Download Linear Equations in Two Variables HCF and LCM of

Quick Reference, STD: X
1
Linear Equations in Two Variables
p(x)
where p(x), q(x)
qx
The general form of a linear equation in two vari- are polynomials and q(x) 6= 0, is called rational
ables x and y is
expression or rational algebraic expression
ax + by + c = 0, and here a 6= 0, b 6= 0
⇒ if p(x), q(x) and r(x) are the polynomial
Solution of Linear equations in two vari- and q(x) 6= 0, r(x) 6= 0 then
ables by the method of Determinants
p(x) × r(x) p(x)
1.
=
Let us consider two equations
q(x) × r(x) q(x)
ax + by = e . . . (1)
cx + dy = f . . . (2) then
p(x)
p(x)
−p(x)
p(x)
=−
=
=
2. −
q(x)
q(x)
q(x)
−q(x)
Dy
Dx
x=
and y =
Addition & subtraction of two rational exD
D
pressions (Denominator are equal)
Here,
a b p(x) r(x)
Let
&
be any two rational expression then:
D = =a×d−b×c
c d
q(x) q(x)
e b
Dx = f d
a e
Dy = c f
∴ An expression of the form
1.
p(x)
r(x)
p(x) + r(x)
+
=
q(x)
q(x)
q(x)
2.
p(x) r(x)
p(x) − r(x)
=
q(x) q(x)
q(x)
=e×d−b×f
=a×f −e×c
HCF and LCM of Polynomials
An algebric expression of the type a0 + a1 x + a2 x2 +
· · · + an−1 xn−1 + an xn is called a polynomial if
(i) a0 , a1 , a2 · · · an−1 , an are real numbers
(ii) n is an integer which ≥ 0
Addition & subtraction of two rational expressions (Denominator are not equal)
p(x) r(x)
Let
&
be any two rational expression then:
q(x) s(x)
1.
p(x)
r(x)
p(x) × s(x) + r(x) × q(x)
+
=
q(x)
s(x)
q(x) × s(x)
2.
p(x) × s(x) − r(x) × q(x)
p(x) r(x)
−
=
q(x) s(x)
q(x) × s(x)
Multipication of Rational Expressions
Test for divisiblity
If ab and dc are two rational numbers, then we we
Test for (x − 1):(x − 1) is a factor of polynomial know that their product is given by ab × dc = a×c
b×d
in x if sum (additon) of all the coefficent of the
p(x) r(x)
similarly, if
&
are the two rational exprespolynomial is zero
q(x) s(x)
sion there product is given by
Test for (x + 1):(x + 1) is a factor of polynop(x) r(x)
p(x) × r(x)
mial in x if sum (additon) of all the coefficent in ⇒ q(x) × s(x) = q(x) × s(x)
even power of x is equal to sum (additon) of all
Division of Rational Expression
the coefficent in odd power of x
p(x)
r(x)
Let
&
be the two rational expression such
q(x)
s(x)
Relationship between HCF & LCM if there
that r(x) 6= is non-zero rational expression then we
are two polynomial say p(x) and q(x) then
LCM of { p(x) and q(x)}× HCF of { p(x) and q(x)} have
p(x) r(x)
p(x) s(x)
p(x) × s(x)
⇒
÷
=
×
=
= p(x) × q(x)
q(x) s(x)
q(x) r(x)
q(x) × r(x)
Rational Algebraic Expressions
we know that a number of the form
is called a rational number.
m
n
Quadratic Equations
where n 6= 0 The general form of a quadratic equation is
ax2 + bx + c = 0, where a,b,c are real number and
Compiled by metric, for soft copy mail me at [email protected]
Quick Reference, STD: X
2
a 6= 0
n is the number of terms in an AP
sol. by perfect square method
⇒Make sure the coefficent of variable with index 2
d is the common difference between the sucis 1 (eg. 5x2 − 4x − 2 = 0, here the coefficent is 5
cessive terms of an AP
for x2 hence first we shall divide both side by 5)
⇒Find the 3rd term with formula Sum of the first n terms of an AP
2
1
Let a, a + d, a + 2d, a + 3d, . . . be an AP with total
Third term=
× coefficent of x
2
n number of terms
⇒ Add third term on both the side
a denote first term,
tn denote last term of AP
sol. by formula method
d denotes the common diff.
