Find the value
Question: Find the value $(a-3 b)^{3}+(3 b-c)^{3}+(c-a)^{3}$ Solution: Let $a-3 b=x, 3 b-c=y, c-a=z$ $x+y+z=a-3 b+3 b-c+c-a=0$ $(\therefore x+y+z=0) ?$ $x^{3}+y^{3}+z^{3}=3 x y z$ $=3(a-3 b)(3 b-c)(c-a)$ $\therefore(a-3 b)^{3}+(3 b-c)^{3}+(c-a)^{3}$ $=3(a-3 b)(3 b-c)(c-a)$...
Read More →Let A = {3, 6, 12, 15, 18, 21},
Question: LetA= {3, 6, 12, 15, 18, 21},B= {4, 8, 12, 16, 20},C= {2, 4, 6, 8, 10, 12, 14, 16} andD= {5, 10, 15, 20}. Find: (i) $A-B$ (ii) $A-C$ (iii) $A-D$ (iv) $B-A$ (v) $C-A$ (vi) $D-A$ (vii) $B-C$ (viii) $B-D$ Solution: Given: A= {3, 6, 12, 15, 18, 21},B= {4, 8, 12, 16, 20},C= {2, 4, 6, 8, 10, 12, 14, 16} andD= {5, 10, 15, 20} (i) $A-B=\{3,6,15,18,21\}$ (ii) $A-C=\{3,15,18,21\}$ (iii) $A-D=\{3,6,12,18,21\}$ (iv) $B-A=\{4,8,16,20\}$ (v) $C-A=\{2,4,8,10,14,16\}$ (vi) $D-A=\{5,10,20\}$ (vii) $B-C...
Read More →Let A = {3, 6, 12, 15, 18, 21},
Question: LetA= {3, 6, 12, 15, 18, 21},B= {4, 8, 12, 16, 20},C= {2, 4, 6, 8, 10, 12, 14, 16} andD= {5, 10, 15, 20}. Find: (i) $A-B$ (ii) $A-C$ (iii) $A-D$ (iv) $B-A$ (v) $C-A$ (vi) $D-A$ (vii) $B-C$ (viii) $B-D$ Solution: Given: A= {3, 6, 12, 15, 18, 21},B= {4, 8, 12, 16, 20},C= {2, 4, 6, 8, 10, 12, 14, 16} andD= {5, 10, 15, 20} (i) $A-B=\{3,6,15,18,21\}$ (ii) $A-C=\{3,15,18,21\}$ (iii) $A-D=\{3,6,12,18,21\}$ (iv) $B-A=\{4,8,16,20\}$ (v) $C-A=\{2,4,8,10,14,16\}$ (vi) $D-A=\{5,10,20\}$ (vii) $B-C...
Read More →Let A = {3, 6, 12, 15, 18, 21},
Question: LetA= {3, 6, 12, 15, 18, 21},B= {4, 8, 12, 16, 20},C= {2, 4, 6, 8, 10, 12, 14, 16} andD= {5, 10, 15, 20}. Find: (i) $A-B$ (ii) $A-C$ (iii) $A-D$ (iv) $B-A$ (v) $C-A$ (vi) $D-A$ (vii) $B-C$ (viii) $B-D$ Solution: Given: A= {3, 6, 12, 15, 18, 21},B= {4, 8, 12, 16, 20},C= {2, 4, 6, 8, 10, 12, 14, 16} andD= {5, 10, 15, 20} (i) $A-B=\{3,6,15,18,21\}$ (ii) $A-C=\{3,15,18,21\}$ (iii) $A-D=\{3,6,12,18,21\}$ (iv) $B-A=\{4,8,16,20\}$ (v) $C-A=\{2,4,8,10,14,16\}$ (vi) $D-A=\{5,10,20\}$ (vii) $B-C...
