CBSE Study Material
black and 
red balls. Find the probability that one of them is black and other is redAs per the given question,
![\left[\begin{array}{ccc} 0 & 1 & 2 \\ 1 & 2 & 3 \\ 3 & 1 & 1 \end{array}\right]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-9e97cd7932741e795ff74dfcc62f2080_l3.png "Rendered by QuickLaTeX.com")
Solution: We have $A=\left(\begin{array}{lll}0 & 1 & 2 \\ 1 & 2 & 3 \\ 3 & 1 & 1\end{array}\right)$ To get the inverse we will proceed by augmented matrix with elementary row...
![\left[\begin{array}{ll} 6 & 7 \\ 8 & 9 \end{array}\right]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-720c2b2a18ed5a12137848fe6835b85c_l3.png "Rendered by QuickLaTeX.com")
Solution: We have $A=\left(\begin{array}{ll}6 & 7 \\ 8 & 9\end{array}\right)$. To get the inverse we will proceed by augmented matrix with elementary row transformation process is as follow:...
black and red balls. Find the probability that (i) Both balls are red (ii) First ball is black and second is red.As per the given question,
![\left[\begin{array}{ll} 4 & 0 \\ 2 & 5 \end{array}\right]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-00e067d8b3d687152537a9d78c193c7f_l3.png "Rendered by QuickLaTeX.com")
Solution: We have $A=\left(\begin{array}{ll}4 & 0 \\ 2 & 5\end{array}\right)$. To get the inverse we will proceed by augmented matrix with elementary row transformation process is as follow:...
![\left[\begin{array}{cc} 2 & -3 \\ 4 & 5 \end{array}\right]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-6d086db9247d6fa2dd93801088be8f5f_l3.png "Rendered by QuickLaTeX.com")
Solution: We have $A=\left(\begin{array}{cc}2 & -3 \\ 4 & 5\end{array}\right)$ To get the inverse we will proceed by augmented matrix with elementary row transformation process is as follow:...
As per the given question,
![\left[\begin{array}{rr} 2 & 5 \\ -3 & 1 \end{array}\right]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-4e7a3750c7e99f5875f5fe5e581bc3bc_l3.png "Rendered by QuickLaTeX.com")
Solution: We have $A=\left(\begin{array}{cc}2 & 5 \\ -3 & 1\end{array}\right)$ To get the inverse we will proceed by augmented matrix with elementary row transformation process is as follow:...
to 
and the odds in favour of another event, independent to the former are 6 to 5 . Find the probability that (a) at least one of the events will occur, and (b) none of the events will occur.As per the given question,
![\left[\begin{array}{cc} 1 & 2 \\ 2 & -1 \end{array}\right]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-918a8e2a33fbf14a8141e6d365aa3df9_l3.png "Rendered by QuickLaTeX.com")
Solution: We have $A=\left(\begin{array}{cc}1 & 2 \\ 2 & -1\end{array}\right)$ To get the inverse we will proceed by augmented matrix with elementary row transformation process is as follow:...
![\left[\begin{array}{ll} 1 & 2 \\ 3 & 7 \end{array}\right]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-6a1968e1a662e8c4373a2fec80dd99d1_l3.png "Rendered by QuickLaTeX.com")
Solution: We have $A=\left(\begin{array}{ll}1 & 2 \\ 3 & 7\end{array}\right)$ To get the inverse we will proceed by augmented matrix with elementary row transformation process is as follow:...
, Three independent trials of the experiment are performed. What is the probability that E occurs at least once?
The probability of occurrence of an event $E$ in one trial is $0.4$ Using Bernoulli's Trial we have, $\begin{array}{l} P(\text { Success }=x)={ }^{n} C_{x} \cdot p^{x} \cdot q^{(n-x)} \\ x=0,1,2,...
at an enemy plane moving away from it. The probabilities of hitting the plane at the first, second, third and fourth shot are 
and 
respectively. What is the probability that the gun hits the plane?As per the given question,
. What is the probability of the safe arrival of at least 3 ships?
