The difference of the squares of two positive integers is 180.
Question: The difference of the squares of two positive integers is 180. The square of the smaller number is 8 times the larger, find the numbers. Solution: Let the larger numberbex. Then according to the question, Square of the smaller number = 8x, then $x^{2}-8 x=180$ $\Rightarrow x^{2}-8 x-180=0$ $\Rightarrow x^{2}-18 x+10 x-180=0$ $\Rightarrow x(x-18)+10(x-18)=0$ $\Rightarrow(x+10)(x-18)=0$ $\Rightarrow x+10=0$ or $x-18=0$ $\Rightarrow x=-10$ or $x=18$ Since,xbeing a positive integer so,xcan...
Read More →Solve the following systems of linear inequations graphically:
Question: Solve the following systems of linear inequations graphically: (i) 2x+ 3y 6, 3x+ 2y 6,x 0,y 0 (ii) 2x+ 3y 6,x+ 4y 4,x 0,y 0 (iii)xy 1,x+ 2y 8, 2x+y 2,x 0,y 0 (iv)x+y 1, 7x+ 9y 63,x 6,y 5,x 0,y 0 (v) 2x+ 3y 35,y 3,x 2,x 0,y 0 Solution: (i) Converting the inequations to equations, we obtain: 2x+ 3y= 6, 3x+ 2y= 6,x= 0,y= 0 2x+ 3y=6: This line meets thex-axis at (3,0) and the y-axis at (0, 2). Draw a thick line joining these points. We see that the origin (0, 0) satisfies the inequation 2x...
Read More →A two digit number is 4 times the sum of its digits
Question: A two digit number is 4 times the sum of its digits and twice the product of its digits. Find the number. Solution: Let the require digit be $=(10 x+y)$ Then according to question $(10 x+y)=4(x+y)$ $(10 x+y)=4 x+4 y$ $10 x+y-4 x-4 y=0$ $6 x-3 y=0$ $2 x-y=0$ $2 x=y \ldots \ldots(1)$ And, $(10 x+y)=2 x y$......(2) Now putting the value ofyin equation (2) from (1) $(10 x+2 x)=2 x \times 2 x$ $4 x^{2}-12 x=0$ $4 x(x-3)=0$ $x(x-3)=0$ So, either $x=0$ Or $(x-3)=0$ $x=3$ So, the digit can nev...
Read More →A function f from the set of natural numbers to integers defined by
Question: A functionf from the set of natural numbers to integers defined by $f(n)=\left\{\begin{aligned} \frac{n-1}{2}, \text { when } n \text { is odd } \\-\frac{n}{2}, \text { when } n \text { is even } \end{aligned}\right.$ is (a) neither one-one nor onto(b) one-one but not onto(c) onto but not one-one(d) one-one and onto both Solution: (d) one-one and onto both Injectivity: Letxandybe any two elements in the domain (N). Case-1: Both $x$ and $y$ are even. Let $f(x)=f(y)$ $\Rightarrow \frac{-...
Read More →Factorise
Question: Factorise $8 a^{3}+27 b^{3}+36 a^{2} b+54 a b^{2}$ Solution: $8 a^{3}+27 b^{3}+36 a^{2} b+54 a b^{2}=(2 a)^{3}+(3 b)^{3}+3(2 a)^{2}(3 b)+3(2 a)(3 b)^{2}$ $=(2 a+3 b)^{3}$ Hence, factorisation of $8 a^{3}+27 b^{3}+36 a^{2} b+54 a b^{2}$ is $(2 a+3 b)^{3}$....
Read More →Two number differ by 4 and their product is 192.
Question: Two number differ by 4 and their product is 192. Find the numbers. Solution: Let two required numbers be $x$ and $(x+4)$ Then according to question $x(x+4)=192$ $x^{2}+4 x-192=0$ $x^{2}+16 x-12 x-192=0$ $x(x+16)-12(x+16)=0$ $(x+16)(x-12)=0$ $(x+16)=0$ $x=-16$ Or $(x-12)=0$ $x=12$ Since,xbeing a number, Therefore, When $x=-16$ then $x+4=-16+4$ $=-12$ And when $x=12$ then $x+4=12+4$ $=16$ Thus, two consecutive number be either 12,16 or $-16,-12$...
