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.
Answer/Explanation
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.
Answer/Explanation
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
Answer/Explanation
<p
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.