Class TABLE OF CONTENT RELATIONS AND FUNCTIONS---------------------------------------------------- 1 TRIGONOMETRY FUNCTION------------------------------------------------------ 1 - 2 INVERSE TRIGONOMETRY FUNCTION---------------------------------------- 2 - 3 QUADRATIC EQUATIONS AND INEQUALITIES----------------------------- 3 - 4 COMPLEX NUMBERS---------------------------------------------------------------- 4 PERMUTATION AND COMBINATIONS------------------------------------------ 5
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BINOMIAL THEOREM---------------------------------------------------------------- 5 SEQUENCE AND SERIES----------------------------------------------------------- 6
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STRAIGHT LINES---------------------------------------------------------------------- 6 CONIC SECTIONS--------------------------------------------------------------------- 7 THREE DIMENSIONAL GEOMETRY--------------------------------------------- 8 DIFFERENTIAL CALCULUS-------------------------------------------------------- 9 CONTINUITY OF FUNCTIONS----------------------------------------------------- 10 DIFFERENTIAL AND APPLICATION--------------------------------------------- 10 - 11 INTEGRAL CALCULUS-------------------------------------------------------------- 12 - 13 PROBABILITY-------------------------------------------------------------------------- 14 MATRICES------------------------------------------------------------------------------- 15 - 16 DETERMINANT------------------------------------------------------------------------- 16 VECTORS-------------------------------------------------------------------------------- 17 - 18 STATISTICS----------------------------------------------------------------------------- 18 DIFFERENTIAL EQUATIONS------------------------------------------------------ 19
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Class
1
RELASTIONS AND FUNCTIONS 1. A function f : X®Y is one-one (or injective) if f ( x1 ) = f (x 2 ) Þ x1 = x 2" x1 , x 2 Î X .
2. A function f: X®Y is onto (or surjective) if given any yÎY, $xÎX such that f (x) = y 3. A function f: X®Y is one – one and onto (or bijective), if f is both one- one and onto 4. The composition of functions f: A®B and g: B®C is the function gof : A®C given by gof (x ) = g ( f ( x ))"x Î A
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5. A function f: X®Y is invertible if $g : Y®X such that gof = I X and fog = IY 6. A function f: X®Y is invertible if and only if f is one - one and onto.
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TRIGONOMETRY FUNCTION 1. sin(a m b ) = sin a cos b m cos a sin b tan a m tan b
3. tan(a m b ) = 1 ± tan a tan b 5. sin A - sin B = 2cos
4. sin A + sin B = 2sin
A+ B A- B sin 2 2 A+ B
A- B
7. cos A - cos B = - 2sin 2 sin 2 9. Cosine rule: cos A = æAö è ø
11.sin ç 2 ÷ =
b2 + c 2 - a2 2bc
( s - b)(s - c) bc
2. cos(a ± b ) = cos a cos b m sin a sin b A+ B A- B cos 2 2 A+ B
A- B
6. cos A + cos B = 2 cos 2 cos 2 a
b
c
8. Sine Rule: sin A = sin B = sin C = 2R æ B- C ö æ b -c ö æ Aö ÷=ç ÷ cot ç ÷ è 2 ø è b +c ø è 2 ø
10. tan ç
æAö
12. tan ç 2 ÷ = è ø
( s - b)(s - c) s( s - a )
é æ n - 1 ö ù é æ nb sin êa + ç ÷ b ú ê sin ç è 2 ø ûë è 2 ë 13. sin a + sin(a + b ) + sin(a + 2b ) + ......tonterms = sin(b / 2)
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öù ÷ú øû
; b ¹ 2np
Class
2
é æ n - 1 ö ù é æ n b öù cos êa + ç ÷ b ú ê sin ç ÷ú è 2 ø û ë è 2 øû ë 14.cosa + cos(a + b ) + cos(a + 2b ) + ......tonterms = ; b ¹ 2 np sin( b / 2) 1 2
1 2
1 2
15.D = ab sin C = bc sin A = ac sin B = æ Aö è ø
æBö è ø
æC ö è ø
17. r = 4 R sin ç ÷ .sin ç ÷ .sin ç ÷ 2 2 2
abc = rs 16. D = s( s - a )(s - b)( s - c ) 4R
18. a = c cos B + b cos C
INVERSE TRIGONOMETRY FUNCTION
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Class
3
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QUADRATIC EQUATIONS AND INEQUALITIES 2
-b ± b - 4ac 1. For the quadratic equation ax + bx + c = 0 , x = 2a 2
(i) (ii) (iii) (iv) (v)
If b = 0 Þ roots are of equal magnitude but of opposite sign If c = 0 Þ one root is zero other is –b /a If b = c =0 Þ both roots are zero. If a = c Þ roots are reciprocal to each other If a > 0, c > 0 üý Þ Roots are of opposite signs. a < 0, c < 0 þ
(vi)
If aa >< 0,0, bb 0,0cc> 0, c < 0 ýþ Þ both roots are positive If signs of a = sign of b ¹ sign of c ÞGreater root in magnitude is negative
þ
a > 0, b < 0, c > 0ü
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Class
4
2. If p +iq (p and q being real) is root of the quadratic equation, where i = -1 , then p –iq is also a root of the quadratic equation. 3. Every equation of nth degree (n³1) has exactly n roots and if the equation has more than n roots, it is an identity. 4. An inequality of the form loga f (x) > b is equivalent to the following systems of inequalities : f(x) > 0, f (x) > ab for a > 1. F(x)>0, f(x) < ab for a < 1.
