Find the probability that a number selected at random from the numbers 3, 4, 4,4 5, 5, 6, 6, 6, 7 will be their mean.
Question: Find the probability that a number selected at random from the numbers 3, 4, 4,4 5, 5, 6, 6, 6, 7 will be their mean. Solution: The number are 3, 4, 4, 4, 5, 5, 6, 6, 6, 7. One number can be selected at random from the given 10 numbers in 10 ways.Total number of outcomes = 10 Mean of the given numbers $=\frac{3+4+4+4+5+5+6+6+6+7}{10}=\frac{50}{10}=5$ Now, there are two 5's among the given numbers. So, there are 2 ways to select the number 5 i.e. the mean of the given numbers.Favourable...
Read More →Two schools P and Q want to award their selected students on the values of Tolerance,
Question: Two schoolsPandQwant to award their selected students on the values of Tolerance, Kindness and Leadership. The schoolPwants to award₹xeach,₹yeach and₹zeach for the three respective values to 3, 2 and 1 students respectively with a total award money of₹2,200.SchoolQwants to spend₹3,100 to award its 4, 1 and 3 students on the respective values (by giving the same award money to the three values as schoolP). If the total amount of award for one prize on each values is₹1,200, using matrice...
Read More →A card is drawn at random form a well-shuffled deck of playing cards.
Question: A card is drawn at random form a well-shuffled deck of playing cards. Find the probability that the card drawn is(i) a card of spades of an ace(ii) a red king(iii) either a king or a queen(iv) neither a king nor a queen. Solution: Total number of all possible outcomes= 52(i) Number of spade cards = 13Number of aces = 4 (including 1 of spade) Therefore, number of spade cards and aces = (13 + 4 1) = 16 $\therefore P($ getting a spade or an ace card $)=\frac{16}{52}=\frac{4}{13}$ (ii) Num...
Read More →Two schools P and Q want to award their selected students on the values of Tolerance,
Question: Two schoolsPandQwant to award their selected students on the values of Tolerance, Kindness and Leadership. The schoolPwants to award₹xeach,₹yeach and₹zeach for the three respective values to 3, 2 and 1 students respectively with a total award money of₹2,200.SchoolQwants to spend₹3,100 to award its 4, 1 and 3 students on the respective values (by giving the same award money to the three values as schoolP). If the total amount of award for one prize on each values is₹1,200, using matrice...
Read More →Find the sum of the lengths of the bases of a trapezium whose area is 4.2 m
Question: Find the sum of the lengths of the bases of a trapezium whose area is 4.2 m2and whose height is 280 cm. Solution: Given: Area of the trapezium $=4.2 \mathrm{~m}^{2}$ Height $=280 \mathrm{~cm}=\frac{280}{100} \mathrm{~m}=2.8 \mathrm{~m}$ Area of trapezium $=\frac{1}{2} \times($ Sum of the parallel bases $) \times($ Height $)$ $4.2=\frac{1}{2} \times($ Sum of the parallel bases $) \times 2.8$ $4.2 \times 2=$ (Sum of the parallel bases) $\times 2.8$ Sum of the parallel bases $=\frac{8.4}{...
Read More →Find the altitude of a trapezium whose area is 65 cm
Question: Find the altitude of a trapezium whose area is 65 cm2and whose bases are 13 cm and 26 cm. Solution: Given: Area of the trapezium $=65 \mathrm{~cm}^{2}$ The lengths of the opposite parallel sides are $13 \mathrm{~cm}$ and $26 \mathrm{~cm}$. Area of trapezium $=\frac{1}{2} \times($ Sum of parallel bases $) \times($ Altitude $)$ On putting $g$ the values: $65=\frac{1}{2} \times(13+26) \times($ Altitude $)$ $65 \times 2=39 \times$ Altitude Altitude $=\frac{130}{39}=\frac{10}{3} \mathrm{~cm...
Read More →Five cards − the ten, jack, queen, king and ace of diamonds are well shuffled with their faces downwards.
