How many numbers of four digits can be formed with the digits 1, 2, 3, 4,
Question: How many numbers of four digits can be formed with the digits 1, 2, 3, 4, 5 if the digits can be repeated in the same number? Solution: The thousand's place can be filled by any of the 5 digits. Number of ways of filling the thousand's place = 5 Since the digits can repeat in the number, the hundred's place, the ten's place and the unit's place can each be filled in 5 ways. $\therefore$ Total numbers $=5 \times 5 \times 5 \times 5=625$...
Read More →Factorise:
Question: Factorise: $27 x^{3}-y^{3}-z^{3}-9 x y z$ Solution: $27 x^{3}-y^{3}-z^{3}-9 x y z$ $=(3 x)^{3}-y^{3}-z^{3}-3 \times(3 x) \times(-y) \times(-z)$ We know, $a^{3}+b^{3}+c^{3}-3 a b c=(a+b+c)\left(a^{2}+b^{2}+c^{2}-a b-b c-c a\right)$ $a=3 x, b=-y, c=-z$ $(3 x)^{3}-y^{3}-z^{3}-3 \times(3 x) \times(-y) \times(-z)=(3 x-y-z)\left(9 x^{2}+y^{2}+z^{2}+3 x y-y z+3 x z\right)$...
Read More →A coin is tossed three times and the outcomes are recorded.
Question: A coin is tossed three times and the outcomes are recorded. How many possible outcomes are there? How many possible outcomes if the coin is tossed four times? Five times?ntimes? Solution: Total number of outcomes when a coin is tossed once = 2 (Heads, Tails) Number of outcomes when the coin is tossed for the second time = 2 $\therefore$ Number of outcomes when the coin is tossed thrice $=2 \times 2 \times 2=8$ Similarly, the number of outcomes when the coin is tossed four times $=2 \ti...
Read More →If two pipes function simultaneously, a reservoir will be filled in 12 hours.
Question: If two pipes function simultaneously, a reservoir will be filled in 12 hours. One pipe fills the reservoir 10 hours faster than the other. How many hours will the second pipe take to fill the reservoir? Solution: Let the first pipe takes $x$ hours to fill the reservoir. Then the second pipe will takes $=(x+10)$ hours to fill the reservoir. Since, the faster pipe takes $x$ hours to fill the reservoir. Therefore, portion of the reservoir filled by the faster pipe in one hour $=\frac{1}{x...
Read More →A coin is tossed three times and the outcomes are recorded.
Question: A coin is tossed three times and the outcomes are recorded. How many possible outcomes are there? How many possible outcomes if the coin is tossed four times? Five times?ntimes? Solution: Total number of outcomes when a coin is tossed once = 2 (Heads, Tails) Number of outcomes when the coin is tossed for the second time = 2 $\therefore$ Number of outcomes when the coin is tossed thrice $=2 \times 2 \times 2=8$ Similarly, the number of outcomes when the coin is tossed four times $=2 \ti...
Read More →Factorize:
Question: Factorize: $2 \sqrt{2} a^{3}+16 \sqrt{2} b^{3}+c^{3}-12 a b c$ Solution: $2 \sqrt{2} a^{3}+16 \sqrt{2} b^{3}+c^{3}-12 a b c=(\sqrt{2} a)^{3}+(2 \sqrt{2} b)^{3}+c^{3}-3 \times(\sqrt{2} a) \times(2 \sqrt{2} b) \times(c)$ $=(\sqrt{2} a+2 \sqrt{2} b+c)\left[(\sqrt{2} a)^{2}+(2 \sqrt{2} b)^{2}+c^{2}-(\sqrt{2} a) \times(2 \sqrt{2} b)-(2 \sqrt{2} b) \times(c)-(\sqrt{2} a) \times(c)\right]$ $=(\sqrt{2} a+2 \sqrt{2} b+c)\left(2 a^{2}+8 b^{2}+c^{2}-4 a b-2 \sqrt{2} b c-\sqrt{2} a c\right)$...
Read More →If three six faced die each marked with numbers 1 to 6 on six faces,
Question: If three six faced die each marked with numbers 1 to 6 on six faces, are thrown find the total number of possible outcomes. Solution: Number of possible outcomes on one dice = 6 {1,2,3,4,5,6} Number of possible outcomes on both the other two dice = 6 $\therefore$ Total number of outcomes when three dice are thrown $=6 \times 6 \times 6=216$...
Read More →If three six faced die each marked with numbers 1 to 6 on six faces,
Question: If three six faced die each marked with numbers 1 to 6 on six faces, are thrown find the total number of possible outcomes. Solution: Number of possible outcomes on one dice = 6 {1,2,3,4,5,6} Number of possible outcomes on both the other two dice = 6 $\therefore$ Total number of outcomes when three dice are thrown $=6 \times 6 \times 6=216$...
Read More →How many numbers of six digits can be formed from the digits 0, 1, 3, 5, 7 and 9 when no digit is repeated?
