SHORT CUT METHODS IN +2 MATHEMATICS
SHORT CUT METHODS @ ALTERNATIVE METHODS IN +2 MATHS
Note: This methods are only to make interest and also based on
entrance exams.
1. Find the square root of 8 – 6 i .
8 6 1 9 6
Solution:
1 3 6
1 3
1 3
√8 6 1 3 !" 1 3.
2. $%& '() *+,-") "!!' !. 7 24 i
solution 1 -7 24 i 9 16 24i
3 24i 4i
3 4i
3 4i
√7 24i 3.If 2 0,1 find Z
Solution:
2 =i ,2 √ ,2 5 5
4. A ladder of length
the vertical wall and
equation of the locus
from the end of the
6
3 4i or 3 4i.
7869
786
√
15m moves with its ends always touching
the horizontal floor. Determine the
of a point P on the ladder, which is 6m
ladder in contact with the floor.
solution:
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SHORT CUT METHODS IN +2 MATHEMATICS
By the similar triangle property
."!: ";(' -%;<
-%;<)& '"-%;<) KL?,
."!: ";(' -%;<)& '"-%;<) ?NO,
cos M sin M 1
sin M ?L A7
?K
6
cos M ?N QL @7
9
?O ?O
@7 A7 RS T S T 1
9
6
R
= '() <!>,* !. ?@7 , A7 is
E9
U7
D9
FV
@7 A7
1
81 36
1,
Which is an ellipse.
5. Use differentials to find the an approximate value for the
given number
Solution:
X
X
y = 3 1.02 + 4 1.02 A .@
H√@ W√@ @ H @ W
CD
CE
7
9
7
H
. ′ @ F @ GH I @ GW
@ 1, &@ ∆@ 0.02
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SHORT CUT METHODS IN +2 MATHEMATICS
9
7
H
7
&A S @ GH @ GW T &@
F
I
@ 1, &@ ∆@ 0.02; .1 2
7
7
&A SF IT 0.02 S
=
I8F
T 0.02
Z.Z[
0.01167
√1.02 √1.02 2 0.01167 2.011
H
W
6.A bag contains 3 types of coins namely Re.1, Rs. 2 and Rs. 5.
There are 30 coins amounting to Rs. 100 in total. Find the
number of coins in each category.
Solution :(Rank method)
By given,x+2y+z=30,x+2y+5z=100,we have two equations but three
unknown
Let z=k
1 1
\K, O] ^1 2
0 0
1 1
^0 1
0 0
1 30
5 100 `
1 _
1
~ ^0
0
@
1
30
4 ` dAf ^70`
e
1
_
1 1
1 4
0 1
30
70` b c b -b7
_
Z=k,y+4z=70 , y+4k=70,we get y=70-4k
x+y+z=30,x+70-4k+k=30,we get x=3k-40
(x,y,z)=(3k-40,70-4k,k), where k=14,15,16,17
Solution:(matrix inversion method)
By given,x+2y+z=30,x+2y+5z=30,we have two equations but three
unknown
Let z=k
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SHORT CUT METHODS IN +2 MATHEMATICS
X+2y=30-k,x+2y=100-5k
The solution is x=KG7 O
KG7 g
@
2
gA h g
1
2 1
h
1 1
1
h
1
g
30 _
h
100 5_
@
3_ 40
gA h g
h
70 4_
(x,y,z)=(3k-40,70-4k,k), where k=14,15,16,17
7.A small seminar hall can hold 100 chairs. Three different
colours(red, blue and green) of chairs are available. The cost
of a
red chair is Rs. 240, cost of blue chair is Rs. 260 and
the cost of a green chair is Rs. 300. The total cost of chair
is Rs. 25,000. Find atleast 3 different solution of the number
of chairs in each colour to be purchased.
Solution :(Rank method)
By given,x+y+z=100,240x+260y+300z=25000 c12x+13y+15z=1250
we have two equations but three unknown ,
1
\K, O] ^12
0
1
\K, O] ^0
0
1
^0
0
1
1
0
1
13
0
1
1
0
1 100
15 1250 `
1 _
Let z=k
1 100
3 50 ` b c b -12b7
1 _
@
1
100
3 ` dAf ^ 50 `
e
1
_
Z=k,y+3z=50,we get y =50-3k
x+y+z =100,x+50-3k+k=100, we get x=50+2k
(x,y,z)=(50+2k,50-3k,k), where k=0.......16
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SHORT CUT METHODS IN +2 MATHEMATICS
Solution:(matrix inversion method)
By given, x+y+z=100,240x+260y+300z=25000 c12x+13y+15z=1250
we have two equations but three unknown ,
x+y=100-k, 12x+13y=1250-15k
KG7 g
The solution is x=KG7 O
@
13
gAh g
12
Let z=k
13 1
h
12 1
1
h
1
100 _
g
h
1250 15_
@
50 2_
gA h g
h
50 3_
(x,y,z)=(50+2k,50-3k,k), where k=0.......16
8. Derive the equation of the plane in the intercept form.
Let the vertices be A (a,0,0),(0,b,0) and C(0,0,C)
Required Equation of the plane is Ax+By+Cz+D =0
………(1)
It passes A(a,0,0),
Aa+D=0
It passes B(0,b,0),
Bb+D=0
It passes C(0,0,c),
Cc+D=0
Putting the values of A,B and
iE
j
iD
k
im
l
, K
, O
, N
C in
Gi
j
Gi
k
Gi
l
(1)
L 0
On dividing throughout by –D
E
j
D
m
k l 1
Which is the required equation of the plane in the intercept
form.
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SHORT CUT METHODS IN +2 MATHEMATICS
9.Points A and B are 10km apart and it is determined from the
sound of an explosion heard at those points at different times
that the location of the explosion is 6km closer to A than B.
Show that the location of the explosion is restricted to a
particular curve and find an equation of it.
Given that 2a=6 , a=3
2ae=10, (6)e=10
For hyperbola n - ) 1 ,n 9 S
o
p
1T,
, e=5/3
n 16
The required equation of hyperbola is
@ A
1
9 16
10.The tangent at any point of the rectangular hyperbola
makes intercepts a, b and the normal at the point makes
intercepts p,q on the axes. Prove that ap + bq = 0
E
j
By given equation of the tangent is
X
Slope is :7 = qX
r
k
:7 j
By given equation of the normal is
Slope is : =
X
u
X
v
E
s
xy = c 2
D
k 1
D
t 1
t
:7 s,
By Condition :7 x: =-1 ,
we get ap + bq = 0
11.Find c , µ and σ 2 of the normal distribution whose
probability function is given by
f ( x ) = ce − x
2
+ 3x
−∞ < X < ∞ .
w
>! )..>)%' !. @
3
>! )..>)%' !. 2@
2
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SHORT CUT METHODS IN +2 MATHEMATICS
x 1
1
>! )..>)%' !. 2@
2
.@ >)
GE 9 8FE
1
x√2y
GEGz9
) {9
Putting x=0 then we get the value of C
x 7
,x 7
√
,N 7
√|
N
}~
)W
1
x√2y
Gz 9
) {9
12. Obtain k , µ and σ 2 of the normal distribution whose
probability distribution function is given by
f ( x ) = ke
− 2 x 2 + 4x
−∞ < X < ∞.
w
>! )..>)%' !. @
4
1
>! )..>)%' !. 2@
4
x 1
1
>! )..>)%' !. 2@
4
.@ ) GE
9 8IE
1
x√2y
)
GEGz9
{9
Putting x=0 then we get the value of K

7
7
x I ,x , √
√|
) G
1
x√2y
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Gz 9
) {9
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