Read More →Expand
Question: Expand (i) $(5 a-3 b)^{3}$ (ii) $\left(3 x-\frac{5}{x}\right)^{3}$ (iii) $\left(\frac{4}{5} a-2\right)^{3}$ Solution: (i) $(5 a-3 b)^{3}=(5 a)^{3}-(3 b)^{3}-3(5 a)^{2}(3 b)+3(5 a)(3 b)^{2}$ $=125 a^{3}-27 b^{3}-225 a^{2} b+135 a b^{2}$ (ii) $\left(3 x-\frac{5}{x}\right)^{3}=(3 x)^{3}-\left(\frac{5}{x}\right)^{3}-3(3 x)^{2}\left(\frac{5}{x}\right)+3(3 x)\left(\frac{5}{x}\right)^{2}$ $=27 x^{3}-\frac{125}{x^{3}}-135 x+\frac{225}{x}$ (iii) $\left(\frac{4}{5} a-2\right)^{3}=\left(\frac{4}{...
Read More →The range of the function $f(x)={ }^{7-x} P_{x-3}$ is
Question: The range of the function $f(x)={ }^{7-x} P_{x-3}$ is (a) {1, 2, 3, 4, 5}(b) {1, 2, 3, 4, 5, 6}(c) {1, 2, 3, 4}(d) {1, 2, 3} Solution: We know that $7-x0 ; x-3 \geq 0$ and $7-x \geq x-3$ $\Rightarrow x7 ; x \geq 3$ and $2 x \leq 10$ $\Rightarrow x7 ; x \geq 3$ and $x \leq 5$ So, $x=\{3,4,5\}$ Range of $f$ $=\left\{{ }^{(7-3)} P_{(3-3)},{ }^{(7-4)} P_{(4-5)},{ }^{(7-5)} P(5-3)\right\}$ $=\left\{4 P_{0}, 3 P_{1}, 2 P_{2}\right\}$ $=\{1,3,2\}$ $=\{1,2,3\}$ So, the answer is (d)....
Read More →Expand
Question: Expand (i) $(3 x+2)^{3}$ (ii) $\left(3 a+\frac{1}{4 b}\right)^{3}$ (iii) $\left(1+\frac{2}{3} a\right)^{3}$ Solution: (i) $(3 x+2)^{3}=(3 x)^{3}+3 \times(3 x)^{2} x 2+3 \times 3 x \times(2)^{2}+(2)^{3}$ $=27 x^{3}+54 x^{2}+36 x+8$ (ii) $\left(3 a+\frac{1}{4 b}\right)^{3}=(3 a)^{3}+\left(\frac{1}{4 b}\right)^{3}+3(3 a)^{2}\left(\frac{1}{4 b}\right)+3(3 a)\left(\frac{1}{4 b}\right)^{2}$ $=27 a^{3}+\frac{1}{64 b^{3}}+\frac{27 a^{2}}{4 b}+\frac{9 a}{16 b^{2}}$ (iii) $\left(1+\frac{2}{3} a\...
Read More →Represent to solution set of each of the following inequations graphically in two dimensional plane:
Question: Represent to solution set of each of the following inequations graphically in two dimensional plane: 10. 3x 2yx+y 8 Solution: Converting the inequation to equation, we obtain $3 x-2 y-x-y+8=0$, i.e $2 x-3 y+8=0$ Putting $y=0$ and $x=0$ in this equation, we obtain $x=-4$ and $y=8 / 3$ respectively. So, this line meets the $x$-axis at $(-4,0)$ and $y$-axis at $(0,8 / 3)$. We plot these points and join them by a thick line. This divides thexyplane into two parts. To determine the region r...
Read More →Two number differ by 3 and their product is 504.
Question: Two number differ by 3 and their product is 504. Find the numbers. Solution: Let two required numbers be $x$ and $(x+3)$ Then according to question $x(x+3)=504$ $x^{2}+3 x-504=0$ $x^{2}+24 x-21 x-504=0$ $x(x+24)-21(x+24)=0$ $(x+24)(x-21)=0$ $(x+24)=0$ $x=-24$ Or $(x-21)=0$ $x=21$ Since,xbeing a number, Therefore, When $x=-24$ then $x+3=-24+3$ $=-21$ And when $x=21$ then $x+3=21+3$ $=24$ Thus, two consecutive number be either 21,24 or $-21,-24$...