The probability of safe arrival of the ship is $1 / 5$ Using Bernoulli's Trial we have, $\begin{array}{l} P(\text { Success }=x)={ }^{n} C_{x} \cdot p^{x} \cdot q^{(n-x)} \\ x=0,1,2, \ldots \ldots...
hits a target is 
and the probability that 
hits it, is 
What is the probability that the target will be hit, if each one of and shoots at the target?As per the given question,
. He tries five times. What is the probability that he will hit the target at least 3 times?
The probability that the man hits the target is $3 / 4$ Using Bernoulli's Trial we have, $\begin{array}{l} P(\text { Success }=x)={ }^{n} C_{x} \cdot p^{x} \cdot q^{(n-x)} \\ x=0,1,2, \ldots \ldots...
and 
. In the process of manufacture of the part 
out of 
parts may be defective. Similarly, out of are likely to be defective in the manufacture of part . Calculate the probability that the assembled product will not be defective.As per the given question,
Using Bernoulli's Trial $P($ Success $=x)={ }^{n} C_{x} \cdot p^{x} \cdot q^{(n-x)}$ $x=0,1,2, \ldots \ldots . . n$ and $q=(1-p)$, here $n=7$ As we know that the favourable outcomes of getting at...
As per the given question,
and 
balls. One ball is drawn from it at random. Its color is noted and then it is put back in the bag. A second draw is made and the same procedure is repeated. Find the probability of drawing first red and second black ball.As per the given question,
Using Bernoulli's Trial $P($ Success $=x)={ }^{n} C_{x} \cdot p^{x} \cdot q^{(n-x)}$ $x=0,1,2, \ldots \ldots . . n \text { and } q=(1-p)$ As we know that the favourable outcomes of getting at least...
and balls. One ball is drawn from it at random. Its color is noted and then it is put back in the bag. A second draw is made and the same procedure is repeated. Find the probability of drawing (i) two red balls, (ii) two black ballsAs per the given question,
Using Bernoulli's Trial $P($ Success $=x)={ }^{n} C_{x} \cdot p^{x} \cdot q^{(n-x)}$ $x=0,1,2, \ldots \ldots . . n$ and $q=(1-p)$ As the die is thrown 5 times the total number of outcomes will be...
on the first toss and 
or 4 on the second toss.As per the given question,
Using Bernoulli's Trial $P($ Success $=x)={ }^{n} C_{x} \cdot p^{x} \cdot q^{(n-x)}$ $x=0,1,2, \ldots \ldots . . \mathrm{n}$ and $\mathrm{q}=(1-\mathrm{p})$ As the coin is tossed 8 times the total...
can solve a problem is 
and the probability that can solve the same problem is 
. Find the probability that none of the two will be able to solve the problem.As per the given question,
Using Bernoulli's Trial $P($ Success $=x)={ }^{n} C_{x} \cdot p^{x} \cdot q^{(n-x)}$ $x=0,1,2, \ldots \ldots \ldots . .$ and $q=(1-p)$ As the coin is tossed 5 times the total number of outcomes will...
on each toss.As per the given question,
Using Bernoulli's Trial $P($ Success $=x)={ }^{n} C_{x} \cdot p^{x} \cdot q^{(n-x)}$ $x=0,1,2, \ldots \ldots . . n$ and $q=(1-p)$ As the coin is tossed 5 times the total number of outcomes will be...
![A=[123]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-10126ffe9d6b60c09e88f578316796d6_l3.png "Rendered by QuickLaTeX.com")
, write 
.Solution: Given that $A=\left[\begin{array}{ll}1 & 23\end{array}\right]$ We will find $A$ ' to calculate AA': $\mathrm{A}^{\prime}=\left[\begin{array}{l} 1 \\ 2 \\ 3 \end{array}\right]$ Now...
![A=\left[\begin{array}{cc}\cos \alpha & \sin \alpha \\ -\sin \alpha & \cos \alpha\end{array}\right]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-b5c8c8d44a123b24c17830efc625a77b_l3.png "Rendered by QuickLaTeX.com")
, show that 
.Solution: Given that $A=\left[\begin{array}{cc}\cos \alpha & \sin \alpha \\ -\sin \theta & \cos \alpha\end{array}\right]$. We wil find $A$ $A^{\prime}=\left[\begin{array}{cc}\cos \alpha...
