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A bag contains 5 red, 8 white and 7 black balls. A ball is drawn at random form the bag.

Question: A bag contains 5 red, 8 white and 7 black balls. A ball is drawn at random form the bag. Find the probability that the drawn ball is (i) red or white (ii) not black (iii) neither white nor black. Solution: GIVEN: A bag contains 5 red, 7 black and 8 white balls and a ball is drawn at random TO FIND: Probability of getting a (i) red or white ball (ii) not black ball (iii) neither white nor black Total number of balls (i) Total number red and white balls are We know that PROBABILITY = Hen...

There are 30 cards, of same size, in a bag on which numbers 1 to 30 are written

Question: There are 30 cards, of same size, in a bag on which numbers 1 to 30 are written. One card is taken out of the bag at random. Find the probability that the number on the selected card is not divisible by 3. Solution: GIVEN: Cards are marked with one of the numbers 1 to 30 are placed in a bag and mixed thoroughly. One card is picked at random. TO FIND: Probability of getting a number not divisible by 3 on the picked card. Total number of cards is 30 Cards marked number not divisible by 3...

In the given figure ABCD is a trapezium such that AL ⊥ DC and BM ⊥ DC. If AB = 7 cm,

Question: In the given figureABCDis a trapezium such thatALDCandBMDC. IfAB= 7 cm,BC=AD= 5 cm andAL=BM= 4 cm, then ar(trap.ABCD)= ? (a) $24 \mathrm{~cm}^{2}$ (b) $40 \mathrm{~cm}^{2}$ (c) $55 \mathrm{~cm}^{2}$ (d) $27.5 \mathrm{~cm}^{2}$ Solution: (b) $40 \mathrm{~cm}^{2}$ In right angledtriangleMBC, we have: $M C=\sqrt{5^{2}-4^{2}}=\sqrt{9}=3 \mathrm{~cm}$ In right angledtriangleADL, we have: $D L=\sqrt{5^{2}-4^{2}}=\sqrt{9}=3 \mathrm{~cm}$ Now,CD = ML + MC + LD= 7 + 3 + 3 = 13 cm $\therefore$ A...

The probability of selecting a green marble at random from a jar that contains only green,

Question: The probability of selecting a green marble at random from a jar that contains only green, white and yellow marbles is 1/4. The probability of selecting a white marble at random from the same jar is 1/3. If this jar contains 10 yellow marbles. What is the total number of marbles in the jar? Solution: GIVEN: A bag contains green, white and yellow marbles. (i) Probability of selecting green marbles = (ii) Probability of selecting white marbles = (iii) The jar contains 10 yellow marbles. ...

The faces of a red cube and a yellow cube are numbered from 1 to 6.

Question: The faces of a red cube and a yellow cube are numbered from 1 to 6. Both cubes are rolled. What is the probability that the top face of each cube will have the same number? Solution: GIVEN: The face of red cube and a yellow cube are marked 1 to 6 TO FIND: Probability of getting the same number on both the cubes Let us first write the all possible events that can occur (1,1), (1,2), (1,3), (1,4), (1,5), (1,6), (2,1), (2,2), (2,3), (2,4), (2,5), (2,6), (3,1), (3,2), (3,3), (3,4), (3,5), ...

What is the probability that a number selected at random

Question: What is the probability that a number selected at random from the number 1, 2, 2, 3, 3, 3, 4, 4, 4, 4 will be their average? Solution: GIVEN: A number is selected from the numbers 1,2,2,3,3,3,4,4,4,4 TO FIND: Probability that the selected number is the average of the numbers Total numbers are 10 Average of numbers is $=\frac{1+2+2+3+3+3+4+4+4+4}{10}$ $=\frac{30}{10}$ $=3$ Total numbers of numbers which are average of these numbers are 3 We know that PROBABILITY = Hence Probability that...

Two parallel sides of a trapezium are 12 cm and 8 cm long and the distance between them is 6.5 cm.

