MATHEMATICS FORMULA BOOKLET - GYAAN SUTRA INDEX 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 Page # 1 MATHEMATICS FORMULA BOOKLET - GYAAN SUTRA STRAIGHT LINE 1. Distance Formula: d (x1 – x2 )2 (y1 – y2 )2 . 2. Section Formula : x= 3. my 2 ny1 mx 2 nx1 ;y= . mn mn Centroid, Incentre & Excentre: x1 x 2 x 3 y1 y 2 y 3 , 3 3 Centroid G , ax1 bx 2 cx 3 ay1 by 2 cy 3 , a b c abc Incentre I ax1 bx 2 cx 3 ay1 by 2 cy 3 , abc abc Excentre I1 4. Area of a Triangle: x1 1 ABC = 2 x 2 x3 5. y1 1 y2 1 y3 1 Slope Formula: y1 y 2 x1 x 2 Condition of collinearity of three points: Line Joining two points (x1 y1) & (x2 y2), m = 6. x1 7. x2 y1 1 y2 1 = 0 x3 y3 1 Angle between two straight lines : tan = m1 m 2 . 1 m1m 2 Page # 2 8. Two Lines : ax + by + c = 0 and ax + b y + c = 0 two lines a a 1. parallel if = b c . b c c 1 c 2 2. Distance between two parallel lines = . a 2 b 2 3 Perpendicular : If aa + bb = 0. 9. A point and line: 1. Distance between point and line = a x1 b y 1 c . a 2 b2 2. Reflection of a point about a line: xx1 yy1 ax 1 by 1 c 2 a b a 2 b 2 3. Foot of the perpendicular from a point on the line is x x1 y y1 ax 1by1 c a b a 2 b 2 10. Bisectors of the angles between two lines: ax by c 2 a b 11. 12. 2 =± ax by c a 2 b2 Condition of Concurrency : a1 of three straight lines ai x+ b i y + c i = 0, i = 1,2,3 is a 2 b1 b2 c1 c 2 = 0. a3 b3 c3 A Pair of straight lines through origin: ax² + 2hxy + by² = 0 If is the acute angle between the pair of straight lines, then tan 2 h 2 ab = ab . Page # 3 CIRCLE 1. Intercepts made by Circle x2 + y2 + 2gx + 2fy + c = 0 on the Axes: (a) 2 g2 c on x -axis 2. (b) 2 f 2 c on y - aixs Parametric Equations of a Circle: x = h + r cos ; y = k + r sin 3. Tangent : (a) Slope form : y = mx ± a 1 m2 (b) Point form : xx1 + yy1 = a2 or T = o (c) Parametric form : x cos + y sin = a. 4. Pair of Tangents from a Point: SS 1 = T². 5. Length of a Tangent : Length of tangent is S1 6. Director Circle: x2 + y2 = 2a2 for x2 + y2 = a2 7. Chord of Contact: T = 0 1. Length of chord of contact = 2 LR R 2 L2 2. Area of the triangle formed by the pair of the tangents & its chord of contact = R L3 R 2 L2 3. Tangent of the angle between the pair of tangents from (x1, y1) 2R L 2 2 L R = 4. Equation of the circle circumscribing the triangle PT 1 T 2 is : (x x1) (x + g) + (y y1) (y + f) = 0. 8. Condition of orthogonality of Two Circles: 2 g 1 g 2 + 2 f1 f2 = c 1 + c 2. 9. Radical Axis : S 1 S 2 = 0 i.e. 2 (g 1 g 2) x + 2 (f1 f2) y + (c 1 c 2) = 0. 10. Family of Circles: S 1 + K S 2 = 0, S + KL = 0. Page # 4 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 + a (m 0) 2. Parametric form ty = x + at2 m 3. Point form T = 0 4. Normals to the parabola y² = 4ax : y y1 = y1 (x x1) at (x1, y1) ; y = mx 2am am 3 at (am 2 2am) ; 2a y + tx = 2at + at3 at (at2, 2at). ELLIPSE 1. Standard Equation Eccentricity: e = 1 : x2 a2 b2 a 2 y2 b2 = 1, where a > b & b² = a² (1 e²). , (0 < e < 1), Directrices : x = ± a . e Focii : S (± a e, 0). Length of, major axes = 2a and minor axes = 2b Vertices : A ( a, 0) & A (a, 0) . Latus Rectum : = 2b 2 2a 1 e 2 a 2. Auxiliary Circle : x² + y² = a² 3. 4. Parametric Representation : x = a cos & y = b sin Position of a Point w.r.t. an Ellipse: The point P(x1, y1) lies outside, inside or on the ellipse according as; x12 a2 y12 b2 1 > < or = 0. Page # 5 5. Line and an Ellipse: The line y = mx + c meets the ellipse 2 x2 y = 1 in two points real, 2 2 a b coincident or imaginary according as c² is < = or > a²m² + b². 6. Tangents: a 2m 2 b 2 , Point form : Slope form: y = mx ± Parametric form: 7. x x1 a 2 yy 1 b2 1, xcos ysin 1 a b Normals: a2 x b2 y a2 b2 m = a² b², ax. sec by. cosec = (a² b²), y = mx . x1 y1 a 2 b 2m 2 8. Director Circle: x² + y² = a² + b² HYPERBOLA 1. Standard Equation: Standard equation of the hyperbola is Focii : x2 y2 1 , where b 2 = a2 (e2 1). a2 b2 S (± ae, 0) Directrices : x = ± a e Vertices : A (± a, 0) 2b2 = 2a (e2 1). a Conjugate Hyperbola : Latus Rectum ( ) : = 2. x2 y2 1 a2 b 2 & x2 y2 1 are conjugate hyperbolas of each. a 2 b2 3. Auxiliary Circle : x2 + y2 = a2. 4. Parametric Representation : x = a sec & y = b tan Page # 6 5. Position of A Point 'P' w.r.t. A Hyperbola : 2 S 1 x1 a2 2 y1 1 >, = or < 0 according as the point (x1, y1) lies inside, on b2 or outside the curve. 6. 7. Tangents : (i) Slope Form : y = m x a 2m 2 b 2 (ii) Point Form : at the point (x1, y1) is (iii) Parametric Form : yy 1 b2 1. x sec y tan 1 . a b a2 x b2 y = a2 + b 2 = a2 e2 . x1 y1 (a) at the point P (x1, y1) is (b) at the point P (a sec , b tan ) is (c) Equation of normals in terms of its slope 'm' are y Asymptotes : a 2 b2 m 2 2 a b m2 x y 0 a b Pair of asymptotes : 9. a 2 Normals : = mx 8. xx 1 ax by = a2 + b 2 = a2 e2 . sec tan . and x y 0 . a b x2 y2 0 . a2 b2 Rectangular Or Equilateral Hyperbola : xy = c2, eccentricity is 2 . Vertices : (± c , ±c) ; Focii : Latus Rectum (l ) : = 2 2c, 2 c . Directrices : x + y = 2 c 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 x y x = 2 & at P (t) is + t y = 2 c. x1 y 1 t 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 # 7 LIMIT OF FUNCTION 1. Limit of a function f(x) is said to exist as x Limit f (a h) Limit f (a + h) = (Left hand limit) (Right hand limit) Indeterminant Forms: 0, 0 3. = some finite value M. h0 h 0 2. a when, , 0 , º, 0º,and 1 . Standard Limits: Limit x 0 1 sin x Limit tanx Limit tan x = = x 0 x 0 x x x x sin 1x Limit e 1 Limit n(1 x ) Limit = x 0 = x 0 =1 x x = x 0 x x x 1 Limit (1 + x)1/x = Limit 1 = e, Limit a 1 = log a, a > 0, x 0 x x 0 e x x n n Limit x a = nan – 1. x a x a 4. Limits Using Expansion x ln a x 2 ln2 a x 3 ln3 a .........a 0 1! 