The dimensions of an oil tin are 26 cm × 26 cm × 45 cm.
Question: The dimensions of an oil tin are 26 cm 26 cm 45 cm. Find the area of the tin sheet required for making 20 such tins. If 1 square metre of the tin sheet costs Rs 10, find the cost of tin sheet used for these 20 tins. Solution: Dimensions of the oil tin are $26 \mathrm{~cm} \times 26 \mathrm{~cm} \times 45 \mathrm{~cm} .$ So, the area of tin sheet required to make one tin $=2 \times($ length $\times$ breadth $+$ breadth $\times$ height $+$ length $\times$ height $)$ $=2 \times(26 \times ...
Read More →The existence of the unique solution of the system of equations:
Question: The existence of the unique solution of the system of equations: $x+y+z=\lambda$ $5 x-y+\mu z=10$ $2 x+3 y-z=6$ depends on (a) $\mu$ only (b) $\lambda$ only (c) $\lambda$ and $\mu$ both (d) neither $\lambda$ nor $\mu$ Solution: (a) $\mu$ only For a unique solution, $|A| \neq 0$ $\Rightarrow\left|\begin{array}{ccc}1 1 1 \\ 5 -1 \mu \\ 2 3 -1\end{array}\right| \neq 0$ $\Rightarrow 1(1-3 \mu)-1(-5-2 \mu)+1(15+2) \neq 0$ $\Rightarrow 1-3 \mu+5+2 \mu+17 \neq 0$ $\Rightarrow-\mu+23 \neq 0$ $...
Read More →The existence of the unique solution of the system of equations:
Question: The existence of the unique solution of the system of equations: $x+y+z=\lambda$ $5 x-y+\mu z=10$ $2 x+3 y-z=6$ depends on (a) $\mu$ only (b) $\lambda$ only (c) $\lambda$ and $\mu$ both (d) neither $\lambda$ nor $\mu$ Solution: (a) $\mu$ only For a unique solution, $|A| \neq 0$ $\Rightarrow\left|\begin{array}{ccc}1 1 1 \\ 5 -1 \mu \\ 2 3 -1\end{array}\right| \neq 0$ $\Rightarrow 1(1-3 \mu)-1(-5-2 \mu)+1(15+2) \neq 0$ $\Rightarrow 1-3 \mu+5+2 \mu+17 \neq 0$ $\Rightarrow-\mu+23 \neq 0$ $...
Read More →Find the surface area of a wooden box whose shape is of a cube,
Question: Find the surface area of a wooden box whose shape is of a cube, and if the edge of the box is 12 cm. Solution: It is given that the side of the cubical wooden box is $12 \mathrm{~cm}$. $\therefore$ Surface area of the cubical box $=6 \times(\text { side })^{2}=6 \times(12)^{2}=864 \mathrm{~cm}^{2}$...
Read More →Justify whether it is true to say that
Question: Justify whether it is true to say that $-1, \frac{-3}{2},-2, \frac{5}{2} \ldots$ forms an AP as $a_{2}-a_{1}=a_{3}-a_{2}$ Solution: False Here, $a_{1}=-1, a_{2}=\frac{-3}{2}, a_{3}=-2$ and $a_{4}=\frac{5}{2}$ $a_{2}-a_{1}=\frac{-3}{2}+1=-\frac{1}{2}$ $a_{3}-a_{2}=-2+\frac{3}{2}=-\frac{1}{2}$ $a_{4}-a_{3}=\frac{5}{2}+2=\frac{9}{2}$ Clearly, the difference of successive terms is not same, all though, a2 a1= a3-a2 but a3 a2a4 a3, therefore it does not form an AP....
Read More →Find the area of the cardboard required to make a closed box of length 25 cm,
Question: Find the area of the cardboard required to make a closed box of length 25 cm, 0.5 m and height 15 cm. Solution: Length of the box $=25 \mathrm{~cm}$ Width of the box $=0.5 \mathrm{~m}$ $=0.5 \times 100 \mathrm{~cm}(\because 1 \mathrm{~m}=100 \mathrm{~cm})$ $=50 \mathrm{~cm}$ Height of the box $=15 \mathrm{~cm}$ $\therefore$ Surface are $a$ of the box $=2 \times($ length $\times$ breadth $+$ breadth $\times$ height $+$ length $\times$ height $)$ $=2 \times(25 \times 50+50 \times 15+25 \...
