Stark

7.B] Find the remainder when (349 × 74 × 36) is divided by 3

7.a] Find the least positive values of x such thati) 71 ≡ 𝑥(𝑚𝑜𝑑8)ii) 78 + 𝑥 ≡ 3(𝑚𝑜𝑑5)iii) 89 ≡ (𝑥 + 3)(𝑚𝑜𝑑4)

6.C] Find the general solution of the equation (𝑝𝑥 − 𝑦)(𝑝𝑦 + 𝑥) = 𝑎2𝑝 by reducing into Clairaut’s form by taking the substitution 𝑋 = 𝑥2, 𝑌 = 𝑦2

6.B] When a switch is closed in a circuit containing a battery E, a resistance R and an inductance L, the current i build up at a rate given by L di dt Ri = E. Find i as a…

6.A] Solve (𝑥2 + 𝑦2 + 𝑥)𝑑𝑥 + 𝑥𝑦𝑑𝑦 = 0

5.C] Solve 𝑥𝑦𝑝2−(𝑥2 + 𝑦2)𝑝+𝑥𝑦 = 0

5.B] Find the orthogonal trajectories of 𝑥2 /𝑎2 +𝑦2/𝑏2+𝜆= 1, where λ is a parameter.

5.A] Solve 𝑑𝑦/𝑑𝑥 + 𝑦/𝑥= 𝑥2𝑦6

4.C] If 𝑥 + 𝑦 + 𝑧 = 𝑢, 𝑦 + 𝑧 = 𝑣 𝑎𝑛𝑑 𝑧 = 𝑢𝑣𝑤, find the values of 𝜕(𝑥,𝑦,𝑧)/𝜕(𝑢,𝑣,𝑤).

4.B] If 𝑢 = 𝑓(𝑥 − 𝑦, 𝑦 − 𝑧, 𝑧 − 𝑥) show that 𝜕𝑢/𝜕𝑥 + 𝜕𝑢/𝜕𝑦 +𝜕𝑢/𝜕𝑧 = 0