚«è£´ BE¡ c÷‹ = 22 )41()02( −+− = 94 + = = 13
ï´‚«è£´ CF¡ c÷‹ = 22
323
521
−+
+ = 22
23
211
−+
=
49
4121+ =
4130
Ýè«õ ΔABC¡ ï´‚«è£´èO¡ c÷ƒèœ 2
130, 13 ,
2130
Ý°‹.
5. (6, 9), (7, 4), (4,2) ñŸÁ‹ (3,7) ÝAòõŸ¬ø º¬ùè÷£è‚ ªè£‡ì èóƒèO¡ ðóŠð÷¾è¬÷‚裇è.
ªè£´‚èŠð†ì ¹œOè¬÷ è®è£ó º¡«ù£†ì F¬ê‚° âF˜F¬êJ™ ܬñ»ñ£Á õK¬êò£è ðìˆF™°P‚辋A(4,2), B(7,4), C (6, 9), D(3,7).èó‹ ABCD¡ ðóŠ¹
=
27942
43674
21
= ( ) ( )[ ]28272414642631621 +++−+++
= [ ]9312721 −
= 3421×
= 17Ýè«õ èó‹ ABCD ¡ ðóŠ¹ 17ê.Üô°èœ
y
x
C(6,9)
B(7,4)
D(3,7)
A(4,2)
www.mathstimes.com
8 1
6. (-4, 5) (0,7) (5,-5) ñŸÁ‹ (-4,-2) º¬ùè÷£è‚ ªè£‡ì èóƒèO¡ ðóŠð÷¾ 裇.
ªè£´‚èŠð†ì ¹œOè¬÷ è®è£ó ºœ«÷£†ì F¬ê‚° âF˜F¬êJ™ ܬñ»ñ£Á õK¬êò£è ðìˆF™
°P‚辋.
A(-4, 5) B(0,7) C(5,-5) D(-4,-2) â¡è. èó‹ ABCD ¡ ðóŠ¹.
=
−−−−−−
25752
44054
21
= 21
[(20+35+0+8) - (-10 + 0 - 28 - 20)]
= 21
[63 + 58]
= 21
x 121
= 60.5Ýè«õ, èó‹ ABCD¡ ðóŠ¹ 60.5 ê.Üô°èœ.
7. (2, -5) (3, -4) ñŸÁ‹ (9, k) å¼ «è£ì¬ñ âQ™ k ¡ ñFŠ¬ð‚ 裇.
A(2, -5) B(3, -4) C (9, k) â¡ðù ªè£´‚èŠð†ì ¹œOèœ â¡è.
Í¡Á ¹œOèÀ‹ å«ó «ï˜‚«è£†®™ ܬñõ ΔABC ¡ ðóŠ¹ Ì„Cò‹ Ý°‹.
Δ = 21
−−− 5k45
2932 = 0
Δ = 21
[(-8 + 3k - 45) - (-15 - 36 + 2k)] = 0
Δ = 21
[-53 + 3k + 51 - 2k] = 0
Δ = -2 + k = 0k - 2 = 0
k = 2
Ýè«õ k ¡ ñFŠ¹ 2.
8. å¼ º‚«è£íˆF¡ º¬ùèœ (0,- 1), (2,1) ñŸø‹ (0,3) âQ™ Þî¡ ð‚èƒèO¡ ï´Š¹œOè¬÷ެ툶 à¼õ£‚°‹ º‚«è£íˆF¡ ðóŠ¹è£‡. «ñ½‹ Þ„CPò º‚«è£íˆF¡ ðóŠð÷MŸ°‹ªè£´‚èŠð†ì º‚«è£íˆF¡ ðóŠð÷MŸ° àœ÷ MAîˆ¬î‚ è£‡.
A (0,- 1) B(2,1) ñŸÁ‹ C(0,3) â¡ðù º‚«è£íˆF¡ à„Cèœ «ñ½‹ D, E, F â¡ðù º¬ø«ò BC,
CA ñŸÁ‹ AB ¡ ï´Š¹œOèœ â¡è.
BC ¡ ï´Š¹œO D =
++2
31,
202
= D (1, 2)
AC ¡ ï´Š¹œO E =
−+2
13,
200
= E (0, 1)
AB ¡ ï´Š¹œO F =
+−+2
11,
220
= F (1, 0)
ΔDEF¡ ðóŠ¹ =
2012
1101
21
= ( ) ( )[ ]01020121 ++−++ = 1 ê.Üô°èœ
D(-4,5) D(0,7)
D(5,-5)A(-4,-2)
O
y
x
www.mathstimes.com
8 2
ΔABC¡ ðóŠ¹ =
− 1131
2002
21
= ( ) ( )[ ]20000621 −+−++
= 4 ê.Üô°èœÝè«õ ΔDEF ñŸÁ‹ ΔABC¡ ðóŠ¹èO¡ MAî‹ 1 : 4 Ý°‹.
9. ΔABC ¡ º¬ùèœ A(1, 8) B (-2, 4), C (8, -5). «ñ½‹ M , N â¡ðù º¬ø«ò AB , AC ÞõŸP¡ï´Š¹œOèœ âQ™ MN ¡ ꣌¬õ‚ 裇. ެ裇´ MN ñŸÁ‹ BC ÝAò «ï˜‚«è£´èœÞ¬í âù‚ 裆´è.
A(1, 8) B (-2, 4), C (8, -5) â¡ðù º‚«è£íˆF¡ º¬ùèœ
AB ¡ ï´Š¹œO M =
+−2
48,
221
=
−6,
21
AC¡ ï´Š¹œO N =
+−+2
85,
281
N =
23
,29
MN ¡ ꣌¾ M1 =
21
29
623
+
−
=
2102123 −
= 10
9− ----- I
BC ¡ ꣌¾ M2
= 2845
+−−
= 10
9−--- II
I ñŸÁ‹ II L¼‰¶ M1 = M
2
Ýè«õ, «ï˜‚«è£´èœ BC ñŸÁ‹ MN ÝAò¬õ Þ¬íò£°‹.
10. (6 , 7), (2 , -9) ñŸÁ‹ (-4 , 1) ÝAòù å¼ º‚«è£íˆF¡ º¬ùèœ âQ™ º‚«è£íˆF¡ ï´‚«è£´èO¡
꣌¾è¬÷‚ 裇è.
A(6 , 7), B(2 , -9) ñŸÁ‹ C(-4 , 1) ÝAò¬õ º‚«è£íˆF¡ º¬ùèœ
«ñ½‹ D, E,F â¡ðù º¬ø«ò BC, CA ñŸÁ‹ AB ¡ ï´Š¹œOèœ â¡è. Ýè«õ AD, BE ñŸÁ‹ CF
ÝAò¬õ ΔABC ï´‚«è£´è÷£°‹.
BC ¡ ï´Š¹œO D =
+−−2
19,
242
= (-1, -4)
CA ¡ ï´Š¹œO E
++−2
71,
264
E = (1, 4)
AB ¡ ï´Š¹œO F
−+2
97,
226
F = (4, -1)
âù«õ AD¡ ꣌¾ = 6174
−−−−
= 711
−−
= 711
A(1,8)
N
C (8,-5)B (-2,4)
M
C(-4,1)
E
A(6,7) C(2,-9)
D
D
www.mathstimes.com
8 3
BE¡ ꣌¾ = 2194
−+
= 1
13−
= -13
CF¡ ꣌¾ = 4411
+−−
= 82−
= 41−
Ýè«õ, ï´‚«è£´èO¡ ꣌¾èœ 711
, -13, 41−
.
11. A (-2, 3), B (a, 5) ÝAò ¹œOè¬÷ ެ킰‹ «ï˜‚«è£´ ñŸÁ‹ C (0, 5) D (-2, 1) ÝAò ¹œOè¬÷ެ킰‹ «ï˜‚«è£´ ÝAòù Þ¬í«è£´èœ âQ™ ‘a’ ¡ ñFŠ¹ 裇.