Question: Five cards the ten, jack, queen, king and ace of diamonds are well shuffled with their faces downwards. One card is then picked up at random.(a) What is the probability that the drawn card is the queeen?(b) If the queen is drawn and put aside and a second card is drawn, find the probability that the second card is (i) an ace, (ii) a queen. Solution: Total number of cards = 5.(a) Number of queens = 1. $\therefore \mathrm{P}($ getting a queen $)=\frac{\text { Number of favourable outcome...
Read More →Find the height of a trapezium,
Question: Find the height of a trapezium, the sum of the lengths of whose bases (parallel sides) is 60 cm and whose area is 600 cm2. Solution: Given: Sum of the parallel sides of a trapezium $=60 \mathrm{~cm}$ Area of the trapezium $=600 \mathrm{~cm}^{2}$ Area of trapezium $=\frac{1}{2} \times($ Sum of the parallel sides $) \times($ Height $)$ On putting the values : $600=\frac{1}{2} \times 60 \times($ Height $)$ $600=30 \times($ Height $)$ Height $=\frac{600}{30}=20 \mathrm{~cm}$...
Read More →Find the area of a trapezium whose parallel sides are of length 16 dm
Question: Find the area of a trapezium whose parallel sides are of length 16 dm and 22 dm and whose height is 12 dm. Solution: Given: Lengths of the parallel sides are $16 \mathrm{dm}$ and $22 \mathrm{dm}$. And, height between the parallel sides is $12 \mathrm{dm}$. Area of trapezium $=\frac{1}{2} \times($ Sum of the parallel sides $) \times($ Height $)$ $=\frac{1}{2} \times(16+22) \times(12)$ $=228 \mathrm{dm}^{2}$ $=228 \times \mathrm{dm} \times \mathrm{dm}$ $=228 \times \frac{1}{10} \mathrm{~...
Read More →A card is drawn at random from a well-shuffled pack of 52 cards.
Question: A card is drawn at random from a well-shuffled pack of 52 cards. Find the probability that the card drawn is neither a red card nor a queen. Solution: Total number of all possible outcomes= 52There are 26 red cards (including 2 queens) and apart from these, there are 2 more queens.Number of cards, each one of which is either a red card or a queen = 26 + 2 = 28LetEbe the event that the card drawn is neither a red card nor a queen. Then, the number of favourable outcomes = (52 28) = 24 $...
Read More →Find the area of trapezium with base 15 cm and height 8 cm,
Question: Find the area of trapezium with base 15 cm and height 8 cm, if the side parallel to the given base is 9 cm long. Solution: Given: Lengths of the parallel sides are $15 \mathrm{~cm}$ and $9 \mathrm{~cm}$. Height $=8 \mathrm{~cm}$ Area of trapezium $=\frac{1}{2} \times($ Sum of the opposite sides $) \times($ Distance between the parallel sides $)$ $=\frac{1}{2} \times(15+9) \times(8)$ $=96 \mathrm{~cm}^{2}$...
Read More →All kings, queens and aces are removed from a pack of 52 cards.
Question: All kings, queens and aces are removed from a pack of 52 cards. The remaining cards are well-shuffled and then a card is drawn from it. Find the probability that the drawn card is(i) a black face card,(ii) a red card. Solution: There are 4 kings, 4 queens and 4 aces. These are removed.Thus, remaining number of cards = 52 4 4 4= 40.(i) Number of black face cards now = 2 (only black jacks). $\therefore \mathrm{P}($ getting a black face card) $)=\frac{\text { Number of favourable outcomes...
Read More →Prove the following
Question: (x2+1)2 x2= 0 has (a) four real roots (b) two real roots (c) no real roots (d) one real root Solution: (c) Given equation is $\left(x^{2}+1\right)^{2}-x^{2}=0$ $\Rightarrow \quad x^{4}+1+2 x^{2}-x^{2}=0 \quad\left[\because(a+b)^{2}=a^{2}+b^{2}+2 a b\right]$ $\Rightarrow \quad x^{4}+x^{2}+1=0$ Let $x^{2}=y$ $\therefore \quad\left(x^{2}\right)^{2}+x^{2}+1=0$ $y^{2}+y+1=0$ On comparing with $a y^{2}+b y+c=0$, we get $a=1, b=1$ and $c=1$ Discriminant, $D=b^{2}-4 a c$ $=(1)^{2}-4(1)(1)$ $=1...