2! 3! (i) a x 1 (ii) ex 1 (iii) ln (1+x) = x (iv) sin x x x x2 x3 ...... 1 ! 2! 3! x2 x3 x4 .........for 1 x 1 2 3 4 x3 x5 x7 ..... 3! 5! 7! Page # 8 cosx 1 (vi) tan x = x (vii) for |x| < 1, n R (1 + x)n = 1 + nx + 5. x2 x4 x6 ..... 2! 4! 6! (v) x 3 2x 5 ...... 3 15 n(n 1) 2 n(n 1)(n 2) 3 x + x + ............ 1. 2 . 3 1. 2 Limits of form 1 , 00, 0 Also for (1) type of problems we can use following rules. lim (1 + x)1/x = e, lim [f(x)]g(x) , x 0 x a lim [ f ( x )1] g( x ) g(x) as x a = e x a where f(x) 1 ; 6. Sandwich Theorem or Squeeze Play Theorem: If f(x) g(x) h(x) x & Limit f(x) = = Limit h(x) then Limit g(x) = . x a x a x a METHOD OF DIFFERENTIATION 1. Differentiation of some elementary functions 1. d n (x ) = nxn – 1 dx 2. d x (a ) = ax n a dx 3. d 1 (n |x|) = dx x 4. 1 d (log ax) = x na dx d (cos x) = – sin x dx 5. d (sin x) = cos x dx 6. 7. d (sec x) = sec x tan x dx 8. 9. d (tan x) = dx 10. sec 2 x d (cosec x) = – cosec x cot x dx d (cot x) = – cosec 2 x dx Page # 9 2. Basic Theorems 1. d (f ± g) = f(x) ± g(x) dx d d (k f(x)) = k f(x) dx dx d (f(x) . g(x)) = f(x) g(x) + g(x) f(x) dx 3. 4. 2. d dx f (x) g( x ) f ( x ) f ( x ) g( x ) = g2 ( x ) g( x ) 5. d (f(g(x))) = f(g(x)) g(x) dx Derivative Of Inverse Trigonometric Functions. d sin –1 x = dx 1 1 x 2 , d cos –1 x =– dx 1 1 x2 , for – 1 < x < 1. 1 d tan –1 x d cot –1 x 1 = =– (x R) 2 , dx dx 1 x 1 x2 1 d sec –1 x d cos ec –1x = , dx dx | x | x2 1 =– 3. 1 | x | x2 1 , for x (– , – 1) (1, ) Differentiation using substitution Following substitutions are normally used to simplify these expression. (i) x 2 a 2 by substituting x = a tan , where – < 2 2 (ii) a 2 x 2 by substituting x = a sin , where – 2 2 (iii) x 2 a 2 by substituting x = a sec , where [0, ], (iv) xa ax x = a cos , where (0, ]. by substituting 2 Page # 10 4. Parametric Differentiation If y = f( ) & x = g( ) where is a parameter, then 5. Derivative of one function with respect to another Let y = f(x); z = g(x) then 6. dy dy / d . dx dx / d dy dy / dx f '( x ) . dz dz / dx g'( x ) f(x) g( x ) h( x ) If F(x) = l( x ) m( x ) u( x ) v( x ) n( x ) , where f, g, h, l, m, n, u, v, w are differentiable w( x ) f ' ( x ) g' ( x) h' ( x) f ( x ) g( x ) h( x ) l ( x ) m ( x ) n ( x ) functions of x then F (x) = + l' ( x ) m' ( x ) n' ( x ) + u( x ) v( x ) w( x ) u( x ) v( x ) w ( x ) f ( x) g( x ) h( x ) l( x ) m( x ) n( x ) u' ( x ) v' ( x ) w' ( x ) 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) = – 1 (x – x1), when f(x1) is nonzero f ( x1 ) 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) = f (h) b ha Page # 11 And equation of tangent is y – b = 3. f (h) b (x – a) ha Length of tangent, normal, subtangent, subnormal 1 (i) PT = | k | 1 (ii) PN = | k | 1 m (iii) TM = (iv) MN = |km| = Length of subnormal. m2 = Length of Tangent p(h,k) 4. 2 = Length of Normal k = Length of subtangent m N M T 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 = 5. m1 m 2 1 m1m2 Shortest distance between two curves Shortest distance between two non-intersecting differentiable curves is always along their common normal. (W herever 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 # 12 7. Lagrange’s Mean Value Theorem (LMVT) : If a function f defined on [a, b] is (i) continuous on [a, b] and (ii) derivable on (a, b) then there exists at least one real numbers between a and b (a < c < b) such that 8. f (b ) f ( a ) = f(c) ba Useful Formulae of Mensuration to Remember : 1. 2. 3. 4. Volume of a cuboid = bh. Surface area of cuboid = 2(b + bh + h). Volume of cube = a3 Surface area of cube = 6a2 5. Volume of a cone = 6. 7. 8. Curved surface area of cone = r ( = slant height) Curved surface area of a cylinder = 2rh. Total surface area of a cylinder = 2rh + 2r2. 9. Volume of a sphere = 10. Surface area of a sphere = 4r2. 11. Area of a circular sector = 12. Volume of a prism = (area of the base) × (height). 13. Lateral surface area of a prism = (perimeter of the base) × (height). 14. Total surface area of a prism = (lateral surface area) + 2 (area of the base) (Note that lateral surfaces of a prism are all rectangle). 15. Volume of a pyramid = 16. Curved surface area of a pyramid = 1 2 r h. 3 4 3 r . 3 1 2 r , when is in radians. 2 1 (area of the base) × (height). 3 1 (perimeter of the base) × 2 (slant height). (Note that slant surfaces of a pyramid are triangles). Page # 13 INDEFINITE INTEGRATION 1. If f & g are functions of x such that g (x) = f(x) then, f(x) dx = g(x) + c d {g(x)+c} = f(x), where c is called the constant of dx integration. 