Find the value
Question: Find the value $8 x^{2} y^{3}-x^{5}$ Solution: $=x^{2}\left((2 y)^{3}-x^{3}\right)$ $=x^{2}(2 y-x)\left((2 y)^{2}+2 y x x+x^{2}\right)$ $\left[\therefore x^{3}-y^{3}=(x-y)\left(x^{2}+x y+y^{2}\right)\right]$ $=x^{2}(2 y-x)\left(4 y^{2}+2 x y+x^{2}\right)$ $\therefore 8 x^{2} y^{3}-x^{5}=x^{2}(2 y-x)\left(4 y^{2}+2 x y+x^{2}\right)$...
Read More →Solve the following systems of equations graphically:
Question: Solve the following systems of equations graphically: $3 x+y+1=0$ $2 x-3 y+8=0$ Solution: The given equations are $3 x+y+1=0 \quad \ldots \ldots .(i)$ $2 x-3 y+8=0 \quad \ldots \ldots \ldots($ ii $)$ Putting $x=0$ in equation $(i)$, we get: $\Rightarrow 3 \times 0+y=-1$ $\Rightarrow y=-1$ $x=0, \quad y=-1$ Putting $y=0$ in equation $(i,$, we get: $\Rightarrow 3 x+0=-1$ $\Rightarrow x=-1 / 3$ $x=-1 / 3, \quad y=0$ Use the following table to draw the graph. Draw the graph by plotting the...
Read More →For all real values of
Question: For all real values of $x$, the minimum value of $\frac{1-x+x^{2}}{1+x+x^{2}}$ is (A) 0 (B) 1 (C) 3 (D) $\frac{1}{3}$ Solution: Let $f(x)=\frac{1-x+x^{2}}{1+x+x^{2}}$ $\therefore f^{\prime}(x)=\frac{\left(1+x+x^{2}\right)(-1+2 x)-\left(1-x+x^{2}\right)(1+2 x)}{\left(1+x+x^{2}\right)^{2}}$ $=\frac{-1+2 x-x+2 x^{2}-x^{2}+2 x^{3}-1-2 x+x+2 x^{2}-x^{2}-2 x^{3}}{\left(1+x+x^{2}\right)^{2}}$ $=\frac{2 x^{2}-2}{\left(1+x+x^{2}\right)^{2}}=\frac{2\left(x^{2}-1\right)}{\left(1+x+x^{2}\right)^{2...
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Question: Find the value $(x+2)^{3}+(x-2)^{3}$ Solution: $=(x+2+x-2)\left((x+2)^{2}-(x+2)(x-2)+(x-2)^{2}\right)$ $\therefore\left[a^{3}+b^{3}=(a+b)\left(a^{2}-a b+b^{2}\right)\right]$ $=2 x\left(x^{2}+4 x+4-(x+2)(x-2)+x^{2}-4 x+4\right)$ $=2 x\left(2 x^{2}+8-\left(x^{2}-2^{2}\right)\right)$ $\left[\therefore(a+b)(a-b)=a^{2}-b^{2}\right]$ $=2 x\left(2 x^{2}+8-x^{2}+4\right)$ $=2 x\left(x^{2}+12\right)$ $\therefore(x+2)^{3}+(x-2)^{3}=2 x\left(x^{2}+12\right)$...
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Question: Find the value $(a+b)^{3}-8(a-b)^{3}$ Solution: $=(a+b)^{3}-[2(a-b)]^{3}$ $=(a+b)^{3}-[2 a-2 b]^{3}$ $=(a+b-(2 a-2 b))\left((a+b)^{2}+(a+b)(2 a-2 b)+(2 a-2 b)^{2}\right)$ $\therefore\left[a^{3}-b^{3}=(a-b)\left(a^{2}+a b+b^{2}\right)\right]$ $=(a+b-2 a+2 b)\left(a^{2}+b^{2}+2 a b+(a+b)(2 a-2 b)+(2 a-2 b)^{2}\right)$ $=(a+b-2 a+2 b)\left(a^{2}+b^{2}+2 a b+2 a^{2}-2 a b+2 a b-2 b^{2}+(2 a-2 b)^{2}\right)$ $=(3 b-a)\left(3 a^{2}+2 a b-b^{2}+(2 a-2 b)^{2}\right)$ $=(3 b-a)\left(3 a^{2}+2 a...