Read More →The dimensions of a cuboid are in the ratio 5 : 3 : 1 and its total surface area is 414 m
Question: The dimensions of a cuboid are in the ratio 5 : 3 : 1 and its total surface area is 414 m2. Find the dimensions. Solution: It is given that the sides of the cuboid are in the ratio $5: 3: 1$. Suppose that its sides are $x$ multiple of each other, then we have: Length $=5 x \mathrm{~m}$ Breadth $=3 x \mathrm{~m}$ Height $=x \mathrm{~m}$ Also, total surface area of the cuboid $=414 \mathrm{~m}^{2}$ Surface area of the cuboid $=2 \times($ length $\times$ breadth $+$ breadth $\times$ heigh...
Read More →For the system of equations:
Question: For the system of equations:x+ 2y+ 3z= 12x+y+ 3z= 25x+ 5y+ 9z= 4(a) there is only one solution(b) there exists infinitely many solution(c) there is no solution(d) none of these Solution: $(\mathrm{a})$ there is only one solution The given system of equations can be written in matrix form as follows: $\left[\begin{array}{lll}1 2 3 \\ 2 1 3 \\ 5 5 9\end{array}\right]\left[\begin{array}{l}x \\ y \\ z\end{array}\right]=\left[\begin{array}{l}1 \\ 2 \\ 4\end{array}\right]$ Here, $A=\left[\be...
Read More →Which of the following form of an AP ?
Question: Which of the following form of an AP ? Justify your answer. (i) $-1,-1,-1,-1, \ldots$ (ii) $0,2,0,2, \ldots$ (iii) $1,1,2,2,3,3, \ldots$ (iv) $11,22,33, \ldots$ (v) $\frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \ldots$ (vi) $2,2^{2}, 2^{3}, 2^{4}$ (vii) $\sqrt{3}, \sqrt{12}, \sqrt{27}, \sqrt{48}, \ldots$ Solution: (i) Here, $t_{1}=-1, t_{2}=-1, t_{3}=-1$ and $t_{4}=-1$ $t_{2}-t_{1}=-1+1=0$ $t_{3}-t_{2}=-1+1=0$ $t_{4}-t_{3}=-1+1=0$ Clearly, the difference of successive terms is same, therefor...
Read More →One card is drawn at random from a well-shuffled deck of 52 cards.
Question: One card is drawn at random from a well-shuffled deck of 52 cards. What is the probability of getting a 6? (a) $\frac{3}{26}$ (b) $\frac{1}{52}$ (C) $\frac{1}{13}$ (d) none of these Solution: (C) $\frac{1}{13}$ Explanation:Total number of all possible outcomes = 52Number of 6 in a deck of 52 cards = 4 $\therefore P($ getting a 6$)=\frac{4}{52}=\frac{1}{13}$...
Read More →One card is drawn at random from a well-shuffled deck of 52 cards.
Question: One card is drawn at random from a well-shuffled deck of 52 cards. What is the probability of getting a 6? (a) $\frac{3}{26}$ (b) $\frac{1}{52}$ (C) $\frac{1}{13}$ (d) none of these Solution: (C) $\frac{1}{13}$ Explanation:Total number of all possible outcomes = 52Number of 6 in a deck of 52 cards = 4 $\therefore P($ getting a 6$)=\frac{4}{52}=\frac{1}{13}$...
Read More →Find the volume of a cube whose surface area is
Question: Find the volume of a cube whose surface area is (i) 96 cm2 (ii) 150 m2 Solution: (i) Surface area of the given cube $=96 \mathrm{~cm}^{2}$ Surface area of a cube $=6 \times(\text { side })^{2}$ $\Rightarrow 6 \times(\text { side })^{2}=96$ $\Rightarrow(\text { side })^{2}=\frac{96}{6}=16$ i.e., side of the cube $=\sqrt{16}=4 \mathrm{~cm}$ $\therefore$ Volume of the cube $=(\text { side })^{3}=(4)^{3}=64 \mathrm{~cm}^{3}$ (ii) Surface area of the given cube $=150 \mathrm{~m}^{2}$ Surfac...
