Factorise:

Question: Factorise: $x^{2}-3 \sqrt{5} x-20$ Solution: $x^{2}-3 \sqrt{5} x-20$ $=x^{2}-4 \sqrt{5} x+\sqrt{5} x-20$ $=x(x-4 \sqrt{5})+\sqrt{5}(x-4 \sqrt{5})$ $=(x-4 \sqrt{5})(x+\sqrt{5})$...

Let A = {1, 2, 3, 4}; B = {3, 5, 7, 9}; C = {7, 23, 47, 79} and

Question: Let $A=\{1,2,3,4\} ; B=\{3,5,7,9\} ; C=\{7,23,47,79\}$ and $f: A \rightarrow B, g: B \rightarrow C$ be defined as $f(x)=2 x+1$ and $g(x)=x^{2}-2$. Express $(g \circ f)^{-1}$ and $f^{-1} \circ g^{-1}$ as the sets of ordered pairs and verify that $(g \circ f)^{-1}=f^{-1} o g^{-1}$. Solution: $f(x)=2 x+1$ $\Rightarrow f=\{(1,2(1)+1),(2,2(2)+1),(3,2(3)+1),(4,2(4)+1)\}=\{(1,3),(2,5),(3,7),(4,9)\}$ $g(x)=x^{2}-2$ $\Rightarrow g=\left\{\left(3,3^{2}-2\right),\left(5,5^{2}-2\right),\left(7,7^{...

Factorise:

Question: Factorise: $x^{2}-2 \sqrt{3} x-24$ Solution: $x^{2}-2 \sqrt{3} x-24$ $=x^{2}-4 \sqrt{3} x+2 \sqrt{3} x-24$ $=x(x-4 \sqrt{3})+2 \sqrt{3}(x-4 \sqrt{3})$ $=(x-4 \sqrt{3})(x+2 \sqrt{3})$...

Solve each of the following system of equations in R. 13. 2 (x − 6) < 3x − 7, 11 − 2x < 6 − x

Question: Solve each of the following system of equations in R. 13. 2 (x 6) 3x 7, 11 2x 6 x Solution: $2(x-6)3 x-7$ $\Rightarrow 2 x-123 x-7$ $\Rightarrow 3 x-72 x-12$ $\Rightarrow 3 x-2 x-12+7$ $\Rightarrow x-5$ $\Rightarrow x \in(-5, \infty) \quad \ldots$ (i) Also, $11-2 x6-x$ $\Rightarrow 6-x11-2 x$ $\Rightarrow 2 x-x11-6$ $\Rightarrow x5$ $\Rightarrow x \in(5, \infty) \quad \ldots$ (ii) Hence, the solution of the given inequation is the intersection of (i) and (ii). $(-5, \infty) \cap(5, \in...

Factorise:

Question: Factorise: $x^{2}-26 x+133$ Solution: $x^{2}-26 x+133$ $=x^{2}-19 x-7 x+133$ $=x(x-19)-7(x-19)$ $=(x-19)(x-7)$...

Factorise:

Question: Factorise: $40+3 x-x^{2}$ Solution: $-x^{2}+3 x+40$ $=-x^{2}+8 x-5 x+40$ $=-x(x-8)-5(x-8)$ $=(x-8)(-x-5)$ $=(8-x)(x+5)$...

Factorise:

Question: Factorise: $x^{2}-\sqrt{3} x-6$ Solution: $x^{2}-\sqrt{3} x-6$ $=x^{2}-2 \sqrt{3} x+\sqrt{3} x-6$ $=x(x-2 \sqrt{3})+\sqrt{3}(x-2 \sqrt{3})$ $=(x-2 \sqrt{3})(x+\sqrt{3})$...

Factorise:

Question: Factorise: $6-x-x^{2}$ Solution: $-x^{2}-x+6$ $=-x^{2}-3 x+2 x+6$ $=-x(x+3)+2(x+3)$ $=(x+3)(-x+2)$ $=(x+3)(2-x)$...

Consider f : {1, 2, 3} → {a, b, c} and g : {a, b, c} → {apple, ball, cat} defined as

Question: Consider $f:\{1,2,3\} \rightarrow\{a, b, c\}$ and $g:\{a, b, c\} \rightarrow\{$ apple, ball, catt defined as $f(1)=a, f(2)=b, f(3)=c, g(\mathrm{a})=$ apple, $g(\mathrm{~b})=$ ball and $g(\mathrm{c})=$ cat. Show that $f, g$ and goo are invertible. Find $f^{-1}, g^{-1}$ and gof $^{-1}$ and show that $(g \circ f)^{-1}=f^{-1} \circ g^{-1}$. Solution: $f=\{(1, a),(2, b),(3, c)\}$ and $g=\{(a$, apple $),(b$, ball $),(c$, cat $)\}$ Clearly, $f$ and $g$ are bijections. So, $f$ and $g$ are inve...