Question: How many numbers of six digits can be formed from the digits 0, 1, 3, 5, 7 and 9 when no digit is repeated? How many of them are divisible by 10? Solution: The first digit of the number cannot be zero. Thus, it can be filled in 5 ways. The number of ways of filling the second digit = 5 (as the repetition of digits is not allowed) The number of ways of filling the third digit = 4 The number of ways of filling the fourth digit = 3 The number of ways of filling the fifth digit = 2 The num...
Read More →Factorize:
Question: Factorize: $125-8 x^{3}-27 y^{3}-90 x y$ Solution: $125-8 x^{3}-27 y^{3}-90 x y=5^{3}+(-2 x)^{3}+(-3 y)^{3}-3 \times 5 \times(-2 x) \times(-3 y)$ $=[5+(-2 x)+(-3 y)]\left[5^{2}+(-2 x)^{2}+(-3 y)^{2}-5 \times(-2 x)-(-2 x)(-3 y)-5 \times(-3 y)\right]$ $=(5-2 x-3 y)\left(25+4 x^{2}+9 y^{2}+10 x-6 x y+15 y\right)$...
Read More →A takes 10 days less than the time taken by B to finish a piece of work.
Question: A takes 10 days less than the time taken by B to finish a piece of work. If both A and B together can finish the work in 12 days, find the time taken by B to finish the work. Solution: Let $B$ alone takes $x$ days to finish the work. Then, $B$ 's one day's work $=\frac{1}{x}$. Similarly, $A$ alone can finish it in $(x-10)$ days to finish the work. Then, $A$ 's one day's work $=\frac{1}{x-10}$. It is given that A's one day's work $+B$ 's one day's work $=(A+B)$ 's one day's work $\frac{...
Read More →Factorize:
Question: Factorize: $8-27 b^{3}-343 c^{3}-126 b c$ Solution: $8-27 b^{3}-343 c^{3}-126 b c=(2)^{3}+(-3 b)^{3}+(-7 c)^{3}-3 \times(2) \times(-3 b) \times(-7 c)$ $=[2+(-3 b)+(-7 c)]\left[(2)^{2}+(-3 b)^{2}+(-7 c)^{2}-(2)(-3 b)-(-3 b)(-7 c)-(2)(-7 c)\right]$ $=(2-3 b-7 c)\left(4+9 b^{2}+49 c^{2}+6 b-21 b c+14 c\right)$...
Read More →How many natural numbers not exceeding 4321 can be formed with the digits
Question: How many natural numbers not exceeding 4321 can be formed with the digits 1, 2, 3 and 4, if the digits can repeat? Solution: Case I: Four-digit number Total number of ways in which the 4 digit number can be formed $=4 \times 4 \times 4 \times 4=256$ Now, the number of ways in which the 4-digit numbers greater than 4321 can be formed is as follows: Suppose, the thousand's digit is 4 and hundred's digit is either 3 or 4. $\therefore$ Number of ways $=2 \times 4 \times 4=32$ But $4311,431...
Read More →Factorize:
Question: Factorize: $8 a^{3}+125 b^{3}-64 c^{3}+120 a b c$ Solution: $8 a^{3}+125 b^{3}-64 c^{3}+120 a b c=(2 a)^{3}+(5 b)^{3}+(-4 c)^{3}-3 \times(2 a) \times(5 b) \times(-4 c)$ $=(2 a+5 b-4 c)\left[(2 a)^{2}+(5 b)^{2}+(-4 c)^{2}-(2 a)(5 b)-(5 b)(-4 c)-(2 a) \times(-4 c)\right]$ $=(2 a+5 b-4 c)\left(4 a^{2}+25 b^{2}+16 c^{2}-10 a b+20 b c+8 a c\right)$...
Read More →Sum of the areas of two squares is 640 m2.
Question: Sum of the areas of two squares is $640 \mathrm{~m}^{2}$. If the difference of their perimeters is $64 \mathrm{~m}$. Find the sides of the two squares. Solution: Let the sides of the squares are $x \mathrm{~m}$ and $=y \mathrm{~m}$. Then According to question, Sum of the difference of their perimeter $=64 \mathrm{~m}$ $4 x-4 y=64$ $x-y=16$ $y=x-16 \ldots .(1)$ And sum of the areas of square $=640 \mathrm{~m}^{2}$ $x^{2}+y^{2}=640$.......(2) Putting the value ofxin equation (2) from equ...
Read More →In how many ways can three jobs I, II and III be assigned to three persons A, B and C
Question: In how many ways can three jobs I, II and III be assigned to three personsA,BandCif one person is assigned only one job and all are capable of doing each job? Solution: Number of ways of assigning a job to person A = 3 Number of ways of assigning the remaining jobs to person B = 2 (since one job has already been assigned to person A) The number of ways of assigning the remaining job to person C = 1 Total number of ways of job assignment $=3 \times 2 \times 1=6$...
Read More →A customer forgets a four-digits code for an Automatic Teller Machine (ATM) in a bank.
