Chapter 2 M 1

Practice Questions
Chapter No.: 2
Chapter Title: Basics
Class: BSc
Subject: Mathematics-1
Course Code: 05a
Compiled By: Sir Waseem Mustafa
Cell No.: 0334 8111881
E-Mail: waseem.mustafa.ibd@gmail.com
Page |1
Important Points:
2.1) Introduction: This chapter discusses some of the very basic aspects of the subject,
aspects on which the rest of the subject builds. It is essential to have a firm understanding
of these topics before the more advanced topics can be understood.
2.2) Powers: When n is a positive integer, the n’th power of the number a, 𝒂𝒏 , is simply the
product of n copies of a, that is,
𝒂𝒏 = 𝒂 × 𝒂 × 𝒂 × … × 𝒂 (n times)
The number n is called the power, exponent or index. We have the power rules (or rules
of exponents) as follows:
Law Example
x1 = x 61 = 6
x0 = 1 70 = 1
x-1 = 1/x 4-1 = 1/4
xmxn = xm+n x2x3 = x2+3 = x5
xm/xn = xm-n x6/x2 = x6-2 = x4
(xm)n = xmn (x2)3 = x2×3 = x6
(xy)n = xnyn (xy)3 = x3y3
(x/y)n = xn/yn (x/y)2 = x2 / y2
x-n = 1/xn x-3 = 1/x3
And the law about Fractional Exponents:
Compiled by: Sir Waseem Mustafa (0334 8111881)
2.3) Simple Algebra: Algebra is the branch of mathematics that uses letters in place of some
unknown numbers. In this course we have to learn collecting up terms, multiplication of
variables, and expansion of bracketed terms.
There are three important algebraic identities which you should know, these are as
follows:
i) (𝒂 + 𝒃)𝟐 = 𝒂𝟐 + 𝟐𝒂𝒃 + 𝒃𝟐
ii) (𝒂 − 𝒃)𝟐 = 𝒂𝟐 − 𝟐𝒂𝒃 + 𝒃𝟐
iii) 𝒂𝟐 − 𝒃𝟐 = (𝒂 + 𝒃)(𝒂 − 𝒃)
Page |2
2.4) Solution of Equations: What is a Solution of Equation or Equations?

A Solution is a value you can put in place of a variable (such as x) that would make the
equation true. For Example, in
x-2=4
If we put 6 in place of x we get: 6 - 2 = 4, which is true
So x = 6 is a solution
Note: try another value for x. Say x = 5: you get 5 – 2 = 4 which is not true, so x = 5 is
not a solution. The equation x - 2 = 4 is called single linear equation.
2.5) Quadratic Equations: A common problem is to find the set of solutions of a quadratic
equation
𝒂𝒙𝟐 + 𝒃𝒙 + 𝒄 = 𝟎
where we may as well assume that 𝒂 ≠ 𝟎, because if 𝒂 = 𝟎 the equation reduces to a
linear one. (Note that, by a solution, we mean a value of x for which the equation is true.)
In some cases, the quadratic expression can be factorised, which means that it can be
written as the product of two linear terms (of the form (x – a) for some a).
For example 𝒙𝟐 − 𝟔𝒙 + 𝟓 = (𝒙 − 𝟏)(𝒙 − 𝟓), so the equation 𝒙𝟐 − 𝟔𝒙 + 𝟓 becomes
(𝒙 − 𝟏)(𝒙 − 𝟓) = 𝟎.
Now the only way that two numbers can multiply to give 0 is if at least one of the numbers
is 0, so we can conclude that x - 1 = 0 or x - 5 = 0; that is, the equation has two solutions,
1 and 5. Although factorisation may be difficult, there is a general technique for
determining the solutions to a quadratic equation, as follows.
Suppose we have the quadratic equation 𝒂𝒙𝟐 + 𝒃𝒙 + 𝒄 = 𝟎, where 𝒂 ≠ 𝟎. Then we can
also use the following formula which is called Quadratic Formula, which is as follows:
−𝒃 ± √𝒃𝟐 − 𝟒𝒂𝒄
𝒙=
𝟐𝒂
Where; a = Coefficient of 𝒙𝟐 ,
b = Coefficient of 𝒙, and
c = Constant Term
Note: We can apply this formula even if factorization is possible
2.6) Nature of Roots of Quadratic Equations (The Discriminant): The discriminant is

defined as ∆= 𝒃𝟐 − 𝟒𝒂𝒄,
This is the expression under the square root in the quadratic formula. The discriminant
determines the nature of the roots of a quadratic equation. The word ‘nature’ refers to the
types of number of the roots which can be described as follows:
 if ∆< 𝟎, the equation has no real solutions;
−𝒃
 if ∆= 𝟎, the equation has exactly one solution, 𝒙 = ;
𝟐𝒂
 if ∆> 𝟎, the equation has two solutions,
Page |3
−𝒃−√𝒃𝟐 −𝟒𝒂𝒄 −𝒃+√𝒃𝟐 −𝟒𝒂𝒄