Solve each of the following system of equations in R. 12. x + 5 > 2(x + 1), 2 − x < 3 (x + 2)

Question: Solve each of the following system of equations in R. 12.x+ 5 2(x+ 1), 2 x 3 (x+ 2) Solution: $x+52(x+1)$ $\Rightarrow x+52 x+2$ $\Rightarrow 2 x+2x+5$ $\Rightarrow 2 x-x5-2$ $\Rightarrow x3$ $\begin{array}{ll}\Rightarrow x \in(-\infty, 3) \ldots(\mathrm{i})\end{array}$ Also, $2-x3(x+2)$ $\Rightarrow 2-x3 x+6$ $\Rightarrow 3 x+62-x$ $\Rightarrow 3 x+x2-6$ $\Rightarrow 4 x-4$ $\Rightarrow x-1$ $\Rightarrow x \in(-1, \infty) \quad \ldots$ (ii) Hence, the solution of the given set of ineq...

If the equation (1+m2)x2+2 mcx+(c2−a2)=0 has equal roots,

Question: If the equation $\left(1+m^{2}\right) x^{2}+2 m c x+\left(c^{2}-a^{2}\right)=0$ has equal roots, prove that $c^{2}=a^{2}\left(1+m^{2}\right)$. Solution: The given equation $\left(1+m^{2}\right) x^{2}+2 m c x+\left(c^{2}-a^{2}\right)=0$, has equal roots Then prove that $c^{2}=\left(1+m^{2}\right)$. Here, $a=\left(1+m^{2}\right), b=2 m c$ and,$c=\left(c^{2}-a^{2}\right)$ As we know that $D=b^{2}-4 a c$ Putting the value of $a=\left(1+m^{2}\right), b=2 m c$ and,$c=\left(c^{2}-a^{2}\right)...

Factorise:

Question: Factorise: $x^{2}-11 x-80$ Solution: $x^{2}-11 x-80$ $=x^{2}-16 x+5 x-80$ $=x(x-16)+5(x-16)$ $=(x-16)(x+5)$...

Question: Consider $f:\{1,2,3\} \rightarrow\{a, b, c\}$ and $g:\{a, b, c\} \rightarrow\{$ apple, ball, cat $\}$ defined as $f(1)=a, f(2)=b, f(3)=c, g(a)=$ apple, $g(b)=$ ball and $g(c)=$ cat. Show that $f, g$ and $g o f$ are invertible. Find $f^{-1}, g^{-1}$ and $g \circ f^{-1}$ and show that $(g \circ f)^{-1}=f^{-1} \circ g^{-1}$. Solution: $f=\{(1, a),(2, b),(3, c)\}$ and $g=\{(a$, apple $),(b$, ball $),(c$, cat $)\}$ Clearly, $f$ and $g$ are bijections. So, $f$ and $g$ are invertible. Now, $f...

Solve each of the following system of equations in R. 11. 4x − 1 ≤ 0, 3 − 4x < 0

Question: Solve each of the following system of equations in R. 11. 4x 1 0, 3 4x 0 Solution: We have, $4 x-1 \leq 0$ $\Rightarrow 4 x \leq 1$ $\Rightarrow x \leq \frac{1}{4} \quad$ (Dividing both the sides by 4 ) $\Rightarrow x \in\left(-\infty, \frac{1}{4}\right] \quad \ldots(\mathrm{i})$ Also, $3-4 x0$ $\Rightarrow 03-4 x$ $\Rightarrow 4 x3$ $\Rightarrow x\frac{3}{4} \quad$ Dividing both sides by 4 $\Rightarrow x \in\left(\frac{3}{4}, \infty\right) \quad \ldots$ (ii) Hence, the solution of the...

Factorise:

Question: Factorise: $x^{2}-32 x-105$ Solution: $x^{2}-32 x-105$ $=x^{2}-35 x+3 x-105$ $=x(x-35)+3(x-35)$ $=(x-35)(x+3)$...

Factorise:

Question: Factorise: $x^{2}-24 x-180$ Solution: $x^{2}-24 x-180$ $=x^{2}-30 x+6 x-180$ $=x(x-30)+6(x-30)$ $=(x-30)(x+6)$...

If a, b, c are real numbers such that ac ≠ 0,

Question: If $a, b, c$ are real numbers such that $a c \neq 0$, then show that at least one of the equations $a \times 2+b x+c=0$ and $-a x^{2}+b x+c=0$ has real roots. Solution: The given equations are $a x^{2}+b x+c=0$........(1) $-a x^{2}+b x+c=0$......(2) Roots are simultaneously real Let $D_{1}$ and $D_{2}$ be the discriminants of equation (1) and (2) respectively, Then, $D_{1}=(b)^{2}-4 a c$ $=b^{2}-4 a c$ And $D_{2}=(b)^{2}-4 \times(-a) \times c$ $=b^{2}+4 a c$ Both the given equation wil...

Solve each of the following system of equations in R. 10. 11 − 5x > −4,

Question: Solve each of the following system of equations in R. 10. 11 5x 4, 4x+ 13 11 Solution: We have, $11-5 x-4$ $\Rightarrow-5 x-4-11$ $\Rightarrow-5 x-15$ $\Rightarrow 5 x15 \quad$ [Multiplying both sides by $-1$ ] $\Rightarrow x\frac{15}{5}$ $\Rightarrow x3$ $\Rightarrow x \in(-\infty, 3) \quad \ldots$ (i) Als $o, 4 x+13 \leq-11$ $\Rightarrow 4 x \leq-11-13$ $\Rightarrow 4 x \leq-24$ $\Rightarrow x \leq-6$ $\Rightarrow x \in(-\infty,-6] \quad \ldots$ (ii) Hence, the solution of the given ...

