IB Mathematics AA Higher Level

1. [Maximum mark: 4] Find the solution of log2 x - log2 5 = 2 + log2 3 .

1. [Maximum mark: 5] Find the term independent of x in the binomial expansion of 2 1 2 2 3 10 x x +     .

1. [Maximum mark: 5] Solve the equation log2(x + 3) + log2(x - 3) = 4 .

2. [Maximum mark: 6] Consider the complex numbers z1 1 3 = + i , z2 = 1 + i and w z z = 1 2 . (a) By expressing z1 and z2 in modulus-argument form write down (i) the modulus of w ; (ii) the argument of w . [4] (b) Find the smallest positive integer value of n , such that wn is a real number. [2]

2. [Maximum mark: 6] The function f is defined by f (x) = 2x3 + 5 , -2 ≤ x ≤ 2 . (a) Write down the range of f . [2] (b) Find an expression for f -1(x) . [2] (c) Write down the domain and range of f -1 . [2]

2. [Maximum mark: 6] The points A and B are given by A(0 , 3 , -6) and B(6 , -5 , 11) . The plane Π is defined by the equation 4x - 3y + 2z = 20 . (a) Find a vector equation of the line L passing through the points A and B . [3] (b) Find the coordinates of the point of intersection of the line L with the plane Π . [3]

3. [Maximum mark: 5] Solve the equation sec2 x + 2 tan x = 0 , 0 ≤ x ≤ 2π .

3. [Maximum mark: 5] The 1st, 4th and 8th terms of an arithmetic sequence, with common difference d , d ≠ 0 , are the first three terms of a geometric sequence, with common ratio r . Given that the 1st term of both sequences is 9 find (a) the value of d ; [4] (b) the value of r . [1]

3. [Maximum mark: 6] Consider the polynomial q(x) = 3x3 - 11x2 + kx + 8 . (a) Given that q(x) has a factor (x - 4) , find the value of k . [3] (b) Hence or otherwise, factorize q(x) as a product of linear factors. [3]

4. [Maximum mark: 5] Three girls and four boys are seated randomly on a straight bench. Find the probability that the girls sit together and the boys sit together.

4. [Maximum mark: 7] A particle moves along a straight line. Its displacement, s metres, at time t seconds is given by s = t + cos 2t , t ≥ 0 . The first two times when the particle is at rest are denoted by t1 and t2 , where t1 < t2 . (a) Find t1 and t2 . [5] (b) Find the displacement of the particle when t = t1 . [2]

4. [Maximum mark: 4] Find the coefficient of x8 in the expansion of x x 2 7 2 −    .

5. [Maximum mark: 7] ABCD is a parallelogram, where AB → = − + + i j k 2 3 and AD → = − − 4 2 i j k . (a) Find the area of the parallelogram ABCD. [3] (b) By using a suitable scalar product of two vectors, determine whether AB C is acute or obtuse. [4]

5. [Maximum mark: 4] In the following Argand diagram the point A represents the complex number -1 + 4i and the point B represents the complex number -3 + 0i . The shape of ABCD is a square. Determine the complex numbers represented by the points C and D. y x A D C B

5. [Maximum mark: 5] A particle moves in a straight line such that at time t seconds (t ≥ 0) , its velocity v , in ms-1 , is given by v = 10te-2t . Find the exact distance travelled by the particle in the first half-second.

6. [Maximum mark: 5] Consider the graphs of y = | x | and y = - | x | + b , where b ∈ + . (a) Sketch the graphs on the same set of axes. [2] (b) Given that the graphs enclose a region of area 18 square units, find the value of b . [3]

6. [Maximum mark: 7] (a) Using the substitution x = tan θ show that 1 1 2 2 0 1 2 0 4 x x + ( ) = ∫ ∫ d d À cos θ θ π . [4] (b) Hence find the value of 1 1 2 2 0 1 x x + ( ) ∫ d . [3]

6. [Maximum mark: 9] (a) Sketch the graph of y x x = − − 1 3 2 , showing clearly any asymptotes and stating the coordinates of any points of intersection with the axes. –5 5 10 –10 –5 0 5 10 y x [4] (b) Hence or otherwise, solve the inequality 1 3 2 2 − − < x x . [5]

7. [Maximum mark: 7] An arithmetic sequence u1 , u2 , u3… has u1 = 1 and common difference d ≠ 0 . Given that u2 , u3 and u6 are the first three terms of a geometric sequence (a) find the value of d . [4] Given that uN = -15 (b) determine the value of 1 N r r u =∑ . [3]

7. [Maximum mark: 7] (a) The random variable X has the Poisson distribution Po(m) . Given that P(X > 0) = 3 4 , find the value of m in the form ln a where a is an integer. [3] (b) The random variable Y has the Poisson distribution Po(2m) . Find P(Y > 1) in the form b c c - ln where b and c are integers. [4]