Solution: (i) One-One but not Onto $\mathrm{f}: \mathrm{N} \rightarrow \mathrm{N}$ be a mapping given by $\mathrm{f}(\mathrm{x})=\mathrm{x} 2$ For one-one $\begin{array}{l} f(x)=f(y) \\ x_{2}=y z \\...
Give an example of a function which is
Define each of the following:
(i) into function
Give an example of each type of functions.
Solution: (i) Into Function: It is is a function where there is atleast one element is Set B who is not the image of any element in set A. For Example: $f(x) = 2x - 1$ from the set of Integers to...
Define each of the following:
(i) bijective function
(ii) many – one function
Give an example of each type of functions.
Solution: (i)Bijective function: It is, also known as one-one onto function and is a function where for every element of set A, there is exactly one image in set B, such that no element is set B is...
Define each of the following:
(i) injective function
(ii) surjective function
Give an example of each type of functions.
Solution: (i) Injective function: It is, also known as one-one function and is a type of function where every element in set A has an image in set B. Hence, f: A β B is one-one or injection function...
Define a function. What do you mean by the domain and range of a function? Give examples.
Solution: A function is stated as the relation between the two sets, where there is exactly one element in set B, for every element of set A. A function is represented as f: A β B, which means βfβ...
Let A = {β3, β1}, B = {1, 3) and C = {3, 5). Find:
(iii)B Γ C
(iv)A Γ (B Γ C)
(iii) Given: B = {1, 3} and C = {3, 5} To find: B Γ C By the definition of the Cartesian product, Given two non β empty sets P and Q. The Cartesian product P Γ Q is the set of all ordered pairs of...
Let A = {β3, β1}, B = {1, 3) and C = {3, 5). Find:
(i) A Γ B
(ii) (A Γ B) Γ C
Answer : (i) Given: A = {-3, -1} and B = {1, 3} To find: A Γ B By the definition of the Cartesian product, Given two non β empty sets P and Q. The Cartesian product P Γ Q is the set of all ordered...
If A = {5, 7), find (i) A Γ A Γ A.
Answer : We have, A = {5, 7} So, By the definition of the Cartesian product, Given two non β empty sets P and Q. The Cartesian product P Γ Q is the set of all ordered pairs of elements from P and Q,...
Let A = {β2, 2} and B = (0, 3, 5). Find:
(iii)A Γ A
(iv) B Γ B
(iii) Given: A = {-2, 2} To find: A Γ A By the definition of the Cartesian product, Given two non β empty sets P and Q. The Cartesian product P Γ Q is the set of all ordered pairs of elements from P...
Let A = {β2, 2} and B = (0, 3, 5). Find:
(i) A Γ B
(ii) B Γ A
Answer : (i) Given: A = {-2, 2} and B = {0, 3, 5} To find: A Γ B By the definition of the Cartesian product, Given two non β empty sets P and Q. The Cartesian product P Γ Q is the set of all ordered...
Let A and B be two sets such that n(A) = 3 and n(B) = 2. If a β b β c and (a, 0), (b, 1), (c, 0) is in A Γ B, find A and B.
Answer : Since, (a, 0), (b, 1), (c, 0) are the elements of A Γ B. β΄ a, b, c Π A and 0, 1 Π B It is given that n(A) = 3 and n(B) = 2 β΄ a, b, c Π A and n(A) = 3 β A = {a, b, c} and 0, 1 Π B and n(B) =...
Let A Γ B = {(a, b): b = 3a β 2}. if (x, β5) and (2, y) belong to A Γ B, find the values of x and y.
Answer : Given: A Γ B = {(a, b): b = 3a β 2} and {(x, -5), (2, y)} Π A Γ B For (x, -5) Π A Γ B b = 3a β 2 β -5 = 3(x) β 2 β -5 + 2 = 3x β -3 = 3x β x = -1 For (2, y) Π A Γ B b = 3a β 2 β y = 3(2) β...
Let A = {2, 3} and B = {4, 5}. Find (A Γ B). How many subsets will (A Γ B) have?
Answer : Given: A = {2, 3} and B = {4, 5} To find: A Γ B By the definition of the Cartesian product, Given two non β empty sets P and Q. The Cartesian product P Γ Q is the set of all ordered pairs...
