Which of the following sets are equal?
Question: Which of the following sets are equal? A= {x:xN,x, 3}, B= {1, 2} C= {3, 1} D= {x:xN,xis odd,x 5}, E= {1, 2, 1, 1}F= {1, 1, 3}. Solution: A= {1, 2} B= {1, 2} C= {3, 1} D= {1, 3} E= {1, 2, 1, 1} = {1, 2} F= {1, 1, 3} = {1, 3} A = B = EandC = D = F...
Read More →From the sets given below, select equal sets and equivalent sets.
Question: From the sets given below, select equal sets and equivalent sets. A= {0,a},B= {1, 2, 3, 4}C= {4, 8, 12},D= {3, 1, 2, 4}, E= {1, 0},F= {8, 4, 12}G= {1, 5, 7, 11},H= {a,b}. Solution: (a)Band D, because every element ofBis a member ofD every element ofDis a member ofB. (b) C andF,because every element ofCis a member ofF every element ofFis a member ofCEquivalent sets: (a)A,Eand H {∵∵n(A) = n(E) =n(H) = 2} (b) B,Dand G {∵∵n(B) = n(D) =n(G) = 4} (c) C andF {∵∵n(C) = n(F) = 3}...
Read More →Are the following pairs of sets equal? Give reasons.
Question: Are the following pairs of sets equal? Give reasons. (i) $A=\{2,3\}, B=\left\{x: x\right.$ is a solution of $\left.x^{2}+5 x+6=0\right\}$; (ii) $A=\{x: x$ is a letter of the word " WOLF" $\}$; B= {x:xis a letter of the word " FOLLOW"}. Solution: (i)A= {2, 3} B= {-2,-3} Ais not equal toBbecause every element ofAis not a member ofB every element ofBis not a member ofA.(ii)A= {W, O, L, F} B= {F, O, L, W} Here,A = Bbecause every element ofAis a member ofB every element ofBis a member ofA....
Read More →Find the value
Question: Find the value $2(x+y)^{2}-9(x+y)-5$ Solution: Let $x+y=z$ $=2 z^{2}-9 z-5$ Splitting the middle term, $=2 z^{2}-10 z+z-5$ $=2 z(z-5)+1(z-5)$ $=(z-5)(2 z+1)$ Substitutingz = x + y = (x + y 5)(2(x + y) + 1) = (x + y 5)(2x + 2y + 1) 2(x + y)2 9(x + y) 5 = (x + y 5)(2x + 2y + 1)...
Read More →From the sets given below, pair the equivalent sets:
Question: From the sets given below, pair the equivalent sets: $A=\{1,2,3\}, B=\{t, p, q, r, s\}, C=\{\alpha, \beta, \gamma\}, D=\{a, e, i, o, u\}$ Solution: Two setsABare equivalent if their cardinal numbers are equal, i.e.,n(A) =n(B). n(A) = 3 n(B) = 5 n(C) = 3 n(D) =5 Therefore, equivalent sets are(AandC) and (BandD)....
Read More →The cost of 2kg of apples and 1 kg of grapes on a day was found to be Rs 160.
Question: The cost of 2kg of apples and 1 kg of grapes on a day was found to be Rs 160. After a month, the cost of 4 kg of apples and 2kg of grapes is Rs. 300 Represent th situation algebraically and geometrically. Solution: Let the cost of 1 kg of apples be Rsx. And, cost of 1 kg of grapes = Rsy According to the question, the algebraic representation is $2 x+y=160$ $4 x+2 y=300$ For $2 x+y=160$, $y=160-2 x$ The solution table is For $4 x+2 y=300$ $y=\frac{300-4 x}{2}$ The solution table is The ...
Read More →Given the linear equation 2x + 3y − 8 = 0,
Question: Given the linear equation 2x+ 3y 8 = 0, write another linear equation in two variables such that the geometrical representation of the pair so formed is : (i) intersecting lines(ii) parallel lines(iii) coincident lines Solution: (i) Given the linear equation are: $2 x+3 y-8=0$ We know that intersecting condition: $\frac{a 1}{a 2} \neq \frac{b 1}{b 2}$ Where $a_{1}=2, b_{1}=3, c_{1}=-8$ Hence the equation of other line is $x+2 y-4=0$ (ii) We know that parallel line condition is: $\frac{...
