Find the amount of Rs 4096 for 18 months at
Question: Find the amount of Rs 4096 for 18 months at $12 \frac{1}{2} \%$ per annum, the interest being compounded semi-annually. Solution: Given: $\mathrm{P}=\mathrm{Rs} 4,096$ $\mathrm{R}=12.5 \%$ p. a. $\mathrm{n}=18$ months $=1.5$ years We have : $\mathrm{A}=\mathrm{P}\left(1+\frac{\mathrm{R}}{100}\right)^{\mathrm{n}}$$=\operatorname{Rs} 4,096\left(1+\frac{12.5}{200}\right)^{3}$ $=\operatorname{Rs} 4,096(1.0625)^{3}$ $=\operatorname{Rs} 4,913$ Thus, the required amount is Rs 4,913 ....
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Question: If $A=\left[\begin{array}{ll}2 -1 \\ 3 -2\end{array}\right]$, then $A^{n}=$ (a) $A=\left[\begin{array}{ll}1 0 \\ 0 1\end{array}\right]$, if $n$ is an even natural number (b) $A=\left[\begin{array}{ll}1 0 \\ 0 1\end{array}\right]$, if $n$ is an odd natural number (c) $A=\left[\begin{array}{cc}-1 0 \\ 0 1\end{array}\right]$, if $n \in N$ (d) none of these Solution: Disclaimer: In all option, the power of $A$ (i.e. $n$ is missing) (a) $A^{n}=\left[\begin{array}{ll}1 0 \\ 0 1\end{array}\ri...
Read More →The curved surface area of a sphere is 5544 cm2.
Question: The curved surface area of a sphere is 5544 cm2. Find its volume. Solution: Let the radius of the sphere be $r$. As, Curved surface area of the sphere $=5544 \mathrm{~cm}^{2}$ $\Rightarrow 4 \pi r^{2}=5544$ $\Rightarrow 4 \times \frac{22}{7} \times r^{2}=5544$ $\Rightarrow r^{2}=5544 \times \frac{7}{4 \times 22}$ $\Rightarrow r^{2}=441$ $\Rightarrow r=\sqrt{441}$ $\Rightarrow r=21 \mathrm{~cm}$ Now, Volume of the sphere $=\frac{4}{3} \pi r^{3}$ $=\frac{4}{3} \times \frac{22}{7} \times ...
Read More →Can two numbers have 18 as their HCF
Question: Can two numbers have 18 as their HCF and 380 as their LCM? Give reasons. Solution: No, because HCF is always a factor of LCM but here 18 is not a factor of 380....
Read More →Amit borrowed Rs 16000 at
Question: Amit borrowed Rs 16000 at $17 \frac{1}{2} \%$ per annum simple interest. On the same day, he lent it to Ashu at the same rate but compounded annually. What does he gain at the end of 2 years? Solution: Amount to be paid by Amit: $\mathrm{SI}=\frac{\mathrm{PRT}}{100}$ $=\frac{16000 \times 17.5 \times 2}{100}$ $=\operatorname{Rs} 5,600$ Amount gained by Amit: $\mathrm{A}=\mathrm{P}\left(1+\frac{\mathrm{R}}{100}\right)^{\mathrm{n}}$ $=\operatorname{Rs} 16,000\left(1+\frac{17.5}{100}\right...
Read More →Explain why3 x 5 x 7 + 7 is a
Question: Explain why3 x 5 x 7 + 7 is a composite number, Solution: We have, $\quad 3 \times 5 \times 7+7=105+7=112$ Now, $112=2 \times 2 \times 2 \times 2 \times 7=2^{4} \times 7$ So, it is the product of prime factors 2 and 7. i.e., it has more than two factors. Hence, it is a composite number....
Read More →The volume of a sphere is 4851 cm3.
