**Qurstion**

**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 2x^{2} 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 2x ^{3} − x^{2} + 6x + 3 is divided by x^{2} + 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**

**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 8x ^{3} + 4x^{2} + 2x + 7 is divided by 4x^{2} + 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 x^{2}+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 – x^{2}

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 4*x*^{3} + *ax* + 2, where *a* is a constant, is denoted by p(*x)*. It is given that( 2*x*+ 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 (2*x +* 1) and equate constant remainder to zero

Obtain *a* = 3

**(ii) (a)** Commence division by (2*x* + 1) reaching a partial quotient of 2*x*^{2} + *kx*

Obtain factorisation (2*x* + 1)(2*x ^{2}* –

*x*+ 2)

[The M1 is earned if inspection reaches an unknown factor 2*x ^{2}* +

*Bx*+

*C*and an equation in

*B*and/or

*C*, or an unknown factor

*Ax*+

^{2}*Bx*+ 2and an equation in A and/or B.]

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

Show that 2*x ^{2}* –

*x*+ 2 is always positive, or that the gradient of 4

*x*+ 3

^{3}*x*+ 2 is always positive

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