Read More →Find the value
Question: Find the value $[x / 2+y+z / 3]^{3}+[x / 3-2 y / 3+z]^{3}+[-5 x / 6-y / 3-4 z / 3]^{3}$ Solution: Let $[x / 2+y+z / 3]=a,[x / 3-2 y / 3+z]=b,[-5 x / 6-y / 3-4 z / 3]=c$ $a+b+c=x / 2+y+z / 3+x / 3-2 y / 3+z-5 x / 6-y / 3-4 z / 3$ $a+b+c=(x / 2+x / 3-5 x / 6)+(y-2 y / 3-y / 3)+(z / 3+z-4 z / 3)$ $a+b+c=3 x / 6+2 x / 6-5 x / 6+3 y / 3-2 y / 3-y / 3+z / 3+3 z / 3-4 z / 3$ $a+b+c=\frac{5 x-5 x}{6}+\frac{3 y-3 y}{3}+\frac{4 z-4 z}{3}$ $a+b+c=0$ $\because a+b+c=0 \therefore a^{3}+b^{3}+c^{3}=...
Read More →Let A=
Question: Let $A=\{x: x \in N\}, B=\{x: x-2 n, n \in N\}, C=\{x: x=2 n-1, n \in N\}$ and $D=\{x: x$ is a prime natural number\}. Find: (i) $A \cap B$ (ii) $A \cap C$ (iii) $A \cap D$ (iv) $B \cap C$ (v) $B \cap D$ (vi) $C \cap D$ Solution: $A=\{x: x \in N\}=\{1,2,3, \ldots\}$ $B=\{x: x-2 n, n \in N\}=\{2,4,6,8, \ldots\}$ $C=\{x: x=2 n-1, n \in N\}=\{1,3,5,7, \ldots\}$ D= {x:xis a prime natural number.} = {2, 3, 5, 7,...} (i) $A \cap B=B$ (ii) $A \cap C=C$ (iii) $A \cap D=D$ (iv) $B \cap C=\phi$ ...
Read More →Find the value
Question: Find the value $(2 x-3 y)^{3}+(4 z-2 x)^{3}+(3 y-4 z)^{3}$ Solution: Let $2 x-3 y=a, 4 z-2 x=b, 3 y-4 z=c$ $\therefore a+b+c=2 x-3 y+4 z-2 x+3 y-4 z=0$ $\because a+b+c=0 \therefore a^{3}+b^{3}+c^{3}=3 a b c$ $=3(2 x-3 y)(4 z-2 x)(3 y-4 z)$ $\therefore(2 x-3 y)^{3}+(4 z-2 x)^{3}+(3 y-4 z)^{3}$ $=3(2 x-3 y)(4 z-2 x)(3 y-4 z)$...
Read More →A tank with rectangular base and rectangular sides,
Question: A tank with rectangular base and rectangular sides, open at the top is to be constructed so that its depth is 2 m and volume is 8 m3. If building of tank costs Rs 70 per sq meters for the base and Rs 45 per square metre for sides. What is the cost of least expensive tank? Solution: Letl,b, andhrepresent the length, breadth, and height of the tank respectively. Then, we have height (h)= 2 m Volume of the tank = 8m3 Volume of the tank =lbh 8 =lb2 $\Rightarrow l b=4 \Rightarrow b=\frac{4}...
Read More →Let A=
Question: Let $A=\{x: x \in N\}, B=\{x: x-2 n, n \in N\}, C=\{x: x=2 n-1, n \in N\}$ and $D=\{x: x$ is a prime natural number\}. Find: (i) $A \cap B$ (ii) $A \cap C$ (iii) $A \cap D$ (iv) $B \cap C$ (v) $B \cap D$ (vi) $C \cap D$ Solution: $A=\{x: x \in N\}=\{1,2,3, \ldots\}$ $B=\{x: x-2 n, n \in N\}=\{2,4,6,8, \ldots\}$ $C=\{x: x=2 n-1, n \in N\}=\{1,3,5,7, \ldots\}$ D= {x:xis a prime natural number.} = {2, 3, 5, 7,...} (i) $A \cap B=B$ (ii) $A \cap C=C$ (iii) $A \cap D=D$ (iv) $B \cap C=\phi$ ...
