Mathematics Tes...

If sin α + cos α = p, then what is cos² (2α) equal to?

Let X be a non-empty set and let A, B, C be subsets of X, consider the following statements:

1. A ⊂ C ⇒ (A ∩ B) ⊂ (C ∩ B), (A ∪ B) ⊂ (C ∪ B)

2. (A ∩ B) ⊂ (C ∪ B) for all sets B ⇒ A ⊂ C

3. (A ∪ B) ⊂ (C ∪ B) for all sets B ⇒ A ⊂ C

Which of the above statements is/are correct?

The probability that when a hand of 7 cards is drawn from a well-shuffled deck of 52 cards, it contains 3 Kings is:

Evaluate \(\int_{0}^{\frac{\pi}{2}} \frac{\cos x}{1+\sin ^{2} x} d x\):

If \(x-i y=\sqrt{\frac{a-i b}{c-i d}}\), then \(\left(x^{2}+y^{2}\right)^{2}\) is equal to:

If \(f: R \rightarrow R\) is a function defined by \(f(x)=[x-1] \cos \left(\frac{2 x-1}{2}\right) \pi\). where, [.] denotes the greatest integer function, then \(f\) is:

The integrating factor of the differential equation \(2y\frac{{d} x}{{d} y}+ x = 5y^{2}\) is: (y≠0)

What is the acute angle between the lines \(x-2=0\) and \(\sqrt{3} x-y-2=0\)?

If \(A=\left[\begin{array}{cc}\cos 2 \theta & -\sin 2 \theta \\ \sin 2 \theta & \cos 2 \theta\end{array}\right]\) and \(A + A ^{T}= I\) Where \(I\) is the unit matrix of \(2 \times 2\) and  \(A ^{T}\) is the transpose of \(A\), then the value of \(\theta\) is equal to:

If \(\alpha\) and \(\beta\) are the roots of the equation \(x^{2}-2 x+4=0\), then what is the value of \(\alpha^{3}+\beta^{3}\) ?