Mathematics formula sheet, Cheat Sheet of Mathematics

Download Mathematics formula sheet and more Cheat Sheet Mathematics in PDF only on Docsity! Page # 1 S.No. Topic Page No. 1. Straight Line 2 – 3 2. Circle 4 3. Parabola 5 4. Ellips 5 –6 5. Hyperbola 6 – 7 6. Limit of Function 8 – 9 7. Method of Differentiation 9 – 11 8. Application of Derivatves 11 – 13 9. Indefinite Intedration 14 – 17 10. Definite Integration 17 – 18 11. Fundamental of Mathematics 19 – 21 12. Quadratic Equation 22 – 24 13. Sequence & Series 24 – 26 14. Binomial Theorem 26 – 27 15. Permutation & Combinnation 28 – 29 16. Probability 29 – 30 17. Complex Number 31 – 32 18. Vectors 32 – 35 19. Dimension 35 – 40 20. Solution of Triangle 41 – 44 21. Inverse Trigonometric Functions 44 – 46 22. Statistics 47 – 49 23. Mathematical Reasoning 49 – 50 24. Sets and Relation 50 – 51 INDEX MATHEMATICS FORMULA BOOKLET - GYAAN SUTRA Page # 2 MATHEMATICS FORMULA BOOKLET - GYAAN SUTRA STRAIGHT LINE 1. Distance Formula: 2 2 1 2 1 2d (x – x ) (y – y )  . 2. Section Formula : x = nm xnxm 12   ; y = nm ynym 12   . 3. Centroid, Incentre & Excentre: Centroid G        3 yyy, 3 xxx 321321 , Incentre I           cba cybyay, cba cxbxax 321321 Excentre I1           cba cybyay, cba cxbxax 321321 4. Area of a Triangle:  ABC = 1yx 1yx 1yx 2 1 33 22 11 5. Slope Formula: Line Joining two points (x1 y1) & (x2 y2), m = 21 21 xx yy   6. Condition of collinearity of three points: 1yx 1yx 1yx 33 22 11 = 0 7. Angle between two straight lines : tan  = 21 21 mm1 mm   . Page # 5 PARABOLA 1. Equation of standard parabola : y2 = 4ax, Vertex is (0, 0), focus is (a, 0), Directrix is x + a = 0 and Axis is y = 0. Length of the latus rectum = 4a, ends of the latus rectum are L(a, 2a) & L’ (a,  2a). 2. Parametric Representation: x = at² & y = 2at 3. Tangents to the Parabola y² = 4ax: 1. Slope form y = mx + m a (m  0) 2. Parametric form ty = x + at2 3. Point form T = 0 4. Normals to the parabola y² = 4ax : y  y1 = a2 y1 (x  x1) at (x1, y1) ; y = mx  2am  am3 at (am2 2am) ; y + tx = 2at + at3 at (at2, 2at). ELLIPSE 1. Standard Equation : 2 2 2 2 b y a x  = 1, where a > b & b² = a² (1  e²). Eccentricity: e = 2 2 a b1 , (0 < e < 1), Directrices : x = ± e a . Focii : S  (± a e, 0). Length of, major axes = 2a and minor axes = 2b Vertices : A ( a, 0) & A  (a, 0) . Latus Rectum : =  2 2 e1a2 a b2  2. Auxiliary Circle : x² + y² = a² 3. Parametric Representation : x = a cos  & y = b sin  4. Position of a Point w.r.t. an Ellipse: The point P(x1, y1) lies outside, inside or on the ellipse according as; 1 b y a x 2 2 1 2 2 1  > < or = 0. Page # 6 5. Line and an Ellipse: The line y = mx + c meets the ellipse 2 2 2 2 b y a x  = 1 in two points real, coincident or imaginary according as c² is < = or > a²m² + b². 6. Tangents: Slope form: y = mx ± 222 bma  , Point form : 1 b yy a xx 2 1 2 1  , Parametric form: 1 b siny a cosx     7. Normals: 1 2 1 2 y yb x xa  = a²  b², ax. sec   by. cosec  = (a²  b²), y = mx    222 22 mba mba   . 8. Director Circle: x² + y² = a² + b² HYPERBOLA 1. Standard Equation: Standard equation of the hyperbola is 12b 2y 2a 2x  , where b2 = a2 (e2  1). Focii : S  (± ae, 0) Directrices : x = ± a e Vertices : A (± a, 0) Latus Rectum (  ) : = a b2 2 = 2a (e2  1). 2. Conjugate Hyperbola : 1 b y a x 2 2 2 2  & 1 b y a x 2 2 2 2  are conjugate hyperbolas of each. 3. Auxiliary Circle : x2 + y2 = a2. 4. Parametric Representation : x = a sec & y = b tan  Page # 7 5. Position of A Point 'P' w.r.t. A Hyperbola : S1  1 b y a x 2 2 1 2 2 1  >, = or < 0 according as the point (x1, y1) lies inside, on or outside the curve. 