Read More →Find the value
Question: Find the value $(3 x-2 y)^{3}+(2 y-4 z)^{3}+(4 z-3 x)^{3}$ Solution: Let $(3 x-2 y)=a,(2 y-4 z)=b,(4 z-3 x)=c$ $\therefore a+b+c=3 x-2 y+2 y-4 z+4 z-3 x=0$ $\because a+b+c=0 \therefore a^{3}+b^{3}+c^{3}=3 a b c$ $=3(3 x-2 y)(2 y-4 z)(4 z-3 x)$ $\therefore(3 x-2 y)^{3}+(2 y-4 z)^{3}+(4 z-3 x)^{3}$ $=3(3 x-2 y)(2 y-4 z)(4 z-3 x)$...
Read More →If A = {1, 2, 3, 4, 5}, B = {4, 5, 6, 7, 8},
Question: If $A=\{1,2,3,4,5\}, B=\{4,5,6,7,8\}, C=\{7,8,9,10,11\}$ and $D=\{10,11,12,13,14\}$, find: (i) $A \cup B$ (ii) $A \cup C$ (iii) $B \cup C$ (iv) $B \cup D$ (v) $A \cup B \cup C$ (vi) $A \cup B \cup D$ (vii) $B \cup C \cup D$ (viii) $A \cap(B \cup C)$ (ix) $(A \cap B) \cap(B \cap C)$ (x) $(A \cup D) \cap(B \cup C)$. Solution: Given: A= {1, 2, 3, 4, 5},B= {4, 5, 6, 7, 8},C= {7, 8, 9, 10, 11} andD= {10, 11, 12, 13, 14} (i) $A \cup B=\{1,2,3,4,5,6,7,8\}$ (ii) $A \cup C=\{1,2,3,4,5,7,8,9,10,...
Read More →Find the value
Question: Find the value $125+8 x^{3}-27 y^{3}+90 x y$ Solution: $=(5)^{3}+(2 x)^{3}+(-3 y)^{3}-3 \times 5 \times 2 x \times(-3 y)$ $=(5+2 x+(-3 y))\left(5^{2}+(2 x)^{2}+(-3 y)^{2}-5(2 x)-2 x(-3 y)-(-3 y) 5\right)$ $=(5+2 x-3 y)\left(25+4 x^{2}+9 y^{2}-10 x+6 x y+15 y\right)$ $\therefore 125+8 x^{3}-27 y^{3}+90 x y$ $=(5+2 x-3 y)\left(25+4 x^{2}+9 y^{2}-10 x+6 x y+15 y\right)$...
Read More →If A = {1, 2, 3, 4, 5}, B = {4, 5, 6, 7, 8},
Question: If $A=\{1,2,3,4,5\}, B=\{4,5,6,7,8\}, C=\{7,8,9,10,11\}$ and $D=\{10,11,12,13,14\}$, find: (i) $A \cup B$ (ii) $A \cup C$ (iii) $B \cup C$ (iv) $B \cup D$ (v) $A \cup B \cup C$ (vi) $A \cup B \cup D$ (vii) $B \cup C \cup D$ (viii) $A \cap(B \cup C)$ (ix) $(A \cap B) \cap(B \cap C)$ (x) $(A \cup D) \cap(B \cup C)$. Solution: Given: A= {1, 2, 3, 4, 5},B= {4, 5, 6, 7, 8},C= {7, 8, 9, 10, 11} andD= {10, 11, 12, 13, 14} (i) $A \cup B=\{1,2,3,4,5,6,7,8\}$ (ii) $A \cup C=\{1,2,3,4,5,7,8,9,10,...
