Qurstion

(i)It is given that 2 ln(4x − 5 )+ ln(x + 1 )3 ln 3.(i) Show that \(16x^{3}-24x^{2}-15x-2=0\)
(ii) By first using the factor theorem, factorise \(16x^{3}-24x^{2}-15x-2=0\) completely. 

(iii) Hence solve the equation 2 ln (4x − 5 )+ ln(x + 1) = 3 ln 3.

Answer/Explanation

(i) Use law for the logarithm for a product or quotient or exponentiation AND for a power

Obtain\( (4x-5)^{2}(x+1)=27\)

Obtain given equation correctly \(16x^{3}-24x^{2}-15x-2=0\)

(ii) Obtain x = 2 is root or (x – 2) is a factor, or likewise with\( x=-\frac{1}{4}\)

Divide by (x – 2) to reach a quotient of the form\((16x^{3}+kx)\)

Obtain quotient \(16x^{3}+8x+1\)

(iii) State x = 2 only

Question

Find the quotient and remainder when \(x^{4}  is divided  by x^{2} + 2x − 1\)

Answer/Explanation

.

Commence division and reach a partial quotient  \(x^{2} + kx \)

Obtain quotient\( x^{2}-2x+5\)
Obtain remainder −12 x + 5

Question

Find the quotient and remainder when 2x2 is divided by x + 2. [3]

Answer/Explanation

Ans:

Carry out division or equivalent at least as far as two terms of quotient 
   Obtain quotient 2x-4
   Obtain remainder 8 

Question

(a) Find the quotient and remainder when n 2x3 − x2 + 6x + 3 is divided by x2 + 3.       

(b) Using your answer to part (a), find the exact value of \(\int_{1}^{3}\frac{2x^{3}-x^{2}+6x+3}{x^{2}+3}dx\). 

Answer/Explanation

(a) Commence division and reach quotient of the form 2x + k

          Obtain quotient 2x – 1 

          Obtain remainder 6 

(b) Obtain terms x2 – x
         (FT on quotient of the form 2x + k)

         Obtain term of the form \(a \tan ^{-1}\left ( \frac{x}{\sqrt{3}} \right )\)

        Obtain term \(\frac{6}{\sqrt{3}}\tan ^{-1}\left ( \frac{x}{\sqrt{3}} \right )\)

        (FT on a constant remainder) 

        Use x = 1 and x = 3 as limits in a solution containing a term of the form \(a\tan ^{-1}(bx)\)

        Obtain final answer \(\frac{1}{\sqrt{3}}\pi +6,\) or exact equivalent

Question

The polynomial \(ax^3+5x^2-4x+b\), where a and b are constants, is denoted by p(x). It is given that (x+2) is a factor of p(x) and that when p(x) is divided by (x+1) the remainder is 2.
Find the values of and b.

Answer/Explanation

Ans:

Substitute x = -2, equate result to zero and obtain a correct equation,
e.g. -8a+20+8+b=0
Substitute x = -1 and equate result to 2
Obtain a correct equation, e.g. -a+5+4+b=2
Solve for a or for b
Obtain a = 3 and b = -4

Question

(a)Find the quotient and remainder when 8x3 + 4x2 + 2x + 7 is divided by 4x2 + 1.

(b) Hence find the exact value of \( \int_{0}^{\frac{1}{2}} \frac{8x^{3}+4x^{2}+2x+7}{4x^{2}+1}dx.\) 

Answer/Explanation

Ans:

(a)Commence division and reach quotient of the form 2x ± 1

Obtain (quotient) 2x + 1

Obtain (remainder) 6

(b)

Obtain terms x2+x

Obtain term of the form a tan-1 2x

Obtain term 3tan-1 2x

Use x = 0 and x = \(\frac{1}{2}\) as limits in a solution containing a term of the form a tan-1 2x

Obtain final answer \(\frac{3}{4}\left ( 1+\pi \right )\) , or exact equivalent 

Question

(a) By sketching a suitable pair of graphs, show that the equation 4 − x 2 = sec \(\frac{1}{2}x\) has exactly one root in the interval 0 ≤ x < π.

(b)Verify by calculation that this root lies between 1 and 2.

(c)Use the iterative formula \(x_{n+1} = \sqrt{4-sec\frac{1}{2}x_{n}}\) to determine the root correct to 2 decimal places.
Give the result of each iteration to 4 decimal places.

Answer/Explanation

Ans:

(a)Sketch a relevant graph, e.g.  y = 4 – x2

Sketch a second relevant graph, e.g. y = sec \(\frac{1}{2}x\) , and justify the give statement

(b)Calculate the value of a relevant expression or values of a pair of relevant expressions at x = 1 and x = 2

Complete the argument with correct calculated values

(c)

Use the iterative process correctly at least twice

Obtain final answer 1.60

Show sufficient iterations to 4 d.p.to justify 1.60 to 2 d.p. or show there is a sign change in the interval (1.595, 1.605)

f

Question

The polynomial p(x) is defined by\(p(x)=x^{3}-3ax+4a\),where a is a constant.
(i) Given that (x − 2) is a factor of p(x), find the value of a. 
(ii) When a has this value,
(a) factorise p(x)completely, 
(b) find all the roots of the equation\( p(x^{2})\)=0

Answer/Explanation

(i) Substitute x = 2 and equate to zero, or divide by x – 2 and equate constant remainder to zero, or equivalent 
Obtain a = 4

(ii) (a) Find further (quadratic or linear) factor by division, inspection or factor theorem or equivalent

Obtain x^{2}+ 2x – 8 or x + 4 

State (x – 2)^{2}

(x + 4) or equivalent

(b) State any two of the four (or six) roots 
State all roots \((\pm \sqrt{2},\pm 2i) \)provided two are purely imaginary

Question

The polynomial 4x3 + ax + 2, where a is a constant, is denoted by p(x). It is given that( 2x+ 1) is a factor of p(x).

   (i) Find the value of a. [2]
   (ii) When a has this value,
         (a) factorise p(x), [2]
         (b) solve the inequality p(x) > 0, justifying your answer. [3]

Answer/Explanation

Ans:

4 (i) Substitute \(x=-\frac{1}{2}\) and equate to zero, or divide by (2x + 1) and equate constant remainder to zero

          Obtain a = 3 

    (ii) (a) Commence division by (2x + 1) reaching a partial quotient of 2x2 + kx

                   Obtain factorisation (2x + 1)(2x2x + 2)

                   [The M1 is earned if inspection reaches an unknown factor 2x2 + Bx + C and an equation in B and/or C, or an unknown factor Ax2 + Bx + 2and an equation in A and/or B.]

            (b) State or imply critical value \(x=-\frac{1}{2}\)

                   Show that 2x2x + 2 is always positive, or that the gradient of 4x3 + 3x + 2 is always positive 

                   Justify final answer \(x>-\frac{1}{2}\)