# Free Mathematics Notes for 9th Class

These notes are according to the syllabus of KPK text book. Other board notes will also be uploaded time to time.

## Class 9 Mathematics Notes

### Mathematics Class 9 Notes (KPK) Chapter # 1

1. A ____________ is a rectangular array of real numbers enclosed in square brackets.
O Algebra
O Real number
O Matrices
O None

2. Each number in a matrix is called ____________ of the matrix.
O Row
O Entry
O Element
O Both b & c

3. Matrices are mostly denoted by ____________ letter.
O Small
O Capital
O Both a & b
O None of these

4. The ____________ of a matrix run horizontally.
O Row
O Column
O Determinant
O None of these

5. In \left[\begin{array}{ll}2 & 5 \\ I & 3\end{array}\right] 2,5, I, 3 all are the ____________ of a matrix.
O Row
O Column
O Elements
O None of these

6. In \left[\begin{array}{lll}1 & 2 & 3 \\ a & b & c \\ x & y & z \end{array}\right] 2, b \ and \ y are ____________ of a matrix.
O Row
O Column
O Equal
O None of these

7. In \left[\begin{array}{lll}1 & 2 & 3 \\ a & b & c \\ x & y & z \end{array}\right] , a, b \ and \ c are ____________ of a matrix.
O Row
O Column
O Equal
O None of these

8. In \left[\begin{array}{ll}1 & 2 \\ 3 & 4\end{array}\right], 1,3 \ and \ 2, 4 are the ____________ of a matrix.
O Rows
O Columns
O Equal
O None of these

9. The matrix with m rows and n columns has order ____________
O m \times n
O m-by-n
O both a & b
O None of these

10. A matrix with represents m is ____________
O Row
O Column
O Both a & b
O None of these

11. A matrix with represents n is ____________
O Row
O Column
O Both a & b
O None of these

12. Order of matrix can be written as ____________
O Column by row
O Row by Row
O Row by Column
O All of them

13. The order of matrix \left[\begin{array}{lll}1 & 3 & 5 \end{array}\right] is ____________.
O 2-by-2
O 3-by-3
O 1-by-3
O 3-by-1

Explanation:
The Matrix consists of One Row and 3 Columns

14. The order of matrix \left[\begin{array}{ll}1 & 2 \\ 3 & 4\end{array}\right] is ____________.
O 2-by-2
O 3-by-3
O 1-by-3
O 3-by-1

Explanation:
The matrix consists of 2 Rows and 2 Columns

15. Order of a matrix is also called _______
O Dimension
O Size
O Both a & b
O None of these

Explanation:
Order of a matrix is also called Dimension or Size of a matrix.

16. m \times n means _______
O m \ multiply \ n

O order of a matrix
O column of a matrix
O Row of a matrix

Explanation:
m \times n does not mean the multiplication.
It shows the order of a matrix having rows and columns.

17. Both \left[\begin{array}{ll}2 & 5 \\ 4 & 3\end{array}\right] and \left[\begin{array}{ll}1+1 & 3+2 \\ 3+1 & 2+1\end{array}\right] are _______
O Equal
O Not equal
O Zero
O None of these

Explanation:
\left[\begin{array}{ll}1+1 & 3+2 \\ 3+1 & 2+1\end{array}\right]=\left[\begin{array}{ll}2 & 5 \\ 4 & 3\end{array}\right]

18. In \left[\begin{array}{ll}3 & 2 \\ 4 & 1\end{array}\right] \ then \ a_{21} is _______
O 3
O 2
O 4
O 1

Explanation:
a_{21} means that 2nd Row and First Column. Thus the position of a_{21} is 4.

19. In \left[\begin{array}{ll}3 & 2 \\ 4 & 1\end{array}\right] \ then \ a_{22} is _______
O 3
O 2
O 4
O 1

Explanation:
a_{22} means that 2nd Row and 2nd Column. Thus the position of a_{22} is 1.

20. In \left[\begin{array}{ll}3 & 2 \\ 4 & 1\end{array}\right] \ then \ a_{12} is _______
O 3
O 2
O 4
O 1

Explanation:
a_{12} means that 1st Row and 2nd Column. Thus the position of a_{12} is 2.

21. In \left[\begin{array}{cc}2 & -3 \\ 4 & v\end{array}\right]=\left[\begin{array}{ll}2 & w \\ 6 & 6\end{array}\right] \ then \ w= ________
O 2
O 5
O 6
O -3

Explanation:
By comparing the corresponding element of w \ which \ is \ -3

22. If \left[\begin{array}{ll}x-1 & 4 \\ y+3 & 7\end{array}\right]=\left[\begin{array}{cc}0 & 4 \\ -2 & -7\end{array}\right] then \ x= \ ?
O -1
O 0
O 1
O 2

Explanation:
By comparing the corresponding element of x \ which \ is
x-1=0
x=1

23. If \left[\begin{array}{cc}x-1 & 4 \\ y+3 & -7\end{array}\right]=\left[\begin{array}{cc}0 & 4 \\ -2 & -7\end{array}\right] \ then\ y= \ ?
O -5
O 5
O 0
O None

Explanation:
By comparing the corresponding element of y \ which \ is
y+3=-2
y=-2-3
y=-5

1. \left[\begin{array}{lll}1 & 3 & 5\end{array}\right] is ______ matrix.
O Row
O Column
O Square
O None

Explanation:
Because the matrix consists of only one row.

2. \left[\begin{array}{lll}1 \\ 3 \\ 5\end{array}\right] is ______ matrix.
O Row
O Column
O Square
O None

Explanation:
Because the matrix consists of only one column.

3. A matrix in which number of Rows equal to number of Columns is called ______ matrix.
O Row
O Column
O Square
O None

Explanation:
Definition of square matrix.

4. \left[\begin{array}{lll}1 & 3 & 5 \\ a & b & c \\ 3 & 5 & 7 \end{array}\right] is ______ matrix.
O Row
O Column
O Square
O None

Explanation:
As number of rows and columns are same. Thus, it is square matrix.

5. \left[\frac{1}{2}\right] is a ______ matrix.
O Row
O Column
O Square
O All of these

Explanation:
Here the matrix consists of one row and also one column and also same number of row and column which is square matrix. So, all of these is correct option.
\frac{1}{2}

6. [3] is ______ matrix.
O Row
O Column
O Square
O All of them

Explanation:
Here the matrix consists of one row and also one column and also same number of row and column which is square matrix. So, all of them is correct option.

7. A matrix in which number of row and number of columns are not equal is called _______ matrix.
O Square
O Rectangular
O Null
O None of these

Explanation:
Definition of Rectangular matrix.

8. \left[\begin{array}{lll}1 & 3 & 5 \\ a & b & c \end{array}\right] is ______ matrix.
O Row
O Column
O Square
O Rectangular

Explanation:
As number of rows and columns are not equal, so it is Rectangular matrix.

9. \left[\begin{array}{ll}0 & 0 \\ 0 & 0\end{array}\right] is ______ matrix.
O Row
O Zero
O Null
O Both b and c

Explanation:
As Zero or null matrix is same. So Both b and c is the correct option.

10. \left[\begin{array}{lll}0 \\ 0 \\ 0\end{array}\right] is ______ matrix.
O Row
O Zero
O Square
O None

Explanation:
As all the elements are zero in the matrix. That is why, its zero matrix.

11. The zero matrix for \left[\begin{array}{ll}1 & 2 \\ 2 & 4\end{array}\right] is ______.
O \left[\begin{array}{l}0 \end{array}\right]
O \left[\begin{array}{ll}0 & 0 \\ 0 & 0\end{array}\right]
O 0
O None of these

Answer: \left[\begin{array}{ll}0 & 0 \\ 0 & 0\end{array}\right]
Explanation:
The order zero matrix must be same according to the given matrix which is 2-by-2 .

12. The product of any matrix and zero matrix is a ______
O Identity
O Scalar
O Row
O Zero

Explanation:
When Zero (0) is multiplied to any number or matrix, the answer must be zero.

13. \left[\begin{array}{lll}1 & 0 & 0 \\ 0 & 5 & 0 \\ 0 & 0 & 9 \end{array}\right] is ______ matrix.
O Diagonal
O scalar
O Null
O None of these

Explanation:
A square matrix on which all elements are zero except diagonal elements is known as diagonal matrix.

14. Scalar matrix is a special case of ______ matrix.
O Identity
O Diagonal
O Both a & b
O None of these

Explanation:
If the diagonal elements in diagonal matrix are same, then it is called scalar matrix.

15. \left[\begin{array}{ll}2 & 0 \\ 0 & 3\end{array}\right] is ______ matrix.
O Diagonal
O scalar
O Null
O None of these

Explanation:
Here all the elements are zero except the diagonal elements.

16. The matrix \left[\begin{array}{cc}-1 & 0 \\ 0 & -1\end{array}\right] is ______ matrix.
O Row
O Scalar
O Null
O None of these

Explanation:
As the diagonal elements are same, so it is called scalar matrix.

17. Identity matrix is represented by ______.
O A
O I
O B
O None

18. Identity matrix is also called ______ matrix.
O Transpose
O Unit
O Symmetric
O None of these

19. \left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right] is ______ matrix.
O Rectangle
O Row
O Identity
O None

Explanation:
When the diagonal elements are “1” then matrix is called Identity matrix.

20. I_3 means ______
O \left[\begin{array}{l}1 \end{array}\right]
O \left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right]
O \left[\begin{array}{lll}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}\right]
O All of them

Answer: \left[\begin{array}{lll}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}\right]
Explanation:
I_3 means, Identity matrix having 3 rows and 3 columns.

21. I_3 is the identity matrix of order ______
O 3
O 3 -by-3
O 3 \times 3
O All of them

Explanation:
I_3 means, Identity matrix of 3 rows and columns. Here all the options show the same 3 rows and columns.

22. A=\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right] is ______ matrix.
O Diagonal
O Scalar
O Identity
O All of these

Explanation:
It is the diagonal matrix having same diagonal elements whcih is scalar matrix and also the diagonal element is 1 which is Identity matrix.
So all are correct options.

23. \left[\begin{array}{lll}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}\right] is ______ matrix
O Diagonal
O Scalar
O Identity
O All of these

Explanation:
It is the diagonal matrix having same diagonal elements whcih is scalar matrix and also the diagonal element is 1 which is Identity matrix.
So all are correct options.

24. The Transpose of \left[\begin{array}{ll}1 & -2 \\ 3 & \ 4\end{array}\right] is ______.
O \left[\begin{array}{ll}1 & -2 \\ 3 & \ 4\end{array}\right]
O \left[\begin{array}{ll}1 & 3 \\ -2 & \ 4\end{array}\right]
O \left[\begin{array}{ll}1 & 2 \\ 3 & \ 4\end{array}\right]
O All of these

O \left[\begin{array}{ll}1 & 3 \\ -2 & \ 4\end{array}\right]
Explanation:
When rows and columns are interchanged with each other, then it is called transpose of a matrix.

25. The transpose of \left[\begin{array}{l}1 \\ 2\end{array}\right] is ______.
O \left[\begin{array}{lll}1 & 2 & 3\end{array}\right]
O \left[\begin{array}{ll}1 & 2\end{array}\right]
O \left[\begin{array}{l}2 \\ 1\end{array}\right]
O None

Explanation:
When rows and columns are interchanged with each other, then it is called transpose of a matrix.

26. If A^t=A , then it is a ______ matrix
O Symmetric
O Skew-symmetric
O Diagonal
O None

Explanation:
When the transpose of matrix comes again that matrix is called symmetric matrix.

27. If D^t=-D then it is a ______ matrix.
O Symmetric
O Skew-symmetric
O Diagonal
O None

Explanation:
When the transpose of a matrix is equal to the negative of that matrix.