6. Tangents : (i) Slope Form : y = m x 222 bma  (ii) Point Form : at the point (x1, y1) is 1 b yy a xx 2 1 2 1  . (iii) Parametric Form : 1 b anty a secx     . 7. Normals : (a) at the point P (x1, y1) is 1 2 1 2 y yb x xa  = a2 + b2 = a2 e2. (b) at the point P (a sec , b tan ) is    tan yb sec xa = a2 + b2 = a2 e2. (c) Equation of normals in terms of its slope 'm' are y = mx    222 22 mba mba   . 8. Asymptotes : 0 b y a x  and 0 b y a x  . Pair of asymptotes : 0 b y a x 2 2 2 2  . 9. Rectangular Or Equilateral Hyperbola : xy = c2, eccentricity is 2 . Vertices : (± c, ±c) ; Focii :  c2,c2  . Directrices : x + y =  c2 Latus Rectum (l ) :  = 2 2 c = T.A. = C.A. Parametric equation x = ct, y = c/t, t  R – {0} Equation of the tangent at P (x1 , y1) is 11 y y x x  = 2 & at P (t) is t x + t y = 2 c. Equation of the normal at P (t) is x t3  y t = c (t4  1). Chord with a given middle point as (h, k) is kx + hy = 2hk. Page # 10 2. Basic Theorems 1. dx d (f ± g) = f(x) ± g(x) 2. dx d (k f(x)) = k dx d f(x) 3. dx d (f(x) . g(x)) = f(x) g(x) + g(x) f(x) 4. dx d       )x(g )x(f = )x(g )x(g)x(f)x(f)x(g 2  5. dx d (f(g(x))) = f(g(x)) g(x) Derivative Of Inverse Trigonometric Functions. dx xsind 1– = 2x1 1  , dx xcosd 1– = – 2x1 1  , for – 1 < x < 1. dx xtand 1– = 2x1 1  , dx xcotd 1– = – 2x1 1  (x  R) dx xsecd 1– = 1x|x| 1 2  , dx xeccosd 1– = – 1x|x| 1 2  , for x  (– , – 1)  (1, ) 3. Differentiation using substitution Following substitutions are normally used to simplify these expression. (i) 22 ax  by substituting x = a tan , where – 2  <  2  (ii) 22 xa  by substituting x = a sin , where – 2    2  (iii) 22 ax  by substituting x = a sec , where [0, ],  2  (iv) xa ax   by substituting x = a cos , where (0, ]. Page # 11 4. Parametric Differentiation If y = f() & x = g() where  is a parameter, then    d/xd d/yd dx dy . 5. Derivative of one function with respect to another Let y = f(x); z = g(x) then )x('g )x('f xd/zd xd/yd zd yd  . 6. If F(x) = )x(w)x(v)x(u )x(n)x(m)x(l )x(h)x(g)x(f , where f, g, h, l, m, n, u, v, w are differentiable functions of x then F (x) = )x(w)x(v)x(u )x(n)x(m)x(l )x('h)x('g)x('f + )x(w)x(v)x(u )x('n)x('m)x('l )x(h)x(g)x(f + )x('w)x('v)x('u )x(n)x(m)x(l )x(h)x(g)x(f APPLICATION OF DERIVATIVES 1. Equation of tangent and normal Tangent at (x1, y1) is given by (y – y1) = f(x1) (x – x1) ; when, f(x1) is real. And normal at (x1 , y1) is (y – y1) = – )x(f 1 1 (x – x1), when f(x1) is nonzero real. 2. Tangent from an external point Given a point P(a, b) which does not lie on the curve y = f(x), then the equation of possible tangents to the curve y = f(x), passing through (a, b) can be found by solving for the point of contact Q. f(h) = ah b)h(f   Page # 12 And equation of tangent is y – b = ah b)h(f   (x – a) 3. Length of tangent, normal, subtangent, subnormal (i) PT = 2m 11|k|  = Length of Tangent p(h,k) N M T (ii) PN = 2m1|k|  = Length of Normal (iii) TM = m k = Length of subtangent (iv) MN = |km| = Length of subnormal. 4. Angle between the curves Angle between two intersecting curves is defined as the acute angle between their tangents (or normals) at the point of intersection of two curves (as shown in figure). tan  = 21 21 mm1 mm   5. Shortest distance between two curves Shortest distance between two non-intersecting differentiable curves is always along their common normal. (Wherever defined) 6. Rolle’s Theorem : If a function f defined on [a, b] is (i) continuous on [a, b] (ii) derivable on (a, b) and (iii) f(a) = f(b), then there exists at least one real number c between a and b (a < c < b) such that f(c) = 0 Page # 15 (xi)  secx dx = n (secx + tanx) + c OR n tan  4 2     x + c (xii)  cosec x dx = n (cosecx  cotx) + c OR n tan x 2 + c OR  n (cosecx + cotx) + c (xiii)  d x a x2 2 = sin1 x a + c (xiv)  d x a x2 2 = 1 a tan1 x a + c (xv)  22 ax|x| xd  = 1 a sec1 x a + c (xvi)   22 ax xd = n  x x a 2 2 + c (xvii)  d x x a2 2 = n  x x a 2 2 + c (xviii)  d x a x2 2 = 1 2a n xa xa   + c (xix)  d x x a2 2 = 1 2a n ax ax   + c (xx)  a x2 2 dx = x 2 a x2 2 + a2 2 sin1 x a + c (xxi)  x a2 2 dx = x 2 x a2 2 + a2 2 n          a axx 22 + c (xxii)  x a2 2 dx = x 2 x a2 2  a2 2 n          a axx 22 + c Page # 16 3. Integration by Subsitutions If we subsitute f(x) = t, then f (x) dx = dt 4. Integration by Part :   )x(g)x(f dx = f(x)   )x(g dx –     dxdx)x(g)x(f dx d         5. Integration of type 2 2 2 dx dx, , ax bx c dx ax bx c ax bx c          Make the substitution bx t 2a   6. Integration of type 2 2 2 px q px qdx, dx, (px q) ax bx c dx ax bx c ax bx c             Make the substitution x + b 2a = t , then split the integral as some of two integrals one containing the linear term and the other containing constant term. 7. Integration of trigonometric functions (i) 2 dx a bsin x OR 2 dx a bcos x OR 2 2 dx asin x bsinx cosx ccos x  put tan x = t. (ii) dx a bsinx OR dx a bcos x OR dx a bsinx c cos x  put tan x 2 = t (iii) a.cos x b.sinx c .cos x m.sin x n      dx. Express Nr  A(Dr) + B d dx (Dr) + c & proceed. Page # 17 8. Integration of type 2 4 2 x 1 dx x Kx 1    where K is any constant. Divide Nr & Dr by x² & put x  1 x = t. 9. Integration of type dx (ax b) px q  OR 2 dx (ax bx c) px q   ; put px + q = t2. 10. Integration of type 2 dx (ax b) px qx r    , put ax + b = 1 t ; 2 2 dx (ax b) px q   , put x = 1 t DEFINITE INTEGRATION Properties of definite integral 1.  b a )x(f dx =  b a )t(f dt 2.  b a )x(f dx = –  a b )x(f dx 3.  b a )x(f dx =  c a )x(f dx +  b c )x(f dx 4.   a a )x(f dx =   a 0 ))x(f)x(f( dx =        )x(f–)x(–f,0 )x(f)x(–f,dx)x(f2 a 0 5.  b a )x(f dx =   b a )xba(f dx Page # 20 Factorisation of the Sum or Difference of Two Sines or Cosines: (a) sinC + sinD = 2 sin 2 DC cos 2 DC (b) sinC  sinD = 2 cos 2 DC sin 2 DC (c) cosC + cosD = 2 cos 2 DC cos 2 DC (d) cosC  cosD =  2 sin 2 DC sin 2 DC Multiple and Sub-multiple Angles : (a) cos 2A = cos²A  sin²A = 2cos²A  1 = 1  2 sin²A; 2 cos² 2  = 1 + cos , 2 sin² 2  = 1  cos . (b) sin 2A = Atan1 Atan2 2 , cos 2A = Atan1 Atan1 2 2   (c) sin 3A = 3 sinA  4 sin3A (d) cos 3A = 4 cos3A  3 cosA (e) tan 3A = Atan31 AtanAtan3 2 3   Important Trigonometric Ratios: (a) sin n  = 0 ; cos n  = 1 ; tan n  = 0, where n   (b) sin 15° or sin 12  = 22 13 = cos 75° or cos 12 5 ; cos 15° or cos 12  = 22 13 = sin 75° or sin 12 5 ; tan 15° = 13 13   = 32 = cot 75° ; tan 75° = 13 13   = 32 = cot 15° (c) sin 10  or sin 18° = 4 15 & cos 36° or cos 5  = 4 15 Page # 21 Range of Trigonometric Expression: – 22 ba  a sin  + b cos   22 ba  Sine and Cosine Series : sin  + sin (+) + sin ( + 2 ) +...... + sin   1n = 2 2 n sin sin   sin          2 1n cos  + cos (+) + cos ( + 2 ) +...... + cos   1n = 2 2 n sin sin   cos          2 1n Trigonometric Equations Principal Solutions: Solutions which lie in the interval [0, 2) are called Principal solutions. General Solution : (i) sin  = sin    = n  + (1)n  where         2 2 , , n  . (ii) cos  = cos    = 2 n  ±  where   [0, ], n  . (iii) tan  = tan    = n  +  where        2 2 , , n  . (iv) sin²  = sin² , cos²  = cos² , tan²  = tan²    = n  ±  Page # 22 QUADRATIC EQUATIONS 1. Quadratic Equation : a x 2 + b x + c = 0, a  0 x = a2 ca4bb 2  , The expression b 2  4 a c  D is called discriminant of quadratic equation. If ,  are the roots, then (a)  +  =  a b (b)  = a c A quadratic equation whose roots are  & , is (x ) (x ) = 0 i.e. x2  ( +  ) x +  = 0 2. Nature of Roots: Consider the quadratic equation, a x 2 + b x + c = 0 having ,  as its roots; D  b2  4 a c D = 0 D  0 Roots are equal =  =  b/2a Roots are unequal a, b, c  R & D > 0 a, b, c  R & D < 0 Roots are real Roots are imaginary  = p + i q,  = p  i q a, b, c  Q & a, b, c  Q & D is a perfect square D is not a perfect square  Roots are rational  Roots are irrational  i.e.  = p + q ,  = p  q a = 1, b, c   & D is a perfect square  Roots are integral. Page # 25 n  Arithmetic Means Between Two Numbers: If a, b are any two given numbers & a, A1, A2,...., An, b are in A.P. then A1, A2,... An are the n A.M.’s between a & b. A1 = a + b a n   1 , A2 = a + 2 1 ( )b a n   ,......, An = a + n b a n ( )  1 r n   1 Ar = nA where A is the single A.M. between a & b. Geometric Progression: a, ar, ar2, ar3, ar4,...... is a G.P. with a as the first term & r as common ratio. (i) nth term = a rn1 (ii) Sum of the first n terms i.e. Sn =           1r,na 1r, 1r 1ra n (iii) Sum of an infinite G.P. when r < 1 is given by S =  a r r 1 1   . Geometric Means (Mean Proportional) (G.M.): If a, b, c > 0 are in G.P., b is the G.M. between a & c, then b² = ac nGeometric Means Between positive number a, b: If a, b are two given numbers & a, G1, G2,....., Gn, b are in G.P.. Then G1, G2, G3,...., Gn are n G.M.s between a & b. G1 = a(b/a)1/n+1, G2 = a(b/a)2/n+1,......, Gn = a(b/a)n/n+1 Harmonic Mean (H.M.): If a, b, c are in H.P., b is the H.M. between a & c, then b = ca ac2  . H.M. H of a1, a2 , ........ an is given by H 1 = n 1        n21 a 1....... a 1 a 1 Page # 26 Relation between means : G² = AH, A.M.  G.M.  H.M. (only for two numbers) and A.M. = G.M. = H.M. if a1 = a2 = a3 = ...........= an Important Results (i) r n   1 (ar ± br) = r n   1 ar ± r n   1 br. (ii) r n   1 k ar = k r n   1 ar. (iii) r n   1 k = nk; where k is a constant. (iv) r n   1 r = 1 + 2 + 3 +...........+ n = n n( )1 2 (v) r n   1 r² = 12 + 22 + 32 +...........+ n2 = n n n( ) ( ) 1 2 1 6 (vi) r n   1 r3 = 13 + 23 + 33 +...........+ n3 = n n2 21 4 ( ) BINOMIAL THEOREM 1. Statement of Binomial theorem : If a, b  R and n  N, then (a + b)n = nC0 a nb0 + nC1 a n–1 b1 + nC2 a n–2 b2 +...+ nCr a n–r br +...+ nCn a 0 bn =    n 0r rrn r n baC 2. Properties of Binomial Theorem : (i) General term : Tr+1 = nCr a n–r br (ii) Middle term (s) : (a) If n is even, there is only one middle term, which is        2 2n th term. (b) If n is odd, there are two middle terms, which are        2 1n th and         1 2 1n th terms. Page # 27 3. Multinomial Theorem : (x1 + x2 + x3 + ........... xk) n =   nr...rr k21k21 !r!...r!r !n k21 r k r 2 r 1 x...x.x Here total number of terms in the expansion = n+k–1Ck–1 4. Application of Binomial Theorem : If n)BA(  =  + f where  and n are positive integers, n being odd and 0 < f < 1 then ( + f) f = kn where A – B2 = k > 0 and A – B < 1. If n is an even integer, then ( + f) (1 – f) = kn 5. Properties of Binomial Coefficients : (i) nC0 + nC1 + nC2 + ........+ nCn = 2n (ii) nC0 – nC1 + nC2 – nC3 + ............. + (–1)n nCn = 0 (iii) nC0 + nC2 + nC4 + .... = nC1 + nC3 + nC5 + .... = 2n–1 (iv) nCr + nCr–1 = n+1Cr (v) 1r n r n C C  = r 1rn  6. Binomial Theorem For Negative Integer Or Fractional Indices (1 + x)n = 1 + nx + !2 )1n(n  x2 + !3 )2n)(1n(n  x3 + .... + !r )1rn).......(2n)(1n(n  xr + ....,| x | < 1. Tr+1 = !r )1rn().........2n)(1n(n  xr Page # 30 3. Conditional Probability : P(A/B) = P(B) B)P(A  . 