Read More →Represent to solution set of each of the following inequations graphically in two dimensional plane:
Question: Represent to solution set of each of the following inequations graphically in two dimensional plane: 9.y 2x 8 Solution: Converting the inequation to equation, we obtain $2 x-y-8=0$ Putting $y=0$ and $x=0$ in this equation, we obtain $x=4$ and $y=-8$ respectively. So, this line meets the $x$-axis at $(4,0)$ and $y$-axis at $(0,-8)$. We plot these points and join them by a thick line.This divides thexyplane into two parts. To determine the region represented by the given inequality, cons...
Read More →Represent to solution set of each of the following inequations graphically in two dimensional plane:
Question: Represent to solution set of each of the following inequations graphically in two dimensional plane: 9.y 2x 8 Solution: Converting the inequation to equation, we obtain $2 x-y-8=0$ Putting $y=0$ and $x=0$ in this equation, we obtain $x=4$ and $y=-8$ respectively. So, this line meets the $x$-axis at $(4,0)$ and $y$-axis at $(0,-8)$. We plot these points and join them by a thick line.This divides thexyplane into two parts. To determine the region represented by the given inequality, cons...
Read More →A two-digit number is such that the product of the digits is 16.
Question: A two-digit number is such that the product of the digits is 16. When 54 is subtracted from the number, the digits are interchanged. Find the number. Solution: Let the tens digit be $x$ then the unit digits $=\frac{16}{x}$ Therefore, number $=\left(10 x+\frac{16}{x}\right)$ And number obtained by interchanging the digits $=\left(10 \times \frac{16}{x}+x\right)$ Then according to question $\left(10 x+\frac{16}{x}\right)-\left(10 \times \frac{16}{x}+x\right)=54$ $\left(10 x+\frac{16}{x}\...
Read More →Represent to solution set of each of the following inequations graphically in two dimensional plane:
Question: Represent to solution set of each of the following inequations graphically in two dimensional plane: 8. 3y 6 2x Solution: Converting the inequation to equation, we obtain $3 y+2 x-6=0$ Puttingy= 0 andx= 0 in this equation, we obtainx= 3 andy= 2 respectively. So. this line meets thex-axis at (3,0) andy-axis at (0,2). We plot these points and join them by a thick line. This divides thexyplane into two parts. To determine the region represented by the given inequality, consider pointO(0,0...
Read More →Evaluate
Question: Evaluate (i) $(99)^{2}$ (ii) $(995)^{2}$ (iii) $(107)^{2}$ Solution: (i) $(99)^{2}=(100-1)^{2}$ $=[(100)+(-1)]^{2}$ $=(100)^{2}+2 \times(100) \times(-1)+(-1)^{2}$ $=10000-200+1$ $=9801$ (ii) $(995)^{2}=(1000-5)^{2}$ $=[(1000)+(-5)]^{2}$ $=(1000)^{2}+2 \times(1000) \times(-5)+(-5)^{2}$ $=1000000-10000+25$ $=990025$ (iii) $(107)^{2}=(100+7)^{2}$ $=(100)^{2}+2 \times(100) \times(7)+(7)^{2}$ $=10000+1400+49$ $=11449$...
Read More →The function
Question: The function $f:[0, \infty) \rightarrow R$ given by $f(x)=\frac{x}{x+1}$ is (a) one-one and onto(b) one-one but not onto(c) onto but not one-one(d) onto but not one-one Solution: Injectivity:Letxandybe two elements in the domain, such that $f(x)=f(y)$ $\Rightarrow \frac{x}{x+1}=\frac{y}{y+1}$ $\Rightarrow x y+x=x y+y$ $\Rightarrow x=y$ So,fis one-one. Surjectivity:Letybe an element in the co domainR, such that $y=f(x)$ $\Rightarrow y=\frac{x}{x+1}$ $\Rightarrow x y+y=x$ $\Rightarrow x(...
Read More →Represent to solution set of each of the following inequations graphically in two dimensional plane:
Question: Represent to solution set of each of the following inequations graphically in two dimensional plane: 7. 0 2x 5y+ 10 Solution: Converting the inequation to equation, we obtain $2 x-5 y+10=0$ Putting $y=0$ and $x=0$ in this equation, we obtain $x=-5$ and $y=2$ respectively. So, this line meets the $x$-axis at $(-5,0)$ and the $y$-axis at $(0,2)$. We plot these points and join them by a thick line. This divides thexyplane into two parts. To determine the region represented by the given in...