(a) (b)
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COMPLEX NUMBERS 1. Let z = a+ib and z = c+id Then 1
2
z1 + z2 = (a + c) + i (b + d ) (b) z z = (ac - bd ) + i (ad + bc) 1 2 (a)
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2. For any non-zero complex number z = a + ib (a ¹ 0, b ¹ 0), there exists the complex number 2 a 2 + i 2 -b 2 , denoted by 1 or z -1 called the multiplicative inverse of z such a +b a +b z that (a + ib )æç 2 a 2 + i 2- b 2 ö÷ = 1 + i 0 = 1 a +b ø è a +b
3. For any integer k, i 4 k = 1, i 4 k +1 = i, i 4 k + 2 = -1, i 4 k +3 = - i 4. | z - z1 | + | z - z2 | = l , represents an ellipse if | z1 - z2 |< l , having the points z and z as its foci. And if | z1 - z2 |= l , then z lies on a line segment connecting z and z . 1
1
1
2
2
1
5. Cube root of unity The three cube root of unity are 1, 2 (-1 + - 3 ), 2 (-1 - -3 ) , which are the same as 1,cos æç 2p ö÷ + i sin æç 2p ö÷ and cos æ 4p ö + i sin æ 4p ö è 3 ø
è 3 ø
ç 3 ÷ è ø
ç 3 ÷ è ø
6. De Moivre's Theorem: for all real values of n, (cosq + i sin q )n = cos nq + i sin nq If z = cosq + i sin q , using De Moivre's Theorem n
z +
1 = 2 cos nq ; zn
n
z -
1 = 2i sin nq zn
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Class
5
PERMUTATION AND COMBINATIONS 1.
The number of permutations of n different things, taken r at a time, where repetition is allowed, is denoted by np and given by n = n! , where 0 £ r £ n r
pr
( n - r)!
2.
The number of permutations of n objects taken all at a time. Where repetition is allowed, is nr.
3.
The number of permutations of n objects where p1objects are of first kind, p2 objects are of the second kind, ..., pk objects are of the kth kind and rest, if any, l are all n! different is p1 ! p2 !..... pk !
4.
The number of combinations of n different things taken r at a time, denoted by nc , 0£r £ n , is given by n = n ! cr
5.
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r !(n - r)!
r
Number of circular permutations of n things when p alike and the rest different taken all at a time distinguish clockwise and anticlockwise arrangement is (n -1)!
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p!
