A binary operation, \(\Delta\), is defined on the set of real numbers by \(a \Delta b = a + b + 4\). Find the identity element.

• A. 4
• B. 2
• C. \(\frac{1}{4}\)
• D. -4

Solve for x in the equation \(5^{x} \times 5^{x + 1} = 25\).

• A. \(-2\)
• B. \(\frac{-1}{2}\)
• C. \(\frac{1}{2}\)
• D. \(2\)

The coefficient of the 7th term in the binomial expansion of \((2 - \frac{x}{3})^{10}\) in ascending powers of x is

• A. \(\frac{560}{243}\)
• B. \(\frac{841}{243}\)
• C. \(\frac{1120}{243}\)
• D. \(\frac{4481}{243}\)

The table shows the scores obtained by a group of artistes in Vocal (X) and the instrument (Y) musical competition.

Calculate the spearman's rank correlation coefficient between the scores.

The parallelogram PQRS has vertices P(-2, 3), Q(1, 4), R(2, 6) and S(-1,5). Find the coordinates of the point of intersection of the diagonals.

• A. \((-1, 5)\)
• B. \((-\frac{1}{2}, 3\frac{1}{2})\)
• C. \((0, 4\frac{1}{2})\)
• D. \((1\frac{1}{2}, 5)\)

A particle is acted upon by two forces 6N and 3N inclined at an angle of 120° to each other. Find the magnitude of the resultant force.

• A. \(18\sqrt{3}\) N
• B. \(27\) N
• C. \(24\) N
• D. \(3\sqrt{3}\) N

For what values of m is \(9y^{2} + my + 4\) a perfect square?

• A. \(\pm {2}\)
• B. \(\pm {3}\)
• C. \(\pm {6}\)
• D. \(+12\)

(a) The speed of a moving bus reduced from 45m/s to 5m/s with a uniform retardation of 10m/s\(^2\). Calculate the distance covered.

(b) A bucket full of water with mass 16kg is pulled out of a well with a light inextensible rope. Find its acceleration when the tension in the rope is 240N. [Take g= 10m/s\(^2\)]

Find the minimum value of \(y = 3x^{2} - x - 6\).

• A. \(-6\frac{1}{6}\)
• B. \(-6\frac{1}{12}\)
• C. \(-6\)
• D. \(0\)

Express \(\frac{1}{1 - \sin 45°}\) in surd form.

• A. \(2 + \sqrt{2}\)
• B. \(2 + \sqrt{3}\)
• C. \(2 - \sqrt{2}\)
• D. \(1 + 2\sqrt{2}\)