Write down all possible proper subsets each of the following sets:
Question: Write down all possible proper subsets each of the following sets: (i) {1, 2}, (ii) {1, 2, 3} (iii) {1}. Solution: (i) {1}, {2} (ii) {1}, {2}, {3}, {1, 2}, {2, 3}, {1, 3} (iii) No proper subsets are there in this set....
Read More →Write down all possible proper subsets each of the following sets:
Question: Write down all possible proper subsets each of the following sets: (i) {1, 2}, (ii) {1, 2, 3} (iii) {1}. Solution: c (ii) {1}, {2}, {3}, {1, 2}, {2, 3}, {1, 3} (iii) No proper subsets are there in this set....
Read More →Solve the following systems of equations graphically:
Question: Solve the following systems of equations graphically: $x+y=6$ $x-y=2$ Solution: The given equations are $x+y=6 \quad \ldots \ldots .(i)$ $x-y=2 \quad \ldots \ldots \ldots(i i)$ Putting $x=0$ in equation $(i)$, we get: $\Rightarrow x+0=6$ $\Rightarrow x=6$ $x=0, \quad y=6$ Putting $y=0$ in equation $(i,$, we get: $\Rightarrow x+0=6$ $\Rightarrow x=6$ $x=6, \quad y=0$ Use the following table to draw the graph. Draw the graph by plotting the two points $A(0,6)$ and $B(6,0)$ from table. Gr...
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Question: Find the value i. $\frac{173 \times 173 \times 173+127 \times 127 \times 127}{173 \times 173-173 \times 127+127 \times 127}$ ii. $\frac{1.2 \times 1.2 \times 1.2-0.2 \times 0.2 \times 0.2}{1.2 \times 1.2+1.2 \times 0.2+0.2 \times 0.2}$ iii. $\frac{155 \times 155 \times 155-55 \times 55 \times 55}{155 \times 155+155 \times 55+55 \times 55}$ Solution: i. $\frac{173 \times 173 \times 173+127 \times 127 \times 127}{173 \times 173-173 \times 127+127 \times 127}$ $=\frac{173^{3}+127^{3}}{173...
Read More →Write down all possible subsets of each of the following sets:
Question: Write down all possible subsets of each of the following sets: (i) $\{a\}$, (ii) $\{0,1\}$, (iii) $\{a, b, c\}$, (iv) $\{1,\{1\}\}$ (v) $\{\phi\}$. Solution: (i) $\phi,\{a\}$ (ii) $\phi,\{0\},\{1\},\{0,1\}$ (iii) $\phi,\{\mathrm{a}\},\{\mathrm{b}\},\{\mathrm{c}\},\{\mathrm{a}, \mathrm{b}\},\{\mathrm{b}, \mathrm{c}\},\{\mathrm{a}, \mathrm{c}\},\{\mathrm{a}, \mathrm{b}, \mathrm{c}\}$ (iv) $\phi,\{1\},\{\{1\}\},\{1,\{1\}\}$ (v) $\phi,\{\phi\}$...
Read More →Let A =
Question: Let $A=\{\phi,\{\phi\}, 1,\{1, \phi\}, 2\}$. Which of the following are true? (i) $\phi \in A$ (ii) $\{\phi\} \in A$ (iii) $\{1\} \in A$ (iv) $\{2, \phi\} \subset A$ (v) $2 \subset A$ (vi) $\{2\{1\}\} \not \subset A$ (vii) $\{\{2\},\{1\}\} \not \subset A$ (viii) $\{\phi,\{\phi\},\{1, \phi\}\} \subset A$ (ix) $\{\{\phi\}\} \subset A$ Solution: (i) True (ii) True (iii) False The correct form would be $\{1\} \subset A$. (iv) True (v) False The correct form would be $1 \in A$. (vi) True (v...
Read More →Let $A=
Question: Let $A=\{\phi,\{\phi\}, 1,\{1, \phi\}, 2\}$. Which of the following are true? (i) $\phi \in A$ (ii) $\{\phi\} \in A$ (iii) $\{1\} \in A$ (iv) $\{2, \phi\} \subset A$ (v) $2 \subset A$ (vi) $\{2\{1\}\} \not \subset A$ (vii) $\{\{2\},\{1\}\} \not \subset A$ (viii) $\{\phi,\{\phi\},\{1, \phi\}\} \subset A$ (ix) $\{\{\phi\}\} \subset A$ Solution: (i) True (ii) True (iii) False The correct form would be $\{1\} \subset A$. (iv) True (v) False The correct form would be $1 \in A$. (vi) True (v...
