Maths Test-1 Number System

1) If a set contain 6 elements then its power set contains -------------- elements.

32

64

12

46

The power set of a set with n elements contains 2^n elements. In this case, the set contains 6 elements, so its power set will contain 2^6 = 64 elements.

Explanation

The power set of a set with n elements contains 2^n elements. In this case, the set contains 6 elements, so its power set will contain 2^6 = 64 elements.

2) Set of integers satisfy closure property under:

Addition

Multiplication

Both a & b

None

The set of integers satisfies closure property under both addition and multiplication. This means that when two integers are added or multiplied, the result is always an integer. For addition, if you add two integers, the sum will always be an integer. Similarly, for multiplication, if you multiply two integers, the product will always be an integer. Therefore, both addition and multiplication satisfy closure property for the set of integers.

Explanation

The set of integers satisfies closure property under both addition and multiplication. This means that when two integers are added or multiplied, the result is always an integer. For addition, if you add two integers, the sum will always be an integer. Similarly, for multiplication, if you multiply two integers, the product will always be an integer. Therefore, both addition and multiplication satisfy closure property for the set of integers.

3) According to this diagram which statement is true:

A and B are overlapping sets

C is not a subset of B

A and C are disjoints sets

All of these

The diagram shows that A and B overlap, meaning they have some elements in common. It also shows that C is not a subset of B, meaning there are elements in C that are not in B. Additionally, the diagram shows that A and C are disjoint sets, meaning they have no elements in common. Therefore, all of the statements are true based on the given diagram.

Explanation

The diagram shows that A and B overlap, meaning they have some elements in common. It also shows that C is not a subset of B, meaning there are elements in C that are not in B. Additionally, the diagram shows that A and C are disjoint sets, meaning they have no elements in common. Therefore, all of the statements are true based on the given diagram.

4) If set A = {0} and W is set of whole number than W-A is:

Z

N

1

{0}

The set A is defined as {0}, which means it only contains the number 0. The set W is defined as the set of whole numbers, which includes all positive integers and zero. When we subtract set A from set W (W-A), we are removing the elements of set A from set W. Since set A only contains the number 0, subtracting it from set W will not change the set W. Therefore, the answer is N, which represents the set of all whole numbers.

Explanation

The set A is defined as {0}, which means it only contains the number 0. The set W is defined as the set of whole numbers, which includes all positive integers and zero. When we subtract set A from set W (W-A), we are removing the elements of set A from set W. Since set A only contains the number 0, subtracting it from set W will not change the set W. Therefore, the answer is N, which represents the set of all whole numbers.

5) π is:

Rational number

Whole number

Irrational number

Integer

The symbol π represents the mathematical constant pi, which is the ratio of a circle's circumference to its diameter. Pi is an irrational number because it cannot be expressed as a fraction or a finite decimal. Its decimal representation goes on forever without repeating or terminating. Therefore, the correct answer is "Irrational number."

Explanation

The symbol π represents the mathematical constant pi, which is the ratio of a circle's circumference to its diameter. Pi is an irrational number because it cannot be expressed as a fraction or a finite decimal. Its decimal representation goes on forever without repeating or terminating. Therefore, the correct answer is "Irrational number."

6) Set {x/x E A or x E B or x E both A and B} shows:

Intersection of A and B

A only

Union of A and B

B only

The correct answer is the Union of A and B. This is because the set {x/x E A or x E B or x E both A and B} includes all elements that are in either A or B or both. Therefore, it represents the union of A and B, which is the set that contains all elements that are in either A or B or both.

Explanation

The correct answer is the Union of A and B. This is because the set {x/x E A or x E B or x E both A and B} includes all elements that are in either A or B or both. Therefore, it represents the union of A and B, which is the set that contains all elements that are in either A or B or both.

7) {a=b and b = c <=> a = c} is a?

Symmetric property

Reflexive property

Transitive Property

None of these

The given statement "{a=b and b = c a = c}" is an example of the transitive property. The transitive property states that if two quantities are equal to a third quantity, then they are also equal to each other. In this case, it is stated that a is equal to b and b is equal to c, which implies that a is also equal to c. Therefore, the transitive property applies to this statement.

