Download Practice Questions for Complex Numbers, Sets, Functions, and Quadratic Equations and more Exercises Mathematics in PDF only on Docsity!
Practice questions Lecture 1 to 5
Question 1
Write the following expression in the form of a + ib.
i i
,^4
i i
Solution
a)^3 2
i i
In order to simplify the above fraction, we will multiply and divide it by the conjugate of its denominator
3 2 2 2 6 3 2 ( 1) 4 ( 1) (^5 5 ) 5
i i i i i i
i (^) i
= +^ +^ + −
b) Part two will be done in the similar way. The simplified form will be (^1) ( 17 59 ) 65
− (^) − + i
Question 2
z = − + 5 8 i then find
z − z *
Solution
z i z i z z i i i i i
Question 3
If z = 3 + 6 i then find
z z .*
Solution
z i z i z z i i i i
Question 4
Evaluate
4 7 5 6
i i
Solution
2 2 2
2 2
(^4 7) * 5 6 (4 7 )( 5 6i) 5 6 5 6 ( 5 6 )( 5 6 ) 20 24 35 42 ( 5) (6 ) 22 59 22 59 22 59 25 36 61 61 61 (^4 7) ( 22 ) ( 59 ) 484 3481 3965 5 6 61 61 3721 3721
i i i i i i i i i i i i i (^) i
i i
= −^ −^ −^ −
= −^ = − = −
Question 5
Write in set builder notation
- A= Set of first 100 odd numbers
- A= Set of Natural numbers starting from 2 to 40.
- A= Set of all even numbers greater than 50.
Solution
A = {x = 2k 1| 0+ k 99} A = {x : x N | 2 x 40} A = {x : x E^ x 50}
Question 6
List the elements of the set
- A = { | x x E + x 10}
- ( 1− − i )( 2 − − 2 ) i
- i^3
Solution
6 2i 3i 1 5 5i
- It will be simplified in the similar pattern as above, the answer will be
4i
The Venn diagram given below shows the sets A, B, and C. Shade the following sets
i. A ( B C ) ii. A ( B C ) c iii. A ( B C )
Question 10
What is the cardinality of these sets?
A = Set of even integers
B = Set of prime numbers less than 30
Solution
Cardinality of A is equal to the cardinality of set of Natural Numbers Cardinality of B is equal to 11
Question 11
i) z = − 7 + 5 i
ii) z = 3 i
Solution i) Domain = R = all real numbers Range = R = all real numbers ii) Domain {x R : x 3 } 2
= ^ −
Range = {y R : y 0} iii) {x^ R : x^ 3} {y R : y 0}
Domain Range
iv) {x R : x 5 } 2 {y R : y 0}
Domain Range
Question 14
Write the domain and range of f x ( ) = 6 x + 9
Solution
Domain of f(x)= All real no’s
Range of f(x)= All real no’s
Question 15
Write the domain and range ofg( ) 5 6
x x
Solution
Domain of g(x)= R − −{ 6}
Range of f(x)= All real numbers except zero.
Question 17
Determine whether the function is even, odd, or none.
i. f ( ) x = x^2 + 3 x ii. f ( ) x = x^2 + 9 iii. f ( ) x = x^2 + 4 iv. ( ) (^2) 1
f x x x
v. f ( ) x = 6 x^5 − x^3 vi. f ( ) x = 2 x^4
Solution We know that a function is even if f(x) = f(-x) and odd if f(-x)=-f(x), so
i) f ( ) x = x^2 + 3 x Neither even nor odd ii) f ( ) x = x^2 + 9 Even function iii) f ( ) x = x^2 + 4 Even function iv) ( ) (^2) 1 f x x x
Odd function v) f ( ) x = 6 x^5 − x^3 Odd function vi) f ( ) x = 2 x^4 Even function
Question 18
Prove that f^^ (x)^ =^ x^^4 +^5 x^2 +^3 is an even function
Solution
4 2 4 2 4 2
(x) 5 3 ( ) ( ) 5( ) 3 ( ) 5 3 ( )
f x x f x x x f x x x f x
Question 19
Prove that f (x) = x^3 − 8 x is an odd function
Solution
3 3 3 3
(x) 8 ( ) ( ) 8( ) ( ) 8 ( 8 ) ( )
f x x f x x x f x x x x x f x
Question 20
Express the quadratic polynomial in complete square form.
- 2 x^2 + 8 x + 6
- x^2^ + 2 x − 7
If the functions f is defined by
f ( ) x 1 , 0 x 5, x
then find the range of f.
Solution
Since x ≤ 5 ,1/x ≥ 1/
i.e.,
f(x) ≥ 1/5.
Question 24
If the functions f and g are defined by
( ) 3 , 0 3, ( ) 5 1, ,
f x x x g x x x R
then Calculate gf ( 2 .)
Solution
( ) (^) ( ( )) ( ) ( ) ( )
gf x g f x g x x x So gf
Question 25
If the functions g is defined by g x ( ) = 8 x − 5, x R ,
then find an expression in terms of x for
g −^1 ( x )
Solution
To find g-^1 (x) , we write y = 8x – 5
y + 5 = 8x x = (y + 5)/8. Therefore, g-^1 (x) = (x + 5)/
Question 26
Let f x ( ) = 6 x +1 and g x ( ) = − 9 x +5. Find theproduct of f &g. i e ( f g. )( ). x
Solution
f g x f x g x x x x x Ans
Practice Questions for Lecture 6 to 10
Question 1
Determine whether the function is even, odd, neither even nor odd.