Read More →Represent to solution set of each of the following inequations graphically in two dimensional plane:
Question: Represent to solution set of each of the following inequations graphically in two dimensional plane: 6.x 8 4y Solution: Converting the inequation to equation, we obtain $x+4 y-8=0$ Puttingy= 0 andx= 0 in this equation, we obtainx= 8 andy= 2 respectively. So, this line meets thex-axis at (8,0) andy-axis at (0,2). We plot these points and join them by a thick line. This divides thexyplane into two parts. To determine the region represented by the given inequality, consider pointO(0,0). C...
Read More →A two-digit number is such that the product of digit is 12.
Question: A two-digit number is such that the product of digit is 12. When 36 is added to the number the digits interchange their places. Determine the number. Solution: Let the tens digit be $x$ then, the unit digits $=\frac{12}{x}$ Therefore, number $=\left(10 x+\frac{12}{x}\right)$ And number obtained by interchanging the digits $=\left(10 \times \frac{12}{x}+x\right)$ Then according to question $\left(10 x+\frac{12}{x}\right)+36=\left(10 \times \frac{12}{x}+x\right)$ $\left(10 x+\frac{12}{x}...
Read More →Solve this
Question: $16 x^{2}+4 y^{2}+9 z^{2}-16 x y-12 y z+24 x z$ Solution: $16 x^{2}+4 y^{2}+9 z^{2}-16 x y-12 y z+24 x z$ $=(4 x)^{2}+(-2 y)^{2}+(3 z)^{2}+2(4 x)(-2 y)+2(-2 y)(3 z)+2(3 z)(4 x)$ $=(4 x-2 y+3 z)^{2} \quad\left[\right.$ using $\left.a^{2}+b^{2}+c^{2}+2 a b+2 b c+2 c a=(a+b+c)^{2}\right]$ Hence, $16 x^{2}+4 y^{2}+9 z^{2}-16 x y-12 y z+24 x z=(4 x-2 y+3 z)^{2}$...
Read More →Let M be the set of all 2 × 2 matrices with entries from the set R of real numbers.
Question: Let $M$ be the set of all $2 \times 2$ matrices with entries from the set $R$ of real numbers. Then, the function $f: M \rightarrow R$ defined by $f(A)=|A|$ for every $A \in M$, is (a) one-one and onto(b) neither one-one nor onto(c) one-one but-not onto(d) onto but not one-one Solution: $M=\left\{A=\left[\begin{array}{ll}a b \\ c d\end{array}\right]: a, b, c, d \in R\right\}$ $f: M \rightarrow R$ is given by $f(A)=|A|$ Injectivity: $f\left(\left[\begin{array}{ll}0 0 \\ 0 0\end{array}\r...
Read More →Represent to solution set of each of the following inequations graphically in two dimensional plane:
Question: Represent to solution set of each of the following inequations graphically in two dimensional plane: 5. 3x+ 2y 6 Solution: Converting the inequation to equation, we obtain $-3 x+2 y=6$, i.e $-3 x+2 y-6=0$ Puttingy= 0 andx= 0 in this equation, we obtainx=-2 andy= 3 respectively. So, this line meets thex-axis at (-2,0) and they-axis at (0,3). We plot these points and join them by a thick line.This divides the xy plane in two parts. To determine the region represented by the given inequal...
Read More →Factorize:
Question: Factorize: $25 x^{2}+4 y^{2}+9 z^{2}-20 x y-12 y z+30 x z$ Solution: We have : $25 x^{2}+4 y^{2}+9 z^{2}-20 x y-12 y z+30 x z$ $=(5 x)^{2}+(-2 y)^{2}+(3 z)^{2}+2(5 x)(-2 y)+2(-2 y)(3 z)+2(3 z)(5 x)$ $=[(5 x)+(-2 y)+(3 z)]^{2}$ $=(5 x-2 y+3 z)^{2}$...
Read More →Factorize:
Question: Factorize: $9 x^{2}+16 y^{2}+4 z^{2}-24 x y+16 y z-12 x z$ Solution: We have : $9 x^{2}+16 y^{2}+4 z^{2}-24 x y+16 y z-12 x z$ $=(-3 x)^{2}+(4 y)^{2}+(2 z)^{2}+2(-3 x)(4 y)+2(4 y)(2 z)+2(2 z)(-3 x)$ $=[(-3 x)+(4 y)+(2 z)]^{2}$ $=(-3 x+4 y+2 z)^{2}$...
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