𝒙𝟏 = and 𝒙𝟐 =
𝟐𝒂 𝟐𝒂
2.7) Simultaneous Equations: Simultaneous Equations have the following types:
a) Two linear equations (we can solve these equations by elimination method or
substitution method)
b) One Linear and one non-linear equation (we can solve these equations by
substitution method)
c) Two non-linear equations (we can solve these equations by substitution method)
2.8) Exponentials: An exponential-type function is one of the form 𝒇(𝒙) = 𝒂𝒙 for some
number a.
(Do not confuse it with the function which raises a number to the power a. An
exponential-type function has the form 𝒇(𝒙) = 𝒂𝒙 , whereas the a’th power function has
the form 𝒇(𝒙) = 𝒙𝒂 ).
If a > 1 then 𝒂𝒙 becomes larger and larger, without bound, as x increases. We say that 𝒂𝒙
tends to infinity as x tends to infinity. Also, for such an a, as x becomes more and more
negative, the function 𝒂𝒙 gets closer and closer to 0.
In other words, 𝒂𝒙 tends to 0 as x tends to `minus infinity' and if a < 1 the behaviour is
completely opposite.
We now define the exponential function. This is the most important exponential-type
function. It is defined to be 𝒇(𝒙) = 𝒆𝒙 , where 𝒆 is the special number 2.71828…
(The function 𝒆𝒙 is also sometimes written as exp(x).) The most important facts about 𝒆𝒙
to remember from this section are the shape of its graph, and its properties.
2.9) The Natural Logarithm: Formally, the natural logarithm of a positive number x,
denoted ln x (or, sometimes, log x), is the number y such that 𝒆𝒚 = 𝒙. In other words, the
natural logarithm function is the inverse of the exponential function 𝒆𝒙 (regarded as a
function from the set of all real numbers to the set of positive real numbers). Sometimes
ln x is called the logarithm to base 𝒆.
The reason for this is that we can, more generally, consider the inverse of the exponential-
type function 𝒂𝒙 . This inverse function is called the logarithm to base 𝒂 and we use the
notation 𝐥𝐨𝐠 𝒂 𝒙. Thus, 𝐥𝐨𝐠 𝒂 𝒙 is the answer to the question `What is the number y such
that 𝒂𝒚 = 𝒙?'.
The two most common logarithms, other than the natural logarithm, are logarithms to
base 2 and 10. For example, since 𝟐𝟑 = 𝟖, we have 𝐥𝐨𝐠 𝟐 𝟖 = 𝟑. It may seem awkward
to have to think of a logarithm as the inverse of an exponential-type function, but it is
really not that strange. Confronted with the question `What is 𝐥𝐨𝐠 𝒂 𝒙?', we simply turn it
around so that it becomes, as above, `What is the number y such that 𝒂𝒚 = 𝒙?'.
There is often some confusion caused by the notations used for logarithms. Some texts
use log to mean natural logarithm, whereas others use it to mean log10. In this guide, ln
will be used to mean natural logarithm.
Page |4
2.10) Properties of Exponentials and the Natural Logarithm:

a) ln (xy) = ln x + lny
b) ln (x/y) = ln x – lny
c) ln e = 1
d) ln 1 = 0
e) ln x r = r ln x
f) ln ex = x
g) 𝑒 ln 𝑥 = 𝑥
h) ln x = Positive No.; If 𝑥 > 1
i) ln x = Negative No.; If 0 < 𝑥 < 1
j) ln (Negative No.) = Undefined
2.11) Trigonometric Functions: The trigonometric functions, sin x; cos x; tan x are very
important in mathematics and they will occur later in this subject (in differentiation &
integration).
It is important to realise that, throughout this subject, angles are measured in Radians
rather than Degrees. The conversion between Degree and Radian Measure is
π Radian = 180 º, where 𝝅 is the number 3.141…. It is good practice not to expand 𝝅 or
multiples of 𝝅 as decimals, but to leave them in terms of the 𝝅. For example, since 60
𝝅
degrees is one third of 180 degrees, it follows that, in radians, 60 degrees is .
𝟑
2.12) Important values of the Trigonometric Functions (Ratios):
θ 0º 30 º 45 º 60 º 90 º
ratios (0 Rad.) (𝝅⁄𝟔 Rad.) (𝝅⁄𝟒 Rad.) (𝝅⁄𝟑 Rad.) (𝝅⁄𝟐 Rad.)
𝟏⁄ 𝟏⁄ √𝟑⁄ 1
Sin 0
𝟐 √𝟐 𝟐
1 √𝟑⁄ 𝟏⁄ 𝟏⁄ 0
Cos
𝟐 √𝟐 𝟐
𝟏⁄ 1 √𝟑 undefined
tan 0
√𝟑
2.13) Important Identities of Trigonometric Ratios:

sin 𝑥
(a) tan 𝑥 ≡ (b) Cos2 x + Sin2 x ≡ 1
cos 𝑥
2.14) Supply and Demand Functions: Supply and demand functions describe the relationship
between the price of a good, the quantity supplied to the market by the manufacturer, and
the amount the consumers wish to buy.
(Topic continues on next page)

