A two-digit number is obtained by either
Question: A two-digit number is obtained by either multiplying the sum of the digits by 8 and then subtracting 5 or by multiplying the difference of the digits by 16 and then adding 3. Find the number. Solution: Let the two-digit number $=10 x+y$ Case I Multiplying the sum of the digits by 8 and then subtracting $5=$ two-digit number $\Rightarrow \quad 8 \times(x+y)-5=10 x+y$ $\Rightarrow \quad 8 x+8 y-5=10 x+y$ $\Rightarrow \quad 2 x-7 y=-5$...(i) Case II Multiplyina the difference of the digit...
Read More →Can a polyhedron have 10 faces,
Question: Can a polyhedron have 10 faces, 20 edges and 15 vertices? Solution: No, because every polyhedron satisfies Euler's formula, given below: $\mathrm{F}+\mathrm{V}=\mathrm{E}+2$ Here, number of faces $\mathrm{F}=10$ Number of edges $E=20$ Number of vertices $\mathrm{V}=15$ So, by Euler's formula: LHS : $10+15=25$ RHS : $20+2=22$, which is not true because $25 \neq 22$ Hence, Eulers formula is not satisfied and no polyhedron may be formed....
Read More →Is a square prism same as a cube?
Question: Is a square prism same as a cube? Solution: Yes, a square prism and a cube are the same. Both of them have 6 faces, 8 vertices and 12 edges. The only difference is that a cube has 6 equal faces, while a square prism has a shape like a cuboid with two square faces, one at the top and the other at the bottom and with, possibly, 4 rectangular faces in between....
Read More →If the probability of winning a game is 0.7, what is the probability of losing it?
Question: If the probability of winning a game is 0.7, what is the probability of losing it? Solution: For any event E, P(E) + P(not E) = 1Let probability of winning a game = P(E) = 0.7 P(winning a game) + P(losing a game) = 1⇒ P(losing a game) = 1 0.7= 0.3Thus, the probability of losing a game is 0.3....
Read More →Is it possible to have a polyhedron
Question: Is it possible to have a polyhedron with any given number of faces? Solution: Yes, it is possible to have a polyhedron with any number of faces. The only condition is that there should be at least four faces. This is because there is no possible polyhedron with 3 or less faces....
Read More →250 lottery tickets were sold and there are 5 prizes on these tickets.
Question: 250 lottery tickets were sold and there are 5 prizes on these tickets. If Kunal has purchased one lottery ticket, what is the probability that he wins a prize? Solution: Total number of tickets = 250Kunal wins a prize if he gets a ticket that assures a prize.Number of tickets on which prizes are assured = 5 $\therefore P($ Kunal wins a prize $)=\frac{5}{250}=\frac{1}{50}$...
Read More →Can a polyhedron have for its faces:
Question: Can a polyhedron have for its faces: (i) 3 triangles? (ii) 4 triangles? (iii) a square and four triangles? Solution: (i) No, because in order to complete a polyhedron, we need at least four triangular faces. (ii) Yes, a polyhedron with 4 triangular faces is a tetrahedron. (iii) Yes, with the help of a square bottom and four triangle faces, we can form a pyramid....
Read More →A motorboat can travel 30 km upstream
Question: A motorboat can travel 30 km upstream and 28 km downstream in 7 h. It can travel 21 km upstream and return in 5 h. Find the speed of the boat in still water and the speed of the stream. Solution: Let the speed of the motorboat in still water and the speed of the stream are u km/h and v km/h, respectively. Then, a motorboat speed in downstream = (u + v) km/h and a motorboat speed in upstream = (u -v) km/h. Motorboat has taken time to travel 30 km upstream, $t_{1}=\frac{30}{u-v} h$ and m...
Read More →In a lottery there are 10 prizes and 25 blanks.