- f ( ) x = x^2 + 3 x
- f ( ) x = x^2 + 9
- f ( ) x = x^2 + 4
•^2
f x x x
f ( ) x = 6 x^5^ − x^3
Question 2
Express the quadratic polynomial in complete square form
2 x^2 + 8 x + 6
Question 3
If f ( ) x = 4 x − 3 g x ( ) = x^3 + 3
x^2^ + 3 x + 6 = 0
2
solution a b c b ac
So there will be no real solutions.
Question 8
If the following equation has one solution then find the value of a
ax^2^ + 5 x + 2 = 0
Question 9
The equation tx^2^ + 5 x − 6 = 0 has two real solutions. What can be deduced about the value of t?
Solution
(^2 5 6 ) , 5, 6
tx x a t b c
For two real solutions the discriminant should be >0 i.e (^2 4 ) 25 24 0 24 25 25 24
b ac t t t
Question 10
The equation kx^2^ − 8 x + 9 = 0 has exactly one solution. What can be deduced about the value of k?
Solution
kx x a k b c
For exactly one solution the discriminant should be equal to 0 i.e (^2 4 ) 64 36 0 64 36 64 36 32 18
b ac k k k
k
Question 11
Given matrices A and B such that
1 2 1 1 , 3 6 2 4
A = ^ ^ B =^
− −^ −^ −
Find 2 A − 3 B.
Question 12
Check whether the matrix
A and B
= ^ = ^
^
can be multiplied? Justify your
answer.
Question 13
Find the inverse^1 4 5
A = ^
Question 14
Multiply the matrix:
1 0 0 0 0 1 0 1 0 0 1 0 0 0 1 1 0 0
A and B
= ^ ^ =^
Question 1 5
Find the transpose of matrix
A B
^ − −^ −
Evaluate A-3B.
Answer
Question 5
Check whether the matrix
A
= ^ −
and
B = ^
can be multiplied? Justify
your answer.
Solution
order of A=3*
Order of B=2*
Inner dimensions do not match so this multiplication is not possible.
Question 6
Find the inverse of A=
Answer
4 / 2 2 / 2 3 / 2 1/ 2
^ −
Question 7
Find the determinant of the following matrix
A
^ −
= ^ − −
Answer: 143
Question 8
Given the sequence, 150, 300, 450, 600, …, find the 15th^ term.
Solution
a = 150 , d = 150, n = 15
15 150 (^ )
(n 1) 150 14 225
d 0
n a
a a = + =
Question 9
What is the general term of the sequence 1,8,27, 64, 125 ….
Solution
an=n^3
Question 10
The 3rd^ and the 4th^ terms of a geometric progression are 18 and 6 respectively. Find the sum to infinity of the progression.
Solution
Given that ar^2 = 18 and ar^3 = 6
r = 6/18 = 1/3.
a(1/9) = 18
a = 162
S a (^) r
Question 11
Common difference = d = 2
The difference between the consecutive terms of the sequence is constant. The constant difference is called common difference. Hence the sequence is arithmetic.
Question 15
Find the 5th term of the geometric sequence with a 1 = 20 and a 2 = 40.
Solution
1
4 5
nd st n n
r term term a ar
a
−
Question 16
If a 0 (^) = 3, a 1 (^) = 4, a 2 (^) = −2, and a 3 = 1 , then compute the summation:
2 0
(^2) i i
a
Solution
0 1 2 3 2 0 0 1 2
3 4, 2, and 1 2 2
2 3 4 2 2 5 10
i^ i
a a a a a a a a =
Question 17
What is the sigma notation for the series : 2(1)^2 + 2(2)^2 + 2(3)^2 + ...+2(10)^2?
Question 18
If (^) a 0 (^) = 3, a 1 (^) = 2, a 2 (^) = 5, a 3 (^) = 4 and a 4 = − 9 , then compute the summation:
3 0
(^3) i i
a
Question 19
Compute the summation: ( )
(^3 ) 0
n
n
Question 20 Formula for the sum of infinite geometric series is ……….. Whose first term is ‘a’ and common ratio is ‘r’.
Question 21 Find the sum of first 'n-1' natural numbers.
Practice Questions for Lecture 16 to 22
Question 1
A teacher is making a multiple-choice quiz. She wants to give each student the same questions, but have each student's questions appear in a different order. If there are 32 students in the class, what is the least number of questions the quiz must contain?
Solution
If there were two questions on the quiz, we could prepare two quizzes with the questions in different order, 2•1 = 2. If there were three questions, we could get 3•2•1 = 6 different orders. If there were four questions, we could get 4•3•2•1 = 24 different orders -- not quite enough for the class of 27 students. If there were five questions, we could get 5•4•3•2•1 = 120 different orders. The teacher will need at least 5 questions on the quiz.
Question 2
The local Family Restaurant has a daily breakfast special in which the customer may choose one item from each of the following groups:
Breakfast Sandwich Accompaniments Juice
