Elective Mathematics — 1995
WASSCE · 801 questions · Answers included
801 questions
If \(log_{y}\frac{1}{8}\) = 3, find the value of y.
A binary operation \(\Delta\) is defined on the set of real numbers, R, by \(a \Delta b = \frac{a+b}{\sqrt{ab}}\), where a\(\neq\) 0, b\(\neq\) 0. Evaluate \(-3 \Delta -1\).
Simplify \(\frac{1}{(1-\sqrt{3})^{2}}\)
If \(x^{2} - kx + 9 = 0\) has equal roots, find the values of k.
Find the coordinates of the centre of the circle \(3x^{2}+3y^{2} - 4x + 8y -2=0\)
The function f: x \(\to \sqrt{4 - 2x}\) is defined on the set of real numbers R. Find the domain of f.
Given that \(f(x) = \frac{x+1}{2}\), find \(f^{1}(-2)\).
Given that \(\frac{6x+m}{2x^{2}+7x-15} \equiv \frac{4}{x+5} - \frac{2}{2x-3}\), find the value of m.
Find the coefficient of \(x^{4}\) in the expansion of \((1-2x)^{6}\).
Find the 21st term of the Arithmetic Progression (A.P.): -4, -1.5, 1, 3.5,...
How many ways can 6 students be seated around a circular table?
If \(\begin{pmatrix} 2 & 1 \\ 4 & 3 \end{pmatrix}\)\(\begin{pmatrix} 5 \\ 4 \end{pmatrix}\) = k\(\begin{pmatrix} 17.5 \\ 40.0 \end{pmatrix}\), find the value of k.
Express cos150° in surd form.
A straight line 2x+3y=6, passes through the point (-1,2). Find the equation of the line.
\(\alpha\) and \(\beta\) are the roots of the equation \(2x^{2} - 3x + 4 = 0\). Find \(\alpha + \beta\).
\(\alpha\) and \(\beta\) are the roots of the equation \(2x^{2} - 3x + 4 = 0\). Find \(\frac{\alpha}{\beta} + \frac{\beta}{\alpha}\)
If \(B = \begin{pmatrix} 2 & 5 \\ 1 & 3 \end{pmatrix}\), find \(B^{-1}\).
Given that \(\sin x = \frac{5}{13}\) and \(\sin y = \frac{8}{17}\), where x and y are acute, find \(\cos(x+y)\).
A circle with centre (4,5) passes through the y-intercept of the line 5x - 2y + 6 = 0. Find its equation.
Given that \(f(x) = 5x^{2} - 4x + 3\), find the coordinates of the point where the gradient is 6.
If \(y = \frac{1+x}{1-x}\), find \(\frac{dy}{dx}\).
Evaluate \(\int_{-1}^{0} (x+1)(x-2) \mathrm{d}x\)
Simplify \(\frac{\sqrt{128}}{\sqrt{32} - 2\sqrt{2}}\)
There are 7 boys in a class of 20. Find the number of ways of selecting 3 girls and 2 boys
The 3rd and 7th term of a Geometric Progression (GP) are 81 and 16. Find the 5th term.
Differentiate \(\frac{5x^{3} + x^{2}}{x}, x\neq 0\) with respect to x.
A curve is given by \(y = 5 - x - 2x^{2}\). Find the equation of its line of symmetry.
In a class of 10 boys and 15 girls, the average score in a Biology test is 90. If the average score for the girls is x, find the average score for the boys in terms of x.
A fair die is tossed twice. What is its smple size?
Given that \( a = \begin{pmatrix} 2 \\ 3 \end{pmatrix}\) and \(b = \begin{pmatrix} -1 \\ 4 \end{pmatrix}\), evaluate \((2a - \frac{1}{4}b)\).
Face 1 2 3 4 5 6 Frequency 12 18 y 30 2y 45 Given the table above as the results of tossing a fair die 150 times. Find the probability of obtaining a 5.
Face 1 2 3 4 5 6 Frequency 12 18 y 30 2y 45 Given the table above as the result of tossing a fair die 150 times, find the mode.
Given that a = 5i + 4j and b = 3i + 7j, evaluate (3a - 8b).
A force (10i + 4j)N acts on a body of mass 2kg which is at rest. Find the velocity after 3 seconds.
Solve \(3^{2x} - 3^{x+2} = 3^{x+1} - 27\)
Find the magnitude and direction of the vector \(p = (5i - 12j)\)
The velocity, V, of a particle after t seconds, is \(V = 3t^{2} + 2t - 1\). Find the acceleration of the particle after 2 seconds.
Given that \(f(x) = 2x^{2} - 3\) and \(g(x) = x + 1\) where \(x \in R\). Find g o f(x).
If P = \({n^{2} + 1: n = 0,2,3}\) and Q = \({n + 1: n = 2,3,5}\), find P\(\cap\) Q.
If \((2x^{2} - x - 3)\) is a factor of \(f(x) = 2x^{3} - 5x^{2} - x + 6\), find the other factor
Simplify \(\frac{\sqrt{3}}{\sqrt{3} -1} + \frac{\sqrt{3}}{\sqrt{3} + 1}\)
Find the domain of \(g(x) = \frac{4x^{2} - 1}{\sqrt{9x^{2} + 1}}\)
Given that \(f(x) = 3x^{2} - 12x + 12\) and \(f(x) = 3\), find the values of x.
A binary operation * is defined on the set of real numbers, by \(a * b = \frac{a}{b} + \frac{b}{a}\). If \((\sqrt{x} + 1) * (\sqrt{x} - 1) = 4\), find the value of x.
If \(4x^{2} + 5kx + 10\) is a perfect square, find the value of k.
If the polynomial \(f(x) = 3x^{3} - 2x^{2} + 7x + 5\) is divided by (x - 1), find the remainder.
\(P = {1, 3, 5, 7, 9}, Q = {2, 4, 6, 8, 10, 12}, R = {2, 3, 5, 7, 11}\) are subsets of \(U = {1, 2, 3, ... , 12}\). Which of the following statements is true?
If \(\log_{3}a - 2 = 3\log_{3}b\), express a in terms of b.
If \(\alpha\) and \(\beta\) are the roots of \(2x^{2} - 5x + 6 = 0\), find the equation whose roots are \((\alpha + 1)\) and \((\beta + 1)\).
Resolve \(\frac{3x - 1}{(x - 2)^{2}}, x \neq 2\) into partial fractions.
If \(\alpha\) and \(\beta\) are the roots of the equation \(2x^{2} + 5x + n = 0\), such that \(\alpha\beta = 2\), find the value of n.
Solve \(\log_{2}(12x - 10) = 1 + \log_{2}(4x + 3)\).
Find the coefficient of \(x^{3}\) in the binomial expansion of \((x - \frac{3}{x^{2}})^{9}\).
The general term of an infinite sequence 9, 4, -1, -6,... is \(u_{r} = ar + b\). Find the values of a and b.
If \(\begin{vmatrix} k & k \\ 4 & k \end{vmatrix} + \begin{vmatrix} 2 & 3 \\ -1 & k \end{vmatrix} = 6\), find the value of the constant k, where k > 0.
How many numbers greater than 150 can be formed from the digits 1, 2, 3, 4, 5 without repetition?
The first term of a Geometric Progression (GP) is \(\frac{3}{4}\), If the product of the second and third terms of the sequence is 972, find its common ratio.
If \(\sin\theta = \frac{3}{5}, 0° < \theta < 90°\), evaluate \(\cos(180 - \theta)\).
Find the radius of the circle \(x^{2} + y^{2} - 8x - 2y + 1 = 0\).
In how many ways can the letters of the word 'ELECTIVE' be arranged?
If the determinant of the matrix \(\begin{pmatrix} 2 & x \\ 3 & 5 \end{pmatrix} = 13\), find the value of x.
Express \(\frac{13}{4}\pi\) radians in degrees.
Find the equation to the circle \(x^{2} + y^{2} - 4x - 2y = 0\) at the point (1, 3).
Given that \(y = x(x + 1)^{2}\), calculate the maximum value of y.
The midpoint of M(4, -1) and N(x, y) is P(3, -4). Find the coordinates of N.
Find the stationary point of the curve \(y = 3x^{2} - 2x^{3}\).
Evaluate \(\int_{\frac{1}{2}}^{1} \frac{x^{3} - 4}{x^{3}} \mathrm {d} x\).
Calculate the standard deviation of 30, 29, 25, 28, 32 and 24.
Evaluate \(\int_{-1}^{1} (x + 1)^{2}\mathrm {d} x\).
Out of 70 schools, 42 of them can be attended by boys and 35 can be attended by girls. If a pupil is selected at random from these schools, find the probability that he/ she is from a mixed school.
The marks scored by 4 students in Mathematics and Physics are ranked as shown in the table below Mathematics 3 4 2 1 Physics 4 3 1 2 Calculate the Spearmann's rank correlation coefficient.
Given that \(a = i - 3j\) and \(b = -2i + 5j\) and \(c = 3i - j\), calculate \(|a - b + c|\).
What is the probability of obtaining a head and a six when a fair coin and and a die are tossed together?
If \(\overrightarrow{OX} = \begin{pmatrix} -7 \\ 6 \end{pmatrix}\) and \(\overrightarrow{OY} = \begin{pmatrix} 16 \\ -11 \end{pmatrix}\), find \(\overrightarrow{YX}\).
A body of mass 28g, initially at rest is acted upon by a force, F Newtons. If it attains a velocity of \(5.4ms^{-1}\) in 18 seconds, find the value of F.
Find the angle between forces of magnitude 7N and 4N if their resultant has a magnitude of 9N.
Find the constant term in the binomial expansion \((2x^{2} + \frac{1}{x})^{9}\)
A particle starts from rest and moves through a distance \(S = 12t^{2} - 2t^{3}\) metres in time t seconds. Find its acceleration in 1 second.
A car is moving at 120\(kmh^{-1}\). Find its speed in \(ms^{-1}\).
Two functions f and g are defined on the set of real numbers by \(f : x \to x^{2} + 1\) and \(g : x \to x - 2\). Find f o g.
If \(P = {x : -2 < x < 5}\) and \(Q = {x : -5 < x < 2}\) are subsets of \(\mu = {x : -5 \leq x \leq 5}\), where x is a real number, find \((P \cup Q)\).
Express \(\frac{8 - 3\sqrt{6}}{2\sqrt{3} + 3\sqrt{2}}\) in the form \(p\sqrt{3} + q\sqrt{2}\).
An operation * is defined on the set, R, of real numbers by \(p * q = p + q + 2pq\). If the identity element is 0, find the value of p for which the operation has no inverse.
Consider the statements: p : Musa is short q : Musa is brilliant Which of the following represents the statement "Musa is short but not brilliant"?
If \(f(x) = \frac{4}{x} - 1, x \neq 0\), find \(f^{-1}(7)\).
If \(y = 4x - 1\), list the range of the domain \({-2 \leq x \leq 2}\), where x is an integer.
Factorize completely: \(x^{2} + x^{2}y + 3x - 10y + 3xy - 10\).
If the solution set of \(x^{2} + kx - 5 = 0\) is (-1, 5), find the value of k.
The remainder when \(x^{3} - 2x + m\) is divided by \(x - 1\) is equal to the remainder when \(2x^{3} + x - m\) is divided by \(2x + 1\). Find the value of m.
If (2t - 3s)(t - s) = 0, find \(\frac{t}{s}\).
Solve for x in the equation \(5^{x} \times 5^{x + 1} = 25\).
If \(\log_{10}y + 3\log_{10}x \geq \log_{10}x\), express y in terms of x.
Simplify \(\frac{^{n}P_{5}}{^{n}C_{5}}\).
Given n = 3, evaluate \(\frac{1}{(n-1)!} - \frac{1}{(n+1)!}\)
Find the coefficient of \(x^{3}\) in the expansion of \([\frac{1}{3}(2 + x)]^{6}\).
Find the fourth term in the expansion of \((3x - y)^{6}\).
The 3rd and 6th terms of a geometric progression (G.P.) are \(\frac{8}{3}\) and \(\frac{64}{81}\) respectively, find the common ratio.
Given that \(-6, -2\frac{1}{2}, ..., 71\) is a linear sequence , calculate the number of terms in the sequence.