Page |5
The demand function 𝒒𝑫 of the price p describes the demand quantity: 𝒒𝑫 (𝒑) is the
quantity which would be sold if the price were p.
Similarly, the supply function 𝒒𝑺 is such that 𝒒𝑺 (𝒑) is the amount supplied when the
market price is p.
In this course, the supply and demand functions will be given and we have to find
equilibrium price and quantity for which we assume 𝒒𝑫 (𝒑) = 𝒒𝑺 (𝒑) = 𝒒.and solve both
equations simultaneously and in some questions we also have to sketch their graphs,
which we’ll discuss in the following topic.
2.15) Further Applications of Functions: Suppose that the demand equation for a good is of
the form p = ax + b where x is the quantity produced. Then, at equilibrium, the quantity
x is the amount supplied and sold, and hence the total revenue TR at equilibrium is price
times quantity, which is
𝑻𝑹 = (𝒂𝒙 + 𝒃)𝒙 = 𝒂𝒙𝟐 + 𝒃𝒙;
a quadratic function which may be maximised either by completing the square, or by
using the techniques of calculus (discussed later).
Another very important function in applications is the total cost function of a firm. In the
simplest model of a cost function, a firm has a fixed cost, that remain fixed independent
of production or sales, and it has variable costs which, for the sake of simplicity, we will
assume for the moment vary proportionally with production.
That is, the variable cost is of the form Vx for some constant V, where x represents the
production level. The total cost TC is then the sum of these two: TC = F + V x. For a
limited range of x this very simplistic relationship often holds well but more complicated
models (for instance, involving quadratic and exponential functions) often occur.
Combining the total cost and revenue functions on one graph enables us to perform break-
even analysis. The break-even output is that for which total cost equals total revenue
(TR = TC). In simplified, linear, models the break-even point (should it exist) is unique.
When non-linear relationships are used, a number of break-even points are possible.
2.16) Graphs: In this section, we consider the graphs of functions. The graphing of functions
is very important in its own right, and familiarity with graphs of common functions and
the ability to produce graphs systematically is a necessary and important aspect of the
subject.
A diagram showing the relation between variable quantities, typically of two variables,
each measured along one of a pair of axes at right angles, as shown below:
Page |6
The graph of a function f (x) is the set of all points in the plane of the form (x; f(x)).
Sketches of graphs can be very useful. To sketch a graph, we start with the x-axis and
y-axis, as in Figure below:. (This figure only shows the region in which x and y are both
non-negative, but the x-axis extends to the left and the y-axis extends downwards.)

We then plot all points of the form (x; f(x)). Thus, at x units from the origin (the point
where the axes cross), we plot a point whose height above the x-axis (that is, whose
y-coordinate) is f (x). This is shown in Figure above.
The graph is sometimes described as the graph y = f (x) to signify that the y-coordinate
represents the function value f (x). Joining together all points of the form (x; f (x))
results in a curve, called the graph of
f (x). This is often described as the curve with equation y = f (x). Figure below gives an
example of what this curve might look like.
We shall discuss the graphs of some standard important functions as we progress. We

start with the easiest of all: the graph of a linear function.
Page |7
2.17) Sketch of Linear Function: The linear functions are those of the form y = mx + c and
their graphs are straight lines, with gradient, or slope, m, which cross the y-axis at the
point (0; c).
2.18) Sketch of Quadratic Function: In general, the graph of a quadratic equation

𝒚 = 𝒂𝒙𝟐 + 𝒃𝒙 + 𝒄 is called a parabola.
a) If a > 0, then the parabola has a minimum point and it opens upwards (U-shaped)
or it has a smiley face and if a < 0, then the parabola has a maximum point and it
opens downwards (n-shaped) or it has a sad face.
b) The Vertex or The Stationary Point: The x-coordinate of the minimum point
𝒃
(or maximum point) is given by 𝒙 = − , for y-coordinate of the minimum point
𝟐𝒂
(or maximum point), we substitute this x-value into our quadratic function (the y
expression). Then we will have the (x, y) coordinates of the minimum (or
maximum) point. This is called the vertex of the parabola.
c) The coordinates of the y-intercept is constant term of standard equation or
substitute x = 0.
d) For the coordinates of the x-intercepts substitute y = 0 and solve the quadratic
equation, (if it exists)
2.19) Sketch of Reciprocal Function: The General form of Reciprocal Function is as

𝒌
y = , where k is a real number and x ≠ 0. To sketch this type of graph, we have to find
𝒙
(in this course)
(i) Vertical Asymptote (by substituting denominator = 0)
(ii) Y-intercept (If exists, by substituting x = 0)
𝟏
Graph of y = is given below: (This is called standard graph of Reciprocal Function)
𝒙
Page |8
𝟏
Similarly, graph of y = is given shown on next page:
𝒙−𝟐

Page |9
Exercise:
Q 1) Expand the following:
a) (2x − 3y)(x + 4y) b) (x 2 − 1)(x + 2) c) (𝑥 + 2)2 − (𝑥 − 2)2
3𝑎 10𝑐 3
d) (2x + 3) (𝑥 2 – x − 5) e) 2
× f) 𝑛2 − (𝑛 − 2)(𝑛 + 2)
5𝑐 𝑎2
3𝑚2 𝑝3 7𝑚2 𝑝3 𝑥 𝑥2𝑧 𝑥𝑧 2

g) ÷ h) × ÷
14𝑚3 𝑝 2 12𝑚𝑝 𝑦 𝑦2 2𝑦
Q 2) Write an algebraic expression for each of the following:

a) Seven times ‘x’ plus three times ‘y’
b) Four times the square of ‘l’ minus twice the cube of ‘k ‘
c) Three ‘x’ cubed plus two ‘y’ squared
d) Five times ‘a’ plus ‘b’ multiplied by the square root of ‘c’
Q 3) Simplify each of the following:

1 1 1 1 1
a) 5a + 4b − 3c + (3 a + 2 b − 3 c) − (2a − 1 b + 1 c)
2 2 2 2 2
5x−9y 4x−7y 6x−5y
b) − +
2a 4a 6a
c) Subtract the sum of (2𝑥 2 – 7𝑥 + 4) and (5𝑥– 4𝑥 3 + 7) from the sum of
(3𝑥 2 − 8𝑥 3 + 9) and (15𝑥– 4𝑥 2 – 3𝑥 3 )

Q 4) Evaluate the following:
1 1 1 1
9 ⁄3 × 3 ⁄3 1 1 1 3 ⁄3 × 30 × 9 ⁄3 4
a) b) 25 ⁄4 × 5 ⁄3 × 5 ⁄6 c) 2 d) (2√3)
6 27 ⁄3
Q 5) Solve the following equations:

a) 32x = 27 b) (1/9)x + 2 = 3 c) 4x (52x) = 10

2𝑥 6
a) 5x + 2 = -8 b) 3y - 7 = 26 c) +3=
𝑥−3 𝑥−3
d) 8x - 2 = -9 + 7x e) a + 5 = -5a + 5 f) 4m – 4 = 4m

a) (x + 2) (2x -1) = 0 b) (7 - 2y) (5 + y) = 0 c) (3x + 5) (x -1) = 0
P a g e | 10
Q 8) Solve the following equations by Factorization:
a) 𝑥 2 − 5𝑥 = 0 b) 𝑥 2 − 4 = 0 c) 𝑥 2 + 5𝑥 + 6 = 0
d) 5x² + 6x + 1 = 0 e) 𝑥 2 − 7𝑥 + 12 = 0 f) 𝑥 2 − 3 = 2𝑥
g) 6x2 + 5x – 6 = 0 h) 6x2 + x – 2 = 0 i) 8x2 + 14x - 15 = 0
Q 9) Solve the following equations by using any method also state if an equation has no
real roots:
a) 𝑥² − 4𝑥 − 8 = 0 b) 5x² + 2x + 1 = 0 c) 𝑥 2 + 4𝑥 + 1 = 0
d) 𝑥 2 + 5𝑥 − 2 = 0 e) 5𝑥 2 + 15𝑥 + 3 = 0 f) 10𝑥 2 + 17𝑥 − 4 = 0
g) 𝑥 4 − 5𝑥 2 + 4 = 0 h) 𝑥 4 − 10𝑥 2 + 9 = 0 i) 𝑥 6 − 7𝑥 3 − 8 = 0
12
j) 𝑥 6 + 𝑥 3 − 12 = 0 k) √𝑡 = 4 + l) √𝑡 (√𝑡 − 6) = −9
√𝑡
Q 10) Use the discriminant to determine the nature of the roots of the following
quadratic equations.
a) 𝑥 2 – 2𝑥– 5 = 0 b) 4𝑥 2 + 4𝑥 + 1 = 0 c) 𝑥(1– 3𝑥) = 2
d) 𝑥 2 – 3𝑥– 5 = 0 e) 𝑥 2 + 2𝑥 + 1 = 0 f) 𝑥 2 – 3𝑥 + 4 = 0
Q 11) The following equations have the number of roots shown in brackets, Using the
discriminant, deduce the value or range of values of K.
a) 𝐾x 2 = 2x– 𝐾 (1) b) x 2 – 2x + 2 = 𝐾 (2)
c) 𝐾(𝑥 + 1) (𝑥– 3) = 𝑥– 4𝐾– 2 (0) d) 𝑥 2 + 3𝑥 + 𝐾 = 0 (2)
e) 𝐾𝑥 2 − 3𝑥 + 5 = 0 (2) f) 𝑥 2 − 4𝑥 + 3𝐾 = 0 (1)
P a g e | 11
Q 12) Find the point of Intersection of the following equations:

1
a) y = 2x – 3 and y=2– 𝑥
2
b) 2x + 3y = 7 and 3x – 4y = 2
c) 5x + 2y = 9 and 3x +y = 8
d) y2 + (2x + 3)2 = 10 and 2x +y = 1
2 𝑥
e) y – x = 3 and − =1
𝑥 𝑦
Q 13) Simplify and express each of the following as a single logarithm:

a) ln 8 – 2 ln 4 b) 2 ln 5 + 3 ln 2 - ln 4
c) ln (8/75) – 2 ln (3/5) + 4 ln (3/2) d) 2 ln (x+2) + ln (x+1) – ln (x2 + 3x +2)
Q 14) Suppose the demand function is q D (p) = 20 – 2p and that the supply function is
𝟐
q s (p) = p – 4. Find the equilibrium price p* and equilibrium quantity q*.
𝟑
Q 15) Find the equilibrium price and quantity of the following Demand functions and
Supply functions
a) Demand function: q + 5 p = 40
Supply function: 2q – 15 p = - 20
b) Demand function: q - 6 p = - 16
Supply function: q + 2 p = 40
c) Demand function: q = 12 p – 4
Supply function: q=8-4p
Q 16) Find the break-even points in the case where the total cost function is TC = 2 + 5x + x2
and the total revenue function is TR = 12 + 8x
Q 17) Suppose that the demand relationship for a product is 𝐩 = 𝟔⁄(𝐪 + 𝟏) and that the
supply relationship is p = q + 2. Determine the equilibrium Price and Quantity.
P a g e | 12
Q 18) If the curve y = k x (x+2) meets the line y = x - k, find the range of values of k. State
the value of k for which the line is a tangent.
Q 19) Sketch the following:

a) y = 5 b) x = 2 c) y = x + 3
d) y = - 3x – 2 e) y = 2x – 8 f) y = - 2x – 8
g) y = x2 – 6x + 5 h) y = 2x2 – 8x + 11
Q 20) Find the equilibrium price and quantity of the following demand and supply
functions also sketch both function on same graph.
a) Demand function: p = 4 – q – q2
Supply function: p = 1 + 4q + q2
b) Demand equation: q = 8 – p2 - 2 p
Supply equation: q = p2 + 2 p – 8
c) Demand equation: P = Q2 – 10Q + 25
Supply equation: P = Q2 + 6Q + 9
P a g e | 13
Questions from Past Papers:
Q 21) (Paper-2003, Zone-A, Section-A, Question-1)

The supply equation for a good is 𝐪 = 𝐩𝟐 + 𝟕𝐩 − 𝟐 and the demand equation is
𝐪 = −𝐩𝟐 − 𝐩 + 𝟒𝟎, where p is the price. Sketch the supply and demand functions for p≥ 𝟎.
Determine the equilibrium price and quantity.
Q 22) (Paper-2003, Zone-B, Section-A, Question-1)

The supply equation for a good is 𝐪 = 𝟒𝐩 − 𝟐 and the demand equation is
𝐪 = −𝟐𝐩𝟐 − 𝟔𝐩 + 𝟗𝟖, where p is the price. Sketch the supply and demand functions for p≥ 𝟎.

The functions f(x) and g(x) are given by f(x) = 𝟒𝐱 𝟐 − 𝟖𝐱 − 𝟏, g(x) = −𝟒𝐱 𝟐 − 𝟐𝐱 + 𝟒. Sketch
the graphs of y = f(x) and y = g(x) for x > 0 on the same diagram, and determine the positive
value of x at which these two graphs intersect.

The functions f(x) and g(x) are given by f(x) = 𝐱 𝟐 + 𝟒𝐱 + 𝟏, g(x) = −𝐱 𝟐 − 𝐱 + 𝟔. Sketch the
graphs of y = f(x) and y = g(x) for x > 0 on the same diagram, and determine the positive value
of x at which these two graphs intersect.

On the same diagram, sketch the curves with equations 𝒚 = 𝟐𝒙𝟐 − 𝒙 − 𝟑 and
𝒚 = 𝟏 + 𝒙 − 𝟐𝒙𝟐 , indicating where each curve crosses each of axes. Determine the values of x
for which the two curves intersect.

On the same diagram, sketch the curves with equations 𝒚 = 𝟐𝐱 𝟐 + 𝟑𝐱 − 𝟓 and
𝒚 = 𝟔𝒙 + 𝟒 − 𝟒𝒙𝟐 , indicating where each curve crosses each of axes. Determine the values of
x for which the two curves intersect.
P a g e | 14

The demand equation for a good is given by 𝒑 = 𝒒𝟐 + 𝟒𝒒 + 𝟐𝟎. Sketch the demand curve for
q ≥ 𝟎. If the supply equation is 𝒑 = −𝒒𝟐 − 𝟏𝟎𝒒 + 𝟏𝟕𝟔, determine the equilibrium price and
quantity.

The demand equation for a good is given by 𝐩 = 𝐪𝟐 + 𝟔𝐪 + 𝟐𝟒. Sketch the demand curve for
q ≥ 𝟎. If the supply equation is 𝐩 = −𝐪𝟐 − 𝟖𝐪 + 𝟏𝟖𝟎, determine the equilibrium price and
quantity.

Show that the graphs of the functions 𝒇(𝒙)=𝒙𝟐 − 𝟐𝒙 − 𝟒 and 𝒈(𝒙) = 𝒙 − 𝟖 do not intersect,
and sketch both graphs on the same diagram. Determine the positive values of the constant a
such that the graph of the function 𝒉(𝒙) = 𝒂𝒙 − 𝟖 does intersect the graph of f.

Show that the graphs of the functions 𝒇(𝒙)=𝒙𝟐 − 𝒙 − 𝟔 and 𝒈(𝒙) = 𝟐𝒙 − 𝟏𝟎 do not intersect,
and sketch both graphs on the same diagram. Determine the positive values of the constant a
such that the graph of the function 𝒉(𝒙) = 𝒂𝒙 − 𝟏𝟎 does intersect the graph of f.