Using Bernoulli's Trial $P($ Success $=x)={ }^{n} C_{x} \cdot p^{x} \cdot q^{(n-x)}$ $x=0,1,2, \ldots \ldots \ldots$ and $q=(1-p)$ As the coin is thrown 6 times the total number of outcomes will be...
and are two independent events such that 
and 
find 
As per the given question,
and , verify that 
: ![A=\left[\begin{array}{rrr}-1 & 2 & -3 \\ 4 & -5 & 6\end{array}\right]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-4b8a6371313bb6ec237c7ac8ff4e7f9d_l3.png "Rendered by QuickLaTeX.com")
and ![B=\left[\begin{array}{cc}3 & -4 \\ 2 & 1 \\ -1 & 0\end{array}\right]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-d5b55b58779f0457b2c521b7bc825595_l3.png "Rendered by QuickLaTeX.com")
Solution: Take $\mathrm{C}=\mathrm{AB}$ $\begin{array}{l} C=\left[\begin{array}{ccc} -1 & 2 & -3 \\ 4 & -5 & 6 \end{array}\right]\left[\begin{array}{cc} 3 & -4 \\ 2 & 1 \\ -1...
and , verify that 
: ![\mathrm{A}=\left[\begin{array}{c} -1 \\ 2 \\ 3 \end{array}\right] \text { and } \mathrm{B}=[-2 \cdot 1-4]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-86cfd44f3d05fdc26f7ec6f9a464ad20_l3.png "Rendered by QuickLaTeX.com")
Solution: Take $C=A B$ $\begin{array}{l} C=\left[\begin{array}{c} -1 \\ 2 \\ 3 \end{array}\right]\left[\begin{array}{lll} -2 & -1 & -4 \end{array}\right] \\...
or 4 on the second toss?
A die is tossed twice, The probability of getting a 4,5 or 6 in the first trial is $3 / 6=\mathrm{P}(\mathrm{A})$ The probability of getting a $1,2,3$ or 4 in the second trial is $4 / 6=P(B)$ As the...
and are two independent events. The probability that and occur is 
and the probability that neither of them oocurs is . Find the probability of occurrence of two events.As per the given question, ......(i)
and , verify that : ![A=\left[\begin{array}{rr} 3 & -1 \\ 2 & -2 \end{array}\right] \text { and } B=\left[\begin{array}{rr} 1 & -3 \\ 2 & -1 \end{array}\right]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-3f4483ea07db74116ba0bc5d55287e5b_l3.png "Rendered by QuickLaTeX.com")
Solution: Take $C=A B$ $\begin{array}{l} C=\left[\begin{array}{rr} 3 & -1 \\ 2 & -2 \end{array}\right]\left[\begin{array}{ll} 1 & -3 \\ 2 & -1 \end{array}\right] \\...
and , verify that : ![A=\left[\begin{array}{ll} 1 & 3 \\ 2 & 4 \end{array}\right] \text { and } B=\left[\begin{array}{ll} 1 & 4 \\ 2 & 5 \end{array}\right]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-c3667d21de9c5134248177485655ffeb_l3.png "Rendered by QuickLaTeX.com")
Solution: Take $\mathrm{C}=\mathrm{A} 8$ $\begin{array}{l} C=\left[\begin{array}{ll} 1 & 3 \\ 2 & 4 \end{array}\right]\left[\begin{array}{ll} 1 & 4 \\ 2 & 5 \end{array}\right] \\...
The couple has two children and one is known to be boy, The probability that the other is boy will be $=$ $\frac{Favourable-outcome}{Total-outcome}$ Total outcomes are 3 , The first child is a boy,...
![A=\left[\begin{array}{lll}3 & 2 & 5 \\ 4 & 1 & 3 \\ 0 & 6 & 7\end{array}\right]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-7aaa077b9058e56f42a554ca8fcb171f_l3.png "Rendered by QuickLaTeX.com")
as sum af two matrices such that and is symmetric and the other is skew-symmetric.Solution: Given that $A=\left[\begin{array}{lll}3 & 2 & 5 \\ 4 & 1 & 3 \\ 0 & 6 & 7\end{array}\right]$, to express as sum of symmetric matrix $P$ and skew symmetric matrix...