«ï˜‚«è£´èœ AB ñŸÁ‹ CD Þ¬í â¡ð Üî¡ ê£Œ¾èœ êñ‹.Ýè«õ AB ¡ ꣌¾ = CD ¡ ꣌¾
AB¡ ꣌¾ = 2a
22a35
+=
+−
CD ¡ ꣌¾ = 24
0251
−−=
−−−
= 2
AB¡ ꣌¾ = CD ¡ ꣌¾
2a2+ = 2
a + 2 = 1a = 1 - 2a = -1
Ýè«õ, a ¡ ñFŠ¹ -112. (2, 2) â¡ø ¹œO õN„ ªê™õ¶‹, ªõ†´ˆ¶‡´èO¡ Ã´î™ 9 Ý辋 ªè£‡ì «ï˜‚«è£´èO¡
êñ¡ð£´è¬÷‚ 裇.b˜¾:a, b â¡ðù º¬ø«ò «ï˜‚«è£†®¡ x ñŸÁ‹ y ¡ ªõ†´ˆ¶‡´èœ â¡è.Ýè«õ a + b = 9 Ü™ô¶ b = 9 - a
ªõ†´ˆ¶‡´ ܬñŠHô£ù «ï˜‚«è£†®¡ êñ¡ð£†®¡ ð® by
a+x
= 1 ---- I
Þ‚«è£´ (2, 2) â¡ø ¹œO õN„ªê™õ  ªðÁð¶ a9
2a2
−+ = 1
a2 - 9a + 18 = 0
(a - 6) (a - 3) = 0 a = 6 Ü™ô¶ a = 3
a = 3 â‹ «ð£¶ 6y
3+x
= 1 2x + y - 6 = 0
a = 6 â‹ «ð£¶ 3y
6+x
=1 x + 2y - 6 = 0
13. A(-2, 6) B (3, -4) ÝAò ¹œOè¬÷ ެ킰‹ «ï˜‚«è£†´ˆ¶‡¬ì P â¡ø ¹œO à†¹øñ£è 2:3
â¡ø MAîˆF™ HK‚A¡ø¶. ¹œO P õNò£è„ ªê™½‹ ꣌¾ 23 à¬ìò «ï˜‚«è£†®¡ êñ¡ð£†¬ì‚
裇.ABä à†¹øñ£è 2 : 3 â¡ø MAîˆF™ HK‚°‹ ¹œO P â¡è.
Ýè«õ ¹œO P
++−
+−+
32)6(3)4(2
,32
)2(3)3(2
= (2, 0) www.mathstimes.com
8 4
꣌¾ 23
Ý辋 (0,2) â¡ø ¹œO õNò£è¾‹ ªê™½‹ «ï˜‚«è£†®¡ êñ¡ð£´
y - 2 = 23
(x - 0)
2y - 4 = 3x3x - 2y + 4 = 0
14. 3x - y + 9 = 0, x + 2y = 4 ÝAò «ï˜‚«è£´èœ ªõ†´‹ ¹œO»ì¡, 2x + y - 4 = 0 , x - 2y + 3 = 0 ÝAò«ï˜‚«è£´èœ ªõ†´‹ ¹œO¬ò ެ킰‹ «ï˜‚«è£†®¡ êñ¡ð£†¬ì‚ 裇.
3x - y = -9 ----- (I) x + 2y = 4 ---- (II)
2x + y = 4 ---- (III) x - 2y = -3 --- (IV)
(I x 2) + II3x - y = -9x + 2y = 4
(I x 2) + II6x - 2y = -18 Put x = -2 we get x - 2y = -4 -2 +2y = 4 7x = -14 2y = 4 + 2
x = -2 y = 6/2 y = 3ªõ†´‹ ¹œO (-2, 3)III, IVä b˜‚è 2x + y = 4 ---- (III)
x - 2y = -3 ---- (IV)
(III x 2) + IV 4x + 2y = 8 Put x = 1
x - 2y = -3 1 - 2y = -35x = 5 x = 1 -2y = -4
y = 2ªõ†´‹ ¹œO (1, 2)(-2, 3) ñŸÁ‹ (1, 2) ä ެ킰‹ «ï˜‚«è£†®¡ êñ¡ð£´
212
323y
++=
−− x
32
13y +=
−− x
x + 3y - 7 = 0
M =
+−−2
82,
253
= (-1, 3)15. ΔABC ¡ º¬ùèœ A(2, -4), B(3, 3) ñŸÁ‹ C(-1, 5) âQ™ BL¼‰¶ õ¬óòŠð´‹ °ˆ¶‚«è£†´ õN„
ªê™½‹ «ï˜‚«è£†®¡ êñ¡ð£†¬ì‚ 裇.
3x-y
+9 =
0
x+2y
= 4
2x+y
-4 =
0
x-2y
+3 =
0
A(2,-4)
B(3,3) C(-1,5)
D
www.mathstimes.com
8 5
A(2, -4), B(3, 3) ñŸÁ‹ C(-1, 5) â¡ðù º‚«è£íˆF¡ à„Cèœ BD â¡ð¶ º‚«è£íˆF¡ à„CBJL¼‰¶ õ¬óòŠð´‹ °ˆ¶‚«è£´ â¡è.
AC ¡ ꣌¾ 2145
−−+
= 3
9− = -3
Ýè«õ, °ˆ¶‚«è£´ BD ¡ ꣌¾ 31
(AC ⊥ BD)
꣌¾ 31Ý辋 (3, 3) â¡ø ¹œO õN„ªê™õ¶ñ£ù «ï˜‚«è£†®¡ êñ¡ð£´.
y - 3 = 31
(x - 3)
3y - 9 = x - 3 x - 3y + 6 = 0
16. ΔABC ¡ º¬ùèœ A (-4, 4) B(8,4) C (8,10) âQ™ A JL¼‰¶ õ¬óòŠð´‹ ï´‚«è£†´ õN„ ªê™½‹«ï˜‚«è£†®¡ êñ¡ð£†¬ì‚ 裇.
º¬ù AJL¼‰¶ õ¬óòŠð´‹ ï´‚«è£´ AD â¡è.
BC ¡ ¬ñòŠ¹œO D
++2104
,2
88
D = (8, 7)
A (-4, 4) D (8,7) ÝAò ¹œOè¬÷ ެ킰‹ ï´‚«è£´ AD ¡ êñ¡ð£´
484
474y
++=
−− x
4y - 16 = x + 4
x - 4y + 20 = 0
6. õ®Mò™õ®Mò™õ®Mò™õ®Mò™õ®Mò™
(50 ñFŠªð‡èÀ‚° «ñ™ ªðø M¼‹¹ðõ˜èÀ‚°50 ñFŠªð‡èÀ‚° «ñ™ ªðø M¼‹¹ðõ˜èÀ‚°50 ñFŠªð‡èÀ‚° «ñ™ ªðø M¼‹¹ðõ˜èÀ‚°50 ñFŠªð‡èÀ‚° «ñ™ ªðø M¼‹¹ðõ˜èÀ‚°50 ñFŠªð‡èÀ‚° «ñ™ ªðø M¼‹¹ðõ˜èÀ‚°)1. å¼ õ†ìˆF¡ ¹œO A ™ õ¬óòŠð´‹ ªî£´«è£´ PQ â¡è. AB â¡ð¶ õ†ìˆF¡  â¡è.
«ñ½‹ ∠ BAC = 54o ñŸÁ‹ ∠ BAQ = 62o â¡Á ܬñ»ñ£Á õ†ìˆF¡ «ñ™ àœ÷ ¹œO C âQ™
∠ ABC 裇è,
PQ å¼ ªî£´«è£´ AB .
∠ BAQ = ∠ ACB = 62o («îŸø‹)
∠ BAC = ∠ ABC + ∠ ACB = 180o
(º‚«è£íˆF¡ Í¡Á «è£íƒèœ)
54 + ∠ ABC+62o = 180o
∠ ABC +- 116 = 180
∠ ABC = 180 - 116 = 64o
∠ ABC = 64o
2. ðìˆF™ TP å¼ ªî£´«è£´, A , Bõ†ìˆF¡ e¶œ÷ ¹œOèœ ∠ BTP = 72o ñŸÁ‹ ∠ ATB = 43o âQ™
∠ ABT 裇è. (Ap. 13)
TP å¼ ªî£´«è£´
TB 
∠ BTP = ∠ BAT = 72o («îŸø‹)
∠ BTP + ∠ ABT + ∠ BAT = 180o
(º‚«è£íˆF¡ Í¡Á «è£íƒèœ)
43o + ∠ ABT + 72o = 180o
C(8,10)
A(-4,4) B(-4,4)
D
T P
72o43o
?