Question: The volume of a sphere is 4851 cm3. Find its curved surface area. Solution: Let the radius of the sphere be $r$. As, Volume of the sphere $=4851 \mathrm{~cm}^{3}$ $\Rightarrow \frac{4}{3} \pi r^{3}=4851$ $\Rightarrow \frac{4}{3} \times \frac{22}{7} \times r^{3}=4851$ $\Rightarrow r^{3}=4851 \times \frac{3 \times 7}{4 \times 22}$ $\Rightarrow r^{3}=\frac{9261}{8}$ $\Rightarrow r=\sqrt[3]{\frac{9261}{8}}$ $\Rightarrow r=\frac{21}{2} \mathrm{~cm}$ Now, Curved surface area of the sphere $=...
Read More →If A is an invertible matrix,
Question: If $A$ is an invertible matrix, then $\operatorname{det}\left(A^{-1}\right)$ is equal to (a) det $(A)$ (b) $\frac{1}{\operatorname{det}(A)}$ (c) 1 (d) none of these Solution: (b) $\frac{1}{\operatorname{det}(A)}$ We know that for any invertible matrix $A,\left|A^{-1}\right|=\frac{1}{|A|}$....
Read More →The numbers 525 and 3000 are both divisible only by 3,
Question: The numbers 525 and 3000 are both divisible only by 3, 5, 15, 25 and 75. What is HCF (525, 3000)? Justify your answer. Solution: Since, the HCF (525, 3000) = 75 By Euclid's Lemma, $3000=525 \times 5+375[\because$ dividend $=$ divisor $\times$ quotient $+$ remainder $]$ $525=375 \times 1+150$ $375=150 \times 2+75$ $150=75 \times 2+0$ and the numbers 3, 5, 15, 25 and 75 divides the numbers 525 and 3000 that mean these terms are common in both 525 and 3000. So, the highest common factor a...
Read More →A right cylindrical vessel is full of water.
Question: A right cylindrical vessel is full of water. How many right cones having the same radius and height as those of the right cylinder will be needed to store that water? Solution: Let the radius and height of the cone be $r$ and $h$, respectively. Then, Radius of the cylindrical vessel $=r$ and Height of the cylindrical vessel $=h$ Now, The number of cones $=\frac{\text { Volume of the cylindrical vessel }}{\text { Volume of a cone }}$ $=\frac{\pi r^{2} h}{\left(\frac{1}{3} \pi r^{2} h\ri...
Read More →If a matrix A is such that
Question: If a matrix $A$ is such that $3 A^{3}+2 A^{2}+5 A+I=0$, then $A^{-1}$ equal to (a) $-\left(3 A^{2}+2 A+5\right)$ (b) $3 A^{2}+2 A+5$ (c) $3 A^{2}-2 A-5$ (d) none of these Solution: (d) none of these $3 A^{3}+2 A^{2}+5 A+I=0$ $\Rightarrow 3 A^{3}+2 A^{2}+5 A=-I$ $\Rightarrow A^{-1}\left(3 A^{3}+2 A^{2}+5 A\right)=-I A^{-1}$ $\Rightarrow 3 A^{2}+2 A+5 I=-A^{-1}$ $\Rightarrow A^{-1}=-3 A^{2}-2 A-5 I$...
Read More →A positive integer is of the form 3q +1,
Question: A positive integer is of the form 3q +1, q being a natural number. Can you write its square in any form other than 3m+1, i.e., 3m or 3m + 2 for some integer ml Justify your answer. Solution: No, by Euclid's Lemma, $b=a q+r, 0 \leq ra$ Here, $b$ is any positive integer $a=3, b=3 q+r$ for $0 \leq r3$ So, this must be in the form $3 q, 3 q+1$ or $3 q+2$. $\begin{array}{lll}\text { Now, } (3 q)^{2}=9 q^{2}=3 m {\left[\text { here, } m=3 q^{2}\right]}\end{array}$ and $\quad(3 q+1)^{2}=9 q^{...
Read More →Find the difference between the compound interest and simple interest.
Question: Find the difference between the compound interest and simple interest. On a sum of Rs 50,000 at 10% per annum for 2 years. Solution: Given: $\mathrm{P}=\mathrm{Rs} 50,000$ $\mathrm{R}=10 \%$ p. $\mathrm{a} .$ $\mathrm{n}=2$ years We know that amount $\mathrm{A}$ at the end of $\mathrm{n}$ years at the rate $\mathrm{R} \%$ per annum when the interest is compounded annually is given by $\mathrm{A}=\mathrm{P}\left(1+\frac{\mathrm{R}}{100}\right)$. $\therefore \mathrm{A}=\mathrm{Rs} 50,000...