If \(\begin{vmatrix} m-2 & m+1 \\ m+4 & m-2 \end{vmatrix} = -27\), find the value of m.
If \(P = \begin{pmatrix} 1 & 2 \\ 5 & 1 \end{pmatrix}\) and \(Q = \begin{pmatrix} 0 & 1 \\ 1 & 3 \end{pmatrix}\), find PQ.
Evaluate \(\cos 75°\), leaving the answer in surd form.
Given that \(\tan x = \frac{5}{12}\), and \(\tan y = \frac{3}{4}\), Find \(\tan (x + y)\).
Find the equation of the line which passes through (-4, 3) and parallel to line y = 2x + 5.
Points E(-2, -1) and F(3, 2) are the ends of the diameter of a circle. Find the equation of the circle.
The lines \(2y + 3x - 16 = 0\) and \(7y - 2x - 6 = 0\) intersect at point P. Find the coordinates of P.
Find \(\lim\limits_{x \to 3} \frac{2x^{2} + x - 21}{x - 3}\).
Find the gradient to the normal of the curve \(y = x^{3} - x^{2}\) at the point where x = 2.
Find the minimum value of \(y = 3x^{2} - x - 6\).
The radius of a circle increases at a rate of 0.5\(cms^{-1}\). Find the rate of change in the area of the circle with radius 7cm. \([\pi = \frac{22}{7}]\)
Find an expression for y given that \(\frac{\mathrm d y}{\mathrm d x} = x^{2}\sqrt{x}\)
Given that \(n = 10\) and \(\sum d^{2} = 20\), calculate the Spearman's rank correlation coefficient.
Find the variance of 11, 12, 13, 14 and 15.
A fair coin is tossed 3 times. Find the probability of obtaining exactly 2 heads.
A box contains 14 white balls and 6 black balls. Find the probability of first drawing a black ball and then a white ball without replacement.
Given that \(r = 3i + 4j\) and \(t = -5i + 12j\), find the acute angle between them.
Find the unit vector in the direction of \(-2i + 5j\).
A body of mass 10kg moving with a velocity of 5\(ms^{-1}\) collides with another body of mass 15kg moving in the same direction as the first with a velocity of 2\(ms^{-1}\). After collision, the two bodies move together with a common velocity v\(ms^{-1}\).
A force 10N acts in the direction 060° and another force 6N acts in the direction 330°. Find the y component of their resultant force.
A man of mass 80kg stands in a lift. If the lift moves upwards with acceleration 0.5\(ms^{-2}\), calculate the reaction from the floor of the lift on the man. \([g = 10ms^{-2}]\)
A ball falls from a height of 18m above the ground. Find the speed with which the ball hits the ground. \([g = 10ms^{-2}]\)
Simplify \(\frac{1 - 2\sqrt{5}}{2 + 3\sqrt{2}}\).
Solve: \(2\cos x - 1 = 0\).
Solve: \(4(2^{x^2}) = 8^{x}\)
If \(\log_{3} x = \log_{9} 3\), find the value of x.
Find the 3rd term of \((\frac{x}{2} - 1)^{8}\) in descending order of x.
Given that \(f : x \to x^{2}\) and \(g : x \to x + 3\), where \(x \in R\), find \(f o g(2)\).
Given that \(\frac{2x}{(x + 6)(x + 3)} = \frac{P}{x + 6} + \frac{Q}{x + 3}\), find P and Q.
Given that \(P = \begin{pmatrix} -2 & 1 \\ 3 & 4 \end{pmatrix}\) and \(Q = \begin{pmatrix} 5 & -3 \\ 2 & -1 \end{pmatrix}\), find PQ - QP.
Which of the following is a factor of the polynomial \(6x^{4} + 2x^{3} + 15x + 5\)?
Given that \(f : x \to \frac{2x - 1}{x + 2}, x \neq -2\), find \(f^{-1}\), the inverse of f.
If \(36, p, \frac{9}{4}, q\) are consecutive terms of an exponential sequence (G.P.). Find the sum of p and q.
Find the minimum value of \(y = x^{2} + 6x - 12\).
A line passes through the origin and the point \((1\frac{1}{4}, 2\frac{1}{2})\), what is the gradient of the line?
A line passes through the origin and the point \((1\frac{1}{4}, 2\frac{1}{2})\). Find the y-coordinate of the line when x = 4.
In how many ways can a committee of 5 be selected from 8 students if 2 particular students are to be included?
If \(x = i - 3j\) and \(y = 6i + j\), calculate the angle between x and y.
The gradient of a curve at the point (-2, 0) is \(3x^{2} - 4x\). Find the equation of the curve.
If \(\alpha\) and \(\beta\) are the roots of \(x^{2} + x - 2 = 0\), find the value of \(\frac{1}{\alpha^{2}} + \frac{1}{\beta^{2}}\).
Given that \(x^{2} + 4x + k = (x + r)^{2} + 1\), find the value of k and r.
Given the statements: p : the subject is difficult q : I will do my best Which of the following is equivalent to 'Although the subject is difficult, I will do my best'?
Given that \(r = 2i - j\), \(s = 3i + 5j\) and \(t = 6i - 2j\), find the magnitude of \(2r + s - t\).
Marks 0 1 2 3 4 5 Number of candidates 6 4 8 10 9 3 The table above shows the distribution of marks scored by students in a test. How many candidates scored above the median score?
Marks 0 1 2 3 4 5 Number of candidates 6 4 8 10 9 3 The table above shows the distribution of marks scored by students in a test. Find the interquartile range of the distribution.
A mass of 75kg is placed on a lift. Find the force exerted by the floor of the lift on the mass when the lift is moving up with constant velocity. \([g = 9.8ms^{-2}]\)
Each of the 90 students in a class speak at least Igbo or Hausa. If 56 students speak Igbo and 50 speak Hausa, find the probability that a student selected at random from the class speaks Igbo only.
If \(\begin{vmatrix} 1+2x & -1 \\ 6 & 3-x \end{vmatrix} = -3 \), find the values of x.
Find \(\int \frac{x^{3} + 5x + 1}{x^{3}} \mathrm {d} x\)
Find the coordinates of the point which divides the line joining P(-2, 3) and Q(4, 6) internally in the ratio 2 : 3.
A particle starts from rest and moves in a straight line such that its acceleration after t seconds is given by \(a = (3t - 2) ms^{-2}\). Find the other time when the velocity would be zero.
A particle starts from rest and moves in a straight line such that its acceleration after t secs is given by \(a = (3t - 2) ms^{-2}\). Find the distance covered after 3 secs.
Given that \(y = 4 - 9x\) and \(\Delta x = 0.1\), calculate \(\Delta y\).
Four fair coins are tossed once. Calculate the probability of having equal heads and tails.
In calculating the mean of 8 numbers, a boy mistakenly used 17 instead of 25 as one of the numbers. If he obtained 20 as the mean, find the correct mean.
Simplify: \(^{n}C_{r} ÷ ^{n}C_{r-1}\)
If \(2\sin^{2} \theta = 1 + \cos \theta, 0° \leq \theta \leq 90°\), find the value of \(\theta\).
A 24N force acts on a body such that it changes its velocity from 5m/s to 9m/s in 2 secs.If the body is travelling in a straight line, calculate the distance covered in the period.
The sum, \(S_{n}\), of a sequence is given by \(S_{n} = 2n^{2} - 5\). Find the 6th term.
Forces \(F_{1} = (8N, 030°)\) and \(F_{2} = (10N, 150°)\) act on a particle. Find the horizontal component of the resultant force.
Forces of magnitude 8N and 5N act on a body as shown above. Calculate, correct to 2 d.p., the resultant force acting at O.
Forces of magnitude 8N and 5N act on a body as shown above. Calculate, correct to 2 dp, the angle that the resultant makes with the horizontal.
If \(\frac{1}{5^{-y}} = 25(5^{4-2y})\), find the value of y.
Simplify: \((1 - \sin \theta)(1 + \sin \theta)\).
Given that \(3x + 4y + 6 = 0\) and \(4x - by + 3 = 0\) are perpendicular, find the value of b.
Given that \(x * y = \frac{x + y}{2}, x \circ y = \frac{x^{2}}{y}\) and \((3 * b) \circ 48 = \frac{1}{3}\), find b, where b > 0.
If \(f(x) = 3x^{3} + 8x^{2} + 6x + k\) and \(f(2) = 1\), find the value of k.
If \(8^{x} ÷ (\frac{1}{4})^{y} = 1\) and \(\log_{2}(x - 2y) = 1\), find the value of (x - y).
Simplify \(\frac{1 + \sqrt{8}}{3 - \sqrt{2}}\).
Using the binomial expansion \((1+x)^{6} = 1 + 6x + 15x^{2} + 20x^{3} + 15x^{4} + 6x^{5} + x^{6}\), find, correct to 3 dp, the value of \((1.98)^{6}\).
If \((x + 2)\) and \((3x - 1)\) are factors of \(6x^{3} + x^{2} - 19x + 6\), find the third factor.
If \(2, (k+1), 8,...\) form an exponential sequence (GP), find the values of k.
A box contains 5 red and k blue balls. A ball is selected at random from the box. If the probability of selecting a blue ball is \(\frac{2}{3}\), find the value of k.
If \(\frac{x + P}{(x - 1)(x - 3)} = \frac{Q}{x - 1} + \frac{2}{x - 3}\), find the value of (P + Q).
Find the derivative of \(\sqrt[3]{(3x^{3} + 1}\) with respect to x.
If \(T = \begin{pmatrix} -2 & -5 \\ 3 & 8 \end{pmatrix}\), find \(T^{-1}\), the inverse of T.
A function is defined by \(h : x \to 2 - \frac{1}{2x - 3}, x \neq \frac{3}{2}\). Find \(h^-1\), the inverse of h.
A function is defined by \(h : x \to 2 - \frac{1}{2x - 3}, x \neq \frac{3}{2}\). Find \(h^{-1}(\frac{1}{2})\).
The radius of a sphere is increasing at a rate \(3cm s^{-1}\). Find the rate of increase in the surface area, when the radius is 2cm.
Age in years 10 - 14 15 - 19 20 - 24 25 - 29 30 - 34 Frequency 6 8 14 10 12 What is the class mark of the median class?
Age in years 10 - 14 15 - 19 20 - 24 25 - 29 30 - 34 Frequency 6 8 14 10 12 In which group is the upper quartile?
Age in years 10 - 14 15 - 19 20 - 24 25 - 29 30 - 34 Frequency 6 8 14 10 12 Find the mean of the distribution.
If \(Px^{2} + (P+1)x + P = 0\) has equal roots, find the values of P.
Integrate \((x - \frac{1}{x})^{2}\) with respect to x.
Given that \(AB = \begin{pmatrix} 4 \\ 3 \end{pmatrix}\) and \(AC = \begin{pmatrix} 2 \\ -3 \end{pmatrix}\), find |BC|.
Find the angle between \((5i + 3j)\) and \((3i - 5j)\).
Find the coefficient of \(x^3\) in the binomial expansion of \((3x + 4)^4\) in ascending powers of x.
If a fair coin is tossed four times, what is the probability of obtaining at least one head?
Forces 90N and 120N act in the directions 120° and 240° respectively. Find the resultant of these forces.
The deviations from the mean of a set of numbers are \((k+3)^{2}, (k+7), -2, \text{k and (} k+2)^{2}\), where k is a constant. Find the value of k.
Find the equation of a circle with centre (2, -3) and radius 2 units.
The first term of a linear sequence is 9 and the common difference is 7. If the nth term is 380, find the value of n.
For what values of m is \(9y^{2} + my + 4\) a perfect square?
A particle accelerates at 12\(ms^{-2}\) and travels a distance of 250m in 6 seconds. Find the initial velocity of the particle.
In how many ways can 9 people be seated on a bench if only 3 places are available?
Find the variance of 1, 2, 0, -3, 5, -2, 4.
If the points (-1, t -1), (t, t - 3) and (t - 6, 3) lie on the same straight line, find the values of t.
A ball is thrown vertically upwards with a velocity of 15\(ms^{-1}\). Calculate the maximum height reached. \([g = 10ms^{-2}]\)
Find the distance between the points (2, 5) and (5, 9).
