A solid metallic sphere of diameter 8 cm is melted and drawn into a cylindrical wire of uniform width.
Question: A solid metallic sphere of diameter 8 cm is melted and drawn into a cylindrical wire of uniform width. If the length of the wire is 12 m, then find its width. Solution: We have, Radius of the metallic sphere, $R=\frac{8}{2}=4 \mathrm{~cm}$ and Height of the cylindrical wire, $h=12 \mathrm{~m}=1200 \mathrm{~cm}$ Let the radius of the base be $r$. Now, Volume of the cylindrical wire $=$ Volume of the metallic sphere $\Rightarrow \pi r^{2} h=\frac{4}{3} \pi R^{3}$ $\Rightarrow r^{2}=\frac...
Read More →Use Euclid’s division algorithm
Question: Use Euclids division algorithm to find the HCF of 441, 567 and 693. Solution: Let a = 693, b = 567 and c = 441 By Euclids division algorithms, $a=b q+r$ $\ldots(1)$ $[\because$ dividend $=$ divisor $\times$ quotient $+$ remainder $]$ First we take, $a=693$ and $b=567$ and find their HCF. $693=567 \times 1+126$ $567=126 \times 4+63$ $126=63 \times 2+0$ $\therefore \quad \operatorname{HCF}(693,567)=63$ Now, we take $c=441$ and say $d=63$, then find their HCF. Again, using Euclid's divisi...
Read More →If A is an invertible matrix of order 3
Question: If $A$ is an invertible matrix of order 3 and $|A|=3$, then $|a d j A|=$___________ Solution: Given: $A$ is an invertible matrix of order 3 $|A|=3$ As we know, $|\operatorname{adj} A|=|A|^{n-1}$, where $n$ is the order of $A$ $\Rightarrow|\operatorname{adj} A|=|A|^{3-1} \quad(\because$ Order of $A$ is 3$)$ $\Rightarrow|\operatorname{adj} A|=|A|^{2}$ $\Rightarrow|\operatorname{adj} A|=(3)^{2} \quad(\because|A|=3)$ $\Rightarrow|\operatorname{adj} A|=9$ Hence, $|\operatorname{adj} A|=\und...
Read More →Find the compound interest at the rate of 5%
Question: Find the compound interest at the rate of 5% for three years on that principal which in three years at the rate of 5% per annum gives Rs 12000 as simple interest. Solution: $\mathrm{P}=\frac{\mathrm{SI} \times 100}{\mathrm{RT}}$ According to the given values, we have: $=\frac{12,000 \times 100}{5 \times 3}$ $=80,000$ The principal is to be compounded annually. So, $\mathrm{A}=\mathrm{P}\left(1+\frac{\mathrm{R}}{100}\right)^{\mathrm{n}}$ $=80,000\left(1+\frac{5}{100}\right)^{3}$ $=80,00...
Read More →Three metallic cubes whose edges are 3 cm, 4 cm and 5 cm,
Question: Three metallic cubes whose edges are 3 cm, 4 cm and 5 cm, are melted and recast into a single large cube. Find the edge of the new cube formed. Solution: We have, Edges of the cubes: $a_{1}=3 \mathrm{~cm}, a_{2}=4 \mathrm{~cm}$ and $a_{3}=5 \mathrm{~cm}$ Let the edge of the new cube be $a$. Now, Volume of the new cube $=a_{1}^{3}+a_{2}^{3}+a_{3}{ }^{3}$ $\Rightarrow a^{3}=3^{3}+4^{3}+5^{3}$ $\Rightarrow a^{3}=27+64+125$ $\Rightarrow a^{3}=216$ $\Rightarrow a=\sqrt[3]{216}$ $\therefore ...
Read More →If A is a square matrix of order 2 such that A (adj A)
Question: If $A$ is a square matrix of order 2 such that $A(\operatorname{adj} A)=\left[\begin{array}{cc}10 0 \\ 0 10\end{array}\right]$, then $|A|=$__________ Solution: As we know that, $A(\operatorname{adj} A)=|A| I$. But it is given that $A(\operatorname{adj} A)=\left[\begin{array}{cc}10 0 \\ 0 10\end{array}\right]$ $\Rightarrow A(\operatorname{adj} A)=10\left[\begin{array}{ll}1 0 \\ 0 1\end{array}\right]$ $\Rightarrow A(\operatorname{adj} A)=10 I$ $\Rightarrow|A| I=10 I$ $\Rightarrow|A|=10$ ...