Read More →Two schools P and Q want to award their selected students on the values of Discipline,
Question: Two schoolsPandQwant to award their selected students on the values of Discipline, Politeness and Punctuality. The schoolPwants to award₹xeach,₹yeach and₹zeach the three respectively values to its 3, 2 and 1 students with a total award money of₹1,000.SchoolQ wants to spend₹1,500 to award its 4, 1 and 3 students on the respective values (by giving the same award money for three values as before). If the total amount of awards for one prize on each value is₹600, using matrices, find the ...
Read More →Find the area, in square metres, of the trapezium whose bases and altitudes are as under:
Question: Find the area, in square metres, of the trapezium whose bases and altitudes are as under: (i) bases = 12 dm and 20 dm, altitude = 10 dm (ii) bases = 28 cm and 3 dm, altitude = 25 cm (iii) bases = 8 m and 60 dm, altitude = 40 dm (iv) bases = 150 cm and 30 dm, altitude = 9 dm. Solution: (i) Given: Bases: $12 \mathrm{dm}=\frac{12}{10} \mathrm{~m}=1.2 \mathrm{~m}$ And, $20 \mathrm{dm}=\frac{20}{10} \mathrm{~m}=2 \mathrm{~m}$ Altitude $=10 \mathrm{dm}=\frac{10}{10} \mathrm{~m}=1 \mathrm{~m}...
Read More →Two schools P and Q want to award their selected students on the values of Discipline,
Question: Two schoolsPandQwant to award their selected students on the values of Discipline, Politeness and Punctuality. The schoolPwants to award₹xeach,₹yeach and₹zeach the three respectively values to its 3, 2 and 1 students with a total award money of₹1,000.SchoolQ wants to spend₹1,500 to award its 4, 1 and 3 students on the respective values (by giving the same award money for three values as before). If the total amount of awards for one prize on each value is₹600, using matrices, find the ...
Read More →All red face cards are removed from a pack of playing cards.
Question: All red face cards are removed from a pack of playing cards. The remaining cards are well shuffled and then a card is drawn at random from them. Find the probability that the drawn card is(i) a red card,(ii) a face card,(iii) a card of clubs. Solution: There are 6 red face cards. These are removed.Thus, remaining number of cards = 52 6 = 46.(i) Number of red cards now = 26 6 = 20. $\therefore P($ getting a red card $)=\frac{\text { Number of favourable outcomes }}{\text { Number of all...
Read More →Which of the following equations
Question: Which of the following equations has no real roots? (a) x2 4x + 32 =0 (b)x2+4x-32=0 (c) x2 4x 32 = 0 (d) 3x2+ 43x + 4=0 Solution: (a) (a) The given equation is $x^{2}-4 x+3 \sqrt{2}=0$. On comparing with $a x^{2}+b x+c=0$, we get $a=1, b=-4$ and $c=3 \sqrt{2}$ The discriminant of $x^{2}-4 x+3 \sqrt{2}=0$ is $D=b^{2}-4 a c$ $=(-4)^{2}-4(1)(3 \sqrt{2})=16-12 \sqrt{2}=16-12 \times(1.41)$ $=16-16.92=-0.92$ $\Rightarrow \quad b^{2}-4 a c0$ (b) Tho givan onuation is $x^{2}+4 x-3 \sqrt{2}=0$ ...
Read More →The king, the jack and the 10 of spades are lost from a pack of 52 cards and a card is drawn from the remaining cards after shuffling.
Question: The king, the jack and the 10 of spades are lost from a pack of 52 cards and a card is drawn from the remaining cards after shuffling. Find the probability of getting a(i) red card(ii) black jack(iii) red king(iv) 10 of hearts. Solution: Total number of cards in a deck is 52.The number of cards left after loosing the King, the Jack and the 10 of spade = 523 = 49.(i) Red card left after loosing King, the Jack and the 10 of spade will be 13 hearts + 13 diamond = 26 cards $P($ red card $)...