Find \(\lim\limits_{x \to 3} (\frac{x^{3} + x^{2} - 12x}{x^{2} - 9})\)
Evaluate \(\frac{\tan 120° + \tan 30°}{\tan 120° - \tan 60°}\)
Express (14N, 240°) as a column vector.
A binary operation * is defined on the set of real numbers, R, by \(x * y = x + y - xy\). If the identity element under the operation * is 0, find the inverse of \(x \in R\).
Solve: \(\sin \theta = \tan \theta\)
Given that \(a^{\frac{5}{6}} \times a^{\frac{-1}{n}} = 1\), solve for n.
Express \(\log \frac{1}{8} + \log \frac{1}{2}\) in terms of \(\log 2\).
If \(f(x) = x^{2}\) and \(g(x) = \sin x\), find g o f.
Find the third term in the expansion of \((a - b)^{6}\) in ascending powers of b.
If \(\sqrt{x} + \sqrt{x + 1} = \sqrt{2x + 1}\), find the possible values of x.
If \(\alpha\) and \(\beta\) are the roots of the equation \(2x^{2} - 6x + 5 = 0\), evaluate \(\frac{\beta}{\alpha} + \frac{\alpha}{\beta}\).
Given that \(f(x) = 2x^{3} - 3x^{2} - 11x + 6\) and \(f(3) = 0\), factorize f(x).
Find the equation of the line that is perpendicular to \(2y + 5x - 6 = 0\) and bisects the line joining the points P(4, 3) and Q(-6, 1).
Differentiate \(x^{2} + xy - 5 = 0\).
The fourth term of an exponential sequence is 192 and its ninth term is 6. Find the common ratio of the sequence.
Find the range of values of x for which \(x^{2} + 4x + 5\) is less than \(3x^{2} - x + 2\)
Given that \(\frac{\mathrm d y}{\mathrm d x} = \sqrt{x}\), find y.
Given that \(P = \begin{pmatrix} y - 2 & y - 1 \\ y - 4 & y + 2 \end{pmatrix}\) and |P| = -23, find the value of y.
An object is thrown vertically upwards from the top of a cliff with a velocity of \(25ms^{-1}\). Find the time, in seconds, when it is 20 metres above the cliff. \([g = 10ms^{-2}]\).
Evaluate \(\int_{0}^{2} (8x - 4x^{2}) \mathrm {d} x\).
Find the coordinates of the point which divides the line joining P(-2, 3) and Q(4, 9) internally in the ratio 2 : 3.
The angle subtended by an arc of a circle at the centre is \(\frac{\pi}{3} radians\). If the radius of the circle is 12cm, calculate the perimeter of the major arc.
The function \(f : F \to R\) = \(f(x) = \begin{cases} 3x + 2 : x > 4 \\ 3x - 2 : x = 4 \\ 5x - 3 : x < 4 \end{cases}\). Find f(4) - f(-3).
A committee consists of 5 boys namely: Kofi, John, Ojo, Ozo and James and 3 girls namely: Rose, Ugo and Ama. In how many ways can a sub-committee consisting of 3 boys and 2 girls be chosen, if Ozo must be on the sub-committee?
Forces 50N and 80N act on a body as shown in the diagram. Find, correct to the nearest whole number, the horizontal component of the resultant force.
The sales of five salesgirls on a certain day are as follows; GH¢ 26.00, GH¢ 39.00, GH¢ 33.00, GH¢ 25.00 and GH¢ 37.00. Calculate the standard deviation if the mean sale is GH¢ 32.00.
A circular ink blot on a piece of paper increases its area at the rate \(4mm^{2}/s\). Find the rate of the radius of the blot when the radius is 8mm. \([\pi = \frac{22}{7}]\).
Express \(\frac{x^{2} + x + 4}{(1 - x)(x^{2} + 1)}\) in partial fractions.
Two bodies of masses 3kg and 5kg moving with velocities 2 m/s and V m/s respectively in opposite directions collide. If they move together after collision with velocity 3.5 m/s in the direction of the 5kg mass, find the value of V.
The equation of a circle is \(x^{2} + y^{2} - 8x + 9y + 15 = 0\). Find its radius.
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.
If \(s = 3i - j\) and \(t = 2i + 3j\), find \((t - 3s).(t + 3s)\).
If \(2\sin^{2}\theta = 1 + \cos \theta, 0° \leq \theta \leq 90°\), find \(\theta\).
Find the upper quartile of the following scores: 41, 29, 17, 2, 12, 33, 45, 18, 43 and 5.
Given that \(P = \begin{pmatrix} 3 & 4 \\ 2 & x \end{pmatrix}; Q = \begin{pmatrix} 1 & 3 \\ -2 & 4 \end{pmatrix}; R = \begin{pmatrix} -5 & 25 \\ -8 & 26 \end{pmatrix}\) and PQ = R, find the value of x.
Two out of ten tickets on sale for a raffle draw are winning tickets. If a guest bought two tickets, what is the probability that both tickets are winning tickets?
P and Q are the points (3, 1) and (7, 4) respectively. Find the unit vector along PQ.
If \(g(x) = \frac{x + 1}{x - 2}, x \neq -2\), find \(g^{-1}(2)\).
Calculate the mean deviation of 1, 2, 3, 4, 5, 5, 6, 7, 8, 9.
If \(V = \begin{pmatrix} -2 \\ 4 \end{pmatrix}\) and \(U = \begin{pmatrix} -1 \\ 5 \end{pmatrix}\), find \(|U + V|\).
Find the equation of the straight line that passes through (2, -3) and perpendicular to the line 3x - 2y + 4 = 0.
If \(\frac{^{n}C_{3}}{^{n}P_{2}} = 1\), find the value of n.
A body is kept at rest by three forces \(F_{1} = (10N, 030°), F_{2} = (10N, 150°)\) and \(F_{3}\). Find \(F_{3}\).
Which of the following sets is equivalent to \((P \cup Q) \cap (P \cup Q')\)?
Simplify: \(\frac{\cos 2\theta - 1}{\sin 2\theta}\)
Solve the inequality \(x^{2} - 2x \geq 3\)
Given that \(\sqrt{6}, 3\sqrt{2}, 3\sqrt{6}, 9\sqrt{2},...\) are the first four terms of an exponential sequence (G.P), find in its simplest form the 8th term.
Given that \(\sin x = \frac{-\sqrt{3}}{2}\) and \(\cos x > 0\), find x.
Evaluate \(\log_{10}(\frac{1}{3} + \frac{1}{4}) + 2\log_{10} 2 + \log_{10} (\frac{3}{7})\)
QRS is a triangle such that \(\overrightarrow{QR} = (3i + 2j)\) and \(\overrightarrow{SR} = (-5i + 3j)\), find \(\overrightarrow{SQ}\).
If (x + 1) is a factor of the polynomial \(x^{3} + px^{2} + x + 6\). Find the value of p.
A polynomial is defined by \(f(x + 1) = x^{3} + px^{2} - 4x + 2\), find f(2).
The equation of a circle is \(3x^{2} + 3y^{2} + 24x - 12y = 15\). Find its radius.
If the midpoint of the line joining (1 - k, -4) and (2, k + 1) is (-k, k), find the value of k.
Evaluate \(\int_{-2}^{3} (3x^{2} - 2x - 12) \mathrm {d} x\)
If \(y = x^{3} - x^{2} - x + 6\), find the values of x at the turning point.
Given that \(P = \begin{pmatrix} 2 & 1 \\ 5 & -3 \end{pmatrix}\) and \(Q = \begin{pmatrix} 4 & -8 \\ 1 & -2 \end{pmatrix}\), Find (2P - Q).
A binary operation, \(\Delta\), is defined on the set of real numbers by \(a \Delta b = a + b + 4\). Find the identity element.
The marks obtained by 10 students in a test are as follows: 3, 7, 6, 2, 8, 5, 9, 1, 4 and 10. Find the mean mark.
The marks obtained by 10 students in a test are as follows: 3, 7, 6, 2, 8, 5, 9, 1, 4 and 10. Find the variance.
If r denotes the correlation coefficient between two variables, which of the following is always true?
A stone is dropped from a height of 45m. Find the time it takes to hit the ground. \([g = 10 ms^{-2}]\)
Differentiate \(\frac{x}{x + 1}\) with respect to x.
Two forces 10N and 6N act in the directions 060° and 330° respectively. Find the x- component of their resultant.
Find the unit vector in the direction of the vector \(-12i + 5j\).
In computing the mean of 8 numbers, a boy mistakenly used 17 instead of 25 as one of the numbers and obtained 20 as the mean. Find the correct mean
Given that \(^{n}P_{r} = 90\) and \(^{n}C_{r} = 15\), find the value of r.
Which of the following is nor a measure of central tendency?
A fair die is tossed twice. Find the probability of obtaining a 3 and a 5.
If P(x - 3) + Q(x + 1) = 2x + 3, find the value of (P + Q).
Find the values of x at the point of intersection of the curve \(y = x^{2} + 2x - 3\) and the lines \(y + x = 1\).
Find the constant term in the binomial expansion of \((2x - \frac{3}{x})^{8}\).
A straight line makes intercepts of -3 and 2 on the x- and y- axes respectively. Find the equation of the line.
Find the number of different arrangements of the word IKOTITINA.
Find the acute angle between the lines 2x + y = 4 and -3x + y + 7 = 0.
A box contains 4 red and 3 blue identical balls. If two are picked at random, one after the other without replacement, find the probability that one is red and the other is blue.
The distance s in metres covered by a particle in t seconds is \(s = \frac{3}{2}t^{2} - 3t\). Find its acceleration.
The angle of a sector of a circle is 0.9 radians. If the radius of the circle is 4cm, find the length of the arc of the sector.
From the diagram above, which of the following represents the vector V in component form?
From the diagram above, \(h[g(3)]\) is
\(g \circ h\) is
The diagram above is a velocity- time graph of a moving object. Calculate the distance travelled when the acceleration is zero.
Simplify \(\frac{x^{3n + 1}}{x^{2n + \frac{5}{2}}(x^{2n - 3})^{\frac{1}{2}}}\)
A binary operation * is defined on the set of real numbers R, by a* b = -1. Find the identity element under the operation *.
Express 75° in radians, leaving your answer in terms of \(\pi\).
If \(\log_{9} 3 + 2x = 1\), find x.
Evaluate \(\cos (\frac{\pi}{2} + \frac{\pi}{3})\)
Find the remainder when \(5x^{3} + 2x^{2} - 7x - 5\) is divided by (x - 2).
A function is defined by \(f(x) = \frac{3x + 1}{x^{2} - 1}, x \neq \pm 1\). Find f(-3).
Simplify \(\sqrt[3]{\frac{8}{27}} - (\frac{4}{9})^{-\frac{1}{2}}\)
Solve \(3x^{2} + 4x + 1 > 0\)
The equation of a circle is \(3x^{2} + 3y^{2} + 6x - 12y + 6 = 0\). Find its radius
\(f(x) = p + qx\), where p and q are constants. If f(1) = 7 and f(5) = 19, find f(3).
The sum and product of the roots of a quadratic equation are \(\frac{4}{7}\) and \(\frac{5}{7}\) respectively. Find its equation.
\(f(x) = (x^{2} + 3)^{2}\) is defines on the set of real numbers, R. Find the gradient of f(x) at x = \(\frac{1}{2}\).
Find \(\lim \limits_{x \to 3} \frac{x + 3}{x^{2} - x - 12}\)
If \(y^{2} + xy - x = 0\), find \(\frac{\mathrm d y}{\mathrm d x}\).
A line is perpendicular to \(3x - y + 11 = 0\) and passes through the point (1, -5). Find its equation.
Solve \(9^{2x + 1} = 81^{3x + 2}\)
The inverse of a function is given by \(f^{-1} : x \to \frac{x + 1}{4}\).
If \(\begin{pmatrix} 3 & 2 \\ 7 & x \end{pmatrix} \begin{pmatrix} 2 \\ 3 \end{pmatrix} = \begin{pmatrix} 12 \\ 29 \end{pmatrix} \), find x.
The fourth term of a geometric sequence is 2 and the sixth term is 8. Find the common ratio.