The demand equation for a good is q (p+3) = 4 and the supply equation is 2q - p + 4 = 0 where
p is the price and q is the quantity. Sketch the supply and demand functions for p ≥0. Determine
the equilibrium price and quantity.

The demand equation for a good is q (2p + 3) = 8 and the supply equation is q - 2p + 4 = 0
where p is the price and q is the quantity. Sketch the supply and demand functions for p≥0.

Functions f and g are as follows: f(x) =𝐱 𝟒 + 𝟐𝐱 𝟑 + 𝟐𝐱 𝟐 + 𝟐, g(x) =−𝐱 𝟒 + 𝟐𝐱 𝟑 + 𝟏𝟖𝐱 𝟐 + 𝟐𝟎.
Show that the curves y = f(x) and y = g(x) intersect for exactly two values of x. Find these values
of x. (Do not attempt to sketch the curves.)
P a g e | 15

Functions f and g are as follows: f(x) =𝐱 𝟒 + 𝐱 𝟑 + 𝟐𝐱 𝟐 + 𝟐, g(x) =−𝐱 𝟒 + 𝐱 𝟑 + 𝟖𝐱 𝟐 + 𝟏𝟎. Show
that the curves y = f(x) and y = g(x) intersect for exactly two values of x. Find these values of
x. (Do not attempt to sketch the curves.)

Show that the curve with equation y = 𝐱 𝟐 − 𝐱 − 𝟐 does not intersect the line with equation
y =𝐱 − 𝟒. Sketch the two curves on the same diagram. For which values of the number a will
the curve with equation y =𝐱 𝟐 − 𝐱 − 𝟐 intersect the line with equation y = 𝐱 − 𝐚? For which
particular value of a will there be precisely one point at which the curve and the line intersect?

The functions f and g are defined by f(x) = 𝐱 𝟐 + 5x + 6; g(x) = 2 - 𝐱 𝟐 + x: Show that the curves
y = f(x) and y = g(x) do not intersect and sketch the curves. For which values of c will the graph
of f intersect the graph of the function h(x) = g(x) + c? For which particular value of c will there
be precisely one point at which they intersect?

The demand equation for a good is q (p + 3) = 8 and the supply equation is q - p + 4 = 0, where
p is the price and q is the quantity. Sketch the supply and demand functions for p ≥ 0. Determine
the equilibrium price and quantity.

The demand equation for a good is q (p + 4) = 10 and the supply equation is q - p + 5 = 0,
where p is the price and q is the quantity. Sketch the supply and demand functions for p ≥ 0.
Q 39) (Paper-2013, Zone-A & B, Section-A, Question-1)

Let 𝒇 and 𝒈 be functions with 𝒇(𝒙) = 𝟐𝒙𝟐 + 𝟒𝒙 + 𝟐 and 𝒈(𝒙) = 𝒙𝟐 + 𝟐𝒙 − 𝟑.
(a) Show that the graphs of the functions 𝒇 and 𝒈 do not intersect.
(b) Sketch the graphs of both functions on the same diagram.
(c) For a constant 𝒄, let 𝒉 be the function with 𝒉(𝒙) = 𝒙𝟐 + 𝟐𝒙 + 𝒄. Find the values of 𝒄 for
which the graphs of 𝒇 and 𝒉 intersect.
P a g e | 16


P a g e | 17
Answers:
Q 1) a) 2x 2 + 5xy − 12y 2 b) x 3 + 2x 2 − x − 2 c) 8x
6𝑐
d) 2x3 + x2− 13x− 15 e) f) 4
𝑎
18 2𝑥 2
g) h)
49𝑚2 𝑝 𝑦2𝑧
Q 2) a) 7x + 3y b) 4l2 − 2k 3 c) 3x 3 + 2y 2
d) 5a + b√c
1 30x−43y
Q 3) a) 6 a + 8b − 8c b)
2 12a
c) – 3x2 – 7 x3 + 17x - 2 = - 7x3 – 3x2 + 17 x - 2
Q 4) a) ½ b) 5 c) 1/3 d) 144
Q 5) a) 3/2 b) - 5/2 c) ½
Q 6) a) -2 b) 11 c) 3 d) -7
e) 0 f) No solution
1 7 −5
Q 7) a) -2 or b) -5 or c) 1 or
2 2 3

Q 8) a) 0 or 5 b) -2 or 2 c) -2 or -3
𝟏
d) -1 or − e) 3 or 4 f) 3 or -1
𝟓
2 3 2 1 5 3
g) or − h) − or i) − or
3 2 3 2 2 4
Q 9) a) 2 ± 2 √3 b) No real roots c) −2 ± √3
−5 ± √33 −15 ± √165 −17 ± √449
d) e) f)
2 10 20
g) ±1, ±2 h) ±1, ±3 i) -1, 2

P a g e | 18
3 3
j) √3 , − √ 4 k) 36 l) 9
Q 10) a) 2 real and distinct roots b) 2 real and equal roots or 1 root
c) no real roots d) 2 real and distinct roots
e) 2 real and equal roots or 1 root f) no real roots
Q 11) a) K = ±1 b) K > 1 c) K > 1⁄4
d) K < 9⁄4 e) K > 9⁄20 f) K = 4⁄3

Q 12) a) x = 2, y=1 b) x = 2, y=1
c) x = 7, y = −13 d) {(0,1), (-1,3)}
e) {(-2,1), (3⁄2, 9⁄2)}
Q 13) a) − ln 2 b) ln 50 c) ln( 3⁄2) d) ln(x + 2)
Q 14) p∗ = 9, q∗ = 2
Q 15) a) p = 4, q = 20 b) p = 7, q = 26
3
c) p= , q=5
4
Q 16) x = 5, OR x = -2
Q 17) p = 3, q = 1, OR p = -2, q = -4
NOTE: In exams, we ignore negative values of price and quantity.

