The document contains 7 multi-part math problems involving polynomials, quadratics, differentiation, integration, trigonometry, and calculus concepts like finding derivatives, integrals, stationary/critical points, and sketching graphs. The problems cover factorizing polynomials, solving equations, finding rates of change, evaluating integrals, solving trigonometric equations, and applying calculus techniques like differentiation and integration to functions.

1. Higher Formal Homework - Polynomials
1. Prove that x  2 is a factor of x x x3 2
10 8   and hence find
the other factors. (4)
2. Solve y = 4 8 33 2
x x x   (5)
3. Find the value of k so that x x x k3 2
5 4   is exactly divisible
by x  2. (3)
4. For what values of a and b are x  2 and 2 1x  both factors
of 4 3
x ax b  ? (5)
5. Find an expression for f(x) from the following graph :
(5)

2. Higher Formal Homework - Quadratics
1. Write each of the following quadratic expressions in the form
a x b c( ) 2
: (6)
222
84)(462)(36)( xxcxxbxxa 
2. Show that the function f x x x( )   2 16 72
can be written in
the form f x a x p q( ) ( )  2
and write down the values of a ,
p and q.
Hence state the minimum value of the function and the
corresponding value of x . (4)
3.(a) Find the value of k which results in the equation
k x kx2
2 1 0   having equal roots, given that k  0 ?
(b) A quadratic equation is given as x p x p2 1
43 3 0    ( ) ( ) .
For what values of p will the above equation have
i) equal roots ;
ii) no real roots.
(c) Show that the roots of the equation ( )t x tx   1 2 4 02
are
real for all values of t .

3. Higher Formal Homework – Differentiation 1
1. Differentiate each of the following with respect to x :
(a) 3 62
x x (b) x x4 1
2 4 
(c) ( )2 3 2
x  (5)
2. Differentiate each of the following functions with respect to
the relevant variable :
(a) f x x x x( ) ( ) 3 2
(b) h x
x x
( )  
1 1
3 2
(c) f t t t t( ) ( ) 
1
2
3
22
(d) h u u
u
( )  
1
2
(e) f v
v
v
( ) 
5 2
(14)
3. Calculate the rate of change of :
(a) f x x x( )  3 2
2 at x  2 (b) h t t t( ) ( )( )  1 2 3 at t  1
(7)
Higher Formal Homework – Differentiation 1
1. Differentiate each of the following with respect to x :
(a) 3 62
x x (b) x x4 1
2 4 
(c) ( )2 3 2
x  (5)
2. Differentiate each of the following functions with respect to
the relevant variable :
(a) f x x x x( ) ( ) 3 2
(b) h x
x x
( )  
1 1
3 2
(c) f t t t t( ) ( ) 
1
2
3
22
(d) h u u
u
( )  
1
2
(e) f v
v
v
( ) 
5 2
(14)
3. Calculate the rate of change of :
(a) f x x x( )  3 2
2 at x  2 (b) h t t t( ) ( )( )  1 2 3 at t  1
(7)

4. Higher Formal Homework – Differentiation 2
1. Find the equation of the tangent to the curve with equation
y x x 3 22
at the point where x  1 . (5)
2. A function f is given by f x x x( )  3 22 3
. Determine the
interval on which f is increasing. (5)
3. Find the stationary points of the curve y x x x  2 2
12 4 3( ) and
determine their nature. (7)
4. A curve has as its equation y x x ( )2 2
.
(a) Find the stationary points of the curve and determine the
nature of each.
(b) Write down the coordinates of the points where the
curve meets the coordinate axes.
(c) Sketch the curve. (10)
5. For each graph below y f x ( ) draw a sketch of the graph of
the derivative, y f x ( ) , of each function.
(a) (b) (5)
-4
-2
x
y
2
3
2-3 x
y

5. Higher Formal Homework - Integration
1. Find: (a)  (3x – x2
)2
dx (b)  dx
x
x
2
)1( 
2. Evaluate each of the following definite integrals:
(a)
x x
x
dx
2
1
4
2
 (b)  1
3 3
1
1
1
3
v
v v dv 

3. For the curve y f x x
dy
dx x
   ( ) ,where 0 6
24
2
If the minimum turning point of the function is (-2, 20), find
f (x).
4. In the diagram opposite the curve has
as its equation 2
10 xxy  and the line
.4xy 
Calculate the shaded area in square units.
y
x

6. Higher Formal Homework - Trigonometry
1. Find the exact value of :
(a) sin 315º (b) tan
7𝜋
6
(c) cos (-150)º
2. Solve each of the following equations :
(a) 
1800for,12tan3  xx
(b)  20for,1sin4 2

(c) 
3600for,012sin2  xx
(d) 20for,033cos2  xx
3. Write down an equation which describes the relationship
between x and y in each graph below :
4. The sketch represents part of the graph of a trigonometric
function of the form y = psin(x + r)° + q. It crosses the axes at
(0, s) and (t, 0) and has turning points at (50, -2) and (u, 4).
(a) Write down values of p, q, r -and u.
(b) Find the values for s and t.
x x
x
y y y(a) (b) (c)
4
-4
o
20
-20
6
2
-2
o
o
360
o
360
o
180o

7. x
fxexd
sin
1
)()23(cos)(3sin2)( 
Higher Formal Homework – Further Calculus
1. Differentiate, expressing your answers with positive indices :
)7(
2
)()4()()23()(
2
24 2
3


x
cxbxa
2. A curve has as its equation y
x


5
4 1
, where x   1
4
.
Find the equation of the tangent to this curve at the point
where x  1 .
3. Integrate :
(𝑎) ∫(6𝑥 + 1)11
𝑑𝑥 (𝑏) ∫
5
(3𝑥+1)4
1
3
0
𝑑𝑥 (𝑐) ∫ 3 cos (
𝜋
2
− 2𝑥) 𝑑𝑥
𝜋
2
𝜋
4
4. In manufacturing, the cost per item is a variable dependent on
how many items are made.
This is called the marginal cost.
If C(x) represents the cost in £s of producing x items, then
𝑑𝐶
𝑑𝑥
represents the marginal cost per item after x items have been
made.
A particular cottage industry makes luxury booklets.
They model their marginal cost by
𝑑𝐶
𝑑𝑥
=
196
(2𝑥+8)
2
3
.
(a) Find an expression for C(x), given that the initial outlay
was £600.
(b) What is the cost of making 1000 booklets to the nearest
£.

8. Higher Formal Homework – Mixed Unit 1
1. Find the value(s) of t if the equation 2
16
 t
x
x has
equal roots ?
2. Evaluate dxx
x

1
1 2
1
3. A curve has as its equation
4
1


x
y .
Find the equation of the tangent to this curve at the point
where 3x .
4. Given that f x x x( ) sin cos 4 2 2
evaluate f ( )
4
.
5. The diagram below shows a sketch of the graph of y f x ( )
where f x x x x( )    3 2
2 5 6 .
(a) Show that x 1 is a factor of this function and hence
fully factorise f x( ) .
(b) Calculate the shaded area.
x
y
o 1

9. 6. A function is defined as f x x x( )   3
3 2 , for x R .
(a) Show that x  2 is a root of the equation x x3
3 2 0  
and hence find the other real root.
(b) Find the coordinates of the stationary points of the
graph y f x ( ) and determine their natures.
(c) Sketch the graph of y f x ( ) marking clearly all relevant
points.
7. The sketch shows y = 2sin2
2x and y = 3sin2x – 1 between 0 and
𝜋.
(a) Where do the two curves intersect at this interval?
(b) Where do they intersect between π and 2π?