Trigonometric Ratios & Identities – Problems 1) Find the value of i) sin (5π/4) ii) tan (855°) iii) tan [-23 π/3] iv) sec (13 π/3) Solutions: i) Given that sin (5π/4) = sin [2π – (π/3)] (since sin (180 – x) = – sinx) = – sin (π/3) = (√3/2) ii) Given that tan (855°) Read more about Trigonometric Ratios & Identities – Problems[…]
Division Algorithm If 0 ≠ a, b ϵ Z then ∃ q, r ϵ Z uniquely such that b = aq + r where 0 ≤ r < |a|. A nonzero integer a is said to divide an integer b if there exist q ϵ z such that b = aq. It is denoted by Read more about Division Algorithm[…]
Kepler’s Second Law of Planetary Motion Kepler’s second law states that the orbiting satellite will speed up when it gets closer to the object at the focus. This is caused by the increased effect of gravity on the orbiting object as it gets nearer to what pulling it into the orbital path. The mathematical statement Read more about Kepler’s Second Law of Planetary Motion[…]
Kepler’s First Law of Planetary Motion Kepler’s laws of planetary motion, in astronomy and classical physics, laws describing the motions of the planets in the solar system. They were derived by the German astronomer Johannes Kepler, whose analysis of the observation of the 16th century Danish astronomer Tycho Brahe enabled him to announce his first Read more about Kepler’s First Law of Planetary Motion[…]
Inverse of the Function – Problems 1. The inverse of the function is Solution: , ⇒ x = ½ log₁₀ , Let y = f (x) ⇒ x = f⁻¹ (y) ⇒ f⁻¹ (y) = ½ log₁₀ , ⇒ f⁻¹ (x) = ½ log₁₀ . 2. If f: R → R is defined by f Read more about Inverse of the Function – Problems[…]
Isentropic Process Isentropic means constant entropy. A process during which the entropy remains constant is called an isentropic process, which is characterized by ΔS = 0 Or S₁ = S₂ for a process 1 – 2. If a process is both reversible and adiabatic, then it is an isentropic process. An isentropic process is an Read more about Isentropic Process[…]
Functions (Range) – Problems 1. The range of the function =? Solution: , , ∴ Range of f (x) is {-1, 1}. 2. If x is real then value of the expression lies between ? Solution: , Let us y = . ⇒ x² + 14x + 9 = yx² + 2xy + 3y ⇒ Read more about Functions (Range) – Problems[…]
Functions – Problems 1. f: R → R defined by f (x) = sinx and g: R → R defined by g (x) = x², fog (x) =? Solution: Given that f (x) = sin x g (x) = x² fog (x) = f [g (x)] = f [x²] = sin x². 2. If f: Read more about Functions – Problems[…]
Isobaric Process An Isobaric Process is a thermodynamic process, in which the pressure of the system remains constant (P = Constant). The heat transfer into or out of the system does work, but also changes the internal energy of the system. Since there are changes in internal energy (dU) and changes in system volume (ΔV) engineers Read more about Isobaric Process[…]
Circle Problems 1) Show that the line x + y + 1 = 0 touches the circle x² + y² – 3x + 7y + 14 = 0 and find the point of contact. Solution: Centre of the circle is C (3/2, -7/8) Radius of the circle (r) = Perpendicular distance from the centre to Read more about Circle Problems[…]