PERMUTATIONS AND COMBINATION 1.2 3. n. nCr nCn-r. nCr + nCr-l = (n + l) Cr. (m + n)Cr = m!n! BINOMIAL THEOREM (x xn + nCl xn 1 a + nC2 x n 2 a2+ nC3 x n 3 a3+ + nCn an. nth term, Tr+l = nCr xn-r ar . PARTIAL FRACTIONS l. 2. 3. f(x) is a proper fraction if the deg (g(x)) > deg (f(x)). g(x) f(x) is a improper fraction if the deg (g(x)) deg (f(x)). g(x) Linear non- repeated factors f(x) (ax + + d) ax+b (cx + d) Linear repeated factors f(x) c (ax + + ax+b (cx + d) (cx + Non-linear(quadratic which can not be factorized) f(x) Ax+B Cx+D (ax 2 + d) ax 2 + b (cx2 + d) • ANALYTICAL GEOMETRY 1. Distance between the two points (Xl, YD and (x2, Y2) in the plane is (x 2 —Xl)2 + (Y2 —Yl)2 OR (Xl —x +(YI —Y2)2 . 2. Section formula mx + nxt nix 2 —nxl myt2 + nyl (for internal division), my2 s— nyl (for external division).
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3. 4. 5. Mid point formula +X2 + Y2 2 2 Centriod formula 3 3 Area of triangle when their vertices are given, 1 2 1 — —l Xl(Y2 Y 3) 4- X2(Y3 — X3(Y1 Y2)l 2 STRAIGHT LINE Slope (or Gradient) of a line — tangent of an inclination — tan0. Slope of a X- axis = O Slope of a line parallel to X-axis = 0 Slope of a Y- axis = oo Slope of a line parallel to Y-axis = oo Slope of a line joining (Xl, x2) and (Yl, Y2) = l. If two lines are parallel, then their slopes are equal (m. m2) If two lines are perpendicular, then their product of slopes is -l (ml -l) EQUATIONS OF STRAIGHT LINE y = mx + c (slope-intercept form) y - = m(x-xl) (point-slope form) (x Xl) (two POInt form) Y-YI = — + — = I (intercept form) x cosa +y Sina = P (normal form) Equation of a straight line in the general form is ax2+ bx + c = 0 a Slope ofax2+ bx + c = 0 is b ml —m 2 2. Angle between two straight lines is given by, tano — Length of the perpendicular from a point (Xl,X2) and the straight line ax2 + bx + c ax 1 + byl + c —0 is a 2 + b2
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Equation of a straight line passing through intersection of two lines alX2 + blX + = () and a2X2 + b2X + c2 = O is ape + blX + Cl + K(a2X2+ b2X + c2 ) = (), where K is any constant. Two lines meeting a point are called intersecting lines. More than two lines meeting a point are called concurrent lines. Equation of bisector of angle between the lines aux + blY+ Cl 0 and apc+bly+cl + a2x+b2Y2 + a2X+ b2Y + c2=o is al 2 + b12 PAIR OF STRAIGHT LINES a22 + b22 l. An equation ax2 +2hxy +by2 — 0, represents a pair of lines passing through origin generally called as homogeneous equation of degree2 in x and y and 2 If-ab angle between these is given by tano = a+b ax2 +2hxy +by2 = 0, represents a pair of coincident lines, if 112 = ab and the same represents a pair of perpendicular lines, if a + b — 0. If ml and are the slopes of the lines ax 2 +2hxy +by2 O,then ml + b a and ml 2. An equation ax2 +2hxy +by2+2gx +2fy +c 0 is called second general second order equation represents a pair of lines if it satisfies the the condition abc + 2fgh —af— bg2 — ch2 0. The angle between the lines ax2 +2hxy +by2+2gx +2fy +c = 0 is given by 2 112 -ab tano = ax2 +2hxy +by2+2gx +2fy +c = 0, represents a pair of parallel lines, if 112 = ab and af— bg2 and the distance between the parallel lines is 2 g2—ac a(a+b) ax2 +2hxy +by2+2gx +2fy +c = 0, represents a pair of perpendicular lines ,ifa + b — O.
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