28. The matrix \left[\begin{array}{ll}1 & 2 \\ 2 & 4\end{array}\right] is ______ matrix.
O Symmetric
O Skew-symmetric
O scalar
O Diagonal

Explanation:
Take the transpose of
\left[\begin{array}{ll}1 & 2 \\ 2 & 4\end{array}\right]^t
\left[\begin{array}{ll}1 & 2 \\ 2 & 4\end{array}\right]
Thus, A^t=A

29. The matrix \left[\begin{array}{cc}0 & 1 \\ -1 & 0\end{array}\right] is ______ matrix.
O Symmetric
O Skew-symmetric
O scalar
O Diagonal

Explanation:
\left[\begin{array}{cc}0 & 1 \\ -1 & 0\end{array}\right]
Take the transpose, we get
\left[\begin{array}{cc}0 & -1 \\ 1 & 0\end{array}\right]
-\left[\begin{array}{cc}0 & 1 \\ -1 & 0\end{array}\right]
Thus the matrix is skew-symmetric.

1. Two matrices are conformable for addition/subtraction, if they are of the ________ order.
O Row
O Column
O Same
O Different

Explanation:
When the number of rows and columns of both the matrices are same, then both the matrices can be added/subtracted.

2. Addition of two matrices is obtained by adding the ________ elements of the matrices.
O Row
O Column
O Corresponding
O Different

Explanation:
For Addition of matrices, it must be noted to add the corresponding elements.

3. The addition of A=\left[\begin{array}{lll}1 & -2 & 3\end{array}\right] \ and \ B=\left[\begin{array}{lll}a & b & c\end{array}\right] is ________.
O Possible
O Not possible
O Identity matrix
O None of these

Explanation:
The addition of Matrix A and B is possible, because the order of both the matrices are same, that is:
1-by-3

4. The addition of A=\left[\begin{array}{lll}1 & 2 \end{array}\right] \ and \ B=\left[\begin{array}{lll}a & b & c\end{array}\right] is ________
O Possible
O Not possible
O Identity matrix
O None of these

Explanation:
The addition of Matrix A and B is not possible, because the order of both the matrices are not same.

5. The real number is multiplying to ________ elements of the matrix.
O Each
O Row
O Column
O All of these

Explanation:
The real number is multiplied to each elements of the matrix. Example is in MCQs No. 6

6. If A=\left[\begin{array}{lll}1 & -2 & 3\end{array}\right] \ then \ -3A= ________
O \left[\begin{array}{lll}-3 & -2 & -9\end{array}\right]
O \left[\begin{array}{lll}-1 & 2 & 3\end{array}\right]
O \left[\begin{array}{lll}-3 & 6 & -9\end{array}\right]
O None of the above

Answer: \left[\begin{array}{lll}-3 & 6 & -9\end{array}\right]
Explanation:
A=\left[\begin{array}{lll}1 & -2 & 3\end{array}\right]
-3A=-3\left[\begin{array}{lll}1 & -2 & 3\end{array}\right]
-3A=\left[\begin{array}{lll}-3 & 6 & -9\end{array}\right]

7. A+B=B+A is ________ property under addition.
O Commutative
O Associative
O Distributive
O None of these

8. A+(B+C)=(A+B)+C is ________ law under addition.
O Commutative
O Associative
O Distributive
O None of these

9. A+0 \ or \ 0+A = ________
O B
O A
O 0
O All of these

Explanation:
When zero (0) is added to any number, the answer should be that mumber.
0+3=3

10. \left[\begin{array}{lll}0 & 0 & 0 \\ 0 & 0 & 0\end{array}\right] is ________ matrix.
O Zero
O Null
O All of these

O Identity
O Scalar
O Null
O None of these

Explanation:
When a null matrix is added to any matrix, the result is that matrix which shows the additive identity.

O Identity
O Scalar
O Null
O None of these

Explanation:
When a null matrix is added to any matrix, the result is that matrix which shows the additive identity.

13. The additive identity of \left[\begin{array}{ll}1 & 3 \\ 2 & 4\end{array}\right] is ________
O \left[\begin{array}{ll}0 & 0 \\ 0 & 0\end{array}\right]
O \left[\begin{array}{ll}1 & 3 \\ 2 & 4\end{array}\right]
O \left[\begin{array}{ll}1 & 2 \\ 3 & 4\end{array}\right]
O \left[\begin{array}{ll}-1 & -3 \\ -2 & -4\end{array}\right]

Answer: \left[\begin{array}{ll}0 & 0 \\ 0 & 0\end{array}\right]
Explanation:

14. The additive identity of \left[\begin{array}{ll}1 & 3 \\ 2 & 4\end{array}\right] is ________
O {\left[\begin{array}{ll}0 \end{array}\right] }
O {\left[\begin{array}{ll}0 & 0 \\ 0 & 0\end{array}\right] }
O {\left[\begin{array}{ll}1 & 3 \\ 2 & 4\end{array}\right] }
O \left[\begin{array}{ll}-1 & -3 \\ -2 & -4\end{array}\right]

Answer: {\left[\begin{array}{ll}0 & 0 \\ 0 & 0\end{array}\right] }
Explanation:
The order of null matrix is same of that matrix.

15. In additive inverse, the sum of two matrices are ________
O Identity
O Zero
O a & b
O None of these

16. If A+B=0 then B is the _______ inverse of A
O Multiplicative
O Zero
O None of these

Explanation:
When the sum of two numbers is zero, then they are the additive inverse of each other.

17. If P+Q=0 , then P \ and \ Q are the ________ inverse of each other.
O Multiplicative
O Identity
O All of these

Explanation:
When the sum of two numbers is zero, then they are the additive inverse of each other.

18. The additive inverse of \left[\begin{array}{ll}1 & -3 \\ 2 & 4\end{array}\right] is ________
O \left[\begin{array}{ll}0 & 0 \\ 0 & 0\end{array}\right]
O \left[\begin{array}{ll}1 & 3 \\ 2 & 4\end{array}\right]
O \left[\begin{array}{ll}1 & 2 \\ 3 & 4\end{array}\right]
O \left[\begin{array}{ll}-1 & 3 \\ -2 & -4\end{array}\right]

Answer: \left[\begin{array}{ll}-1 & 3 \\ -2 & -4\end{array}\right]
Explanation:
when the sum of two matrices is zero OR the matrices of opposite signs.
\left[\begin{array}{ll}1 & -3 \\ 2 & 4\end{array}\right]+\left[\begin{array}{ll}-1 & 3 \\ -2 & -4\end{array}\right] =\left[\begin{array}{ll}0 & 0 \\ 0 & 0\end{array}\right]

1. When number of columns of First matrix equal to number of rows of Second matrix is ________ for multiplication.
O Conformable
O Not conformable
O Both a & b
O None of these

Explanation:
Rules for Multiplication

2. If A is m \times p \ \& \ B \ is \ p \times n then AB is ________ for multiplication.
O Conformable
O Not conformable
O Both a & b
O None of these.

Explanation:
This shows that A and B are conformable for multiplication because number of columns of first matrix equal to number of rows of second matrix. In short p=p

3. If the order of A \ is \ m \times p \ \& \ B \ is \ p \times n then the order of AB is ________
O m \times p
O p \times m
O m \times n
O p \times n

Explanation:
This shows the order of product AB. Thus
A_{m \times p} \times B_{p \times n}=AB_{m \times n}

4. If A \ is \ p \times n \ \& \ B \ is \ m \times p , then AB is ________ for multiplication.
O Conformable
O Not conformable
O Both a & b
O None of these.

Explanation:
Here No. of rows of first matrix equal to number of columns of second matrix. Thus it is not conformable for multiplication.

5. If A=\left[\begin{array}{ll}2 & 3 \\ 1 & 4\end{array}\right], B=\left[\begin{array}{l}3 \\ 5\end{array}\right] , then AB is ________
O Possible
O Not possible
O None of these

Explanation:
Here number of columns of A matrix equal to number of rows of B matrix. Thus, multiplication is possible.

6. If A=\left[\begin{array}{l}3 \\ 2\end{array}\right], B=\left[\begin{array}{l}1 \\ 2\end{array}\right] then multiplication of matrices are ________
O Possible
O Not possible
O Correct
O None

Explanation:
Here number of columns of A matrix not equal to number of rows of B matrix. Thus, multiplication is not possible.

7. Commutative law of multiplication of matrices may be ________
O A B=B A
O A B \neq B A
O Both a & b
O None of these

Explanation:
In multiplication of matrices, Sometime Commutative law is possible but mostly not possible.
See Example 10 and 11 in KPK book Page No. 23

8. A(BC)=(AB) C is called ________ law of multiplication.
O Commutative
O Associative
O Distributive
O None of these

9. Identity matrix is also known as ________ identity.
O Multiplicative
O All of these
O None of these

Explanation:
When Identity matrix is multiplied to any matrix, the answer will be that matrix.

10. (A+B)C=AC+AB is called ________ law of multiplication over addition.
O Commutative
O Associative
O Distributive
O None of these

11. A I=I A= ________
O A
O Null
O None

Explanation:
When Identity matrix “I” is multiplied to matrix “A”, the answer will be that matrix.

12. B I= ________
O A
O B
O I
O None

Explanation:
When Identity matrix “I” is multiplied to matrix “B”, the answer will be that matrix.

13. Multiplicative identity of A=\left[\begin{array}{ll}1 & 3 \\ 4 & 5\end{array}\right] is ________
O \left[\begin{array}{ll}1 & 4 \\ 3 & 5\end{array}\right]
O \left[\begin{array}{ll}1 & 4 \\ 3 & 5\end{array}\right]
O \left[\begin{array}{ll}-1 & -3 \\ -4 & -5\end{array}\right]
O \left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right]

Answer: \left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right]
Explanation:
The multiplicative identity for every matrix is:
\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right]

14. If A=\left[\begin{array}{lll}1 & 2 & 3 \\ 4 & 5 & 6\end{array}\right] \ and \ I=\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right] \ then \ AI= ________
O A
O I
O AI
O Not possible

Explanation:
Here number of columns of A matrix not equal to number of rows of I matrix. Thus, multiplication is not possible.

15. Transpose of \left[\begin{array}{ll}1 & 3 \\ 4 & 5\end{array}\right] is ________
O \left[\begin{array}{ll}1 & 4 \\ 3 & 5\end{array}\right]
O \left[\begin{array}{ll}1 & 6 \\ 5 & 7\end{array}\right]
O \left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right]
O All of these

Answer: \left[\begin{array}{ll}1 & 4 \\ 3 & 5\end{array}\right]
Explanation:
In transpoes, we interchanged the rows and columns.

16. \left(A^t\right)^t= ________
O A
O A^2
O All of these

Explanation:
If we take the transpose of a matrix and then again took the transpose, as a result the answer will be that matrix.

17. \left(c^t\right)^t= ________
O c^t
O -c
O I
O C

Explanation:
If we take the transpose of a matrix and then again took the transpose, as a result the answer will be that matrix.

18. (A B)^t= ________
O A^t B^t
O B^t A^t
O (B A)^t
O None of these

Explanation:
The transpose of the product of the matrices is equal to the product of their transposes but in the reverse order.

19. (B A)^t= ________
O A^t B^t
O B^t A^t
O (A B)^t
O None of these

Explanation:
The transpose of the product of the matrices is equal to the product of their transposes but in the reverse order.

20. (A+B)^t= ________
O A^t+B^t
O A^t-B^t
O (A B)^t
O A^t B^t

Explanation:

21 . (A-B)^t= ________
O A^t+B^t
O A^t-B^t
O (A B)^t
O A^t B^t

Explanation:

1. |A| is called as
O Determinant
O A symmetric
O None

Explanation:
This is the way to write Determinant

2. |A|=\left|\begin{array}{ll}a & b \\ c & d\end{array}\right|=
O a b+c d
O a b+a d
O abcd
O a d-b c

Explanation:
In this way, find the Determinant

3. \left|\begin{array}{cc}-2 & 2 \\ 3 & 5\end{array}\right|=
O 16
O -6
O -16
O None of these

Explanation:
(-2)(5)-(3)(2)
-10-6
-16

4. \left|\begin{array}{cc}4 & -2 \\ -2 & 1\end{array}\right|=
O 16
O 0
O 2
O 4

Explanation:
(4)(1)-(-2)(-2)
4-4
0

5. The determinant is a
O number
O Transpose
O Matrix

Explanation:

6. \left|\begin{array}{cc}4 & -2 \\ -2 & 1\end{array}\right|=
O Singular
O Non-singular
O None

Explanation:
(4)(1)-(-2)(-2)
4-4
0
As Determinant is 0. Thus, it is singular matrix.