Read More →Which of the following statements are correct?
Question: Which of the following statements are correct? Write a correct form of each of the incorrect statements. (i) $a \subset\{a, b, c\}$ (ii) $\{a\} \in\{a, b, c\}$ (iii) $a \in\{\{a\}, b\}$ (iv) $\{a\} \subset\{\{a\}, b\}$ (v) $\{b, c\} \subset\{a,\{b, c\}\}$ (vi) $\{a, b\} \subset\{a,\{b, c\}\}$ (vii) $\phi \in\{a, b\}$ (viii) $\phi \subset\{a, b, c\}$ (ix) $\{x: x+3=3\}=\phi$ Solution: Here, (viii) is correct.The correct forms of each of the incorrect statements are: (i) $a \in\{a, b, c\...
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Question: Find the value $(a-2 b)^{3}-512 b^{3}$ Solution: $=(a-2 b)^{3}-(8 b)^{3}$ $=(a-2 b-8 b)\left((a-2 b)^{2}+(a-2 b) 8 b+(8 b)^{2}\right)$ $\therefore\left[a^{3}-b^{3}=(a-b)\left(a^{2}+a b+b^{2}\right)\right]$ $=(a-10 b)\left(a^{2}+4 b^{2}-4 a b+8 a b-16 b^{2}+64 b^{2}\right)$ $=(a-10 b)\left(a^{2}+52 b^{2}+4 a b\right)$ $\therefore(a-2 b)^{3}-512 b^{3}=(a-10 b)\left(a^{2}+52 b^{2}+4 a b\right)$...
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Question: Find the value $32 a^{3}+108 b^{3}$ Solution: $=4\left(8 a^{3}+27 b^{3}\right)$ $=4\left((2 a)^{3}+(3 b)^{3}\right)$ $=4\left[(2 a+3 b)\left((2 a)^{2}-2 a \times 3 b+(3 b)^{2}\right.\right.$ $\therefore\left[a^{3}+b^{3}=(a+b)\left(a^{2}-a b+b^{2}\right)\right]$ $=4(2 a+3 b)\left(4 a^{2}-6 a b+9 b^{2}\right)$ $\therefore 32 a^{3}+108 b^{3}=4(2 a+3 b)\left(4 a^{2}-6 a b+9 b^{2}\right)$...
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Question: Find the value $54 x^{6} y+2 x^{3} y^{4}$ Solution: $=2 x^{3} y\left(27 x^{3}+y^{3}\right)$ $=2 x^{3} y\left((3 x)^{3}+y^{3}\right)$ $=2 x^{3} y(3 x+y)\left((3 x)^{2}-3 x x y+y^{2}\right)$ $\therefore\left[a^{3}+b^{3}=(a+b)\left(a^{2}-a b+b^{2}\right)\right]$ $=2 x^{3} y(3 x+y)\left(9 x^{2}-3 x y+y^{2}\right)$ $\therefore 54 x^{6} y+2 x^{3} y^{4}=2 x^{3} y(3 x+y)\left(9 x^{2}-3 x y+y^{2}\right)$...
Read More →Write which of the following statements are true?
Question: Write which of the following statements are true? Justify your answer. (i) The set of all integers is contained in the set of all set of all rational numbers. (ii) The set of all crows is contained in the set of all birds. (iii) The set of all rectangle is contained in the set of all squares. (iv) The set of all real numbers is contained in the set of all complex numbers. (v) The setsP= {a} andB= {{a}} are equal.\ (vi) The sets $A=\{x: x$ is a letter of the word "LITTLE" $\}$ and, $B=\...
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Question: Find the value $10 x^{4} y-10 x y^{4}$ Solution: $=10 x y\left(x^{3}-y^{3}\right)$ $=10 x y(x-y)\left(x^{2}+x y+y^{2}\right)$ $\therefore\left[x^{3}-y^{3}=(x-y)\left(x^{2}+x y+y^{2}\right)\right]$ $\therefore 10 x^{4} y-10 x y^{4}=10 x y(x-y)\left(x^{2}+x y+y^{2}\right)$...