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Question: Find the value $8 a^{3}-b^{3}-4 a x+2 b x$ Solution: $=(2 a)^{3}-b^{3}-2 x(2 a-b)$ $=(2 a-b)\left((2 a)^{2}+2 a \times b+b^{2}\right)-2 x(2 a-b)$ $\left[\therefore a^{3}-b^{3}=(a-b)\left(a^{2}+a b+b^{2}\right)\right]$ $=(2 a-b)\left(4 a^{2}+2 a b+b^{2}-2 x\right)$ $\therefore 8 a^{3}-b^{3}-4 a x+2 b x=(2 a-b)\left(4 a^{2}+2 a b+b^{2}-2 x\right)$...
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Question: Find the value $a^{3}+3 a^{2} b+3 a b^{2}+b^{3}-8$ Solution: $=(a+b)^{3}-8$ $\left[\therefore a^{3}+3 a^{2} b+3 a b^{2}+b^{3}=(a+b)^{3}\right]$ $=(a+b)^{3}-23$ $=(a+b-2)\left((a+b)^{2}+(a+b) \times 2+2^{2}\right)$ $=(a+b-2)\left(a^{2}+2 a b+b^{2}+2 a+2 b+4\right)$ $\therefore a^{3}+3 a^{2} b+3 a b^{2}+b^{3}-8=(a+b-2)\left(a^{2}+2 a b+b^{2}+2 a+2 b+4\right)$...
Read More →Show that the function given by
Question: Show that the function given by $f(x)=\frac{\log x}{x}$ has maximum at $x=e$. Solution: The given function is $f(x)=\frac{\log x}{x}$. $f^{\prime}(x)=\frac{x\left(\frac{1}{x}\right)-\log x}{x^{2}}=\frac{1-\log x}{x^{2}}$ Now, $f^{\prime}(x)=0$ $\Rightarrow 1-\log x=0$ $\Rightarrow \log x=1$ $\Rightarrow \log x=\log e$ $\Rightarrow x=e$ Now, $f^{\prime \prime}(x)=\frac{x^{2}\left(-\frac{1}{x}\right)-(1-\log x)(2 x)}{x^{4}}$ $=\frac{-x-2 x(1-\log x)}{x^{4}}$ $=\frac{-x-2 x(1-\log x)}{x^{...
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Question: Find the value $a^{3}-1 / a^{3}-2 a+2 a$ Solution: $=\left(a^{3}-1 / a^{3}\right)-2(a-1 / a)$ $=\left(a^{3}-(1 / a)^{3}\right)-2(a-1 / a)$ $=(a-1 / a)\left(a^{2}+a \times 1 / a+(1 / a)^{2}\right)-2(a-1 / a)$ $\left[\therefore a^{3}-b^{3}=(a-b)\left(a^{2}+a b+b^{2}\right)\right]$ $=(a-1 / a)\left(a^{2}+1+1 / a^{2}\right)-2(a-1 / a)$ $=(a-1 / a)\left(a^{2}+1+1 / a^{2}-2\right)$ $=(a-1 / a)\left(a^{2}+1 / a^{2}-1\right)$ $\therefore a^{3}-1 / a^{3}-2 a+2 a=(a-1 / a)\left(a^{2}+1 / a^{2}-1...
Read More →Let A = {{1, 2, 3}, {4, 5}, {6, 7, 8}}.
Question: LetA= {{1, 2, 3}, {4, 5}, {6, 7, 8}}. Determine which of the following is true or false: (i) $1 \in A$ (ii) $\{1,2,3\} \subset A$ (iii) $\{6,7,8\} \in A$ (iv) $\{\{4,5\}\} \subset A$ (v) $\phi \in A$ (vi) $\phi \subset A$ϕA Solution: (i) False If it could be $1 \notin A$, then it would be true . (ii) False The correct form would be $\{1,2,3\} \in A$ or $\{\{1,2,3\}\} \subset A$. (iii) True(iv) True(v) False' A null set is a subset of every set. Therefore, the correct form would be $\ph...
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Question: Find the value $a^{3}+b^{3}+a+b$ Solution: $=\left(a^{3}+b^{3}\right)+1(a+b)$ $=(a+b)\left(a^{2}-a b+b^{2}\right)+1(a+b)$ $\left[\therefore a^{3}+b^{3}=(a+b)\left(a^{2}-a b+b^{2}\right)\right]$ $=(a+b)\left(a^{2}-a b+b^{2}+1\right)$ $\therefore a^{3}+b^{3}+a+b=(a+b)\left(a^{2}-a b+b^{2}+1\right)$...
