What sum will amount to Rs 4913 in 18 months,
Question: What sum will amount to Rs 4913 in 18 months, if the rate of interest is $12 \frac{1}{2} \%$ per annum, compounded half-yearly? Solution: Let the sum be Rs $\mathrm{x}$. Given: $\mathrm{A}=\mathrm{Rs} 4913$ $\mathrm{R}=12.5 \%$ $\mathrm{n}=18$ months $=1.5$ years We know that: $\mathrm{A}=\mathrm{P}\left(1+\frac{\mathrm{R}}{200}\right)^{2 \mathrm{n}}$ $4,913=\mathrm{P}\left(1+\frac{\mathrm{R}}{200}\right)^{2 \mathrm{n}}$ $4,913=\mathrm{x}\left(1+\frac{12.5}{200}\right)^{3}$ $4,913=\mat...
Read More →A metallic solid right circular cone is of height 84 cm and the radius of its base is 21 cm.
Question: A metallic solid right circular cone is of height 84 cm and the radius ofits base is 21 cm. It is melted and recast into a solid sphere. Find thediameter of the sphere. Solution: We have, Height of the cone, $h=84 \mathrm{~cm}$ and Base radius of the cone, $r=21 \mathrm{~cm}$ Let the radius of the solid sphere be $R$. Now, Volume of the solid sphere $=$ Volume of the solid cone $\Rightarrow \frac{4}{3} \pi R^{3}=\frac{1}{3} \pi r^{2} h$ $\Rightarrow R^{3}=\frac{r^{2} h}{4}$ $\Rightarro...
Read More →A sum amounts to Rs 756.25 at 10% per annum in 2 years,
Question: A sum amounts to Rs 756.25 at 10% per annum in 2 years, compounded annually. Find the sum. Solution: Let the sum be Rs $\mathrm{x}$. Then, $\mathrm{A}=\mathrm{P}\left(1+\frac{\mathrm{R}}{100}\right)^{\mathrm{n}}$ $=\mathrm{P}\left[\left(1+\frac{\mathrm{R}}{100}\right)^{\mathrm{n}}\right]$ $756.25=\mathrm{x}\left[\left(1+\frac{10}{100}\right)^{2}\right]$ $756.25=\mathrm{x}\left[(1.10)^{2}\right]$ $\mathrm{x}=\frac{756.25}{1.21}$ $=625$ Thus, the required sum is Rs 625 ....
Read More →If A is a square matrix such that
Question: If $A$ is a square matrix such that $A(\operatorname{adj} A)=10 I$, then $|A|=$_____________ Solution: Given: $A$ is a square matrix $A(\operatorname{adj} A)=10 I$ As we know, $A(\operatorname{adj} A)=|A| I$ $\Rightarrow 10 I=|A| I$ $\Rightarrow|A|=10$ Hence, $|A|=\underline{10}$....
Read More →The interior of a building is in the form of a right circular cylinder of diameter 4.2 m
Question: The interior of a building is in the form of a right circular cylinder ofdiameter 4.2 m and height 4 m surmounted by a cone of same diameter.The height of the cone is 2.8 m. Find the outer surface area of thebuilding. Solution: We have, Radius of the cylinder $=$ Radius of the cone $=r=\frac{4.2}{2}=2.1 \mathrm{~m}$, Height of the cylinder, $H=4 \mathrm{~m}$ and Height of the cone, $h=2.8 \mathrm{~m}$ Also, The slant height of the cone, $l=\sqrt{r^{2}+h^{2}}$ $=\sqrt{2.1^{2}+2.8^{2}}$ ...
Read More →Prove that √p + √q is irrational,
Question: Prove that p + q is irrational, where p and q are primes. Solution: Let us suppose that $\sqrt{p}+\sqrt{q}$ is rational. Again, let $\sqrt{p}+\sqrt{q}=a$, where $a$ is rational. Therefore, $\sqrt{q}=a-\sqrt{p}$ On squaring both sides, we get $q=a^{2}+p-2 a \sqrt{p}$ $\left[\because(a-b)^{2}=a^{2}+b^{2}-2 a b\right]$ Therefore, $\sqrt{p}=\frac{a^{2}+p-q}{2 a}$, which is a contradiction as the right hand side is rational number while $\sqrt{p}$ is irrational, since $\rho$ and $q$ are pri...