Read More →Prove that, if x and y are both odd positive integers,
Question: Prove that, if x and y are both odd positive integers, then x2+ yzis even but not divisible by 4. Solution: Let x = 2m + 1 and y = 2m + 3 are odd positive integers, for every positive integer m. Then, $\quad x^{2}+y^{2}=(2 m+1)^{2}+(2 m+3)^{2}$ $=4 m^{2}+1+4 m+4 m^{2}+9+12 m \quad\left[\because(a+b)^{2}=a^{2}+2 a b+b^{2}\right]$ $=8 m^{2}+16 m+10=$ even $=2\left(4 m^{2}+8 m+5\right)$ or $4\left(2 m^{2}+4 m+2\right)+1$ Hence, $x^{2}+^{\prime} y^{2}$ is even for every positive integer $m...
Read More →If A is a square matrix of order 2 such that A (adj A)
Question: If $A$ is a square matrix of order 2 such that $A(\operatorname{adj} A)=\left[\begin{array}{cc}10 0 \\ 0 10\end{array}\right]$, then $|A|=$_)___________ Solution: As we know that, $A(\operatorname{adj} A)=|A| I$. But it is given that $A(\operatorname{adj} A)=\left[\begin{array}{cc}10 0 \\ 0 10\end{array}\right]$ $\Rightarrow A(\operatorname{adj} A)=10\left[\begin{array}{ll}1 0 \\ 0 1\end{array}\right]$ $\Rightarrow A(\operatorname{adj} A)=10 I$ $\Rightarrow|A| I=10 I$ $\Rightarrow|A|=1...
Read More →What will Rs 125000 amount to at the rate of 6%,
Question: What will Rs 125000 amount to at the rate of 6%, if the interest is calculated after every 3 months? Solution: Because interest is calculated after every 3 months, it is compounded quarterly. Given : $\mathrm{P}=\mathrm{Rs} 125,000$ $\mathrm{R}=6 \%$ p. a. $=\frac{6}{4} \%$ quarterly $=1.5 \%$ quarterly $\mathrm{n}=4$ So, $\mathrm{A}=\mathrm{P}\left(1+\frac{\mathrm{R}}{100}\right)^{\mathrm{n}}$ $=125,000\left(1+\frac{1.5}{100}\right)^{4}$ $=125,000(1.015)^{4}$ $=132,670($ approx $)$ Th...
Read More →Let A be a square matrix of order 3 such that
Question: Let $A$ be a square matrix of order 3 such that $|A|=11$ and $B$ be the matrix of confactors of elements of $A$. Then, $|B|^{2}=$_________ Solution: Given: $A$ be a square matrix of order 3 $|A|=11$ $B$ be the matrix of cofactors of elements of $A$ Since, $B$ be the matrix of cofactors of elements of $A$ $\Rightarrow B=(\operatorname{adj} A)^{T}$ $\Rightarrow|B|=\left|(\operatorname{adj} A)^{T}\right|$ $\Rightarrow|B|=|\operatorname{adj} A|$ $\Rightarrow|B|^{2}=|\operatorname{adj} A|^{...
Read More →If n is an odd integer,
Question: If n is an odd integer, then show that n2 1 is divisible by 8. Solution: Let $\quad a=n^{2}-1 \quad \ldots$ (i) Given that, $n$ is an odd integer. $\therefore$ $n=1,3,5, \ldots$ From Eq. (i), at $n=1, a=(1)^{2}-1=1-1=0$, which is divisible by $8 .$ From Eq. (i), at $n=3, a=(3)^{2}-1=9-1=8$, which is divisible by 8 . From Eq. (i), at $n=5, a=(5)^{2}-1=25-1=24=3 \times 8$, which is divisible by 8 . From Eq. (i), at $n=7, a=(7)^{2}-1=49-1=48=6 \times 8$, which is divisible by 8 . Hence, $...
Read More →Let A be a square matrix of order 3 such that
Question: Let $A$ be a square matrix of order 3 such that $|A|=11$ and $B$ be the matrix of confactors of elements of $A$. Then, $|B|^{2}=$_________ Solution: Given: $A$ be a square matrix of order 3 $|A|=11$ $B$ be the matrix of cofactors of elements of $A$ Since, $B$ be the matrix of cofactors of elements of $A$ $\Rightarrow B=(\operatorname{adj} A)^{T}$ $\Rightarrow|B|=\left|(\operatorname{adj} A)^{T}\right|$ $\Rightarrow|B|=|\operatorname{adj} A|$ $\Rightarrow|B|^{2}=|\operatorname{adj} A|^{...
Read More →Two cubes each of volume 125 cm3 are joined end to end to form a solid.