Explanation

The given statement "{a=b and b = c a = c}" is an example of the transitive property. The transitive property states that if two quantities are equal to a third quantity, then they are also equal to each other. In this case, it is stated that a is equal to b and b is equal to c, which implies that a is also equal to c. Therefore, the transitive property applies to this statement.

8) Set of {R} is closed under

ADDITION

MULTIPLICATION

BOTH A AND B

NONE

The set {R} refers to the set of real numbers. The statement "closed under addition" means that if you add any two real numbers, the result will also be a real number. Similarly, the statement "closed under multiplication" means that if you multiply any two real numbers, the result will also be a real number. Since the set of real numbers satisfies both of these conditions, it is closed under both addition and multiplication. Therefore, the correct answer is "BOTH A AND B".

Explanation

The set {R} refers to the set of real numbers. The statement "closed under addition" means that if you add any two real numbers, the result will also be a real number. Similarly, the statement "closed under multiplication" means that if you multiply any two real numbers, the result will also be a real number. Since the set of real numbers satisfies both of these conditions, it is closed under both addition and multiplication. Therefore, the correct answer is "BOTH A AND B".

9) Which of the following statement is true:

W⊂N⊂ Z⊂R

R⊂Z⊂ W⊂N

N⊂W⊂ Z⊂R

All of these

The correct answer is N⊂W⊂ Z⊂R. This is because the given statement shows a nested relationship between the sets. N is a subset of W, which is a subset of Z, which is a subset of R.

Explanation

The correct answer is N⊂W⊂ Z⊂R. This is because the given statement shows a nested relationship between the sets. N is a subset of W, which is a subset of Z, which is a subset of R.

10) As N is a sub set of W then N' ∩N is equal to:

{ }

{0 }

W

NONE

Since N is a subset of W, it means that every element in N is also in W. Therefore, the intersection of N' (complement of N) and N will result in an empty set because there are no elements that are in both N' and N. Hence, the correct answer is {}.

Explanation

Since N is a subset of W, it means that every element in N is also in W. Therefore, the intersection of N' (complement of N) and N will result in an empty set because there are no elements that are in both N' and N. Hence, the correct answer is {}.

11) If z= cosθ+i sin θ then arg (z^2+z) =_______

3θ/2

θ

θ/2

3θ

When we raise z to the power of 2, we double the angle θ, resulting in 2θ. Then, when we add z to the result, we simply add θ to the angle. Therefore, the argument of (z^2 + z) is θ + 2θ, which simplifies to 3θ. However, since we are asked for the argument of (z^2 + z), we divide 3θ by 2 to get the final answer of θ/2.

Explanation

When we raise z to the power of 2, we double the angle θ, resulting in 2θ. Then, when we add z to the result, we simply add θ to the angle. Therefore, the argument of (z^2 + z) is θ + 2θ, which simplifies to 3θ. However, since we are asked for the argument of (z^2 + z), we divide 3θ by 2 to get the final answer of θ/2.

12) Set of rational numbers is only proper sub set of?

Set of real number

Set of integers

Set of rational number

None of these

The set of rational numbers is a proper subset of the set of real numbers because the set of real numbers includes both rational and irrational numbers, while the set of rational numbers only includes numbers that can be expressed as a fraction. Therefore, the set of real numbers is a larger set that encompasses the set of rational numbers.

Explanation

The set of rational numbers is a proper subset of the set of real numbers because the set of real numbers includes both rational and irrational numbers, while the set of rational numbers only includes numbers that can be expressed as a fraction. Therefore, the set of real numbers is a larger set that encompasses the set of rational numbers.

13) A set of natural number is the sub set of whole number then N∩W is:

N

W

IRRATIONAL

R

The set of natural numbers is a subset of the set of whole numbers. The intersection of the set of natural numbers (N) and the set of whole numbers (W) would therefore be the set of natural numbers itself (N).

Explanation

The set of natural numbers is a subset of the set of whole numbers. The intersection of the set of natural numbers (N) and the set of whole numbers (W) would therefore be the set of natural numbers itself (N).

