Basic algebra mathematics formula sheet, Cheat Sheet of Algebra

Download Basic algebra mathematics formula sheet and more Cheat Sheet Algebra in PDF only on Docsity! SOME IMPORTANT MATHEMATICAL FORMULAE Circle : Area = π r2 ; Circumference = 2 π r. Square : Area = x2 ; Perimeter = 4x. Rectangle: Area = xy ; Perimeter = 2(x+y). Triangle : Area = 1 2 (base)(height) ; Perimeter = a+b+c. Area of equilateral triangle = 3 4 a2 . Sphere : Surface Area = 4 π r2 ; Volume = 4 3 π r3 . Cube : Surface Area = 6a2 ; Volume = a3 . Cone : Curved Surface Area = π rl ; Volume = 1 3 π r2 h Total surface area = . π r l + π r2 Cuboid : Total surface area = 2 (ab + bh + lh); Volume = lbh. Cylinder : Curved surface area = 2 π rh; Volume = π r2 h Total surface area (open) = 2 π rh; Total surface area (closed) = 2 π rh+2 π r2 . SOME BASIC ALGEBRAIC FORMULAE: 1.(a + b)2 = a2 + 2ab+ b2 . 2. (a - b)2 = a2 - 2ab+ b2 . 3.(a + b)3 = a3 + b3 + 3ab(a + b). 4. (a - b)3 = a3 - b3 - 3ab(a - b). 5.(a + b + c)2 = a2 + b2 + c2 +2ab+2bc +2ca. 6.(a + b + c)3 = a3 + b3 + c3+3a2b+3a2c + 3b2c +3b2a +3c2a +3c2a+6abc. 7.a2 - b2 = (a + b)(a – b ) . 8.a3 – b3 = (a – b) (a2 + ab + b2 ). 9.a3 + b3 = (a + b) (a2 - ab + b2 ). 10.(a + b)2 + (a - b)2 = 4ab. 11.(a + b)2 - (a - b)2 = 2(a2 + b2 ). 12.If a + b +c =0, then a3 + b3 + c3 = 3 abc . INDICES AND SURDS 1. am an = am + n 2. ma m na na −= . 3. m n mn(a ) a= . 4. m m m(ab) a b= . 5. m ma a mb b   =   . 6. 0a 1, a 0= ≠ . 7. 1ma ma − = . 8. yxa a x y= ⇒ = 9. x xa b a b= ⇒ = 10. a 2 b x y± = ± , where x + y = a and xy = b. S B SATHYANARAYANA M. Sc., M.I.E ., M Phil . 9481477536 1 LOGARITHMS xa m log m x a = ⇒ = (a > 0 and a ≠ 1) 1. loga mn = logm + logn. 2. loga m n      = logm – logn. 3. loga mn = n logm. 4. logba = log a log b . 5. logaa = 1. 6. loga1 = 0. 7. logba = a 1 log b . 8. loga1= 0. 9. log (m +n) ≠ logm +logn. 10. e logx = x. 11. logaax = x. PROGRESSIONS ARITHMETIC PROGRESSION a, a + d, a+2d,-----------------------------are in A.P. nth term, Tn = a + (n-1)d. Sum to n terms, Sn = [ ]n 2a (n 1)d 2 + − . If a, b, c are in A.P, then 2b = a + c. GEOMETRIC PROGRESSION a, ar, ar2 ,--------------------------- are in G.P. Sum to n terms, Sn = na(1 r ) 1 r − − if r < 1 and Sn = na(r 1) r 1 − − if r > 1. Sum to infinite terms of G.P, a S 1 r∞ = − . If a, b, c are in A.P, then b2 = ac. HARMONIC PROGRESSION Reciprocals of the terms of A.P are in H.P 1 1 1 , , , a a d a 2d+ + ----------------- are in H.P If a, b, c are in H.P, then b = 2ac a c+ . MATHEMATICAL INDUCTION 1 + 2 + 3 + -----------------+n = n(n 1) n 2 +=∑ . 12+22 +32 + -----------------+n2 = 2 n(n 1)(2n 1) n 6 + +=∑ . S B SATHYANARAYANA M. Sc., M.I.E ., M Phil . 9481477536 2 Equation of a straight line passing through intersection of two lines a1x2 + b1x + c1 = 0 and a2x2 + b2x + c2 = 0 is a1x2 + b1x + c1 + K(a2x2 + b2x + c2 ) = 0, 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 a1x + b1y+ c1 = 0 and a2x + b2y + c2 = 0 is 1 1 1 2 2 2 2 2 2 2 2 1 1 2 2 a x b y c a x b y c a b a b + + + += ± + + PAIR OF STRAIGHT LINES 1. 