Read More →Write whether the square of any positive
Question: Write whether the square of any positive integer can be of the form 3m + 2, where m is a natural number. Justify your answer. Solution: No, by Euclid's lemma, $b=a q+r, 0 \leq r \leq a$ Here, $b$ is any positive integer, $a=3, b=3 q+r$ for $0 \leq r \leq 2$ So, any positive integer is of the form $3 k, 3 k+1$ or $3 k+2$. $\begin{array}{lll}\text { Now, } (3 k)^{2}=9 k^{2}=3 m \text { [where, } m=3 k^{2} \text { ] }\end{array}$ Now, $(3 k)^{2}=9 k^{2}=3 m$ [where, $m=3 k^{2}$ ] and $\qu...
Read More →A cylinder with base radius 8 cm and height 2 cm is melted to form a cone of height 6 cm.
Question: A cylinder with base radius 8 cm and height 2 cm is melted to form a cone of height 6 cm. Calculate the radius of the base of the cone. Solution: We have, Base radius of the cylinder, $r=8 \mathrm{~cm}$, Height of the cylinder, $h=2 \mathrm{~cm}$ and Height of the cone, $H=6 \mathrm{~cm}$ Let the base radius of the cone be $R$. Now, Volume of the cone = Volume of the cylinder $\Rightarrow \frac{1}{3} \pi R^{2} H=\pi r^{2} h$ $\Rightarrow R^{2}=\frac{3 r^{2} h}{H}$ $\Rightarrow R^{2}=\f...
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Question: If $\left[\begin{array}{cc}1 -\tan \theta \\ \tan \theta 1\end{array}\right]\left[\begin{array}{cc}1 \tan \theta \\ -\tan \theta 1\end{array}\right]-1=\left[\begin{array}{cc}a -b \\ b a\end{array}\right]$, then (a) $a=1, b=1$ (b) $a=\cos 2 \theta, b=\sin 2 \theta$ (c) $a=\sin 2 \theta, b=\cos 2 \theta$ (d) none of these Solution: (b) $a=\cos 2 \theta, b=\sin 2 \theta$ $\left[\begin{array}{cc}1 \tan \theta \\ -\tan \theta 1\end{array}\right]^{-1}=\frac{1}{\sec ^{2} \theta}\left[\begin{a...
Read More →Meera borrowed a sum of Rs 1000 from Sita for two years.
Question: Meera borrowed a sum of Rs 1000 from Sita for two years. If the rate of interest is 10% compounded annually, find the amount that Meera has to pay back. Solution: Given: $\mathrm{P}=\mathrm{Rs} 1,000$ $\mathrm{R}=10 \%$ p. $\mathrm{a} .$ $\mathrm{n}=2$ years We know that amount $\mathrm{A}$ at the end of n years at the rate $\mathrm{R} \%$ per annum when the interest is compounded annually is given by $\mathrm{A}=\mathrm{P}\left(1+\frac{\mathrm{R}}{100}\right)$. $\therefore \mathrm{A}=...
Read More →If the area of the base of a right circular cone is 3850 cm2
Question: If the area of the base of a right circular cone is 3850 cm2and its height is 84 cm, then find the slant height of the cone. Solution: We have, Height $=84 \mathrm{~cm}$ Let the radius and the slant height of the cone be $r$ and $l$, respectively. As, Area of the base of the cone $=3850 \mathrm{~cm}^{2}$ $\Rightarrow \pi r^{2}=3850$ $\Rightarrow \frac{22}{7} \times r^{2}=3850$ $\Rightarrow r^{2}=3850 \times \frac{7}{22}$ $\Rightarrow r^{2}=1225$ $\Rightarrow r=\sqrt{1225}$ $\therefore ...