What percentage increase in the radius of a sphere will cause its volume to increase by 45%?
Evaluate \(\frac{1}{1 - \sin 60°}\), leaving your answer in surd form.
Find the equation of a circle with centre (-3, -8) and radius \(4\sqrt{6}\).
Determine the coefficient of \(x^{2}\) in the expansion of \((a + 3x)^{6}\).
The mean of 2, 5, (x + 2), 7 and 9 is 6. Find the median.
The probability that Kofi and Ama hit a target in a shooting competition are \(\frac{1}{6}\) and \(\frac{1}{9}\) respectively. What is the probability that only one of them hit the target?
In how many ways can 3 prefects be chosen out of 8 prefects?
Find the standard deviation of the numbers 3,6,2,1,7 and 5.
Marks 5-7 8-10 11-13 14-16 17-19 20-22 No of students 4 7 26 41 14 8 The table above shows the distribution of marks of students in a class. Find the upper class boundary of the modal class.
If \(^{3x}C_{2} = 15\), find the value of x?
Four doctors and two nurses are to sit round a circular table. In how many ways can this be done if the nurses are to sit together?
A basket contains 3 red and 1 white identical balls. A ball is drawn from the basket at random. Calculate the probability that it is either white or red.
A force of 200N acting on a body of mass 20kg initially at rest causes it to move a distance of 320m along a straight line for t secs. Find the value of t.
Two forces 10N and 15N act on an object at an angle of 120° to each other. Find the magnitude of the resultant.
A body of mass 25kg changes its speed from 15m/s to 35m/s in 5 seconds by the action of an applied force F. Find the value of F.
A particle starts from rest and moves in a straight line such that its velocity, v, at time t seconds is given by \(v = (3t^{2} - 2t) ms^{-1}\). Calculate the distance covered in the first 2 seconds.
A particle starts from rest and moves in a straight line such that its velocity, v, at time t seconds is given by \(v = (3t^{2} - 2t) ms^{-1}\). Determine the acceleration when t = 2 secs.
Given that \(q = 9i + 6j\) and \(r = 4i - 6j\), which of the following statements is true?
The functions f and g are defined on the set, R, of real numbers by \(f : x \to x^{2} - x - 6\) and \(g : x \to x - 1\). Find \(f \circ g(3)\).
Find the unit vector in the direction of (-5i + 12j).
Find, correct to two decimal places, the acute angle between \(p = \begin{pmatrix} 13 \\ 14 \end{pmatrix}\) and \(q = \begin{pmatrix} 12 \\ 5 \end{pmatrix}\).
Find the domain of \(f(x) = \frac{x}{3 - x}, x \in R\), the set of real numbers.
Find the value of \(\cos(60° + 45°)\) leaving your answer in surd form.
If \(\frac{5}{\sqrt{2}} - \frac{\sqrt{8}}{8} = m\sqrt{2}\), where m is a constant. Find m.
If \(16^{3x} = \frac{1}{4}(32^{x - 1})\), find the value of x.
Simplify \(\frac{\log_{5} 8}{\log_{5} \sqrt{8}}\).
The coefficient of the 7th term in the binomial expansion of \((2 - \frac{x}{3})^{10}\) in ascending powers of x is
The roots of a quadratic equation are \((3 - \sqrt{3})\) and \((3 + \sqrt{3})\). Find its equation.
If (x - 3) is a factor of \(2x^{2} - 2x + p\), find the value of constant p.
If \(\sin x = -\sin 70°, 0° < x < 360°\), determine the two possible values of x.
For what values of x is \(\frac{x^{2} - 9x + 18}{x^{2} + 2x - 35}\) undefined?
Calculate, correct to one decimal place, the length of the line joining points X(3, 5) and Y(5, 1).
If \(y = 2(2x + \sqrt{x})^{2}\), find \(\frac{\mathrm d y}{\mathrm d x}\).
Calculate, correct to one decimal place, the acute angle between the lines 3x - 4y + 5 = 0 and 2x + 3y - 1 = 0.
Evaluate \(\int_{1}^{2} \frac{4}{x^{3}} \mathrm {d} x\)
If \(\begin{vmatrix} 3 & x \\ 2 & x - 2 \end{vmatrix} = -2\), find the value of x.
Given that \(P = {x : \text{x is a factor of 6}}\) is the domain of \(g(x) = x^{2} + 3x - 5\), find the range of x.
The third of geometric progression (G.P) is 10 and the sixth term is 80. Find the common ratio.
Find the axis of symmetry of the curve \(y = x^{2} - 4x - 12\).
Find the equation of the tangent to the curve \(y = 4x^{2} - 12x + 7\) at point (2, -1).
The mean age of 15 pupils in a class is 14.2 years. One new pupil joined the class and the mean changed to 14.1 years. Calculate the age of the new pupil.
The distance s metres of a particle from a fixed point at time t seconds is given by \(s = 7 + pt^{3} + t^{2}\), where p is a constant. If the acceleration at t = 3 secs is \(8 ms^{-2}\), find the value of p.
The probabilities that a husband and wife will be alive in 15 years time are m and n respectively. Find the probability that only one of them will be alive at that time.
In a class of 50 pupils, 35 like Science and 30 like History. What is the probability of selecting a pupil who likes both Science and History?
P, Q, R, S are points in a plane such that PQ = 8i - 5j, QR = 5i + 7j, RS = 7i + 3j and PS = xi + yj. Find (x, y).
Find the least value of n for which \(^{3n}C_{2} > 0, n \in R\).
If \(\overrightarrow{OA} = 3i + 4j\) and \(\overrightarrow{OB} = 5i - 6j \) where O is the origin and M is the midpoint of AB, find OM.
Find the direction cosines of the vector \(4i - 3j\).
Yomi was asked to label four seats S, R, P, Q. What is the probability he labelled them in alphabetical order?
Two forces (2i - 5j)N and (-3i + 4j)N act on a body of mass 5kg. Find in \(ms^{-2}\), the magnitude of the acceleration of the body.
Two particles are fired together along a smooth horizontal surface with velocities 4 m/s and 5 m/s. If they move at 60° to each other, find the distance between them in 2 seconds.
Two forces \(F_{1} = (7i + 8j)N\) and \(F_{2} = (3i + 4j)N\) act on a particle. Find the magnitude and direction of \(F_{1} - F_{2}\).
A stone is thrown vertically upwards and its height at any time t seconds is \(h = 45t - 9t^{2}\). Find the maximum height reached.
Given that \(\frac{\mathrm d y}{\mathrm d x} = 3x^{2} - 4\) and y = 6 when x = 3, find the equation for y.
If \(h(x) = x^{3} - \frac{1}{x^{3}}\), evaluate \(h(a) - h(\frac{1}{a})\).
A company took delivery of 12 vehicles made up of 7 buses and 5 saloon cars for two of its departments; Personnel and General Administration. If the Personnel department is to have at least 3 saloon cars, in how many ways can these vehicles be distributed equally between the departments?
A bicycle wheel of diameter 70 cm covered a distance of 350 cm in 2 seconds. How many radians per second did it turn?
The initial velocity of an object is \(u = \begin{pmatrix} -5 \\ 3 \end{pmatrix} ms^{-1}\). If the acceleration of the object is \(a = \begin{pmatrix} 3 \\ -4 \end{pmatrix} ms^{-2}\) and it moved for 3 seconds, find the final velocity.
Find the maximum value of \(2 + \sin (\theta + 25)\).
Simplify \((1 + 2\sqrt{3})^{2} - (1 - 2\sqrt{3})^{2}\)
What is the angle between \(a = (3i - 4j)\) and \(b = (6i + 4j)\)?
Solve \(x^{2} - 2x - 8 > 0\).
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.
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.
The coordinates of the centre of a circle is (-2, 3). If its area is \(25\pi cm^{2}\), find its equation.
Given \(\sin \theta = \frac{\sqrt{3}}{2}, 0° \leq \theta \leq 90°\), find \(\tan 2\theta\) in surd form.
Find the coefficient of \(x^{4}\) in the binomial expansion of \((2 + x)^{6}\).
Which of the following binary operations is not commutative?
Express \(\frac{2}{3 - \sqrt{7}} \text{ in the form} a + \sqrt{b}\), where a and b are integers.
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.
Given that \(2^{x} = 0.125\), find the value of x.
The gradient of point P on the curve \(y = 3x^{2} - x + 3\) is 5. Find the coordinates of P.
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.
The first term of a geometric progression is 350. If the sum to infinity is 250, find the common ratio.
p and q are statements such that \(p \implies q\). Which of the following is a valid conclusion from the implication?
The roots of a quadratic equation are -3 and 1. Find its equation.
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).
Simplify \((216)^{-\frac{2}{3}} \times (0.16)^{-\frac{3}{2}}\)
Given that \(\log_{3}(x - y) = 1\) and \(\log_{3}(2x + y) = 2\), find the value of x.
If \(\frac{^{8}P_{x}}{^{8}C_{x}} = 6\), find the value of x.
Evaluate \(\int_{1}^{2} [\frac{x^{3} - 1}{x^{2}}] \mathrm {d} x\).
If \(P = \begin{vmatrix} 1 & 1 \\ 2 & 1 \end{vmatrix}\), find \((P^{2} + P)\).
Which of the following is the semi- interquartile range of a distribution?
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. \([g = 10 ms^{-2}]\).
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.
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.
The position vectors of A and B are (2i + j) and (-i + 4j) respectively; find |AB|.
Two fair dices, each numbered 1, 2, ..., 6, are tossed together. Find the probability that they both show even numbers.
Calculate, correct to the nearest degree, the angle between the vectors \(\begin{pmatrix} 13 \\ 1 \end{pmatrix}\) and \(\begin{pmatrix} 1 \\ 4 \end{pmatrix}\).
Simplify \(2\log_{3} 8 - 3\log_{3} 2\)
Evaluate \(\begin{pmatrix} 2 & 3 \\ 4 & 1 \end{pmatrix} \begin{pmatrix} 2 \\ 3 \end{pmatrix}\).
If the mean of -1, 0, 9, 3, k, 5 is 2, where k is a constant, find the median of the set of numbers.
Eight football clubs are to play in a league on home and away basis. How many matches are possible?
Two balls are drawn, from a bag containing 3 red, 4 white and 5 black identical balls. Find the probability that they are all of the same colour.
A force F acts on a body of mass 12kg increases its speed from 5 m/s to 35 m/s in 5 seconds. Find the value of F.
Express the force F = (8 N, 150°) in the form (a i + b j) where a and b are constants.
Three defective bulbs got mixed up with seven good ones. If two bulbs are selected at random, what is the probability that both are good?
The ages, in years, of 5 boys are 5, 6, 6, 8 and 10. Calculate, correct to one decimal place, the standard deviation of their ages.
A body is acted upon by forces \(F_{1} = (10 N, 090°)\) and \(F_{2} = (6 N, 180°)\). Find the magnitude of the resultant force.
In the diagram, a ladder PS leaning against a vertical wall PR makes angle x° with the horizontal floor. The ladder slides down to a point QT such that angle QTR = 30° and SNT = y°. Find the relation between x and y.
In the diagram, a ladder PS leaning against a vertical wall PR makes angle x° with the horizontal floor. The ladder slides down to a point QT such that angle QTR = 30° and SNT = y°. Find an expression for tan y.
Simplify \(\frac{\sqrt{3} + \sqrt{48}}{\sqrt{6}}\)
Find the range of values of x for which \(2x^{2} + 7x - 15 > 0\).
A function f is defined on R, the set of real numbers, by: \(f : x \to \frac{x + 3}{x - 2}, x \neq 2\), find \(f^{-1}\).
The sum of the first n terms of a linear sequence is \(S_{n} = n^{2} + 2n\). Find the common difference of the sequence.
The sum of the first n terms of a linear sequence is \(S_{n} = n^{2} + 2n\). Determine the general term of the sequence.
If \(f(x) = 2x^{2} - 3x - 1\), find the value of x for which f(x) is minimum.
The polynomial \(2x^{3} + x^{2} - 3x + p\) has a remainder of 20 when divided by (x - 2). Find the value of constant p.
If \(2\log_{4} 2 = x + 1\), find the value of x.