Read More →Once card is drawn at random from a well-shuffled deck of 52 cards.
Question: Once card is drawn at random from a well-shuffled deck of 52 cards. What is the probability of getting a black face card? (a) $\frac{1}{26}$ (b) $\frac{3}{26}$ (c) $\frac{3}{13}$ (d) $\frac{3}{14}$ Solution: (b) $\frac{3}{26}$ Explanation:Total number of all possible outcomes= 52Number of black face cards ( 2 kings + 2 queens + 2 jacks) = 6 $\therefore P($ getting a black face card $)=\frac{6}{52}=\frac{3}{26}$...
Read More →Once card is drawn at random from a well-shuffled deck of 52 cards.
Question: Once card is drawn at random from a well-shuffled deck of 52 cards. What is the probability of getting a black face card? (a) $\frac{1}{26}$ (b) $\frac{3}{26}$ (c) $\frac{3}{13}$ (d) $\frac{3}{14}$ Solution: (b) $\frac{3}{26}$ Explanation:Total number of all possible outcomes= 52Number of black face cards ( 2 kings + 2 queens + 2 jacks) = 6 $\therefore P($ getting a black face card $)=\frac{6}{52}=\frac{3}{26}$...
Read More →Let a, b, c be positive real numbers.
Question: Let $a, b, c$ be positive real numbers. The following system of equations in $x, y$ and $z$ (a) no solution(b) unique solution(c) infinitely many solutions(d) finitely many solutions Solution: $(\mathrm{b})$ unique solution The given system of equations can be written in matrix form as follows: $\left[\begin{array}{ccc}\frac{1}{a^{2}} \frac{1}{b^{2}} \frac{-1}{c^{2}} \\ \frac{1}{a^{2}} \frac{-1}{b^{2}} \frac{1}{c^{2}} \\ \frac{-1}{a^{2}} \frac{1}{b^{2}} \frac{1}{c^{2}}\end{array}\right...
Read More →One card is drawn at random from a well-shuffled deck of 52 cards.
Question: One card is drawn at random from a well-shuffled deck of 52 cards. What is the probability of getting a face card? (a) $\frac{1}{26}$ (b) $\frac{3}{26}$ (C) $\frac{3}{13}$ (d) $\frac{4}{13}$ Solution: (c) $\frac{3}{13}$ Explanation:Total number of all possible outcomes= 52Number of face cards ( 4 kings + 4 queens + 4 jacks) = 12 $\therefore P($ getting a face card $)=\frac{12}{52}=\frac{3}{13}$...
Read More →One card is drawn at random from a well-shuffled deck of 52 cards.
Question: One card is drawn at random from a well-shuffled deck of 52 cards. What is the probability of getting a face card? (a) $\frac{1}{26}$ (b) $\frac{3}{26}$ (C) $\frac{3}{13}$ (d) $\frac{4}{13}$ Solution: (c) $\frac{3}{13}$ Explanation:Total number of all possible outcomes= 52Number of face cards ( 4 kings + 4 queens + 4 jacks) = 12 $\therefore P($ getting a face card $)=\frac{12}{52}=\frac{3}{13}$...
Read More →From a well-shuffled deck of 52 cards, one card is drawn at random.
Question: From a well-shuffled deck of 52 cards, one card is drawn at random. What is the probability of getting a queen? (a) $\frac{1}{13}$ (b) $\frac{1}{26}$ (c) $\frac{4}{39}$ (d) none of these Solution: (a) $\frac{1}{13}$ Explanation:Total number of all possible outcomes= 52Number of queens = 4 $\therefore P($ getting a queen $)=\frac{4}{52}=\frac{1}{13}$...
Read More →Find the surface area of a cube whose volume is
Question: Find the surface area of a cube whose volume is (i) 343 m3 (ii) 216 dm3 Solution: (i) Volume of the given cube $=343 \mathrm{~m}^{3}$ We know that volume of a cube $=(\text { side })^{3}$ $\Rightarrow(\text { side })^{3}=343$ i. e., side $=\sqrt[3]{343}=7 \mathrm{~m}$ $\therefore$ Surface area of the cube $=6 \times(\text { side })^{2}=6 \times(7)^{2}=294 \mathrm{~m}^{2}$ (ii) Volume of the given cube $=216 \mathrm{dm}^{3}$ We know that volume of a cube $=(\text { side })^{3}$ $\Righta...