Question: In a lottery there are 10 prizes and 25 blanks. What is the probability of getting a prize? Solution: Total number of lottery tickets = 10 + 25 = 35Number of prizes = 10Let E be the event of getting a prize. $\therefore \mathrm{P}($ getting a prize $)=\mathrm{P}(\mathrm{E})=\frac{\text { Number of outcomes favourable to } \mathrm{E}}{\text { Number of all possible outcomes }}$ $=\frac{10}{35}=\frac{2}{7}$ Thus, the probability of getting a prize is $\frac{2}{7}$....
Read More →What is the least number of planes that can enclose a solid?
Question: What is the least number of planes that can enclose a solid? What is the name of the solid? Solution: The least number of planes that can enclose a solid is 4 . Tetrahedron is a solid with four planes (faces)....
Read More →What is the least number of planes that can enclose a solid?
Question: What is the least number of planes that can enclose a solid? What is the name of the solid? Solution: Theleastnumberofplanesthatcanencloseasolidis4.Tetrahedronisasolidwithfourplanes(faces)....
Read More →12 defective pens are accidentally mixed up the 132 good ones.
Question: 12 defective pens are accidentally mixed up the 132 good ones. It is not possible to just look at a pen and tell whether it is defective or not. One pen is taken out at random from the lot. Find the probability that the pen taken out is ] (i) a good one, (ii) defective Solution: Number of defective pens in the lot = 12 Number of good pens in the lot = 132Total number of pens in the lot = 12 + 132 = 144 Total number of outcomes = 144(i) There are 132 good pens in the lot. Out of these p...
Read More →Match the following figures:
Question: Match the following figures: Solution: (a)The given figure is a cuboid with sides 4,4 and 6 units.Area of a rectangle $=$ length $\times$ width $\therefore$ Area of the rectangular face sith sides 4 and $4=4 \times 4=16$ And, area of the other face with sides (b) The given figure is a cuboid with sides 3,3 and 8 . Area of a rectangle $=$ length $\times$ width $\therefore$ Area of the rectangular face sith sides 3 and $3=3 \times 3=9$ And the area of the other face with sides 3 and $8=3...
Read More →A child has die whose 6 faces show the letters given below.
Question: A child has die whose 6 faces show the letters given below. The die is thrown once. What is the probability of getting (i) A, (ii) B? Solution: When the die is thrown once, then any one of the six faces can show up. Total number of outcomes = 6(i) There are 3 faces on the die showing the letter A. $\therefore \mathrm{P}($ Getting the letter $\mathrm{A})=\frac{\text { Favourable number of outcomes }}{\text { Total number of outcomes }}=\frac{3}{6}=\frac{1}{2}$ (ii) There are 2 faces on ...
Read More →A letter of English alphabet is chosen at random.
Question: A letter of English alphabet is chosen at random. Determine the probability that the chosen letter is a consonant. Solution: Let E be the event of getting a consonant.Out of 26 letters of English alphabets, there are 21 consonants. $\therefore \mathrm{P}$ (getting a consonant) $=\mathrm{P}(\mathrm{E})=\frac{\text { Number of outcomes favourable to } \mathrm{E}}{\text { Number of all possible outcomes }}$ $=\frac{21}{26}$ Thus, the probability of getting a consonant is $\frac{21}{26}$....
Read More →Draw nets for each of the following polyhedrons:
Question: Draw nets for each of the following polyhedrons: Solution: (i)(ii)(iii)(iv)...
Read More →Dice are cubes where the numbers on the opposite faces must total 7.
Question: Dice are cubes where the numbers on the opposite faces must total 7. Which of the following are dice? Solution: Among the given figures, only figure (i) is a dice. This is because if we fold the given net from the edges, we'll get a cube in which the sum of the opposite faces is 7 ....
Read More →A die is thrown once. Find the probability of getting
Question: A die is thrown once. Find the probability of getting(i) an even number(ii) a number less than 5(iii) a number greater than 2(iv) a number between 3 and 6(v) a number other than 3(vi) the number 5. Solution: In a single throw of a die, the possible outcomes are 1, 2, 3, 4, 5 and 6.Total number of possible outcomes = 6 (i)Let E be the event of getting an even number.Then, the favourable outcomes are 2, 4 and 6.Number of favourable outcomes = 3 $\therefore$ Probability of getting an even...