Question: Two parallel sides of a trapezium are 12 cm and 8 cm long and the distance between them is 6.5 cm. The area of the trapezium is (a) $74 \mathrm{~cm}^{2}$ (b) $32.5 \mathrm{~cm}^{2}$ (c) $65 \mathrm{~cm}^{2}$ (d) $130 \mathrm{~cm}^{2}$ Figure Solution: (c) $65 \mathrm{~cm}^{2}$ Area of the trapezium $=\frac{1}{2} \times($ sum of parallel sides $) \times$ distance between them $=\frac{1}{2} \times(12+8) \times 6.5$ $=65 \mathrm{~cm}^{2}$...

Why is tossing a coin considered to be a fair way of deciding which team should choose ends in a game of cricket?

Question: Why is tossing a coin considered to be a fair way of deciding which team should choose ends in a game of cricket? Solution: When we toss a coin then the outcomes have the same probability for its occurrence they are equally likely events. So, the result of an individual coin toss is completely unpredictable....

then show that a, b, c and d are in G.P.

Question: If $\frac{a+b x}{a-b x}=\frac{b+c x}{b-c x}=\frac{c+d x}{c-d x}(x \neq 0)$, then show that $a, b, c$ and $d$ are in G.P. Solution: Given: $\frac{a+b x}{a-b x}=\frac{b+c x}{b-c x}=\frac{c+d x}{c-d x}$ Now, $\frac{a+b x}{a-b x}=\frac{b+c x}{b-c x}$ Applying componendo and dividendo $\Rightarrow \frac{(a+b x)+(a-b x)}{(a+b x)-(a-b x)}=\frac{(b+c x)+(b-c x)}{(b+c x)-(b-c x)}$ $\Rightarrow \frac{2 a}{2 b x}=\frac{2 b}{2 c x}$ $\Rightarrow \frac{a}{b}=\frac{b}{c}$ Similiarly, $\frac{(b+c x)+...

In a class, there are 18 girls and 16 boys.

Question: In a class, there are 18 girls and 16 boys. The class teacher wants to choose one pupil for class monitor. What she does, she writes the name of each pupil on a card and puts them into a basket and mixes thoroughly. A child is asked to pick one card from the basket. What is the probability that the name written on the card is:(i) the name of a girl(ii) the name of a boy? Solution: GIVEN: In a class there are 18 girls and 16 boys, the class teacher wants to choose one name. The class te...

Two customers are visiting a particular shop in the same week

Question: Two customers are visiting a particular shop in the same week (Monday to Saturday). Each is equally likely to visit the shop on any one day as on another. What is the probability that both will visit the shop on:(i) the same day?(ii) Different days?(iii) consecutive days? Solution: GIVEN: Two customers are visiting a particular shop in the same week (Monday to Saturday). Each is equally likely to visit the shop on any one day as on another. TO FIND: Probability that both will visit the...

The lengths of the diagonals of a rhombus are 12 cm and 16 cm.

Question: The lengths of the diagonals of a rhombus are 12 cm and 16 cm. The area of the rhombus is (a) $192 \mathrm{~cm}^{2}$ (b) $96 \mathrm{~cm}^{2}$ (c) $64 \mathrm{~cm}^{2}$ (d) $80 \mathrm{~cm}^{2}$ Solution: (b) $96 \mathrm{~cm}^{2}$ Area of the rhombus $=\frac{1}{2} \times$ product of diagonals $=\frac{1}{2} \times 12 \times 16=96 \mathrm{~cm}^{2}$...