4. Binomial Probability Theorem If an experiment is such that the probability of success or failure does not change with trials, then the probability of getting exactly r success in n trials of an experiment is nCr p r qn – r, where ‘p’ is the probability of a success and q is the probability of a failure. Note that p + q = 1. 5. Expectation : If a value Mi is associated with a probability of p i , then the expectation is given by  piMi. 6. Total Probability Theorem : P(A) =   n 1i ii )B/A(P.)B(P 7. Bayes’ Theorem : If an event A can occur with one of the n mutually exclusive and exhaustive events B1, B2 , ....., Bn and the probabilities P(A/B1), P(A/B2) .... P(A/Bn) are known, then P(Bi / A) =   n 1i ii ii )B/A(P.)B(P )B/A(P.)B(P B1, B2, B3,........,Bn A = (A  B1)  (A  B2)  (A  B3)  ........  (A  Bn) P(A) = P(A  B1) + P(A B2) + ....... + P(A Bn) =    n 1i i )BA(P 8. Binomial Probability Distribution : (i) Mean of any probability distribution of a random variable is given by : µ = i ii p xp   =  pi xi = np n = number of trials p = probability of success in each probability q = probability of failure (ii) Variance of a random variable is given by, 2 = (xi – µ)2 . pi = pi xi 2 – µ2 = npq Page # 31 COMPLEX NUMBER 1. The complex number system z = a + ib, then a – ib is called congugate of z and is denoted by z . 2. Equality In Complex Number: z1 = z2  Re(z1) = Re(z2) and m (z1) = m (z2). 3. Properties of arguments (i) arg(z1z2) = arg(z1) + arg(z2) + 2m for some integer m. (ii) arg(z1/z2) = arg (z1) – arg(z2) + 2m for some integer m. (iii) arg (z2) = 2arg(z) + 2m for some integer m. (iv) arg(z) = 0  z is a positive real number (v) arg(z) = ± /2  z is purely imaginary and z  0 4. Properties of conjugate (i) |z| = | z | (ii) z z = |z|2(iii) 21 zz  = 1z + 2z (iv) 21 zz  = 1z – 2z (v) 21zz = 1z 2z (vi)       2 1 z z = 2 1 z z (z2  0) (vii) |z1 + z2|2 = (z1 + z2) )zz( 21  = |z1|2 + |z2|2 + z1 2z + 1z z2 (viii) )z( 1 = z (ix) If w = f(z), then w = f( z ) (x) arg(z) + arg( z ) 5. Rotation theorem If P(z1), Q(z2) and R(z3) are three complex numbers and PQR = , then         21 23 zz zz = 21 23 zz zz   ei 6. Demoivre’s Theorem : If n is any integer then (i) (cos  + i sin  )n = cos n + i sin n (ii) (cos 1 + i sin 1) (cos 2 + i sin 2) (cos3 + i sin 2) (cos 3 + i sin 3) .....(cos n + i sin n) = cos (1 + 2 + 3 + ......... n) + i sin (1 + 2 + 3 + ....... + n) Page # 32 7. Cube Root Of Unity : (i) The cube roots of unity are 1,  1 3 2 i ,  1 3 2 i . (ii) If  is one of the imaginary cube roots of unity then 1 +  + ² = 0. In general 1 + r + 2r = 0; where r  I but is not the multiple of 3. 8. Geometrical Properties: Distance formula : |z1 – z2|. Section formula : z = nm nzmz 12   (internal division), z = nm nzmz 12   (external division) (1) amp(z) =  is a ray emanating from the origin inclined at an angle  to the x axis. (2) z  a = z  b is the perpendicular bisector of the line joining a to b. (3) If 2 1 zz zz   = k  1, 0, then locus of z is circle. VECTORS . Position Vector Of A Point: let O be a fixed origin, then the position vector of a point P is the vector OP . If a  and b  are position vectors of two points A and B, then, AB = ab   = pv of B  pv of A. DISTANCE FORMULA : Distance between the two points A )a(  and B )b(  is AB = ba   SECTION FORMULA : nm bmanr      . Mid point of AB = 2 ba   . Page # 35  The positon vector of the centroid of a tetrahedron if the pv’s of its vertices are     a b c d, , & are given by 1 4 [ ]     a b c d   . V. Vector Triple Product:    a x b x c( ) = ( . ) ( . )       a c b a b c , ( )    a x b x c = ( . ) ( . )       a c b b c a  ( ) ( )       a x b x c a x b x c , in general 3-DIMENSION 1. Vector representation of a point : Position vector of point P (x, y, z) is x î + y ĵ + z k̂ . 