![A=\left[\begin{array}{ccc}3 & -1 & 0 \\ 2 & 0 & 3 \\ 1 & -1 & 2\end{array}\right]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-1b1e9a45433f3d96b64a151e3b79bd27_l3.png "Rendered by QuickLaTeX.com")
.Solution: Given that $A=\left[\begin{array}{ccc}3 & -1 & 0 \\ 2 & 0 & 3 \\ 1 & -1 & 2\end{array}\right]$, to express as sum of symmetric matrix $P$ and skew symmetric matrix...
and are two independent events such that 
and 
, then find .As per the given question,
![\mathrm{A}=\left[\begin{array}{ccc}-1 & 5 & 1 \\ 2 & 3 & 4 \\ 7 & 0 & 9\end{array}\right]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-7350e822cb973005882ae131c2364e63_l3.png "Rendered by QuickLaTeX.com")
as the sum of a symmetric and a skew-symmetric matrix.Solution: Given that $\mathrm{A}=\left[\begin{array}{ccc}-1 & 5 & 1 \\ 2 & 3 & 4 \\ 7 & 0 & 9\end{array}\right]$, to express as sum of symmetric matrix $\mathrm{P}$ and skew...
![\mathrm{A}=\left[\begin{array}{rr}3 & -4 \\ 1 & -1\end{array}\right]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-4c224184d619670975e11f1ad1098530_l3.png "Rendered by QuickLaTeX.com")
as the sum of a symmetric matrix and a skew-symmetric matrix.Solution: Given that $\mathrm{A}=\left[\begin{array}{rr}3 & -4 \\ 1 & -1\end{array}\right]$,to express as the sum of symmetric matrix $\mathrm{P}$ and skew symmetric matrix Q. $A=P+Q$ Where...
of the students read mathematics, 
biology and 
both mathematics and biology. One student is selected at random. What is the probability that he reads mathematics if it is known that he reads biology?
Given: $60 \%$ of the students read mathematics, $25 \%$ biology and $15 \%$ both mathematics and biology That means, Let the event A implies students reading mathematics, Let the event B implies...
![\mathrm{A}=\left[\begin{array}{cc}2 & 3 \\ -1 & 4\end{array}\right]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-0327e38dfafec712de874ce9c4347484_l3.png "Rendered by QuickLaTeX.com")
as the sum of a symmetric matrix and a skew-symmetric matrix.Solution: Given that $A=\left[\begin{array}{cc}2 & 3 \\ -1 & 4\end{array}\right]$, As for a symmetric matrix $A^{\prime}=A$ hence $\begin{array}{l} A+A^{\prime}=2 A \\...
, and and are independent events, then find 
.As per the given question,
If the sum if even, what is the probability that both numbers are odd?
The sum will be even when; both numbers are either even or odd, i.e. for both numbers to be even, the total cases ${ }^{5} \mathrm{C}_{1} \mathrm{X}^{4} \mathrm{C}_{1}$ (Both the numbers are odd)...
![\mathrm{A}=\left[\begin{array}{ccc}0 & \mathrm{a} & \mathrm{b} \\ -\mathrm{a} & 0 & \mathrm{c} \\ -\mathrm{b} & -\mathrm{c} & 0\end{array}\right]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-46d4fca5a1e4ccd3daf4e4683a818d9d_l3.png "Rendered by QuickLaTeX.com")
is skew-symmetric.Solution: We have $A=\left(\begin{array}{ccc}0 & a & b \\ -a & 0 & c \\ -b & -c & 0\end{array}\right)$. The transpose of the matrix is an operation of making interchange of...
![A=\left[\begin{array}{ll}3 & -4 \\ 1 & -1\end{array}\right]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-16f776216ed4662f25a9fa186c592e2f_l3.png "Rendered by QuickLaTeX.com")
, show that 
is skew-symmetric.Solution: We have $A=\left(\begin{array}{ll}3 & -4 \\ 1 & -1\end{array}\right)$ The transpose of the matrix is an operation of making interchange of elements by the rule on positioned...
and such that 
and 
, Find 
Given that $A\;and\;B$ are independent events and $P(A)\;=\;0.3,\; P(B)\;=\;0.6$
![A=\left[\begin{array}{ll}4 & 1 \\ 5 & 8\end{array}\right]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-2cd8607c366b843d16c49d92af8d1f55_l3.png "Rendered by QuickLaTeX.com")
, show that is symmetric.Solution: We have $A=\left(\begin{array}{ll}4 & 1 \\ 5 & 8\end{array}\right)$. The transpose of the matrix is an operation of making interchange of elements by the rule on positioned element...
and such that and , Find (i) 
(ii) 
Given that $A\;and\;B$ are independent events and $P(A)\;=\;0.3,\; P(B)\;=\;0.6$ (i) (ii)
The die is thrown twice, So the favourable outcomes that the sum appears to be 7 are $(1,6),(2,5),(3,4),(4,3),(5,2)$ and $(6,1)$ Out of these 2 appears twice, So the probability that 2 appears at...