7. |A|=0
O Singular
O Non-singular
O None

Explanation:
When the Determinant of matrix is zero, then the matrix is singular matrix.

8. |A| \neq 0
O Singular
O Non-singular
O None

Explanation:
When the Determinant of matrix is not equal zero, then the matrix is Non-singular matrix.

9. If A=\left[\begin{array}{ll}7 & 8 \\ 3 & 2\end{array}\right] \ then \ adj \ A=
O \left[\begin{array}{cc}2 & 8 \\ -3 & 7\end{array}\right]
O \left[\begin{array}{cc}2 & -8 \\ -3 & 7\end{array}\right]
O \left[\begin{array}{cc}7 & 8 \\ -3 & 2\end{array}\right]
O None

Answer: \left[\begin{array}{cc}2 & -8 \\ -3 & 7\end{array}\right]
Explanation:
In Adjoint, change the places of a and d with each other and change the signs of b and c.

10. A=\left[\begin{array}{ll}a & b \\ c & d\end{array}\right] \ the \ adjA \ =
O \left[\begin{array}{cc}d & -b \\ -c & a\end{array}\right]
O \left[\begin{array}{ll}a & c \\ d & a\end{array}\right]
O a d-b c
O All of these

Answer: \left[\begin{array}{cc}d & -b \\ -c & a\end{array}\right]
Explanation:
In Adjoint, change the places of a and d with each other and change the signs of b and c.

11. Multiplicative inverse of A is
O AB
O B
O A
O A^{-1}

Explanation:

12. A \cdot A^{-1}=A^{-1} \cdot A
O I
O A
O A^{-1}
O None

Explanation:
A matrix and its multiplicative inverse is equal to Multiplicative Identity “I”.

O B
O I
O A
O None

Explanation:
Definition of Multiplicative inverse

O A
O I
O F
O None

Explanation:
Definition of Multiplicative inverse

15. If A=\left[\begin{array}{cc}1 & 3 \\ 2 & -2\end{array}\right] \ then \ A^{-1}=
O -\frac{1}{8}\left[\begin{array}{cc}-1 & -3 \\ 2 & 2 \end{array}\right]
O -\frac{1}{8}\left[\begin{array}{cc} -2 & -3 \\ -2 & 1 \end{array}\right]
O \frac{1}{8}\left[\begin{array}{cc} -2 & -3 \\ -2 & 1 \end{array}\right]
O None of these

Explanation:
A=\left[\begin{array}{cc}1 & 3 \\ 2 & -2\end{array}\right]
We have
|A|=\left|\begin{array}{ll}1 & 3 \\ 2 & -2\end{array}\right|
|A|=(1)(-2)-(2)(-3)
|A|=-2-6
|A|=-8
Adj \ A=\left[\begin{array}{cc}-2 & -3 \\ -2 & 1\end{array}\right]
Now
A^{-1}=\frac{1}{-8} \left[\begin{array}{cc}-2 & -3 \\ -2 & 1\end{array}\right]

1. \left[\begin{array}{ll}0 & 0 \\ 0 & 0\end{array}\right] is
a. An identity matrix w.r.t multiplication
b. A column matrix
c. An identity matrix w.r.t addition
d. A row matrix

Explanation:
Zero (0) is called additive identity. Thus Zero or Null matrix is additive identity matrix.

2. The matrix \left[\begin{array}{cc}4 & 0 \\ 0 & -12\end{array}\right] is
a. A scalar matrix
b. \quad 2 \times 3 matrix
c. A diagonal matrix
d. None of these

Explanation:
A square matrix on which all elements are zero except diagonal elements is known as diagonal matrix.

3. If A=\left[\begin{array}{cc}-1 & -2 \\ 3 & 1\end{array}\right] , then adj A is equal to
a. \quad\left[\begin{array}{cc}-1 & -2 \\ 3 & 1\end{array}\right]
b. \quad\left[\begin{array}{cc}1 & 2 \\ -3 & -1\end{array}\right]
c. \left[\begin{array}{cc}-1 & 2 \\ 3 & 1\end{array}\right]
d. \quad\left[\begin{array}{cc}1 & -2 \\ 3 & 1\end{array}\right]

Explanation:
Let A=\left[\begin{array}{ll}a & b \\ c & d\end{array}\right]
As change the places of a \ and \ d with each other and change the signs of b \ and \ c . So
Adj \ A=\left[\begin{array}{cc}d & -b \\ -c & a\end{array}\right]

4. If A =\left[\begin{array}{ll}2 & 3 \\ 3 & 4\end{array}\right] then A^{-1}=
a. \quad\left[\begin{array}{cc}4 & 3 \\ -3 & 2\end{array}\right]
b. \quad\left[\begin{array}{cc}4 & -3 \\ -3 & 2\end{array}\right]
c. \quad\left[\begin{array}{cc}-2 & 3 \\ 3 & -4\end{array}\right]
d. \quad\left[\begin{array}{cc}-4 & 3 \\ 3 & -2\end{array}\right]

Explanation:
A=\left[\begin{array}{ll} 2 & 3 \\ 3 & 4 \end{array}\right]

|A|=\left|\begin{array}{ll}2 & 3 \\ 3 & 4\end{array}\right|
|A|=8-9
|A|=-1 \neq 0
{Adj} A=\left[\begin{array}{cc}4 & -3 \\ -3 & 2\end{array}\right]
Put the values in equation
A^{-1}=\frac{1}{-1}\left[\begin{array}{cc} 4 & -3 \\ -3 & 2 \end{array}\right]

A^{-1}=-\left[\begin{array}{cc} 4 & -3 \\ -3 & 2 \end{array}\right]

A^{-1}=\left[\begin{array}{cc} -4 & 3 \\ 3 & -2 \end{array}\right]

5. For what value of d is the 2 \times 2 matrix \left[\begin{array}{cc}1 & 1.5 \\ 2 & d\end{array}\right] not invertible?
a. -0.6
b. 0
c. 0.6
d. 3

Explanation:
Singular matrix is also called NOT invertible.
Thus |A|=0
\left|\begin{array}{ll}5 & 1.5 \\ 2 & d\end{array}\right|=0
5 \times d-2 \times 1.5=0
5d-3=0
5d=3
d=\frac{3}{5}
d=0.6

6. Suppose A and B are 2 \times 5 matrices, which of the following are the dimensions of the matrix A+B ?
a. 2 \times 5
b. 10 \times 10

Explanation:
For Addition of two matrices, the dimensions of the matrices must be same. Thus A+B have the dimensions 2 \times 5

7.Which of the following is the multiplicative inverse of \left[\begin{array}{ll}1 & 2 \\ 0 & 1\end{array}\right] is
a. \quad\left[\begin{array}{ll}1 & 2 \\ 0 & 1\end{array}\right]
b. \quad\left[\begin{array}{cc}1 & -2 \\ 0 & 1\end{array}\right]
c. \quad\left[\begin{array}{cc}-1 & 2 \\ 0 & 1\end{array}\right]
d. \quad\left[\begin{array}{ll}1 & -2 \\ 0 & -1\end{array}\right]

Explanation:
Let \ A=\left[\begin{array}{ll} 1 & 2 \\ 0 & 1 \end{array}\right]

Let \ |A|=\left|\begin{array}{ll}1 & 2 \\ 0 & 1\end{array}\right|
|A|=1-0
|A|=1 \neq 0
{Adj} A=\left[\begin{array}{cc}1 & -2 \\ 0 & 1\end{array}\right]
Put the values in equation
A^{-1}=\frac{1}{1}\left[\begin{array}{cc} 1 & -2 \\ 0 & 1 \end{array}\right]

A^{-1}=\left[\begin{array}{cc} 1 & -2 \\ 0 & 1 \end{array}\right]

8. The determinant of \left[\begin{array}{cc}4 & -1 \\ -9 & 2\end{array}\right] is
a. 17
b. 1
c. -1

Explanation:
Let \ |A|=\left|\begin{array}{ll}4 & -1 \\ -9 & 2\end{array}\right|
|A|=(4)(2)-(-9)(-1)
|A|=8-9
|A|=-1

### Mathematics Class 9 Notes (KPK) Chapter # 2

1. The number that can be expressed in the form of \frac{p}{q} \ where \ p \ and \ q are integers and q \neq 0 is called _______ numbers.
O Rational
O Irrational
O Imaginary
O None of these

Explanation:
Definition of rational number.

2. A rational in the form of \frac{p}{q} where \ p \ and q are _______
O Imaginary
O Complex
O Integers
O All of them

Explanation:

3. A rational in the form of \frac{p}{q} where q _______ 0.
O Equal
O Not equal
O Both a and b
O None of these

Explanation:
if q=0 then it becomes infinite.

4. The word Rational is derived from _______
O Ratio
O Not Ratio
O lota
O All of them

Explanation:

5. The rules of rational number for p \ and \ q are _______.
O \frac{p}{q}
O Integers
O q \neq 0
O All of them

Explanation:
These all are the rules for ratinal number.

6. Rational number is denoted by _______.
O Q
O Q^{\prime}
O Both a and b
O None of these

Explanation:

7. Irrational number is denoted by _______
O Q
O Q^{\prime}
O Both a and b
O None of these

Explanation:

8. The word irrational means_______
O Ratio
O Not Ratio
O lota
O None of these

Explanation:

9. Irrational numbers consist of all the numbers which are_______
O Rational
O Not rational
O lota
O None of these

Explanation:
Numbers other than rational numbers are called Irrational numbers.

10. 1\frac{3}{4} is_______
O Rational
O Irrational
O Complex
O None of these

Explanation:
1\frac{3}{4}=\frac{7}{4}=1.75
Terminating decimals are Rational numbers.

11. \sqrt{3} is_______
O Rational
O Irrational
O Complex
O None of these

Explanation:
\sqrt{3}=1.7320508076 \dots
Non-terminating and non-recurring (repeating) decimals are irrational numbers.

12. \pi is_______
O Rational
O Irrational
O Complex
O None of these

Explanation:
The approximate value of \pi \ is \ 3.1415926 \dots which is Non-terminating and non-recuring decimal. Thus \pi is an irrational number.

13. R is the symbol of _______ number.
O Rational
O Irrational
O Real
O Both a & b

Explanation:

14. The union of rational and irrational numbers is the set of _______ numbers.
O Complex
O lota
O Prime
O Real

Explanation:
Rational and irrational numbers are collectively called real number.

15. Q \cup Q^{\prime}= _______
O i
O Prime
O R
O N

Explanation:
The union of rational and irrational numbers is the set of Real numbers.

16. All the numbers on the number line are _______ numbers.
O i
O Complex
O Real
O None of these

Explanation:
Every number there is a point on the line. The number associated with a point is called the coordinate of that point on the line and the point is called the graph of the number.

17. All the numbers on the number line are _______ numbers.
O Rational
O Irrationa|
O Real
O All of them

Explanation:
Rational, Irrational and Real all are shown by number line.

18. -17 is _______ numbers.
O Whole
O Natural
O Integers
O All of them

Explanation:
Natural and whole number is always be positive.

19. A whole number is a number that does not contain_______
O Decimal
O Negative
O Fraction
O All of them

Explanation:
Whole number does not contain Decimal, Negative and Fraction.

20. All terminating and repeating decimals are_______
O Rational
O Irrational
O Complex
O All of them

Explanation:
This is the rule for rational number.

21. _______ decimals are Rational numbers.
O Terminating
O Repeating
O Both a & b
O None of these

Explanation:
Terminating and repeating decimals are rational numbers.

22. \frac{3}{8}=0.375 is _______ decimal.
O Terminating
O Repeating
O Both a & b
O None of these

Explanation:
A decimal number that contains a finit number of digits after the decimal point is called terminting decimal.

23. \frac{2}{15}=0.133 \ldots is _______ decimal
O Terminating
O Repeating
O Both a & b
O None of these

Explanation:
When some digits are repeated in same order after decimal point is called repeating decimal.

24. 0.1 \overline{3} is _______ decimal
O Terminating
O Repeating
O Both a & b
O None of these

Explanation:
The bar over the digit 3 means that this digit repeat forever.

