If (x + 3) is a factor of the polynomial \(x^{3} + 3x^{2} + nx - 12\), where n is a constant, find the value of n.
პასუხი
(D)
-4
3
The line \(y = mx - 3\) is a tangent to the curve \(y = 1 - 3x + 2x^{3}\) at (1, 0). Find the value of the constant m.
პასუხი
(C)
3
4
The coordinates of the centre of a circle is (-2, 3). If its area is \(25\pi cm^{2}\), find its equation.
პასუხი
(D)
\(x^{2} + y^{2} + 4x - 6y - 12 = 0\)
5
Given \(\sin \theta = \frac{\sqrt{3}}{2}, 0° \leq \theta \leq 90°\), find \(\tan 2\theta\) in surd form.
პასუხი
(A)
\(- \sqrt{3}\)
6
Find the coefficient of \(x^{4}\) in the binomial expansion of \((2 + x)^{6}\).
პასუხი
(C)
60
7
Which of the following binary operations is not commutative?
პასუხი
(D)
\(a * b = a - b + ab\)
8
Express \(\frac{2}{3 - \sqrt{7}} \text{ in the form} a + \sqrt{b}\), where a and b are integers.
პასუხი
(B)
\(3 + \sqrt{7}\)
9
The roots of the quadratic equation \(2x^{2} - 5x + m = 0\) are \(\alpha\) and \(\beta\), where m is a constant. Find \(\alpha^{2} + \beta^{2}\) in terms of m.
პასუხი
(A)
\(\frac{25}{4} - m\)
10
Given that \(2^{x} = 0.125\), find the value of x.
პასუხი
(D)
-3
11
The gradient of point P on the curve \(y = 3x^{2} - x + 3\) is 5. Find the coordinates of P.
პასუხი
(A)
(1, 5)
12
An arc of length 10.8 cm subtends an angle of 1.2 radians at the centre of a circle. Calculate the radius of the circle.
პასუხი
(D)
9.0 cm
13
The first term of a geometric progression is 350. If the sum to infinity is 250, find the common ratio.
პასუხი
(B)
\(-\frac{2}{5}\)
14
p and q are statements such that \(p \implies q\). Which of the following is a valid conclusion from the implication?
პასუხი
(C)
\(\sim q \implies \sim p\)
15
The roots of a quadratic equation are -3 and 1. Find its equation.
პასუხი
(C)
\(x^{2} + 2x - 3 = 0\)
16
The derivative of a function f with respect to x is given by \(f'(x) = 3x^{2} - \frac{4}{x^{5}}\). If \(f(1) = 4\), find f(x).
A stone is projected vertically with a speed of 10 m/s from a point 8 metres above the ground. Find the maximum height reached. \([g = 10 ms^{-2}]\).
პასუხი
(A)
13 metres
24
The velocity \(v ms^{-1}\) of a particle moving in a straight line is given by \(v = 3t^{2} - 2t + 1\) at time t secs. Find the acceleration of the particle after 3 seconds.
პასუხი
(D)
\(16 ms^{-2}\)
25
Three men, P, Q and R aim at a target, the probabilities that P, Q and R hit the target are \(\frac{1}{2}\), \(\frac{1}{3}\) and \(\frac{3}{4}\) respectively. Find the probability that exactly 2 of them hit the target.
პასუხი
(C)
\(\frac{5}{12}\)
26
The position vectors of A and B are (2i + j) and (-i + 4j) respectively; find |AB|.
პასუხი
(A)
\(3\sqrt{2}\)
27
Two fair dices, each numbered 1, 2, ..., 6, are tossed together. Find the probability that they both show even numbers.
პასუხი
(B)
\(\frac{1}{4}\)
28
Calculate, correct to the nearest degree, the angle between the vectors \(\begin{pmatrix} 13 \\ 1 \end{pmatrix}\) and \(\begin{pmatrix} 1 \\ 4 \end{pmatrix}\).