AB
www.mathstimes.com
8 6
∠ ABT + 115 = 180
∠ ABT = 180 - 115
= 65o
∠ ABT = 65o
3. E ñŸÁ‹ F â¡ø ¹œOèœ º¬ø«ò ΔPQR ¡ ð‚èƒèœ PQ ñŸÁ‹ PR ÝAòõŸP¡ e¶ ܬñ‰¶œ÷ù.EF||QR â¡ð¬î„ êKð£˜. PE = 3.9 ªê.e, EQ = 3 ªê.e, PF = 3.6 ªê.e ñŸÁ‹ FR = 2.4 ªê.e
FRPF
EQPE =
4.26.3
39.3 =
3.9 x 2.4 = 3 x 3.6
9.36 ≠ 10.8
âù«õ EF || QR (Þ¬í Ü™ô)
4. E ñŸÁ‹ F â¡ø ¹œOèœ º¬ø«ò ΔPQR ¡ ð‚èƒèœ PQ ñŸÁ‹ PR ÝAòõŸP¡ e¶ ܬñ‰¶œ÷ù.EF || QR â¡ð¬î„ êKð£˜. PE = 4 ªê.e, QE = 4.5 ªê.e, PF = 8 ªê.e ‹Á‹ RF = 9 ªê.e
(ºòŸC ªêŒè)
5. AD â¡ð¶ ΔABC J™ ∠ A ¡ «è£í Þ¼êñ ªõ†® Ý°ñ£ âù„ «ê£F‚è, AB = 4 ªê.e, AC = 6 ªê.e
BD = 1.6 ªê.e ñŸÁ‹ CD = 2.4 ªê.e
ACAB
DCBD =
64
4.26.1 =
1.6 x 6 = 2.4 x 4
9.6 = 9.6
âù«õ AD â¡ð¶ ∠ A ¡ «è£í Þ¼êñªõ†®
6. AD â¡ð¶ ΔABC J™ ∠ A ¡ «è£í Þ¼êñ ªõ†® Ý°ñ£ âù„ «ê£F‚è. AB = 6 ªê.e, AC = 8 ªê.e,
BD = 1.5 ªê.e ñŸÁ‹ CD = 3 ªê.e (ºòŸC ªêŒè)
1. ð®‚è «õ‡®ò «îŸøƒèœ
1. Ü®Šð¬ì MAî îêñ «îŸø‹ (Ü™ô¶) «î™v «îŸø‹ (Oct.14, Ap. 14, Ju. 13)
2. «è£í Þ¼êñªõ†® «îŸø‹ (Oct. 13, Ap. 12)
3. Hî£èóv «îŸø‹. (Ap. 13, Ju. 12)
2. å¼ CÁõ¡ ¬õóˆF¡ °Á‚° ªõ†´ «îŸøõ®M™ ðìˆF™ 裆®òõ£Á å¼ ð†ì‹ ªêŒî£¡. Þƒ°AE = 16 ªê.e, EC = 81ªê.e. Üõ¡ BD â¡Ÿ °Á‚°‚ °„CJ¬ùŠ ðò¡ð´ˆî M¼‹¹Aø£¡. °„CJ¡
c÷‹ âšõ÷õ£è Þ¼‚è «õ‡´‹. (Govt. Model Question)
ΔEAD ~ ΔEDC
âù«õ EDEA
= ECED
ED2 = EA x EC ED2 = 16 x 81
ED = 8116×
= 4 x 9 ED = 36 ED = BE = 36 cm
P
F
RQ
E
3
3.9 3.6
2.4
C
B
A
D
E
81
16
A
6
CB
4
D1.6 2.4
www.mathstimes.com
8 7
âù«õ BD = 36 + 36 = 72
BD = 72 ªê.e
°„CJ¡ c÷‹ = 72 ªê.e
3. å¼ î£ñ¬óŠ Ìõ£ù¶ î‡a˜ ñ†ìˆFŸ° «ñ™ 20 ªê.e àòóˆF™ àœ÷¶. ¡ eFŠð°F î‡a˜ñ†ìˆFŸ° W«ö àœ÷¶. 裟Á i²‹ «ð£¶  îœ÷Šð†´ î£ñ¬óŠ Ìõ£ù¶ ¡ Ýó‹ðG¬ôJL¼‰¶ 40 ªê.e ÉóˆF™ î‡a¬óˆ ªî£´Aø¶. Ýó‹ð G¬ôJ™ î‡a˜ ñ†ìˆFŸ° W«öàœ÷ ¡ c÷‹ 裇è.
b˜¾:
BD = î‡aK¡ W«ö ¡ c÷‹ = x ªê.e â¡è.
ΔBCD ™ Hî£èóv «îŸøˆF¡ð®
BC2 = AB2 + AC2
(x + 20)2 = x2 + 402
x2 + 40x + 400 = x2 + 1600
x2 + 40x + 400 = x2 + 1600
40x = 1600 - 400
40 = 1200
x = 40
1200 = 30
x = 30 cm
î‡a˜ ñ†ìˆFŸ° W«ö àœ÷ ¡ c÷‹ = 30 ªê.e
4. ªêšõè‹ ABCD¡ à†¹ø ¹œO O ML¼‰¶ ªêšõèˆF¡ º¬ùèœ A, B, C, D Þ¬í‚èŠð†´œ÷ù
âQ™ OA2 + OC2 + OB2 + OD2 âù GÁ¾è. (Ju. 14)
5. å¼ Þ¬íèóˆF¡ â™ô£Š ð‚èƒèÀ‹ å¼ õ†ìˆF¬ù ªî£´ñ£ù£™ ÜšM¬íèó‹ å¼ ê£Œê¶óñ£°‹
âù GÁ¾è. (March 15)
6. ABCD â¡ø èó‹, Üî¡ â™ô£ ð‚èƒèÀ‹ å¼ õ†ìˆ¬î ªî£´ñ£Á ܬñ‰¶œ÷¶. AB = 6 ªê.e,
BC = 6.5 ªê.e ñŸÁ‹ CD = 7ªê.e âQ™ AD¡ c÷ˆ¬î‚ 裇è.
7. å¼ GöŸðì‚ è¼MJ½œ÷ ðì„ ²¼O™ å¼ ñóˆF¡ H‹ðˆF¡ c÷‹ 35 I.e, ªô¡²‚°‹ ð섲¼À‚°‹Þ¬ìŠð†ì Éó‹ 42 I.e. «ñ½‹, ªô¡RL¼‰¶ ñ󈶂° àœ÷ Éó‹ 6 e âQ™, GöŸðì‹â´‚èŠð´‹ ñóˆF¡ ð°FJ¡ c÷‹ 裇è. (Score Model Question)
7. º‚«è£íMò™º‚«è£íMò™º‚«è£íMò™º‚«è£íMò™º‚«è£íMò™
1. «ï˜‚°ˆî£ù å¼ ñóˆF¡ «ñ™ð£è‹ 裟Pù£™ ºPŸ¶, Ü‹ ºP‰î ð°F W«ö M¿‰¶ Mì£ñ™ñóˆF¡ à„C î¬ó»ì¡ 30o «è£íˆ¬î ãŸð´ˆ¶Aø¶. ñóˆF¡ à„C Üî¡ Ü®JL¼‰¶ 30 eªî£¬ôM™ î¬ó¬òˆ ªî£´Aø¶ âQ™ ñóˆF¡ º¿ àòóˆ¬î‚ 裇è.
ñóˆF¡ àòó‹ x e + ye â¡è (Oct-12, July-14)âF˜ð‚è‹
ΔABC ™, tanθ = ---------------------Ü´ˆ¶œ÷ ð‚è‹
tan 30o = 30x
3
1=
30x
3 x = 1 x 30
x = 3
30
A
40cm C
B
x cm
A
20cm
x+20
30m
A30o
C
B30m
x
y
www.mathstimes.com
8 8
= 3.3
330 =
3330
= 10 3 e
x = 10 3 e
Ü´ˆ¶œ÷ ð‚è‹ Cosθ = ----------------------------
è˜í‹
cos30o = y30
23
= y30
3 y= 2 x 30
y = 3.3
3302 ×× =
33302 ××
= 20 3
y = 20 3 e
ñóˆF¡ àòó‹ = x + y
= 10 3 + 20 3 e
ñóˆF¡ àòó‹ = 30 3 e
2. æ˜ ÜF«õèŠ «ð£˜ Mñ£ù‹ î¬ó ñ†ìˆFL¼‰¶ 3000 e àòóˆF™ ñŸªø£¼ ÜF«õèŠ «ð£˜Mñ£ùˆ¬î «ï˜ «ñô£è‚ èì‚Aø¶. Üšõ£Á èì‚°‹ «ð£¶ î¬óñ†ìˆF™ å¼ °PŠH†ì ¹œOJL¼‰¶Üî¡ ãŸø‚«è£íƒèœ º¬ø«ò 60o ñŸÁ‹ 45o âQ™ Ü‰î «ïóˆF™ 2õ¶ «ð£˜ Mñ£ù‹ ñŸÁ‹ ºî™
«ð£˜ Mñ£ù‹, ÝAòõŸPŸ° Þ¬ìŠð†ì Éó‹ èí‚A´è. ( 3 =1.732) (Ju. 13)
Þ¬ìŠð†ì Éó‹ = h e â¡èΔOAC ™
âF˜ð‚è‹tanθ = ------------------------
Ü´ˆ¶œ÷ ð‚è‹
tan 60o = OC
3000
3 = OC
3000
3 OC = 3000
OC = 3
3000
= 33
33000
×
× =
333000
= 1000 3
OC = 1000 3 e
âF˜ ð‚è‹
ΔOCB ™ tanθ = ------------------------Ü´ˆ¶œ÷ ð‚è‹
tan45o = OC
h3000 −
30
00 m
www.mathstimes.com
8 9
1 = 31000
3000 h−
1 x 1000 3 = 3000 - h h = 3000 - 1000 3
= 3000 - 1000 x 1.732
= 3000 - 1732
h = 1268 e
Þ¬ìŠð†ì Éó‹ = 1268e
3. 500 e àòóˆF™ ðø‰¶ ªè£‡®¼‚°‹ ªýL裊ìK™ å¼õ˜ æ˜ ÝŸP™ Þ¼ è¬óèO™ «ïªóFó£è
àœ÷ Þ¼ ªð£¼†è¬÷ 30o , 45o Þø‚è‚ «è£íƒèO™ 裇Aø£˜ âQ™ ÝŸP¡ Üèô‹ 裇è.