2. Standard Formula: ax bn 1 dx = a n 1 (i) (ax + b)n (ii) dx 1 = n (ax + b) + c ax b a (iii) eax+b dx = 1 ax+b e +c a (iv) apx+q dx = 1 a pxq + c; a > 0 p n a (v) sin (ax + b) dx = (vi) cos (ax + b) dx = (vii) tan(ax + b) dx = (viii) cot(ax + b) dx = (ix) sec² (ax + b) dx = (x) cosec²(ax + b) dx = + c, n 1 1 cos (ax + b) + c a 1 a sin (ax + b) + c 1 n sec (ax + b) + c a 1 n sin(ax + b)+ c a 1 a tan(ax + b) + c 1 cot(ax + b)+ c a Page # 14 (xi) secx dx = n (secx + tanx) + c (xii) cosec x dx = n (cosecx cotx) + c x 2 OR n tan (xiii) (xiv) (xv) (xvi) (xvii) 2 a x dx a x2 = sin 1 x a +c 1 a x a +c 2 = 2 tan 1 dx 2 | x | x a 2 x a 2 = 2 = n dx 2 x a 2 1 a x a sec 1 +c x x a x x a 2 2 2 = n (xviii) dx a x2 = 1 2a n ax ax +c (xix) dx x a2 = 1 2a n xa xa +c dx = x 2 a2 x2 2 2 a2 x2 (xxi) x x a dx = 2 (xxii) (xx) 2 2 2 2 x a x + c 4 2 + c OR n (cosecx + cotx) + c dx dx n tan OR dx = x 2 +c 2 + a2 2 a2 x a + 2 2 2 2 2 x a a2 2 +c sin 1 x a +c x x2 a2 n a +c x x2 a 2 a +c n Page # 15 3. Integration by Subsitutions If we subsitute f(x) = t, then f (x) dx = dt 4. Integration by Part : d f( x) g(x) dx = f(x) g(x) dx – dx f (x ) g(x ) dx dx 5. Integration of type ax Make the substitution x 6. 2 dx dx , , ax 2 bx c dx 2 bx c ax bx c b t 2a Integration of type ax px q px q dx, dx, (px q) ax 2 bx c dx bx c ax 2 bx c 2 Make the substitution x + b = t , then split the integral as some of two 2a integrals one containing the linear term and the other containing constant term. 7. Integration of trigonometric functions (i) a b sin dx 2 OR dx x a sin 2 a b sin x (iii) x dx OR a bcos x dx OR 2 dx put tan x = t. x b sin x cos x c cos 2 x dx (ii) a bcos OR a b sin x c cos x a.cos x b.sin x c .cos x m.sin x n put tan x =t 2 dx. Express Nr A(Dr) + B d (Dr) + c & proceed. dx Page # 16 8. Integration of type x2 1 x 4 Kx2 1 dx where K is any constant. Divide Nr & Dr by x² & put x 9. Integration of type dx (ax b) 10. 1 = t. x px q OR (ax dx 2 bx c) px q ; put px + q = t2. Integration of type (ax b) (ax dx 2 px qx r , put ax + b = dx 2 2 b) px q , put x = 1 ; t 1 t DEFINITE INTEGRATION Properties of definite integral b 1. b f ( x ) dx = a b 3. a a 4. 2. a b f ( x ) dx = – f (x ) dx b a b f ( x ) dx + a f (x ) dx c a f (x ) dx = (f (x ) f ( x )) dx a 5. c f ( x ) dx = a b f (t ) dt 0 a 2 f ( x) dx , f (– x ) f ( x) = 0 0 , f (– x ) – f ( x ) b f (x ) dx = f (a b x) dx a a Page # 17 a a f (x ) dx = f (a x ) dx 6. 0 0 a 2 f ( x ) dx , f (2a – x ) f ( x ) f ( x ) dx = ( f ( x ) f (2a x )) dx = 7. 0 0 0 0 , f (2a – x ) – f ( x) 2a a 8. If f(x) is a periodic function with period T, then T nT f ( x ) dx = n a nT f ( x ) dx, n z, 0 0 a f (x ) dx, n z, a R 0 T nT T f ( x ) dx = n a nT a f (x ) dx = (n – m) f (x ) dx, m, n z, f (x ) dx = f (x ) dx, n z, a R 0 mT b nT 0 a f ( x ) dx = a nT 9. nT f (x ) dx, n z, a, b R a If (x) f(x) (x) a x b, b b then for ( x ) dx b f ( x ) dx a a (x ) dx a b 10. If m f(x) M for a x b, then m (b – a) f (x ) dx M (b – a) a b 11. If f(x) 0 on [a, b] then f (x) dx 0 a h( x ) Leibnitz Theorem : If F(x) = f (t ) dt , then g( x ) dF( x ) = h(x) f(h(x)) – g(x) f(g(x)) dx Page # 18 FUNDAMENTAL OF MATHEMATICS Intervals : Intervals are basically subsets of R and are commonly used in solving inequalities or in finding domains. If there are two numbers a, b R such that a < b, we can define four types of intervals as follows : Symbols Used (i) Open interval : (a, b) = {x : a < x < b} i.e. end points are not included. ( ) or ] [ (ii) Closed interval : [a, b] = {x : a x b} i.e. end points are also included. [] This is possible only when both a and b are finite. (iii) Open-closed interval : (a, b] = {x : a < x b} ( ] or ] ] (iv) Closed - open interval : [a, b) = x : a x < b} [ ) or [ [ The infinite intervals are defined as follows : (i) (a, ) = {x : x > a} (ii) (iii) (– , b) = {x : x < b} (iv) (v) (– ) = {x : x R} [a, ) = {x : x a} (, b] = {x : x b} Properties of Modulus : For any a, b R |a| 0, |a| = |–a|, |a| a, |a| –a, |ab| = |a| |b|, a |a| = , |a + b| |a| + |b|, |a – b| ||a| – |b|| b |b| Trigonometric Functions of Sum or Difference of Two Angles: (a) sin (A ± B) = sinA cosB ± cosA sinB 2 sinA cosB = sin(A+B) + sin(AB) and and 2 cosA sinB = sin(A+B) sin(AB) (b) (c) (d) cos (A ± B) = cosA cosB sinA sinB 2 cosA cosB = cos(A+B) + cos(AB) and 2sinA sinB = cos(AB) cos(A+B) sin²A sin²B = cos²B cos²A = sin (A+B). sin (A B) cos²A sin²B = cos²B sin²A = cos (A+B). cos (A B) (e) cot (A ± B) = (f) tan (A + B + C) = cot A cot B 1 cot B cot A tan A tan B tanCtan A tan B tan C . 1 tan A tan B tan B tan C tan C tan A Page # 19 Factorisation of the Sum or Difference of Two Sines or Cosines: CD CD cos 2 2 (a) sinC + sinD = 2 sin (b) sinC sinD = 2 cos (c) CD CD sin 2 2 CD CD cosC + cosD = 2 cos cos 2 2 CD CD sin 2 2 Multiple and Sub-multiple Angles : (d) cosC cosD = 2 sin (a) cos 2A = cos²A sin²A = 2cos²A 1 = 1 2 sin²A; 2 cos² = 1 + cos , 2 sin² 2 tan A = 1 cos . 2 , cos 2A = 1tan 2 A (b) sin 2A = (c) (d) sin 3A = 3 sinA 4 sin 3A cos 3A = 4 cos3A 3 cosA (e) tan 3A = (a) sin n = 0 ; cos n = 1 ; tan n = 0, (b) sin 15° or sin 1 tan 2 A 2 1 tan2 A 3 tan A tan3 A 1 3 tan 2 A Important Trigonometric Ratios: 5 3 1 = = cos 75° or cos ; 12 12 2 2 cos 15° or cos tan 15° = = (c) sin where n 5 3 1 = = sin 75° or sin ; 12 12 2 2 3 1 3 1 3 1 3 1 = 2 3 = cot 75° ; tan 75° = 2 3 = cot 15° or sin 18° = 10 5 1 & cos 36° or cos = 5 4 5 1 4 Page # 20 Range of Trigonometric Expression: – a 2 b 2 a sin + b cos a2 b2 Sine and Cosine Series : sin + sin (+) + sin ( + 2 ) +...... + sin n 1 n n1 2 sin 2 sin 2 sin = cos + cos (+) + cos ( + 2 ) +...... + cos n 1 n n 1 2 = cos 2 sin 2 sin Trigonometric Equations Principal Solutions: Solutions which lie in the interval [0, 2) are called Principal solutions. General Solution : (ii) sin = sin = n + (1)n where , , n . 