Read More →The area of a rhombus is equal to the area of a triangle whose base and the corresponding altitude are 24.8 cm and 16.5 cm respectively.
Question: The area of a rhombus is equal to the area of a triangle whose base and the corresponding altitude are 24.8 cm and 16.5 cm respectively. If one of the diagonals of the rhombus is 22 cm, find the length of the other diagonal. Solution: Given: Area of the rhombus = Area of the triangle with base $24.8 \mathrm{~cm}$ and altitude $16.5 \mathrm{~cm}$ Area of the triangle $=\frac{1}{2} \times$ base $\times$ altitude $=\frac{1}{2} \times 24.8 \times 16.5=204.6 \mathrm{~cm}^{2}$ $\therefore$ A...
Read More →A game consists of tossing a one-rupee coin three times, and noting its outcome each time.
Question: A game consists of tossing a one-rupee coin three times, and noting its outcome each time. If getting the same result in all the tosses is a success, find the probability of losing the game. Solution: When a coin is tossed three times, the possible outcomes areHHH, HHT, HTH, THH, HTT, THT, TTH, TTTTotal number of outcomes = 8A game can be lost if the if all the tosses do not give the same result. The outcomes where tosses do not give the same result areHHT, HTH, THH, HTT, THT, TTHFavou...
Read More →Which of the following equations
Question: Which of the following equations has two distinct real roots? (a) $2 x^{2}-3 \sqrt{2} x+\frac{9}{4}=0$ (b) $x^{2}+x-5=0$ (c) $x^{2}+3 x+2 \sqrt{2}=0$ (d) $5 x^{2}-3 x+1=0$ Solution: (b) The given equation is $x^{2}+x-5=0$ On comparing with $a x^{2}+b x+c=0$, we get $a=1, b=1$ and $c=-5$ The discriminant of $x^{2}+x-5=0$ is $D=b^{2}-4 a c=(1)^{2}-4(1)(-5)$ $=1+20=21$ $\Rightarrow \quad b^{2}-4 a c0$ So, $x^{2}+x-5=0$ has two distinct real roots. (a) Given equation is, $2 x^{2}-3 \sqrt{2...
Read More →A field in the form of a rhombus has each side of length 64 m and altitude 16 m.
Question: A field in the form of a rhombus has each side of length 64 m and altitude 16 m. What is the side of a square field which has the same area as that of a rhombus? Solution: Given: Each side of a rhombus shaped field $=64 \mathrm{~m}$ Altitude $=16 \mathrm{~m}$ We know: Area of rhombus $=$ Side $\times$ Altitude $\therefore$ Area of the field $=64 \times 16=1024 \mathrm{~m}^{2}$ Given: Area of the square field $=$ Area of the rhombus We know: Area of a square $=(\text { Side })^{2}$ $\th...
Read More →A letter is chosen at random from the letters of the word ASSOCIATION.
Question: A letter is chosen at random from the letters of the word ASSOCIATION. Find the probability that the chosen letter is a(i) vowel(ii) consonant(iii) an S. Solution: Total numbers of letters in the given word ASSOCIATION = 11(i) Number of vowels ( A, O, I, A, I, O) in the given word = 6 $\therefore P($ getting a vowel $)=\frac{6}{11}$ (ii) Number of consonants in the given word ( S, S, C, T, N) = 5 $\therefore P$ (getting a consonant) $=\frac{5}{11}$ (iii) Number of S in the given word =...
Read More →A garden is in the form of a rhombus whose side is 30 metres and the corresponding altitude is 16 m.
Question: A garden is in the form of a rhombus whose side is 30 metres and the corresponding altitude is 16 m. Find the cost of levelling the garden at the rate of Rs 2 per m2. Solution: Given: Side of the rhombus shaped garden $=30 \mathrm{~m}$ Altitude $=16 \mathrm{~m}$ Now, area of a rhombus $=$ side $\times$ altitude $\therefore$ Area of the given garden $=30 \times 16=480 \mathrm{~m}^{2}$ Also, it is given that the rate of levelling the garden is Rs 2 per $1 \mathrm{~m}^{2}$. $\therefore$ T...
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