( 3 = 1.732 ) (Ap. 14)
ÝŸP¡ Üèô‹ = x e + y e
ΔABC J™
âF˜ð£‚è‹
tanθ = ------------------------- Ü´ˆ¶œ÷ ð‚è‹
tan 30o = x
500
3
1 =
x500
x = 500 3
x = 500 x 1.732= 866.00
x = 866 e
âF˜ð£‚è‹
In ΔACD, tanθ = ------------------------- Ü´ˆ¶œ÷ ð‚è‹
tan45o = y500
1 = y500
1 x y = 500
y = 500 e
ÝŸP¡ Üèô‹ = x e + y e
= 866 + 500
= 1366 e4. 700 e àòóˆF™ ðø‰¶ ªè£‡®¼‚°‹ ªýL裊ìK™ å¼õ˜ æ˜ ÝŸP™ Þ¼ è¬óèO™ «ïªóFó£è
àœ÷ Þ¼ ªð£¼†è¬÷ 30o , 45o Þø‚è‚ «è£íƒèO™ 裇Aø£˜ âQ™ ÝŸP¡ Üèô‹ 裇è.
( 3 = 1.732 )
Mù£ ⇠3äŠ «ð£ô«õ b˜¾ è£íô£‹
60 ñFŠªð‡èÀ‚° «ñ™ ªðø M¼‹¹‹ ñ£íõ˜èœ ðJŸC ªêŒò «õ‡®ò Cô èí‚°èœ1. ð‚è‹ 210, â.è£. 7.6 (Ap.13)2. ð‚è‹ 213, â.è£. 7.12 (Oct. 13, Ap. 15)3. ð‚è‹ 214, ðJŸC 7.1™ 5 (Score model question III)
B DC
45o30o
A
30o 45o
500m
xm ym
www.mathstimes.com
9 0
4. ð‚è‹ 220, â.è£. 7.20 (Oct. 14)5. ð‚è‹ 221, â.è£. 7.22 (Score model question V)6. ð‚è‹ 226 ðJŸC 7.2™ 10 (10) (Ap. 13)7. ð‚è‹ 226 ðJŸC 7.2™ 12 (Ap. 12)8. ð‚è‹ 227, ðJŸC 7.2™ 17 (Ju. 12)9. ð‚è‹ 227, ðJŸC 7.2™ 16 (Score model question IV)
8. Ü÷Mò™Ü÷Mò™Ü÷Mò™Ü÷Mò™Ü÷Mò™
1. 120 ªê.e c÷º‹, 84 ªê.e M†ìº‹ ªè£‡ì å¼ ê£¬ô¬ò êñŠð´ˆ¶‹ ༬÷¬ò‚ ªè£‡´ å¼M¬÷ò£†´ˆ Fì™ êñŠð´ˆîŠð´Aø¶. M¬÷ò£†´ˆ Fì¬ô êñŠð´ˆî Þš¾¼¬÷ 500 º¿„²ŸÁ‚蜲öô «õ‡´‹. M¬÷ò£†´ˆFì¬ô êñŠð´ˆî å¼ ê.e†ì¼‚° 75 ¬ðê£ iî‹, Fì¬ô„ êñŠð´ˆî
Ý°‹ ªêô¬õ‚ 裇è. ( π = 722
)
b˜¾:Þƒ° 2r = 84 ªê.e r = 42 ªê.e, h = 120 ªê.e༬÷J¡ å¼ º¿„²ŸPù£™ êñŠð´ˆîŠð´‹ FìL¡ ðóŠ¹ = 2πrh
= 2 x722
x 42 x 120
= 31680 ªê.e2
500 º¿„²ŸÁèO™ êñŠð´ˆîŠð´‹ FìL¡ ðóŠ¹ = 31680 x 500= 15840000 ªê.e2
= 10000
15840000 e2
= 1584 e2
1 ê.e†ì¼‚° êñŠð´ˆî Ý°‹ ªêô¾ = 75 ¬ðê£âù«õ M¬÷ò£†´ˆ Fìì¬ô êñŠð´ˆî Ý°‹ ªñ£ˆî ªêô¾ = 1584 x 0.75
= Ï. 11882. å¼ F‡ñ «ï˜õ†ì ༬÷J¡ ªñ£ˆîŠ¹øŠðóŠ¹ 660ê.ªê.e Üî¡ M†ì‹ 14 ªê.e âQ™, Üš¾¼¬J¡
àòóˆ¬î»‹, õ¬÷ðóŠ¬ð»‹ 裇è.b˜¾:
Þƒ° 2r = 14 ªê.e r = 7 ªê.e
ªñ£ˆîŠ¹øŠðóŠ¹ = 660 ê.ªê.e
2πr (h+r) = 660
2 x 722
x 7 (h + 7) = 660
h = 222
660×
- 7
= 15 - 7
= 8 ªê.e
༬÷J¡ õ¬÷ðóŠ¹ = 2πrh = 2 x 722
x 7 x 8 = 352 ªê.e2
3. å¼ F‡ñ‚ ËH¡ Ýó‹ ñŸÁ‹ ꣻòó‹ º¬ø«ò 20 ªê.e ñŸÁ‹ 29 ªê.e âQ™ ܈F‡ñ‚ ËH¡èù Ü÷¬õ‚ 裇è.b˜¾:
Þƒ° r = 20 ªê.e , l = 29 ªê.e
h = 22 r− www.mathstimes.com
9 1
= 22 2029 −
= 400841−
= 441 h = 21 ªê.e
ËH¡ èùÜ÷¾ = 31
πr2h
= 31
x 722
x 20 x 20 x 21
= 8800 ªê.e3.4. å¼ «ï˜õ†ì‚ ËH¡ Þ¬ì‚è‡ìˆF¡ Þ¼¹øº‹ ܬñ‰î õ†ì MO‹¹èO¡ ²Ÿø÷¾èœ º¬ø«ò 44
ªê.e ñŸÁ‹ 8.4π ªê.e â¡è. Üî¡ àòó‹ 14 ªê.e âQ™ ÜšM¬ì‚è‡ìˆF¡ èù Ü÷¬õ‚ 裇è.b˜¾:
Þƒ° 2πR = 44 ªê.e ñŸÁ‹ 2πr = 8.4 π ªê.e
2 x 722
x R = 44 2r = 8.4
R = 222744
××
r = 4.2 ªê.e
R = 7 ªê.e
Þ¬ì‚è‡ìˆF¡ èùÜ÷¾ = 31
πh (R2 + r2 + Rr)
= 14722
31 ×× (72 + 4.22 + 7 x 4.2)
= 344
(49 + 29.4 +17.64)
= 344
x 96.04
= 1408.58 ªê.e3
5. å¼ F‡ñ ñóŠªð£‹¬ñò£ù¶ ܬó‚«è£÷ˆF¡ «ñ™ ˹ Þ¬í‰î õ®M™ àœ÷¶. ܬó‚«è£÷‹ñŸÁ‹ ˹ ÝAòõŸP¡ Ýó‹ 3.5 ªê.e «ñ½‹ ªð£‹¬ñJ¡ ªñ£ˆî àòó‹ 17.5 ªê.e âQ™
ÜŠªð£‹¬ñ îò£K‚èŠðò¡ð´ˆîŠð†ì ñóˆF¡ èùÜ÷¬õ 裇è. ( π = 722
)
b˜¾:Þƒ°Ü¬ó‚«è£÷‹ ˹
r = 3.5 ªê.e h = 3. 5ªê.eh = 17.5 - 3.5
èù Ü÷¾ = 32
πr3 = 14 ªê.e
= 32
π x 3.5 x 3.5 x 3.5 èù Ü÷¾ = 31
πr2h
= 31
π x 3.5 x 3.5 x 14
ñóˆF¡ èùÜ÷¾ = ܬó‚«è£÷ˆF¡ èùÜ÷¾ + ËH¡ èùÜ÷¾
= 32
π x 3.5 x 3.5 x 3.5 + 31
π x 3.5 x 3.5 x 14www.mathstimes.com
9 2
= 31
π x 3.5 x 3.5 [2 x 3.5 + 14]
= 31
x 722
x 3.5 x 3.5 [7+14]
= 31
x 722
x 3.5 x 3.5 x 21
= 22 x 3.5 x 3.5= 296.5 è.ªê.e
6. å¼ Ãì£óñ£ù¶ ༬÷J¡ e¶ ˹ Þ¬íˆî õ®M™ àœ÷¶. Ãì£óˆF¡ ªñ£ˆî àòó‹ 13.5 eñŸÁ‹ M†ì‹ 28e. «ñ½‹ ༬÷Š ð£èˆF¡ àòó‹ 3 e âQ™ Ãì£óˆF¡ ªñ£ˆî ¹øŠðóŠ¬ð‚裇è.