1 1
Q 18) 𝑘 ≤ , 𝑘=
4 4
P a g e | 19
Q 19) a) y
y=5
O x

b)
y
O x=2 x
c)
y y = x+3
-3 O x
P a g e | 20
d)
y = -3x-2 y
−𝟐⁄ O x
𝟑
-2

e)
y y = 2x-8
O 4 x
-8
P a g e | 21
f)
y = -2x-8 y
-4 O x
-8

g)
y
𝟐
𝒚 = 𝒙 − 𝟔𝒙 + 𝟓
O 1 5 x
P a g e | 22
h)
y
11 𝒚 = 𝟐𝒙𝟐 − 𝟖𝒙 + 𝟏𝟏
(2, 3)
O x
Q 20) a) p = 2, q=1 OR p = -2, q = -3

p
𝟏 𝟏𝟕
(− 𝟐 , 𝟒
)
𝒑 = 𝟏 + 𝟒𝒒 + 𝒒𝟐
O q
(-2, -3) 𝒑 = 𝟒 − 𝒒 − 𝒒𝟐
P a g e | 23
b) p = 2, q=0 OR p = -4, q=0
𝒒 = 𝒑𝟐 + 𝟐𝒑 − 𝟖
8
-4 O 2 p
-8 𝒒 = 𝟖 − 𝒑𝟐 − 𝟐𝒑

c)
P = 16, Q=1
P 𝑷 = 𝑸𝟐 − 𝟏𝟎𝑸 + 𝟐𝟓
25
𝑷 = 𝑸𝟐 + 𝟔𝑸 + 𝟗
16
-3 O 1 5 Q
P a g e | 24
Q 21)
q
40 𝒒 = 𝒑𝟐 + 𝟕𝒑 − 𝟐
−𝟕+√𝟓𝟕 −𝟏+√𝟏𝟔𝟏
O p
𝟐 𝟐
-2
𝒒 = −𝒑𝟐 − 𝒑 + 𝟒𝟎
Equilibrium Price = 3, and Equilibrium Quantity = 28
Q 22)
q
98 𝒒 = 𝟒𝒑 − 𝟐
𝟏 −𝟑+√𝟐𝟎𝟓
O p
𝟐 𝟐
-2
𝒒 = −𝟐𝒑𝟐 − 𝟔𝒑 + 𝟗𝟖
P a g e | 25
Q 23)
y 𝒚 = 𝟒𝒙𝟐 − 𝟖𝒙 − 𝟏
−𝟏+√𝟏𝟕 𝟐+√𝟓
O x
𝟒 𝟐
-1
(1, -5) 𝒚 = −𝟒𝒙𝟐 − 𝟐𝒙 + 𝟒
The value of x at the point of intersection is x = 5⁄4
Q 24)
y 𝒚 = 𝒙𝟐 + 𝟒𝒙 + 𝟏
O 𝟐 x
𝒚 = −𝒙𝟐 − 𝒙 + 𝟔
√𝟔𝟓−𝟓
The value of x at the point of intersection is x =
𝟒
P a g e | 26
Q 25)
y
1 𝐲 = 𝟐𝐱 𝟐 − 𝐱 − 𝟑
-1 − 𝟏⁄𝟐 O 1 𝟑⁄ x
𝟐
𝐲 = 𝟏 + 𝐱 − 𝟐𝐱 𝟐
𝟏±√𝟏𝟕
The values of x at the points of intersection are x =
𝟒
Q 26)
y 𝐲 = 𝟐𝐱 𝟐 + 𝟑𝐱 − 𝟓
− 𝟓⁄𝟐 − 𝟏⁄𝟐 O 1 𝟐 x
𝐲 = 𝟔𝐱 + 𝟒 − 𝟒𝐱 𝟐
𝟑
The values of x at the points of intersection are x = Or x = -1
𝟐
P a g e | 27
Q 27) The graph of demand equation 𝒑 = 𝒒𝟐 + 𝟒𝒒 + 𝟐𝟎 is
20
O q
Q 28) The graph of demand equation 𝒑 = 𝒒𝟐 + 𝟔𝒒 + 𝟐𝟒 is
24
O q

P a g e | 28
Q 29) The graph is as follows:
𝑎≥2
𝑎≥3
P a g e | 29
Equilibrium Price = 5, and Equilibrium Quantity = 1/2
Equilibrium Price = 5/2, and Equilibrium Quantity = 1

