Further Maths question (see attached image). 2(i) Using a suitable algorithm and without performing any division, determine whether 13 306 617 is divisible by 9. 2(ii) The group G = {1, 3, 7, 9, 11, 13, 17, 19} has multiplication modulo 20 as its operation. (a) Complete the Cayley table for G. (b) State the inverse of the element 7. (c) Determine the order of the element 13. (d) Write down a subgroup of G of order 4.
A-Level statistics question (see attached image). The graph of the probability density function f(x) of the continuous uniform random variable X is shown. The PDF is constant (height 0.2) from x = -3 to x = 2, and zero elsewhere. (a) Write down P(X = 1). (b) Find E(X). (c) Find Var(X). (d) Sketch the cumulative distribution function of X for -3 <= x <= 2, labelling points where it touches or crosses the axes. (e) Find P(X^2 > 1.96).
A-Level question (see attached image). Let z = sqrt(3) - 3i. (a) Write z in the form r(cos theta + i sin theta) where -pi < theta <= pi. (b) Show and label on a single Argand diagram the point P representing z and the point Q representing iz. (c) Describe the geometrical transformation that maps P onto Q.
Prove that the set of real numbers in the interval [0,1] is uncountable, using Cantor's diagonal argument.
(a) Solve the quadratic equation x^2 - 5x + 6 = 0. (b) Hence solve the equation x^4 - 5x^2 + 6 = 0, giving all real roots.
Explain how gradient descent trains a simple neural network. Describe the loss, the gradient and the weight update rule.
Derive the backpropagation weight update rule for the output layer of a 2-layer neural network with sigmoid activation and a mean squared error loss.
Explain how the binary search algorithm works on a sorted array, and state its time complexity in big-O notation.
Solve the recurrence relation T(n) = 2 T(n/2) + n for n a power of 2, with T(1) = 1. Find a closed form for T(n) and state the asymptotic complexity.
Use integration by parts to find the integral of ln(x) with respect to x, and hence evaluate the definite integral from 1 to e.
Find the Thevenin equivalent voltage and resistance of a circuit consisting of a 12 V source in series with a 4 ohm resistor, which is in parallel with a 6 ohm resistor. Explain the method.
Derive the quantised energy levels for a particle of mass m in a one-dimensional infinite potential well of width L, by solving the time-independent Schrodinger equation with appropriate boundary conditions.
A simply supported beam of length 4 m carries a point load of 10 kN at its centre. Determine the maximum bending moment and draw the bending moment diagram.
A ball is thrown from ground level at 20 m/s at an angle of 30 degrees above the horizontal. Find the horizontal range of the ball. Take g = 9.8 m/s^2.
A car starts from rest and accelerates at 2 m/s^2 for 5 seconds. Calculate its final speed and the distance it travels in that time.
Find the integral of x e^(2x) with respect to x, using integration by parts. Show full working.
Prove by mathematical induction that 1^3 + 2^3 + ... + n^3 = [n(n+1)/2]^2 for all positive integers n.
Find the eigenvalues and a corresponding eigenvector for each, of the matrix [[2,1],[1,2]].
Solve the equation 3x + 7 = 22 and explain each step.
What is 3/4 of 24? Show me how to work it out with fractions.
A circle has radius 7 cm. Calculate its area, leaving your answer in terms of pi.
solve
solve
explain to me the connection between a rubix cube and group theory.
explain to me backpropogation very in depth at uni level, clear visuals
explain to me backpropogation
explain to me how backpropogation works
Explain the chain rule with a worked example and a diagram of the composite function.
EXPLAIN HOW GRADIENT DESCENT WORKS
Explain integration by parts with a worked example, and include a diagram where it helps.
explain to me how a kalman filter works
Explain why the derivative of sin(x) is cos(x), intuitively.
Explain gradient descent and how it learns, with a visual analogy.
Explain why the derivative of sin(x) is cos(x), intuitively.