Question: A customer forgets a four-digits code for an Automatic Teller Machine (ATM) in a bank. However, he remembers that this code consists of digits 3, 5, 6 and 9. Find the largest possible number of trials necessary to obtain the correct code. Solution: Assuming that the code of an ATM has all distinct digits. Number of ways for selecting the first digit = 4 Number of ways for selecting the second digit = 3 Number of ways for selecting the third digit = 2 Number of ways for selecting the fo...
Read More →Is it possible to design a rectangular park of perimeter 80 m
Question: Is it possible to design a rectangular park of perimeter $80 \mathrm{~m}$ and area $400 \mathrm{~m}^{2}$. If so, find its length and breadth. Solution: Let the breadth of the rectangle be $=x$ metres . Then Perimeter $=80$ metres $2($ length $+$ breadth $)=80$ $($ length $+x)=40$ length $=40-x$ And area of the rectangle length $\times$ breadth $=400$ $(40-x) x=400$ $40 x-x^{2}=400$ $x^{2}-40 x+400=0$ $x^{2}-20 x-20 x+400=0$ $x(x-20)-20(x-20)=0$ $(x-20)(x-20)=0$ $(x-20)^{2}=0$ $(x-20)=0...
Read More →A number lock on a suitcase has 3 wheels each labelled with ten digits 0 to 9.
Question: A number lock on a suitcase has 3 wheels each labelled with ten digits 0 to 9. If opening of the lock is a particular sequence of three digits with no repeats, how many such sequences will be possible? Also, find the number of unsuccessful attempts to open the lock. Solution: The digits in the sequence do not repeat. Number of ways of selecting the first digit = 10 Number of ways of selecting the second digit = 9 Number of ways of selecting the third digit = 8 Total number of possible ...
Read More →Serial numbers for an item produced in a factory are to be made using two letters followed by four digits (0 to 9).
Question: Serial numbers for an item produced in a factory are to be made using two letters followed by four digits (0 to 9). If the letters are to be taken from six letters of English alphabet without repetition and the digits are also not repeated in a serial number, how many serial numbers are possible? Solution: Number of ways of selecting the first letter = 6 Number of ways of selecting the second letter = 5 (as repetition of letters is not allowed) Number of ways of selecting the digit in ...
Read More →Is it possible to design a rectangular mango grove whose length
Question: Is it possible to design a rectangular mango grove whose length is twice its breadth and the area is $800 \mathrm{~m}^{2} ?$ If so, find its length and breadth. Solution: Let the breadth of the rectangular mango grove be $x$ meter and the length $=2 x$ metres. Then Area of the rectangle length $\times$ breadth $=800$ $x \times 2 x=800$ $2 x^{2}=800$ $x^{2}=400$ $x=\sqrt{400}$ $=\pm 20$ Sides of the rectangular hall never be negative Therefore, length $=2 x$ $=2 \times 20$ $=40$ Yes, it...
Read More →How many four digit different numbers, greater than 5000 can be formed with the digits 1, 2, 5, 9, 0
Question: How many four digit different numbers, greater than 5000 can be formed with the digits 1, 2, 5, 9, 0 when repetition of digits is not allowed? Solution: As the number has to be greater than 5000, the first digit can either be 5 or 9. Hence, it can be filled only in two ways. Number of ways for filling the second digit = 4 Number of ways for filling the third digit = 3 (as repetition is not allowed) Number of ways for filling the fourth digit = 2 Total numbers $=2 \times 4 \times 3 \tim...
Read More →How many different numbers of six digits can be formed from the digits 3, 1, 7, 0, 9, 5
Question: How many different numbers of six digits can be formed from the digits 3, 1, 7, 0, 9, 5 when repetition of digits is not allowed? Solution: The first digit cannot be zero. Thus, the first digit can be filled in 5 ways. Number of ways for filling the second digit = 5 (as repetition of digits is not allowed) Number of ways for filling the third digit = 4 Number of ways for filling the fourth digit = 3 Number of ways for filling the fifth digit = 2 Number of ways for filling the sixth dig...
Read More →The area of a right angled triangle is 165 m2.
Question: The area of a right angled triangle is $165 \mathrm{~m}^{2}$. Determine its base and altitude if the latter exceeds the former by $7 \mathrm{~m}$. Solution: Let the base of the right triangle be $=x$ metres and the altitude $=(x+7)$ metres Then According to question, Areas of the right triangle $=165 \mathrm{~m}^{2}$ And as we know that the area of the right triangle $=\frac{1}{2} \times$ base $\times$ height $\frac{1}{2} \times x \times(x+7)=165$ $x^{2}+7 x=330$ $x^{2}+7 x-330=0$ $x^{...
Read More →How many different numbers of six digits each can be formed from the digits
Question: How many different numbers of six digits each can be formed from the digits 4, 5, 6, 7, 8, 9 when repetition of digits is not allowed? Solution: Number of ways of filling the first digit = 6 Number of ways of filling the second digit = 5 (as repetition is not allowed) Number of ways of filling the third digit = 4 Number of ways of filling the fourth digit =3 Number of ways of filling the fifth digit = 2 Number of ways of filling the sixth digit = 1 Total numbers $=6 \times 5 \times 4 \...
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