11th Maths Monthly Test-4 ( Sets, Relations and Functions )

St. Britto Hr. Sec. School - Madurai

11th Maths Monthly Test-4 ( Sets, Relations and Functions )-Aug 2020

Time : 2.30 Hours

Part - A

Generic 2 - Answer All Question

Find the largest possible domain of the real valued function\(f\left( x \right) =\frac { \sqrt { 4-{ x }^{ 2 } } }{ \sqrt { { x }^{ 2 }-9 } } \)

Write the following in roster form.
{x\(\in \)N:x²<121 and x is a prime}.

On the set of natural number let R be the relation defined by aRb if 2a + 3b = 30. Write down the relation by listing all the pairs. Check whether it is transitive

Show that the relation xy=-2 is a function for a suitable domain. Find the domain and the range of the function.

On the set of natural number let R be the relation defined by aRb if a + b \(\le\) 6. Write down the relation by listing all the pairs. Check whether it is reflexive

If f: R ⟶ R is an bijection given by f(x)= x³+3, then find f^-1(x)

On the set of natural numbers let R be the relation defined by xRy if x + 2y = 21. It is better to write the relation explicitly. The relation R is the set

If A = {1, 2, 3, 4} and B = {3, 4, 5, 6}, find \((n(A\cup B)\times(A\cap B)\times(A \triangle B)).\)

Check whether the following sets are disjoint where p={x:x is a prime<15} and Q={x:x is a multiple of 2 and x<16}

State whether the following relations are functions or not. If it is a function check for one-to-oneness and ontoness. If it is not a function state why?
If A={a,b,c} and f={(a, c) (b, c) (c, b)} : (f : A \(\rightarrow\)A).

Part - B

Generic 3 - Answer All Question

In the set Z of integers, define mRn if m - n is divisible by 7. Prove that R is an equivalence relation.

By taking suitable sets A, B, C, verify the following results:
(A\(\times\) B)\(\cap \)(B\(\times\)A) = (A\(\cap \)B) \(\times\) (B\(\cap \)A)

By taking suitable sets A, B, C, verify the following results:
(B-A) \(\cup\) C = (B\(\cup\)C) - (A-C)

By using the same concept applied in previous example, graphs of y = sin x and y = sin 2x, and also their combined graphs are given figures (a), (b) and (c). The minimum and maximum values of sin x and sin 2x are the same. But they have different x-intercepts. The x-intercepts for y = sin x are \(\pm n\pi\) and for y = sin 2x are \(\pm{1\over 2}n\pi,\ n\in Z.\)

Find the domain and range of the function f(x)=\(\frac { 1 }{ \sqrt { x-5 } } \).

Part - C

Generic 5 - Answer All Question

Check whether the following for one-to-oneness and ontoness.
\(f:R-\{0\}\rightarrow R\) defined by f \(f(x)={1\over x}.\)

If the function is defined as \(f(x)= \begin{cases} \begin{matrix} 3x-2, & x>3 \end{matrix} \\ \begin{matrix} { x }^{ 2 }-2, & -2\le x\le 2 \end{matrix} \\ \begin{matrix} 2x+1, & x<2 \end{matrix} \end{cases}\)
Then find the values, if exists f(4),f(-4),f(0),f(-7).

Check the following functions for one-to-oneness and ontoness.
\(f:R\rightarrow R\) defined by f(n) = n².

Discuss the following relations for reflexivity, symmetricity and transitivity :
The relation R defined on the set of all positive integers by "mRn if m divided n".

A relation R is defined on the set z of integers as follows:
(x, Y) ∈ R⇔x² + y²= 25. Express R and R^-1 as the set of ordered pairs and hence find their respective domains.

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11th Maths Monthly Test-4 ( Sets, Relations and Functions )

St. Britto Hr. Sec. School - Madurai

11th Maths Monthly Test-4 ( Sets, Relations and Functions )-Aug 2020

Part - A

Generic 2 - Answer All Question

Find the largest possible domain of the real valued function\(f\left( x \right) =\frac { \sqrt { 4-{ x }^{ 2 } } }{ \sqrt { { x }^{ 2 }-9 } } \)

Write the following in roster form. {x\(\in \)N:x2<121 and x is a prime}.