2. Distance formula : 2 21 2 21 2 21 )zz()yy()xx(  , AB = | OB – OA | 3. Distance of P from coordinate axes : PA = 22 zy  , PB = 22 xz  , PC = 22 yx  4. Section Formula : x = nm nxmx 12   , y = nm nymy 12   , z = nm nzmz 12   Mid point : 2 zzz, 2 yyy, 2 xxx 212121       5. Direction Cosines And Direction Ratios (i) Direction cosines: Let    be the angles which a directed line makes with the positive directions of the axes of x, y and z respectively, then cos , cos cos  are called the direction cosines of the line. The direction cosines are usually denoted by (, m, n). Thus  = cos , m = cos , n = cos . (ii) If , m, n be the direction cosines of a line, then 2 + m2 + n2 = 1 (iii) Direction ratios: Let a, b, c be proportional to the direction cosines , m, n then a, b, c are called the direction ratios. (iv) If , m, n be the direction cosines and a, b, c be the direction ratios of a vector, then 222222222 cba cn, cba bm, cba a       Page # 36 (vi) If the coordinates P and Q are (x1, y1, z1) and (x2, y2, z2) then the direction ratios of line PQ are, a = x2  x1, b = y2  y1 & c = z2  z1 and the direction cosines of line PQ are   = |PQ| xx 12  , m = |PQ| yy 12  and n = |PQ| zz 12  6. Angle Between Two Line Segments: cos  = 2 2 2 2 2 2 2 1 2 1 2 1 212121 cbacba ccbbaa   . The line will be perpendicular if a1a2 + b1b2 + c1c2 = 0, parallel if 2 1 a a = 2 1 b b = 2 1 c c 7. Projection of a line segment on a line If P(x1, y1, z1) and Q(x2, y2, z2) then the projection of PQ on a line having direction cosines , m, n is )zz(n)yy(m)xx( 121212  8. Equation Of A Plane : General form: ax + by + cz + d = 0, where a, b, c are not all zero, a, b, c, d  R. (i) Normal form : x + my + nz = p (ii) Plane through the point (x1, y1, z1) : a (x  x1) + b( y  y1) + c (z  z1) = 0 (iii) Intercept Form: 1 c z b y a x  (iv) Vector form: ( r  a  ). n = 0 or r . n = a  . n (v) Any plane parallel to the given plane ax + by + cz + d = 0 is ax + by + cz +  = 0. Distance between ax + by + cz + d1 = 0 and ax + by + cz + d2 = 0 is = 222 21 cba |dd|   Page # 37 (vi) Equation of a plane passing through a given point & parallel to the given vectors: r = a  + b  +  c  (parametric form) where  &  are scalars. or r . )cb(   = a . )cb(   (non parametric form) 9. A Plane & A Point (i) Distance of the point (x, y, z) from the plane ax + by + cz+ d = 0 is given by 222 cba d'cz'by'ax   . (ii) Length of the perpendicular from a point ( a  ) to plane r . n = d is given by p = |n| |dn.a|    . (iii) Foot (x, y, z) of perpendicular drawn from the point (x1, y1, z1) to the plane ax + by + cz + d = 0 is given by c z'z b y'y a x'x 111      = – 222 111 cba )dczbyax(   (iv) To find image of a point w.r.t. a plane: Let P (x1, y1, z1) is a given point and ax + by + cz + d = 0 is given plane Let (x, y, z) is the image point. then c z'z b y'y a x'x 111      = – 2 222 111 cba )dczbyax(   10. Angle Between Two Planes: cos  = 2'2'2'222 cbacba 'cc'bb'aa   Planes are perpendicular if aa + bb + cc = 0 and planes are parallel if 'a a = 'b b = 'c c Page # 40 4. Skew Lines: (i) The straight lines which are not parallel and noncoplanar i.e. nonintersecting are called skew lines. lines x – y – z – m n       & x – ' y – ' z – ' ' m' n'       If  = 'n'm' nm '''     0, then lines are skew. (ii) Shortest distance formula for lines r  = 1a  +  1b  and r  = 2a  +  2b  is    2 1 1 2 1 2 a – a . b b d b b         (iii) Vector Form: For lines r  = 1a  +  1b  and r  = 2a  +  2b  to be skew ( 1b  x 2b  ). ( 2a  1a )  0 (iv) Shortest distance between parallel lines r  = 1a  +  b  & r = 2a +  b  is d = |b| b)aa( 12    . (v) Condition of coplanarity of two lines r  = a  +  b  & r = c  +  d  is [a – c b d] 0    5. Sphere General equation of a sphere is x2 + y2 + z2 + 2ux + 2vy + 2wz + d = 0. (u, – v, w) is the centre and dwvu 222  is the radius of the sphere. Page # 41 SOLUTION OF TRIANGLE 1. Sine Rule: Csin c Bsin b Asin a  . 