Which of the following quadratic curves will not intersect with the x- axis?
What is the coordinate of the centre of the circle \(5x^{2} + 5y^{2} - 15x + 25y - 3 = 0\)?
Evaluate \(\int_{1}^{2} (2 + 2x - 3x^{2}) \mathrm {d} x\).
A rectangle has a perimeter of 24m. If its area is to be maximum, find its dimension.
Express \(\frac{7\pi}{6}\) radians in degrees.
If \(P = \begin{pmatrix} 1 & -2 \\ 3 & 4 \end{pmatrix}\) and \(Q = \begin{pmatrix} -2 & 3 \\ 1 & 0 \end{pmatrix}\), find PQ.
Two statements are represented by p and q as follows: p : He is brilliant; q : He is regular in class Which of the following symbols represent "He is regular in class but dull"?
Find the locus of points which is equidistant from P(4, 5) and Q(-6, -1).
A binary operation ,*, is defined on the set R, of real numbers by \(a * b = a^{2} + b + ab\). Find the value of x for which \(5 * x = 37\).
Find the derivative of \(3x^{2} + \frac{1}{x^{2}}\)
The coefficient of the 5th term in the binomial expansion of \((1 + kx)^{8}\), in ascending powers of x is \(\frac{35}{8}\). Find the value of the constant k.
Given that \(f '(x) = 3x^{2} - 6x + 1\) and f(3) = 5, find f(x).
Express \(\frac{1}{1 - \sin 45°}\) in surd form.
If \(\begin{vmatrix} 4 & x \\ 5 & 3 \end{vmatrix} = 32\), find the value of x.
If events A and B are independent and \(P(A) = \frac{7}{12}\) and \(P(A \cap B) = \frac{1}{4}\), find P(B).
Given that \(\overrightarrow{AB} = 5i + 3j\) and \(\overrightarrow{AC} = 2i + 5j\), find \(\overrightarrow{BC}\).
The probability of Jide, Atu and Obu solving a given problem are \(\frac{1}{12}\), \(\frac{1}{6}\) and \(\frac{1}{8}\) respectively. Calculate the probability that only one solves the problem.
Two forces \(F_{1} = (10N, 020°)\) and \(F_{2} = (7N, 200°)\) act on a particle. Find the resultant force.
Marks 2 3 4 5 6 7 8 No of students 5 7 9 6 3 6 4 The table above shows the distribution of marks by some candidates in a test. What is the median score?
Marks 2 3 4 5 6 7 8 No of students 5 7 9 6 3 6 4 The table above shows the distribution of marks by some candidates in a test. Find, correct to one decimal place, the mean of the distribution.
Marks 2 3 4 5 6 7 8 No of students 5 7 9 6 3 6 4 The table above shows the distribution of marks by some candidates in a test. If a student is selected at random, what is the probability that she scored at least 6 marks?
Express \(r = (12, 210°)\) in the form \(a i + b j\).
A test consists of 12 questions out of which candidates are to answer 10. If the first 6 are compulsory, in how many ways can each candidate select her questions?
A body starts from rest and moves in a straight line with uniform acceleration of \(5 ms^{-2}\). How far, in metres, does it go in 10 seconds?
If n items are arranged two at a time, the number obtained is 20. Find the value of n.
If \(p = \begin{pmatrix} 2 \\ -2 \end{pmatrix} \) and \(q = \begin{pmatrix} 3 \\ 4 \end{pmatrix}\), find \(|q - \frac{1}{2}p|\).
Find the value of the constant k for which \(a = 4 i - k j\) and \(b = 3 i + 8 j\) are perpendicular.
The initial and final velocities of an object of mass 5 kg are \(u = \begin{pmatrix} 1 \\ 3 \end{pmatrix}\) and \(v = \begin{pmatrix} 4 \\ 7 \end{pmatrix}\) respectively. Find the magnitude of its change in momentum.
If \(y = x^{2} - 6x + 11\) is written in the form \(y = a(x - h)^{2} + k\), find the value of \((a + h + k)\).
The distance between P(x, 7) and Q(6, 19) is 13 units. Find the values of x.
In the diagram above, forces P, Q and 50N are acting on a body at M. If the system is in equilibrium, calculate, in terms of \(\theta\), the magnitude of P.
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.
Given that \(\alpha\) and \(\beta\) are the roots of an equation such that \(\alpha + \beta = 3\) and \(\alpha \beta = 2\), find the equation.
Which of the following is the same as \(\sin (270 + x)°\)?
The sum of the first three terms of an Arithmetic Progression (A.P) is 18. If the first term is 4, find their product.
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 gradient of the line passing through the points P(4, 5) and Q(x, 9) is \(\frac{1}{2}\). Find the value of x.
Simplify \(\frac{\sqrt{3}}{\sqrt{3} - 1} + \frac{\sqrt{3}}{\sqrt{3} +1}\)
Simplify \(\frac{\tan 80° - \tan 20°}{1 + \tan 80° \tan 20°}\)
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.
Find the fourth term of the binomial expansion of \((x - k)^{5}\) in descending powers of x.
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.
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.
Given that \(\log_{2} y^{\frac{1}{2}} = \log_{5} 125\), find the value of y.
Two vectors m and n are defined by \(m = 3i + 4j\) and \(n = 2i - j\). Find the angle between m and n.
Find the area of the circle whose equation is given as \(x^{2} + y^{2} - 4x + 8y + 11 = 0\).
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.
Calculate in surd form, the value of \(\tan 15°\).
Evaluate \(\lim \limits_{x \to 3} \frac{x^{2} - 2x - 3}{x - 3}\)
If \(f(x) = mx^{2} - 6x - 3\) and \(f'(1) = 12\), find the value of the constant m.
A bag contains 2 red and 4 green sweets of the same size and shape. Two boys pick a sweet each from the box, one after the other, without replacement. What is the probability that at least a sweet with green wrapper is picked?
A body is acted upon by two forces \(F_{1} = (5 N, 060°)\) and \(F_{2} = (10 N, 180°)\). Find the magnitude of the resultant force.
The equation of a curve is given by \(y = 2x^{2} - 5x + k\). If the curve has two intercepts on the x- axis, find the value(s) of constant k.
Find the value of p for which \(x^{2} - x + p\) becomes a perfect square.
The polynomial \(2x^{3} + 3x^{2} + qx - 1\) has the same reminder when divided by \((x + 2)\) and \((x - 1)\). Find the value of the constant q.
Marks 5 - 7 8 - 10 11 - 13 14 - 16 17 - 19 20 - 22 Frequency 4 7 26 41 14 8 The table above shows the marks obtained by 100 pupils in a test. Find the upper class boundary of the class containing the third quartile.
Marks 5 - 7 8 - 10 11 - 13 14 - 16 17 - 19 20 - 22 Frequency 4 7 26 41 14 8 The table above shows the marks obtained by 100 pupils in a test. Find the probability that a student picked at random scored at least 14 marks.
How many ways can 12 people be divided into three groups of 2, 7 and 3 in that order?
Given that \(P = \begin{pmatrix} 4 & 9 \end{pmatrix}\) and \(Q = \begin{pmatrix} -1 & -2 \\ 3 & 2 \end{pmatrix}\). Which of the following operations is possible?
Given that \(P = \begin{pmatrix} 4 & 9 \end{pmatrix}\) and \(Q = \begin{pmatrix} -1 & -2 \\ 3 & 2 \end{pmatrix}\). Evaluate \(|Q|P\).
The equation of a circle is given by \(x^{2} + y^{2} - 4x - 2y - 3\). Find the radius and the coordinates of its centre.
Simplify \(\frac{^{n}P_{3}}{^{n}C_{2}} + ^{n}P_{0}\)
X and Y are two independent event. If \(P(X) = \frac{1}{5}\) and \(P(X \cap Y) = \frac{2}{15}\), find \(P(Y)\).
Given that \(p = 4i + 3j\), find the unit vector in the direction of p.
A particle is projected vertically upwards with a speed of 40 m/s. At what times will it be 35m above its point of projection? \(\text{Take g} = 10 ms^{-2}\)
Three students are working independently on a Mathematics problem. Their respective probabilities of solving the problem are 0.6, 0.7 and 0.8. What is the probability that at least one of them solves the problem?
Given that \(R = (4, 180°)\) and \(S = (3, 300°)\), find the dot product.
Calculate, correct to one decimal place, the standard deviation of the numbers: -1, 5, 0, 2 and 9.
A group of 5 boys and 4 girls is to be chosen from a class of 8 boys and 6 girls. In how many ways can this be done?
A force of 30 N acts at an angle of 60° on a body of mass 6 kg initially at rest on a smooth horizontal plane. Find the distance covered in 10 seconds.
Three forces \(F_{1} = (8 N, 300°), F_{2} = (6 N, 090°)\) and \(F_{3} = (4 N, 180°)\) act on a particle. Find the vertical component of the resultant force.
\(P = {x : 1 \leq x \leq 6}\) and \(Q = {x : 2 < x < 9}\) where \(x \in R\), find \(P \cap Q\).
Solve the inequality \(2x^{2} + 5x - 3 \geq 0\).
Simplify \(\sqrt{(\frac{-1}{64})^{\frac{-2}{3}}}\).
A binary operation ♦ is defined on the set R, of real numbers by \(a ♦ b = \frac{ab}{4}\). Find the value of \(\sqrt{2} ♦ \sqrt{6}\).
If \((x - 3)\) is a factor of \(2x^{3} + 3x^{2} - 17x - 30\), find the remaining factors.
Two functions f and g are defined by \(f : x \to 3x - 1\) and \(g : x \to 2x^{3}\), evaluate \(fg(-2)\).
Given that \(\frac{1}{8^{2y - 3y}} = 2^{y + 2}\).
Given that \((\sqrt{3} - 5\sqrt{2})(\sqrt{3} + \sqrt{2}) = p + q\sqrt{6}\), find q.
If \(f(x) = \frac{1}{2 - x}, x \neq 2\), find \(f^{-1}(-\frac{1}{2})\).
Find the coefficient of \(x^{4}\) in the binomial expansion of \((1 - 2x)^{6}\).
Find the equation of the line passing through (0, -1) and parallel to the y- axis.
The roots of the equation \(2x^{2} + kx + 5 = 0\) are \(\alpha\) and \(\beta\), where k is a constant. If \(\alpha^{2} + \beta^{2} = -1\), find the values of k.
Find the sum of the exponential series \(96 + 24 + 6 +...\)
Evaluate \(\log_{0.25} 8\)
Evaluate \(\lim \limits_{x \to 1} \frac{1 - x}{x^{2} - 3x + 2}\)
The mean age of n men in a club is 50 years. Two men aged 55 and 63 years left the club, and the mean age reduced by 1 year. Find the value of n.
A committee of 4 is to be selected from a group of 5 men and 3 women. In how many ways can this be done if the chairman of the committee must be a man?
Simplify \(\frac{^{n}P_{4}}{^{n}C_{4}}\)
Which of the following is a singular matrix?
Simplify \(8^{n} \times 2^{2n} \div 4^{3n}\)
The area of a sector of a circle is 3\(cm^{2}\). If the sector subtends an angle of 1.5 radians at the centre, calculate the radius of the circle.
A particle of mass 2.5 kg is moving at a speed of 12 m/s. If a force of magnitude 10 N acts against it, find how long it takes to come to rest.
Age(in years) 1 - 5 6 - 10 11 - 15 Frequency 3 5 2 Calculate the standard deviation of the distribution.
In a firing contest, the probabilities that Kojo and Kwame hit the target are \(\frac{2}{5}\) and \(\frac{1}{3}\) respectively. What is the probability that none of them hit the target?
The equation of the line of best fit for variables x and y is \(y = 19.33 + 0.42x\), where x is the independent variable. Estimate the value of y when x = 15.
Find the coordinates of the point on the curve \(y = x^{2} + 4x - 2\), where the gradient is zero.
Find the least value of the function \(f(x) = 3x^{2} + 18x + 32\).
A force of 32 N is applied to an object of mass m kg which is at rest on a smooth horizontal surface. If the acceleration produced is 8\(ms^{-2}\), find the value of m.
