A die is thrown once. Find the probability of getting a number less than 3.
Question: A die is thrown once. Find the probability of getting a number less than 3. Solution: GIVEN: A dice is thrown once TO FIND: Probability of getting a number less than 3 Total number on a dice is 6. Number less than 3 are1 and 2 Total number of numbers less than 3 is 2 We know that PROBABILITY = Hence probability of getting a number less than 3 is equal to Hence probability of getting a number less than 3 =...
Read More →The ratio of the sum of first three terms is to that of first 6 terms of a G.P. is 125 : 152.
Question: The ratio of the sum of first three terms is to that of first 6 terms of a G.P. is 125 : 152. Find the common ratio. Solution: Letabe the first term andrbe the common ratio of the G.P. $\therefore S_{3}=a\left(\frac{r^{3}-1}{r-1}\right)$ and $S_{6}=a\left(\frac{r^{6}-1}{r-1}\right)$ Then, according to the question $\frac{S_{3}}{S_{6}}=\frac{a\left(\frac{r^{3}-1}{r-1}\right)}{a\left(\frac{r^{6}-1}{r-1}\right)}$ $\Rightarrow \frac{125}{152}=\frac{r^{3}-1}{r^{6}-1}$$\Rightarrow \frac{125}...
Read More →From a well shuffled pack of cards, a card is drawn at random.
Question: From a well shuffled pack of cards, a card is drawn at random. Find the probability of getting a black queen. Solution: GIVEN: One card is drawn from a well shuffled deck of 52 playing cards TO FIND: Probability of getting a black queen Total number of cards are52. Cards which are black queen are 1 from 2 black suits Total number of black queen cards is We know that PROBABILITY = Hence probability of getting an black queen= Hence probability of getting an black queen =...
Read More →Tickets numbered 1 to 20 are mixed up and then a ticket is drawn
Question: Tickets numbered 1 to 20 are mixed up and then a ticket is drawn at random. What is the probability that the ticket drawn bears a number which is a multiple of 3? Solution: GIVEN: Tickets are marked from 1 to 20 are mixed up. One ticket is picked at random. TO FIND: Probability that the ticket bears a multiple of 3 Total number of cards is 20 Cards marked multiple of 3 number are 3,6,9,12,15,18 Total number of cards marked multiple of 3 are 6 We know that PROBABILITY = Hence probabilit...
Read More →Two coins are tossed simultaneously.
Question: Two coins are tossed simultaneously. What is the probability of getting at least one head? Solution: GIVEN: Two coins are tossed simultaneously. TO FIND: Probability of getting at least one head. When two coins are tossed then the outcome will be TT, HT, TH, HH. Hence total number of outcome is 4. At least one head means 1H or 2H. Hence total number of favorable outcome i.e. at least one head is 3 We know that PROBABILITY = Hence probability of getting at least one head =...
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 an ace? Solution: GIVEN: One card is drawn from a well shuffled deck of 52 playing cards TO FIND: Probability of getting an Ace Total number of cards are52. Cards which are Ace are 1 from each suit Total number of Ace cards is We know that PROBABILITY = Hence probability of getting an Ace= Hence probability of getting an Ace =...
Read More →if
Question: If $A=\left[\begin{array}{lll}0 1 0 \\ 0 0 1 \\ p q r\end{array}\right]$, and $I$ is the identity matrix of order 3 , show that $A^{3}=p \mid+q A+r A^{2}$. Solution: Given : $A=\left[\begin{array}{lll}0 1 0 \\ 0 0 1 \\ p q r\end{array}\right]$ Now, $A^{2}=A A$ $\Rightarrow A^{2}=\left[\begin{array}{lll}0 1 0 \\ 0 0 1 \\ p q r\end{array}\right]\left[\begin{array}{lll}0 1 0 \\ 0 0 1 \\ p q r\end{array}\right]$ $\Rightarrow A^{2}=\left[\begin{array}{lll}0+0+0 0+0+0 0+1+0 \\ 0+0+p 0+0+q 0+...
Read More →The common ratio of a G.P. is 3 and the last term is 486.
Question: The common ratio of a G.P. is 3 and the last term is 486. If the sum of these terms be 728, find the first term. Solution: Here, common ratio,r= 3 nthterm,an= 486 Sn= 728 $a_{n}=486$ $\Rightarrow a r^{n-1}=486$ $\Rightarrow a(3)^{n-1}=486$ $\Rightarrow a(3)^{n}=486 \times 3$ $\Rightarrow a(3)^{n}=1458$ ....(i) Now, $S_{n}=728$ $\Rightarrow 728=a\left(\frac{3^{n}-1}{3-1}\right)$ $\Rightarrow 728=\left\{\frac{a(3)^{n}-a}{2}\right\}$ $\Rightarrow 1456=a(3)^{n-1}-a$ $\Rightarrow 1456=1458-...
Read More →If E¯¯¯ denote the complement or negation of an even E, what is the value of P(E) + P(E¯¯¯) ?
