Prove that is an orthogonal matrix.
To prove that the matrix \[ A = \begin{bmatrix} \cos 2\alpha & -\sin 2\alpha \\ \sin 2\alpha & \cos 2\alpha \end{bmatrix} \] is an orthogonal matrix, we need to show that \( A \) satisfies the condition: \[ A^T A = I \] where \( A^T \) is ______ ________ _________ _______ _________ __________.
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In a factory there are three machines A, B, C which produce 10%, 40% and 50% items respectively. Past experience shows that percentage of defective items produced by machines A, B, C are 5%, 4%, 2% respectively. An item from the production of these machines is selected at random and it is found defective. What is the probability (6) that it is produced by machine A?
Solve the following system of equations using Cramer’s rule. x + 3y + 2z = 6, – x + 4y + 5z = 8, 2x + 5y + 3z = 10
State whether the following statement are TRUE or FALSE. Give reason in support of your answer.
a) Derivative of 1/x with respect to x is 1.
b) If A is a matrix of order 2 by 3 and B is a matrix of order 3 by 2, then order of the matrix A + B is 2 by 3.
c) If and
they are perpendicular to each other.
d) If probability of an event E is 1/2 and probability of the event is 1/6, then probability of the event F is 1/3, where events E and F are independent.
e) If the first term of an AP is 5 and 101 term of the AP is 1005 then common difference of the AP will be 105
If and
then find
Evaluate
Solve the differential equation
In an iron determination (taking 1 g sample every time) the following four replicate results were obtained: 24.8, 25.2, 23.6 and 24.7 mg iron. Calculate the coefficient of variation and relative standard deviation in ppm of the given data.