Given that \(\begin{pmatrix} 1 & -3 \\ 1 & 4 \end{pmatrix} \begin{pmatrix} -6 \\ P \end{pmatrix} = \begin{pmatrix} 3 \\ -26 \end{pmatrix}\), find the value of P.
Find the coordinates of the centre of the circle \(4x^{2} + 4y^{2} - 5x + 3y - 2 = 0\).
A and B are two independent events such that \(P(A) = \frac{2}{5}\) and \(P(A \cap B) = \frac{1}{15}\). Find \(P(B)\).
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.
Find, in surd form, the value of \(\cos 165\).
The mean and median of integers x, y, z and t are 5 and z respectively. If x < y < z < t and y = 4, find (x + t).
If \(a = \begin{pmatrix} 3 \\ 2 \end{pmatrix}\) and \(b = \begin{pmatrix} -3 \\ 5 \end{pmatrix}\), find a vector c such that \(4a + 3c = b\).
A lift moving upwards with a uniform acceleration of 5\(ms^{-2}\) carries a body of mass p kg. If the reaction on the floor is 480 N, find the value of p. [Take g = \(10 ms^{-2}\)].
Calculate, correct to one decimal place, the angle between 5i + 12j and -2i + 3j.
A particle is projected vertically upwards from a height 45 metres above the ground with a velocity of 40 m/s. How long does it take it to hit the ground? [Take g = \(10 ms^{-2}\)].
Two forces, each of magnitude 16 N, are inclined to each other at an angle of 60°. Calculate the magnitude of their resultant.
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.
Solve: 8\(^{x - 2}\) = 4\(^{3x}\)
Solve; \(\frac{P}{2} + \frac{k}{3}\) = 5 and 2p = k = 6 simultaneously
Evaluate tan 75\(^o\); leaving the answer in surd form (radicals)
Rationalize; \(\frac{1}{\sqrt{2 + 1}}\)
If \(^nC_2\) = 15, find the value of n
An operation (*) is defined on the set T = {-1, 0, ...., 5} by x * y = x + y - xy. Which of the following operation(s) will give an image that is an element of T? I. 2(*)5 II. 3(*)2 III. 3(*)4
Given that g ; x \(\to\) 3x and f ; x \(\to\) cos x. Find the value of g\(^o\) f(20\(^o\))
A linear transformation is defined by T: (x, y) \(\to\) (-x + y, -4y). Find the image, Q`, of Q(-3, 2) under T
If g : r \(\to\) 5 - 2r, r is a real number, find the image of -3
Consider the following statements: p: Birds fly q: The sky is blue r: The grass is green What is the symbolic representation of "If the grass is green and the sky is not blue, then the birds do not fly"?
Given that \(\frac{1}{x^2 - 4} = \frac{p}{(x + 2)} + \frac{Q}{(x - 2})\) x \(\neq \pm 2\) Find the value of (P + Q)
Find the sum of the first 20 terms of the sequence -7-3, 1, ......
Find the value of x for which 6\(\sqrt{4x^2 + 1}\) = 13x, where x > 0
Calculate the distance between points (-2, -5) and (-1, 3)
If P = \(\begin {pmatrix} 2 & 3\\ -4 & 1 \end {pmatrix}\), Q = \(\begin{pmatrix} 6 \\ 8 \end {pmatrix}\) and PQ = k \(\begin {pmatrix} 45\\ -20 \end {pmatrix}\). Find the value of k.
The second and fourth terms of an exponential sequence (G.P) are \(\frac{2}{9}\) and \(\frac{8}{81}\) respectively. Find the sixth term of the sequence
Point X and Y are on the same horizontal base as the foot of a building such that X is 96m due east of the building and Y is due west. If the angle of elevation of the top of that building from X is 30\(^o\) and that of Y is 50\(^o\), calculate the distance of Y from the base of the building.
Find the coordinates of the point in the curve y = 3x\(^2\) - 2x - 5 where the tangent is parallel to the line y = - 5 = 8x
If the mean of 2, 5, (x + 1), (x + 2), 7 and 9 is 6, find the median.
Calculate the mean deviation of 5, 8, 2, 9 and 6
A particle starts from rest and moves in a straight line such that its velocity, V ms\(^{-1}\), at time t second is given by V = 3t\(^2\) - 6t. Calculate the acceleration in the 3rd second.
A particle starts from rest and moves in a straight line such that its velocity, V ms\(^{-1}\), at time t second is given by V = 3t\(^2\) - 6t. Calculate the acceleration in the 3rd second.
Find the constant term in the binomial expansion of (2x\(^2\) + \(\frac{1}{x^2}\))\(^4\)
Which of these inequalities is represented by the shaded portion of the graph?
A 35 N force acts on a body of mass 5 kg for 2 seconds. Calculate the change in momentum of the body.
Solve, correct to three significant figures, (0.3)\(^x\) = (0,5)\(^8\)
Given that P and Q are non-empty subsets of the universal set, U. Find P \(\cap\) (Q U Q`).
Find the coefficient of the third term in the binomial expansion of [2x + \(\frac{3y}{4}\)]\(^3\) in descending powers of x.
Find the coordinates of the centre of the circle 3x\(^2\) + 3y\(^2\) - 6x + 9y - 5 = 0
Evaluate \(\int^9_0 \sqrt{x} dx\)
The function f : x \(\to\) x\(^2\) + px + q has turning point when x = -3 and remainder of -6 when divided by (x + 2). Find the value of q.
If y = (5 - x)\(^{-3}\), and \(\frac{dy}{dx}\)
Which of the following vectors is perpendicular to \(\begin{pmatrix} -1 & 3 \end{pmatrix}\)?
Find correct to the nearest degree,5 the angle between p = 12i - 5j and q = 4i +3j
Find the area between line y = x + 1 and the x-axis from x = -2 to x = 0.
How many numbers greater than 200 can be formed from the digits 1,2,3,4, 5 if no digit is to be repeated in any particular number?
The probabilities that John and Jane will pass an examination are 0.9 and 0.7 respectively. Find the probability that at least one of them will pass the examination.
Given that X and Y are independent events such that P(X) = 0.5, P(Y) = m and P(X U Y) = 0.75, find the value of m.
A uniform beam, PQ. is 100 m long and weighs 35 N. It is placed on a support at a point 40 cm from P. If weights of 54 N and FN are attached at P and Q respectively in order to keep it in a horizontal position, calculate, correct to the nearest whole number, the value of F.
Evaluate: \(^{lim}_{x \to 1} \begin{pmatrix} \frac{1 - x}{x^2 - 3x + 2} \end {pmatrix}\)
A binary operation * is defined on the set of real number, R, by x*y = x\(^2\) - y\(^2\) + xy, where x, \(\in\) R. Evaluate (\(\sqrt{3}\))*(\(\sqrt{2}\)) \({\color{red}2x} \times 3\)
Find the inverse of \(\begin{pmatrix} 3 & 5 \\ 1 & 2 \end{pmatrix}\)
If cos x = -0.7133, find the values of x between 0\(^o\) and 360\(^o\)
If \(\int^3_0(px^2 + 16)dx\) = 129. Find the value of p.
If \(\begin{pmatrix} p+q & 1\\ 0 & p-q \end {pmatrix}\) = \(\begin{pmatrix} 2 & 1 \\ 0 & 8 \end{pmatrix}\) Find the values of p and q
Given that X : R \(\to\) R is defined by x = \(\frac{y + 1}{5 - y}\) , y \(\in\) R, find the domain of x.
Simplify; \(\frac{\sqrt{5} + 3}{4 - \sqrt{10}}\)
If \(\frac{6x + k}{2x^2 + 7x - 15}\) = \(\frac{4}{x + 5} - \frac{2}{2x - 3}\). Find the value of k.
Differentiate \(\frac{x}{x + 1}\) with respect to x.
Given that 2x + 3y - 10 = 0 and 3x = 2y - 11, calculate the value of (x - y).
If V = plog\(_x\), (M + N), express N in terms of X, P, M and V
Determine the coefficient of x\(^3\) in the binomial expansion of ( 1 + \(\frac{1}{2}\)x)
Given that P = {x : 1 \(\geq\) x \(\geq\) 6} and Q = {x : 2 < x < 10}. Where x are integers, find n(p \(\cap\) Q)
If X = \(\frac{3}{5}\) and cos y = \(\frac{24}{25}\), where X and Y are acute, find the value of cos (X + Y).
Find the median of the numbers 9,7, 5, 2, 12,9,9, 2, 10, 10, and 18.
Calculate the probability that the product of two numbers selected at random with replacement from the set {-5,-2,4, 8} is positive
Find the angle between i + 5j and 5i - J
Given that F = 3i - 12j, R = 7i + 5j and N = pi + qj are forces acting on a body, if the body is in equilibrium. find the values of p and q.
A stone was dropped from the top of a building 40m high. Find, correct to one decimal place, the time it took the stone to reach the ground. [Take g = 9.8ms\(^{-2}\)]
In which of the following series can be the formula S = \(\frac{a}{1 - r}\) where a is the first term and r is the common ratio, be used to find the sum of all the terms?
If the binomial expansion of (1 + 3x)\(^6\) is used to evaluate (0.97)\(^6\), find the value of x.
Find the nth term of the linear sequence (A.P) (5y + 1), ( 2y + 1), (1- y),...
A circle with centre (5,-4) passes through the point (5, 0). Find its equation.
Calculate, correct to two decimal places, the area enclosed by the line 3x - 5y + 4 = 0 and the axes.
In how many ways can the letters of the word MEMBER be arranged?
Which of the following is not an equation of a circle?
A function f defined by f : x -> x\(^2\) + px + q is such that f(3) = 6 and f(3) = 0. Find the value of q.
In what interval is the function f : x -> 2x - x\(^2\) increasing?
A force of 230N acts in its direction 065\(^o\). Find its horizontal component.
Calculate the variance of \(\sqrt{2}\), (1 + \(\sqrt{2}\)) and (2 + \(\sqrt{2}\))
A three-digit odd number less than 500 is to be formed from 1,2,3,4 and 5. If repetition of digits is allowed, in how many ways can this be done?
The variables x and y are such that y =2x\(^3\) - 2x\(^2\) - 5x + 5. Calculate the corresponding change in y and x changes from 2.00 to 2.05.
A bag contains 5 red and 5 blue identical balls. Three balls are selected at random without replacement. Determine the probability of selecting balls alternating in color.
The distance(s) in metres covered by a particle in motion at any time, t seconds, is given by S =120t - 16t\(^2\). Find in metres, the distance covered by the body before coming to rest.
P(3,4) and Q(-3, -4) are two points in a plane. Find the gradient of the line that is normal to the line PQ.
Find the unit vector in the direction opposite to the resultant of forces. F\(_1\) = (-2i - 3j) and F\(_2\) = (5i - j)
If the sum of the roots of 2x\(^2\) + 5mx + n = 0 is 5, find the value of m.
If log 5(\(\frac{125x^3}{\sqrt[ 3 ] {y}}\) is expressed in the values of p, q and k respectively.
Consider the statements: x: Birds fly y: The sky is blue Which of the following statements can be represented as x \(\to\) y?
Simplify ( \(\frac{1}{2 - √3}\) + \(\frac{2}{2 + √3}\) )\(^{-1}\)
For what range of values of x is x\(^2\) - 2x - 3 ≤ 0
Given that M = \(\begin{pmatrix} 3 & 2 \\ -1 & 4 \end{pmatrix}\) and N = \(\begin{pmatrix} 5 & 6 \\ -2 & -3 \end{pmatrix}\), calculate (3M - 2N)
Simplify \(\frac{1}{3}\) log8 + \(\frac{1}{3}\) log 64 - 2 log6
Solve (\(\frac{1}{9}\))\(^{x + 2}\) = 243\(^{x - 2}\)
g(x) = 2x + 3 and f(x) = 3x\(^2\) - 2x + 4 find f {g (-3)}.
Using binomial expansion of ( 1 + x)\(^6\) = 1 + 6x + 15x\(^2\) + 20x\(^3\) + 6x\(^5\) + x)\(^6\), find, correct to three decimal places, the value of (1.998))\(^6\)
In how many ways can 8 persons be seated on a bench if only three seats are available?