b˜¾:
༬÷ ˹
h = 3e h = 14 m
2r = 28e h = 13.5 - 3
r = 14e = 10.5 m
õ¬÷ðóŠ¹ = 2πrh l = 22 hr +
= 2π x 14 x 3 = 22 5.1014 +
= 84π = 25.110196 +
= 25.306
= 17.5 mõ¬÷ðóŠ¹
= πrl= π x 14 x 17.5 m= 245π
Ãì£óˆF¡ ªñ£ˆî ¹øŠðóŠ¹ = ༬÷J¡ õ¬÷ðóŠ¹ + ËH¡ õ¬÷ðóŠ¹= 84 π + 245 π= 329 π
= 329 x 722
= 1034 ê.ªê.e7. èOñ‡¬íŠ ðò¡ð´ˆF å¼ ñ£íõ¡ 48 ªê.e àòóº‹ 12 ªê.e Ýóº‹ ªè£‡ì «ï˜õ†ì F‡ñ‚
Ë¬ð„ ªêŒî£˜. ܂ˬð ñŸªø£¼ ñ£íõ˜ å¼ F‡ñ‚ «è£÷ñ£è ñ£ŸPù£˜. Üšõ£Á ñ£ŸøŠð†ì¹Fò «è£÷ˆF¡ Ýóˆ¬î‚ 裇è.
b˜¾:
˹ «è£÷‹ r1 = 12 ªê.e r = ?h = 48 ªê.e«è£÷ˆF¡ èùÜ÷¾ = ËH¡ èùÜ÷¾
3 m
28 m
13.5
m
r
12 cm
48 c
m
www.mathstimes.com
9 3
ie 34
πr3 = 31
πr12 h
4r3 = r12 h
4r3 = 12 x 12 x 48
r3 = 12
481212 ××
r3 = 12 x 12 x 12 r = 12 ªê.e
8. 8 ªê.e M†ìº‹ 12 ªê.e àòóº‹ ªè£‡ì å¼ «ï˜õ†ì F‡ñ Þ¼‹¹‚ Ëð£ù¶ ༂èŠð†´ 4 I.eÝóºœ÷ F‡ñ‚ «è£÷ õ®õ °‡´è÷£è õ£˜‚èŠð†ì£™ A¬ì‚°‹ «è£÷ õ®õ °‡´èO¡â‡E‚¬è¬ò‚ 裇è.b˜¾:
r , h â¡ð¶ ËH¡ Ýó‹, àòó‹ Ý°‹. r1 â¡ð¶ «è£÷ˆF¡ Ýó‹ Ý°‹.˹ «è£÷‹ h = 12 cm = 120 I.e r1 = 4I.e
2r = 8 ªê.e r = 4ªê.e
= 40 I.e ËH¡ èùÜ÷¾
A¬ì‚°‹ «è£÷ õ®õ °‡´èO¡ â‡E‚¬è = ------------------------------«è£÷ˆF¡ èùÜ÷¾
= 31
2
r34
hr31
π
π
= 31
2
r4
hr
= 4444
1204040×××
××
= 7509. ñíô£™ GóŠðŠð†ì å¼ à¼¬÷ õ®õ õ£OJ¡ àòó‹ 32 ªê.e ñŸÁ‹ Ýó‹ 18 ªê.e Ü‹ñí™
º¿õ¶‹ î¬óJ™ å¼ «ï˜õ†ì‚ ˹ õ®M™ ªè£†ìŠð´Aø¶. Üšõ£Á ªè£†ìŠð†ì ñíŸ Ã‹H¡
àòó‹ 24 ªê.e âQ™ ܂ËH¡ Ýó‹ ñŸÁ‹ ꣻòóˆ¬î‚ 裇è.
b˜¾:
༬÷ (õ£O) ˹ (ñí™)
h = 32 ªê.e h = 24 ªê.e
r = 18 ªê.e r = ?
༬÷J¡ è÷Ü÷¾ = πr2h ËH¡ èùÜ÷¾ = 31
πr2h
= π x 18 x 18 x 32 = 31
π x r2 x 24
ËH¡ èù Ü÷¾ = ༬÷J¡ èùÜ÷¾
31
π x r2 x 24 = π x 18 x 18 x 32
r2 x 8 = 18 x 18 x 32
r2 = 8
321818 ××
r2 = 18 x 18 x 4r = 18 x 2
= 36 ªê.e
www.mathstimes.com
9 4
l = 22 rh +
= 22 3624 +
= 1296576 +
= 1872
= 1312 ªê.e
10. 14 e M†ìº‹ 20 e Ýöºœ÷ å¼ AíÁ ༬÷ õ®M™ ªõ†ìŠð´Aø¶. Üšõ£Á ªõ†´‹«ð£¶«î£‡®ªò´‚èŠð†ì ñ‡ Yó£è ðóŠðŠð†´ 20 e x 14 e Ü÷¾èO™ Ü®Šð‚èñ£è‚ ªè£‡ì 弫ñ¬ìò£è ܬñ‚èŠð†ì£™, Ü‹«ñ¬ìJ¡ àòó‹ 裇è.b˜¾:
༬÷ (AíÁ) èù„ªêšõè‹ («ñ¬ì)
h = 20 e l = 20 e 2r = 14 e b = 14 e r = 7 e h1 = ?
èùÜ÷¾ = πr2 h èùÜ÷¾ = l bh
= 722
x 7 x 7 x 20 = 20 x 14 x h
= 22 x 7 x 20èù„ªêšõèˆF¡ èùÜ÷¾ = ༬÷J¡ èùÜ÷¾
20 x 14 x h = 22 x 7 x 20
h = 142020722
×××
= 11 e
11. ¹œOJò™¹œOJò™¹œOJò™¹œOJò™¹œOJò™
1. 18, 20, 15, 12, 25 ñ£Á𣆴‚ªè¿ 裇è.n = 5
x = 5
2512152018 ++++=
590
= 18
x d = x - x d2
18 0 020 2 415 -3 912 -6 3625 7 49
98
σ = n
2d = 5
98= 6.19 ~ 4.428
C.V.= x
σ x 100% =
18428.4
x 100
C.V. = 24.6%
www.mathstimes.com
9 5
2. 20, 18, 32, 24, 26 ñ£Á𣆴‚ ªè¿ 裇è. n = 5
x = 5
2624321820 ++++
= 5
120= 24
x d = x - x d2
20 -4 1618 -6 3632 8 6424 0 026 2 4
120
σ = n
2d = 5
120= 24 ~ 4.9
C.V = x
σ x 100% =
249.4
x 100
C.V. = 20.4%3. 20, 14, 16, 30, 21, 25 F†ìMô‚è‹ è£‡è.
x = 6
252130161420 +++++=
6126
= 21
x d = x - x d2
20 -1 114 -7 4916 -5 2530 9 8121 0 025 4 16
172
σ = n
2d = 6
172= 6.28
σ ~ 5.34. 62, 58, 53, 50, 63, 52, 55 F†ìMô‚è‹ è£‡è.
n = 7
x =7
55526350535862 ++++++
= 7
393 = 56
x d = x - x d2
62 6 3658 2 453 -3 950 -6 3663 7 4952 -4 1655 -1 1
151
www.mathstimes.com
9 6
σ = n
2d = 7
151= 5.21
σ ~ 4.95. 10, 20, 15, 8, 3, 4 F†ìMô‚è‹ èí‚A´è.
n = 6
x = 6
438152010 +++++ =
660
= 10
x d = x - 10 d2
10 0 020 10 10015 5 258 -2 43 -7 494 -6 36
214
σ = n
2d=
6214
= 6.35
σ ~ 5.96. 38, 40, 34, 31, 28, 26, 34 F†ìMô‚è‹ èí‚A´è.
n = 7
x = 7
34262831344038 ++++++=
7231
= 33
x d = x - x d2
38 5 2540 7 4934 1 131 -2 428 -5 2526 -7 4934 1 1
154
σ = n
2d=
7154
= 22
σ ~ 4.69
12. Gè›îè¾Gè›îè¾Gè›îè¾Gè›îè¾Gè›îè¾
1. Í¡Á ï£íòƒèœ ²‡´‹ªð£¿¶ êKò£è Þ¼ Ì‚èœ Ü™ô¶ °¬ø‰îð†ê‹ å¼ î¬ô A¬ì‚èGè›îè¾ ò£¶?
