Prove the following
Question: $\hat{\eta}^{2}-1$ is divisible by 8, if $n$ is (a) an integer (b) a natural number (c) an odd integer (d) an even integer Solution: Let $a=n^{2}-1$ Here $n$ can be ever or odd. Case I $n=$ Even $i . e ., n=2 k$, where $k$ is an integer. $\Rightarrow \quad a=(2 k)^{2}-1$ $\Rightarrow \quad a=4 k^{2}-1$ At $k=-1,=4(-1)^{2}-1=4-1=3$, which is not divisible by 8 . At $k=0, a=4(0)^{2}-1=0-1=-1$, which is not divisible by 8 , which is not Case II $n=$ Odd i.e., $n=2 k+1$, where $k$ is an od...
Read More →List price of a cooler is Rs 2563. The rate of VAT is 10%.
Question: List price of a cooler is Rs 2563. The rate of VAT is 10%. The customer requests the shopkeeper to allow a discount in the price of the cooler to such an extent that the price remains Rs 2563 inclusive of VAT. Find the descount in the price of the cooler. Solution: Let the reduced price of the cooler, excluding VAT, be Rs. $x$. VAT $=10 \%$ of Rs $x=\frac{10}{100} \times x$ $=R s . \frac{x}{10}$ Therefore, SP of the cooler will be $=$ Rs. $\left(x+\frac{x}{10}\right)=$ Rs. $\frac{11 x}...
Read More →For non-singular square matrix A, B and C of the same order
Question: For non-singular square matrix $A, B$ and $C$ of the same order $\left(A B^{-1} C\right)=$ (a) $A^{-1} B C^{-1}$ (b) $C^{-1} B^{-1} A^{-1}$ (c) $C B A^{-1}$ (d) $C^{-1} B A^{-1}$ Solution: Disclaimer: In Quesion, We are to find the inverse of $\left(A B^{-1} C\right)$. The inverse is missing in the question. (d) $C^{-1} B A^{-1}$ We have, $\left(A B^{-1} C\right)^{-1}=C^{-1}\left(B^{-1}\right)^{-1} A^{-1}$ $=C^{-1} B A^{-1}$...
Read More →Shruti bought a set of cosmetic items for Rs 345 including 15% value added tax and a purse for Rs 110 including 10% VAT.
Question: Shruti bought a set of cosmetic items for Rs 345 including 15% value added tax and a purse for Rs 110 including 10% VAT. What percent is the VAT charged on the whole transaction? Solution: Let the price of the cosmetic items be Rs. $x$ and $t h e p$ rice of the purse be Rs. $y$. $\mathrm{VAT}=15 \%$ of $x=\frac{15 x}{100}=R s \cdot \frac{3 x}{20}$ $\mathrm{VAT}=10 \%$ of $y=\frac{10 y}{100}=R s . \frac{y}{10}$ So, SP of cosmetic items $=x+\frac{3 x}{20}=R s . \frac{23 x}{20}$ $\mathrm{...
Read More →Solve the following equations
Question: If $A=\left[\begin{array}{lll}2 0 0 \\ 0 2 0 \\ 0 0 2\end{array}\right]$, then $A^{5}=$ (a) $5 \mathrm{~A}$ (b) $10 \mathrm{~A}$ (c) $16 \mathrm{~A}$ (d) $32 \mathrm{~A}$ Solution: (c) $16 \mathrm{~A}$ $A=\left[\begin{array}{lll}2 0 0 \\ 0 2 0 \\ 0 0 2\end{array}\right]$ $\Rightarrow A=2\left[\begin{array}{lll}1 0 0 \\ 0 1 0 \\ 0 0 1\end{array}\right]$ $\Rightarrow A=2 I$ $\Rightarrow A^{5}=(2 I)^{5}$ $\Rightarrow A^{5}=16 \times 2 I$ $\Rightarrow A^{5}=16\left[\begin{array}{lll}2 0 0 ...
Read More →For some integer q,
Question: For some integer q, every odd integer is of the form (a) q (b) q +1 (c) 2g (d) 2q +1 Solution: (d) We know that, odd integers are $1,3,5, \cdots$ So, it can be written in the form of $2 q+1$. where, $q=$ integer $=Z$ or $\quad q=\cdots,-1,0,1,2,3, \ldots$ $\therefore \quad 2 q+1=\cdots,-3,-1,1,3,5, \ldots$ Alternate Method Let ' $a$ ' be given positive integer. On dividing ' $a$ ' by 2 , let $q$ be the quotient and $r$ be the remainder. Then, by Euclid's division algorithm, we have $a=...