Question: Two cubes each of volume 125 cm3are joined end to end to form a solid. Find the surface area of the resulting cuboid. Solution: Let the edge of the cube be $a$. As, Volume of the cube $=125 \mathrm{~cm}^{3}$ $\Rightarrow a^{3}=125$ $\Rightarrow a=\sqrt[3]{125}$ $\Rightarrow a=5 \mathrm{~cm}$ So, Length of the resulting cuboid, $l=2 \times 5=10 \mathrm{~cm}$, Breadth of the resulting cuboid, $b=5 \mathrm{~cm}$ and Height of the resulting cuboid, $h=5 \mathrm{~cm}$ Now, Surface area of t...
Read More →Ramu borrowed Rs 15625 from a finance company to buy a scooter.
Question: Ramu borrowed Rs 15625 from a finance company to buy a scooter. If the rate of interest be $16 \%$ per annum compounded annually, what payment will he have to make after $2 \frac{1}{4}$ years? Solution: Given: $\mathrm{P}=\mathrm{Rs} 15,625$ $\mathrm{R}=16 \%$ p. a. $\mathrm{n}=2 \frac{1}{4}$ years $\therefore$ Amount after $2 \frac{1}{4}$ years $=\mathrm{P}\left(1+\frac{\mathrm{R}}{100}\right)^{2}\left(1+\frac{\frac{1}{4}(\mathrm{R})}{100}\right)$ $=$ Rs $15,625\left(1+\frac{16}{100}\...
Read More →If A is a singular matrix,
Question: If $A$ is a singular matrix, then $A(\operatorname{adj} A)=$____________ Solution: As we know that, $A(\operatorname{adj} A)=|A| I$. But it is given that $A$ is a singular matrix Thus, $|A|=0$ Therefore, $A(\operatorname{adj} A)=0 I=O$, where $O$ is the zero matrix. Hence, if $A$ is a singular matrix, then $A(\operatorname{adj} A)=\underline{O}$....
Read More →Find the ratio of the volumes of a cylinder, a cone and a sphere,
Question: Find the ratio of the volumes of a cylinder, a cone and a sphere, if each has the same diameter and same height? Solution: Let the radius of the sphere be $r$. We have, The radius of the cone $=$ The radius of the cylinder $=$ The radius of the sphere $=r$ and The height of the cylinder = The height of the cone = The height of the sphere $=2 r$ Now, Volume of the cylinder $=\pi r^{2}(2 r)=2 \pi r^{3}$, Volume of the cone $=\frac{1}{3} \pi r^{2}(2 r)=\frac{2}{3} \pi r^{3}$ and Volume of...
Read More →Find the amount of Rs 12500 for 2 years compounded annually,
Question: Find the amount of Rs 12500 for 2 years compounded annually, the rate of interest being 15% for the first year and 16% for the second year. Solution: Given: $\mathrm{P}=\mathrm{Rs} 12,500$ $\mathrm{R}_{1}=15 \%$ p. $\mathrm{a} .$ $\mathrm{R}_{2}=16 \%$ p. $\mathrm{a} .$ $\therefore$ Amount after two years $=\mathrm{P}\left(1+\frac{\mathrm{R}_{1}}{100}\right)\left(1+\frac{\mathrm{R}_{2}}{100}\right)$ $=$ Rs $12,500\left(1+\frac{15}{100}\right)\left(1+\frac{16}{100}\right)$ $=$ Rs $12,50...
Read More →Show that the square of any odd integer
Question: Show that the square of any odd integer is of the form 4m + 1, for some integerm Solution: By Euclids division algorithm, we have a = bq + r, where 0 r 4 (i) On putting b = 4 in Eq. (i), we get $a=4 q+r_{1}$ where $0 \leq r4$ i.e., $r=0,1,2,3 \quad$...(ii) If $r=0 \Rightarrow a=4 q, 4 q$ is divisible by $2 \Rightarrow 4 q$ is even. If $r=1 \Rightarrow a=4 q+1,(4 q+1)$ is not divisible by 2 . If $r=2 \Rightarrow a=4 q+2,2(2 q+1)$ is divisible by $2 \Rightarrow 2(2 q+1)$ is even. If $r=3...
Read More →Solve this
Question: If the matrix $A=\left[\begin{array}{lll}1 a 2 \\ 1 2 5 \\ 2 1 1\end{array}\right]$ is not invertible, than $a=$ Solution: Given: $A=\left[\begin{array}{lll}1 a 2 \\ 1 2 5 \\ 2 1 1\end{array}\right]$ $A$ is not invertible if $|A|=0$. $\left|\begin{array}{lll}1 \alpha 2 \\ 1 2 5 \\ 2 1 1\end{array}\right|=0$ $\Rightarrow 1(2-5)-1(\alpha-2)+2(5 \alpha-4)=0$ $\Rightarrow 1(-3)-1 \alpha+2+10 \alpha-8=0$ $\Rightarrow-3-\alpha+2+10 \alpha-8=0$ $\Rightarrow 9 \alpha-9=0$ $\Rightarrow 9 \alpha...