2 2 cos = cos = 2 n ± where [0, ], n . (iii) tan = tan = n + where (iv) sin² = sin² , cos² = cos² , tan² = tan² = n ± (i) , , n . 2 2 Page # 21 QUADRATIC EQUATIONS 1. Quadratic Equation : a x2 + b x + c = 0, a 0 x= b b2 4 a c 2a , T he exp ression b 2 4 a c D is c alled discriminant of quadratic equation. If , are the roots, then (a) + = b a (b) = c a A quadratic equation whose roots are & , is (x ) (x ) = 0 i.e. x2 ( + ) x + = 0 2. Nature of Roots: Consider the quadratic equation, a x2 + b x + c = 0 having , as its roots; D b2 4 a c D=0 Roots are equal = = b/2a a, b, c R & D > 0 Roots are real D0 Roots are unequal a, b, c R & D < 0 Roots are imaginary = p + i q, = p i q a, b, c Q & D is a perfect square Roots are rational a, b, c Q & D is not a perfect square Roots are irrational i.e. = p + q ,=p q a = 1, b, c & D is a perfect square Roots are integral. Page # 22 3. Common Roots: Consider two quadratic equations a1 x2 + b 1 x + c 1 = 0 & a2 x2 + b 2 x + c 2 = 0. (i) If two quadratic equations have both roots common, then b1 a1 c1 = = . b2 a2 c2 (ii) If only one root is common, then = 4. b1 c 2 b 2 c1 c1 a 2 c 2 a1 = c1 a 2 c 2 a1 a1 b 2 a 2 b1 Range of Quadratic Expression f (x) = a x 2 + b x + c. Range in restricted domain: (a) If b [x1, x2] then, 2a ( ) ( ) , f(x) min f x1 , f x 2 (b) If Given x [x1, x2] b 2a max f ( x 1) , f ( x 2 ) [x1, x2] then, D f(x) min f ( x1) , f ( x 2 ) , , 4a D max f ( x1) , f ( x 2 ) , 4 a Page # 23 5. Location of Roots: Let f (x) = ax² + bx + c, where a > 0 & a, b , c R. (i) Conditions for both the roots of f (x) = 0 to be greater than a specified number‘x0’ are b² 4ac 0; f (x0) > 0 & ( b/2a) > x0. (ii) Conditions for both the roots of f (x) = 0 to be smaller than a specified number ‘x0’ are b² 4ac 0; f (x0) > 0 & ( b/2a) < x0. (iii) Conditions for both roots of f (x) = 0 to lie on either side of the number ‘x0’ (in other words the number ‘x0’ lies between the roots of f (x) = 0), is f (x0) < 0. (iv) Conditions that both roots of f (x) = 0 to be confined between the numbers x1 and x2, (x1 < x2) are b² 4ac 0; f (x1) > 0 ; f (x2) > 0 & x1 < ( b/2a) < x2. (v) Conditions for exactly one root of f (x) = 0 to lie in the interval (x1, x2) i.e. x1 < x < x2 is f (x1). f (x2) < 0. SEQUENCE & SERIES An arithmetic progression (A.P.) : a, a + d, a + 2 d,..... a + (n 1) d is an A.P. Let a be the first term and d be the common difference of an A.P., then nth term = tn = a + (n – 1) d The sum of first n terms of A.P. are Sn = n [2a + (n – 1) d] 2 = n [a + ] 2 rth term of an A.P. when sum of first r terms is given is tr = Sr – Sr – 1. Properties of A.P. (i) If a, b, c are in A.P. 2 b = a + c & if a, b, c, d are in A.P. a + d = b + c. (ii) Three numbers in A.P. can be taken as a d, a, a + d; four numbers in A.P. can be taken as a 3d, a d, a + d, a + 3d; five numbers in A.P. are a 2d, a d, a, a + d, a + 2d & six terms in A.P. are a 5d, a 3d, a d, a + d, a + 3d, a + 5d etc. (iii) Sum of the terms of an A.P. equidistant from the beginning & end = sum of first & last term. Arithmetic Mean (Mean or Average) (A.M.): If three terms are in A.P. then the middle term is called the A.M. between the other two, so if a, b, c are in A.P., b is A.M. of a & c. Page # 24 n Arithmetic Means Between Two Numbers: If a, b are any two given numbers & a, A 1, A 2,...., A n, b are in A.P. then A 1, A 2,... A n are the n A.M.’s between a & b. A 1 = a + A2 = a + ba , n1 n (b a ) 2 (b a ) ,......, A n = a + n1 n1 n A r = nA where A is the single A.M. between a & b. r1 Geometric Progression: a, ar, ar2, ar3, ar4,...... is a G.P. with a as the first term & r as common ratio. (i) n th term = a rn1 (ii) (iii) a rn 1 Sum of the first n terms i.e. S n = r 1 na , r 1 , r 1 Sum of an infinite G.P. when r < 1 is given by S = a r 1 . 1 r 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 nGeometric Means Between positive number a, b: If a, b are two given numbers & a, G 1, G 2,....., G n, b are in G.P.. Then G 1, G 2, G 3,...., G n are n G.M.s between a & b. G 1 = a(b/a)1/n+1, G 2 = a(b/a)2/n+1,......, G n = 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 = H.M. H of a1, a2 , ........ an is given by 2ac . ac 1 1 1 1 1 ....... = a a a H n 1 2 n Page # 25 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 n (i) n (ar ± b r ) = r1 r1 n n ar ± b r . (ii) n k ar = k r1 r1 ar . r1 n (iii) k = nk; where k is a constant. r1 n (iv) r = 1 + 2 + 3 +...........+ n = r1 n (n 1) 2 n (v) r² = 1 2 + 2 2 + 3 2 +...........+ n 2 = n (n 1) (2n 1) 6 r3 = 1 3 + 2 3 + 3 3 +...........+ n 3 = n 2 (n 1) 2 4 r1 n (vi) r1 BINOMIAL THEOREM 1. Statement of Binomial theorem : If a, b R and n N, then (a + b)n = nC 0 anb 0 + nC 1 an–1 b 1 + nC 2 an–2 b 2 +...+ nC r an–r b r +...