Read More →Solve the following systems of equations graphically:
Question: Solve the following systems of equations graphically: $2 x-3 y+13=0$ $3 x-2 y+12=0$ Solution: The given equations are: $2 x-3 y+13=0$$.(i)$ $3 x-2 y+12=0$..(ii) Putting $x=0$ in equation $(i)$, we get: $\Rightarrow 2 \times 0-3 y=-13$ $\Rightarrow y=13 / 3$ $x=0, \quad y=13 / 3$ Use the following table to draw the graph. Draw the graph by plotting the two points $A(0,13 / 2)$ and $B(-13 / 2,0)$ from table. Graph of the equation.... (ii): $3 x-2 y=-12$ (ii) Putting $x=0$ in equation $($...
Read More →Find the value
Question: Find the value $8 x^{3}+27 y^{3}-216 z^{3}+108 x y z$ Solution: $=(2 x)^{3}+(3 y)^{3}+(-6 y)^{3}-3(2 x)(3 y)(-6 z)$ $=(2 x+3 y+(-6 z))\left((2 x)^{2}+(3 y)^{2}+(-6 z)^{2}-2 x+3 y-3 y(-6 z)-(-6 z) 2 x\right)$ $=(2 x+3 y+(-6 z))\left(4 x^{2}+9 y^{2}+36 z^{2}-6 x y+18 y z+12 z x\right)$ $? 8 x^{3}+27 y^{3}-216 z^{3}+108 x y z$ $=(2 x+3 y+(-6 z))\left(4 x^{2}+9 y^{2}+36 z^{2}-6 x y+18 y z+12 z x\right)$...
Read More →Find the value
Question: Find the value $\frac{1}{27} x^{3}-y^{3}+125 z^{3}+5 x y z$ Solution: $=(x / 3)^{3}+(-y)^{3}+(5 z)^{3}-3 \times x / 3(-y)(5 z)$ $=(x / 3+(-y)+5 z)\left((x / 3)^{2}+(-y)^{2}+(5 z)^{2}-x / 3(-y)-(-y) 5 z-5 z(x / 3)\right)$ $=(x / 3-y+5 z)\left(x^{2} / 9+y^{2}+25 z^{2}+x y / 3+5 y z-5 z x / 3\right)$ $\therefore \frac{1}{27} \mathrm{x}^{3}-\mathrm{y}^{3}+125 \mathrm{z}^{3}+5 \mathrm{xyz}$ $=(x / 3-y+5 z)\left(x^{2} / 9+y^{2}+25 z^{2}+x y / 3+5 y z-5 z x / 3\right)$...
Read More →Find the maximum area of an isosceles triangle inscribed in the ellipse
Question: Find the maximum area of an isosceles triangle inscribed in the ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$ with its vertex at one end of the major axis. Solution: The given ellipse is $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$. Let the major axis be along thexaxis. Let ABC be the triangle inscribed in the ellipse where vertex C is at (a, 0). Since the ellipse is symmetrical with respect to thexaxis andyaxis, we can assume the coordinates of A to be (x1,y1) and the coordinates...
Read More →If A and B are two sets such that
Question: If $A$ and $B$ are two sets such that $A \subset B$, then find: (i) $A \cap B$ (ii) $A \cup B$ Solution: From the Venn diagrams given below, we can clearly say that if $A$ and $B$ are two sets such that $A \subset B$, then(i) Form the given Venn diagram, we can see thatABAB=A (ii) Form the given Venn diagram, we can see thatABAB=B...
Read More →Find the value
Question: Find the value $x^{3}-8 y^{3}+27 z^{3}+18 x y z$ Solution: $=x^{3}-(2 y)^{3}+(3 z)^{3}-3 \times x \times(-2 y)(3 z)$ $=(x+(-2 y)+3 z)\left(x^{2}+(-2 y)^{2}+(3 z)^{2}-x(-2 y)-(-2 y)(3 z)-3 z(x)\right)$ $\left[\therefore a^{3}+b^{3}+c^{3}-3 a b c=(a+b+c)\left(a^{2}+b^{2}+c^{2}-a b-b c-c a\right)\right]$ $=(x+(-2 y)+3 z)\left(x^{2}+4 y^{2}+9 z^{2}+2 x y+6 y z-3 z x\right)$ $\therefore x^{3}-8 y^{3}+27 z^{3}+18 x y z$ $=(x+(-2 y)+3 z)\left(x^{2}+4 y^{2}+9 z^{2}+2 x y+6 y z-3 z x\right)$...