Factorise:

Question: Factorise: $x^{2}+5 \sqrt{5} x+30$ Solution: $x^{2}+5 \sqrt{5} x+30$ $=x^{2}+3 \sqrt{5} x+2 \sqrt{5} x+30$ $=x(x+3 \sqrt{5})+2 \sqrt{5}(x+3 \sqrt{5})$ $=(x+3 \sqrt{5})(x+2 \sqrt{5})$...

Factorise:

Question: Factorise: $x^{2}+6 \sqrt{6} x+48$ Solution: $x^{2}+6 \sqrt{6} x+48$ $=x^{2}+4 \sqrt{6} x+2 \sqrt{6} x+48$ $=x(x+4 \sqrt{6})+2 \sqrt{6}(x+4 \sqrt{6})$ $=(x+4 \sqrt{6})(x+2 \sqrt{6})$...

Solve each of the following system of equations in R. 9. 3x − 1 ≥ 5, x + 2 > −1

Question: Solve each of the following system of equations in R. 9. 3x 1 5,x+ 2 1 Solution: $3 x-1 \geqslant 5$ $\Rightarrow 3 x \geqslant 5+1$ $\Rightarrow x \geq 2$ $\Rightarrow x \in[2, \infty) \quad \ldots$ (i) Also, $x+2-1$ $\Rightarrow x-1-2$ $\Rightarrow x-3$ $\Rightarrow x \in(-3, \infty)$ ...(ii) Hence, the solution of the given set of inequalities is the intersection of (I) and (ii). $[2, \infty) \cap(-3, \infty)=[2, \infty)$ Thus, the solution of the given set of inequalities is $[2, \...

Find f −1 if it exists : f : A → B, where

Question: Find $f^{-1}$ if it exists: $f: A \rightarrow B$, where (i) $A=\{0,-1,-3,2\} ; B=\{-9,-3,0,6\}$ and $f(x)=3 x$. (ii) $A=\{1,3,5,7,9\} ; B=\{0,1,9,25,49,81\}$ and $f(x)=x^{2}$ Solution: (i) $A=\{0,-1,-3,2\} ; B=\{-9,-3,0,6\}$ and $f(x)=3 x$. Given: $f(x)=3 x$ So, $f=\{(0,0),(-1,-3),(-3,-9),(2,6)\}$ Clearly, this is one-one. Range of $f=$ Range of $f=B$ So, $f$ is a bijection and, thus, $f^{-1}$ exists. Hence, $f^{-1}=\{(0,0),(-3,-1),(-9,-3),(6,2)\}$ (ii) $A=\{1,3,5,7,9\} ; B=\{0,1,9,25,...

Prove that both the roots of the equation (x−a) (x−b)+(x−b) (x−c)+(x−c) (x−a)=0

Question: Prove that both the roots of the equation $(x-a)(x-b)+(x-b)(x-c)+(x-c)(x-a)=0$ are real but they are equal only when $a=b=c$. Solution: The quadric equation is $(x-a)(x-b)+(x+b)(x-c)+(x-c)(x-a)=0$ Here, After simplifying the equation $x^{2}-(a+b) x+a b+x^{2}-(b+c) x+b c+x^{2}-(c+a) x+c a=0$ $3 x^{2}-2(a+b+c) x+(a b+b c+c a)=0$ $a=3, b=2(a+b+c)$ and,$c=(a b+b c+c a)$ As we know that $D=b^{2}-4 a c$ Putting the value of $a=3, b=2(a+b+c)$ and, $c=(a b+b c+c a)$ $D=\{2(a+b+c)\}^{2}-4 \time...

Factorise:

Question: Factorise: $x^{2}+3 \sqrt{3} x+6$ Solution: $x^{2}+3 \sqrt{3} x+6$ $=x^{2}+2 \sqrt{3} x+\sqrt{3} x+6$ $=x(x+2 \sqrt{3})+\sqrt{3}(x+2 \sqrt{3})$ $=(x+2 \sqrt{3})(x+\sqrt{3})$...

Solve each of the following system of equations in R. 8. 5x − 1 < 24, 5x + 1 > −24

Question: Solve each of the following system of equations in R. 8. 5x 1 24, 5x+ 1 24 Solution: $5 x-124$ $\Rightarrow 5 x24+1$ $\Rightarrow x5$ $\Rightarrow x \in(-\infty, 5)$ Also, $5 x+1-24$ $\Rightarrow 5 x-24-1$ $\Rightarrow x-5$ $\Rightarrow x \in(-5, \infty) \quad \ldots$ (ii) Hence, the solution of the given set of inequalities is the intersection of (i) and (ii). $(-\infty, 5) \cap(-5, \infty)=(-5,5)$ Thus, the solution of the given set of inequalities is $(-5,5)$....