P a g e | 30
Q 33) x = 3, OR x = -3
Q 34) x = 2, OR x = -2
𝑎≤3 𝑎=3
𝑐≥2 𝑐=2
P a g e | 31

q
q-p+4=0
8/3 q(p + 3) = 8
O 4 p
-4

q
q-p+5=0
5/2 q(p + 4) = 10
O 5 p
-5

P a g e | 32
Q 39) (b)
c) 𝑐 ≥ 1
Q 40) Equilibrium Price = 4, and Equilibrium Quantity = 1
Q 41) Equilibrium Price = 5, and Equilibrium Quantity = 1
Q 42) The intersection points are (2; 5) and (-1; -4)

P a g e | 33
Q 43) The intersection points are (2; -5) and (-1; 4)
The End
Good Luck

Chapter 2 M 1

Uploaded by

Copyright:

Available Formats

Chapter 2 M 1

Uploaded by

Document Information

Original Description:

Original Title

Copyright

Available Formats

Share this document

Share or Embed Document

Sharing Options

Did you find this document useful?

Is this content inappropriate?

Copyright:

Available Formats

Chapter 2 M 1

Uploaded by

Copyright:

Available Formats

Practice Questions

Chapter Title: Basics

Course Code: 05a

Compiled By: Sir Waseem Mustafa

Cell No.: 0334 8111881

x-1 = 1/x 4-1 = 1/4

xmxn = xm+n x2x3 = x2+3 = x5

xm/xn = xm-n x6/x2 = x6-2 = x4

(xm)n = xmn (x2)3 = x2×3 = x6

(xy)n = xnyn (xy)3 = x3y3

(x/y)n = xn/yn (x/y)2 = x2 / y2

x-n = 1/xn x-3 = 1/x3

And the law about Fractional Exponents:

Compiled by: Sir Waseem Mustafa (0334 8111881)

2.4) Solution of Equations: What is a Solution of Equation or Equations?

Compiled by: Sir Waseem Mustafa (0334 8111881)

2.6) Nature of Roots of Quadratic Equations (The Discriminant): The discriminant is

−𝒃−√𝒃𝟐 −𝟒𝒂𝒄 −𝒃+√𝒃𝟐 −𝟒𝒂𝒄

Compiled by: Sir Waseem Mustafa (0334 8111881)

2.10) Properties of Exponentials and the Natural Logarithm:

2.12) Important values of the Trigonometric Functions (Ratios):

Compiled by: Sir Waseem Mustafa (0334 8111881)

2.13) Important Identities of Trigonometric Ratios:

(Topic continues on next page)

Compiled by: Sir Waseem Mustafa (0334 8111881)

Compiled by: Sir Waseem Mustafa (0334 8111881)

We shall discuss the graphs of some standard important functions as we progress. We

2.18) Sketch of Quadratic Function: In general, the graph of a quadratic equation

Compiled by: Sir Waseem Mustafa (0334 8111881)

2.19) Sketch of Reciprocal Function: The General form of Reciprocal Function is as

Compiled by: Sir Waseem Mustafa (0334 8111881)

3𝑚2 𝑝3 7𝑚2 𝑝3 𝑥 𝑥2𝑧 𝑥𝑧 2

Q 2) Write an algebraic expression for each of the following:

Q 3) Simplify each of the following:

Compiled by: Sir Waseem Mustafa (0334 8111881)

Q 5) Solve the following equations:

Q 6) Solve the following equations:

Q 7) Solve the following equations:

Q 8) Solve the following equations by Factorization:

g) 6x2 + 5x – 6 = 0 h) 6x2 + x – 2 = 0 i) 8x2 + 14x - 15 = 0

Compiled by: Sir Waseem Mustafa (0334 8111881)

d) 𝑥 2 + 5𝑥 − 2 = 0 e) 5𝑥 2 + 15𝑥 + 3 = 0 f) 10𝑥 2 + 17𝑥 − 4 = 0

a) 𝑥 2 – 2𝑥– 5 = 0 b) 4𝑥 2 + 4𝑥 + 1 = 0 c) 𝑥(1– 3𝑥) = 2

a) 𝐾x 2 = 2x– 𝐾 (1) b) x 2 – 2x + 2 = 𝐾 (2)

c) 𝐾(𝑥 + 1) (𝑥– 3) = 𝑥– 4𝐾– 2 (0) d) 𝑥 2 + 3𝑥 + 𝐾 = 0 (2)

Q 12) Find the point of Intersection of the following equations:

Q 13) Simplify and express each of the following as a single logarithm:

c) ln (8/75) – 2 ln (3/5) + 4 ln (3/2) d) 2 ln (x+2) + ln (x+1) – ln (x2 + 3x +2)

Compiled by: Sir Waseem Mustafa (0334 8111881)

Q 19) Sketch the following:

Compiled by: Sir Waseem Mustafa (0334 8111881)

Questions from Past Papers:

Q 21) (Paper-2003, Zone-A, Section-A, Question-1)

Q 22) (Paper-2003, Zone-B, Section-A, Question-1)

Q 23) (Paper-2004, Zone-A, Section-A, Question-1)

Q 24) (Paper-2004, Zone-B, Section-A, Question-1)

Compiled by: Sir Waseem Mustafa (0334 8111881)

Q 26) (Paper-2005, Zone-B, Section-A, Question-1)