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 angle between these is given by tanθ = 22 h ab a b − + . ax2 +2hxy +by2 = 0, represents a pair of coincident lines, if h2 = ab and the same represents a pair of perpendicular lines, if a + b = 0. If m1 and m2 are the slopes of the lines ax2 +2hxy +by2 = 0,then m1 + m2 = 2h b − and m1 m2 = a b . 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 –af2 – bg2 – ch2 = 0. The angle between the lines ax2 +2hxy +by2+2gx +2fy +c = 0 is given by tanθ = 22 h ab a b − + . ax2 +2hxy +by2+2gx +2fy +c = 0, represents a pair of parallel lines, if h2 = ab and af2= bg2 and the distance between the parallel lines is 22 g ac a(a b) − + . ax2 +2hxy +by2+2gx +2fy +c = 0, represents a pair of perpendicular lines ,if a + b = 0. S B SATHYANARAYANA M. Sc., M.I.E ., M Phil . 9481477536 5 TRIGNOMETRY Area of a sector of a circle = 21 r 2 θ . Arc length, S = r θ. sinθ = opp hyp ,cosθ = adj hyp ,tanθ = opp adj ,cotθ = adj opp , secθ = hyp adj , cosecθ = hyp opp . Sinθ = 1 cos ecθ or cosecθ = 1 sin θ , cosθ = 1 secθ or secθ = 1 cos θ , tanθ = 1 cot θ or cotθ = 1 tan θ , tanθ = sin cos θ θ , cotθ = cos sin θ θ . sin2θ + cos2θ = 1;⇒ sin2θ = 1- cos2θ; cos2θ = 1- sin2θ; sec2θ - tan2θ = 1; ⇒ sec2θ = 1+ tan2θ; tan2θ = sec2θ – 1; cosec2θ - cot2θ = 1; ⇒ cosec2θ = 1+ cot2θ; cot2θ = cosec2θ – 1. STANDARD ANGLES 00 or 0 030 or 6 π 045 or 4 π 060 or 3 π 090 or 2 π 015 or 12 π 075 or 5 12 π Sin 0 1 2 1 2 3 2 1 3 1 2 2 − 3 1 2 2 + Cos 1 3 2 1 2 1 2 0 3 1 2 2 + 3 1 2 2 − Tan 0 1 3 1 3 ∞ 3 1 3 1 − + 3 1 3 1 + − Cot ∞ 3 1 1 3 0 3 1 3 1 + − 3 1 3 1 − + Sec 1 2 3 2 1 ∞ 2 2 3 1+ 2 2 3 1− Cosec ∞ 2 2 2 3 1 2 2 3 1− 2 2 3 1+ ALLIED ANGLES Trigonometric functions of angles which are in the 2nd, 3rd and 4th quadrants can be obtained as follows : If the transformation begins at 900 or 2700, the trigonometric functions changes as sin ↔ cos tan ↔ cot sec ↔ cosec S B SATHYANARAYANA M. Sc., M.I.E ., M Phil . 9481477536 6 where as the transformation begins at 1800 or 3600, the same trigonometric functions will be retained, however the signs (+ or -) of the functions decides ASTC rule. COMPOUND ANGLES Sin(A+B)=sinAcosB+cosAsinB. Sin(A-B)= sinAcosB-cosAsinB. Cos(A+B)=cosAcosB-sinAsinB. Cos(A-B)=cosAcosB+sinAsinB. tan(A+B)= tan A tan B 1 tan A tan B + − tan(A-B)= tan A tan B 1 tan A tan B − + tan A 4 π +   = 1 tan A 1 tan A + − tan A 4 π −   = 1 tan A 1 tan A − + tan(A+B+C)= tan A tan B tan C tan A tan B tan C 1 (tan A tan B tan B tan C tan C tan A) + + − − + + sin(A+B) sin(A-B)= 2 2 2 2sin A sin B cos B cos A− = − cos(A+B) cos(A-B)= 2 2cos A sin B− MULTIPLE ANGLES 1.sin 2A=2 sinA cosA. 2. sin 2A= 2 2 tan A 1 tan A+ . 3.cos 2A = 2 2cos A sin A− =1-2 2sin A . = 2 2cos A 1− = 2 2 1 tan A 1 tan A − + 4. tan 2A= 2 2 tan A 1 tan A− , 5. 1+cos 2A= 22cos A , 6. 2cos A = 1 (1 cos 2A) 2 + . 7. 1-cos 2A= 22sin A , 8. 2sin A = 1 (1 cos 2A) 2 − , 9.1+sin 2A= 2(sin A cos A)+ , 10. 1-sin 2A= 2(cos A sin A)− = 2(sin A cos A)− , 11.cos 3A= 34cos A 3cos A− , 12. sin 3A= 33sin A 4sin A− , 13.tan 3A= 3 2 3tan A tan A 1 3tan A − − . S B SATHYANARAYANA M. Sc., M.I.E ., M Phil . 9481477536 7