Read More →Let A = {{1, 2, 3}, {4, 5}, {6, 7, 8}}.
Question: LetA= {{1, 2, 3}, {4, 5}, {6, 7, 8}}. Determine which of the following is true or false: (i) $1 \in A$ (ii){1,2,3}A1,2,3A (iii){6,7,8}A6,7,8A (iv){{4,5}}A4,5A (v)ϕAϕA (vi)ϕA Solution: (i) False If it could be $1 \notin A$, then it would be true . (ii) False The correct form would be $\{1,2,3\} \in A$ or $\{\{1,2,3\}\} \subset A$. (iii) True(iv) True(v) False' A null set is a subset of every set. Therefore, the correct form would be $\phi \subset A$. (vi) True...
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Question: Find the value $x^{3}+6 x^{2}+12 x+16$ Solution: $=x^{3}+6 x^{2}+12 x+8+8$ $=x^{3}+3 \times x^{2} \times 2+3 \times x \times 2^{2}+2^{3}+8$ $=(x+2)^{3}+8$ $\left[\therefore a^{3}+3 a^{2} b+3 a b^{2}+b^{3}=(a+b)^{3}\right]$ $=(x+2)^{3}+23$ $=(x+2+2)\left((x+2)^{2}-2(x+2)+2^{2}\right)$ $\therefore\left[a^{3}+b^{3}=(a+b)\left(a^{2}-a b+b^{2}\right)\right]$ $=(x+2+2)\left(x^{2}+4+4 x-2 x-4+4\right)$ $\left[\therefore(a+b)^{2}=a^{2}+b^{2}+2 a b\right]$ $=(x+4)\left(x^{2}+4+2 x\right)$ $\the...
Read More →Using differentials, find the approximate value of each of the following.
Question: Using differentials, find the approximate value of each of the following. (a) $\left(\frac{17}{81}\right)^{\frac{1}{4}}$ (b) $(33)^{-\frac{1}{5}}$ Solution: (a) Consider $y=x^{\frac{1}{4}}$. Let $x=\frac{16}{81}$ and $\Delta x=\frac{1}{81}$. Then, $\Delta y=(x+\Delta x)^{\frac{1}{4}}-x^{\frac{1}{4}}$ $=\left(\frac{17}{81}\right)^{\frac{1}{4}}-\left(\frac{16}{81}\right)^{\frac{1}{4}}$ $=\left(\frac{17}{81}\right)^{\frac{1}{4}}-\frac{2}{3}$ $\therefore\left(\frac{17}{81}\right)^{\frac{1}...
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Question: Find the value $a^{12}+b^{12}$ Solution: $=\left(a^{4}\right)^{3}+\left(b^{4}\right)^{3}$ $=\left(a^{4}+b^{4}\right)\left(\left(a^{4}\right)^{2}-a^{4} \times b^{4}+\left(b^{4}\right)^{2}\right)$ $\therefore\left[a^{3}+b^{3}=(a+b)\left(a^{2}-a b+b^{2}\right)\right]$ $=\left(a^{4}+b^{4}\right)\left(a^{8}-a^{4} b^{4}+b^{8}\right)$ $\therefore a^{12}+b^{12}=\left(a^{4}+b^{4}\right)\left(a^{8}-a^{4} b^{4}+b^{8}\right)$...
Read More →Solve the following systems of equations graphically:
Question: Solve the following systems of equations graphically: $2 x+y-3=0$ $2 x-3 y-7=0$ Solution: The given equations are $\Rightarrow 2 x+y=3$$\ldots . .(i)$ $\Rightarrow 2 x-3 y=7$ $\ldots . .(ii)$ Putting $x=0$ in equation $(i)$, we get: $\Rightarrow 2 \times 0+y=3$ $\Rightarrow y=3$ $x=0, \quad y=3$ Putting $y=0$ in equation $(i,$, we get: $\Rightarrow 2 x+0=3$ $\Rightarrow x=3 / 2$ $x=3 / 2, \quad y=0$ Use the following table to draw the graph. Draw the graph by plotting the two points $A...
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Question: Find the value $x^{4} y^{4}-x y$ Solution: $=x y\left(x^{3} y^{3}-1\right)$ $=x y\left((x y)^{3}-1^{3}\right)$ $=x y(x y-1)\left((x y)^{2}+x y+1+12\right)$ $\therefore\left[x^{3}-y^{3}=(x-y)\left(x^{2}+x y+y^{2}\right)\right]$ $=x y(x y-1)\left(x^{2} y^{2}+x y+1\right)$ $\therefore x^{4} y^{4}-x y=x y(x y-1)\left(x^{2} y^{2}+x y+1\right)$...
