### 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.

(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 $$x^{4} is divided by x^{2} + 2x − 1$$

.

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. 

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 $$\int_{1}^{3}\frac{2x^{3}-x^{2}+6x+3}{x^{2}+3}dx$$.

(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.

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 $$\int_{0}^{\frac{1}{2}} \frac{8x^{3}+4x^{2}+2x+7}{4x^{2}+1}dx.$$

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.

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

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

(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. 
(ii) When a has this value,
(a) factorise p(x), 

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 $$x>-\frac{1}{2}$$

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