Read More →A person,rowing at the rate of 5 km/h in still water,
Question: A person,rowing at the rate of 5 km/h in still water,takes thrice as much time in going 40 km upstream as in going 40 km downstream. Find the speed of the stream. Solution: Let the speed of the stream be v km/h. Given that, a person rowing in still water = 5 km/h The speed of a person rowing in downstream = (5+ v) km/h and the speed of a person has rowing in upstream = (5 v) km/h Now, the person taken time to cover 40 km downstream, $t_{1}=\frac{40}{5+v} \mathrm{~h} \quad\left[\because...
Read More →Name the polyhedron that can be made by folding each net:
Question: Name the polyhedron that can be made by folding each net: Solution: (i) If we fold the given figure from the edges, we'll get a pyramid with a square base. (ii) If we fold the given polyhedron from the edges, we'll get a triangular prism. (iii) If we fold the given polyhedron from the edges, we'll get a triangular prism. (iv) If we fold the given polyhedron from the edges, we'll get a hexagonal prism. (v) If we fold the given net from the edges, we'll get a hexagonal pyramid. (vi) If w...
Read More →Which among the following are nets for a cube?
Question: Which among the following are nets for a cube? Solution: To create a cube, we need six equal faces that enclose a closed box. In the given figure, only (iv), (v) and (vi) are such nets that enclose a box when we fold each face from the edge....
Read More →Two coins are tossed simultaneously. Find the probability of getting
Question: Two coins are tossed simultaneously. Find the probability of getting(i) exactly 1 head(ii) at most 1 head(iii) at least 1 head Solution: When two coins are tossed simultaneously, all possible outcomes areHH, HT, TH andTT. Total number of possible outcomes = 4 (i) Let E be the event of getting exactly one head. Then, the favourable outcomes are HT and TH. Number of favourable outcomes = 2 $\therefore P($ getting exactly 1 head $)=\frac{\text { Number of favourable outcomes }}{\text { T...
Read More →Using Euler's formula find the unknown:
Question: Using Euler's formula find the unknown: Solution: We know that the Euler's formula is: $\mathrm{F}+\mathrm{V}=\mathrm{E}+2$ (i) The number of vertices $\mathrm{V}$ is 6 and the number of edges $\mathrm{E}$ is 12 . Using Euler's formula: $\mathrm{F}+6=12+2$ $\mathrm{~F}+6=14$ $\mathrm{~F}=14-6$ $\mathrm{~F}=8$ So, the number of faces in this polyhedron is 8 . (ii) Faces, $F=5$ Edges, $\mathrm{E}=9$. We have to find the number of vertices. Putting these values in Euler's formula: $5+\mat...
Read More →Ankita travels 14 km to her home partly
Question: Ankita travels 14 km to her home partly by rickshaw and partly by bus. She takes half an hour, if she travels 2 km by rickshaw and the remaining distance by bus. On the other hand, if she travels 4 km by rickshaw and the remaining distance by bus, she takes 9 min longer. Find the speed of the rickshaw and of the bus. Solution: Let the speed of the rickshaw and the bus are $x$ and $y \mathrm{~km} / \mathrm{h}$, respectively. Now, she has taken time to travel $2 \mathrm{~km}$ by rickshaw...
Read More →A coin is tossed once. What is the probability of getting a tail?
Question: A coin is tossed once. What is the probability of getting a tail? Solution: When a coin is tossed once, the possible outcomes are H and T.Total number of possible outcomes = 2Favourable outcome = 1 $\therefore$ Probability of getting a tail $=P(T)=\frac{\text { Number of favo u rable outcomes }}{\text { Total number of possible outcomes }}=\frac{1}{2}$...
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