2. Cosine Formula: (i) cos A = b c a b c 2 2 2 2   (ii) cos B = c a b ca 2 2 2 2   (iii) cos C = a b c a b 2 2 2 2   3. Projection Formula: (i) a = b cosC + c cosB (ii) b = c cosA + a cosC (iii) c = a cosB + b cosA 4. Napier’s Analogy - tangent rule: (i) tan 2 CB  = cb cb   cot 2 A (ii) tan 2 AC  = ac ac   cot B 2 (iii) tan A B 2 = a b a b   cot C 2 5. Trigonometric Functions of Half Angles: (i) sin A 2 = ( ) ( )s b s c b c   ; sin B 2 = ( ) ( )s c s a ca   ; sin C 2 = ( ) ( )s a s b a b   (ii) cos A 2 = s s a b c ( ) ; cos B 2 = s s b ca ( ) ; cos C 2 = s s c a b ( ) (iii) tan A 2 = ( ) ( ) ( ) s b s c s s a    =  s s a( ) where s = a b c  2 is semi perimetre of triangle. (iv) sin A = )cs)(bs)(as(s bc 2  = bc 2 Page # 42 6. Area of Triangle () :  = 2 1 ab sin C = 2 1 bc sin A = 2 1 ca sin B = s s a s b s c( ) ( ) ( )   7. m - n Rule: If BD : DC = m : n, then (m + n) cot      m n n B m C cot cot cot cot   8. Radius of Circumcirlce : R = Csin2 c Bsin2 b Asin2 a  = 4 cba 9. Radius of The Incircle : (i) r =  s (ii) r = (s  a) tan A 2 = (s  b) tan B 2 = (s  c) tan C 2 (iii) r = 4R sin A 2 sin B 2 sin C 2 10. Radius of The Ex- Circles : (i) r1 =  s a ; r2 =  s b ; r3 =  s c (ii) r1 = s tan A 2 ; r2 = s tan B 2 ; r3 = s tan C 2 (iii) r1 = 4 R sin A 2 . cos B 2 . cos C 2 Page # 45 P - 2 (i) sin1 (sin x) = x,      2 2 x (ii) cos1 (cos x) = x; 0  x   (iii) tan1 (tan x) = x;      2 2 x (iv) cot1 (cot x) = x; 0 < x <  (v) sec1 (sec x) = x; 0  x  , x  2  (vi) cosec1 (cosec x) = x; x  0,      2 2 x P - 3 (i) sin1 (x) =  sin1 x, 1  x  1 (ii) tan1 (x) =  tan1 x, x  R (iii) cos1 (x) =   cos1 x, 1  x  1 (iv) cot1 (x) =   cot1 x, x  R P - 5 (i) sin1 x + cos1 x =  2 , 1  x  1 (ii) tan1 x + cot1 x =  2 , x  R (iii) cosec1 x + sec1 x =  2 , x  1 2. Identities of Addition and Substraction: I - 1 (i) sin1 x + sin1 y = sin1 x y y x1 12 2      , x  0, y  0 & (x2 + y2)  1 =   sin1 x y y x1 12 2      , x  0, y  0 & x2 + y2 > 1 (ii) cos1 x + cos1 y = cos1 x y x y      1 12 2 , x  0, y  0 (iii) tan1 x + tan1 y = tan1 x y xy  1 , x > 0, y > 0 & xy < 1 =  + tan1 x y xy  1 , x > 0, y > 0 & xy > 1 =  2 , x > 0, y > 0 & xy = 1 Page # 46 I - 2 (i) sin1 x  sin1 y = sin1      22 x1yy1x , x  0, y  0 (ii) cos1 x  cos1 y = cos1      22 y1x1yx , x  0, y  0, x  y (iii) tan1 x  tan1y = tan1 yx1 yx   , x  0, y  0 I - 3 (i) sin1        2x1x2 =                 2 1 2 1 2 1 xifxsin2 xifxsin2 |x|ifxsin2 1 1 1 (ii) cos1 (2 x2  1) =         0x1ifxcos22 1x0ifxcos2 1 1 (iii) tan1 2x1 x2  =             1xifxtan2 1xifxtan2 1|x|ifxtan2 1 1 1 (iv) sin1 2 1 2 x x =             1xifxtan2 1xifxtan2 1|x|ifxtan2 1 1 1 (v) cos1 2 2 x1 x1   =         0xifxtan2 0xifxtan2 1 1 If tan1 x + tan1 y + tan1 z = tan1         xzzyyx1 zyxzyx if, x > 0, y > 0, z > 0 & (xy + yz + zx) < 1 NOTE: (i) If tan1 x + tan1 y + tan1 z =  then x + y + z = xyz (ii) If tan1 x + tan1 y + tan1 z =  2 then xy + yz + zx = 1 (iii) tan1 1 + tan1 2 + tan1 3 =  Page # 47 STATISTICS 1. Arithmetic Mean / or Mean If x1, x2, x3 ,.......xn are n values of variate xi then their A.M. x is defined as x = n x.......xxx n321  = n x n 1i i  If x1, x2, x3, .... xn are values of veriate with frequencies f1, f2, f3,.........fn then their A.M. is given by x = n321 