+ nC n a0 b n n = n Cr an r b r r 0 2. Properties of Binomial Theorem : (i) (ii) (a) General term : T r+1 = nC r an–r b r Middle term (s) : If n is even, there is only one middle term, n2 th term. 2 which is (b) If n is odd, there are two middle terms, n 1 n1 1 th terms. th and 2 2 which are Page # 26 3. Multinomial Theorem : (x1 + x2 + x3 + ........... xk)n = r1 r2 ... rk n! x 1r1 . x r22 ...x rkk r ! r !... r ! k n 1 2 Here total number of terms in the expansion = 4. n+k–1 C k–1 Application of Binomial Theorem : n If ( A B ) = + f where and n are positive integers, n being odd and 0 < f < 1 then ( + f) f = k n where A – B 2 = k > 0 and If n is an even integer, then ( + f) (1 – f) = k 5. Properties of Binomial Coefficients : (i) n C 0 + nC 1 + nC 2 + ........+ nC n = 2 n (ii) n (iii) n (iv) n C 0 – nC 1 + nC 2 – nC 3 + ............. + (–1)n nC n = 0 C 0 + nC 2 + nC 4 + .... = nC 1 + nC 3 + nC 5 + .... = 2 n–1 n 6. A – B < 1. n n C r + C r–1 = n+1 C r (v) n Cr Cr 1 = n r 1 r Binomial Theorem For Negative Integer Or Fractional Indices (1 + x)n = 1 + nx + n(n 1) 2 n(n 1)(n 2) 3 x + x + .... + 2! 3! n(n 1)(n 2).......(n r 1) r x + ....,| x | < 1. r! T r+1 = n(n 1)(n 2).........(n r 1) xr r! Page # 27 PERMUTATION & COMBINNATION 1. Arrangement : number of permutations of n different things taken r at a time = 2. n P r = n (n 1) (n 2)... (n r + 1) = n! (n r )! Circular Permutation : The number of circular permutations of n different things taken all at a time is; (n – 1)! 3. Selection : Number of combinations of n different things taken r at a time = nC r = 4. The number of permutations of 'n' things, taken all at a time, when 'p' of them are similar & of one type, q of them are similar & of another type, 'r' of them are similar & of a third type & the remaining n (p + q + r) are all different is 5. n! . p! q! r ! Selection of one or more objects (a) (b) (c) 6. n n! Pr = r! (n r )! r! Number of ways in which atleast one object be selected out of 'n' distinct objects is n C 1 + nC 2 + nC 3 +...............+ nC n = 2 n – 1 Number of ways in which atleast one object may be selected out of 'p' alike objects of one type 'q' alike objects of second type and 'r' alike of third type is (p + 1) (q + 1) (r + 1) – 1 Number of ways in which atleast one object may be selected from 'n' objects where 'p' alike of one type 'q' alike of second type and 'r' alike of third type and rest n – (p + q + r) are different, is (p + 1) (q + 1) (r + 1) 2 n – (p + q + r) – 1 Multinomial Theorem : Coefficient of xr in expansion of (1 x)n = 7. n+r1 C r (n N) Let N = p a. q b. rc...... where p, q, r...... are distinct primes & a, b, c..... are natural numbers then : (a) The total numbers of divisors of N including 1 & N is = (a + 1) (b + 1) (c + 1)........ (b) The sum of these divisors is = (p 0 + p 1 + p 2 +.... + p a) (q 0 + q 1 + q 2 +.... + q b) (r0 + r1 + r2 +.... + rc)........ Page # 28 (c) Number of ways in which N can be resolved as a product of two factors is 1 (a 1)(b 1)(c 1).... if N is not a perfect square = 12 (a 1)(b 1)(c 1)....1 if N is a perfect square 2 (d) 8. Number of ways in which a composite number N can be resolved into two factors which are relatively prime (or coprime) to each other is equal to 2 n1 where n is the number of different prime factors in N. Dearrangement : Number of ways in which 'n' letters can be put in 'n' corresponding envelop es su ch t hat no lett er g oes t o co rrec t en velop e is n ! 1 1 1 1 1 1 .......... .. ( 1)n 1 ! 2 ! 3 ! 4 ! n ! PROBABILITY 1. Classical (A priori) Definition of Probability : If an experiment results in a total of (m + n) outcomes which are equally likely and mutually exclusive with one another and if ‘m’ outcomes are favorable to an event ‘A’ while ‘n’ are unfavorable, then the probability of occurrence of the event ‘A’ = P(A) = m n( A ) = . mn n(S) We say that odds in favour of ‘A’ are m : n, while odds against ‘A’ are n : m. P( A ) = 2. n = 1 – P(A) mn Addition theorem of probability : P(AB) = P(A) + P(B) – P(AB) De Morgan’s Laws : (a) (A B)c = A c B c (b) (A B)c = A c B c Distributive Laws : (a) A (B C) = (A B) (A C) (b) A (B C) = (A B) (A C) (i) P(A or B or C) = P(A) + P(B) + P(C) – P(A B) – P(B C) – P(C A) + P(A B C) (ii) P (at least two of A, B, C occur) = P(B C) + P(C A) + P(A B) – 2P(A B C) (iii) P(exactly two of A, B, C occur) = P(B C) + P(C A) + P(A B) – 3P(A B C) (iv) P(exactly one of A, B, C occur) = P(A) + P(B) + P(C) – 2P(B C) – 2P(C A) – 2P(A B) + 3P(A B C) Page # 29 3. Conditional Probability : P(A/B) = 4. Binomial Probability Theorem P(A B) . P(B) 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 nC r p r q n – 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 M i is associated with a probability of p i , then the expectation is given by p i M i . n 6. 7. Total Probability Theorem : P(A) = P(B ) . P(A / B ) i i i 1 Bayes’ Theorem : If an event A can occur with one of the n mutually exclusive and exhaustive events B 1, B 2 , ....., B n and the probabilities P(A/B 1), P(A/B 2) .... P(A/B n) are known, then P(B i / A) = P(B i ) . P( A / B i ) B 1, B 2, B 3,........,B n n P(B i ) . P( A / B i ) i 1 A = (A B 1) (A B 2) (A B 3) ........ (A B n) n P(A) = P(A B 1) + P(A B 2) + ....... + P(A B n) = 8. P(A B ) i i1 Binomial Probability Distribution : (i) Mean of any probability distribution of a random variable is given by : µ= pi x i = p i xi = np pi 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 . p i = p i xi 2 – µ2 = npq Page # 30 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 3. 4. 5. Re(z1) = Re(z2) and m (z1) = m (z2). 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 Properties of conjugate z1 z 2 = z1 + z2 (i) |z| = | z | (iv) z1 z 2 = z1 – z2 (vi) z z1 = 1 z2 z2 (vii) |z1 + z2|2 = (z1 + z2) ( z1 z 2 ) = |z1|2 + |z2|2 + z1 z2 + z1 z2 (viii) ( z1 ) = z (x) arg(z) + arg( z ) (ii) z z = |z|2 (iii) z1z 2 = z1 z2 (v) (z2 0) (ix) If w = f(z), then w = f( z ) Rotation theorem If P(z1), Q(z2) and R(z3) are three complex numbers and PQR = , then z3 z2 = z1 z 2 6. z 3 z 2 i e z1 z 2 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 # 31 7. Cube Root Of Unity : 1 i 3 , 1 i 3 . 2 2 (i) The cube roots of unity are 1, (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 = mz 2 nz1 mz 2 nz1 (internal division), z = (external mn m n 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 z z1 z z2 = 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 = b a = pv of B pv of A. DISTANCE FORMULA : Distance between the two points A (a) and B (b) is AB = a b na m b SECTION FORMULA : r . mn ab Mid point of AB = . 2 Page # 32 4. Scalar Product Of Two Vectors: a . b = | a | | b | cos , where | a |, | are magnitude of a and b respectively and is angle between a and a . b i.i = j.j = k.k = 1; i.j = j.k = k.i = 0 projection of a on b |b| If a = a1i + a2j + a3k & b = b 1i + b 2j + b 3k then a.b = a1b 1 + a2b 2 + a3b 3 a.b , 0 The angle between a& b is given by cos |a| |b| a.b 0 a b (a 0 b 0 ) . Vector Product Of Two Vectors: 1. If a & b are two vectors & is the angle between them then axb a b sin n , . 1. 2. 3. b| b. where n is the unit vector perpendicular to both a & b such that a,b&n forms a right handed screw system. 2. Geometrically a xb = area of the parallelogram whose two adjacent sides are represented by a& b . 3. î î ĵ ĵ k̂ k̂ 0 ; î ĵ k̂, ĵ k̂ î, k̂ î ĵ 4. If a = a1 î +a2 ĵ + a3 k̂ î ĵ a b a1 a 2 b1 b 2 5. & b = b 1 î + b 2 ĵ + b 3 k̂ then k̂ a3 b3 a b o a and b are parallel (collinear) (a 0,b 0) i.e. a K b , where K is a scalar.. Page # 33 6. a xb Unit vector perpendicular to the plane of a & b is n̂ a xb If a,b&c are the pv’s of 3 points A, B & C then the vector area of triangle ABC = 1 axb bxc cxa . 2 & d2 The points A, B & C are collinear if a x b b x c c x a 0 Area of any quadrilateral whose diagonal vectors are d 1 is given by 1 d1 xd2 2 2 2 a.aa.b a.bb.b Lagrange's Identity : (axb)2 a b (a.b)2 V. Scalar Triple Product: The sc alar triple product o f three vectors a ,b&c is defined as: a x b . c a b c sin cos . Volume of tetrahydron V [abc] In a scalar triple product the position of dot & cross can be interchanged i.e. a . ( b x c) (a x b). c OR [ a b c ] [ b c a ] [ c a b ] a . (b x c) a .( c x b) i. e. [ a b c ] [ a c b ] If If a Volume of tetrahedron OABC with O as origin & A( a ), B( b ) and C( c ) a = a1i+a2j+a3k; b a1 a 2 a 3 = b 1i+b 2j+b 3k & c = c 1i+c 2j+c 3k then [abc ] b1 b 2 b 3 . c1 c 2 c 3 , b , c are coplanar [abc]0 . be the vertices = 1 [a b c] 6 Page # 34 The positon vector of the centroid of a tetrahedron if the pv’s of its vertices 1 [a b c d] . , b , c & d are given by 4 are a 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 : ( x 1 x 2 ) 2 ( y1 y 2 ) 2 ( z1 z 2 ) 2 3. AB = | OB – OA | Distance of P from coordinate axes : PA = 4. , 2 2 y 2 z 2 , PB = z x , PC = Section Formula : x = x2 y2 mx 2 nx1 my 2 ny1 mz 2 nz1 ,y= ,z= mn mn mn x1 x 2 y y2 z z2 , y 1 ,z 1 2 2 2 Direction Cosines And Direction Ratios (i) Direction cosines: Let be the angles which a directed line Mid point : x 5. 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 + m 2 + n 2 = 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 a a 2 b2 c 2 ,m b a2 b 2 c 2 ,n c a 2 b2 c 2 Page # 35 (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 = m= 6. x 2 x1 , | PQ | z 2 z1 y 2 y1 and n = | PQ | | PQ | Angle Between Two Line Segments: cos = a1a 2 b1b 2 c1c 2 a12 b12 c12 a 22 b 22 c 22 . The line will be perpendicular if a1a2 + b 1b 2 + c 1c 2 = 0, parallel if 7. c1 b1 a1 = = b2 a2 c2 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 (x 2 x1 ) m( y 2 y1 ) n(z2 z1 ) 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) (ii) Normal form : x + my + nz = p Plane through the point (x1, y1, z1) : a (x x1) + b( y y1) + c (z z1) = 0 (iii) Intercept Form: (iv) (v) x y z 1 a b c Vector form: ( r a ). n = 0 or r . n = a . n Any plane parallel to the given plane ax + by + cz + d = 0 is ax + by + cz + = 0. Distance between ax + by + cz + d 1 = 0 and ax + by + cz + d 2 = 0 is = | d1 d2 | a2 b2 c 2 Page # 36 (vi) Equation of a plane passing through a given point & parallel to the given vectors: r = or 9. a + b + c (parametric form) where & are scalars. r . ( b c) = a . ( b c) (non parametric form) A Plane & A Point (i) Distance of the point (x, y, z) from the plane ax + by + cz+ d = 0 is given by (ii) ax' by' cz' d