Read More →If A and B are square matrices
Question: If $A$ and $B$ are square matrices such that $B=-A^{-1} B A$, then $(A+B)^{2}=$ (a) 0 (b) $A^{2}+B^{2}$ (c) $A^{2}+2 A B+B^{2}$ (d) $A+B$ Solution: (b) $A^{2}+B^{2}$ $B=-A^{-1} B A$ $\Rightarrow A B=-A A^{-1} B A$ $\Rightarrow A B=-B A$ ....(1) $\left(\because A A^{-1}=I\right)$ Now, $(A+B)^{2}=(A+B)(A+B)$ $\Rightarrow(A+B)^{2}=A^{2}+A B+B A+B^{2}$ $\Rightarrow(A+B)^{2}=A^{2}-B A+B A+B^{2}$ [Using (1)] $\Rightarrow(A+B)^{2}=A^{2}+B^{2}$...
Read More →Shikha purchased a car with a marked price of RS 210000 at a discount of 5%.
Question: Shikha purchased a car with a marked price of RS 210000 at a discount of 5%. If VAT is charged at the rate of 10%, find the amount Shikha had paid for purchasing the car. Solution: Marked price of the car $=$ Rs. $2,10,000$ $D$ iscount allowed $=5 \%$ Therefore, discount $=5 \%$ of Rs. $2,10,000=\frac{5}{100} \times 2,10,000=$ Rs. 10,500 So, $S$ hikha get $s$ a discount of Rs. 10,500 . Therefore, cost of $t$ he car will be $=$ Marked price - Discount Rs. $(2,10,000-10,500)$ $=$ Rs. $1,...
Read More →For some integer m,
Question: For some integer m, every even integer is of the form (a) m (b) m +1 (c) 2m +1 (d) 2m Solution: (c) We know that, even integers are 2, 4, 6, So, it can be written in the form of 2m where, $m=$ Integer $=Z \quad[$ since, integer is represented by $Z]$ or $m=\cdots,-1,0,1,2,3, \cdots$ $\therefore \quad 2 m=\cdots,-2,0,2,4,6 \ldots$ Alternate Method Let ' $a$ ' be a positive integer. On dividing ' $a$ ' by 2 , let $m$ be the quotient and f be the remainder. Then, by Euclid's division algo...
Read More →If for the matrix
Question: If for the matrix $A, A^{3}=1$, then $A^{-1}=$ (a) $A^{2}$ (b) $A^{3}$ (c) $A$ (d) none of these Solution: (a) $A^{2}$ Given : $A^{3}=I$ $\Rightarrow A^{3} A^{-1}=I A^{-1} \quad$ [Multiplying both sides by $\left.A^{-1}\right]$ $\Rightarrow A^{2}=A^{-1}$...
Read More →The sales price of a television, inclusive of VAT, is Rs 13,500.
Question: The sales price of a television, inclusive of VAT, is Rs 13,500. If VAT charged at the rate of 8% of the list price, find the list price of the television. Solution: Let the list price of the TV be Rs. $x$. VAT charged $=8 \%$ Given, Cost price of the $T V=R s .13,500$ So, Cost price of the $T V=x+8 \%$ of $x$ $13,500=x+0.08 x$ $13,500=1.08 x$ $x=\frac{13,500}{1.08}$ $x=$ Rs. 12,500 Thus, the list price of the TV is Rs. 12,500 ....
Read More →Solve this
Question: If $A$ satisfies the equation $x^{3}-5 x^{2}+4 x+\lambda=0$ then $A^{-1}$ exists if (a) $\lambda \neq 1$ (b) $\lambda \neq 2$ (c) $\lambda \neq-1$ (d) $\lambda \neq 0$ Solution: (d) $\lambda \neq 0$ $A$ satisfies $x^{3}-5 x^{2}+4 x+\lambda=0$ $\Rightarrow A^{3}-5 A^{2}+4 A=-\lambda$ Assuming $A^{-1}$ exists, we get $A^{-1}\left(A^{3}-5 A^{2}+4 A\right)=-\lambda A^{-1}$ $\Rightarrow A^{2}-5 A+4=-A^{-1} \lambda$ $\Rightarrow A^{-1}=\frac{-\left(A^{2}-5 A+4\right)}{\lambda}$ Thus, $\mathr...
Read More →Sunita purchases a bicycle for Rs 660. She has paid a VAT of 10%.