Question: If $\bar{E}$ denote the complement or negation of an even $E$, what is the value of $P(E)+P(\bar{E})$ ? Solution: Given: $\bar{E}$ denotes the complement or negation of an event $E$. TO FIND: $P(E)+P(\bar{E})$ CALCULATION: We know that sum of probability of occurrence of an event and probability of non occurrence of an event is $1 .$ Hence $P(E)+P(E)=1$ Hence $P(E)+P(E)=1$...
Read More →The common ratio of a G.P. is 3 and the last term is 486.
Question: The common ratio of a G.P. is 3 and the last term is 486. If the sum of these terms be 728, find the first term. Solution: Here, common ratio,r= 3 nthterm,an= 486 Sn= 728 $a_{n}=486$ $\Rightarrow a r^{n-1}=486$ $\Rightarrow a(3)^{n-1}=486$ $\Rightarrow a(3)^{n}=486 \times 3$ $\Rightarrow a(3)^{n}=1458$ ....(i) Now, $S_{n}=728$ $\Rightarrow 728=a\left(\frac{3^{n}-1}{3-1}\right)$ $\Rightarrow 728=\left\{\frac{a(3)^{n}-a}{2}\right\}$ $\Rightarrow 1456=a(3)^{n-1}-a$ $\Rightarrow 1456=1458-...
Read More →A die is thrown once. What is the probability of getting an odd number?
Question: A die is thrown once. What is the probability of getting an odd number? Solution: GIVEN: A dice is thrown once TO FIND: Probability of getting an odd number. Total number on a dice is 6. Odd numbers on a dice are 1, 3 and 5 Total number of odd numbers on dice is 3 We know that PROBABILITY = Hence probability of an odd number = Hence probability of getting an odd number =...
Read More →A die thrown once.
Question: A die thrown once. What is the probability of getting a number lying between 2 and 6? Solution: GIVEN: A dice is thrown once TO FIND: Probability of getting a number lying between 2 and 6. Total number on a dice is 6. Number lying between 2 and 6 are 3, 4 and 5 Total number of numbers lying between 2 and 6 is 3 We know that PROBABILITY = Hence probability of getting a number lying between 2 and 6 = Hence probability of getting a number lying between 2 and 6 =...
Read More →A die is thrown once. What is the probability of getting a prime number?
Question: A die is thrown once. What is the probability of getting a prime number? Solution: GIVEN: A dice is thrown once TO FIND: Probability of getting a prime number. Total number on a dice is 6. Prime numbers on a dice are 2, 3 and 5 Total number of prime numbers on dice is 3 We know that PROBABILITY = Hence probability of getting a prime number = Hence probability of getting a prime number =...
Read More →The sum of n terms of the G.P. 3, 6, 12, ... is 381.
Question: The sum ofnterms of the G.P. 3, 6, 12, ... is 381. Find the value ofn. Solution: Here, a = 3 Common ratio,r= 3 Sum ofnterms,Sn= 381 Sn= 3 + 6 + 12 + ... +nterms $\Rightarrow 381=3\left(\frac{2^{n}-1}{2-1}\right)$ $\Rightarrow 381=3\left(2^{n}-1\right)$ $\Rightarrow 127=2^{n}-1$ $\Rightarrow 2^{n}=128$ $\Rightarrow 2^{n}=2^{7}$ $\therefore n=7$...
Read More →A bag contains 5 red, 8 green and 7 white balls,
Question: A bag contains 5 red, 8 green and 7 white balls, One ball is drawn at random from the bag. What is the probability of getting a white ball or a green ball? Solution: GIVEN: A bag contains5 red, 8 green and 7 white balls TO FIND: Probability of getting a white ball or a green ball. Total number of balls Total number of green or white balls We know that PROBABILITY = Hence probability of getting a green or a white ball = Hence probability of getting an green or white ball =...
Read More →One card is drawn from a well shuffled deck of 52 playing cards.
Question: One card is drawn from a well shuffled deck of 52 playing cards. What is the probability of getting a non-face card? Solution: GIVEN: One card is drawn from a well shuffled deck of 52 playing cards TO FIND: Probability of getting a non-face card Total number of cards are52. Cards which are non-face are cards of numbers from 1 to 10 of each suit and there are four suits Total number of non-face cards is We know that PROBABILITY = Hence probability of getting non-faced card = Hence proba...
Read More →How many terms of the sequence
Question: How many terms of the sequence $\sqrt{3}, 3,3 \sqrt{3}, \ldots$ must be taken to make the sum $39+13 \sqrt{3} ?$ Solution: Here, $a=\sqrt{3}$ Common ratio, $r=\sqrt{3}$ Sum of $n$ terms, $S_{n}=39+3 \sqrt{3}$ $S_{n}=\sqrt{3}\left(\frac{(\sqrt{3})^{n}-1}{\sqrt{3}-1}\right)$ $\Rightarrow 39+13 \sqrt{3}=\frac{\sqrt{3}}{(\sqrt{3}-1)}\left\{(\sqrt{3})^{n}-1\right\}$ $\Rightarrow(\sqrt{3})^{n}-1=\frac{(39+13 \sqrt{3})(\sqrt{3}-1)}{\sqrt{3}}$ $\Rightarrow(\sqrt{3})^{n}=1+26$ $\Rightarrow(\sqr...