BINOMIAL THEOREM 1. The expansion of a binomial for any positive integral n is given by Binomial Theorem. which is (a + b)n = nC an + nC1 an -1 + nC2 a n- 2b 2 + ..... + nCn -1 a.bn -1 + nCnb n 0
The coefficients of the expansions are arranged in an array. This array is called Pascal's triangle. When the index is other than a positive integer such as negative integer or fraction. The number of terms in the expansion of (1 + x)n is infinite and the symbol nCr cannot be used to denote the coefficients of the general term. 2. The general term of an expansion (a + b)n is Tr +1 = nC an - r.br The total number of terms in the expansion of (a + b)n is n +1. r
th
3. In the expansion (a + b)n , if n is even, then the middle term is the æç n + 1 ö÷ term. If n is odd, è2 ø then the middle terms are æ n + 1 öth and æ n + 1 öth terms. ç 2 ÷ è ø
ç 2 +1 ÷ è ø
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Class
6
SEQUENCE AND SERIES 1. The general term or the nth term of the A.P is given by an = a + (n -1)d. n
n
The sum Sn of the first n terms of an A.P is given by Sn = 2 [2a + (n - 1)d ] = 2 (a + l ) 2. The sum Sn of the first n terms of G.P is given by Sn =
a(r n - 1) a (r n - 1) or if r¹1 r -1 r -1
3. A series who's each term is formed, by multiplying corresponding terms of an A.P. and a G.P., is called ar Arithmetic-geometric series. Summation of n terms n -1
Sn =
a dr(1 - r ) [ a + (n -1) d n + .r 1- r (1 - r )2 1- r
4. Harmonical progression is defined as a series in which reciprocal "of its terms are in A.P. The standard from of a H.P. is 1 + 1 + 1 + ...... a
a+d
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a + 2d
STRAIGHT LINES
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1. An acute angle (say q) between lines L1 and L2 with slopes m1 and m2 is given by tan q =
m2 - m2 1 + m1 m2 , 1 + m1m2 ¹ 0
2. Equation of the line passing through the points (x1, y1) and (x2, y2) is given by y - y1= y2 - y1 ( x - x ) . x2 - x1
1
3. Equation of a line making intersects a and b on the x- and y- axis, respectively, is x y + =1 .a b
4. The perpendicular distance (d) of a line Ax + By +C =0 from a point (x1, y1)is given by d=
| Ax1 + By1 + C | A2 + B 2
5. Distance between the parallel lines Ax + By +C1 =0 and Ax + By +C2 =0, is given by . | C1 - C2 | d=
A2 + B2
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Class
7
CONIC SECTIONS 1. The equation of a circle with centre (h, k) and the radius r is (x –h)2 + (y –k)2 = r2. 2. Equation of tangent: xx1 + yy1 +g(x + x1) +f(y + y1) +c =0 3. The equation of the parabola with focus at (a, 0) a > 0 and directrix x = - a is y2 =4ax. 4. The equation of the parabola is a line segment perpendicular to the axis of the parabola, through the focus and whose end points lie on the parabola. 5. Length of the latus rectum of the parabola y2 = 4ax is 4a.
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6. The parametric equation of the parabola is x = at2, y = 2at.
7. An ellipse is the set of all points in a plane, the sum of whose distances from two fixed points in the plane is a constant.
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x y2 8. The equations of an ellipse with foci on the x-axis is a2 + b2 = 1 ; its parametric equation is x = a cos q ; y = b sin q x2 y2 2b2 + = 1 9. Latus rectum of the ellipse a2 b2 is a
2
10.The eccentricity of an ellipse is the ratio between the distances from the centre of the ellipse to one of the foci and to one of the vertices of the ellipse. 11.A hyperbola is the set of all points in a plane, the difference of whose distances from two fixed points in the plane is a constant. x2 y2 12. The equation of a hyperbola with foci on the x-axis is a2 - b2 = 1 2 2 Two asymptotes: x 2 - y2 = 0 a b 2b2 x2 y2 13.Latus rectum of the hyperbola : a2 - b2 = 1 is a
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Class
8