Read More →Rekha deposited Rs 16000 in a foreign bank which pays interest at the rate of 20% per annum compounded quarterly,
Question: Rekha deposited Rs 16000 in a foreign bank which pays interest at the rate of 20% per annum compounded quarterly, find the interest received by Rekha after one year. Solution: Given: $\mathrm{P}=\mathrm{Rs} 16,000$ $\mathrm{R}=20 \%$ p. a. $\mathrm{n}=1$ year We know that: $\mathrm{A}=\mathrm{P}\left(1+\frac{\mathrm{R}}{100}\right)^{\mathrm{n}}$ When compounded quarterly, we have : $\mathrm{A}=\mathrm{P}\left(1+\frac{\mathrm{R}}{400}\right)^{4 \mathrm{n}}$ $=\operatorname{Rs} 16,000\le...
Read More →Find the ratio of the volume of a cube to that of a sphere which will fit inside it.
Question: Find the ratio of the volume of a cube to that of a sphere which will fit inside it. Solution: Let the radius of the shere be $R$ and the edge of the cube be $a$. As, the sphere is fit inside the cube. So, diameter of the sphere = edge of the cube $\Rightarrow 2 R=a \quad \ldots$ (i) Now, The ratio of the volume of the cube to that of the sphere $=\frac{\text { Volume of the cube }}{\text { Volume of the sphere }}$ $=\frac{a^{3}}{\left(\frac{4}{3} \pi R^{3}\right)}$ $=\frac{(2 R)^{3}}{...
Read More →If A and B are square matrices of the same order
Question: If $A$ and $B$ are square matrices of the same order and $A B=3 I$, then $A^{-1}=$____________ Solution: Given: $A$ and $B$ are square matrices of the same order $A B=3 I$ Pre-Multiplying both sides by $A^{-1}$, we get $\Rightarrow A^{-1}(A B)=A^{-1}(3 I)$ $\Rightarrow\left(A^{-1} A\right) B=3\left(A^{-1} /\right)$ $\Rightarrow(I) B=3 A^{-1}$ $\Rightarrow B=3 A^{-1}$ $\Rightarrow \frac{1}{3} B=A^{-1}$ Hence, if $A$ and $B$ are square matrices of the same order and $A B=3 I$, then $A^{-...
Read More →Show that the square of any positive
Question: Show that the square of any positive integer cannot be of the form 6m+ 2 or 6m + 5 for any integer m. Solution: Let a be an arbitrary positive integer, then by Euclids division algorithm, corresponding to the positive integers a and 6, there exist non-negative integers q and r such that a = 6q + r, where 0 r 6 $a=6 q+r$, where $0 \leq r6$ $a^{2}=(6 a+r)^{2}=36 q^{2}+r^{2}+12 q r \quad\left[\because(a+b)^{2}=a^{2}+2 a b+b^{2}\right]$ $\Rightarrow \quad a^{2}=6\left(6 q^{2}+2 q r\right)+...
Read More →Find the compound interest on Rs 15625 for 9 months,
Question: Find the compound interest on Rs 15625 for 9 months, at 16% per annum, compounded quarterly. Solution: Given: $\mathrm{P}=\mathrm{Rs} 15,625$ $\mathrm{R}=16 \%=\frac{16}{4}=4 \%$ quarterly $\mathrm{n}=9$ months $=3$ quarters We know that: $\mathrm{A}=\mathrm{P}\left(1+\frac{\mathrm{R}}{100}\right)^{\mathrm{n}}$ $=$ Rs $15,625\left(1+\frac{4}{100}\right)^{3}$ $=$ Rs $15,625(1.04)^{3}$ $=$ Rs 17,576 Also, $\mathrm{CI}=\mathrm{A}-\mathrm{P}$ $=$ Rs $17,576-$ Rs 15,625 $=$ Rs 1,951 Thus, t...
Read More →If A is a non-singular square matrix
Question: If $A$ is a non-singular square matrix such that $A^{3}=I$, then $A^{-1}=$___________ Solution: Given: $A^{3}=1$ $A^{3}=1$ Multiplying both sides by $A^{-1}$, we get $\Rightarrow A^{3} A^{-1}=I A^{-1}$ $\Rightarrow A^{2}\left(A A^{-1}\right)=I A^{-1}$ $\Rightarrow A^{2}(I)=A^{-1}$ $\Rightarrow A^{2}=A^{-1}$ Hence, if $A$ is a non-singular square matrix such that $A^{3}=l$, then $A^{-1}=\underline{A}^{2}$....
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