25. Bar over the digit means that this digit is_______
O Terminated
O Repeated
O Both a & b
O None of these

Explanation:
The bar over the digit means that this digit repeat forever.

26. A decimal which is non-terminating and non-repeating is called _______ numbers.
O Rational
O Irrational
O Complex
O All of them

Explanation:
Non-terminating and non-recurring (repeating) decimals are irrational numbers.

27. The number line help in visualizing the set of _______ numbers.
O Complex
O Imaginary
O Real
O None of these

Explanation:
We assume that for any point on a line there is a real number.

28. For every real number there is a _______ on the line.
O Point
O Line
O Both a & b
O None of these

Explanation:
For every real number, there must be a point on the line.

29. The number associated with a point on the line is called the of _______ that point.
O Coordinate
O Zero
O Both a & b
O None of these

Explanation:
It is the graphical representation of real number on line.

30. The point on the number line is called the _______ of the number.
O Coordinate
O Zero
O Graph
O None of these

Explanation:

31. The numbers to the right of “0” on a number line are called _______ numbers.
O Positive
O Negative
O Complex
O None of these

Explanation:
The numbers greater than “0” are written on the right side of zero.

32. The numbers to the left of “O” on a number line are called _______ numbers.
O Positive
O Negative
O Complex
O None of these

Explanation:
The numbers less than “0” are written on the left side of zero.

33. 0 is _______
O Positive integers
O Negative integers
O Neither positive nor negative
O Not an integer

Explanation:
0 is the only number which is neither positive nor negative.

34. Repeating decimals are called _______ decimals.
O Recurring
O Non-Recurring
O Both a & b
O None of these

Explanation:
Repeating decimals are also called Recurring decimals.

35. Non-Repeating decimals are called _______ decimals.
O Recurring
O Non-Recurring
O Botha \& b
O None of these

Explanation:
The Non-Repeating decimals are also called Non-Recurring

36. \frac{5}{27}=0.185185185 \dots is called _______ recurring decimals.
O Terminating
O Non-Terminating
O Both a & b
O None of these

Explanation:
Here the number of digits repeated infinitely.

37. \frac{5}{7} is _______ number.
O Rational
O Irrational
O Imaginary
O None of these

Explanation:
\frac{5}{7}=0.714285714285 \dots
Non-terminating recurring (repeating) decimals are rational numbers.

38. \sqrt{36} is _______ .
O Rational
O Whole
O Natural
O All of them

Explanation:
\sqrt{36}=6
Thus 6 shows Natural, Whole and Rational number at a time.

39. \frac{14}{3} is _______ number.
O Rational
O Whole
O Both a & b
O Irrational

Explanation:
\frac{14}{3}=4.66666 \dots Non-terminating recurring (repeating) decimals are rational numbers.
1. The set of Real number is the union of two ________ sets.
O Zero
O New
O Disjoint
O None of these

Explanation:

2. Q \cap Q^{\prime}= ________ ]
O Q
O Q^{\prime}
O \emptyset
O All of them

Explanation:
The intersection of Rational and Irrational set is empty set.

3. The sum of two real number is also a real number is called ________ property w.r.t Addition.
O Closure
O Commutative
O Associative
O None of these

Explanation:
Statement for Closure Property.

4 . The ________ of two real number is also a real number is called closure property.
O Product
O Commutative
O Associative
O None of these

Explanation:
In Closure property the product of two real number is alway be a real number.

5. Example of closure property:
O 7+9=16
O 7 \times 9=63
O Both a & b
O None of these

Explanation:
In Closure property, the sum and product of two real numbers must be the real number. Thus both a & b obey closure property.

6. Commutative property w.r.t addition is ________
O a+b=b+c
O a+c=b+c
O a+b+c=a+b
O a+b=b+c

Explanation:
General form of Commutative property w.r.t Addition.

7. Commutative property w.r.t multiplication is ________
O a b=b c
O a c=b c
O a b c=a b
O a b=b c

Explanation:
General form of Commutative property w.r.t Multiplication.

8. Commutative property is ________
O a+b=b+a
O a b=b a
O Both a & b
O None of these

Explanation:
Both a & b Showed the Commutative property of Addition and Multiplication respectively.

9. Associative property w.rt Addition is ________
O a(bc)=(ab)c
O a+(b+c)=(a+b)+c
O Both a & b
O None of these

Explanation:
General form of Associative property w.r.t Addition.

10. Zero is called ________
O Both a & b
O None of these

Explanation:
Zero (0) is called Additive identity because adding “0” to a number does not change that number.

11. a+0=0+a=a is
O Both a & b
O None of these

Explanation:
Zero (0) is called Additive identity because adding “0” to a number does not change that number.

12. The product of real number and zero is________
O a
O That number
O Imaginary
O Zero

Explanation:
Any number multiplied to zero is always be zero.

13. 1 is called ________ w.r.t multiplication.
O Multiplicative identity
O Imaginary
O Multiplicative inverse
O None of these

Explanation:
1 is called Multiplicative identity because multiplying “1” to a number does not change that number.

14. a \times 1=1 \times a=a is ________ property. O Multiplicative identityg>
O Imaginary
O Multiplicative inverse
O None of these

Explanation:
1 is called Multiplicative identity because multiplying “1” to a number does not change that number.

15. The product of 1 and a number is________
O 10
O Zero
O That number
O None of these

Explanation:
1 is called Multiplicative identity because multiplying “1” to any number does not change that number.

16. The sum of two numbers is zero (0) is called________
O Both a & b
O None of these

Explanation:

17. If a+a^{\prime}=a^{\prime}+a=0 \ then \ a^{\prime} is called ________ of a .
O Both a & b
O None of these

Explanation:
When a real number and its opposite, the result will always be 0.

18. If a+(-a)=-a+a=0 \ then \ -a is called of ________ a .
O Both a & b
O None of these

Explanation:
When a real number and its opposite, the result will always be 0.

19. The product of two numbers is 1 is called________
O Multiplicative identity
O Imaginary
O Multiplicative inverse
O None of these

Explanation:
When a real number is multiplied by its inverse or reciprocal, the result will always be 1.

20. If a \cdot a^{-1}=a^{-1} \cdot a=1 \ then \ a^{-1} is called ________ of a .
O Multiplicative identity
O Imaginary
O Multiplicative inverse
O None of these

Explanation:
When the Product of two numbers is “1” then it is said to be Multiplicative inverse.

21 If a \cdot \frac{1}{a}=\frac{1}{a}, a=1 \ then \ \frac{1}{a} is called of ________ a .
O Multiplicative identity
O Imaginary
O Multiplicative inverse
O None of these

Explanation:
When the Product of two numbers is “1” then it is said to be Multiplicative inverse.

22. Distributive Property of Multiplication over Addition is ________
O a(b+c)=ab+ac
O (b+c)a=ba+ca
O Both a & b
O None of these

Explanation:
Both a & b showed the Distributive Property of Multiplication over Addition

23. If a=a , then it is ________ property.
O Transitive
O Symmetric
O Reflexive
O None of these

Explanation:
Every number is equal to itself is known as Reflexive property.

24. If a=b , then also b=a , it is ________ property.
O Transitive
O Symmetric
O Reflexive
O None of these

Explanation:
By interchanging the sides of an equation doesn’t effect the result is known as symmetric Prperty.

25. If a=b \ and \ b=c then a=c , it is ________ property.
O Transitive
O Symmetric
O Reflexive
O None of these

Explanation:
If a equal to b under a rule and b equal to c under the same rules then  a equal to  c is known as transitive property.

26. If y=x^2 \ then \ also \ x^2=y , it is ________ property.
O Transitive
O Symmetric
O Reflexive
O None of these

Explanation:
By interchanging the sides of an equation doesn’t effect the result is known as symmetric Prperty.

27. I x+y=z \ and \ z=a+b then x+y=a+b , it is ________ property.
O Transitive
O Symmetric
O Reflexive
O None of these

Explanation:
If a equal to b under a rule and b equal to c under the same rules then  a equal to  c is known as transitive property.

28. If 3=3 , then it is ________ property. O Transitive>
O Symmetric
O Reflexive
O None of these

Explanation:
Every number is equal to itself is known as Reflexive property.

29. If a=b , then also a+c=b+c , it is ________ property of equality. O Ad>ditive
O Multiplicative
O Both a & b
O None of these

Explanation:
If we add the same number or expression on both sides of an equation, the equation does not change which means both the sides remain equal.

30. If a=b then also ac=bc , it is ________ property of equality.
O Multiplicative
O Both a & b
O None of these

Explanation:
If we Multiply the same number or expression on both sides of an equation, the equation does not change which means both the sides remain equal.

31. If a+c=b+c then a=b , it is Cancellation property w.r.t ________
O Multiplication
O Both a & b
O None of these

Explanation:
In this, cancelled the non-zero common factor from both side of the equation by Adding or Subtraction.

32. If ac=bc then a=b , it is Cancellation property w.r.t ________
O Multiplication
O Both a & b
O None of these

Explanation:
In this, cancelled the non-zero common factor from both side of the equation by Multiplication or Divison.

33. Trichotomy property is used for ________ two numbers.
O Increasing
O Decreasing
O Comparing
O Equating

Explanation:
See MCQs No. 34

34. Trichotomy property must be true for ________
O a=b
O a>b
O a < b
O All of them

Explanation:
Trichotomy property is used for compare two numbers.

35. Trichotomy property must be true for________
O 5=5
O 3 < 5
O Both a & b
O None of these

Explanation:
Trichotomy property is used for compare two numbers.

36. If a > b and b>c then a > c , it is ________ property of inequality.
O Multiplicative
O Transitive
O All of them

Explanation:
If a greater than b under a rule and b greater than c under the same rule then  a greater than  c is known as transitive property of inequlity.

37. If a < b & b < c then a < c, it is ________ property. O Additive>
O Multiplicative
O Transitive
O All of them

Explanation:
If a less than b under a rule and b less than c under the same rule then  a less than  c is known as transitive property of inequlity.

38. If a > b then a+c > b+c , it is ________ property of inequlity.
O Multiplicative
O Transitive
O All of them

Explanation:
If we add the same number or expression on both sides of an inequality, but the result will remain the same. i.e. left side is greater than right side.

39. If a < b then a+c < b+c , it is ________ property.
O Multiplicative
O Transitive
O All of them

Explanation:
If we add the same number or expression on both sides of an inequality, but the result will remain the same. i.e. left side is less than right side.

40. If x > 5 then ________
O x \times 2 > 5 \times 2
O x \times 2 < 5 \times 2
O Both a & b
O None of these

Answer: x \times 2 > 5 \times 2
Explanation:
If we multiply the same number or expression on both sides of an inequality, but the result will remain the same. i.e. left side is greater than right side.
Note:
The number should be positive.

41. If x>5 then ________
O x \times -2 > 5 \times -2
O x \times -2 < 5 \times -2
O Both a & b
O None of these

Answer: x \times -2 < 5 \times -2
Explanation:
If we multiply the same Negative number to both sides of an inequality, the result will changed. i.e. left side becomes less than right side.

42. For c > 0 and a < b then Multiplicative property ________
O ac < bc
O ac > bc
O Both a & b
O None of these

Explanation:
If we multiply the same Positive number to both sides of an inequality, the result will remain same. i.e. left side is less than right side.

43. For c < 0 and a < b then Multiplicative property ________
O ac < bc
O ac > bc
O Both a & b
O None of these

Explanation:
If we multiply the same Negative number to both sides of an inequality, the result will changed. i.e. left side becomes greaer than right side.

44. For c < 0 and a > b then Multiplicative property ________
O ac < bc
O ac > bc
O Both a & b
O None of these

Explanation:
If we multiply the same Negative number to both sides of an inequality, the result will changed. i.e. left side becomes less than right side.

1. In \sqrt[n]{a},\ then \ \sqrt{ \quad } is called
O Index
O All of them

Explanation:

2. In \sqrt[n]{a}, \ then \ a is called
O Index
O All of them

Explanation:

3. In \sqrt[n]{a}, \ then \ n is called
O Index
O All of them

Explanation:

4. The exponential form of \sqrt[n]{a} is
O a^n
O a^2
O a^{\frac{2}{n}}
O a^{\frac{1}{n}}

Explanation:
It is the General Exponential form of any radical form

5. \sqrt{2} , the index is
O 0
O 1
O 2
O All of them

Explanation:
If the index is not given means it is 2 which we cannot write.