nn332211 f......fff ff......xfxfxf   = N xf n 1i ii  , where N =   n 1i if 2. Properties of Arithmetic Mean : (i) Sum of deviation of variate from their A.M. is always zero that is   xx i  = 0. (ii) Sum of square of deviation of variate from their A.M. is minimum that is   2i xx  is minimum (iii) If x is mean of variate xi then A.M. of (xi + ) = x +  A.M. of i . xi = . x A.M. of (axi + b) = a x + b 3. Median The median of a series is values of middle term of series when the values are written is ascending order or descending order. Therefore median, divide on arranged series in two equal parts For ungrouped distribution : If n be number of variates in a series then Median =                              evenisnwhenterm2 2 nand 2 nofMean oddisnwhen,term 2 1n thth th Page # 50 Fallacy : This is statement which is false for all truth values of its compo- nents. It is denoted by f or c. Consider truth table of p ̂ ~ p FTF FFT p~pp~p  (i) Statements p q p q p q p q Negation (~ p) (~ q) (~ p) (~ q) p (~ q) p –q         (ii) Let p q Then (Contrapositive of p q)is(~ q ~ p)    SETS AND RELATION Laws of Algebra of sets (Properties of sets): (i) Commutative law : (A  B) = B  A ; A  B = B  A (ii) Associative law:(A  B)  C=A  (B  C) ; (A  B)C = A  (B  C) (iii) Distributive law : A (B  C) = (A  B)  (A  C) ; A  (B  C) = (A  B)  (A  C) (iv) De-morgan law : (A  B)' = A'  B' ; (A  B)' = A'  B' (v) Identity law : A  U = A ; A   = A (vi) Complement law : A  A' = U, A  A' = , (A')' = A (vii) Idempotent law : A  A = A, A  A = A Some important results on number of elements in sets : If A, B, C are finite sets and U be the finite universal set then (i) n(A  B) = n(A) + n(B) – n(A  B) (ii) n(A – B) = n(A) – n(A  B) (iii) n(A  B  C) = n(A) + n(B) + n(C) – n(A  B) – n(B  C) – n (A  C) + n(A  B  C) (iv) Number of elements in exactly two of the sets A, B, C = n(A  B) + n(B  C) + n(C  A) – 3n(A  B  C) (v) Number of elements in exactly one of the sets A, B, C = n(A) + n(B) + n(C) – 2n(A  B) – 2n(B  C) – 2n(A  C) + 3n(A  B  C) Page # 51 Types of relations : In this section we intend to define various types of relations on a given set A. (i) Void relation : Let A be a set. Then   A × A and so it is a relation on A. This relation is called the void or empty relation on A. (ii) Universal relation : Let A be a set. Then A × A  A × A and so it is a relation on A. This relation is called the universal relation on A. (iii) Identity relation : Let A be a set. Then the relation IA = {(a, a) : a  A} on A is called the identity relation on A. In other words, a relation IA on A is called the identity relation if every element of A is related to itself only. (iv) Reflexive relation : A relation R on a set A is said to be reflexive if every element of A is related to itself. Thus, R on a set A is not reflexive if there exists an element a  A such that (a, a)  R. Note : Every identity relation is reflexive but every reflexive relation in not identity. (v) Symmetric relation : A relation R on a set A is said to be a symmetric relation iff (a, b)  R  (b ,a)  R for all a, b  A. i.e. a R b  b R a for all a, b  A. (vi) Transitive relation : Let A be any set. A relation R on A is said to be a transitive relation iff (a, b)  R and (b, c)  R  (a, c)  R for all a, b, c  A i.e. a R b and b R c  a R c for all a, b, c  A (vii) Equivalence relation : A relation R on a set A is said to be an equivalence relation on A iff (i) it is reflexive i.e. (a, a)  R for all a  A (ii) it is symmetric i.e. (a, b)  R  (b, a)  R for all a, b  A (iii) it is transitive i.e. (a, b)  R and (b, c)  R  (a, c)  R for all a,bA