THREE DIMENSIONAL GEOMETRY 1. The coordinates of the points R which divides the line segment joining two points P(x1, y1, z1) and Q(x2, y2,z2) internally and externally in the ratio m : n are given by æ mx 2 + nx1 my 2 + ny1 mz 2 + nz1 ö and æ mx 2 - nx1 , my2 - ny1 , mz 2 - nz1 ö , respectively. , , ç m+n è
m +n
ç m +n è
m + n ÷ø
m- n
m -n
÷ ø
2. The coordinates of the centroid of the triangle, whose vertices are (x1, y1, z1), (x2, y2,z2), (x3, y3,z3) are æ x1 + x 2 + x3 , y1 + y2 + y3 , z1 + z2 + z3 ö ç è
3
3
÷ ø
3
3. If l, m ,n are the direction cosines and a, b, c are the direction ratios of a line then l=
±a
a2 + b 2 + c 2
,m =
±b
a 2 + b2 + c
,n = 2
±c
a 2 + b2 + c 2
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4. If l1, m1,n1 and l2, m2,n2 are the direction cosine of two lines; and q is the acute angle between two lines; then cosq =| l1l2 + m1 m2 + n1n2 |
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5. Equation of a line through a point (x1, y1, z1) and having direction cosine, m, n is x - x1 y - y1 z - z1 = = l m n
r ur
ur
r uur
ur uur
uur
6. Shortest distance between r = a1 + l b1 and r = a2 + mb2
1
is
uur ur
. a -a ) (b ´ bur)( uur 2
2
1
b1 ´ b2
r
7. The equation r uur through a point whose position vector is a and perpendicular to uur ofr plane the vector N is (r - a ).N =0 8. Vector equation r ur r uur of a plane r ur that uur passes through the line of intersection of planes r.n1 = d1 and r.n2 = d 2 is r.(n1 + l n2 )= d1 + l d 2 where l is any nonzero constant. r r rr d a .n$ 9. The distance of a point whose position vector is a from the plane r.n = d is
10.Skew line: Two straight line are said to be skew lines if they are parallel nor intersecting. å (x2 - x1 )(m1n2 - m2n1 ) Shortest distance:
å (m n
1 2
- m2 n1 )
2 2 2 2 11. Equation of a sphere: (x - a) + ( y - b) + ( z - c ) = R where centre is (a,b,c) and radius R 2 2 2 General form: (x + y + z ) + 2ux + 2vy + 2 wz + d = 0
- u -v - w
é ù Centre ê a , a , a ú and u 2 + v 2 + w2 - d ë û a a a a 2
2
2
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Class
9
DIFFERENTIAL CALCULUS LIMIT 1. A function f has the limit L as X approaches a if the limit from the left exists and both f ( x) = lim f ( x ) = lim f ( x) remember L im it x a limit are L that is, lim x®a x®a x ®a x® a Let L im f ( x ) = l and L im g ( x ) = m . If l and m are finite then: x ®a
x® a
(i)
L im ( f ( x) ± g ( x ) ) = l ± mf
(ii)
L im f ( x ).g ( x) = l .m
(iii)
L im
(iv)
L im Kf ( x ) = k L im f ( x ) = kl ; where k is a constant.
(v)
L im [ f ( x ) + k ] = L im f ( x ) + k , where k is a constant. x ®a x ®a
(vi)
x ®a
x ®a
x®a
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f ( x ) l , provided m¹0 = g ( x) m
x ®a
x®a
f ( x ) £ L im g ( x) If f ( x ) £ g ( x ), then Lx ®im a x ®a
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2. Limit in case of composite function: ; provided f is continuous at X L im f [g ( x) ] = f L im g ( x )
(
x ®a
x ®a
)
sin x x sin-1 x x L im = L im = L im = L im -1 = 1 3. x®0 x x ®0 sin x x®0 x®0 sin x x
(Where x is measured in radian) x
x
ax
æ 1ö 1/ x æ aö a/ x æ 1ö im ç 1 + ÷ = L im (1 + x ) = e , L im ç 1 + ÷ = L im (1 + x ) = L im ç 1 + ÷ = ea 4. Lx®¥ x ®0 x®¥ è x ®0 x ®¥ è xø xø xø è
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Class
10
CONTINUITY OF FUNCTIONS 1. All Polynomials, Trigonometrically, exponential and logarithmic function are continuous in their domain. 2. If f(x) is continuous and g(x) is discontinuous at x =0 then the product function f ( x ) = f ( x ) g( x ) is not necessarily be discontinuous at x =a. p ,x ¹0 x e.g. f(x) & g(x) 0, x = 0 sin
3. For any positive integer n and any continuous function f, [if (x)]n and n f ( x ) are continuous. When n is even, the inputs of f in n f ( x ) are restricted to inputs x for which f(x) ³ 0
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4. If f(X) and (g(x) are continuous, then so are f(x) + g(x), f(x)- g(X), and f(x). g(x). 5. If f(X) and g(x) are continuous, so is g(x) / f(x), so long as the inputs x do not yield outputs f(x) =0.