S = {(HHH, HHT, HTH, THH, TTT, TTH, THT, HTT}
n(S) = 8
êKò£ù Þ¼ Ì‚èœ: A = {HTT, TTH, THT}, n (A) = 3
P(A) = 83
°¬ø‰î ð†ê‹ å¼ î¬ô: B = {HTT, THT, TTH, HHT, HTH, THH, HHH}
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9 7
n(B) = 7, P(B) = 87
AB = {HTT, TTH, THT}, n(AB) = 3
P(AB) = 83
P(AB) = P(A) + P(B) - P(AB) 83
+87
- 83
= 87
2. å¼ ðè¬ì Þ¼º¬ø ༆ìŠð†ì¶. °¬ø‰î¶ å¼ à¼†ìLô£õ¶ ⇠5 A¬ì‚è Gè›îè¾ è£‡è.S = {(1,1) ..... (6, 6)} n(S) = 36ºî™ ༆ìL™ 5: A = {(5, 1) (5, 2) (5, 3) (5,4) (5,5) (5,6)}
n(A) = 6 , P(A) = 366
Þó‡ì£‹ ༆ìL™ 5: B = {(1, 5) (2, 5) (3, 5) (4, 5) (5, 5) (6, 5)}
n(B) = 6, P(B) = 366
AB = {(5,5)} n (AB) = 1 P(AB) = 361
P(AB) = P(A) + P(B) − P(AB) 366
+366
-361
= 3611
3. ENTERTAINMENT â¡ø ªê£™L™ ݃Aô àJªó¿ˆ¶ Ü™ô¶ T Ýè Þ¼‚è Gè›îè¾?S = {E, N, T, E, R, T, A I, N, M, E, N, T} N(S) = 13݃Aô àJªó¿ˆ¶ : A = {E, E, A, I, E}, n (A) = 5
P(A) = 135
T ⿈¶ : B = {T, T, T}, n(B) = 3
P(B) = 133
n(AB) = 0 , P (AB) = 0
P(AB) = P(A) + P(B) 135
+ 133
= 138
4. 52 Y†´èœ å¼ Þó£ê£ Ü™ô¶ å¼ ý£˜† Y†´ â´‚è Gè›îè¾?n(S) = 52
ý£˜† : A n(A) = 13 , P(A) = 5213
Þó£ê£ : B n(B) = 4, P(B) = 524
n(AB) = 1, P(AB) = 521
P(AB) = P(A) + P(B) - P(AB)
5213
+ 524
- 521
= 5216
= 134
5. å¼ ¬ðJ™ 10 ªõœ¬÷, 5 輊¹, 3 ð„¬ê, 2 CõŠ¹ ð‰¶. å¼ ð‰¶ ªõœ¬÷ Ü™ô¶ 輊¹ Ü™ô¶ð„¬ê Ýè Þ¼‚è Gè›îè¾
n(S) = 10 + 6 + 10 = 26
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9 8
ªõœ¬÷: A n(A) = 10, P(A) = 2610
輊¹ : B n(B) = 6 , P(B) = 266
P(AB) = P(A) + P(B) =2610
+266
= 2616
= 138
6. å¼ ðè¬ì Þ¼º¬ø ༆ìŠð´‹. ºîô£õ¶ ༆ìL™ Þó†¬ìŠð¬ì ⇠ܙô¶ ºè Ã´î™ 8.S = {(1, 1) ...... (1,6) } ; n (S) = 36
Þó†¬ìŠ ð¬ì â‡èœ A¬ì‚è : A = { (2, 1) (2, 2) (2, 3) (2,4) (2,5) (2,6)(4, 1) (4, 2) (4, 3) (4,4) (4,5)(4,6)(6, 1) (6, 2) (6, 3) (6,4) (6,5) (6,6)}
n(S) = 18, P (A) = 3618
Ã´î™ 8 A¬ì‚è : B = {(2, 6) (3,5) (4,4) (5,3) (6,2)}
n(B) = 5, P(B) = 365
AB = {(2, 6) (4, 4) (6, 2)}n(AB) = 3 , P(AB) = 363
P(AB) = P(A) + P(B) −−−−− P(AB)
= 3618
+ 365
−−−−− 363
= 3620
= 95
7. å¼ ¹Fò ñA›‰¶ õ®õ¬ñŠHŸ° M¼¶ Gè›îè¾ 0.25, âKªð£¼œ ðò¡ð£†®¡ Gè›îè¾ 0.35, Þ¼M¼¶èÀ‹ 0.15, i) °¬ø‰î¶ ãî£õ¶ å¼ M¼¶ ii) å«ó å¼ M¼¶ ñ†´‹
P(A) = 0.25, P(B) = 0.35, P(AB) = 0.15
i) P(AB) = P(A) + P(B) −−−−− P (AB) = 0.25 + 0.35 −−−−− 0.15
= 0.45
ii) )BA(P)BA(P + = P(A) −−−−− P(AB) + P(B) −−−−− P(AB)
= 0.25 −−−−− 0.15 + 0.35 −−−−− 0.15
= 0.10 + 0.20
= 0.3
8. å¼ ñ£íM ñ¼ˆ¶õ è™ÖK Gè›îè¾ 0.16, ªð£PJò™ è™ÖK Gè›îè¾ 0.24, Þ¼ è™ÖKèÀ‹ 0.11i) ãî£õ¶ å¼ è™ÖK ii) å«ó å¼ è™ÖK ñ†´‹
P(A) = 0.16, P(B) = 0.24, P(AB) = 0.11
i) P(AB) = P(A) + P(B) −−−−− P (AB) = 0.16 + 0.24 −−−−− 0.11
= 0.40 - 0.11 = 0.29
ii) )BA(P)BA(P + = P(A) −−−−− P(AB) + P(B) −−−−− P(AB)
= 0.16 −−−−− 0.11 + 0.24 −−−−− 0.11
= 0.05 + 0.13 = 0.18
9. 52 Y†´èœ, ܉î Y†´ v«ð´ Ü™ô¶ Þó£ê£ õ£è Ü™ô¶ CõŠ¹ Gø„ Y†ì£è Þ¼Šð Gè›îè¾ ò£¶?
n(S) = 52Þó£ê£ v«ð´ CõŠ¹ Gø„ Y†´
n(A) = 4, n(B) = 13, n(C) = 26,
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9 9
P(A) = 524
P(B) = 5213
P(C) =5226
n(AB) = 1 n(BC) = 13 n(AC) = 2 n(ABC) = 1
P(AB) = 521
P(BC)5213
P(AC) = 522
P(ABC) = 521
P(ABC) = P(A) + P(B) + P(C) −−−−− P(AB) −−−−− P(BC) −−−−− P(AC) + P(ABC)
= 524
+ 5213
+ 5226
−521
−−−−−5213
− − − − −522
+ 521
= 5228
= 137
10. å¼ ¬ðJ™ 10 ªõœ¬÷, 5 èÁŠ¹, 2 CõŠ¹ àœ÷¶. å¼ ð‰¶ ªõœ¬÷ Ü™ô¶ èÁŠ¹ Ü™ô¶ ð„¬êGø‹
n(S) = 10 + 5 + 3 + 2 = 20ªõœ¬÷: A 輊¹: B ð„¬ê: Cn(A) = 10 n(B) = 5 n(C) = 3
P(A) = 2010
P(B) = 205
P(C) = 203
P(ABC) = P(A) + P(B) + P(C)
= 2010
+ 205
+203
= 2018
=109
11. P(A) =54
, P(B) = 32
, P(C) 73
, P(AB)= 158
, P(BC) = 72
, P(AC) =3512
, P(ABC) = 358
,
P(ABC) = ?
P(ABC) = 54
+ 32
+ 73
−−−−−158
−−−−− 72
−−−−− 3512
+ 358
= 105
24363056457084 +−−−++ =
105101
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100
9. ªêŒº¬ø õ®Mò™ªêŒº¬ø õ®Mò™ªêŒº¬ø õ®Mò™ªêŒº¬ø õ®Mò™ªêŒº¬ø õ®Mò™
ð®ð®ð®ð®ð®1: àîMŠðì‹ õ¬óè. Ýó‹ OT , Éó‹ OP ä °P‚è.
ð®ð®ð®ð®ð® 2 : O ¬õ ¬ñòñ£è‚ ªè£‡´ ªè£´‚èŠð†ì ÝóˆF¡ Ü÷M™ õ†ì‹
õ¬óè.
ð®ð®ð®ð®ð® 3: OML¼‰¶ OP™ P â¡ø ¹œO¬ò‚ °P‚è.
ð®ð®ð®ð®ð® 4: OP ¡ ï´‚°ˆ¶‚«è£´ õ¬óè.
ð®ð®ð®ð®ð® 5: Mä ¬ñòñ£è¾‹ OM ä Ýóñ£è¾‹ à¬ìò Þó‡ì£õ¶ õ†ì‹ õ¬óè.
ð®ð®ð®ð®ð® 6 : Þó‡´ õ†ìƒèÀ‹ T, T’ â¡ø Þ¼ ¹œOèO™ ªõ†´A¡øù.
PT , PT’ ä Þ¬í. ÜõŸP¡ c÷ƒè¬÷ Ü÷‰¶ ⿶è.