Read More →Solve the following systems of equations graphically:
Question: Solve the following systems of equations graphically: $x-2 y=5$ $2 x+3 y=10$ Solution: The given equations are $x-2 y=5 \quad \ldots \ldots \ldots(i)$ $2 x+3 y=10 \quad \ldots \ldots \ldots($ ii $)$ Putting $x=0$ in equation $(i)$, we get: $\Rightarrow 0-2 y=5$ $\Rightarrow y=-5 / 2$ $x=0, y=-5 / 2$ Putting $y=0$ in equation $(i)$, we get: $\Rightarrow x+2 \times 0=5$ $\Rightarrow x=5$ $x=5, \quad y=0$ Use the following table to draw the graph. Draw the graph by plotting the two points...
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Question: Find the value $x^{3} / 216-8 y^{3}$ Solution: $=\frac{x^{3}}{6}-(2 y)^{3}$ $=(x / 6-2 y)\left((x / 6)^{2}+x / 6 \times 2 y+(2 y)^{2}\right)$ $\therefore\left[x^{3}-y^{3}=(x-y)\left(x^{2}+x y+y^{2}\right)\right]$ $=(x / 6-2 y)\left(x^{2} / 36+x y / 3+4 y^{2}\right)$ $\therefore x^{3} / 216-8 y^{3}=(x / 6-2 y)\left(x^{2} / 36+x y / 3+4 y^{2}\right)$...
Read More →Decide among the following sets, which are subsets of which:
Question: Decide among the following sets, which are subsets of which: $A=\left\{x: x\right.$ satisfies $\left.x^{2}-8 x+12=0\right\}$ $B=\{2,4,6\}, C=\{2,4,6,8, \ldots\}, D=\{6\}$ Solution: We have: $A=\left\{x: x\right.$ satisfies $\left.x^{2}-8 x+12=0 .\right\}=\{2,6\}$ B = {2, 4, 6} C= {2, 4, 6, 8,...} D= {6} Therefore, we can say that $D \subset \bar{A} \subset \bar{B} \subset \bar{C}$....
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Question: Find the value $64 a^{3}-b^{3}$ Solution: $=(4 a)^{3}-b^{3}$ $=(4 a-b)\left((4 a)^{2}+4 a \times b+b^{2}\right)$ $\therefore\left[a^{3}-b^{3}=(a-b)\left(a^{2}+a b+b^{2}\right)\right]$ $=(4 a-b)\left(16 a^{2}+4 a b+b^{2}\right)$ $\therefore 64 a^{3}-b^{3}=(4 a-b)\left(16 a^{2}+4 a b+b^{2}\right)$...
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Question: Find the value $8 x^{3} y^{3}+27 a^{3}$ Solution: $=(2 x y)^{3}+(3 a)^{3}$ $=(2 x y+3 a)\left((2 x y)^{2}-2 x y \times 3 a+(3 a)^{2}\right)$ $\therefore\left[a^{3}+b^{3}=(a+b)\left(a^{2}-a b+b^{2}\right)\right]$ $=(2 x y+3 a)\left(4 x^{2} y^{2}-6 x y a+9 a^{2}\right)$ $\therefore 8 x^{3} y^{3}+27 a^{3}=(2 x y+3 a)\left(4 x^{2} y^{2}-6 x y a+9 a^{2}\right)$...
Read More →State whether the following statements are true or false:
Question: State whether the following statements are true or false: (i) $1 \in\{1,2,3\}$ (ii) $a \subset\{b, c, a\}$ (iii) $\{a\} \in\{a, b, c\}$ (iv) $\{a, b\}=\{a, a, b, b, a\}$ (v) The set $\{x ; x+8=8\}$ is the null set. Solution: (i) True (ii) False It should be written as $\{a\} \subset\{b, c, a\}$ or $a \in\{b, c, a\}$. (iii) False It should be written as $\{a\} \subset\{b, c, a\}$ or $a \in\{b, c, a\}$. (iv) True (v) False The element of the set $\{x ; x+8=8\}$ is $\{0\}$. Therefore, it ...