If α and β are the roots of 3x\(^2\) - 7x + 6 = 0, find \(\frac{1}{α}\) + \(\frac{1}{β}\)
If f(x) = 4x\(^3\) + px\(^2\) + 7x - 23 is divided by (2x -5), the remainder is 7. find the value of p
For what value of k is 4x\(^2\) - 12x + k, a perfect square?
A binary operation * is defined on the set of real numbers, R, by P * q = \(\frac{q^2 - p^2}{2pq}\). Find 3 * 2
Find the inverse of \(\begin{pmatrix} 4 & 2 \\ -3 & -2 \end{pmatrix}\)
Given that P = { x: 0 ≤ x ≤ 36, x is a factor of 36 divisible by 3} and Q = { x: 0 ≤ x ≤ 36, x is an even number and a perfect square}, find P n Q.
A body of mass 15kg is placed on a smooth plane which is inclined at 60° to the horizontal. If the box is at rest, calculate the normal reaction to the plane. [ Take g = 10m/s\(^2\) ]
A fair die is tossed 60 times and the results are recorded in the table Number of die 1 2 3 4 5 6 Frequency 15 10 14 2 8 11 Find the probability of obtaining a prime number.
If 2y\(^2\) + 7 = 3y - xy, find \(\frac{dy}{dx}\)
Three forces, F\(_1\) (8N, 030°), F\(_2\) (10N, 150° ) and F\(_3\) ( KN, 240° )are in equilibrium. Find the value of N
In △PQR, \(\overline{PQ}\) = 5i - 2j and \(\overline{QR}\) = 4i + 3j. Find \(\overline{RP}\).
A stone is thrown vertically upward and distance, S metres after t seconds is given by S = 12t + \(\frac{5}{2t^2}\) - t\(^3\). Calculate the maximum height reached.
A stone is thrown vertically upward and distance, S metres after t seconds is given by S = 12t + \(\frac{5}{2t^2}\) - t\(^3\). Calculate the distance travelled in the third second.
Given that F\(^1\)(x) = x\(^3\) √x, find f(x)
If ( 1- 2x)\(^4\) = 1 + px + qx\(^2\) - 32x\(^3\) + 16\(^4\), find the value of (q - p)
If sin x = \(\frac{12}{13}\) and sin y = \(\frac{4}{5}\), where x and y are acute angles, find cos (x + y)
The first term of an AP is 4 and the sum of the first three terms is 18. Find the product of the first three terms
A committee consists of 6 boys and 4 girls. In how many ways can a sub-committee consisting of 3 boys and 2 girls be formed if one particular boy and one particular girl must be on the sub-committee?
If √5 cosx + √15sinx = 0, for 0° < x < 360°, find the values of x.
If 2i +pj and 4i -2j are perpendicular, find the value of p.
Consider the following statements: X: Benita is polite y: Benita is neat z: Benita is intelligent Which of the following symbolizes the statement: "Benita is neat if and only if she is neither polite nor intelligent"?
A bag contains 8 red, 4 blue and 2 green identical balls. Two balls are drawn randomly from the bag without replacement. Find the probability that the balls drawn are red and blue. A. 12/91 B. C. D.
The gradient ofy= 3x\(^2\) + 11x + 7 at P(x.y) is -1. Find the coordinates of P.
Find the equation of the normal to the curve y= 2x\(^2\) - 5x + 10 at P(1, 7).
Find the value of the derivative of y = 3x\(^2\) (2x +1) with respect to x at the point x = 2.
Find the radius of the circle 2x\(^2\) - 4x + 2y\(^2\) - 6y -2 = 0.
Given that f: x --> x\(^2\) - x + 1 is defined on the Set Q = { x : 0 ≤ x < 20, x is a multiple of 5}. find the set of range of F.
If \(\frac{15 - 2x}{(x+4)(x-3)}\) = \(\frac{R}{(x+4)}\) + \(\frac{9}{(x-3)}\), find the value of R
The table shows the distribution of marks obtained by some students in a test Marks 0-9 10-19 20-29 30-39 40-49 Frequency 4 12 16 6 2 What is the upper class boundary of the upper quartile class?
The table shows the distribution of marks obtained by some students in a test Marks 0-9 10-19 20-29 30-39 40-49 Frequency 4 12 16 6 2 Find the modal class mark.
A binary operation ∆ is defined on the set of real numbers R, by x∆y = \(\sqrt{x+y - \frac{xy}{4}}\), where x, yER. Find the value of 4∆3
(\(\frac{3\sqrt6 + \sqrt{54}}{\sqrt5(3\sqrt5)})^{-1}\)
If \(log_{10}(3x-1) + log_{10}4 = log_{10}(9x+2)\), find the value of x
Simplify \(\frac{9*3^{n+1} - 3^{n+2}}{3^{n+1} - 3^{n}}\)
Consider the following statement: x: All wrestlers are strong y: Some wresters are not weightlifters. Which of the following is a valid conclusion?
The functions f:x → 2x\(^2\) + 3x -7 and g:x →5x\(^2\) + 7x - 6 are defined on the set of real numbers, R. Find the values of x for which 3f(x) = g(x).
Express \(\frac{4π}{2}\) radians in degrees.
A straight line makes intercepts of -3 and 2 on the x and y axes respectively. Find the equation of the line.
Which of the following is the semi-interquartile range of a distribution?
Evaluate \(∫^0_{-1}\) (x + 1)(x - 2) dx
If 36, p,\(\frac{9}{4}\) and q are consecutive terms of an exponential sequence (G.P), find the sum of p and q.
Differentiate \(\frac{5x^ 3+x^2}{x}\), x ≠ 0 with respect to x.
Given that \(\frac{8x+m}{x^2-3x-4} ≡ \frac{5}{x+1} + \frac{3}{x-4}\)
If \(x^2+y^2+-2x-6y+5 =0\), evaluate dy/dx when x=3 and y=2.
Evaluate\({1_0^∫} x^2(x^3+2)^3\)
Given \(\begin{vmatrix} 2 & -3 \\ 1 & 4 \end{vmatrix} \begin{vmatrix} -6 \\ k \end{vmatrix} \begin{vmatrix} 3 \\ -26 \end{vmatrix} = 15\). Solve for k.
A linear transformation T is defined by T: (x,y) → (3x - y, x + 4y). Find the image of (2, -1) under T.
Evaluate \(4p_2 + 4C_2 - 4p_3\)
Find the coefficient of x\(^2\)in the binomial expansion of \((x + \frac{2}{x^2})^5\)
Given that P = {x: x is a multiple of 5}, Q = {x: x is a multiple of 3} and R = {x: x is an odd number} are subsets of μ = {x: 20 ≤ x ≤ 35}, (P⋃Q)∩R.
A particle moving with a velocity of 5m/s accelerates at 2m/s\(^2\). Find the distance it covers in 4 seconds.
If Un = kn\(^2\) + pn, U\(_1\) = -1, U\(_5\) = 15, find the values of k and p.
In how many ways can six persons be paired?
Solve: \(3^{2x-2} - 28(3^{x-2}) + 3 = 0\)
Given that P = (-4, -5) and Q = (2,3), express →PQ in the form (k,θ). where k is the magnitude and θ the bearing.
If →PQ = -2i + 5j and →RQ = -i - 7j, find →PR
The table shows the distribution of the distance (in km) covered by 40 hunters while hunting. Distance(km) 3 4 5 6 7 8 Frequency 5 4 x 9 2x 1 If a hunter is selected at random, find the probability that the hunter covered at least 6km.
The table shows the distribution of the distance (in km) covered by 40 hunters while hunting. What is the mode of the distribution? Distance(km) 3 4 5 6 7 8 Frequency 5 4 x 9 2x 1
If g(x) = √(1-x\(^2\)), find the domain of g(x)
Find the coefficient of x\(^3\)y\(^2\) in the binomial expansion of (x-2y)\(^5\)
The first, second and third terms of an exponential sequence (G.P) are (x - 4), (x + 2), and (3x + 1) respectively. Find the values of x.
A body of mass 18kg moving with velocity 4ms-1 collides with another body of mass 6kg moving in the opposite direction with velocity 10ms-1. If they stick together after the collision, find their common velocity.
The mean heights of three groups of students consisting of 20, 16 and 14 students each are 1.67m, 1.50m and 1.40m respectively. Find the mean height of all the students.
Find correct to the nearest degree, the acute angle formed by the lines y = 2x + 5 and 2y = x - 6
Solve: 4sin\(^2\)θ + 1 = 2, where 0º < θ < 180º
Find the range of values of x for which 2x\(^2\) + 7x - 15 ≥ 0.
The probability that a student will graduate from college is 0.4. If 3 students are selected from the college, what is the probability that at least one student will graduate?
The equation of a circle is given as 2x\(^2\) + 2y\(^2\) - x - 3y - 41 = 0. Find the coordinates of its centre.
The gradient of a function at any point (x,y) 2x - 6. If the function passes through (1,2), find the function.
A particle of mass 3kg moving along a straight line under the action of a F N, covers a line distance, d, at time, t, such that d = t\(^2\) + 3t. Find the magnitude of F at time t.
If α and β are roots of x\(^2\) + mx - n = 0, where m and n are constants, form the equation whose roots are 1 α and 1 β .
A particle is acted upon by forces F = (10N, 060º), P = (15N, 120º) and Q = (12N, 200º). Express the force that will keep the particle in equilibrium in the form xi + yj, where x and y are scalars.
Evaluate: lim\(_{x→-2}\) \(\frac{x^3+8}{x+2}\).
If f(x-1) = x\(^3\) + 3x\(^2\) + 4x - 5, find f(2)
The length of the line joining points (x,4) and (-x,3) is 7 units. Find the value of x.
Calculate, correct to one decimal place, the angle between 5 i + 12 j and -2 i + 3 j
Find the equation of the normal to the curve y = \(3x^2 + 2\) at point (1, 5).
The distance S metres moved by a body in t seconds is given by \(S = 5t^3 - \frac{19}{2} t^2 + 6t - 4\). Calculate the acceleration of the body after 2 seconds
Evaluate \(\int^1_0 x(x^2-2)^2 dx\)
Given that \(sin x = \frac{4}{5}\) and \(cos y = \frac{12}{13}\), where x is an obtuse angle and y is an acute angle, find the value of sin (x - y).
If\((\frac{1}{9})^{2x-1} = (\frac{1}{81})^{2-3x}\)find the value of x
The table shows the operation * on the set {x, y, z, w}. * X Y Z W X Y Z X W Y Z W Y X Z X Y Z W W W X W Z Find the identity of the element.
Find the radius of the circle \(2x^2 + 2y^2 - 4x + 5y + 1 = 0\)
Given that M is the midpoint of T (2, 4) and Q (-8, 6), find the length of MQ .
A particle began to move at \(27 ms^{-1}\) along a straight line with constant retardation of \(9 ms^{-2}\). Calculate the time it took the particle to come to a stop.
Find the fifth term in the binomial expansion of \((q + x)^7\).
Given that P = {x : 2 ≤ x ≤ 8} and Q = {x : 4 < x ≤ 12} are subsets of the universal set μ = {x : x ∈ R}, find (P ⋂ Q\(^1\)).
Consider the statements: x: The school bus arrived late y: The student walked down to school Which of the following can be represented by y ⇒ x?
\(Differentiate f (x) = \frac{1}{(1 - x^2)^5}\) with respect to \(x\).
Express \(\frac{3}{3 - √6}\) in the form \(x + m√y\)
The table shows the mark obtained by students in a test. Marks 1 2 3 4 5 Frequency 2 k 1 1 2 If the mean mark is 3, find the value of k.
\(Simplify: \frac{log √27 - log √8}{log 3 - log 2}\)
Given that r = (10 N , 200º) and n = (16 N , 020º), find (3r - 2n).
Solve 6 sin 2θ tan θ = 4, where 0º < θ < 90º
An exponential sequence (G.P.) is given by 8√2, 16√2, 32√2, ... . Find the n\(^{th}\) term of the sequence
If \(f : x → 2 tan x\) and \(g : x → √(x^2 + 8), find ( g o f )(45^o)\)
A uniform beam PQ of length 80 cm and weight 60 N rests on a support at X where | PX | = 30 cm. If the body is kept in equilibrium by a mass m kg which is placed at P , calculate the value of m [Take g = 10 ms\(^{-2}\)]
An exponential sequence (G.P.) is given by \(\frac{9}{2},\frac{3}{4},\frac{1}{8},\)....Find its sum to infinity.