Question: Sunita purchases a bicycle for Rs 660. She has paid a VAT of 10%. Find the list price of the bicycle? Solution: Let the sale price of the bicycle be Rs. $x$. VAT charged $=10 \%$ Again, Cost of the bicycle $=x+10 \%$ of $x$ $660=x+0.10 x$ $660=1.10 x$ $x=\frac{660}{1.10}$ $x=R s .600$ Thus, the list price of the bicycle is Rs. 600 ....
Read More →Five identical cubes, each of edge 5 cm, are placed adjacent to each other.
Question: Five identical cubes, each of edge 5 cm, are placed adjacent to each other. Find the volume of the resulting cuboid. Solution: We have, Length of the resulting cuboid, $l=5 \times 5=25 \mathrm{~cm}$, Breadth of the resulting cuboid, $b=5 \mathrm{~cm}$ and Height of the resulting cuboid, $h=5 \mathrm{~cm}$ Now, Volume of the resulting cuboid $=l b h$ $=25 \times 5 \times 5$ $=625 \mathrm{~cm}^{3}$ So, the volume of the resulting cuboid is 625 cm3....
Read More →Solve this
Question: If $A^{5}=O$ such that $A^{n} \neq I$ for $1 \leq n \leq 4$, then $(I-A)^{-1}$ equals (a) $A^{4}$ (b) $A^{3}$ (c) $l+A$ (d) none of these Solution: (d) none of the these $I-A^{5}=(I-A)\left(I+A+A^{2}+A^{3}+A^{4}\right)$ Now, $A^{5}=0$ $\Rightarrow I=(I-A)\left(I+A+A^{2}+A^{3}+A^{4}\right)$ $\Rightarrow \frac{I}{(I-A)}=\left(I+A+A^{2}+A^{3}+A^{4}\right)$ $\Rightarrow(I-A)^{-1}=I+A+A^{2}+A^{3}+A^{4}$...
Read More →Solve this
Question: If $A^{5}=O$ such that $A^{n} \neq I$ for $1 \leq n \leq 4$, then $(I-A)^{-1}$ equals (a) $A^{4}$ (b) $A^{3}$ (c) $l+A$ (d) none of these Solution: (d) none of the these $I-A^{5}=(I-A)\left(I+A+A^{2}+A^{3}+A^{4}\right)$ Now, $A^{5}=0$ $\Rightarrow I=(I-A)\left(I+A+A^{2}+A^{3}+A^{4}\right)$ $\Rightarrow \frac{I}{(I-A)}=\left(I+A+A^{2}+A^{3}+A^{4}\right)$ $\Rightarrow(I-A)^{-1}=I+A+A^{2}+A^{3}+A^{4}$...
Read More →For any 2 × 2 matrix,
Question: For any $2 \times 2$ matrix, if $A(\operatorname{adj} A)=\left[\begin{array}{cc}10 0 \\ 0 10\end{array}\right]$, then $|A|$ is equal to (a) 20 (c) 100 (d) 10 (d) 0 Solution: (c) 10 $A(a d j A)=\left[\begin{array}{ll}10 0\end{array}\right.$ $\left.\begin{array}{ll}0 10\end{array}\right]$ By definition, we have $A(\operatorname{adj} A)=|A| I=(\operatorname{adj} A) A$ (Where $I$ is the identity matrix) $\Rightarrow|A| I=A(\operatorname{adj} A)$ $\Rightarrow|A| I=10\left[\begin{array}{ll}1...
Read More →Three cubes of iron whose edges are 6 cm, 8 cm and 10 cm, respectively are melted and formed into a single cube.
Question: Three cubes of iron whose edges are 6 cm, 8 cm and 10 cm, respectively are melted and formed into a single cube. Find the edge of the new cube formed. Solution: We have, Edges of the cubes: $a_{1}=6 \mathrm{~cm}, a_{2}=8 \mathrm{~cm}$ and $a_{3}=10 \mathrm{~cm}$ Let the edge of the new cube so formed be $a$. As, Volume of the new cube so formed $=a_{1}^{3}+a_{2}^{3}+a_{3}{ }^{3}$ $\Rightarrow a^{3}=6^{3}+8^{3}+10^{3}$ $\Rightarrow a^{3}=216+512+1000$ $\Rightarrow a^{3}=1728$ $\Rightarr...
Read More →Rajeeta purchased a set of cosmetics. She paid Rs 165 for it including VAT.