°PŠ¹: i) M¬ì¬ò êK𣘂è â¡Á «è†ì£™ ñ†´‹ êK𣘂è.
ii) õ†ìˆ¶‚° ªõO«ò àœ÷ ¹œOJL¼‰¶ õ†ìˆ¶‚° Þ¼
ªî£´«è£´èœ õ¬óòô£‹.
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101
1. Ýó‹ 3 ªê.e Ü÷M™ å¼ õ†ì‹ õ¬óè. ¬ñòˆFL¼‰¶ 7 ªê.e ªî£¬ôM½œ÷ å¼ ¹œOJL¼‰¶
õ†ìˆ¶‚° ªî£´«è£´èœ õ¬óè. Üî¡ c÷ƒè¬÷ Ü÷‰¶ ⿶è.
Fair diagram
2. Ýó‹ 4.2 ªê.e Ü÷M™ å¼ õ†ì‹ õ¬óè. õ†ìˆF¡ e¶ ãî£õ¶ å¼ ¹œO¬ò‚ °Pˆ¶, ¬ñòŠ¹œO¬òŠ
ðò¡ð´ˆF ÜŠ¹œOJ™ å¼ ªî£´«è£´ õ¬óè.
Fair diagram
T1
T2
P7 cm
3cm
àîMŠðì‹àîMŠðì‹àîMŠðì‹àîMŠðì‹àîMŠðì‹
O
3 cm
7 cm
6.5 cm
T1
T2
PM
6.5 cm
O
4.2 cmO P
T
T1
àîMŠðì‹àîMŠðì‹àîMŠðì‹àîMŠðì‹àîMŠðì‹T
N
P
LO
T1
M
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102
3. M†ì‹ 10 ªê.e Ü÷M™ å¼ õ†ì‹ õ¬óè. ¬ñòˆFL¼‰¶ 13 ªê.e ªî£¬ôML¼‰¶ õ†ìˆFŸ°PA , PB â¡ø Þ¼ ªî£´«è£´èœ õ¬óè. ÜõŸP¡ c÷ƒè¬÷ Ü÷‰¶ ⿶è.
Fair Diagram
4. Ýó‹ 6 ªê.e Ü÷¾œ÷ õ†ìˆF¡ ¬ñòˆFL¼‰¶ 10 ªê.e ªî£¬ôM™ àœ÷ å¼ ¹œOJL¼‰¶õ†ìˆFŸ° Þ¼ ªî£´«è£´èœ õ¬óè. ÜõŸP¡ c÷ƒè¬÷ Ü÷‰¶ ⿶è.
Fair Diagram
A
B
P10 cm
6cm
àîMŠðì‹àîMŠðì‹àîMŠðì‹àîMŠðì‹àîMŠðì‹
O
A
B
P13 cm
5 cm
àîMŠðì‹àîMŠðì‹àîMŠðì‹àîMŠðì‹àîMŠðì‹
O
13 cm
A
B
MO
12 cm
12 cm
5 cm
P
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103
10 cm P
6cm
B
A
O M
8 cm
8 cm
5. Ýó‹ 3 ªê.e Ü÷¾œ÷ õ†ìˆF¡ ¬ñòˆFL¼‰¶ 9 ªê.e ªî£¬ôM™ àœ÷ å¼ ¹œOJL¼‰¶õ†ìˆFŸ° Þó‡´ ªî£´«è£´èœ õ¬óè.
Fair Diagram
A
B
P9 cm
3cm
àîMŠðì‹àîMŠðì‹àîMŠðì‹àîMŠðì‹àîMŠðì‹
O
3 cm
9 cm
8.5 cm
A
B
PM
8.5 cm
O
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104
º‚«è£íƒèœ õ¬ó‚«è£íƒèœ õ¬ó‚«è£íƒèœ õ¬ó‚«è£íƒèœ õ¬ó‚«è£íƒèœ õ¬óî™1. AB = 6 ªê.e ∠ C = 40o ñŸÁ‹ à„C CJL¼‰¶ AB‚° õ¬óòŠð†ì °ˆ¶‚«è£†®¡ c÷‹ 4.2 ªê.e
ªè£‡ì ΔABC õ¬óè.
Fair Diagram
2. ΔABC ™ BC = 5 ªê.e∠ A = 45o ñŸÁ‹ à„C AL¼‰¶ BC‚° õ¬óòŠð†ì ï´‚«è£†®¡ c÷‹ 4 ªê.eâù Þ¼‚°‹ð® ΔABC õ¬óè.
Fair Diagram
A B
C
40o
6cm
4.2c
m
àîMŠðì‹àîMŠðì‹àîMŠðì‹àîMŠðì‹àîMŠðì‹
M
6cmA B
C C’
O40o
M
Y
X
H
K
4.2
cm
>
>
50o
B C
A
45o
5cm
4cm
àîMŠðì‹àîMŠðì‹àîMŠðì‹àîMŠðì‹àîMŠðì‹
M
B C
A
5cm
4cm
O
X
X
K
M
>
>
45o
www.mathstimes.com
105
3. Ü®Šð‚è‹ BC = 5.5ªê.e ∠ A = 60o ñŸÁ‹ à„C AJL¼‰¶ õ¬óòŠð†ì ï´‚«è£´ AM¡ c÷‹ 4.5 ªê.eªè£‡ì ΔABC õ¬óè.
Fair diagram
4. PQ = 4ªê.e ∠ R = 25o ñŸÁ‹ à„C R L¼‰¶ PQ‚° õ¬óòŠð†ì °ˆ¶‚«è£†®¡ c÷‹ 4.5ªê.e â¡øÜ÷¾èœ ªè£‡ì ΔPQR õ¬óè.
Fair Diagram
B C
A
60o
5.5cm
4.5c
m
àîMŠðì‹àîMŠðì‹àîMŠðì‹àîMŠðì‹àîMŠðì‹
M
B
A
C5.5cm
4.5c
m
60o
X
Y
M
K
A’
O
>
>
P Q
R
25o
4cm
4c
m
àîMŠðì‹àîMŠðì‹àîMŠðì‹àîMŠðì‹àîMŠðì‹
M
P Q
R
4cm
4cm
25 o
R’
X
X
M
H
O
K
>
>
4.5c
m
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106
5. ΔPQR ™ Ü®Šð‚è‹ PQ = 6ªê.e ∠ R = 60o ñŸÁ‹ à„C R L¼‰¶ PQ‚° õ¬óòŠð†ì °ˆ¶‚«è£†®¡c÷‹ 4 ªê.e âù Þ¼‚°ñ£Á ΔPQR õ¬óè.
Fair Diagram
6. BC = 5ªê.e ∠ BAC = 40o ñŸÁ‹ à„C A JL¼‰¶ BC‚° õ¬óòŠð†ì ï´‚«è£†®¡ c÷‹ 6 ªê.e â¡øÜ÷¾èœ ªè£‡ì ΔABC õ¬óè. à„C AJL¼‰¶ õ¬óòŠð†ì °ˆ¶‚«è£†®¡ c÷‹ 裇è.
Fair Diagram
P Q
R
60o
6cm
4c
m
àîMŠðì‹àîMŠðì‹àîMŠðì‹àîMŠðì‹àîMŠðì‹
M
P Q
R
6cm
4cm60
o
O
X
XH
K
M
>
>
30o
R’
B C
A40
o
5cm
6c
màîMŠðì‹àîMŠðì‹àîMŠðì‹àîMŠðì‹àîMŠðì‹
M
B
A
C5cm
X
O6cm
40 o
Y
4.8c
m
A’
M
K >
>
www.mathstimes.com
107
10. õ¬óðìƒèœõ¬óðìƒèœõ¬óðìƒèœõ¬óðìƒèœõ¬óðìƒèœ
1. W›‚裵‹ ܆ìõ¬í‚°ˆ î°‰î õ¬óðì‹ õ¬ó‰¶ ñ£PèO¡ ñ£Á𣆴ˆ ñ¬ò‚ 裇è.x 2 3 5 8 10y 8 12 20 32 40
«ñ½‹, x=4 âQ™ y¡ ñFŠ¬ð‚ 裇è.b˜¾ : x = 4 âQ™ y = 16 Ý°‹
^
^
80
75
70
65
60
55
50
45
40
35
30
25
20
15
10
5
1 2 3 4 5 6 7 8 9 10
y
x′ x^
^
(2,8)
(3,12)
(5,20)
(8,32)
(10,40)
Scalex-axis 1 cm = 1 unity-axis 1 cm = 5 units
y = 4x
^
^
16
O>
>
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108
2. å¼ IFõ‡® 憴ðõ˜ A â¡ø ÞìˆFL¼‰¶ B â¡ø ÞìˆFŸ° å¼ Yó£ù «õèˆF™ å«ó õNJ™ªõš«õÁ èO™ ðòí‹ ªêŒAø£ .̃ Üõ˜ ðòí‹ ªêŒî «õè‹ ÜˆÉóˆF¬ù‚ èì‚è â´ˆ¶‚ªè£‡ì«ïó‹ ÝAòùõŸ¬øŠ ðŸPò Mõóƒèœ («õè&è£ô) H¡õ¼‹ ܆ìõ¬íJ™ ªè£´‚èŠð†´œ÷ù.