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DIFFERENTIAL AND APPLICATION 1. Interpretation of the Derivative: If y = f(x) then, (a) m = f'(a) is the slope of the tangent line to y =f(x) at x =a and the equation of the tangent line at x =a is given by y= f(a) +f'(a) (x-a). (b) f'(a) is the instantaneous rate of change of f(x) at x =a. (c) If f(x) is the position of an object at time x then f'(a) is the velocity of the object at x =a. 2. Basic Properties and Formulas : If f(x) and g(x) are differentiable functions (the derivative exists), c and n are any real numbers 1. (cf') = cf' (x) 2. ( f ± g )' = f '( x ) ± g '( x ) 3. (fg)' = f'g +fg’ '
æ fö f ' g - fg ' 4. ç g ÷ = g2 è ø d 5. (x )n = nx n -1 dx d ( f ( g ( x )) = f '( g ( x )) g '( x ) 6. dx
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Class
11
3. Increasing / Decreasing: (i) If f'(x) > 0 for all x in an interval I then f(x) is increasing on the interval I. (ii) If f'(x) 0 to the left of x =c and f'(x) < 0 to the right of x =c. 2. A rel. max. of f(x) if f'(x) < 0 to the left of x =c and f'(x) > 0 to the right of x =c. 3. Not a relative extrema of f(x) if f'(x) is the same sign on both sides of x=c.
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2nd Derivative Test: If x =c is a critical point of f(x) such that f'© =0 then x =c. 1. Is a relative maximum of f(x) if f''(c ) < 0. 2. Is a relative maximum of f(x) if f''(c ) > 0. 3. F(x) may have a relative maximum and minimum, or neither if f''(c ) =0.
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6. Mean value theorem: If f(x) is continuous on the closed interval [a,b] and differentiable on the open interval (a,b) then there is a number a < c< b such that n
æ ö f (b ) - f (a ) = nx n -1 ÷ ç f '(c ) = b -a è ø
7. Rolle's theorem: If a function f(x) is continuous on the closed interval [a,b] and differentiable in an interval (a,b) and also f(a) = f(b), then there exist at least one value of c of x in the interval (a,b) such that f'© =0. 8. L' Hospital's Rule: lim f ( x) = f '(a ) = f ''(a) x®a
f ( x)
f '(a )
f ''(a)
If f(a) =0 or ¥ , f '(a ) =0 or ¥ 9. Length of Sub- tangent = y1 æç dx ö÷ ( x1 , y1 ) ; sub- normal = . y1 æç dy ö÷ ( x1 , y1 ) è dx ø è dy ø Length of tangent = Length of normal =
ìï æ dx ö 2 üï y1 í1 + ç ÷ ý ïî è dy ø ( x1 , y1 ) ïþ ìï æ dy ö 2 üï y1 í1 + ç ÷ ý îï è dx ø ( x1 , y1 ) þï
10. Orthogonal trajectory: Any curve which cuts every member of a given family of curves at right angle, is called an orthogonal trajectory of the family.
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Class
12
INTEGRAL CALCULUS
x 1. Fundamental theorem of calculus: g ( x ) = Part I: If f(x) is continuous on [a,b] then ò f (t ) dt is also x continuous on [a,b] andg '( x ) = d f (t ) dt = f ( x ) a ò
dx
a
Part II: f(x) is continuous on [a,b], F(x) is an anti- derivative of f(x) i.e. F ( x) = ò f ( x) dx then
b
ò f ( x )dx = F (b) - F ( a) a
2. Integration by Substitution: The substitution u = g(x) will convert b
g (b )
ò
ò
f ( g ( x ))g '( x )dx =
a
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f (u )du using du = g'(x)dx.