Read More →If A is a square matrix of order 3 such that
Question: If $A$ is a square matrix of order 3 such that $|A|=\frac{5}{2}$, then $\left|A^{-1}\right|=$___________ Solution: Given: $A$ is a square matrix of order 3 $|A|=\frac{5}{2}$ As we know, $\left|A^{-1}\right|=|A|^{-1}$ $\Rightarrow\left|A^{-1}\right|=\frac{1}{|A|}$ $\Rightarrow\left|A^{-1}\right|=\frac{1}{\frac{5}{2}}$ $\Rightarrow\left|A^{-1}\right|=\frac{2}{5}$ Hence, $\left|A^{-1}\right|=\underline{\frac{2}{5}}$....
Read More →Find the principal if the interest compounded annually at the rate of 10%
Question: Find the principal if the interest compounded annually at the rate of 10% for two years is Rs 210. Solution: Let the sum be Rs $\mathrm{x}$. We know that: $\mathrm{CI}=\mathrm{A}-\mathrm{P}$ $=\mathrm{P}\left(1+\frac{\mathrm{R}}{100}\right)^{\mathrm{n}}-\mathrm{P}$ $=\mathrm{P}\left[\left(1+\frac{\mathrm{R}}{100}\right)^{\mathrm{n}}-1\right]$ $210=\mathrm{x}\left[\left(1+\frac{10}{100}\right)^{2}-1\right]$ $210=\mathrm{x}\left[(1.10)^{2}-1\right]$ $\mathrm{x}=\frac{210}{0.21}$ $=1,000$...
Read More →Write the denominator of rational number
Question: Write the denominator of rational number $\frac{257}{5000}$ in the form $2^{m} \times 5^{n}$, where $m, n$ are non-negative integers. Hence, write its decimal expansion, without actual division Solution: Denominator of the rational number $\frac{257}{5000}$ is 5000 . Now, factors of $5000=2 \times 2 \times 2 \times 5 \times 5 \times 5 \times 5=(2)^{3} \times(5)^{4}$, which is of the type $2^{m} \times 5^{n}$, where $m=3$ and $n=4$ are non-negative integers. $\therefore \quad$ Rational ...
Read More →A bucket of height 24 cm is in the form of frustum of a cone whose circular ends are of diameter 28 cm and 42 cm.
Question: A bucket of height 24 cm is in the form of frustum of a cone whosecircular ends are of diameter 28 cm and 42 cm. Find the cost of milk atthe rate of₹30 per litre, which the bucket can hold. Solution: We have, Height of the frustum, $h=24 \mathrm{~cm}$, Radius of the open end, $R=\frac{42}{2}=21 \mathrm{~cm}$ and Radius of the close end, $r=\frac{28}{2}=14 \mathrm{~cm}$ Now, Volume of the bucket $=\frac{1}{3} \pi h\left(R^{2}+r^{2}+R r\right)$ $=\frac{1}{3} \times \frac{22}{7} \times 24...
Read More →On what sum will the compound interest at 5% per annum for 2 years
Question: On what sum will the compound interest at 5% per annum for 2 years compounded annually be Rs 164? Solution: Let the sum be Rs $\mathrm{x}$. We know that: $\mathrm{CI}=\mathrm{A}-\mathrm{P}$ $=\mathrm{P}\left(1+\frac{\mathrm{R}}{100}\right)^{\mathrm{n}}-\mathrm{P}$ $=\mathrm{P}\left[\left(1+\frac{\mathrm{R}}{100}\right)^{\mathrm{n}}-1\right]$ $164=\mathrm{x}\left[\left(1+\frac{5}{100}\right)^{2}-1\right]$ $164=\mathrm{x}\left[(1.05)^{2}-1\right]$ $\mathrm{x}=\frac{164}{0.1025}$ $=1,600$...
Read More →On a morning walk, three persons step off together
Question: On a morning walk, three persons step off together and their steps measure 40 cm, 42 cm and 45 cm, respectively.What is the minimum distance each should walk, so that each can cover the same distance in complete steps? Solution: We have to find the LCM of 40 cm, 42 cm and 45 cm to get the required minimum distance. For this, $40=2 \times 2 \times 2 \times 5$, $42=2 \times 3 \times 7$ and$45=3 \times 3 \times 5$ $\therefore$ $\operatorname{LCM}(40,42,45)=2 \times 3 \times 5 \times 2 \ti...
Read More →A hollow sphere of external and internal diameters 8 cm and 4 cm, respectively is melted into a solid cone of base diameter 8 cm.