«õè‹ (A.e/ñE) 2 4 6 10 12«ïó‹ (ñEJ™) 60 30 20 12 10«õè&è£ô õ¬óðì‹ õ¬ó‰¶ ÜFL¼‰¶ i) Üõ˜ ñE‚° 5 A.e «õèˆF™ ªê¡ø£™ Éóˆ¬î‚ èì‚èÝ°‹ ðòí «ïó‹ ii) Üõ˜ Þ‚°PŠH†ì Éóˆ¬î 40 ñE «ïóˆF™ èì‚è â‰î «õèˆF™ ðòE‚è«õ‡´‹. ÝAòùõŸ¬ø‚ 裇è,b˜¾: i) x = 5 âQ™ y = 24, ii) y = 40 âQ™ x = 3
^x′ x1 2 3 4 5 6 7 8 9 10
^
100
90
80
70
60
50
40
30
20
10
y Scalex-axis 1cm = 1 unity-axis 1cm = 10 units
-10
-20
-30
-1 O
^
^
«ïó‹
(ñ£P
)«ïó‹
(ñ£P
)«ïó‹
(ñ£P
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(ñ£P
)
«õè‹ (A.e)«õè‹ (A.e)«õè‹ (A.e)«õè‹ (A.e)«õè‹ (A.e)
xy = 120
^
(2,60)
(3, 40)
(4,30)
(5, 24)(6, 20)
(10,12)
>
>
>
>
> www.mathstimes.com
109
3. õ£ƒèŠð†ì «ï£†´Š ¹ˆîèƒèO¡ â‡E‚¬è ñŸÁ‹ ÜîŸè£ù M¬ô Mõó‹ H¡õ¼‹ ܆ìõ¬íJ™
îóŠð†´œ÷¶.
«ï£†´Š¹ˆîèƒèO¡ â‡E‚¬è (x) 2 4 6 8 10 12
M¬ô Ï. (y) 30 60 90 120 150 180
ÞîŸè£ù õ¬óðì‹ õ¬ó‰¶ Üî¡ Íô‹ i) ã¿ «ï£†´Š¹ˆîèƒèO¡ M¬ô¬ò‚ 裇è.
ii) Ï.165‚° õ£ƒèŠð†ì «ï£†´Š¹ˆîèƒèO¡ â‡E‚¬è¬ò‚ 裇è.
b˜¾: x = 7 âQ™ y = Ï.105 y = Ï.165 âQ™ x = 13
^
^
210
195
180
165
150
135
120
105
90
75
60
45
30
15
2 4 6 8 10 12 14 16 18 20
y Scalex-axis 1 cm = 2 unitsy-axis 1 cm =15units
(2,30)
(4,60)
(6,90)
(8,120)
(10,150)
(12,180)
^
^
^
y=15
x
«ï£†´Š ¹ˆîèƒèO¡ â‡E‚¬è«ï£†´Š ¹ˆîèƒèO¡ â‡E‚¬è«ï£†´Š ¹ˆîèƒèO¡ â‡E‚¬è«ï£†´Š ¹ˆîèƒèO¡ â‡E‚¬è«ï£†´Š ¹ˆîèƒèO¡ â‡E‚¬è
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110
4. x 1 3 5 7 8
y 2 6 10 14 16
«ñŸè‡ì ܆ìõ¬íJ™ àœ÷ MõóˆFŸ° õ¬óðì‹ õ¬ó‰¶ Üî¡ Íô‹
i) x = 4 âQ™ y ¡ ñFŠ¬ð»‹ ii) y = 12 âQ™ x¡ ñFŠ¬ð»‹ 裇è.
b˜¾ : i) x = 4 âQ™ y = 8
ii) y = 12 âQ™ x = 6
^
^
20
18
16
14
12
10
8
6
4
2
1 2 3 4 5 6 7 8 9 10
y^
(1,2)
(3,6)
(5,10)
(8,16)
Scalex-axis 1 cm = 1 unity-axis 1 cm = 2 units
^
^
(7,14)y =
2x
^
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>
>
>
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www.mathstimes.com
111
5. «õ¬ôò£†èO¡ â‡E‚¬è (x) 3 4 6 8 9 16
èO¡ â‡E‚¬è (y) 96 72 48 36 32 18
܆ìõ¬íJ™ ªè£´‚èŠð†´œ÷ MõóˆFŸè£ù õ¬óðì‹ õ¬óè. Üî¡Íô‹ 12 «õ¬ôò£†èœÜš«õ¬ô¬ò º¿õ¶ñ£è ªêŒ¶ º®‚è Ý°‹ èO¡ â‡E‚¬è¬ò‚ 裇è.
b˜¾ : x = 12 âQ™ y = 24
^
^
96
90
84
78
72
66
60
54
48
42
36
30
24
18
12
6
2 4 6 8 10 12 14 16 18 20
y
x′ x
Scalex-axis 1 cm = 2 unitsy-axis 1 cm = 6 units
(3, 96)
(4,72)
(6,48)
(8,36)
(9,32)
(16,18)
(12,24)
^
^
xy=288
O >
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112
1. å¼ õƒA ͈®ñèQ¡ ¬õŠ¹ˆªî£¬è‚° 10% îQõ†® î¼Aø¶. ¬õŠ¹ˆªî£¬è‚°‹ ÜîŸ°æ˜ Ý‡´‚° A¬ì‚°‹ õ†®‚°‹ Þ¬ì«òò£ù ªî£ì˜H¬ù‚ 裆ì å¼ õ¬óðì‹ õ¬óè. Üî¡Íô‹ i) Ï. 650 ¬õŠ¹ˆªî£¬è‚° A¬ì‚°‹ õ†® ñŸÁ‹ii) Ï.45 õ†®ò£è‚ A¬ì‚è õƒAJ™ ªê½ˆîŠðì «õ‡®ò ¬õŠ¹ˆªî£¬è ÝAòùõŸ¬ø‚ 裇è.
¬õŠ¹ˆªî£¬è (x) 100 200 300 400 500 600 700 800
îQõ†® (y) 10 20 30 40 50 60 70 80
õ¬óðìˆFL¼‰¶ x = Ï. 650 âQ™ y = Ï. 65 y = Ï. 45 âQ™ x = Ï. 450
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130
120
110
100
90
80
70
60
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40
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20
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Scalex-axis 1 cm = 100 unitsy-axis 1 cm = 10 units
1000
900
800
700
600
500
400
300
200
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113
2. å¼ «ð¼‰¶ ñE‚° 40 A.e «õèˆF™ ªê™Aø¶ ÞKò Éó è£ô ªî£ì˜H¬ù õ¬óðì‹ õ¬óè.Þ¬îŠðò¡ð´ˆF 3 ñE «ïóˆF™ ÞŠ«ð¼‰¶ ðòEˆî Éóˆ¬î‚ 致H®. (June 13, June 14)b˜¾:
è£ô‹ (ñEJ™) 1 2 3 4 5 6 7
¶ó‹ y (A.e) 40 80 120 160 200 240 280
õ¬óðìˆFL¼‰¶ x = 3 ñE âQ™ y = 120 A.e
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320
280
240
200
160
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80
40
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114
3. å¼ L†ì˜ ð£L¡ M¬ô Ï.15 â¡è. ð£L¡ Ü÷¾‚°‹ M¬ô‚°‹ àœ÷ˆ ªî£ì˜H¬ù‚ 裆´‹õ¬óðì‹ õ¬óè. Üî¬ùŠ ðò¡ð´ˆF
i) MAîêñ ñ£PL¬ò‚ 裇è. ii) 3 L†ì˜ ð£L¡ M¬ô¬ò‚ 裇è.b˜¾:
ð£L¡ Ü÷¾ L†ìK™ (x) 1 2 3 4 5 6 7
M¬ô Ï. y 15 30 45 60 75 90 105
i) MAî êñ ñ£PL 15 ii) 3 L†ì˜ ð£L¡ M¬ô Ï.45
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165
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135
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(4,60)
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115
4. xy = 20, x, y > 0 â¡ø õ¬óðì‹ õ¬óè. õ¬óðìˆFL¼‰¶ x = 5 âQ™ y ¡ ñFŠ¬ð»‹ y = 10 âQ™x¡ ñFŠ¬ð»‹ 裇è.
b˜¾:xy = 20‚è£ù ܆ìõ¬í
x 1 2 4 5 10 20
y 20 10 5 4 2 1
x = 5 âQ™ y = 4, y = 10 âQ™ x = 2
^
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22
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18
16
14
12
10
8
6
4
2
2 4 6 8 10 12 14 16 18 20
y′
y
x′ x
Scalex-axis 1 cm = 2 unitsy-axis 1 cm = 2 units
(2,10)
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(1,20)
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(10,2)(20,1)xy = 20
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