6. \sqrt{1}=
O 1
O 0
O -1
O 2

Explanation:
Square root of 1 will always be 1.

7. \sqrt[3]{1}=
O 1
O 0
O -1
O 2

Explanation:
Cube root of 1 will always be 1

8. \sqrt[3]{1}=
O 1
O 0
O -1
O 2

Explanation:
Forth root of 1 will always be 1.

9. \sqrt{36}=
O 6
O 0
O -6
O 4

Explanation:
\sqrt{36}=\sqrt{6^2}
\sqrt{36}=(6^2)^\frac{1}{2}
\sqrt{36}=6

10. \sqrt[3]{216}=
O 4
O 5
O 6
O 7

Explanation:
\sqrt[3]{216}=\sqrt[3]{6^3}
\sqrt[3]{216}=(6^3)^\frac{1}{3}
\sqrt[3]{216}=6

11. \sqrt[4]{256}=
O 4
O 5
O 6
O 7

Explanation:
\sqrt[4]{256}=\sqrt[4]{4^4}
\sqrt[4]{256}=(4^4)^\frac{1}{4}
\sqrt[4]{256}=4

12. \sqrt[4]{625}=
O 4
O 5
O 6
O 7

Explanation:
\sqrt[4]{625}=\sqrt[4]{5^4}
\sqrt[4]{625}=(5^4)^\frac{1}{4}
\sqrt[4]{625}=5

13. \sqrt[4]{1296}=
O 4
O 5
O 6
O 7

Explanation:
\sqrt[4]{1296}=\sqrt[4]{6^4}
\sqrt[4]{1296}=(6^4)^\frac{1}{4}
\sqrt[4]{1296}=6

14. If x^2=16 , then x=
O 4
O -4
O Both a & b
O None of these

Explanation:
This means what numbers squared becomes 16. Thus x \ can \ be \ 4 \ or \ -4 \ like \ (4)^2=16 \ and \ also \ (-4)^2=16.
Hence the value of x=\pm 4 .

15. If x=\sqrt{16} , then x=
O 4
O -4
O Both a & b
O None of these

Explanation:
Here x is the principal square root of 16, which has always a positive value such is x=4 .

16. \sqrt[4]{1296} is called root
O 4^{th}
O 5^{th}
O 2^{nd}
O Square

Explanation:
Here is the index of 4, thus it is called 4^{th} root.

17. \sqrt[3]{64}=
O 4
O -4
O Imaginary
O None of these

Explanation:
\sqrt[3]{64}=\sqrt[3]{4^3}
\sqrt[3]{64}=(4^3)^\frac{1}{3}
\sqrt[3]{64}=4

18. \sqrt[3]{-64}=
O 4
O -4
O Imaginary
O None of these

Explanation:
If a is negative, then n must be odd for the nth root of a to be a real number.
\sqrt[3]{-64}=\sqrt[3]{(-4)^3}
\sqrt[3]{-64}=\left[(-4)^3\right]^\frac{1}{3}
\sqrt[3]{-64}=-4

19. \sqrt{-64}=
O 4
O -4
O Imaginary
O None of these

Explanation:
If radicand is negative, then index must be odd, here the index is 2 which is even.
Hence, \sqrt{-64}= imaginary

20. \sqrt[n]{0}=
O 1
O 0
O n
O -0

Explanation:
If a is zero, then
\sqrt[n]{0}=0

21. \sqrt[n]{ab}=
O \sqrt[n]{a} \cdot \sqrt[n]{b}
O \sqrt[n]{a}+\sqrt[n]{b}
O \sqrt[n]{a} \cdot \sqrt{b}
O \sqrt{a} \cdot \sqrt[n]{b}

Explanation:
It is the product rule of Radical.

22. \sqrt[n]{\frac{a}{b}}=
O \sqrt[n]{a} \cdot \sqrt[n]{b}
O \sqrt[n]{a}+\sqrt[n]{b}
O \sqrt[n]{a}-\sqrt[n]{b}
O \frac{\sqrt[n]{a}}{\sqrt[n]{b}}

Explanation:
It is the Quotient rule of Radicand.

23. 2 \sqrt{\frac{150 x y}{3 x}}=
O 2 \sqrt{y}
O 10 \sqrt{y}
O 2 \sqrt{2 y}
O 10 \sqrt{2 y}

Explanation:
2 \sqrt{\frac{150 x y}{3 x}}=2 \sqrt{50y}
2 \sqrt{\frac{150 x y}{3 x}}=2 \sqrt{25 \times 2y}
2 \sqrt{\frac{150 x y}{3 x}}=2 \times 5 \sqrt{2y}
2 \sqrt{\frac{150 x y}{3 x}}=10 \sqrt{2y}

24. \sqrt[n]{a}=
O a^{\frac{m}{n}}
O a^{\frac{1}{n}}
O a^{\frac{n}{m}}
O All of them

Explanation:
It is the exponential form of radical.

25. a^{\frac{m}{n}}=
O \sqrt[n]{a}
O \sqrt[n]{a^m}
O Both a & b
O None of these

Explanation:
a^{\frac{m}{n}}=(a^m)^\frac{1}{n}
a^{\frac{m}{n}}=\sqrt[n]{a^m}

26. \sqrt{13} is ________ form
O Exponential
O Cubic

Explanation:
This is the way to represent the radical form.

27. 13^2 is ________ form.
O Exponential
O Cubic

Explanation:
This is the way to represent the Exponential form.

28. 2^4=
O 16
O -16
O Both a & b
O None of these

Explanation:
2^4= 2 \times 2 \times 2 \times 2
2^4= 16

29. -2^4=
O 16
O -16
O Both a & b
O None of these

Explanation:
-2^4= -(2 \times 2 \times 2 \times 2)
-2^4= -16

30. (-2)^4=
O 16
O -16
O Both a & b
O None of these

Explanation:
(-2)^4= -2 \times -2 \times -2 \times -2
(-2)^4= 16

31. \sqrt[n]{a^m}=
O a^{\frac{m}{n}}
O a^{\frac{1}{n}}
O a^{\frac{n}{m}}
O All of them

Explanation:
\sqrt[n]{a^m}=(a^m)^\frac{1}{n}
\sqrt[n]{a^m}= a^{\frac{m}{n}}

1. a^m \cdot a^n=
O a^{\frac{m}{n}}
O a^{m-n}
O a^{\frac{n}{m}}
O a^{m+n}

Explanation:
To multiply powers of the same base, keep the base same and add the exponents.

2. \frac{a^m}{a^n}=
O a^{\frac{m}{n}}
O a^{m-n}
O a^{\frac{n}{m}}
O a^{m+n}

Explanation:
To divide two expressions with the same bases and different exponents, keep the base same and subtract the exponents.

3. \left(a^m\right)^n=
O a^{\frac{m}{n}}
O a^{m-n}
O a^{m n}
O a^{m+n}

Explanation:
To raise an exponential expression to a power, keep the base same and multiply the exponents.

4. (a b)^n=
O a b^n
O a^n b
O a b
O a^n b^n

Explanation:
To multiply different bases with same exponent, keep the exponent same and multiply the expressions with the same exponent.

5. \left(\frac{a}{b}\right)^n=
O \frac{a^n}{b^n}
O \frac{a^n}{b}
O \frac{a}{b^n}
O All of these

Explanation:
To divide the two expressions with the same exponent, keep the exponent same and divide the expressions.

6. a^0=
O a
O 1
O 0
O None of theses

Explanation:
Any non-zero number raised to the zero power equals one.

7. 400^{\circ}=
O 400
O 1
O 0
O None of these

Explanation:
Any non-zero number raised to the zero power equals one.

8. a^{-n}=
O \frac{1}{a^n}
O \frac{1}{a^{-n}}
O a^n
O None of these

Explanation:
If we convert the Numerator having a Negative exponent, moves it to Denominator and the exponent becomes positive.

9. \frac{1}{a^{-n}}=
O \frac{1}{a^n}
O \frac{1}{a^{-n}}
O a^n
O None of these

Explanation:
If we convert the Denominator having a Negative exponent, moves it to Numerator and the exponent becomes positive.

10. (-a)^3 \times(-a)^4=
O a
O a^7
O a^4
O -a^7

Explanation:
(-a)^3 \times(-a)^4=(-a)^{3+4}
(-a)^3 \times(-a)^4=(-a)^7
(-a)^3 \times(-a)^4=-a^7

11. \left(-2 a^2 b^3\right)^3=
O 8 a^6 b^9
O 2 a^2 b^3
O 2 a b
O -8 a^6 b^9

Explanation:
\left(-2 a^2 b^3\right)^3=\left(-2 \right)^3 a^{2 \times 3} b^{3 \times 3}
\left(-2 a^2 b^3\right)^3=-8 a^6 b^9

12. \frac{a^0 \cdot b^0}{2}=
O 1
O 2
O \frac{1}{2}
O 0

Explanation:
\frac{a^0 \cdot b^0}{2}=\frac{1 \times 1}{2}
\frac{a^0 \cdot b^0}{2}=\frac{1}{2}

13. \left(\frac{a^2}{b^4}\right)^{\frac{3}{2}}=
O \frac{a^2}{b^4}
O \frac{a}{b}
O \frac{a^3}{b^6}
O None of these

Explanation:
\left(\frac{a^2}{b^4}\right)^{\frac{3}{2}}=\frac{a^{2 \times \frac{3}{2}}}{b^{4 \times \frac{3}{2}}}
\left(\frac{a^2}{b^4}\right)^{\frac{3}{2}}=\frac{a^{1 \times 3}}{b^{2 \times 3}}
\left(\frac{a^2}{b^4}\right)^{\frac{3}{2}}=\frac{a^3}{b^6}

14. (\sqrt[3]{a})^{\frac{1}{2}}=
O a^{\frac{1}{6}}
O a^{\frac{1}{3}}
O a^{\frac{3}{2}}
O a^{\frac{2}{3}}

Explanation:
See Ex # 2.4 Q No. 3 Part (iv)

15. \sqrt[8]{x^8} \cdot \sqrt[8]{x^8}=
O \sqrt[8]{x^8}
O x^8
O x
O x^2

Explanation:
\sqrt[8]{x^8} \cdot \sqrt[8]{x^8}=\left(x^8 \right)^{\frac{1}{8}} \left(x^8 \right)^{\frac{1}{8}}
\sqrt[8]{x^8} \cdot \sqrt[8]{x^8}=x \cdot x
\sqrt[8]{x^8} \cdot \sqrt[8]{x^8}=x^2

1. A number in the form of a+bi where a \ and \ b are real number is called_______ number.
O Whole
O Natural
O Real
O Complex

Explanation:
Definintion of Complex Number.

2. In complex number a+bi, \ "a" is called part.
O Real
O Imaginary
O Conjugate
O Transpose

Explanation:

3. In complex number a+bi, \ "b" is called part.
O Real
O Imaginary
O Conjugate
O Transpose

Explanation:

4. The conjugate of a+bi is
O -a-b i
O -a+b i
O a-b i
O None of these

Explanation:
A conjugate of a complex number is obtained by changing the sign of imaginary part.

5. \overline{a+bi}=
O -a-b i
O -a+b i
O a-b i
O None of these

Explanation:
The conjugate is denoted by \overline{a+bi}

6. Let Z_1=a+b i \ and \ Z_2=c+di \ then \ Z_1=Z_2 if
O a=c
O b=d
O Both a & b
O None of these

Explanation:
Z_1=Z_2 if real parts are equal i.e. a=c and imaginary parts are equal i.e. b=d .