Read More →Which of the following is a false statement?
Question: Which of the following is a false statement?(a) If the diagonals of a rhombus are 18 cm and 14 cm, then its area is 126 cm2. (b) Area of a $\| \mathrm{gm}=\frac{1}{2} \times$ base $\times$ corresp $o$ nding height. (c) A parallelogram and a rectangle on the same base and between the same parallels are equal in area.(d) If the area of a || gm with one side 24 cm and corresponding heighthcm is 192 cm2, thenh= 8 cm. Solution: (b) Area of a $\| g m=\frac{1}{2} \times$ base $\times$ corresp...
Read More →Cards each marked with one of the numbers 4, 5, 6, ..., 20
Question: Cards each marked with one of the numbers 4, 5, 6, ..., 20 are placed in a box and mixed thoroughly. One card is drawn at random from the box. What is the probability of getting an even number? Solution: GIVEN: Cards are marked with one of the numbers 4, 5, 620 are placed in a box and mixed thoroughly. One card is picked at random. TO FIND: Probability of getting an even number on the picked card. Total number of cards is 20-3=17 Cards marked even number are 4,6,8,10,12,14,16,18,20 Tot...
Read More →Suppose you drop a tie at random on the rectangular region shown
Question: Suppose you drop a tie at random on the rectangular region shown in the given figure. What is the probability that it will land inside the circle with diameter 1 m? Solution: Given:Suppose you drop a tie at random on the rectangular region shown in figure To find:Probability that it will land in inside the circle of diameter 1m Total area of circle with diameter 1 m Area of circle with diameter $1 \mathrm{~m}=\pi\left(\frac{1}{2}\right)^{2}$ $=\frac{\pi}{4} \mathrm{~m}^{2}$ Area of rec...
Read More →Compute the elements a43 and a22 of the matrix:
Question: Compute the elementsa43anda22of the matrix: $A=\left[\begin{array}{lll}0 1 0 \\ 2 0 2 \\ 0 3 2 \\ 4 0 4\end{array}\right]\left[\begin{array}{rr}2 -1 \\ -3 2 \\ 4 3\end{array}\right]\left[\begin{array}{rrrrr}0 1 -1 2 -2 \\ 3 -3 4 -4 0\end{array}\right]$ Solution: We have, Given : $A=\left[\begin{array}{lll}0 1 0 \\ 2 0 2 \\ 0 3 2 \\ 4 0 4\end{array}\right]\left[\begin{array}{cc}2 -1 \\ -3 2 \\ 4 3\end{array}\right]\left[\begin{array}{ccccc}0 1 -1 2 -2 \\ 3 -3 4 -4 0\end{array}\right]$ $...
Read More →Which of the following is a false statement?
Question: Which of the following is a false statement? (a) A median of a triangle divides it into two triangles of equal areas.(b) The diagonals of a || gm divide it into four triangles of equal areas. (c) In a $\triangle A B C$, if $E$ is the midpoint of median $A D$, then $\operatorname{ar}(\Delta B E D)=\frac{1}{4} \operatorname{ar}(\Delta A B C)$. (d) In a trap.ABCD, it is given thatAB||DCand the diagonalsACandBDintersect atO. Then, ar(∆AOB) = ar(∆COD). Solution: (d)In a trap.ABCD, it is giv...
Read More →In the given figure, a square dart board is shown.
Question: In the given figure, a square dart board is shown. The length of a side of the larger square is 1.5 times the length of a side of the smaller square. If a dart is thrown and lands on the larger square. What is the probability that it will land in the interior of the smaller square? Solution: Given:A square dart board is shown. The length of a side of the larger square is 1.5 times the length of a side of the smaller square. If a dart is thrown and lands on the larger square To find:Pro...
Read More →How many terms of the series 2 + 6 + 18 + ... must be taken to make the sum equal to 728?
Question: How many terms of the series 2 + 6 + 18 + ... must be taken to make the sum equal to 728? Solution: Here,a= 2 Common ratio, r= 3 Sum ofnterms, Sn= 728 $S_{n}=2\left(\frac{3^{n}-1}{3-1}\right)$ $\Rightarrow 728=2\left(\frac{3^{n}-1}{2}\right)$ $\Rightarrow 728=3^{n}-1$ $\Rightarrow 3^{n}=729$ $\Rightarrow 3^{n}=3^{6}$ $\therefore n=6$...
Read More →In the given figure, JKLM is a square with sides of length 6 units.
Question: In the given figure,JKLMis a square with sides of length 6 units. PointsAandBare the mid points of sidesKLandLMrespectively. If a point is selected at random from the interior of the square. What is the probability that the point will be chosen from the interior of ΔJAB? Solution: Given:JKLM is a square with sides of length 6units. Points Aand B are the midpoints of sides KL and ML respectively. If a point is selected at random from the interior of the square To find:Probability that t...
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