If a, b, c, d and p are different real numbers such that:

Question: Ifa,b,c,dandpare different real numbers such that: (a2+b2+c2)p2 2 (ab+bc+cd)p+ (b2+c2+d2) 0, then show thata,b,canddare in G.P. Solution: $\left(a^{2}+b^{2}+c^{2}\right) p^{2}-2(a b+b c+c d) p+\left(b^{2}+c^{2}+d^{2}\right) \leq 0$ $\Rightarrow\left(a^{2} p^{2}+b^{2} p^{2}+c^{2} p^{2}\right)-2(a b p+b c p+c d p)+\left(b^{2}+c^{2}+d^{2}\right) \leq 0$ $\Rightarrow\left(a^{2} p^{2}-2 a b p+b^{2}\right)+\left(b^{2} p^{2}-2 b c p+c^{2}\right)+\left(c^{2} p^{2}-2 c d p+d^{2}\right) \leq 0$ ...

A game of chance consists of spinning an arrow which is equally

Question: A game of chance consists of spinning an arrow which is equally likely to come to rest pointing to one of the number, 1, 2, 3, ..., 12 as shown in following figure. What is the probability that it will point to: (i) 10?(ii) an odd number?(iii) a number which is multiple of 3?(iv) an even number? Solution: GIVEN: A game of chance consists of spinning an arrow which is equally likely to come to rest pointing number 1,2,3..12 TO FIND: Probability of following Total number on the spin is 1...

A bag contains cards which are numbered form 2 to 90.

Question: A bag contains cards which are numbered form 2 to 90. A card is drawn at random from the bag. Find the probability that it bears(i) a two digit number(ii) a number which is a perfect square. Solution: GIVEN: Cards are marked with one of the numbers 2 to 90 are placed in a bag and mixed thoroughly. One card is picked at random. TO FIND: Probability of getting (i) a two digit number (ii) a number which is a perfect square Total number of cards is. (i) Cards marked two digit starts from 1...

A bag contains 3 red balls and 5 black balls.

Question: A bag contains 3 red balls and 5 black balls. A ball is drawn at random from the bag. What is the probability that the ball drawn is :(i) red(ii) black Solution: GIVEN: A bag contains 3 red, and 5 black balls. A ball is drawn at random TO FIND: Probability of getting a (i) red ball (ii) white ball Total number of balls (i) Total number red balls are 3 We know that PROBABILITY = Hence probability of getting red ball is equal to (ii) Total number of black ball are 5 We know that PROBABIL...

In a GP the 3rd term is 24 and the 6th term is 192.

Question: In a GP the 3rdterm is 24 and the 6thterm is 192. Find the 10thterm. Solution: Let $a$ be the first term and $r$ be the common ratio. $\therefore a_{3}=24$ and $a_{6}=192$ $\Rightarrow a r^{2}=24$ and $a r^{5}=192$ $\Rightarrow \frac{a r^{5}}{a r^{2}}=\frac{192}{24}$ $\Rightarrow r^{3}=8$ $\Rightarrow r^{3}=2^{3}$ $\Rightarrow r=2$ Putting $r=2$ in $a r^{2}=24$ $a(2)^{2}=24$ $\Rightarrow a=6$ Now, $10^{\text {th }}$ term $=a_{10}=a r^{9}$ Putting $a=6$ and $r=2$ in $a_{10}=a r^{9}$ $\R...

Five cards−ten, jack, queen, king, and an ace of diamonds

Question: Five cardsten, jack, queen, king, and an ace of diamonds are shuffled face downwards. One card is picked at random.(i) What is the probability that the card is a queen?(ii) If a king is drawn first and put aside, what is the probability that the second card picked up is the ace? Solution: GIVEN: Five cards-ten, jack, queen, king and Ace of diamond are shuffled face downwards TO FIND: Probability of following Total number of cards are5 (i) Cards which is a queen Total number of Cards wh...

One card is drawn from a well shuffled deck of 52 cards. Find the probability of getting:

Question: One card is drawn from a well shuffled deck of 52 cards. Find the probability of getting:(i) a king of red suit(ii) a face card(iii) a red face card(iv) a queen of black suit(v) a jack of hearts(vi) a spade Solution: GIVEN: One card is drawn from a well shuffled deck of 52 playing cards TO FIND: Probability of following Total number of cards is 52 (i) Cards which are king of red suit are 2 Total number of Cards which are king of red suit is 2 Number of favorable event i.e. Total number...