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Question: Find the value $1-27 a^{3}$ Solution: $=(1)^{3}-(3 a)^{3}$ $=(1-3 a)\left(1^{2}+1 \times 3 a+(3 a)^{2}\right)$ $\therefore\left[a^{3}-b^{3}=(a-b)\left(a^{2}+a b+b^{2}\right)\right]$ $=(1-3 a)\left(1^{2}+3 a+9 a^{2}\right)$ $\therefore 1-27 a^{3}=(1-3 a)\left(1^{2}+3 a+9 a^{2}\right)$...
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Question: Find the value $y^{3}+125$ Solution: $=y^{3}+5^{3}$ $\therefore\left[a^{3}+b^{3}=(a+b)\left(a^{2}-a b+b^{2}\right)\right]$ $=(y+5)\left(y^{2}-5 y+5^{2}\right)$ $=(y+5)\left(y^{2}-5 y+25\right)$ $\therefore y^{3}+125=(y+5)\left(y^{2}-5 y+25\right)$...
Read More →Which of the following statements are true? Give reason to support your answer.
Question: Which of the following statements are true? Give reason to support your answer. (i) For any two sets $A$ and $B$ either $A \subseteq B$ or $B \subseteq A$; (ii) Every subset of an infinite set is infinite; (iii) Every subset of a finite set is finite; (iv) Every set has a proper subset; (v) {a,b,a,b,a,b, ...} is an infinite set; (vi) {a,b,c} and {1, 2, 3} are equivalent sets; (vii) A set can have infinitely many subsets. Solution: (i) False It is not necessary that for any two sets $A ...
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Question: Find the value $p^{3}+27$ Solution: $=p^{3}+3^{3}$ $\therefore\left[a^{3}+b^{3}=(a+b)\left(a^{2}-a b+b^{2}\right)\right]$ $=(p+3)\left(p^{2}-3 p-9\right)$ $\therefore p^{3}+27=(p+3)\left(p^{2}-3 p-9\right)$...
Read More →What are the possible expression for the cuboid having volume
Question: What are the possible expression for the cuboid having volume $3 x^{2}-12 x$. Solution: Volume $=3 x^{2}-12 x$ =3x(x 4) =3x(x 4) Also volume = Length Breadth Height Possible expression for dimensions of cuboid are = 3, x, (x 4)...
Read More →Solve the following systems of equations graphically:
Question: Solve the following systems of equations graphically:x+y= 32x+ 5y= 12 Solution: The given equations are: $x+y=3 \quad \ldots \ldots(i)$ $2 x+5 y=12 \quad \ldots \ldots(i i)$ Putting $x=0$ in equation $(i)$, we get: $\Rightarrow 0+y=3$ $\Rightarrow y=3$ $x=0, y=3$ Putting $y=0$ in equation $(i)$, we get: $\Rightarrow x+0=3$ $\Rightarrow x=3$ $x=3, \quad y=0$ Use the following table to draw the graph. Draw the graph by plotting the two points $A(0,3)$ and $B(3,0)$ from table. Graph of th...
Read More →Show that the set of letters needed to spell
Question: Show that the set of letters needed to spell "CATARACT" and the set of letters needed to spell "TRACT" are equal. Solution: Letters required to spell CATARACT are {C, A, T, R}. Let this set be denoted as E. E={C, A, T, R} Letters required to spell TRACT are {T, R, A, C}. Let this set be denoted as F. F={T, R, A, C} The two sets E F are equal because every element ofE is a member of F every element of F is a member ofE....
Read More →Give the possible expression for the length & breadth of the rectangle having
Question: Give the possible expression for the length \ breadth of the rectangle having $35 y^{2}-13 y-12$ as its area. Solution: Area is given as $35 y^{2}-13 y-12$ Splitting the middle term, Area $=35 y^{2}+218 y-15 y-12$ $=7 y(5 y+4)-3(5 y+4)$ $=(5 y+4)(7 y-3)$ We also know that area of rectangle= length breadth Possible length =(5y + 4)andbreadth = (7y 3) Or possible length = (7y 3) and breadth = (5y + 4)...
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