Part -1:

Question 1 

A conic with equation

ax² + by² = c

has two focus points, where a = 9, b = 27 and c = 51. One of those focus points has a positive x-coordinate. To two decimal places, what is the value of that positive x-coordinate?

Question 2 

What is the value of the Fibonacci number F_n, where n = 35?

Question 3

A conic has equation

ax² + by² = c

where a = 6, b = -17 and c = 33.

To two decimal places what is the gradient of the asymptote that has a positive gradient?

Question 4

Suppose a second-order difference equation is of the form ay_n+2 + by_n+1 + Cy_n = d

where a = 2, b = 2, c = 13 and d = 14

In radians and to two decimal places, what is the value of

θ

in the complementary function?

Question 5

A conic M, in standard or reflected standard form, is rotated through an angle r about the origin to obtain the conic N with equation ax2 bxy

ax² + bxy² + cy² = d

cy² = d

where a= 12, b = 4, c = 49 and d = 35.

To two decimal places, what is the absolute value of the angle r in degrees? As usual, do not give any units in your answer. Do not include a minus sign in your answer.

Question 6

To two decimal places, what is the value of the eccentricity of the conic with equation

ax² + by² = c

where a = 6, b= 30 and c = 36?

Question 7

A sequence Ln is such that L₁ = 0.2, L₂ = al and

L_n = 2L_n-1 - 1.5L_n-2 (n≥ 3).

Use Mathcad to determine the value of L₄₃, giving your answer to 2 decimal places.

Question 8

If you rationalise the denominator in the expression

1/a√(b-c)

where a = 72, b = 150 and c = 22, and you do not simplify your answer (i.a do not divide top and bottom by some number), then what is the absolute value of the new denominator? There should not be a negative sign in your answer.

Question 9

To two decimal places, what is the value of the eccentricity of the conic with equation

ax² + y² = c

where a = 10 b =-13 and c = 63?

Question 10

Suppose a second-order difference equation is of the form

y_n+2 -by_n, = dn²

where b = 130 and d = 17.

To two decimal places, what is the value of the positive root in the auxiliary equation?

Part -2:

Question 1

The composite transformation q_a O q_b can be expressed as the single transformation r_c, where a = 0.1, b = 1.9 and

0≤ c ≤ 2Π

To 2 decimal places, what is the value of c?

Question 2

f(x) = x² + bx + c, where b = -28 and c = 36. What is the coefficient of x² in the function f(f(x)).?

Question 3

f(x) = ax² + bx + c, where a = -25, b = 12 and c = 14. What is the gradient of the point, with x coordinate 19, on the graph of y = f(x)?

Question 4

To 2 decimal places, what is the coefficient of x⁸ in the expansion of

(ax³ + b/x)⁸

where a = 3.2 and b = 1.5?

Question 5

The composite transformation q_{a O} r_b can be expressed as the single transformation q_c where a = 0.8. b = 5.1 and 0 ≤ c < Π

To 2 decimal places, what is the value of c?

Question 6

f and g are linear transformations. The matrices for f and g are respectively

where a = -8, b = 4, c = 5, d = -8, m - 7, n = -7, p = -2 and q = 9.

What is the lower-left entry in the matrix that represents the composite transformation g o f?

Question 7

What is the area of the triangle with vertices at (a, h), (c, d) and (m, n) if

a = 7,b= -8, c = -13, d = 3, m = 17, and n = 11?

As usual, your answer should be a number only (with no units).

Question 8

f(x) = ax² + bx + c, where a = 6, h = -25 and c = -15. There are two fixed points at x₁ and x₂, where x₁ > x₂. To two decimal places what is the value of x₁?

Question 9

f(x) = ax³+ (b - ad)x² + (c - bd)x - cd, where a = 27, b = -11, c = 25 and d = 12.

What value of x satisfies the equation f(x) = 0?

Question 10

The composite of three translations given by

t_a, O t_c,d O t_h,k

is equivalent to the single translation t_m,n.

What is the value of m if a = -19, b = 12, c = 16, d = -10, h = -6 arid k = -4 ?

Part 3

Question 1

Two sequences y_t and z_t satisfy

y_t= ay_t-1 + bz_t-1

z_t = cy_t-1 + dz_t-1

where a = 6, b = -20, c = -17 and d = -12. From the two given equations above, a second-order difference equation of the following form can be derived: Y_t+1 + ny_t + ry_t-1 = 0

where n and r are constants. The auxiliary equation

m² + nm + r = 0

has two solutions m = h and m = k where h > k. To 2 decimal places, what is the value of h?

Question 2

D is the matrix given by

where h = 36 and k = -4. What is the lower-light entry in the matrix Dⁿ, where n = 6?

Question 3

f(x) = ax² + bx + c, where a = -19, b= 22 and c = 12.

If

-1 < f'(x) < 1

then h < x a k. To 2 decimal places, what is the value of k?

Question 4

The matrix A is given by

where a = 6, b = -21, c = -6 and d = -12. The matrix A has two eigenvalues h and k, where h > k. To 2 decimal places, what is the value of h?

Question 5

where a = -6, b = 8, C = 6 and d = -6.

What is the top-left entry in the matrix A⁴?

Question 6

A conic has equation

ay² + bx = 0

where a = 13 and 13 = -296. If the focus point is at (F, 0) then what is the value of F to 2 decimal places?

Question 7

The matrix A is given by

where a = 2B, b = 13, c = 12 and d = 20.

Iteration sequence: x_{n + 1} = Ax_n (n = 0, 1, 2, ...), where x₀ is the following vector:

In the long run (i.e. as n tends to infinity),

y_n/x_n → L

To 2 decimal places, what is the value of the constant L?

Question 8

An ellipse has equation ax² + by² = c

where a = 19, b= 39 and c = 269. One of the vertices; (0, d), lies on the positive y-axis. To 2 decimal places, what is the value of d?

Question 9 

To 2 decimal places, what is the value of the lower-right entry in the reflection matrix Q_a, if a = 1.18?

Question 10

The matrix A is given by

where a = 3, b = c = -5 and d = -18. The matrix A has two eigenvalues h and Is. where h > k. To 2 decimal places, what is the gradient of the eigenline that corresponds to eigenvalue h?