g (a )
b
b
3. Integration by parts:òudv=uv -òvdu and ò udv = uv | - ò vdu. Choose u and dv from integral b a
a
a
and compute du by differentiating u and compute v using v =òdv
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4. Integration by partial fraction: If integrating
P ( x)
ò Q( x ) dx
where the degree of P(x) is
smaller than the degree of Q(x). Factor denominator as completely as possible and find the partial fraction decomposition of the rational expression. Integrate the partial fraction decomposition (P.F.D). 5. Some properties of definite integral: (a) If an interval [a, b] (a < b), the function f(x) and f ( x) satisfy the a b condition f ( x ) £ f ( x ) , then ò f ( x )dx £ òf ( x) dx b
a
(b) If m and M are the smallest and greatest values of a function f(x) on an interval a [a,b] and a £ b then m(b - a ) £ f ( x )dx £ M (b - a ) b
(c)
ò a
(d)
b
ò a
(e)
na
ò -a
a
(f)
ò
-a
c
b
ò b
f ( x )dx = ò f ( x )dx + ò f ( x ) , where a < c < b a
c
b
f ( x )dx = ò f ( a + b - x )dx a a
f ( x )dx = nò f (x )dx , where a is the period of the function and nÎI 0
ì a ï2 f (x )dx ,if f(x) is even function, i.e. f(-x) =f(x) f ( x )dx = í ò0 ï î0,if f(x) is odd function, i.e. f(-x) =f(x)
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Class 2a
(g)
ò 0
13
ì a ï2 f ( x )dx, f(2a-x)=f(x) f ( x) dx = í ò0 ï î0,if f(2a-x)=-f(x) g (x )
6. Leibnitz Rule: d dx
ò
f (t )dt = g '( x ) F ( g ( x )) - f '(x ) F ( f ( x ))
f (x )
7. If a series can be put in the form 1 r =n -1 æ r ö 1 r =n f ç ÷ or å å n r= 0 è n ø n r =1
0 ærö f ç ÷ , then its limit as n®¥ is ò f ( x )dx ènø 1
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APPLICATION OF INTEGRALS b
8. Net area: ò f ( x )dx represents the area between the curve y =f(x), x –axis and two coa
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ordinates at x =a and x =b (b > a) with area above x –axis positive and area below x –axis negative. 9. Area between curve: b
y = f ( x ) Þ A = ò [upper function]-[lower function]dx And ab
x = f ( y ) Þ A = ò [right function]-[left function]dy a
If the curves intersect then the area of each portion must be found individually. b 10 The volumes of the solid generated by the revolution about the x-axis of the area 2
bounded by the curve y =f(x), the x- axis and the ordinates x =a, x =b is
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ò p y dx a
Class
14
PROBABILITY 1. Probability of an event: For a finite sample space with equally likely outcomes Probability of an event P ( A) = n ( A) , where n (A) = number of elements in the set A, n( S )
n(S) = number of elements in the set S. 2. If A and B are any two events, then P(A or B) = P(A) + P(B)- P(A and B) Equivalently, p ( A U B ) = P ( A) + P( B ) - P( A I B) 3. If A and B are mutually exclusive, then P(A or B) = P(A) + P(B) 4. If A is any event, then P (not A) = 1 – P(A)
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5. The conditional probability of an event E, given the occurrence of the event F is given P( E I E ) by p ( E | F) =
P( F)
, P( F) ¹ 0
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6. Theorem of total Probability: let {E1, E2, ……….,En} be an partition of a sample space and suppose that each of E1, E2, …..,En has nonzero probability. Let a be any event associated with S then P(A) = P(E1) P(A |E1)+P (E2)P(A|E2) +…..+P(En) P(A|En) 7. Bayes' theorem: If E1, E2, ……….,En are events which constitute a partition of sample space S, i.e. E1, E2, ……….,En are pair wise disjoint and E1 U E2 U .... U E n = S and P (E i ) P( Ei | A)) A be any event with non zero probability, then P ( Ei | A) =
n
å P( E ) P ( A | E ) j
j
j =1
8. Let X be a random variable whose possible values x1, x2, x3,….xn occur with probabilities p1, p2, p3,…..,pn respectively. The mean of X, denoted by m is the number n
åx p i
i
i=1
The mean of a random variable X is also called the expectation of X, denoted by E(X). 9. Trails of a random experiment are called Bernoulli trails, if they satisfy the following conditions: (a) There should be a finite number of trails. (b) The trails should be independent. (c) Each trail has exactly two outcomes: success or failure. (d) The probability of success remains the same in each trail. For Binomial distribution B (n, p), P (X = x) = nCx qn-x, x = 0, 1, …, n (q = 1- p)
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Class
15
MATRICES 1. Order of a matrix: A matrix which has in rows and n columns is called a matrix of orderm´n. A = éë aij ùû m´n where i=1,2,…..m J=1,2,……n Here a denotes the element of ith row and jth column. ij
2. Properties of scalar Multiplication : If A, B are Matrices of the same order and p, q are any two scalars then – i. p(A + B) = pA + pB ii. (p + q) A = pA + qA iii. P(qA) = (pqA) = q(PA) iv. (-pA) = -(pA) = p(-A) v. tr(kA) = k tr(A)
s s la
3. Multiplication of matrices: If A and B be any two matrices, then their product AB will be defined only when number of column in A is equal to the number of rows in B. If A = éëaij ùû m´ n and B = éë bij ùû then their product AB = C = éë cij ùû , will be matrix of order m´p, n´ p n
where ( AB )ij = cij = å airbrj r =1
4. Positive Integral Powers of Matrix : For any positive integral m, n i. Am An = Am+n ii. (Am)n = Amn = (An)m iii. In = I, Im = I iv. A0 =In where A is a square of order n.