A black die and a white die are thrown at the same time. Write all the possible outcomes. What is the probability?

Question: A black die and a white die are thrown at the same time. Write all the possible outcomes. What is the probability?(i) that the sum of the two numbers that turn up is 7?(ii) of obtaining a total of 6?(iii) of obtaining a total of 10?(iv) of obtaining the same number on both dice?(v) of obtaining a total more than 9?(vi) that the sum of the two numbers appearing on the top of the dice is 13?(vii) that the sum of the numbers appearing on the top of the dice is less than or equal to 12?(vi...

A bag contains 4 red, 5 black and 6 white balls

Question: A bag contains 4 red, 5 black and 6 white balls. A ball is drawn from the bag at random. Find the probability that the ball drawn is :(a) white(b) red(c) not black(d) red or white Solution: GIVEN: A bag contains 4 red, 5 black and 6white balls and a ball is drawn at random TO FIND: Probability of getting a (i) white ball (ii) red ball (iii) not black ball (iv) red or white Total number of balls (i) Total number white balls are 6 We know that PROBABILITY = Hence probability of getting w...

The 4th term of a G.P. is square of its second term, and the first term is − 3.

Question: The 4th term of a G.P. is square of its second term, and the first term is 3. Find its 7thterm. Solution: Let $r$ be the common ratio of the given G.P.' Then, $a_{4}=\left(a_{2}\right)^{2}$ [Given] Now, $a r^{3}=a^{2} r^{2}$ $\Rightarrow r=a$ $\Rightarrow r=-3$ [Putting $a=-3$ ] $\therefore a_{7}=a r^{6}$ $\Rightarrow a_{7}=(-3)(-3)^{6}$ [Putting $a=-3$ and $r=-3$ ] $\Rightarrow a_{7}=(-3)(-729)$ $\Rightarrow a_{7}=-2187$ Thus, the $7^{\text {th }}$ term of the G.P. is $-2187$....

A bag contains 5 black, 7red and 3 white balls.

Question: A bag contains 5 black, 7red and 3 white balls. A ball is drawn from the bag at random. Find the probability that the ball drawn is :(i) red(ii) black or white(iii) not black Solution: GIVEN: A bag contains 7 red, 5 black and 3 white balls and a ball is drawn at random TO FIND: Probability of getting a (i) Red ball (ii) Black or white ball (iii) Not black ball Total number of balls (i) Total number red balls are 7 We know that PROBABILITY = Hence probability of getting a red ball is eq...

If the probability of winning a game is 0.3,

Question: If the probability of winning a game is 0.3, what is the probability of loosing it? Solution: Given: Probability of winning a game TO FIND: Probability of losing the game CALCULATION:We know that sum of probability of occurrence of an event and probability of non occurrence of an event is 1. $P(E)+P(\bar{E})=1$ $0.3+P(\bar{E})=1$ $P(\bar{E})=1-0.3$ $P(\bar{E})=0.7$ Hence the probability of losing the game is $P(\bar{E})=0.7$...

The midpoints of the sides of a triangle along with any of the vertices as the fourth point makes a parallelogram of area equal to

Question: The midpoints of the sides of a triangle along with any of the vertices as the fourth point makes a parallelogram of area equal to (a) $\frac{1}{2}(\operatorname{ar} \Delta \mathrm{ABC})$ (b) $\frac{1}{3}($ ar $\Delta \mathrm{ABC})$ (c) $\frac{1}{4}($ ar $\Delta \mathrm{ABC})$ Solution: D, E and F are the midpoints of sides BC, AC and AB respectively.On joining FE, we divide△△ABC into 4 triangles of equal area.Also, median of a triangle divides it into two triangles with equal area $\o...

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