Read More →The sum of first five multiples
Question: The sum of first five multiples of 3 is (a) 45 (b) 55 (c) 65 (d) 75 Solution: (a)The first five multiples of 3 are 3, 6, 9,12 and 15. Here, first term, a = 3, common difference, d = 6-3 = 3 and number of terms, n = 5 $\therefore \quad S_{5}=\frac{5}{2}[2 a+(5-1) d] \quad\left[\because S_{n}=\frac{n}{2}\{2 a+(n-1) d\}\right]$ $=\frac{5}{2}[2 \times 3+4 \times 3]$ $=\frac{5}{2}(6+12)=5 \times 9=45$...
Read More →A card is drawn at random from a well-shuffled deck of 52 cards.
Question: A card is drawn at random from a well-shuffled deck of 52 cards. What is the probability of getting a black king? (a) $\frac{1}{13}$ (b) $\frac{1}{26}$ (c) $\frac{2}{39}$ (d) none of these Solution: (b) $\frac{1}{26}$ Explanation:Total number of all possible outcomes = 52Number of black kings = 2 $\therefore P($ getting a black king $)=\frac{2}{52}=\frac{1}{26}$...
Read More →Consider the system of equations:
Question: Consider the system of equations:a1x+b1y+c1z= 0a2x+b2y+c2z= 0a3x+b3y+c3z= 0, if $\left|\begin{array}{lll}a_{1} b_{1} c_{1} \\ a_{2} b_{2} c_{2} \\ a_{3} b_{3} c_{3}\end{array}\right|=0$, then the system has (a) more than two solutions(b) one trivial and one non-trivial solutions(c) no solution(d) only trivial solution (0, 0, 0) Solution: (a) more than two solutions Here, $|A|=0$ and $B=0$ (Given) If $|A|=0$ and (adj $A) B=0$, then the system is consistent and has infinitely many soluti...
Read More →A cuboidal box is 5 cm by 5 cm by 4 cm.
Question: A cuboidal box is 5 cm by 5 cm by 4 cm. Find its surface area. Solution: The dimensions of the cuboidal box are $5 \mathrm{~cm} \times 5 \mathrm{~cm} \times 4 \mathrm{~cm}$. Surface area of the cuboidal box $=2 \times($ length $\times$ breadth $+$ breadth $\times$ height $+$ length $\times$ height $)$ $=2 \times(5 \times 5+5 \times 4+5 \times 4)$ $=2 \times(25+20+20)$ $=130 \mathrm{~cm}^{2}$...
Read More →A card is drawn at random from a well-shuffled deck of 52 cards.
Question: A card is drawn at random from a well-shuffled deck of 52 cards. What is the probability of getting a black king? (a) $\frac{1}{13}$ (b) $\frac{1}{26}$ (c) $\frac{2}{39}$ (d) none of these Solution: (b) $\frac{1}{26}$ Explanation:Total number of all possible outcomes = 52Number of black kings = 2 $\therefore P($ getting a black king $)=\frac{2}{52}=\frac{1}{26}$...
Read More →Find the surface area of a cube whose edge is
Question: Find the surface area of a cube whose edge is (i) 1.2 m (ii) 27 cm (iii) 3 cm (iv) 6 m (v) 2.1 m Solution: (i) Edge of the a cube $=1.2 \mathrm{~m}$ $\therefore$ Surface area of the cube $=6 \times(\text { side })^{2}=6 \times(1.2)^{2}=6 \times 1.44=8.64 \mathrm{~m}^{2}$. (ii) Edge of the a cube $=27 \mathrm{~cm}$ $\therefore$ Surface area of the cube $=6 \times(\text { side })^{2}=6 \times(27)^{2}=6 \times 729=4374 \mathrm{~cm}^{2}$ (iii) Edge of the a cube $=3 \mathrm{~cm}$ $\therefo...
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