If A Γ B = {(β2, 3), (β2, 4), (0, 4), (3, 3), (3, 4), find A and B.
Answer : Here, A Γ B = {(β2, 3), (β2, 4), (0, 4), (3, 3), (3, 4)} To find: A and B Clearly, A is the set of all first entries in ordered pairs in A Γ B β΄ A = {-2, 0, 3} and B is the set of all...
Let A = {x Ο΅ W : x < 2}, B = {x Ο΅ N : 1 < x β€ 4} and C = {3, 5}. Verify that: (i) A Γ (B ???? C) = (A Γ B) ???? (A Γ C) (ii) A Γ (B β© C) = (A Γ B) β© (A Γ C)
Answer : (i) Given: A = {x Ο΅ W : x < 2} Here, W denotes the set of whole numbers (non β negative integers). β΄ A = {0, 1} [β΅ It is given that x < 2 and the whole numbers which are less than 2...
If A = {1, 3, 5) B = {3, 4} and C = {2, 3}, verify that:
(i) A Γ (B ???? C) = (A Γ B) ???? (A Γ C)
(ii) A Γ (B β© C) = (A Γ B) β© (A Γ C)
Answer : (i) Given: A = {1, 3, 5}, B = {3, 4} and C = {2, 3} H. S = A Γ (B β C) By the definition of the union of two sets, (B β C) = {2, 3, 4} = {1, 3, 5} Γ {2, 3, 4} Now, by the definition of the...
If A = {x Ο΅ N : x β€ 3} and {x Ο΅ W : x < 2}, find (A Γ B) and (B Γ A). Is (A Γ B) = (B Γ A)?
Answer : Given: A = {x Ο΅ N: x β€ 3} Here, N denotes the set of natural numbers. β΄ A = {1, 2, 3} [β΅ It is given that the value of x is less than 3 and natural numbers which are less than 3 are 1 and...
If A = {2, 3, 5} and B = {5, 7}, find:
(iii)A Γ A
(iv)B Γ B
(iii) Given: A = {2, 3, 5} and B = {2, 3, 5} To find: A Γ A By the definition of the Cartesian product, Given two non β empty sets P and Q. The Cartesian product P Γ Q is the set of all ordered...
If A = {2, 3, 5} and B = {5, 7}, find:
(i)A Γ B
(ii)B Γ A
Answer : (i) Given: A = {2, 3, 5} and B = {5, 7} To find: A Γ B By the definition of the Cartesian product, Given two non β empty sets P and Q. The Cartesian product P Γ Q is the set of all ordered...
If P = {a, b} and Q = {x, y, z}, show that P Γ Q β Q Γ P.
Answer : Given: P = {a, b} and Q = {x, y, z} To show: P Γ Q β Q Γ P Now, firstly we find the P Γ Q and Q Γ P By the definition of the Cartesian product, Given two non β empty sets P and Q. The...
If A = {9, 1} and B = {1, 2, 3}, show that A Γ B β B Γ A.
Answer : Given: A = {9, 1} and B = {1, 2, 3} To show: A Γ B β B Γ A Now, firstly we find the A Γ B and B Γ A By the definition of the Cartesian product, Given two non β empty sets P and Q. The...
Find the values of a and b, when:(a β 2, 2b + 1 = (b β 1, a + 2)
Since, the ordered pairs are equal, the corresponding elements are β΄, a β 2 = b β 1 β¦(i) & 2b + 1 = a + 2 β¦(ii) Solving eq. (i), we get a β 2 = b β 1 β a β b = -1 + 2 β a β b = 1 β¦ (iii) Solving...
Find the values of a and b, when:
(i) (a + 3, b β2) = (5, 1)
(ii) (a + b, 2b β 3) = (4, β5)
Answer : Since, the ordered pairs are equal, the corresponding elements are equal. β΄, a + 3 = 5 β¦(i) and b β 2 = 1 β¦(ii) Solving eq. (i), we get a + 3 = 5 β a = 5 β 3 β a = 2 Solving eq. (ii), we...
One zero of the polynomial $3 x^{3}+16 x^{2}+15 x-18$ is $\frac{2}{3}$. Find the other zeros of the polynomial.
