### Question

The polynomial p (x) is defined by

p(x) = 4x3 + 16x2 + 9x -15.

(a)Find the quotient when p(x) is divided by (2x+3), and show that the remainder is −6.

(b) Find $$\int \frac{p(x)}{2x+3}dx$$ .

(c)Factorise p (x) +6 completely and hence solve the equation

p (cosec 2θ) + 6 = 0

for 00 < θ < 1350.

Ans:

(a)Carry out division at least as far as 2x2 + kx

Obtain quotient 2x2 + 5x – 3

Confirm remainder is -6.

(b)Integrate to obtain at least k1 x 3 and  k2 ln (2x 3) + terms

Obtain $$\frac{2}{3}x^{3} + \frac{5}{2}x^{2} – 3x – 3 In (2x+3)$$

(c)State or imply p(x) + 6 = (2x + 3) (2x2 + 5x -3)

Conclude (2x + 3) (2x – 1) (x + 3)

State or imply  sin 2θ = $$-\frac{2}{3}$$ or sin 2θ = $$-\frac{1}{3}$$ or both

Carry out correct process to find θ in at least one case

Obtain 99.7 and 110.9

### Question

Given that $$3e^{x}+8e^{-x}=14$$ find the possible values of $$e^{x}$$ and hence solve the equation $$3e^{x}+8e^{-x}=14$$ correct to 3 significant figures.

Rearrange to $$3e^{2x}-14e^{x}+8=0$$ or equivalent involving substitution Solve quadratic equation in$$e^{x}$$ to find two values of $$e^{x}$$ Obtain$$\frac{2}{3} and 4$$

Use natural logarithms to solve equation of form$$e^{x}$$ =k where k > 0 dep on Allow M mark if left in exact form

Obtain −0.405

Obtain 1.39

### Question

Show that ln $$\left ( x^{3} -4x\right )In\left ( x+2 \right )$$Use logarithm subtraction property to produce logarithm of quotient M1

Factorise at least as far as  $$x\left ( x^{2} -4\right )$$ and  $$x\left ( x-2 \right )$$ or use correct algebraic long division to obtain a quotient of x+2 and a remainder5 of 0 from correct working .
Obtain final answer $$\ln \left ( x+2 \right )$$ using correct process.