14) |z+z bar| is equal to ------------------ where z = a + ib

2a

4a^2

2b

4b^2

The expression |z+z bar| represents the absolute value of the sum of z and its complex conjugate. In this case, z is given as a+ib. The complex conjugate of z is a-ib. When we add z and its complex conjugate, we get (a+ib) + (a-ib) = 2a. Therefore, the absolute value of 2a is equal to 2a.

Explanation

The expression |z+z bar| represents the absolute value of the sum of z and its complex conjugate. In this case, z is given as a+ib. The complex conjugate of z is a-ib. When we add z and its complex conjugate, we get (a+ib) + (a-ib) = 2a. Therefore, the absolute value of 2a is equal to 2a.

15) Venn diagram shows

Sets

Operation on sets

Both

None of these

The Venn diagram shows both sets and the operation on sets. It visually represents the relationship between different sets and how they intersect or overlap with each other. By using different shapes or colors, the Venn diagram helps to illustrate the concept of set theory and the various operations that can be performed on sets, such as union, intersection, and complement.

Explanation

The Venn diagram shows both sets and the operation on sets. It visually represents the relationship between different sets and how they intersect or overlap with each other. By using different shapes or colors, the Venn diagram helps to illustrate the concept of set theory and the various operations that can be performed on sets, such as union, intersection, and complement.

16) Convert z=radical3+1i in polar form

(2,30^o)

(radical 2,60^o)

(radical3,30^o)

None

The given complex number z = √3 + i can be converted to its polar form by finding the magnitude and argument. The magnitude can be found using the formula √(a^2 + b^2), where a and b are the real and imaginary parts of the complex number respectively. In this case, the magnitude is √(√3^2 + 1^2) = √(3 + 1) = √4 = 2. The argument can be found using the formula tan^(-1)(b/a), where a and b are the real and imaginary parts of the complex number respectively. In this case, the argument is tan^(-1)(1/√3) = 30°. Therefore, the polar form of the complex number z is (2, 30°).

Explanation

The given complex number z = √3 + i can be converted to its polar form by finding the magnitude and argument. The magnitude can be found using the formula √(a^2 + b^2), where a and b are the real and imaginary parts of the complex number respectively. In this case, the magnitude is √(√3^2 + 1^2) = √(3 + 1) = √4 = 2. The argument can be found using the formula tan^(-1)(b/a), where a and b are the real and imaginary parts of the complex number respectively. In this case, the argument is tan^(-1)(1/√3) = 30°. Therefore, the polar form of the complex number z is (2, 30°).

17) If z, iz and z+iz are the vertices of a triangle whose area is 2 units then the value of z is_____

4

2

-2

None

The value of z is 4 because in a complex plane, the vertices of a triangle can be represented by complex numbers. The area of a triangle formed by three complex numbers z1, z2, and z3 is given by the formula: A = 0.5 * Im((z2 - z1) * (z3 - z1)). In this case, we are given that the area is 2 units, so 2 = 0.5 * Im((iz - z) * (z + iz)). Simplifying this equation, we get Im(-2iz) = 4. Since the imaginary part of -2iz is 4, we can conclude that the value of z is 4.

Explanation

The value of z is 4 because in a complex plane, the vertices of a triangle can be represented by complex numbers. The area of a triangle formed by three complex numbers z1, z2, and z3 is given by the formula: A = 0.5 * Im((z2 - z1) * (z3 - z1)). In this case, we are given that the area is 2 units, so 2 = 0.5 * Im((iz - z) * (z + iz)). Simplifying this equation, we get Im(-2iz) = 4. Since the imaginary part of -2iz is 4, we can conclude that the value of z is 4.

18) If a decimal is non recurring but terminating then it shows?

Natural number

Rational number

Irrational number

None of these

If a decimal is non-recurring but terminating, it means that it has a finite number of digits after the decimal point and the digits do not repeat. This can only happen if the decimal can be expressed as a fraction of two integers. Therefore, the correct answer is "Rational number".

Explanation

If a decimal is non-recurring but terminating, it means that it has a finite number of digits after the decimal point and the digits do not repeat. This can only happen if the decimal can be expressed as a fraction of two integers. Therefore, the correct answer is "Rational number".