$x=\frac{2}{3}$ is one of the zero of $3 x^{3}+16 x^{2}+15 x-18$ Now, we have $\mathrm{x}=\frac{2}{3}$ $\Rightarrow \mathrm{x}-\frac{2}{3}=0$ Now, we divide $3 x^{3}+16 x^{2}+15 x-18$ by...
If $(x+a)$ is a factor of the polynomial $2 x^{2}+2 a x+5 x+10$, find the value of $a$.
$(x+a)$ is a factor of $2 x^{2}+2 a x+5 x+10$ $x+a=0$ $\Rightarrow \mathrm{x}=-\mathrm{a}$ Since, it satisfies the above polynomial. => $2(-a)^{2}+2 a(-a)+5(-a)+10=0$ $\Rightarrow 2 a^{2}-2...
If $x=\frac{2}{3}$ and $x=-3$ are the roots of the quadratic equation $a x^{2}+2 a x+5 x+10$ then find the value of a and $b$.
$a x^{2}+7 x+b=0$ Since, $x=\frac{2}{3}$ is the root of the above quadratic equation Hence, it will satisfy the above equation. => $a\left(\frac{2}{3}\right)^{2}+7\left(\frac{2}{3}\right)+b=0$...
Find the quadratic polynomial, sum of whose zeroes is $\sqrt{2}$ and their product is $\left(\frac{1}{3}\right)$.
Quadratic equation can be foundΒ if we know the sum of the roots and product of the roots by using the formula: $\mathrm{x}^{2}-($ Sum of the roots) $\mathrm{x}+$ Product of roots $=0$ $\Rightarrow...
Find the quadratic polynomial, sum of whose zeroes is $\left(\frac{5}{2}\right)$ and their product is 1 . Hence, find the zeroes of the polynomial.
Let $\alpha$ and $\beta$ be the zeroes of the required polynomial $\mathrm{f}(\mathrm{x})$. =>$(\alpha+\beta)=\frac{5}{2}$ and $\alpha \beta=1$ $\therefore...
Find the quadratic polynomial, sum of whose zeroes is 8 and their product is $12 .$ Hence, find the zeroes of the polynomial.
Let $\alpha$ and $\beta$ be the zeroes of the required polynomial $\mathrm{f}(\mathrm{x})$. Then $(\alpha+\beta)=8$ and $\alpha \beta=12$ $\therefore f(x)=x^{2}-(\alpha+\beta) x+\alpha \beta$...
Find the quadratic polynomial whose zeroes are $\frac{2}{3}$ and $\frac{-1}{4}$. Verify the relation between the coefficients and the zeroes of the polynomial.
Let $\alpha=\frac{2}{3}$ and $\beta=\frac{-1}{4}$. Sum of the zeroes $=(\alpha+\beta)=\frac{2}{3}+\left(\frac{-1}{4}\right)=\frac{8-3}{12}=\frac{5}{12}$ Product of the zeroes, $\alpha...
Find the quadratic polynomial whose zeroes are 2 and $-6 .$ Verify the relation between the coefficients and the zeroes of the polynomial.
Let $\alpha=2$ and $\beta=-6$ Sum of the zeroes, $(\alpha+\beta)=2+(-6)=-4$ Product of the zeroes, $\alpha \beta=2 \times(-6)=-12$ $\therefore$ Required polynomial $=\mathrm{x}^{2}-(\alpha+\beta)...
Find the zeroes of the quadratic polynomial $\left(3 x^{2}-x-4\right)$ and verify the relation between the zeroes and the coefficients.
$$ \begin{aligned} &3 x^{2}-x-4=0 \\ &\Rightarrow 3 x^{2}-4 x+3 x-4=0 \\ &\Rightarrow x(3 x-4)+1(3 x-4)=0 \\ &\Rightarrow(3 x-4)(x+1)=0 \\ &\Rightarrow(3 x-4) \text { or }(x+1)=0 \\ &\Rightarrow...
Find the zeroes of the quadratic polynomial (5y $\left.^{2}+10 \mathrm{y}\right)$ and verify the relation between the zeroes and the coefficients.
f(u)=5u2+10u \mathrm{f}(\mathrm{u})=5 \mathrm{u}^{2}+10 \mathrm{u} It can be written as $5 \mathrm{u}(\mathrm{u}+2)$ ∴f(u)=0⇒5u=0 or u+2=0 \therefore \mathrm{f}(\mathrm{u})=0...