Read More →Find the value
Question: Find the value $7(x-2 y)^{2}-25(x-2 y)+12$ Solution: Let $x-2 y=P$ $=7 P^{2}-25 P+12$ Splitting the middle term, $=7 P^{2}-21 P-4 P+12$ $=7 P(P-3)-4(P-3)$ $=(P-3)(7 P-4)$ Substituting P = x - 2y $=(x-2 y-3)(7(x-2 y)-4)$ $=(x-2 y-3)(7 x-14 y-4)$ $\therefore 7(x-2 y)^{2}-25(x-2 y)+12=(x-2 y-3)(7 x-14 y-4)$...
Read More →Are the following sets equal?
Question: Are the following sets equal? A= {x:xis a letter in the word reap}: B= {x:xis a letter in the word paper}; C= {x:xis a letter in the word rope}. Solution: A= {r, e, a, p} B= {p, a, e, r} C= {r, o, p, e} Here,A = Bbecause every element ofAis a member ofB every element ofBis a member ofA. But every element ofCis not a member ofAB. Also, every element ofAandBis not a member ofC. Therefore, we can say that these sets are not equal....
Read More →On comparing the ratios a1a2, b1b2 and c1c2, and without drawing them,
Question: On comparing the ratios $\frac{a_{1}}{a_{2}}, \frac{b_{1}}{b_{2}}$ and $\frac{c_{1}}{c_{2}}$, and without drawing them, find out whether the lines representing the following pairs of linear equations intersect at a point, are parallel or coincide : (i) $5 x-4 y+8=0$ $7 x+6 y-9=0$ (ii) $9 x+3 y+12=0$ $18 x+6 y+24=0$ (iii) $6 x-3 y+10=0$ $2 x-y+9=0$ Solution: (i) Given equation are: $5 x+4 y+8=0$ 7x+ 6y 9 = 0 Where, $a_{1}=5, b_{1}=-4, c_{1}=8$ $a_{2}=7, b_{2}=6, c_{3}=-9$ We have $\frac...
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Question: Find the value $9(2 a-b)^{2}-4(2 a-b)-13$ Solution: Let $2 a-b=x$ $=9 x^{2}-4 x-13$ Splitting the middle term, $=9 x^{2}-13 x+9 x-13$ $=x(9 x-13)+1(9 x-13)$ $=(9 x-13)(x+1)$ Substitutingx = 2a - b = [9(2a b) 13](2a b + 1) = (18a 9b 13)(2a b + 1) $\therefore 9(2 a-b)^{2}-4(2 a-b)-13=(18 a-9 b-13)(2 a-b+1)$...
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Question: Find the value $2 x^{2}+3 \sqrt{5} x+5$ Solution: Splitting the middle term, $=2 x^{2}+2 \sqrt{5} x+\sqrt{5} x+5$ $=2 x(x+\sqrt{5})+\sqrt{5}(x+\sqrt{5})$ $=(x+\sqrt{5})(2 x+\sqrt{5})$ $\therefore 2 x^{2}+3 \sqrt{5} x+5$ $=(x+\sqrt{5})(2 x+\sqrt{5})$...
Read More →Find the value
Question: Find the value $5 \sqrt{5} x^{2}+20 x+3 \sqrt{5}$ Solution: Splitting the middle term, $=5 \sqrt{5} x^{2}+15 x+5 x+3 \sqrt{5}$ $[\therefore 20=15+5$ and $15 \times 5=5 \sqrt{5} \times 3 \sqrt{5}]$ $=5 x(\sqrt{5} x+3)+\sqrt{5}(\sqrt{5} x+3)$ $=(\sqrt{5} x+3)(5 x+\sqrt{5})$ $\therefore 5 \sqrt{5} x^{2}+20 x+3 \sqrt{5}=(\sqrt{5} x+3)(5 x+\sqrt{5})$...