7. i^2=
O 1
O i
O -1
O All of them

Explanation:
i^2=-1

8. i=
O 1
O i
O -1
O \sqrt{-1}

Explanation:
i=\sqrt{-1}

9. 2i(4-5i)
O 8-5i
O 10-8i
O 10+8i
O None of these

Explanation:
2i(4-5i)=8i-10i^2
2i(4-5i)=8i-10(-1)
2i(4-5i)=8i+10
2i(4-5i)=10+8i

10. (3-2i)(3+2i)=
O 2+3i
O 3-2i
O -13
O 13

Explanation:
(3-2i)(3+2i)=3^2-(2i)^2
(3-2i)(3+2i)=9-4i^2
(3-2i)(3+2i)=9-4(-1)
(3-2i)(3+2i)=9+4
(3-2i)(3+2i)=13
1. The additive inverse of \sqrt{5} is
a. -\sqrt{5}
b. \frac{1}{\sqrt{5}}
c. \sqrt{-3}
d. -5

Explanation:
When the sum of two numbers is zero (0)
OR
Additive inverse is obtained by changing the sign. So, the additive inverse of
\sqrt{5} \ is \ -\sqrt{5}

2. 2(3+4)=2 \times 3+2 \times 4 , here the property used is.
a. Commutative
b. Associative
c. Distributive
d. Closure

Explanation:
Distributive Property is:
a(b+c)=ab+ac

3. \sqrt{-1} \times \sqrt{-1}=
a. 1
b. i
c. -1
d. 0

Explanation:
\sqrt{-1} \times \sqrt{-1}
\sqrt{-1} \times \sqrt{-1}=i \times i \qquad \because \sqrt{-1}=i
\sqrt{-1} \times \sqrt{-1}=i^2
\sqrt{-1} \times \sqrt{-1}=-1

4. Which of the following represents numbers greater than -3 but less than 6 ?

Answer: {x: -3 < x < 6}
Explanation:

5. If n=8 \ and \ 16 \times 2^m=4^{n-8} , then \ m= ?
a. -4
b. -2
c. 0
d. 8

Explanation:
16 \times 2^m=4^{n-8}
Put \ n=8
16 \times 2^m=4^{8-8}
2^4 \times 2^m=4^0
2^4 \times 2^m=1
2^m=\frac{1}{2^4}
2^m=2^{-4}
Thus \ m=-4

6. (i).(-i)=
a. 1
b. -1
c. -i
d. i

Explanation:
(i).(-i)=-i^2
(i).(-i)=-(-1)
(i).(-i)=-1

7. The multiplicative identity of real number is
a. 0
b. 1
c. -1
d. R

Explanation:
1 is called Multiplicative identity because multiplying “1” to a number does not change that number.

8. 0 is
a. a positive integer
b. a negative integer
c. neither positive nor negative
d. not an integer

Explanation:
0 is the only number which is neither positive nor negative.

9. For i=\sqrt{-1} , if \ 3i(2+5i)=x+6i , then \ x= ?
a. 5
b. -15
c. 5i
d. 15i

Explanation:
As we have
3i(2+5i)=x+6i
6i+15i^2=x+6i
-15=x

10. \sqrt{0}=
a. 0
b. 1
c. -1
d. Not defined

11. \sqrt{-(-9)^2}= ?
a. 9
b. 9+i
c. 9-i
d. 9i

Explanation:
\sqrt{-(-9)^2}=\sqrt{-81}
\sqrt{-(-9)^2}=\sqrt{-1 \times 81}
\sqrt{-(-9)^2}=\sqrt{-1} \times \sqrt{81}
\sqrt{-(-9)^2}=i \times 9 \qquad \because \sqrt{-1}=i
\sqrt{-(-9)^2}=9i

### Mathematics Class 9 Notes (KPK) Chapter # 3

1. __________ is a way of writing numbers that are too big or too small to be easily written in decimal form.
O Standard notation
O Binary notation
O Scientific notation
O All of them

Explanation:

2. Scientific notation is a way of writing numbers that are too big or too small to be easily written in _________ form.
O Standard
O Binary
O Decimal
O All of them

Explanation:

3. General form of scientific notation is
O x=a \times 10^n
O x=a \times 10^a
O x=a \times 10^{10}
O None of them

Explanation:
Here "a" is real number greater than or equal to 1 but less than 10 and integer power n of 10. i.e.
x=a \times 10^n

4. In standard to scientific notation, the decimal should place after ____________ non – zero digit.
O Zero
O First
O Second
O All of them

Explanation:
Rules to convert standard to scientific notation.

5. From standard to scientific notation, if decimal moved towards left, then power of 10 will be ____________
O Positive
O Negative
O Both of them
O None of them

Explanation:
Rules to convert standard to scientific notation.

6. From standard to scientific notation, if decimal moved towards right, then power of 10 will be ____________
O Positive
O Negative
O Both of them
O None of them

Explanation:
Rules to convert standard to scientific notation.

7. From scientific to standard notation, it the exponent of 10 is negative, the decimal will move towards ___________
O Left
O Right
O Both of them
O None of them

Explanation:
Rules to convert Scientific to Standard notation.

8. From scientific to standard notation, it the exponent of 10 is positive, the decimal will move towards ___________
O Left
O Right
O Both of them
O None of them

Explanation:
Rules to convert Scientific to Standard notation.

9. The scientific form of 16700000 is
O 1.67 \times 10^7
O 1.67 \times 10^8
O 167.00 \times 10^7
O All of them

Explanation:
Place the Decimal after first non-zero digit which is 1.
Count the digits towards left which are 7
Here the decimal moves towards left, So power of 10 is positive. Thus,
1.67 \times 10^7

10. The scientific form of 0.00000039 is
O 3.9 \times 10^7
O 3.9 \times 10^{-7}
O 39 \times 10^8
O 39 \times 10^{-8}

Explanation:
Place the Decimal after first non-zero digit which is 3.
Count the digits towards right which are 7
Here the decimal moves towards Right, So power of 10 is Negative. Thus,
3.9 \times 10^{-7}

11. The scientific form of 0.05 \times 10^{-3} is
O 5 \times 10^{-5}
O 5.0 \times 10^{-5}
O 5 \times 10^5
O Both a & b

Explanation:
0.05 \times 10^{-3}=5.0 \times 10^{-2} \times 10^{-3}
0.05 \times 10^{-3}=5.0 \times 10^{-2-3}
0.05 \times 10^{-3}=5.0 \times 10^{-5}

12. The standard form of 3.15 \times 10^{-6} is
O 0.0000003
O 0.00000315
O 315
O 3150000

Explanation:
If the Power of 10 is Negative, So Move the decimal towards Left.
Here the power is -6 then move the decimal upto 6 Digits towards left. Thus,
0.00000315

13. The standard form of 3.15 \times 10^{6} is
O 0.0000003
O 0.00000315
O 315
O 3150000

Explanation:
If the Power of 10 is Positive, So Move the decimal towards Right.
Here the power is 6 then move the decimal upto 6 Digits towards Right. Thus,
3150000

14. The standard form of -2.6 \times 10^{6} is
O -2.6
O 0.0000026
O 2600000
O -2600000

Explanation:
If the Power of 10 is Positive, So Move the decimal towards Right.
Here the power is 6 then move the decimal upto 6 Digits towards Right. Thus,
-2600000

1. The exponential form of \log _a y=x is
O a^y=x
O y=x
O a^x=y
O a^x=y

Explanation:
We called \log _a y=x \ like \ log \ of \ y \ to \ the \ base \ a \ equal \ to \ x.

2. The logarithm form of a^x=y is
O \log _a y=x
O \log _a x=y
O \log x
O \log y

Explanation:
If a^x=y then the index x is called the logarithm of y to the base a and wirtthe as:
\log _a y=x

3. The logarithm form of 2^{-6}=\frac{1}{64} is
O \log _{-6} \frac{1}{64}=2
O \log _{64} 2=\frac{1}{64}
O \log _2 \frac{1}{64}=-6
O \log _a y=x

Explanation:
General form of Conversion is:
a^x=y \longleftrightarrow \log _a y=x

4. The logarithm form of 10^{\circ}=1 is
O \log _{10} 1=1
O \log _{10} 1=0
O \log _0 1=1
O \log _a y=x

Explanation:
a^x=y \longleftrightarrow \log _a y=x

5. The logarithm form of x^{\frac{3}{4}}=y is
O \log _y x=\frac{3}{4}
O \log _{\frac{3}{4}} y=x
O \log _x y=\frac{3}{4}
O All of them

Explanation:
a^x=y \longleftrightarrow \log _a y=x

6. The exponential form of \log _2 \frac{1}{128}=-7 is
O 2^{-7}=\frac{1}{128}
O 2^{-7}=128
O -7^2=\frac{1}{128}
O a^x=y

Explanation:
\log _a y=x \longleftrightarrow a^x=y

7. The exponential form of \log _a a=1 is
O 1^a=1
O 1=1
O a^1=1
O None of them

Explanation:
\log _a y=x \longleftrightarrow a^x=y

8. The exponential form of \log_a 1=0 is
O 1^a=0
O 1=1
O a^0=1
O a=1

Explanation:
\log _a y=x \longleftrightarrow a^x=y

9. The exponential form of \log_4 \frac{1}{8}=\frac{-3}{2} is
O 4^{\frac{-3}{2}}=\frac{1}{8}
O 4^{\frac{1}{3}}=\frac{-3}{2}
O Both a & b
O None of them

Explanation:
\log _a y=x \longleftrightarrow a^x=y

10. The exponential form of \log _{\sqrt{5}} 125=x is
O (\sqrt{5})^x=125
O \left(5^{\frac{1}{2}}\right)^x=125
O 5^{\frac{x}{2}}=125
O All of them

Explanation:
\log _a y=x \longleftrightarrow a^x=y

11. The exponential form of \log _3(5 x+1)=2 is
O 3^x=5
O 3^2=5 x+1
O 3^2=5 x
O None of them

Explanation:
\log _a y=x \longleftrightarrow a^x=y

12. In \log _{\sqrt{5}} 125=x , the Value of x is
O 5
O 125
O 6
O None of them

Explanation:
See Ex # 3.2
Q No. 3
Part (i)

13. In log x=-3 , the Value of x is
O \frac{1}{64}
O 64
O 4
O -3

Explanation:
See Ex # 3.2
Q No. 3
Part (ii)

14. In \log _{81} 9=x , the Value of x is
O \frac{1}{2}
O 81
O 9
O -3

Explanation:
See Ex # 3.2
Q No. 3
Part (iii)

15. In \log _3(5 x+1)=2 , the Value of x is
O \frac{5}{8}
O \frac{8}{5}
O 5
O 2

Explanation:
See Ex # 3.2
Q No. 3
Part (iv)

16. In \log _2 x=7 , the Value of x is
O 2
O 7
O 0
O 128

Explanation:
See Ex # 3.2
Q No. 3
Part (v)

17. In \log _x 0.25=2 , the Value of x is
O \frac{5}{10}
O \frac{1}{2}
O 0.5
O All of them

Explanation:
See Ex # 3.2
Q No. 3
Part (vi)

18. In \log _x(0.001)=-3 , the Value of x is
O 1
O 10
O 0
O All of them

Explanation:
See Ex # 3.2
Q No. 3
Part (vii)

19. In \log _x \frac{1}{64}=-2 , the Value of x is
O 64
O 2
O 8
O All of them

Explanation:
See Ex # 3.2
Q No. 3
Part (viii)

20. In \log _{\sqrt{3}} x=16 , the Value of x is
O 6561
O 4
O 3
O None of these

Explanation:
See Ex # 3.2
Q No. 3
Part (ix)

1. Logarithms having base 10 are called________ Logarithms
O Natural
O Common
O Briggs
O Both b & c

Explanation:

2. Common logarithm is also called __________ logarithm
O Natural
O Briggs
O Both a & b
O None of these

Explanation:

3. The digit before the decimal point or integral part is called _____________
O Characteristics
O Mantissa
O Both a & b
O None of these

Explanation:
In 1.5377 Characteristics is 1.

4. The decimal fraction part is called ________
O Characteristics
O Mantissa
O Both a & b
O None of these

Explanation:
In 1.5377 Mantissa \ is \ .5377 .

5. In 1.5377 , characteristics is
O 1
O .5377
O 1.5377
O None of these

Explanation:
The digit before the decimal point or Integral part is called characteristics.

6. In 1.5377 , Mantissa is
O 1
O .5377
O 1.5377
O None of these

Explanation:
The decimal fraction part is Mantissa.

7. The mean difference digits are added to ______________
O Characteristics
O Mantissa
O Both a & b
O None of these

Explanation:
The mean difference is the third part to find the mantissa and it is added to mantissa.

