Find the derivative of the following functions with respect to x: (2+2)
(i) (x⁻¹ᐟ² - x¹ᐟ²)/(x⁻¹ᐟ² + x¹ᐟ²)
(ii) 3x/√(5+2x²)
For function (i), let (f(x) = \frac{x^{-1/2} - x^{1/2}}{x^{-1/2} + x^{1/2}}). Using the quotient rule, ((\frac{u}{v})' = \frac{u'v - uv'}{v^2}), set (u = x^{-1/2} - _____ ______ ___ _______ ____ ______ ___ ____ _____ ____ ________ _______.
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(a) Evaluate the following limit.
(b) If the law of motion of a particle is given as: s = -t³ + 3t² + 25, then (2+1)
i) find its velocity and acceleration.
ii) find the distance covered by the particle in time t = 5 units
Solve the following differential equation:
(1/y²) (dy/dt) = 1 - e⁻³ᵗ
(c) Show that the equation
(y - 2x³)dx = x(1 - xy)dy
becomes exact on multiplication by x⁻² and solve it.
In an iron determination from the same amount of sample, the five replicate results were obtained: 20.1, 19.6, 20.0 and 19.9 and 20.4 mg iron.
Calculate the standard deviation, variance, standard deviation of mean, coefficient of variation and relative standard deviation in ppm of the given data.
(a) Which of the following sets are finite, and which are infinite?
(i) The set of points on the circumference of a circle.
(ii) ]0, 1[
(iii) [-1, 1]
(iv) {1, 2, ..., 100}
(b) If A = {1, 2, 3}, B = {2, 3, 4, 5}, and C = {1}, determine A ∪ B ∪ C and also verify
A ∪ B ∪ C = (A ∪ B) ∪ C = A ∪ (B ∪ C)
1.50 mol of PCl₅(g) is decomposes at room temperature to form PCl₃(g) and Cl₂(g). Determine their concentration at equilibrium, when K_c = 1.80.
∫((1+ln x)³)/x dx
(a) Find the differential equation whose solution is given by
y = eˣ(A cos x + B sin x)
where, A and B are arbitrary constants.