19) Which of the following set doesn't satisfy the closure property under addition:

E

Z

O

W

The set O doesn't satisfy the closure property under addition because if we add any two elements from the set O, the result will not be in the set O.

Explanation

The set O doesn't satisfy the closure property under addition because if we add any two elements from the set O, the result will not be in the set O.

20) If z = x + y i and l3zl = lz-4l then x^2 + y^2 + x = _______

1

-1

2

-2

The given equation is l3zl = lz-4l. This equation represents the absolute value of the complex number z, which is equal to the absolute value of the complex number -4. The absolute value of a complex number is given by the square root of the sum of the squares of its real and imaginary parts. In this case, the real part is x and the imaginary part is y. Therefore, x^2 + y^2 = 16. Since the equation also states that x^2 + y^2 + x = 2, we can substitute the value of x^2 + y^2 into this equation to find that x = 2. Hence, the answer is 2.

Explanation

The given equation is l3zl = lz-4l. This equation represents the absolute value of the complex number z, which is equal to the absolute value of the complex number -4. The absolute value of a complex number is given by the square root of the sum of the squares of its real and imaginary parts. In this case, the real part is x and the imaginary part is y. Therefore, x^2 + y^2 = 16. Since the equation also states that x^2 + y^2 + x = 2, we can substitute the value of x^2 + y^2 into this equation to find that x = 2. Hence, the answer is 2.

21) If a set have n elements then number of its proper subsets is:

2^n

(2^n)-1

1

NONE

The correct answer is (2^n)-1. This is because for a set with n elements, there are 2^n total subsets including the empty set and the set itself. However, the question asks for the number of proper subsets, which excludes the empty set and the set itself. Therefore, the number of proper subsets is 2^n - 1.

Explanation

The correct answer is (2^n)-1. This is because for a set with n elements, there are 2^n total subsets including the empty set and the set itself. However, the question asks for the number of proper subsets, which excludes the empty set and the set itself. Therefore, the number of proper subsets is 2^n - 1.

22) Set of irrational number is the sub set of

Natural number

Whole numbers

Real numbers

All of these

The set of irrational numbers is a subset of the real numbers because irrational numbers cannot be expressed as a fraction or a decimal that terminates or repeats. Real numbers include both rational and irrational numbers, so all irrational numbers are included in the set of real numbers.

Explanation

The set of irrational numbers is a subset of the real numbers because irrational numbers cannot be expressed as a fraction or a decimal that terminates or repeats. Real numbers include both rational and irrational numbers, so all irrational numbers are included in the set of real numbers.

23) And if it is recurring and non terminating then it shows?

Irrational number

Rational number

Real number

B & c

If a number is recurring and non-terminating, it means that its decimal representation repeats indefinitely without ending. This type of number can only be a rational number because irrational numbers cannot be expressed as a fraction of two integers. Therefore, the correct answer is "Rational number."

Explanation

If a number is recurring and non-terminating, it means that its decimal representation repeats indefinitely without ending. This type of number can only be a rational number because irrational numbers cannot be expressed as a fraction of two integers. Therefore, the correct answer is "Rational number."

24) The number of the form √(n) where n is positive and perfect square always belongs to:

Q

W

Z

R

The number of the form √(n) where n is a positive perfect square always belongs to the set of rational numbers (Q). This is because the square root of a perfect square is always a whole number, and whole numbers are a subset of rational numbers. Therefore, the answer is Q.

Explanation

The number of the form √(n) where n is a positive perfect square always belongs to the set of rational numbers (Q). This is because the square root of a perfect square is always a whole number, and whole numbers are a subset of rational numbers. Therefore, the answer is Q.

25) As N is a sub set of W then N'U N is equal to:

W

{}

N

NONE

Since N is a subset of W, it means that all elements of N are also elements of W. Therefore, when we take the union of N' (complement of N) and N, we are essentially taking all the elements that are not in N and adding them to N. Since N already contains all the elements of N, the union of N' and N will still result in N.

Explanation

Since N is a subset of W, it means that all elements of N are also elements of W. Therefore, when we take the union of N' (complement of N) and N, we are essentially taking all the elements that are not in N and adding them to N. Since N already contains all the elements of N, the union of N' and N will still result in N.