On the set of natural number let R be the relation defined by aRb if 2a + 3b = 30. Write down the relation by listing all the pairs. Check whether it is transitive

Show that the relation xy=-2 is a function for a suitable domain. Find the domain and the range of the function.

On the set of natural number let R be the relation defined by aRb if a + b \(\le\) 6. Write down the relation by listing all the pairs. Check whether it is reflexive

If f: R ⟶ R is an bijection given by f(x)= x3+3, then find f-1(x)

On the set of natural numbers let R be the relation defined by xRy if x + 2y = 21. It is better to write the relation explicitly. The relation R is the set

If A = {1, 2, 3, 4} and B = {3, 4, 5, 6}, find \((n(A\cup B)\times(A\cap B)\times(A \triangle B)).\)

Check whether the following sets are disjoint where p={x:x is a prime<15} and Q={x:x is a multiple of 2 and x<16}

State whether the following relations are functions or not. If it is a function check for one-to-oneness and ontoness. If it is not a function state why? If A={a,b,c} and f={(a, c) (b, c) (c, b)} : (f : A \(\rightarrow\)A).

Part - B

Generic 3 - Answer All Question

In the set Z of integers, define mRn if m - n is divisible by 7. Prove that R is an equivalence relation.

By taking suitable sets A, B, C, verify the following results: (A\(\times\) B)\(\cap \)(B\(\times\)A) = (A\(\cap \)B) \(\times\) (B\(\cap \)A)

By taking suitable sets A, B, C, verify the following results: (B-A) \(\cup\) C = (B\(\cup\)C) - (A-C)

Find the domain and range of the function f(x)=\(\frac { 1 }{ \sqrt { x-5 } } \).

Part - C

Generic 5 - Answer All Question

Check whether the following for one-to-oneness and ontoness. \(f:R-\{0\}\rightarrow R\) defined by f \(f(x)={1\over x}.\)

If the function is defined as \(f(x)= \begin{cases} \begin{matrix} 3x-2, & x>3 \end{matrix} \\ \begin{matrix} { x }^{ 2 }-2, & -2\le x\le 2 \end{matrix} \\ \begin{matrix} 2x+1, & x<2 \end{matrix} \end{cases}\) Then find the values, if exists f(4),f(-4),f(0),f(-7).

Check the following functions for one-to-oneness and ontoness. \(f:R\rightarrow R\) defined by f(n) = n2.

Discuss the following relations for reflexivity, symmetricity and transitivity : The relation R defined on the set of all positive integers by "mRn if m divided n".

A relation R is defined on the set z of integers as follows: (x, Y) ∈ R⇔x2 + y2 = 25. Express R and R-1 as the set of ordered pairs and hence find their respective domains.

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Write the following in roster form.
{x\(\in \)N:x²<121 and x is a prime}.

If f: R ⟶ R is an bijection given by f(x)= x³+3, then find f^-1(x)

State whether the following relations are functions or not. If it is a function check for one-to-oneness and ontoness. If it is not a function state why?
If A={a,b,c} and f={(a, c) (b, c) (c, b)} : (f : A \(\rightarrow\)A).

By taking suitable sets A, B, C, verify the following results:
(A\(\times\) B)\(\cap \)(B\(\times\)A) = (A\(\cap \)B) \(\times\) (B\(\cap \)A)

By taking suitable sets A, B, C, verify the following results:
(B-A) \(\cup\) C = (B\(\cup\)C) - (A-C)

Check whether the following for one-to-oneness and ontoness.
\(f:R-\{0\}\rightarrow R\) defined by f \(f(x)={1\over x}.\)

If the function is defined as \(f(x)= \begin{cases} \begin{matrix} 3x-2, & x>3 \end{matrix} \\ \begin{matrix} { x }^{ 2 }-2, & -2\le x\le 2 \end{matrix} \\ \begin{matrix} 2x+1, & x<2 \end{matrix} \end{cases}\)
Then find the values, if exists f(4),f(-4),f(0),f(-7).

Check the following functions for one-to-oneness and ontoness.
\(f:R\rightarrow R\) defined by f(n) = n².

Discuss the following relations for reflexivity, symmetricity and transitivity :
The relation R defined on the set of all positive integers by "mRn if m divided n".

A relation R is defined on the set z of integers as follows:
(x, Y) ∈ R⇔x² + y²= 25. Express R and R^-1 as the set of ordered pairs and hence find their respective domains.