Read More →Gloria is walking along the path joining (−2, 3) and (2, −2),
Question: Gloria is walking along the path joining (2, 3) and (2, 2), while Suresh is walking along the path joining (0, 5) and (4, 0). Represent this situation graphically. Solution: Gloria is walking the path joining $(-2,3)$ and $(2,-2)$ Suresh is walking the path joining $(0,5)$ and $(4,0)$ The graphical representations are...
Read More →Find the value
Question: Find the value $21 x^{2}-2 x+1 / 21$ Solution: $=(\sqrt{21 x})^{2}-2 \sqrt{21} x \times \frac{1}{\sqrt{21}}+\left(\frac{1}{\sqrt{21}}\right)^{2}$ Using the identity $(x-y)^{2}=x^{2}+y^{2}-2 x y$ $=\left(\sqrt{21} x-\frac{1}{\sqrt{21}}\right)^{2}$ $\therefore 21 \mathrm{x}^{2}-2 \mathrm{x}+\frac{1}{21}=\left(\sqrt{21} \mathrm{x}-\frac{1}{\sqrt{21}}\right)^{2}$...
Read More →The path of a train A is given by the equation 3x + 4y − 12 = 0
Question: The path of a train A is given by the equation 3x+ 4y 12 = 0 and the path of another train B is given by the equation 6x+ 8y 48 = 0. Represent this situation graphically. Solution: The given equation are $3 x+4 y-12=0$ and $6 x+8 y-48=0$. In order to represent the above pair of linear equation graphically, we need Two points on the line representing each equation. That is, we find two solutions of each equation as given below: We have, $3 x+4 y-12=0$ Putting $y=0$, we get $3 x+0-12=0$ ...
Read More →Find the value
Question: Find the value $x^{2}+\frac{12}{35} x+\frac{1}{35}$ Solution: Splitting the middle term, $=x^{2}+\frac{5}{35} x+\frac{7}{35} x+\frac{1}{35}$ $\left[\therefore \frac{12}{35}=\frac{5}{35}+\frac{7}{35}\right.$ and $\left.\frac{5}{35} \times \frac{7}{35}=\frac{1}{35}\right]$ $=x^{2}+x / 7+x / 5+1 / 35$ $=x(x+1 / 7)+1 / 5(x+1 / 7)$ $=(x+1 / 7)(x+1 / 5)$ $\therefore \mathrm{x}^{2}+\frac{12}{35} \mathrm{x}+\frac{1}{35}=\left(\mathrm{x}+\frac{1}{7}\right)\left(\mathrm{x}+\frac{1}{5}\right)$...
Read More →Find the value
Question: Find the value $2 x^{2}-\frac{5}{6} x+\frac{1}{12}$ Solution: Splitting the middle term, $=2 x^{2}-x^{2}-x^{3}+1 / 12$ $[\therefore-5 / 6=-1 / 2-1 / 3$ also $-1 / 2 \times-1 / 3=2 \times 1 / 12]$ $=x(2 x-1 / 2)-1 / 6(2 x-1 / 2)$ $=(2 x-1 / 2)(x-1 / 6)$ $\therefore 2 x^{2}-56 x+1 / 12=(2 x-1 / 2)(x-1 / 6)$...
Read More →Aftab tells his daughter,
Question: Aftab tells his daughter, "Seven years ago, I was seven times as old as you were then. Also, three years from now, I shall be three times as old as you will be". Is not this interesting? Represent this situation algebraically and graphically. Solution: Let age of Aftab is $x$ years and age of his daughter is $y$ years. 7 Years ago his age was 7 times older as her daughter was. Then $\Rightarrow y-7=7(x-7)$ $\Rightarrow y-7=7 x-49$ $\Rightarrow y-7 x+42=0$ ...(1) Three years from now, h...