8 The mantissa of 763.5 is
O .8825
O .8828
O 2
O 76

Explanation:
(i). First ignore the decimal point
(ii). Take first two digits e.g. 76 and proceed along this row until we come to column headed by third digit 3 of the number which is 8825
(iii). Now take fourth digit i.e. 5 and proceed along this row in mean difference column which is 5.
Thus Mantissa of 763.5 \ is \ .8828

9. The characteristics of 982.5 is
O 0
O 2
O 3
O 4

Explanation:
First convert 982.5 to Scientific form:
9.825 \times 10^2
Thus Characteristics is 2

10. The characteristics of 7824 is
O 0
O 1
O 2
O 3

Explanation:
First convert 7824 to Scientific form:
7.824 \times 10^3
Thus Characteristics is 3

11. The characteristics of 56.3 is
O 0
O 1
O 2
O 3

Explanation:
First convert 56.3 to Scientific form:
5.63 \times 10^1
Thus Characteristics is 1

12. The characteristics of 7.43 is
O 0
O 1
O 2
O 3

Explanation:
First convert 7.43 to Scientific form:
7.43 \times 10^0
Thus Characteristics is 0

13. The characteristics of 0.71 is
O 1
O -1
O 2
O -2

Explanation:
First convert 0.71 to Scientific form:
7.1 \times 10^{-1}
Thus Characteristics is -1

14. The characteristics of 37300 is
O 0
O 2
O 3
O 4

Explanation:
First convert 37300 to Scientific form:
3.73 \times 10^4
Thus Characteristics is 4

15. The characteristics of 0.00159 is
O 1
O -1
O -3
O -2

Explanation:
First convert 0.00159 to Scientific form:
0.00159 \times 10^{-3}
Thus Characteristics is -3

16. The mantissa of 2476 is
O .3927
O .3938
O 3
O None of these

Explanation:
(i). First ignore the decimal point
(ii). Take first two digits e.g. 24 and proceed along this row until we come to column headed by third digit 7 of the number which is 3927
(iii). Now take fourth digit i.e. 6 and proceed along this row in mean difference column which is 11.
Thus Mantissa of 2476 \ is \ .3938

17. The log of 2.4 is
O 24
O 0.3802
O 2.3802
O None of these

Explanation:
See Ex # 3.3
Q No. 3
Part No. (ii)

18. The log of 482.7 is
O .6836
O 2.6836
O 2.6830
O None of these

Explanation:
See Ex # 3.3
Q No. 3
Part No. (iv)

19. The log of 0.783 is
O .8938
O \overline{1} .8938
O 1.8938
O None of these

Explanation:
See Ex # 3.3
Q No. 3
Part No. (v)

20. The log of 0.09566 is
O \overline{2} .9805
O \overline{2} .9808
O 2.9808
O None of these

Explanation:
See Ex # 3.3
Q No. 3
Part No. (vi)

21. The log of 700 is
O .8451
O 1.8451
O 2.8451
O None of these

Explanation:
See Ex # 3.3
Q No. 3
Part No. (viii)

22. The anti-\log 1.2508 is
O 1.781
O 17.81
O 1781
O None of these

Explanation:
See Ex # 3.4
Q No. 1
Part No. (i)

23. The anti -\log 0.8401 is
O 6.920
O 69.20
O 6920
O None of these

Explanation:
See Ex # 3.4
Q No. 1
Part No. (ii)

24. The anti-\log \overline{2} .2508 is
O 1.781
O 17.81
O 1781
O 0.01781

Explanation:
See Ex # 3.4
Q No. 1
Part No. (iv)

1. \log _a m n=
O \log _a m+\log _a n
O \log _a m-\log _a n
O n \log _a m
O All of them

Answer: \log _a m+\log _a n
Explanation:
Let \log _a m=x \ and \ \log _a n=y
Write them in Exponential form:
a^x=m \ and \ a^y=n
Now multiply these:
a^x \times a^y=mn
Or
mn=a^x \times a^y
mn=a^{x+y}
Taking \log _a on B.S
\log _a m n=\log _a a^{x+y}
\log _a m n=(x+y) \log _a a
\log _a m n=(x+y)(1) \qquad \log _a a=1
\log _a m n=x+y
\log _a m n=\log _a m+\log _a n

2. \log _a \frac{m}{n}=
O \log _a m+\log _a n
O \log _a m-\log _a n
O n \log _a m
O All of them

Answer: \log _a m-\log _a n
Explanation:
Let \log _a m=x \ and \ \log _a n=y
Write them in Exponential form:
a^x=m and a^y=n
Now Divide these:
\frac{a^x}{a^y}=\frac{m}{n}
Or
\frac{m}{n}=\frac{a^x}{a^y}
\frac{m}{n}=a^{x-y}
Taking \log _a on B.S
\log _a \frac{m}{n}=\log _a a^{x-y}
\log _a \frac{m}{n}=(x-y) \log _a a
\log _a \frac{m}{n}=(x-y)(1) \qquad \log _a a=1
\log _a \frac{m}{n}=x-y
Hence \ \log _a \frac{m}{n}=\log _a m-\log _a n

3. \log _a m^n=
O \log _a m+\log _a n
O \log _a m-\log _a n
O n \log _a m
O All of them

Explanation:
Let \log _a m=x
In Exponential form:
a^x=m
Or
m=a^x
Taking power ‘ n ‘ on B.S
m^n=\left(a^x\right)^n
m^n=a^{n x}
Taking \log _a on B.S
\log _a m^n=\log _a a^{n x}
\log _a m^n=n x \log _a a
\log _a m^n=n x(1) \qquad \log _a a=1
\log _a m^n=n x
\log _a m^n=n \log _a m

4. \quad \log _a m+\log _a n
O \log _a \frac{m}{n}
O \log _a m n
O n \log _a m
O All of them

Explanation:
\log _a m n=\log _a m+\log _a n

5. \log _a m-\log _a n
O \log _a \frac{m}{n}
O \log _a m n
O n \log _a m
O All of them

Explanation:
\log _a \frac{m}{n}=\log _a m-\log _a n

6. n \log _a m
O \log _a \frac{m}{n}
O \log _a m n
O \log _a m^n
O All of them

Explanation:
\log _a m^n=n \log _a m

7. \log m n=
O \log m+\log n
O \log m-\log n
O n \log m
O All of them

Explanation:
\log _a m n=\log _a m+\log _a n

8. \log \frac{m}{n}=
O \log m+\log n
O \log m-\log n
O n \log m
O All of them

Explanation:
\log _a \frac{m}{n}=\log _a m-\log _a n

9. \log m^n=
O \log m+\log n
O \log m-\log n
O n \log m
O All of them

Explanation:
\log _a m^n=n \log _a m

10. \log m+\log n
O \log \frac{m}{n}
O \log m n
O \log m^n
O All of them

Explanation:
\log _a m n=\log _a m+\log _a n

11. \log m-\log n
O \log \frac{m}{n}
O \log m n
O \log m^n
O All of them

Explanation:
\log _a \frac{m}{n}=\log _a m-\log _a n

12. n \log m
O \log \frac{m}{n}
O \log m n
O \log m^n
O All of them

Explanation:
\log _a m^n=n \log _a m

13. \log 2 \times 3=
O \log 2+\log 3
O \log 2-\log 3
O 2 \log 3
O All of them

Explanation:
\log _a m n=\log _a m+\log _a n

14. \log \frac{2}{3}=
O \log 2+\log 3
O \log 2-\log 3
O 2 \log 3
O All of them

Explanation:
\log _a \frac{m}{n}=\log _a m-\log _a n

15. \log 3^2=
O \log 2+1
O \log 2-1
O 2 \log 3
O All of them

Explanation:
\log _a m^n=n \log _a m

16. \log 2+\log 3
O \log 2 \times 3
O \log 6
O \log 2
O Both a & b

Explanation:
\log 2+\log 3 =\log 2\times 3
\log 2+\log 3 =\log 6

17. \log 2-\log 3
O \log \frac{2}{3}
O \log 2 \times 3
O \log 3^2
O All of them

Explanation:
\log _a \frac{m}{n}=\log _a m-\log _a n

18. 2 \log 3=
O \log \frac{2}{3}
O \log 2 \times 3
O \log 3^2
O All of them

Explanation:
\log _a m^n=n \log _a m
2 \log 3= \log 3^2
2 \log 3= \log 9

19. If \log _2 6+\log _2 7=\log _2 a \ then \ a=
O 6
O 7
O 24
O 42

Explanation:
\log _2 6+\log _2 7=\log _2 a
As \log _a m n=\log _a m+\log _a n
\log _2 6 \times 7=\log _2 a
\log _2 42=\log _2 a
Thus \ a=42

20. \log _a m \log _m n=
O \log _a n
O \log _a m
O Both a & b
O None of these

Explanation:
Let \log _a m=x and \log _m n=y
Write them in Exponential form:
a^x=m \ and \ m^y=n
Now multiply these:
As a^{x y}=\left(a^x\right)^y
But \left(a^x\right)^y=m
So a^{x y}=(m)^y=n
Then a^{x y}=n
Taking \log _a on B.S
\log _a a^{x y}=\log _a n
(x y) \log _a a=\log _a n
x y(1)=\log _a n \qquad As \ \log _a a=1
Now
\log _a m \log _m n=\log _a n

21. \log _2 3 \log _3 5=
O \log _5 2
O \log _2 5
O Both a & b
O None of these

Explanation:
\log _a m \log _m n=\log _a n

22. \log _2 3 \log _3 4 \log _4 5=
O \log _5 2
O \log _2 5
O Both a & b
O None of these

Explanation:
\log _a m \log _m n=\log _a n

23. \log _m n=\frac{\log _a n}{\log _a m} is called ____________ law
O Logarithm
O Change of Base
O Change of Logarithm
O None of these

Explanation:

24. \frac{\log _a n}{\log _a m}=
O \log _m n
O \log _t r
O \log 10
O None of these

Explanation:

25. \frac{\log _7 r}{\log _7 t}=
O \log _m n
O \log _t r
O \log 10
O None of these

Explanation:

26. \log _a a=
O 0
O 1
O 10
O None of these

Explanation:

27. \log _{10} 10=
O 0
O 1
O 10
O None of these

Explanation:

28. log⁡10= __________
O 0
O 1
O 10
O None of these

Explanation:

29. log_a⁡ 1= __________
O 0
O 1
O 10
O None of these

Explanation:

29. log⁡1= __________
O 0
O 1
O 10
O None of these

Explanation:

1. \log _9 \frac{1}{81}=
O -1
O -2
O 2
O Does not exist

Explanation:
\log _9 \frac{1}{81}=\log _9 \frac{1}{9^2}
\log _9 \frac{1}{81}=\log _9 9^{-2}
\log _9 \frac{1}{81}=-2 \log _9 9
\log _9 \frac{1}{81}=-2(1)
\log _9 \frac{1}{81}=-2

2. If \log _2 8=x \ then \ x=
O 64
O 3^2
O 3
O 2^8

Explanation:

\log _2 8=x
\log _2 2^3=x
3 \log _2 2=x
3(1)=x
3=x

3. Base of common log is:
O 10
O e
O \pi
O 5

Explanation:

4. \log \sqrt{10}=
O -1
O -\frac{1}{2}
O \frac{1}{2}
O 2

Explanation:
\log \sqrt{10} =\log (10)^{\frac{1}{2}}
\log \sqrt{10} =\frac{1}{2} \log 10
\log \sqrt{10} =\frac{1}{2}(1)
\log \sqrt{10} =\frac{1}{2}

5. For any non-zero value of x \cdot x^0=
O 2
O 1
O 0
O 10

Explanation:

6. Rewrite t=\log _b m as an exponent equation
O t=m^b
O b^m=t
O m=b^t
O m^t=b

Explanation:

7. \log _{10} 10=
O 2
O 3
O 0
O 1

Explanation:

8. Characteristics of \log 0.000059 is
O -5
O 5
O -4
O 4

Explanation:

9. Evaluate \log _7 \frac{1}{\sqrt{7}}
O -1
O -\frac{1}{2}
O \frac{1}{2}
O 2

Explanation:
\log _7 \frac{1}{\sqrt{7}} =\log _7 \frac{1}{(7)^{\frac{1}{2}}}
\log _7 \frac{1}{\sqrt{7}} =\log _7 7^{-\frac{1}{2}}
\log _7 \frac{1}{\sqrt{7}} =-\frac{1}{2} \log _7 7
\log _7 \frac{1}{\sqrt{7}} =-\frac{1}{2}(1)
\log _7 \frac{1}{\sqrt{7}} =-\frac{1}{2}

10. Base of natural log is
O 10
O e
O \pi
O 1

Explanation:

11. \log m+\log n=
O \log m\log n
O \log m-\log n
O \log mn
O \log \frac{m}{n}

Explanation:

12. 0.069 can be written in scientific notation as
O 6.9 \times 10^3
O 6.9 \times 10^{-2}
O 0.69 \times 10^3
O 69 \times 10^2

Explanation:

13. \ln x-2 \ln y
O \ln \frac{x}{y}
O \ln x y^2
O \ln \frac{x^2}{y}
O \ln \frac{x}{y^2}

Explanation:

### Mathematics Class 9 Notes (KPK) Chapter # 4

1. (a+b)^2=
a^2+b^2+2 a b

2. (a-b)^2=
a^2+b^2-2 a b

3. \left(x+\frac{1}{x}\right)^2=
x^2+\frac{1}{x^2}+2(x)\left(\frac{1}{x}\right)

4. \left(x-\frac{1}{x}\right)^2=
x^2+\frac{1}{x^2}-2(x)\left(\frac{1}{x}\right)

5. \left(x+\frac{1}{x}\right)^2=
x^2+\frac{1}{x^2}+2

6. \left(x-\frac{1}{x}\right)^2=
x^2+\frac{1}{x^2}-2

7. (a+b)^2+(a-b)^2=
2\left(a^2+b^2\right)

8. (a+b)^2-(a-b)^2=
4 a b

9. (a+b)(a-b)=
a^2-b^2

10. (a+b+c)^2=
a^2+b^2+c^2+2(a b+b c+c a)

11. (a+b)^3=
a^3+b^3+3 a b(a+b)

12. (a-b)^3=
a^3-b^3-3 a b(a-b)

13. \left(x+\frac{1}{x}\right)^3=
x^3+\frac{1}{x^3}+3\left(x+\frac{1}{x}\right)

14. \left(x-\frac{1}{x}\right)^3=
x^3-\frac{1}{x^3}-3\left(x-\frac{1}{x}\right)

15. a^3+b^3=
(a+b)\left(a^2-a b+b^2\right)

16. a^3-b^3=

(a-b)\left(a^2+a b+b^2\right)

17. x^3+\frac{1}{x^3}=
\left(x+\frac{1}{x}\right)\left(x^2-(x)\left(\frac{1}{x}\right)+\frac{1}{x^2}\right)

18. x^3-\frac{1}{x^3}=
\left(x-\frac{1}{x}\right)\left(x^2+(x)\left(\frac{1}{x}\right)+\frac{1}{x^2}\right)

19. x^3+\frac{1}{x^3}=
\left(x+\frac{1}{x}\right)\left(x^2+\frac{1}{x^2}-1\right)

20. x^3-\frac{1}{x^3}=
\left(x-\frac{1}{x}\right)\left(x^2+\frac{1}{x^2}+1\right)

21. a^2+2 a b+b^2=
(a+b)^2

22. a^2-2 a b+b^2=
(a-b)^2

23) x^2+2(x)\left(\frac{1}{x}\right)+\frac{1}{x^2}=
\left(x+\frac{1}{x}\right)^2

24) x^2-2(x)\left(\frac{1}{x}\right)+\frac{1}{x^2}=
\left(x-\frac{1}{x}\right)^2

25) x^2+2+\frac{1}{x^2}=
\left(x+\frac{1}{x}\right)^2

26) x^2-2+\frac{1}{x^2}=
\left(x-\frac{1}{x}\right)^2

27) a^2-b^2=
(a+b)(a-b)

28) (a+b)^2-c^2=
(a+b+c)(c+b-c)

29) (a+b)^3=
a^3+3 a^2 b+3 a b^2+b^3

30) (a-b)^3=
a^3-3 a^2 b+3 a b^2-b^3

31) (a+b)\left(a^2-a b+b^2\right)=
a^3+b^3

32) (a-b)\left(a^2+a b+b^2\right)=
a^3-b^3

1. A number of the form of \sqrt[n]{a} is called ______, where a is a positive rational number.
O Conjugate
O Surd
O Rational
O None of these
Surd

2. The rules for surd is/are _________
O It is irrational
O It is a root
O A root of a rational number.
O All of these
All of these

3. In a surd, \sqrt[n]{a}, n is called ______ of the surd.
O type
O order
O conjugate
O None of these
Order

4. \sqrt{3} is
O a surd
O not a surd
O conjugate of a surd
O None of these
Explanation: Here is the root of rational number, so it is a surd

5. \sqrt{5+\sqrt{3}} is
O a surd
O not a surd
O conjugate of a surd
O None of these
Explanation: Here the root of irrational number, so it is not a surd.

6. \sqrt[3]{8} is
O a surd
O not a surd
O conjugate of a surd
O None of these
Explanation:
\sqrt[3]{8} is not a surd because its value is 2 which is rational.

7. \sqrt{3} is called _____ surd.
O Cubic
O None of these
Explanation: Here the surd index order is 2, so it is called quadratic.

8. \sqrt[3]{3} is called _____ surd.
O Cubic
O None of these
Explanation:
Here the surd index order is 3, so it is called cubic.

9. \sqrt[4]{3} is called _____ surd.
O Cubic
O None of these
Explanation:
Here the surd index order is 4, so it is called biquadratic.

10. \sqrt[3]{3} \ and \ \sqrt[3]{5} is called _____ surds.
O None of these
Explanation:
Surds of the same order are called equiradical surds which is 3.

11. \sqrt{3} is called _____ surd.
O Monomial
O Binomial
O Trinomial
O None of these
Explanation:
Surd consists of one term is called monomial.

12. \sqrt{3}-2 is called _____ surd.
O Monomial
O Binomial
O Trinomial
O None of these
Explanation:
Surd consists of Two terms is called binomial.

13. 2-\sqrt{3}+\sqrt{5} is called _____ surd.
O Monomial
O Binomial
O Trinomial
O None of these
Explanation:
Surd consists of three terms is called trinomial.

14. Expressions containing two or more surds are called _____ surds.
O Monomial
O Binomial
O Compound
O None of these
Compound

15. \sqrt[3]{5} is called _____ surd.
O an entire
O mixed
O Trinomial
O None of these
Explanation:
If the coefficient of the surd is \pm1 is called an entire surd.

16. -4 \sqrt[3]{5} is called _____ surd.
O an entire
O mixed
O Trinomial
O None of these
Explanation:
If the coefficient of the surd is other than \pm1 is called mixed surd.

17. \sqrt{8} and \sqrt{18} are _____ surds.
O Similar
O Like
O Both a & b
O None of these
Explanation:
\sqrt{8}=\sqrt{4 \times 2}
\sqrt{8}=2\sqrt{2}
AND
\sqrt{18}=\sqrt{9 \times 2}
\sqrt{18}=3\sqrt{2}

18. Conjugate of 3-\sqrt{5} is
O -3-\sqrt{5}
O -3+\sqrt{5}
O 3+\sqrt{5}
O \sqrt{3}+\sqrt{5}
Explanation:
The conjugate of
a\sqrt{x}+b\sqrt{y} is
a\sqrt{x}-b\sqrt{y}

1. m r^2+3 m r^2-5 m r^2=
O -m r^2
O -m r
O -m r
O m r^2
Explanation:
m r^2+3 m r^2-5 m r^2
=4 m r^2-5 m r^2
=-m r^2

2. \left(x^3 y^2\right)\left(x^2 y^3\right)=
O x^5 y^5
O x^5 y^4
O x^4 y^5
O x^4 y^4
Explanation:
\left(x^3 y^2\right)\left(x^2 y^3\right)
=x^{3+2} y^{2+3}
=x^5 y^5

3. \left(4 x y^4\right)^3=
O 64 x^3 y^8
O 64 x^3 y^{10}
O 64 x^3 y^{12}
O 64 x^3 y^7
Explanation:
\left(4 x y^4\right)^3
=4^3 x^3 y^{4 \times 3}
=64 x^3 y^{12}

4. (7x+4 y)-(3 x-6 y)=
O 3 x
O 2 x+10 y
O 4 x+y
O 4 x+10 y
Explanation:
(7 x+4 y)-(3 x-6 y)
=7 x+4 y-3 x+6 y
=7 x-3 x+4 y+6 y
=4 x+10 y

5. (a+b)^2-(a-b)^2=
O 4 a b
O 2\left(a^2+b^2\right)
O a^2-4 a b+2 b^2
O a^4-b^4
4 a b

6. (a+b+c)^2=
O a^2+b^2+c^2
O a^2+b^2+c^2+2(a+b+c)
O a^2+b^2+c^2+2(a b+b c+c a)
O a+b+c+2(a b+b c+c a)
a^2+b^2+c^2+2(a b+b c+c a)

7. a^3+b^3=
O (a+b)^3-2 a b(a+b)
O (a+b)\left(a^2+a b+b^2\right)
O (a-b)\left(a^2-a b+b^2\right)
O (a+b)\left(a^2-a b+b^2\right)
(a+b)\left(a^2-a b+b^2\right)

8. Conjugate of 3-\sqrt{5} is
O -3-\sqrt{5}
O -3+\sqrt{5}
O 3+\sqrt{5}
O \sqrt{3}+\sqrt{5}
Explanation:
The conjugate of
a\sqrt{x}+b\sqrt{y} is
a\sqrt{x}-b\sqrt{y}

9. Which of the following expression is equivalent to the expression \left(m^2+4\right)^{-\frac{1}{2}}
O \frac{m^2+4}{2}
O -\sqrt{m^2+4}
O \frac{1}{\sqrt{m^2+4}}
O \frac{1}{m+2}
Explanation:
\left(m^2+4\right)^{-\frac{1}{2}}
=\frac{1}{(m 2+4)^{\frac{1}{2}}}
=\frac{1}{\sqrt{m^2+4}}

10. For which of the following expression a+b is not a factor?
O a^2-b^2
O a^2+b^2
O a^3+b^3
O a^4+b^4
Explanation:
a^2+b^2 cannot be factorize further, so
a+b is not its factor.

11. \frac{p}{q} \div \frac{r}{q} \cdot \frac{p}{q}=
O \frac{p}{q r}
O \frac{p^2}{q^2 r}
O \frac{p}{q^2 r^2}
O \frac{p^2}{q r}
Explanation:
\frac{p}{q} \div \frac{r}{q} \cdot \frac{p}{q}
=\frac{p}{q} \cdot \frac{q}{r} \cdot \frac{p}{q}
=\frac{p}{q} \cdot \frac{1}{r} \cdot \frac{p}{1}
=\frac{p^2}{q r}

12. Evaluate k^2(2 l-3 m)
when k=-2, l=3, m=4
O 24
O -24
O 6
O 4
Explanation:
k^2(2 l-3 m)
=(-2)^2[2(3)-3(4)]
=4(6-12)
=4(-6)
=-24

13. Evaluate 5 m \sqrt{k^2+l^2}
when k=-2, l=3, m=4.
O 10 \sqrt{13}
O 20
O 20 \sqrt{13}
O \sqrt{13}
Explanation:
5 m \sqrt{k^2+l^2}
=5(4) \sqrt{(-2)^2+(3)^2}
=20 \sqrt{4+9}
=20 \sqrt{13}

### 36 thoughts on “Free Mathematics Notes for 9th Class”

1. Masha’Allah good step, keep it up, start video series if possible so the poor students will get benefits from it.

2. Mashallah…I have never seen such a good notes before congrats to your team efforts towards smart education…thank you tehkals.com whole team….

3. Theorm hain Un mai kiya tou naa samajhna hai 👊👊

1. Dear if there is any mistake or other
Please send the picture on this WhatsApp No. 03336-3747471