C
5. Transpose of a matrix: The matrix obtained from a given matrix A by changing its rows into columns or columns into rows is called transpose of matrix A and is denoted by AT or A'. From the definition it is obvious that if order of A is n´m .Properties of Transpose i. (AT)T = A ii. ( A ± B )T = AT ± BT iii. (AB)T = BT AT iv. (kA)T = k(A)T K is scalar v. IT = I vi. tr (A) = tr (A)T vii.(A1A2A3….An-1 An)T = AnT An-1 T ……A3T A2T A1T
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16
6. Inverse of matrix: If A and B are two matrices such that AB = I = BA then B is called the inverse of A and it is denoted by A-1, thus A-1 = B ÛAB = I = BA To find inverse matrix of a given matrix A we use following formula A-1 =
adj .A | A|
Thus A-1 exists Û |A| ¹ 0
DETERMINANT 1. To find the value of a third order determinant a11
a12
a13
s s la
Let D = a21 a22 a23 a31
a32
a33
Be a third order determinant. To find its value we expand it by any row or column as the sum of three determinants of order 2. If we expand it by first row then D = a11
a22
a23
a32
a33
- a12
a21 a23 a31
a33
+ a13
a21
a22
a31
a32
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2. Minor: The Determinant that is left by cancelling the row and column intersecting at a particular element is called the minor of that element. 3. Cofactor: The cofactor of an element aij is denoted by Fij and is equal to (-1) i+j Mij where M is a minor of element aij. 4. Multiplication of two determinants: Multiplication of two third determinants is defined as follows: a1
b1
l1
m1
n1
a1m1 + b1 m2 + c1m3
a1 n1 + b1 n2 + c1 n3
a2 b2
c2 ´ l2
m2
n2 = a2 l1 + b2 l2 + c2l 3 a2 m1 + b2 m2 + c2 m3
a2 n1 + b2 n2 + c 2 n3
a3
c3
m3
n3
a3 n1 + b3 n2 + c3 n3
b3
c1
l3
a1l1 + b1l 2 + c1 l3 a3 l1 + b3 l2 + c3 l3
a3 m1 + b3 m2 + c3 m3
5. System of linear equation in three unknowns : Using Crammer's rule of determinant we get x = y = z = 1 i.e. x = D1 , y = D 2 , z = D 3 D1
I. ii. ii. iv.
D2
D3
D
D
D
D
Case-I: If D¹0 and If at least one of D1,D2,D3 is not zero then the system of equation is inconsistent i.e. has no solution. If d1 = d2 =d3 = 0 or D1,D2,D3 are all zero then the system is consistent and has infinitely many solutions.
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VECTORS ur
1. Triangle law of vector addition of two vector Magnitude of R : R = A 2 + B 2 + 2 AB cosq ur
ur
uur uur
2. Components of vector: Consider a vector A that lies in xy plane A = A1 + A2 as shown ur uur uur A1 = Ax $i, A2 = Ay $j Þ A = Ax i$ + Ay $j
ur
The quantities Ax and Ay are called x- and y- components of the vector A = Ax is itself not a vector but Ax i$ is a vector and so is Ay $j. Ax = A cosq and Ay = A sin q ur ur
3. Scalar Product : A.B = AB cos q (here q is the angle between the vectors) ur
ur ur
4. Vector product: C = A ´ B = AB cosq $n r
r
r
r
r
r
r
r
r
s s la
r r r
5. Given vectors x1 a + y1 b + z1 c, x2 a + y2 b + z2 c , x3 a + y3 b + z3 c , where a, b, c are non – coplanar vectors, will be coplanar if and only if x y z 6. Scalar rtriple ur product uur uur : (a). If a = a1 $i + a2 $j + a3 r r r
r rr
a1 a2
(a ´ b ).c = [ abc] = b
1
c1
b2 c2
a3
1
1
x2
y2
z2 = 0
x3
y3
z3
r ur uur ur k, b = b i$ + b $j + b k 1 2 3
b3 c3
1
r ur uur ur and c = c1 i$ + c2 $j + c3 k then
C
r r r a, b, c
(b). [a b c] = volume of the parallelepiped whose coterminous edges are formed by r r r
(c). a, b, c
r r r
are coplanar if and only if [ a, b, c ] = 0 r r r ur