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Question: Find the value $x^{2}+2 \sqrt{3} x-24$ Solution: Splitting the middle term, $=x^{2}+4 \sqrt{3} x-2 \sqrt{3} x-24$ $[\therefore 2 \sqrt{3}=4 \sqrt{3}-2 \sqrt{3}$ also $4 \sqrt{3}(-2 \sqrt{3})=-24]$ $=x(x+4 \sqrt{3})-2 \sqrt{3}(x+4 \sqrt{3})$ $=(x+4 \sqrt{3})(x-2 \sqrt{3})$ $\therefore x^{2}+2 \sqrt{3} x-24=(x+4 \sqrt{3})(x-2 \sqrt{3})$...
Read More →Akhila went to a fair in her village.
Question: Akhila went to a fair in her village. She wanted to enjoy rides in the Giant Wheel and play Hoopla (a game in which you throw a rig on the items kept in the stall, and if the ring covers any object completely you get it.) The number of times she played Hoopla is half the number of rides she had on the Giant Wheel. Each ride costs Rs 3, and a game of Hoopla costs Rs 4. If she spent Rs 20 in the fair, represent this situation algebraically and graphically. Solution: Let no. of ride is $x...
Read More →Find the value
Question: Find the value $x^{2}+5 \sqrt{5} x+30$ Solution: Splitting the middle term, $=x^{2}+2 \sqrt{5} x+3 \sqrt{5} x+30$ $[\therefore 5 \sqrt{5}=2 \sqrt{5}+3 \sqrt{5}$ also $2 \sqrt{5} \times 3 \sqrt{5}=30]$ $=x(x+2 \sqrt{5})+3 \sqrt{5}(x+2 \sqrt{5})$ $=(x+2 \sqrt{5})(x+3 \sqrt{5})$ $\therefore x^{2}+5 \sqrt{5} x+30$ $=(x+2 \sqrt{5})(x+3 \sqrt{5})$...
Read More →Find the value
Question: Find the value $x^{2}-\sqrt{3} x-6$ Solution: Splitting the middle term, $=x^{2}-2 \sqrt{3} x+\sqrt{3} x-6$ $[\therefore-\sqrt{3}=-2 \sqrt{3}+\sqrt{3}$ also $-2 \sqrt{3} \times \sqrt{3}=-6]$ $=x(x-2 \sqrt{3})+\sqrt{3}(x-2 \sqrt{3})$ $=(x-2 \sqrt{3})(x+\sqrt{3})$ $\therefore x^{2}-\sqrt{3} x-6=(x-2 \sqrt{3})(x+\sqrt{3})$...
Read More →The area of the triangle formed by the lines
Question: The area of the triangle formed by the lines 2x+ 3y= 12,xy 1 = 0 andx= 0 (as shown in Fig. 3.23), is (a) 7 sq. units (b) $7.5 \mathrm{sq}$. units (c) $6.5 \mathrm{sq}$. units (d) 6 sq. units Solution: Given $2 x+3 y=12, x-y-1=0$ and $x=0$ If $x=0$ We have plotting points as $D(0,-1) B(0,4) P(3,2)$ Therefore, area of $c B P D=\frac{1}{2}($ Base $\times$ Height $)=\frac{1}{2}(B P \times P M)=\frac{1}{2}(5 \times 3)=\frac{1}{2}(15)=7.5$ Area of triangle $A B C$ is $7.5$ square units Hence...
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Question: Find the value $x^{2}-2 \sqrt{2} x-30$ Solution: Splitting the middle term, $=x^{2}-5 \sqrt{2} x+3 \sqrt{2} x-30$ $[\therefore-2 \sqrt{2}=-5 \sqrt{2}+3 \sqrt{2}$ also $-5 \sqrt{2} \times 3 \sqrt{2}=-30]$ $=x(x-5 \sqrt{2})+3 \sqrt{2}(x-5 \sqrt{2})$ $=(x-5 \sqrt{2})(x+3 \sqrt{2})$ $\therefore x^{2}-2 \sqrt{2} x-30$ $=(x-5 \sqrt{2})(x+3 \sqrt{2})$...
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