a 2 b2 c 2 . Length of the perpendicular from a point ( a ) to plane r .n = d | a .n d | is given by p = . |n| (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 =– (iv) x ' x 1 y ' y 1 z ' z 1 a b c (ax1 by1 cz1 d) a2 b2 c 2 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 (ax1 by1 cz1 d) x ' x 1 y ' y 1 z ' z 1 =–2 a b c a2 b2 c 2 10. Angle Between Two Planes: cos = aa'bb'cc ' a2 b 2 c 2 a '2 b ' 2 c ' 2 Planes are perpendicular if aa + bb + cc = 0 and planes are parallel if a b c = = a' b' c ' Page # 37 The angle between the planes = r . n1 = d 1 and r . n2 = d 2 is given by, cos n1 . n 2 | n1 | . | n2 | Planes are perpendicular if n1 . n 2 = 0 & planes are parallel if n1 11. = n 2 , is a scalar Angle Bisectors (i) The equations of the planes bisecting the angle between two given planes a1x + b 1y + c 1z + d 1 = 0 and a2x + b 2y + c 2z + d 2 = 0 are a1x b1y c 1z d1 a12 (ii) b12 =± a 22 b 22 c 22 Bisector of acute/obtuse angle: First make both the constant terms positive. Then a1 a2 + b 1 b 2 + c 1 c 2 > 0 a1 a2 + b 1 b 2 + c 1 c 2 < 0 12. a 2 x b 2 y c 2 z d2 c 12 origin lies on obtuse angle origin lies in acute angle Family of Planes (i) Any plane through the intersection of a1x + b 1y + c 1z + d 1 = 0 & a2x + b 2y + c 2z + d 2 = 0 is a1x + b 1y + c 1z + d 1 + (a2x + b 2y + c 2z + d 2) = 0 (ii) The equation of plane passing through the intersection of the planes r . n1 = d1 & r . n2 = d 2 is r . (n 1 + n 2 ) = d 1 d 2 where is arbitrary scalar 13. Volume Of A Tetrahedron: Volume of a tetrahedron with vertices A (x1, y1, z1), B( x2, y2, z2), C (x3, y3, z3) and D (x4, y4, z4) is given by V = x1 x2 y1 y2 4 4 z1 1 z2 1 1 x 3 y 3 z3 1 6 x y z 1 4 Page # 38 A LINE 1. Equation Of A Line (i) A straight line is intersection of two planes. it is reprsented by two planes a1 x + b 1y + c 1 z + d 1 = 0 and a2x + b2y + +c2z + d2 =0. (ii) Symmetric form : (iii) (iv) Vector equation: r = a + b Reduction of cartesion form of equation of a line to vector form & vice versa x x1 a = = y y1 b z z1 c = r.. x x 1 y y1 z z 1 = = r =(x1 î +y1 ĵ + z1 k̂ ) + (a î + b ĵ + c k̂ ). b c a 2. Angle Between A Plane And A Line: (i) x x1 If is the angle between line = y y1 m = z z1 n and the plane ax + by + cz + d = 0, then sin = (ii) a bm c n (a 2 b 2 c 2 ) . 2 m 2 n2 Vector form: If is the angle between a line r = ( a + b ) and . = d then sin = b . n . r n (iii) (iv) 3. | b | | n | Condition for perpendicularity a Condition for parallel = m b = n c , a + bm + cn = 0 b xn b.n =0 =0 Condition For A Line To Lie In A Plane (i) Cartesian form: Line x x1 = y y1 m = z z1 n would lie in a plane ax + by + cz + d = 0, if ax 1 + by 1 + cz 1 + d = 0 & a + bm + cn = 0. (ii) Vector form: Line r = a + b would lie in the plane r . n = d if b.n = 0 &a .n = d Page # 39 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 – ' x– y– z– & ' m' n' m n ' ' ' If = (ii) m n ' m' n' 0, then lines are skew.. Shortest distance formula for lines a2 – a1 . b1 b2 r = a 1 + b1 and r = a 2 + b 2 is d b1 b 2 (iii) a1 ) 0 (a 2 a1 ) b d= . |b| Condition of coplanarity of two lines r = a + b & [a – c b d] 0 5. Shortest distance between parallel lines r = a 1 + b & r = a 2 + b is (v) Vector Form: For lines r = a 1 + b1 and r = a 2 + b 2 to be skew ( b1 x b 2 ). ( a 2 (iv) r = c + d is Sphere General equation of a sphere is x2 + y2 + z2 + 2ux + 2vy + 2wz + d = 0. (u, – v, w) is the centre and u 2 v 2 w 2 d is the radius of the sphere. Page # 40 SOLUTION OF TRIANGLE 1. 2. a b c . sin A sin B sin C Cosine Formula: Sine Rule: (i) cos A = b 2 c2 a 2 2b c c2 a 2 b 2 2 ca a 2 b 2 c2 2a b (iii) cos C = 3. (ii) cos B = 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 BC A bc = cot 2 2 bc (iii) tan 5. (ii) tan c a CA B = cot ca 2 2 AB ab C = cot ab 2 2 Trigonometric Functions of Half Angles: (i) sin (s b) (s c) A B (s c) (s a ) = ; sin = ; bc 2 2 ca sin C = 2 (s a ) (s b) ab (ii) cos A B C s (s a ) s (s b ) = ; cos = ; cos = 2 2 2 bc ca (iii) tan A = 2 s (s c) ab (s b) (s c) abc = where s = is semi s (s a ) s (s a ) 2 perimetre of triangle. (iv) sin A = 2 bc s(s a )(s b)(s c) = 2 bc Page # 41 6. Area of Triangle () = 7. : 1 1 1 ab sin C = bc sin A = ca sin B = s (s a ) (s b) (s c) 2 2 2 m - n Rule: If BD : DC = m : n, then (m + n) cot m cot n cot n cot B m cot C 8. Radius of Circumcirlce : R= 9. a b c a bc = 2 sinA 2 sinB 2 sinC 4 Radius of The Incircle : (i) r = s (ii) r = (s a) tan (iii) r = 4R sin 10. A B C = (s b) tan = (s c) tan 2 2 2 A B C sin sin 2 2 2 Radius of The Ex- Circles : (i) r1 = ; ; r2 = r3 = sa sb sc (ii) r1 = s tan A 2 (iii) r1 = 4 R sin ; r2 = s tan B 2 ; r3 = s tan C 2 B C A . cos . cos 2 2 2 Page # 42 11. Length of Angle Bisectors, Medians & Altitudes : (i) Length of an angle bisector from the angle A = a = (ii) Length of median from the angle A = m a = 1 2 & (iii) Length of altitude from the angle A = A a = 12. 2 bc cos A 2 bc ; 2 b2 2 c2 a 2 2 a Orthocentre and Pedal Triangle: The triangle KLM which is formed by joining the feet of the altitudes is called the Pedal Triangle. (i) Its angles are 2A, 2B and 2C. (ii) Its sides are a cosA = R sin 2A, b cosB = R sin 2B and c cosC = R sin 2C (iii) Circumradii of the triangles PBC, PCA, PAB and ABC are equal. Page # 43 13. Excentral Triangle: The triangle formed by joining the three excentres 1, 2 and 3 of ABC is called the excentral or excentric triangle. (i) ABC is the pedal triangle of the 1 2 3. 