, c , d rur respectively (d). Four points A,uuuB, C, D with position vectors ra, brr are coplanar if ruuuruuur r and only if [ AB AC AD ] =0 i.e. if and only if [b - ac - ad - a] = 0
r r r
1 rrr 6 r r r 1 r rr with three coterminous edges a, b, c = |[ abc ]| 2 r ur r r r ur r ur r r a.d r ur = (a.c )(b.d ) - (a.d )(b.c) b.d
(e) Volume of a tetrahedron with three coterminous edges a, b, c = | [ abc] | (f) Volume of prism on a triangle base rr r r r ur a.c 7. Lagrange's identity: (a ´ b ).( c ´ d ) = r r b.c
8. Reciprocal system of vectors: rrr r rr If abc be any three non coplanar vector so that [abc] ¹ 0 then the three vectors
r r r r r r r r rur b ´ c ur c ´ a r a ´b a ' b' c ' defined by the equations a ' = r rr , b ' = rrr c ' = rrr are called the reciprocal [ abc ] [abc ] [abc ] r r r system of vectors to the given vector a, b, c
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9. Application of vector in geometry: r
r
r
r
r r
(a)
Vector equation of a straight line passing through two points a and b is r = a + t (b - a)
(b)
Vector equation of a plane passing through the points a, b, c
r r r
r r r r r = (1 - s - t )a + sb + tc
r r r r r
r r
is
rrr
or r.(b ´ c + c ´ a + a ´ b) = [abc] r
r
( c ) Vector of a plane passing through the point a and perpendicular to n r r requation r is r.n = a.n r
(d)
Perpendicular distance of a point P(r) from a line passing through a and parallel r to b is given by 1/ 2
r r r | (r - a ) ´ b | uur PM = |b |
(e)
r r r 2 ér r ìï (r - a).b üï ù 2 = ê(r - a ) - í uur ý ú ê ïî | b | ïþ úû ë
s s la
r
Perpendicular distance of a point P® from a plane passing through the points a r r r r r r r r r r (r - a).(b ´ c + c ´ a + a ´ b) b and c is given by PM = r r r r r r b ´ c + c ´ a + a´ b
C
STATISTICS 1. Mean deviation for ungrouped data M .D( x) = 2.
å| x
f Mean deviation for grouped data M .D( x) = å
i
i
- x|
n | xi - x | N
M .D ( M ) = å
| xi - M | n
, where N=åfi
2 3. Variance and standard deviation for ungrouped data s =
2 1 2 1 ( xi - x ) , s = ( xi - x ) å å n n
4. Variance and standard deviation of a distance frequency distribution 2
s =
2 2 1 1 f i ( xi - x) f i ( xi - x ) , s = å å n n
5. Variance and standard deviation of a continuous frequency distribution 2 1 æ å f i xi ö 1 f i ( xi - x) 2 å , s = å fi xi - çç ÷÷ n n è N ø s 6. Coefficient of variation (C.V) = ´ 100 , x ¹ 0 x 2
s =
For series with equal means, the series with lesser standard deviation is more consistent or less scattered.
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19
DIFFERENTIAL EQUATIONS dy
1. A differential equation of the form dx + Py = Q , where P and Q are constants or functions of x only is called a first order linear differential equation. (a)Differential equation of the form dy = f ( x) : y = ò f ( x) dx + c dx
dy dy = f ( x ) g( y ) Þ ò = f ( x )dx + c dx g ( y) ò (c) dy = f (ax + by + c) : ò dv = ò dx dx a + bf (v )
(b)
s s la
(d)Differential equation of homogeneous type: An equation in x and y is said to be homogeneous if it can be put in the form dy f ( x, y ) where f(x,y) and g(x,y) are both dx
=
g ( x, y)
homogeneous functions of the same degree in x & y.
C
dx dv ò x = ò f (v) - v + c
(e)Differential Equation reducible to homogeneous form: A differential equation of the form dy = a1 x + b1 x + c1 , where a1 ¹ b1 can be reduced to a2 b2 dx a2 x + b2 x + c2 homogeneous form by using X = X + h, y = Y + k so that dY dy dX
=
dx
2. Linear differential equations: dy + Py = Q; dx
ye ò
pdx
= ò Qe ò pdx + c
pdx is called the integrating factor for this equation. eò
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