Let ax2 + bx + c = 0 be a quadratic equation, Let us denote sum upto n terms by S , we have
n
n
where where a,b,c are real numbers and a 6= 0 then
⇒ Sn = (a + tn )
solution by√formula method is given by
2
⇒ tn = a + (n − 1)d
−b ± b2 − 4ac
n
x=
∴ ⇒ Sn = [2a + (n − 1)d]
2a
2
Case1:⇒ b2 − 4ac = 0 i.e b2 = 4ac
b
b
x = − , x = −
→ both roots are equal
2a
2a
Probability
2
Case2:⇒ b − 4ac > 0 i.e b2 > 4ac then the
equation has
√ two distinct root √
Probability of an event
−b + b2 − 4ac
−b − b2 − 4ac
x=
, x=
probability of an even A,written as P(A), is defined
2a
√ 2a
2
2
Case 3:⇒ b − 4ac < 0 then b − 4ac is not real as
Number of outcomes favourable to A
number and hence quadractic equation cannot have P(A)= Total number of possible outcomes
probability of an impossible event is 0
any real roots
probability of a sure event is 1. Probability of any
event will lie between 0 and 1, In general for any
Imp. Result
2
event A,we have
1
1
⇒P(A)=1-P(A0 )
1. x2 + 2 = x +
−2
x
x
⇒P(A)+P(A0 )=1
2
⇒P(A0 )=1-P(A)
1
1
2
2. x + 2 = x −
+2
Here P(A0 )denotes probability of not happening of
x
x
an event A
Arthmetic Progressions(AP)
The general form of AP is
t, t + d, t + 2d, t + 3d, . . ., here t is the first term
and d is common difference,
The AP with first term 100 and common difference
50 is
100, 150, 200, 250, 300 . . .
Statistics
Mean of Raw Data
Mean of the values x1 , x2 , x3 . . . , xn is denoted by x̄
and is given by
n
x 1 , x 2 , x3 . . . , xn
1X
x̄ =
=
xi
n
n i=1
nth (or the general) term of an AP, nth of an
arthemetic progression t, t + d, t + 2d, t + 3d, . . . Assumed Mean Method for Calculating
the Mean
is given by
A=an arbitary constant(usually A is chosen some
where in the middle part of the given value) A is
tn = t + (n − 1)d , here
also called assumed mean
tn is the nth term
di =The reduced value, di = xi − A and is called
t is the first term
deviation of xi from A
Compiled by metric, for soft copy mail me at [email protected]
Quick Reference, STD: X
n
X
d¯ =
3
Similarity
fi di
i=1
n
X
& Mean = x̄ = A + d¯
fi
Two 4 0 s are said to be similar if
1. Their corresponding angles are equal
i=1
2. Their corresponding sides are in the same ratio
(i.e sides are propotional)
Mean of Grouped Data
n
X
Ratio of area of 40 s
Let A1 and A2 be the area, b1 , b2 be the bases and
h1 , h2 be the heights of any two 40 s Then the ratio
of there area is given as
fi xi
1. Direct method x̄ = i=1
n
X
fi
i=1
2. Assumed mean method x̄ = A + d¯
1.
b1 × h1
A1
=
A2
b2 × h2
3. Step-deviation method x̄ = A + h.d¯
(h is the width of the class intervals)
2.
A1
b1
= , if heights of two 40 s are equal
A2
b2
3.
A1
h1
= , if bases of two 40 s are equal
A2
h2
n
X
Remember d¯ is calculated as d¯ =
fi di
i=1
n
X
fi
i=1
Median
Formula for computing Median from grouped data
N
− c.f.
× h, where
Median=L + 2
f
L =Lower boundary of a median class
Trigonometry
Trigonometric ratios of an acute angle
Sin θ
Opposite side of angle θ
Hypothenuse
Cos θ
Adjacent side of angle θ
Hypothenuse
Tan θ
Opposite side of angle θ
Adjacent side of angle θ
Cosec θ
Hypotenus
Opposite side of angle θ
Sec θ
Hypotenus
Adjacent side of angle θ
Cot θ
Adjacent side of angle θ
Opposite side of angle θ
N =Total frequency
c.f =cumulative f requency of the class preceding the median class
f =f requency of the median class
h=Width of the median class
Mode
Formula for computing
Modefrom grouped data
fm − f1
×h
Mode = L +
2fm − f1 − f2
L =Lower boundary of a modal class
fm =Frequency of the modal class
f1 =Frequency of the class coming before the modal class
f2 =Frequency fo the class coming after the modal class
h=Width of the modal class
Instalments
⇒ x n = Pn 1 +
R n
100
Pn = Amount of nth instalment.