Find the zeroes of the quadratic polynomial $\left(x^{2}-5\right)$ and verify the relation between the zeroes and the coefficients.
$$ \begin{aligned} &f(x)=x^{2}-5 \\ &\text { It can be written as } x^{2}+0 x-5 . \\ &=\left(x^{2}-(\sqrt{5})^{2}\right) \\ &=(x+\sqrt{5})(x-\sqrt{5}) \\ &\therefore f(x)=0...
Find the zeroes of the quadratic polynomial $4 \mathrm{x}^{2}-4 \mathrm{x}+1$ and verify the relation between the zeroes and the coefficients.
$$ \begin{aligned} &4 x^{2}-4 x+1=0 \\ &\Rightarrow(2 x)^{2}-2(2 x)(1)+(1)^{2}=0 \end{aligned} $$ $$ \begin{aligned} &\Rightarrow(2 \mathrm{x}-1)^{2}=0 \quad\left[\because \mathrm{a}^{2}-2...
Find the zeroes of the polynomial $f(x)=2 \sqrt{3} x^{2}-5 x+\sqrt{3}$ and verify the relation between its zeroes and coefficients.
$$ \begin{aligned} &2 \sqrt{3} x^{2}-5 x+\sqrt{3} \\ &\Rightarrow 2 \sqrt{3} x^{2}-2 x-3 x+\sqrt{3} \\ &\Rightarrow 2 x(\sqrt{3} x-1)-\sqrt{3}(\sqrt{3} x-1)=0 \\ &\Rightarrow(\sqrt{3} x-1) \text {...
Find the zeroes of the quadratic polynomial $f(x)=5 x^{2}-4-8 x$ and verify the relationship between the zeroes and coefficients of the given polynomial.
$f(x)=5 x^{2}-4-8 x$ $=5 x^{2}-8 x-4$ $=5 x^{2}-(10 x-2 x)-4$ $=5 x^{2}-10 x+2 x-4$ $=5 x(x-2)+2(x-2)$ $=(5 x+2)(x-2)$ $\therefore \mathrm{f}(\mathrm{x})=0 \Rightarrow(5...
Find the zeroes of the quadratic polynomial $f(x)=4 x^{2}-4 x-3$ and verify the relation between its zeroes and coefficients.
$f(x)=4 x^{2}-4 x-3$ $=4 x^{2}-(6 x-2 x)-3$ $=4 x^{2}-6 x+2 x-3$ $=2 x(2 x-3)+1(2 x-3)$ $=(2 x+1)(2 x-3)$ $\therefore \mathrm{f}(\mathrm{x})=0 \Rightarrow(2 \mathrm{x}+1)(2 \mathrm{x}-3)=0$...
Find the zeroes of the quadratic polynomial $f(x)=x^{2}+3 x-10$ and verify the relation between its zeroes and coefficients.
$f(x)=x^{2}+3 x-10$ $=x^{2}+5 x-2 x-10$ $=x(x+5)-2(x+5)$ $=(x-2)(x+5)$ $\therefore \mathrm{f}(\mathrm{x})=0 \Rightarrow(\mathrm{x}-2)(\mathrm{x}+5)=0$ $\Rightarrow x-2=0$ or $x+5=0$ $\Rightarrow...
Find the zeroes of the polynomial $f(x)=x^{2}-2 x-8$ and verify the relation between its zeroes and coefficients.
$x^{2}-2 x-8=0$ $\Rightarrow \mathrm{x}^{2}-4 \mathrm{x}+2 \mathrm{x}-8=0$ $\Rightarrow x(x-4)+2(x-4)=0$ $\Rightarrow(x-4)(x+2)=0$ $\Rightarrow(x-4)=0$ or $(x+2)=0$ $\Rightarrow x=4$ or $x=-2$ Sum...
Find the zeros of the polynomial $f(x)=x^{2}+7 x+12$ and verify the relation between its zeroes and coefficients.
$x^{2}+7 x+12=0$ $\Rightarrow x^{2}+4 x+3 x+12=0$ $\Rightarrow x(x+4)+3(x+4)=0$ $\Rightarrow(x+4)(x+3)=0$ $\Rightarrow(x+4)=0$ or $(x+3)=0$ $\Rightarrow \mathrm{x}=-4$ or $\mathrm{x}=-3$ Sum of...