Question: Rajeeta purchased a set of cosmetics. She paid Rs 165 for it including VAT. If the rate of VAT is 10%, find the sale price of the set. Solution: Let the sale price of the set be Rs. $x$ Given that the VAT charged is $10 \%$. Cost of the set $=x+10 \%$ of $x$ $165=x+0.10 x$ $165=1.10 x$ $x=\frac{165}{1.10}$ $x=R s .150$ Thus, the sale price of the set is Rs. 150 ....
Read More →If B is a non-singular matrix and A is a square matrix,
Question: If $B$ is a non-singular matrix and $A$ is a square matrix, then $\operatorname{det}\left(B^{-1} A B\right)$ is equal to (a) Det $\left(A^{-1}\right)$ (b) Det $\left(B^{-1}\right)$ (c) Det (A) (d) Det (B) Solution: (c) $\operatorname{Det}(A)$ $B$ is non-singular. This implies that $|B| \neq 0$, that $B$ is invertible and that $B^{-1}$ exists. Here, $B$ is invertible. $\therefore\left|B^{-1}\right|=|B|^{-1}=\frac{1}{|B|}$ $\Rightarrow\left|B^{-1} A B\right|=\left|B^{-1}\right||A B|$ $\R...
Read More →Rakesh goes to a departmental store and purchases the following articles:
Question: Rakesh goes to a departmental store and purchases the following articles: (i) biscuits and bakery products costing Rs 50, VAT @ 5%, (ii) medicines costing Rs 90, VAT @ 10%, (iii) clothes costing Rs 400, VAT @ 1%, and (iv) cosmetics costing Rs 150, VAT @ 10%. Calculate the total amount to be paid by Rakesh to the store. Solution: (i) Cost of $b$ iscuits and bakery products $=$ Rs. 50 VAT charged $=5 \%$ So, VAT $=5 \%$ of Rs. $50=\frac{5}{100} \times 50=$ Rs. $2.50$ So, the total amount...
Read More →How many cubes of 10 cm edge can be put in a cubical box of 1 m edge?
Question: How many cubes of 10 cm edge can be put in a cubical box of 1 m edge? Solution: We have, Edge of the cube, $a=10 \mathrm{~cm}$ and Edge of the cubical box, $l=1 \mathrm{~m}=100 \mathrm{~cm}$ Now, The number of cubes that can be put in the box $=\frac{\text { Volume of the cubical box }}{\text { Volume of the cube }}$ $=\frac{l^{3}}{a^{3}}$ $=\frac{100^{3}}{10^{3}}$ $=10^{3}$ $=1000$ So, the number of cubes that can be put in the cubical box is 1000....
Read More →Solve this
Question: If $A=\left[\begin{array}{ccc}1 2 -1 \\ -1 1 2 \\ 2 -1 1\end{array}\right]$, then $\operatorname{ded}(\operatorname{adj}(\operatorname{adj} A))$ is (a) $14^{4}$ (b) $14^{3}$ (c) $14^{2}$ (d) 14 Solution: (a) $14^{4}$ Given : $A=\left[\begin{array}{lll}1 2 -1\end{array}\right.$ $\begin{array}{lll}-1 1 2\end{array}$ $\left.\begin{array}{lll}2 -1 1\end{array}\right]$ $\therefore|A|=\mid 1 \quad 2 \quad-1$ $\begin{array}{lll}-1 1 2\end{array}$ $2-1 \quad 1 \mid=1(1+2)-2(-1-4)-1(1-2)=3+10+1...
Read More →Manoj buys a lather coat costing Rs 900 at Rs 990 after paying the VAT.
Question: Manoj buys a lather coat costing Rs 900 at Rs 990 after paying the VAT. Calculate the rate of VAT charge on the coat. Solution: Let the rate of VAT be $x \%$. Then, VAT $=$ Rs. $\left(\frac{x}{100} \times 900\right)$ $=$ Rs. $9 x$ So, $900+9 x=990$ $9 x=90$ $x=10$ Thus, Manoj was charged $10 \%$ VAT on the leather jacket....
Read More →Ajit buys a motorcycle for Rs 17600 including value added tax.
Question: Ajit buys a motorcycle for Rs 17600 including value added tax. If the rate of VAT is 10%, what is the sale price of the motorcycle? Solution: Let the sale price of the motorcycle be Rs $x$. Cost including VAT $=10 \%$ of $x+x$ $17600=\frac{10}{100} \times x+x$ $17600=0.10 x+x$ $1.10 \mathrm{x}=17600$ $x=\frac{17600}{1.10}$ $=16000$ Thu $s$, the sale price of the motorcycle is Rs 16000 ....
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