Read More →Let A = {a, b, {c, d}, e}. Which of the following statements are false and why?
Question: LetA= {a,b, {c,d},e}. Which of the following statements are false and why? (i) $\{c, d\} \subset A$ (ii) $\{c, d\} \in A$ (iii) $\{\{c, d\}\} \subset A$ (iv) $a \in A$ (v) $a \subset A$ (vi) $\{a, b, e\} \subset A$ (vii) $\{a, b, e\} \in A$ (viii) $\{a, b, c\} \subset A$ (ix) $\phi \in A$ (x) $\{\phi\} \subset A$ Solution: A= {a,b, {c,d},e} (i) False The correct statement would be $\{\{c, d\}\} \subset A$. (ii) True (iii) True (iv) True (v) False The correct statement would be $\{a\} \...
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Question: Find the value $x^{3} y^{3}+1$ Solution: $=(x y)^{3}+1^{3}$ $=(x y+1)\left((x y)^{2}+x y+1^{2}\right)$ $\left[\therefore x^{3}+y^{3}=(x+y)\left(x^{2}-x y+y^{2}\right)\right]$ $=(x y+1)\left(x^{2} y^{2}-x y+1\right)$ $\therefore x^{3} y^{3}+1=(x y+1)\left(x^{2} y^{2}-x y+1\right)$...
Read More →Find the value
Question: Find the value $x^{6}+y^{6}$ Solution: $=\left(x^{2}\right)^{3}+\left(y^{2}\right)^{3}$ $=\left(x^{2}+y^{2}\right)\left(\left(x^{2}\right)^{2}-x^{2} y^{2}+\left(y^{2}\right)^{2}\right)$ $=\left(x^{2}+y^{2}\right)\left(x^{4}-x^{2} y^{2}+y^{4}\right)$ $\left[\therefore a^{3}+b^{3}=(a+b)\left(a^{2}-a b+b^{2}\right)\right]$ $\therefore x^{6}+y^{6}=\left(x^{2}+y^{2}\right)\left(x^{4}-x^{2} y^{2}+y^{4}\right)$...
Read More →The maximum value of
Question: The maximum value of $[x(x-1)+1]^{\frac{1}{3}}, 0 \leq x \leq 1$ is (A) $\left(\frac{1}{3}\right)^{\frac{1}{3}}$ (B) $\frac{1}{2}$ (C) 1 (D) 0 Solution: Let $f(x)=[x(x-1)+1]^{\frac{1}{3}}$. $\therefore f^{\prime}(x)=\frac{2 x-1}{3[x(x-1)+1]^{\frac{2}{3}}}$ Now, $f^{\prime}(x)=0 \Rightarrow x=\frac{1}{2}$ Then, we evaluate the value of $f$ at critical point $x=\frac{1}{2}$ and at the end points of the interval $[0,1]\{$ i.e., at $x=0$ and $x=1\}$. $f(0)=[0(0-1)+1]^{\frac{1}{3}}=1$ $f(1)...
Read More →Let A = {a, b, {c, d}, e}. Which of the following statements are false and why?
Question: LetA= {a,b, {c,d},e}. Which of the following statements are false and why? (i) $\{c, d\} \subset A$ (ii) $\{c, d\} \in A$ (iii) $\{\{c, d\}\} \subset A$ (iv) $a \in A$ (v) $a \subset A$ (vi) $\{a, b, e\} \subset A$ (vii) $\{a, b, e\} \in A$ (viii) $\{a, b, c\} \subset A$ (ix) $\phi \in A$ (x) $\{\phi\} \subset A$ Solution: A= {a,b, {c,d},e} (i) False The correct statement would be{{c,d}}A{c,d}A. (ii) True (iii) True (iv) True (v) False The correct statement would be {a}AoraA. (vi) True...
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Question: Find the value $1029-3 x^{3}$ Solution: $=3\left(343-x^{3}\right)$ $=3\left((7)^{3}-x^{3}\right)$ $=3(7-x)\left(72+7 x+x^{2}\right)$ $\left[\because a^{3}-b^{3}=(a-b)\left(a^{2}+a b+b^{2}\right)\right]$ $=3(7-x)\left(49+7 x+x^{2}\right)$ $\therefore 1029-3 x^{3}=3(7-x)\left(49+7 x+x^{2}\right)$...
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