P =Amount borrowed,cash price.
R=Rate of interest
A=Principle + interest=Amount due
x=Amount of installment
Relationship between the trignometric ratios
Compiled by metric, for soft copy mail me at [email protected]
Quick Reference, STD: X
4
sinAcosecA=1
1
sinA= cosecA
1
cosecA= sinA
cosAsecA=1
1
secA= cosA
1
cosA= secA
tanAcotA=1
1
tanA= cotA
1
cotA= tanA
sinA
tanA= cosA
cotA= cosA
sinA
Ratios of angle
M. of ∠s →
0◦
s
↓ Ratios of ∠
sin θ
0
cos θ
tan θ
cosec θ
sec θ
cot θ
1
0
ND
1
ND
3. Right circular cylinder
r:radius of the base, h:height
(a) Curved Surface Area=2πrh
(b) Total Surface Area=2πr(h + r)
(c) Volume=πr2 h
4. Cone r:radius of the base
h:height, l:slant height
◦
30
45
1
√2
3
2
√1
3
√1
2
√1
2
2
√2
√3
3
◦
1
√
2
√
2
1
◦
90
3
2
1
√2
1
60
◦
√
3
√2
3
2
√1
3
0
ND
1
ND
0
Trigonometric Identities
1. sin2 θ + cos2 θ = 1
2. 1 + tan2 θ = sec2 θ
2
(a)
(b)
(c)
(d)
Curved Surface Area=πrl
Total Surface Area=2πr(l + r)
Volume= 13 πr2 h
By pythagoras theorem:l2 = h2 + r2
5. Sphere r:radius
(a) Surface Area=4πr2
(b) Volume= 43 πr3
6. Hemisphere r:radius
(a) Surface Area=2πr2
(b) Volume= 23 πr3
Co-ordinate Geometry
2
3. 1 + cot θ = cosec θ
Distance formula
Trigonometric Ratios of Complementary An- Let P (x , y ), Q(x , y ) be any two points
1 1
2 2p
gles
Distance formula= (x2 − x1 )2 + (y2 − y1 )2
The distance
1. sin(90◦ − θ) = cos θ
p of P (x, y) from origin O(0, 0) is given
by
OP
=
(x2 + y 2 )
2. cos(90◦ − θ) = sin θ
Section formula for internal division
◦
coordinates of the point P (x, y) which divides a line3. tan(90 − θ) = cot θ
segment joining the point A(x1 , y1 ), B(x2 , y2 ) inter4. cot(90◦ − θ) = tan θ
nallyin a given ratio m : n are given
by
mx2 + nx1
my2 + ny1
5. cosec(90◦ − θ) = sec θ
P ≡
,
m+n
m+n
6. sec(90◦ − θ) = cosec θ
Section formula for external division
Coordinates of the point P (x, y) dividing the seg.joining
Surface Areas & Volumes
1. Cuboid l:length, b:breadth, h:heigth
(a) Curved Surface Area=2h(l + b)
(b) Total Surface Area=2(lb + bh + hl)
(c) Volume=lbh
2. Cube a: side of the cube
(a) Curved Surface Area=4a2
(b) Total Surface Area=6a2
(c) Volume=a3
points
A(x1 , y1 ), B(x2 , y2 ) externally
are given by
P
≡
mx2 − nx1
my2 − ny1
,
m−n
m−n
Mid-point formula
Point P (x, y) is the midpoint of segment joining points
B(x2 , y2 ) its coordinates are given by P
≡
x1 + x2
y1 + y2
,
2
2
Coordinates of Centroid
If A(x1 , y1 ), B(x2 , y2 ), C(x3 , y3 ) are the vertices of
4ABC,
G(x, y) is the centroid then
x1 + x2 + x3 y1 + y2 + y3
G≡
,
3
3
A(x
1 , y1 ),
Compiled by metric, for soft copy mail me at [email protected]