Title Handbook - Physics.pdf ✅

QUADRATIC EQUATION Roots of ax2 + bx + c = 0 are x = 2 b b 4ac 2a −± − Sum of roots x1 + x2 = b a − ; Product of roots x1 x2 = c a BINOMIAL APPROXIMATION If x << 1, then (1 + x)n ≈ 1 + nx & (1 – x)n ≈ 1 – nx LOGARITHM log mn = log m + log n log m/n = log m – log n log mn = n log m loge m = 2.303 log10m log 2 = 0.3010 COMPONENDO AND DIVIDENDO LAW If p a pq ab then q b pq ab + + = = − − ARITHMETIC PROGRESSION-AP FORMULA a, a + d, a + 2d, a + 3d, ..., a + (n – 1)d, here d = common difference Sum of n terms Sn = n 2 [2a + (n – 1)d], an = a1 + (n – 1)d Basic Mathematics Applied in Physics
Hand Book (Physics) 2 NOTES (i) 1 + 2 + 3 + 4 + 5 ... + n = n(n 1) 2 + (ii) 12 + 22 + 32 + ... + n2 = n(n 1)(2n 1) 6 + + GEOMETRICAL PROGRESSION-GP FORMULA a, ar, ar2 , ... here, r = common ratio Sum of n terms Sn = n a(1 r ) 1 r − − Sum of ∞ terms S∞ = a 1 r − , (r < 1) TRIGONOMETRY • 2p radian = 360° ⇒ 1 rad = 57.3° • cosec q = 1 sin θ • sec q = 1 cos θ • cot q = 1 tan θ • sin2 q + cos2 q = 1 • 1 + tan2 q = sec2 q • 1 + cot2 q = cosec2 q • sin (A ± B) = sin A cos B ± cos A sin B • cos (A ± B) = cos A cos B  sin A sin B • tan (A ± B) = tan A tan B 1 tan A tan B ±  • sin 2A = 2 sin A cos A • cos 2A = cos2 A – sin2 A = 1 – 2 sin2 A = 2 cos2 A – 1 • tan 2A = 2 2 tan A 1 tan A − , If A = B sine law sin A sin B sin C abc = =
Basic Mathematics Applied in Physics 3 cosine law 222 22 2 2 22 bca cab abc cos A , cos B , cos C 2bc 2ca 2ab +− +− +− = = = A A B B C C c b a Approximation for small q • sin q ≈ q • cos q ≈ 1 • tan q ≈ q ≈ sin q Differentiation • n dy n 1 y x nx dx − =→ = • dy 1 y lnx dx x =→= • dy y sin x cos x dx = →= • dy y cos x sin x dx = → =− • x x dy ye e dx α +β α +β = → =α • dy dv du y uv u v (Product rule) dx dx dx =→ = + • dy df (g(x)) d(g(x)) y f (g(x)) (Chain rule) dx dg(x) dx = →= × • y = k = constant ⇒ dy dx = 0 • 2 du dv v u u dy dx dx y (Division Rule) v dx v − =⇒ =
Hand Book (Physics) 4 Integration • n 1 n x x dx C, n 1 n 1 + = + ≠− + ∫ • 1 dx nx C x = + ∫  • sin xdx cos x C =− + ∫ • cos xdx sin x C = + ∫ • x x 1 e dx e C α +β α +β = + α ∫ • n 1 n (x ) ( x ) dx C (n 1) + α +β α +β = + α + ∫ Maxima and Minima of a Function y = f(x) • For maxima 2 2 dy d y 0 & ve dx dx = = − • For minima 2 2 dy d y 0 & ve dx dx = = + Average of a Varying Quantity If y = f(x) then < y > = 2 2 1 1 2 1 x x x x x 2 1 x ydx ydx y x x dx = = − ∫ ∫ ∫ Formulae for Calculation of Area • Area of a square = (side)2 • Area of rectangle = length × breadth • Area of a triangle = 1/2 × base × height • Area of a trapezoid = 1/2 × (distance between parallel sides) × (sum of parallel sides) • Area enclosed by a circle = p r2 , (r = radius) • Surface area of a sphere = 4p r2 , (r = radius) • Area of a parallelogram = base × height