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If \(f(x) = mx^{2} - 6x - 3\) and \(f'(1) = 12\), find the value of the constant m.
Evaluate \(\lim \limits_{x \to 3} \frac{x^{2} - 2x - 3}{x - 3}\)
Calculate in surd form, the value of \(\tan 15°\).
Two bodies of masses 8 kg and 5 kg travelling in the same direction with speeds x m/s and 2 m/s respectively collide. If after collision, they move together with a speed of 3.85 m/s, find, correct to the nearest whole number, the value of x.
Find the area of the circle whose equation is given as \(x^{2} + y^{2} - 4x + 8y + 11 = 0\).
Two vectors m and n are defined by \(m = 3i + 4j\) and \(n = 2i - j\). Find the angle between m and n.
Given that \(\log_{2} y^{\frac{1}{2}} = \log_{5} 125\), find the value of y.
The tangent to the curve \(y = 4x^{3} + kx^{2} - 6x + 4\) at the point P(1, m) is parallel to the x- axis, where k and m are constants. Determine the coordinates of P.
The tangent to the curve \(y = 4x^{3} + kx^{2} - 6x + 4\) at the point P(1, m) is parallel to the x- axis, where k and m are constants. Find the value of k.
Find the fourth term of the binomial expansion of \((x - k)^{5}\) in descending powers of x.
The binary operation * is defined on the set of R, of real numbers by \(x * y = 3x + 3y - xy, \forall x, y \in R\). Determine, in terms of x, the identity element of the operation.
Simplify \(\frac{\tan 80° - \tan 20°}{1 + \tan 80° \tan 20°}\)
Simplify \(\frac{\sqrt{3}}{\sqrt{3} - 1} + \frac{\sqrt{3}}{\sqrt{3} +1}\)
The gradient of the line passing through the points P(4, 5) and Q(x, 9) is \(\frac{1}{2}\). Find the value of x.
Two functions f and g are defined on the set R of real numbers by \(f : x \to 2x - 1\) and \(g : x \to x^{2} + 1\). Find the value of \(f^{-1} \circ g(3)\).
The sum of the first three terms of an Arithmetic Progression (A.P) is 18. If the first term is 4, find their product.
Which of the following is the same as \(\sin (270 + x)°\)?
Given that \(\alpha\) and \(\beta\) are the roots of an equation such that \(\alpha + \beta = 3\) and \(\alpha \beta = 2\), find the equation.
Given that the straight lines \(kx - 5y + 6 = 0\) and \(mx + ny - 1 = 0\) are parallel, find a relationship connecting the constants m, n and k.
ABCD is a square. Forces of magnitude 14N, 4N, 2N and \(2\sqrt{2} N\) act along the sides AB, BC, CD and DA respectively. Find in Newtons, the magnitude of the resultant of the forces.