Question: A hollow sphere of external and internal diameters 8 cm and 4 cm, respectively is melted into a solid cone of base diameter 8 cm. Find the height of the cone. Solution: We have, External radius of the hollow sphere, $R_{1}=\frac{8}{2}=4 \mathrm{~cm}$, Internal radius of the hollow sphere, $R_{2}=\frac{4}{2}=2 \mathrm{~cm}$ and Base radius of the cone, $r=\frac{8}{2}=4 \mathrm{~cm}$ Let the height of the cone be $h$. Now, Volume of the cone $=$ Volume of the hollow sphere $\Rightarrow \...
Read More →Solve this
Question: If $A=\operatorname{diag}(1,2,3)$, then $|\operatorname{adj}(\operatorname{adj} A)|=$ Solution: Given: $A=\operatorname{diag}(1,2,3)$ $\Rightarrow|A|=1 \times 2 \times 3=6$ As we know, $|\operatorname{adj}(\operatorname{adj} A)|=|A|^{(n-1)^{2}}$, where $n$ is the order of $A$ $\Rightarrow|\operatorname{adj}(\operatorname{adj} A)|=|A|^{(3-1)^{2}} \quad(\because$ Order of $A$ is 3$)$ $\Rightarrow|\operatorname{adj}(\operatorname{adj} A)|=|A|^{(2)^{2}}$ $\Rightarrow|\operatorname{adj}(\op...
Read More →Find the compound interest at the rate of 5% per annum for 3 years
Question: Find the compound interest at the rate of 5% per annum for 3 years on that principal which in 3 years at the rate of 5% per annum gives Rs 1200 as simple interest. Solution: We know that: $P=\frac{S I \times 100}{R T}$ $\therefore \mathrm{P}=\frac{1200 \times 100}{5 \times 3}$ = 8,000 Now, $\mathrm{A}=\mathrm{P}\left(1+\frac{\mathrm{R}}{100}\right)^{\mathrm{I}}$ $=8,000\left(1+\frac{5}{100}\right)^{3}$ $=8,000(1.05)^{3}$ $=9,261$ Now, $\mathrm{CI}=\mathrm{A}-\mathrm{P}$ $=9,261-8,000$ ...
Read More →Prove the following
Question: Show that $12^{n}$ cannot end with the digit 0 or 5 for any natural number $n$. Solution: If any number ends with the digit 0 or 5 , it is always divisible by 5 . If $12^{n}$ ends with the digit zero it must be divisible by $5 .$ This is possible only if prime factorisation of $12^{n}$ contains the prime number $5 .$ Now, $12=2 \times 2 \times 3=2^{2} \times 3$ $\Rightarrow$ $12^{n}=\left(2^{2} \times 3\right)^{n}=2^{2 n} \times 3^{n}[$ since, there is no term contains 5$]$ Hence, ther...
Read More →Three cubes of a metal whose edges are in the ratio 3 : 4 : 5
Question: Three cubes of a metal whose edges are in the ratio $3: 4: 5$ are melted and converted into a single cube whose diagonal is $12 \sqrt{3} \mathrm{~cm}$.Find theedges of the three cubes. Solution: Let the edge of the metal cubes be $3 x, 4 x$ and $5 x$. Let the edge of the single cube be $a$. As, Diagonal of the single cube $=12 \sqrt{3} \mathrm{~cm}$ $\Rightarrow a \sqrt{3}=12 \sqrt{3}$ $\Rightarrow a=12 \mathrm{~cm}$ Now, Volume of the single cube $=$ Sum of the volumes of the metallic...
Read More →If A is an invertible matrix of order 3 and
Question: If $A$ is an invertible matrix of order 3 and $|A|=4$, then $|\operatorname{adj}(\operatorname{adj} A)|=$____________ Solution: Given: $A$ is an invertible matrix of order 3 $|A|=4$ As we know, $|\operatorname{adj}(\operatorname{adj} A)|=|A|^{(n-1)^{2}}$, where $n$ is the order of $A$ $\Rightarrow|\operatorname{adj}(\operatorname{adj} A)|=|A|^{(3-1)^{2}} \quad(\because$ Order of $A$ is 3$)$ $\Rightarrow|\operatorname{adj}(\operatorname{adj} A)|=|A|^{(2)^{2}}$ $\Rightarrow|\operatorname...
Read More →Simple interest on a sum of money for 2 years at
Question: Simple interest on a sum of money for 2 years at $6 \frac{1}{2} \%$ per annum is Rs 5200 . What will be the compound interest on the sum at the same rate for the same period? Solution: P=\frac{S I \times 100}{R T} $\therefore P=\frac{5,200 \times 100}{6.5 \times 2}$ = 40,000 Now, $\mathrm{A}=\mathrm{P}\left(1+\frac{\mathrm{R}}{100}\right)^{\mathrm{n}}$ $=40,000\left(1+\frac{6.5}{100}\right)^{2}$ $=40,000(1.065)^{2}$ $=45,369$ Also, $\mathrm{CI}=\mathrm{A}-\mathrm{P}$ $=45,369-40,000$ $...