Read More →The product of three consecutive positive
Question: The product of three consecutive positive integers is divisible by 6 Is this statement true or false? Justify your answer Solution: yes, three consecutive integers can be n, (n + 1)and (n + 2). So, one number of these three must be divisible by 2 and another one must be divisible by 3. Hence, product of numbers is divisible by 6....
Read More →The product of two consecutive positive
Question: The product of two consecutive positive integers is divisible by 2. Is this statement true or false? Give reasons. Solution: yes, two consecutive integers can be n, (n +1). So, one number of these two must be divisible by 2. Hence, product of numbers is divisible by 2....
Read More →Rahman lent Rs 16000 to Rasheed at the rate of
Question: Rahman lent Rs 16000 to Rasheed at the rate of $12 \frac{1}{2} \%$ per annum compound interest. Find the amount payable by Rasheed to Rahman after 3 years. Solution: Given, $\mathrm{P}=\mathrm{Rs} 16,000$ $\mathrm{R}=12.5 \%$ p. a $\mathrm{n}=3$ years We know that amount $\mathrm{A}$ at the end of $\mathrm{n}$ years at the rate $\mathrm{R} \%$ per annum when the interest is compounded annually is given by $\mathrm{A}=\mathrm{P}\left(1+\frac{\mathrm{R}}{100}\right)^{\mathrm{n}}$. $\ther...
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Question: If $A=\left[\begin{array}{lll}1 0 1 \\ 0 0 1 \\ a b 2\end{array}\right]$, then $a I+b A+2 A^{2}$ equals (a) $A$ (b) $-A$ (c) $a b A$ (d) none of these Solution: (d) none of these $A=\left[\begin{array}{lll}1 0 1 \\ 0 0 1 \\ a b 2\end{array}\right]$ $\Rightarrow A^{2}=\left[\begin{array}{ccc}1+a b 3 \\ a b 2 \\ 3 a 2 b a+b+4\end{array}\right]$ Now, $a I+b A+2 A^{2}=\left[\begin{array}{ccc}a 0 0 \\ 0 a 0 \\ 0 0 a\end{array}\right]+\left[\begin{array}{ccc}b 0 b \\ 0 0 b \\ a b b^{2} 2 b\e...
Read More →Write whether every positive integer
Question: Write whether every positive integer can be of the form 4q + 2, where q is an integer. Justify your answer. Solution: No, by Euclids Lemma, b = aq + r,0,ra [-.-dividend = divisor x quotient + remainder] Here, b is any positive integer a = 4, b = Aq + r for 0 ^ r 4 i.e., r = 0,1,2, 3 So, this must be in the form Aq, 4q + 1, Aq + 2 or 4q + 3....
Read More →The decimal expansion of the rational
Question: The decimal expansion of the rational number $\frac{14587}{1250}$ will terminate after (a) one decimal place (b) two decimal places (c) three decimal places (d) four decimal places Solution: (d) Rational number $=\frac{14587}{1250}=\frac{14587}{2^{1} \times 5^{4}}$ $=\frac{14587}{10 \times 5^{3}} \times \frac{(2)^{3}}{(2)^{3}}$ $=\frac{14587 \times 8}{10 \times 1000}$ $=\frac{116696}{10000}=11.6696$ Hence, given rational number will terminate after four decimal places....
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Question: If $A=\frac{1}{3}\left[\begin{array}{ccc}1 1 2 \\ 2 1 -2 \\ x 2 y\end{array}\right]$ is orthogonal, then $x+y=$ (a) 3 (b) 0 (c) $-3$ (d) 1 Solution: We have, $A=\frac{1}{3}\left[\begin{array}{ccc}1 1 2 \\ 2 1 -2 \\ x 2 y\end{array}\right]$ $\Rightarrow A^{T}=\frac{1}{3}\left[\begin{array}{ccc}1 2 x \\ 1 1 2 \\ 2 -2 y\end{array}\right]$ Now, $A^{T} A=I$ $\Rightarrow\left[\begin{array}{ccc}x^{2}+5 2 x+3 x y-2 \\ 3+2 x 6 2 y \\ x y-6 2 y y^{2}+8\end{array}\right]=\left[\begin{array}{ccc}9...
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