14. C A B . , & 2 2 2 2 2 2 (ii) Its angles are (iii) Its sides are 4 R cos (iv) 1 = 4 R sin (v) Incentre of ABC is the orthocentre of the excentral 1 2 3. A 2 A 2 , 4 R cos ; 2 = 4 R sin B 2 & 4 R cos C . 2 C B ; 3 = 4 R sin . 2 2 Distance Between Special Points : (i) Distance between circumcentre and orthocentre OH2 = R2 (1 – 8 cosA cos B cos C) (ii) Distance between circumcentre and incentre O2 = R2 (1 – 8 sin A C B sin sin ) = R2 – 2Rr 2 2 2 (iii) Distance between circumcentre and centroid OG2 = R2 – 1 2 (a + b2 + c2) 9 INVERSE TRIGONOMETRIC FUNCTIONS 1. Principal Values & Domains of Inverse Trigonometric/Circular Functions: Function Domain Range (i) y = sin 1 x where 1x1 (ii) y = cos1 x where 1x1 0y (iii) y = tan 1 x where xR (iv) y = cosec 1 x where x 1 or x 1 (v) y = sec 1 x where x 1 or x 1 (vi) y = cot1 x where xR y 2 2 y 2 2 y ,y0 2 2 0 y ; y 2 0<y< Page # 44 x 2 2 (i) sin 1 (sin x) = x, (ii) cos1 (cos x) = x; 0 x (iii) tan 1 (tan x) = x; (iv) cot1 (cot x) = x; 0<x< (v) sec 1 (sec x) = x; 0 x , x 2 (vi) cosec 1 (cosec x) = x; x 2 2 P-3 (i) (ii) (iii) (iv) sin 1 (x) = sin 1 x, 1 x 1 tan 1 (x) = tan 1 x, xR cos1 (x) = cos1 x, 1 x 1 cot1 (x) = cot1 x, x R P-5 (i) sin1 x + cos1 x = , 1 x 1 2 (ii) tan1 x + cot 1 x = , xR 2 P-2 x 2 2 x 0, , x 1 2 Identities of Addition and Substraction: (i) sin1 x + sin1 y (iii) 2. I-1 cosec1 x + sec1 x = 2 2 = sin1 x 1 y y 1 x , x 0, y 0 & (x 2 + y2) 1 2 2 = sin1 x 1 y y 1 x , x 0, y 0 & x 2 + y2 > 1 (ii) 2 cos1 x + cos1 y = cos1 x y 1 x (iii) tan1 x + tan1 y = tan1 xy , x > 0, y > 0 & xy < 1 1 xy = + tan1 = 1 y 2 , x 0, y 0 xy , x > 0, y > 0 & xy > 1 1 xy , x > 0, y > 0 & xy = 1 2 Page # 45 I-2 (i) (ii) (iii) I-3 (i) (ii) (iii) (iv) (v) 2 2 sin1 x sin1 y = sin1 x 1 y y 1 x , x 0, y 0 2 cos1 x cos1 y = cos1 x y 1 x x 0, y 0, x y tan1 x tan1y = tan1 sin 2 x 1 x 2 1 cos1 (2 x 2 1) 2x 1 tan 1 x2 2x sin1 1 x2 cos1 1 x2 1x2 1 y2 , xy , x 0, y 0 1 xy 1 2 sin x 1 = 2 sin x 2 sin 1 x if | x | 1 if if x 1 2 2 x 1 2 2 cos 1 x if 0 x 1 = 1 2 2 cos x if 1 x 0 2 tan 1x 1 = 2 tan x 2 tan 1x 2 tan 1x = 2 tan 1x 2 tan 1x if | x | 1 if x 1 if x 1 if | x | 1 if x 1 if x 1 2 tan 1x if x 0 = 1 2 tan x if x 0 x y z xyz If tan1 x + tan1 y + tan1 z = tan1 1 xy yz zx if, x > 0, y > 0, z > 0 & (xy + yz + zx) < 1 NOTE: (i) If tan1 x + tan1 y + tan1 z = then x + y + z = xyz (ii) If tan1 x + tan1 y + tan1 z = (iii) tan1 1 + tan1 2 + tan1 3 = then xy + yz + zx = 1 2 Page # 46 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 n x1 x 2 x 3 ....... x n = x = n x i i1 n If x1, x2, x3, .... xn are values of veriate with frequencies f1, f2, f 3,.........fn then their A.M. is given by n f1x1 f2 x 2 f3 x 3 ......fn fn = x = f1 f2 f3 ...... fn 2. f x i i i1 n , where N = N f i i 1 Properties of Arithmetic Mean : (i) Sum of deviation of variate from their A.M. is always zero that is (ii) x i x = 0. Sum of square of deviation of variate from their A.M. is minimum that is x i x 2 is minimum (iii) 3. 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 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 n 1 th term, when n is odd 2 th th Median = n n Mean of and 2 term when n is even 2 2 Page # 47 4. Mode If a frequency distribution the mode is the value of that variate which have the maximum frequency. Mode for For ungrouped distribution : The value of variate which has maximum frequency. For ungrouped frequency distribution : The value of that variate which have maximum frequency. Relationship between mean, median and mode. (i) (ii) In symmetric distribution, mean = mode = median In skew (moderately asymmetrical) distribution, median divides mean and mode internally in 1 : 2 ratio. 5. median = 2Mean Mode 3 Range difference of extreme values = sum of extreme values LS LS where L = largest value and S = smallest value 6. Mean deviation : n Mean deviation = | x A | i i 1 n n Mean deviation = f | x A | i i i 1 (for frequency distribution) N 7. Variance : Standard deviation = + var iance formula 2 x2 = x i x n Page # 48 n 2 x = x 2 i i1 n – n 2 xi i1 n = n x 2 i i1 – x 2 n 2 d2 = d2i di – , where di = xi – a , where a = assumed mean n n (ii) coefficient of S.D. = x coefficient of variation = × 100 (in percentage) x Properties of variance : (i) var(xi + ) = var(xi) (ii) var(.xi) = 2(var xi) (iii) var(a xi + b) = a2(var xi) where , a, b are constant. MATHEMATICAL REASONING Let p and q are statements p q p q pvq p q q p p q q p T T T T T T T T T F F T F F F F F T F F F T T T F T F F T F F T Tautology : This is a statement which is true for all truth values of its components. It is denoted by t. Consider truth table of p v ~ p p ~ p p v ~p T F T F T T Page # 49 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 p ~ p p ~p T F F F T F pq pq pq pq (i) Statements Negation (~ p) (~ q) (~ p) (~ q) p (~ q) p –q Let p q Then (ii) (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 # 50 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,bA Page # 51