Read More →A wooden toy was made by scooping out a hemisphere of same radius from each end of a solid cylinder.
Question: A wooden toy was made by scooping out a hemisphere of same radiusfrom each end of a solid cylinder. If the height of the cylinder is 10 cmand its base is of radius 3.5 cm, then find the volume of wood in the toy. Solution: We have, Radius of the cylinder $=$ Radius of the hemispher $=r=3.5 \mathrm{~cm}$ and Height of the cylinder, $h=10 \mathrm{~cm}$ Now, Volume of the toy $=$ Volume of the cylinder $-$ Volume of the two hemispheres $=\pi r^{2} h-2 \times \frac{2}{3} \pi r^{3}$ $=\pi r...
Read More →Prove that√3 + √5 is irrational.
Question: Prove that3 + 5 is irrational. Solution: Let us suppose that 3 + 5 is rational. Let 3 + 5 = a, where a is rational. Therefore,$\sqrt{3}=a-\sqrt{5}$ On squaring both sides, we get $(\sqrt{3})^{2}=(a-\sqrt{5})^{2}$ $\Rightarrow$ $3=a^{2}+5-2 a \sqrt{5}$ $\left[\because(a-b)^{2}=a^{2}+b^{2}-2 a b\right]$ $\Rightarrow$ $2 a \sqrt{5}=a^{2}+2$ Therefore, $\sqrt{5}=\frac{a^{2}+2}{2 a}$ which is contradiction. As the right hand side is rational number while V5 is irrational. Since, 3 and 5 are...
Read More →If A is an invertible matrix of order 3 and |A|
Question: If $A$ is an invertible matrix of order 3 and $|A|=5$, then $\operatorname{adj}(\operatorname{adj} A)=$___________ Solution: Given: $A$ is an invertible matrix of order 3 $|A|=5$ As we know, $\operatorname{adj}(\operatorname{adj} A)=|A|^{n-2} A$, where $n$ is the order of $A$ $\Rightarrow \operatorname{adj}(\operatorname{adj} A)=|A|^{3-2} A \quad(\because$ Order of $A$ is 3$)$ $\Rightarrow \operatorname{adj}(\operatorname{adj} A)=|A|^{1} A$ $\Rightarrow \operatorname{adj}(\operatorname...
Read More →A sum of money was lent for 2 years at 20% compounded annually.
Question: A sum of money was lent for 2 years at 20% compounded annually. If the interest is payable half-yearly instead of yearly, then the interest is Rs 482 more. Find the sum. Solution: $\mathrm{A}=\mathrm{P}\left(1+\frac{\mathrm{R}}{100}\right)^{\mathrm{n}}$ Also, $\mathrm{P}=\mathrm{A}-\mathrm{CI}$ Let the sum of money be Rs $\mathrm{x}$. If the interest is compounded annually, then: $\mathrm{A}_{1}=\mathrm{x}\left(1+\frac{20}{100}\right)^{2}$ $=1.44 \mathrm{x}$ $\therefore \mathrm{CI}=1.4...
Read More →A 5-m-wide cloth is used to make a conical tent of base diameter 14 m and height 24 m.
Question: A 5-m-wide cloth is used to make a conical tent of base diameter 14 mand height 24 m. Find the cost of cloth used, at the rate of₹25 per metre. Solution: We have, Width of the cloth, $B=5 \mathrm{~m}$, Radius of the conical tent, $r=\frac{14}{2}=7 \mathrm{~m}$ and Height of the conical tent, $h=24 \mathrm{~m}$ Let the length of the cloth used for making the tent be $L$. Also, The slant height of the conical tent, $l=\sqrt{r^{2}+h^{2}}$ $=\sqrt{7^{2}+24^{2}}$ $=\sqrt{49+576}$ $=\sqrt{62...
Read More →Using Euclid’s division algorithm,
Question: Using Euclids division algorithm, find the largest number that divides 1251, 9377 and 15628 leaving remainders 1, 2 and 3, respectively. Solution: Since, 1, 2 and 3 are the remainders of 1251, 9377 and 15628, respectively. Thus, after subtracting these remainders from the numbers. We have the numbers, 1251-1 = 1250,9377-2 = 9375 and 15628-3 = 15625 which is divisible by the required number. Now, required number = HCF of 1250,9375 and 15625 [for the largest number] By Euclids division a...
Read More →