Adu's scores in five subjects in an examination are 85 , 84 , 83 , 86 and 87 . Calculate the standard deviation.
In how many ways can a committee of 3 women and 2 men be chosen from a group of 7 men and 5 women?
Evaluate: \(\int(2x + 1)^3 dx\)
If α and β are the roots of \(7x2 +12x - 4 = 0\),find the value of \(\frac{αβ}{(α + β)^2}\)
If \(3x^2 + p x + 12 = 0\) has equal roots, find the values of p .
Given that \(\frac{3x + 4}{(x - 2)(x + 3)}≡\frac{P}{x + 3}+\frac{Q}{x - 2}\),find the value of Q.
The velocity of a body of mass 4.56 kg increases from \((10 ms^{-1}, 060^o) to (50 ms ^{-1}, 060^o)\) in 16 seconds . Calculate the magnitude of force acting on it.
A linear transformation on the oxy plane is defined by \(P : (x, y) → (2x + y, -2y)\). Find \(P^2\)
Given that \(y^2 + xy = 5,find \frac{dy}{dx}\).
If \(X\) and \(Y\) are two independent events such that \(P (X) = \frac{1}{8}\) and \(P (X ⋃ Y) = \frac{5}{8}\), find \(P (Y)\).
A function \(f\) is defined by \(f :x→\frac{x + 2}{x - 3},x ≠ 3\).Find the inverse of \(f\) .
The probabilities that Atta and Tunde will hit a target in a shooting contest are \(\frac{1}{6}\) and \({1}{9}\) respectively. Find the probability that only one of them will hit the target.
Given that \(p = \begin{bmatrix} x&4\\3&7\end{bmatrix} Q =\begin{bmatrix} x&3\\1&2x\end{bmatrix}\) and the determinant of \(Q\) is three more than that of \(P\) , find the values of \(x\).
If m and ( m + 4) are the roots of \(4x^2 - 4x - 15 = 0\), find the equation whose roots are 2 m and (2 m + 8).
Find the coefficient of the \(6^{th}term\) in the binomial expansion of \((1 - \frac{2x}{3})10\) in ascending powers of \(x\).
In how many ways can four Mathematicians be selected from six ?
If \((x - 5)\) is a factor of \(x^3 - 4x^2 - 11x + 30\), find the remaining factors.
If \(\frac{5}{\sqrt{2}}\) - \(\frac{\sqrt{8}}{8}\) = m\(\sqrt{2}\), find the value of m
Given that f: x → \(\sqrt{x}\) and g : x → 25 - x\(^2\), find the value of f o g(3)
\(\sqrt{x}\) - \(\frac{6}{\sqrt{x}}\) = 1, find the value of x
A binary operation * is defined on the set of real numbers, R by x * y = \(\frac{y^2 - x^2}{2xy}\), x, y ≠ 0, where x and y are real numbers. Evaluate -3 * 2
If the n\(^{th}\) term of a linear sequence (A.P) is (5n - 2), find the sum of the first 12 terms of the sequence.
If h(x) = x\(^2\) + px + 2 is divided by (x + 3), the remainder is 5, find p
If 5x + 7 \(\equiv\) P(x + 3) + Q(x - 1), find the value of p
If log\(_2^x\) = 2, evaluate log\(_x^{128}\).
Evaluate: \(\frac{cos^2 300º - 4sin^2 120º}{tan^2 135º}\)
If f(x) = \(\frac{2 - x}{x}\), x ≠ 0, find the inverse of f.
Solve 2\(^{2x}\) - 5(2\(^x\)) + 4 = 0
If p = \(\begin{pmatrix}2 \\ 4 \end{pmatrix}\) and q = \(\begin{pmatrix} 10 \\ -1 \end{pmatrix}\), find a vector, r such that 2p - 3r = q
Given that p = \(\begin{pmatrix} m + 1 & m - 1 \\ m + 4 & m - 8 \end{pmatrix}\) and |p| = - 34, find the value of m.
If r = i + 2j and n = -i + 3j, find |2n - r|.
The gradient of the curve y = mx\(^2\) + 3x - 1 at the point (-1, 1) is 9. Find the value of m
If kx\(^2\) is a term in the binomial expansion of (1 - 2x)\(^4\), find the value of k.
A fair dice is thrown twice. Find the probability that the sum obtained will be a factor of 12.
A body of mass 42 kg increases its speed from 15 ms\(^{-1}\) to 43 ms\(^{-1}\) in 12 seconds. Find the force acting on the body.
Given that M and N are two sets. Which of the following is the same as (M ∩ N) '?
A particle starts from rest accelerates at 4ms\(^{-2}\). Find the distance covered after 4 seconds.
Find the range of values of x for which 9x - 1 > 14x\(^2\)
A particle of mass 40 kg is kept on a smooth plane inclined at an angle of 30º to the horizontal by a force up the plane. find, correct to one decimal place, the magnitude of the normal reaction of the plane of the particle.[Take g = 10 ms\(^{-2}\)]
The point P(-3, 5) lies on a line which is perpendicular to 2x - 4y + 3 = 0. Find the equation of the line.
Find the coefficient of y\(^2\) in the binomial expansion of (y - 2x)\(^5\).
Given that f(x) = x\(^2\) + 3x + 1, find the value of x at the turning point.
How many three-digit numbers can be formed from the digits 2, 3, 4, 5, 6, 7, and 8 if repetition is not allowed?
If \(\begin{pmatrix} 6 & 4 \\ 7 & 5 \end{pmatrix}\) \(\begin{pmatrix} 2 \\ m \end{pmatrix}\) = 2\(\begin{pmatrix} 12 \\ 14.5 \end{pmatrix}\), find the value of m.
A body of mass 80 kg moving with a velocity of 25 ) ms\(^{-1}\) collides with another moving in the opposite direction at 10 ms\(^{-1}\). After collision, both bodies moved with a common velocity of 12.8 ms\(^{-1}\). Calculate, correct to the nearest whole number, the mass of the second body.
In how many ways can 12 people be seated on a bench if only 5 spaces are available?
In triangle XYZ, |XY| = 10cm, |YZ| = 9 cm and |XZ| = 7 cm. If XZY = \(\alpha\), find the value of cos \(\alpha\).
If y\(^2\) + 2xy - 8 = 0, find \(\frac{dy}{dx}\)
The mean of four numbers is 5 and the mean of another three numbers is 12. Find the mean of the seven numbers.
Find, correct to the nearest degree, the acute angle between 3x - y - 5 = 0, and 7x - y - 3 = 0
The gradient of a curve is given by 3x\(^2\) - 8x + 2. If the curve passes through P(0, 4), find the equation of the curve.
Given that y = 2x - 1 and Δx = 0.1, find Δ y
The scores of some students in a class test are 4, 6, 1, 8, 9, 5, and 2. Calculate, correct to one decimal place, the mean deviation of their scores.
The line x + y + 4 = 0 makes an angle of \(\theta\) with the x-axis. Find the value of \(\theta\)
The parents of 7 out of every 10 students in a class are farmers. If 12 students were selected at random, find the probability that the parents of 8 of them will be farmers.
In a truth table, if p is true and q is false, which of the following notations is false
Find the sum of the first 20 terms of the sequences, -7, - 3, 1, . . . . .
If P = {x:1 ≤ x ≤ 6} and Q = {x: 2 < x < 9 },where x ∈ R, find P ∩ Q.
Solve the inequality 2x\(^2\) + 5x - 3 ≥ 0.
simplfy \(\sqrt{(\frac{1}{64}}\))\(^{\frac{-2}{3}}\).
If (x - 3)is a factor of 2x\(^3\) + 3x\(^2\) - 17x - 30, find the remaining factors.
A binary operation * is defined on the set R of real numbers by a*b = \(\frac{\text{ab}}{4}\), find the value of \(\sqrt{2}\) * \(\sqrt{6}\)
Two functions f and g are defined by f:x → 3x - 1, g: x → 2x\(^3\), find fg(- 2).
Given that \(\frac{1}{8^{2-3y}}\) = 2\(^{y + 2}\), find y.
Given that (\(\sqrt{3}\) - 5\(\sqrt{2}\))(\(\sqrt{3}\) + \(\sqrt{2}\)) = p + q\(\sqrt{6}\). Find q
If f(x) = \(\frac{1}{2 - x}\), x \(\neq\) 2. Find f\(^{-1}\)(\(\frac{-1}{2}\))
Find the coefficient of x\(^4\) in the binomial expansion (1 - 2x)\(^6\)
Find the equation of the line passing through (0, -1) and parallel to the y-axis
The roots of the equation 2x\(^2\) + kx + 5 = 0 are α and β, where k is a constant. If α\(^2\) + β\(^2\) = -1, find the values of k
Find the sum of the exponential series 96 + 24 + 6 +......
Evaluate log\(_{0.25}\) 8
Evaluate lim\(_{x→1}\) \(\frac{1-x}{x^2 - 3x + 2}\)
The mean age of n men in a club is 50 years. Two men aged 55 and 63 left the club, and the mean age reduced by 1 year. Find the value of n
A committee of 4 is to be selected from a group of 5 men and 3 women. In how many ways can this be done if the chairman of the committee must be a man?
Simplify \(\frac{^n P_4}{ ^n C_4}\)
Which of the following matrices is a singular matrix?
The area of a sector of a circle is 3cm\(^2\). If the sector subtends an angle of 1.5 radians at the centre. Calculate the radius of the circle
Simplify 8\(^n\) x 2\(^{2n}\) + 4\(^{3n}\)
A particle of mass 2.5kg is moving at a speed of 12 ms\(^{-1}\). If a force of magnitude 10N acts against it, find how long it takes to come to rest.
In a firing contest, the probabilities that Kojo and Kwame hit the target are \(\frac{2}{5}\) and \(\frac{1}{3}\) respectively. What is the probability that none of them will hit the target?
The equation of the line of best fit for variable x and y is y = 19.33 + 0.42x, where x is the independent variable. Find y when x =15
A force of 32 newtons is applied to an object of mass m kg, which is at rest on a smooth horizontal surface. if acceleration produced is 8ms\(^{-2}\), find the value of m
Find the coordinate of the point on the curve y = x\(^2\) + 4x - 2, where the gradient is zero
find the least value of the function f(x)= 3x\(^2\) + 18x + 32
A force of 32N is applied to an object of mass mkg, which is at rest on a smooth horizontal surface. If the acceleration produced is 8ms\(^{-2}\), find the value of m
Given that \(\left| \begin{array}{cc} 2 & -3 \\ 1 & 4 \end{array} \right| \left| \begin{array}{c} 6 \\ p \end{array} \right| = \left| \begin{array}{c} 3 \\ -26 \end{array} \right|\) Find p
Find the coordinates of the centre of the circle 4x\(^2\) + 4y\(^2\) − 5x + 3y −2 = 0
A and B are two independent events such that P(A) = \(\frac{2}{5}\) and P(A∩B) = \(\frac{1}{15}\) find P(B)
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.
Find in surd form, the value of cos165°.
The mean and median of integers x,y, z, and t are 5 and z, respectively. If x< y< z< t and y = 4. Find (x + t)
If a = \(\left| \begin{array}{cc} 3 \\ 2 \end{array} \right|\) and b = \(\left| \begin{array}{cc} -3 \\ 5\end{array} \right|\). Find the vector c such that 4a + 3c = b
A lift moving with a uniform acceleration of 5 ms\(^{-2}\) carries a body of mass p kg. If the reaction on the floor is 480N, find the value of p [take g =10ms\(^{-1}\)]
Calculate correct to one decimal place, the angle between 5i + 12j and -2i + 3J
A particle is projected vertically upward from a height 45 metres above the ground with a velocity of 40m/s. How long does it takes to hit the ground?
Two forces, each of magnitude 16N, are inclined to each other at an angle of 60º
ABCD is a square force of magnitude 14N, 4N, 2N, and 2\(\sqrt{2}\)N act along the sides AB, BC, CD, and DA, respectively. Find in Newton, the magnitude of the resultant of the force.