Read More →If A and B are two sets such that
Question: If $A$ and $B$ are two sets such that $A \subset B$, then find: (i) $A \cap B$ (ii) $A \cup B$ Solution: From the Venn diagrams given below, we can clearly say that ifAandBare two sets such thatABAB, then(i) Form the given Venn diagram, we can see thatABAB=A (ii) Form the given Venn diagram, we can see thatABAB=B...
Read More →Find the value
Question: Find the value $a^{3}+8 b^{3}+64 c^{3}-24 a b c$ Solution: $=(a)^{3}+(2 b)^{3}+(4 c)^{3}-3 \times a \times 2 b \times 4 c$ $=(a+2 b+4 c)\left(a^{2}+(2 b)^{2}+(4 c)^{2}-a \times 2 b-2 b \times 4 c-4 c \times a\right)$ $\left[\therefore a^{3}+b^{3}+c^{3}-3 a b c=(a+b+c)\left(a^{2}+b^{2}+c^{2}-a b-b c-c a\right)\right]$ $=(a+2 b+4 c)\left(a^{2}+4 b^{2}+16 c^{2}-2 a b-8 b c-4 a c\right)$ $\therefore a^{3}+8 b^{3}+64 c^{3}-24 a b c$ $=(a+2 b+4 c)\left(a^{2}+4 b^{2}+16 c^{2}-2 a b-8 b c-4 a ...
Read More →If A = [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10],
Question: IfA= [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], then insert the appropriate symbol or in each of the following blanks spaces: (i) 4 ......A (ii) 4 ......A (iii) 12 ......A (iv) 9 ......A (v) 0 ......A (vi) 2 ......A Solution: (i) $4 \in A$ (ii) $-4 \notin A$ (iii) $12 \notin A$ (iv) $9 \in A$ (v) $0 \in A$ (vi) $-2 \notin A$...
Read More →Find the value
Question: Find the value $a^{3} x^{3}-3 a^{2} b x^{2}+3 a b^{2} x-b^{3}$ Solution: $=(a x)^{3}-3(a x)^{2} \times b+3(a x) \times b^{2}-b^{3}$ $=(a x-b)^{3}$ $\left[\therefore a^{3}-b^{3}-3 a^{2} b+3 a b^{2}=(a-b)^{3}\right]$ $=(a x-b)(a x-b)(a x-b)$ $\therefore a^{3} x^{3}-3 a^{2} b x^{2}+3 a b^{2} x-b^{3}$ $=(a x-b)(a x-b)(a x-b)$...
Read More →Solve the following systems of equations graphically:
Question: Solve the following systems of equations graphically: $2 x+3 y=4$ $x-y+3=0$ Solution: The given equations are: $2 x+3 y=4$$.(i)$ $x-y+3=0$..(ii) Putting $x=0$ in equation $(i)$, we get: $\Rightarrow 2 \times 0+3 y=4$ $\Rightarrow y=4 / 3$ $x=0, \quad y=4 / 3$ Putting $y=0$ in equation $(i,$, we get: $\Rightarrow 2 x+3 \times 0=4$ $\Rightarrow x=2$ $x=2, \quad y=0$ Use the following table to draw the graph. Draw the graph by plotting the two points $A(0,4 / 3)$ and $B(2,0)$ from table G...
Read More →Which of the following collections are sets?
Question: Which of the following collections are sets? Justify your answer: (i) A collection of all natural numbers less than 50. (ii) The collection of good hockey players in India. (iii) The collection of all girls in your class. (iv) The collection of most talented writers of India. (v) The collection of difficult topics in mathematics. (vi) The collection of all months of a year beginning with the letter J. (Vii) A collection of novels written by Munshi Prem Chand. (Viii) The collection of a...
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