Title Basic Maths 4.0.pdf ✅

Basic Mathematics 1 1 Basic Mathematics 1. Algebra 1.1 Quadratic Equation An algebraic equation of second order (highest power of the variable is equal to 2) is called a quadratic equation. The equation ax2 + bx + c = 0 ....(i) Is the general form of quadratic equation where a ≠ 0. The general solution of above equation is 2 b b – 4ac x 2a − ± = If values of x be x1 and x2 then 2 1 –b b – 4ac x 2a + = and 2 2 –b – b – 4ac x 2a = Here x1 and x2 are called roots of equation (i). We can easily see that sum of roots = x1 + x2 = – b a and product of roots = x1 x2 = c a Example 1: Find roots of equation 2x2 – x – 3 = 0. (1) 3/2, –1 (2) 1, 3/2 (3) 2, 3 (4) –3/2, –1 Solution: Compare this equation with standard quadratic equation ax2 + bx + c = 0, we have a=2, b=–1, c=–3. Now from x = 2 –b b – 4ac 2a ± ; x = 2 –(–1) (–1) – 4(2)(–3) 2(2) ± 1 22 6 3 –4 x x x –1 42 4 ⇒ ==⇒ = ⇒ = Example 2: In quadratic equation ax2 + bx + c =0, if discriminant D = b2 – 4ac, then roots of quadratic equation are: (1) real and distinct, if D > 0 (2) real and equal (repeated roots), if D = 0 (3) non-real (imaginary), if D < 0 (4) None of the above Solution: (1,2,3)
2 Basic Mathematics Example 3: Find the sum and product of roots of equation 3x2 – 5x – 12 = 0. (1) 5/3, 4 (2) –5/3, 4 (3) –4, 5/3 (4) –4, –5/3 Solution: α + β = – b/a = 5/3 (α, β are roots of equation) αβ = c/a = –4 1.2 Binomial Approximation If x is very small as compared to 1 (1+x)n ≈ 1 + nx 1.3 Logarithm Common formulae: • loga a = 1 • log mn = log m + log n • log m n       = log m – log n • log mn = n log m • loge m = 2.303 log10m 1.4 Application of ratio (componendo and dividendo) • If p q = a b , then p q p–q + = a b a–b + • If p q = a b , then p q q + = a b b + • If p q = a b , then p–q q = a–b b Example 4: The value of acceleration due to gravity (g) at height h above the surface of earth is given by g' = g –2 R R h       + . What is the value of g' for very small height? Solution: g' = g –2 e h 1 R       +     ⇒ g' = g e 2h 1– R           Example 5: Calculate the value of 0.99 Solution: 0.99 ⇒ 1/2 1 1– 100       ⇒ 1 1– 200       = 0.995

4 Basic Mathematics 2. Trigonometry 2.1 Degrees and Radian Let AOB be a central angle in a circle of radius r, as in figure. Then the angle AOB or θ is defined in radius as- θ = Arc length Radius ⇒ θ = AB r If r = 1 then θ = AB The relation between radians and degrees is given by: π radians = 180° Angle Conversion formulas 1 degree = 180 π (≈ 0.02) radian Degrees to radian : multiply by 180 π 1 radian ≈ 57 degrees 180 Radians to degrees : multiply by π Standard values (1) 30° = 6 π rad (2) 45° = 4 π rad (3) 60° = 3 π rad (4) 90° = 2 π rad (5) 120° = 2 3 π rad (6) 135° = 3 4 π rad (7) 150° = 5 6 π rad (8) 180° = π rad (9) 360° = 2π rad Example 7: (i) Convert 6 π rad to degrees. (ii) Convert 15° to radians. Solution: (i) 6 π × 180° π = 30° (ii) 15° × 180 π ° = 12 π rad B θ A r O