![P=\left[\begin{array}{rr}3 & 4 \\ 2 & -1 \\ 0 & 5\end{array}\right]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-25a68e7dd1de3e3d940408c0f29c6ce9_l3.png "Rendered by QuickLaTeX.com")
and ![P=\left[\begin{array}{cc}7 & -5 \\ -4 & 0 \\ 2 & 6\end{array}\right]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-0609e63fb51547a4b3ca4db5d7e945bf_l3.png "Rendered by QuickLaTeX.com")
, verify that 
Solution: We have $P=\left(\begin{array}{cc}3 & 4 \\ 2 & -1 \\ 0 & 5\end{array}\right)$ and $Q=\left(\begin{array}{cc}7 & -5 \\ -4 & 0 \\ 2 & 6\end{array}\right)$. The...
and are two events such that 
and 
, then the events and B areGiven, $\begin{array}{l} P(A \cup B)=\left(\frac{5}{6}\right), P(A \cap B)=\left(\frac{1}{3}\right) \text { and } \\ P(\bar{B})=\left(\frac{1}{2}\right), P(B)=1-P(\bar{B})=1-\frac{1}{2}=\frac{1}{2}...
![A=\left[\begin{array}{rrr}3 & 2 & -1 \\ -5 & 0 & -6\end{array}\right]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-c32e17bbb774b334da0bf3fafe18b27e_l3.png "Rendered by QuickLaTeX.com")
and ![B=\left[\begin{array}{ccc}-4 & -5 & -2 \\ 3 & 1 & 8\end{array}\right]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-733024809307a4305930fcb87f2b032f_l3.png "Rendered by QuickLaTeX.com")
, verify that 
.Solution: We have $A=\left(\begin{array}{ccc}3 & 2 & -1 \\ -5 & 0 & -6\end{array}\right)$ and $B=\left(\begin{array}{ccc}-4 & -5 & -2 \\ 3 & 1 & 8\end{array}\right)$....
Answer : For the equation of line equidistant from both lines, we will find point through which line passes and is equidistant from both line. As any point lying on x = - 2 line is ( - 2, 0) and on...
![A=\left[\begin{array}{cc}3 & 5 \\ -2 & 0 \\ 4 & -6\end{array}\right]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-1b5da973e1eb1c39298d165a510a5b93_l3.png "Rendered by QuickLaTeX.com")
, verify that 
.Solution: We have $A=\left(\begin{array}{cc}3 & 5 \\ -2 & 0 \\ 4 & -6\end{array}\right)$. Thus $2 A=\left(\begin{array}{cc}6 & 10 \\ -4 & 0 \\ 8 & -12\end{array}\right)$ The...
and such that and , Find 
Given that $A\;and\;B$ are independent events and $P(A)\;=\;0.3,\; P(B)\;=\;0.6$ (i) (ii)
![A=\left[\begin{array}{ccc}2 & -3 & 5 \\ 0 & 7 & -4\end{array}\right]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-6ac0f76ea62d4473fa0142e884721096_l3.png "Rendered by QuickLaTeX.com")
, verify that 
.Solution: We have $A=\left(\begin{array}{ccc}2 & -3 & 5 \\ 0 & 7 & -4\end{array}\right)$. The transpose of the matrix is an operation of making interchange of elements by the rule on...
and are independent events, then 
$\mathrm{P}(\overline{\mathrm{A}} / \overline{\mathrm{B}})=\frac{P(\bar{A} \cap \bar{B})}{P(\bar{B})}=\frac{P(\bar{A}) P(\overline{\bar{B})}}{1-P(B)}=1-P(A)$ Hence, the correct option is a.
and such that and , Find 
Given that $A\;and\;B$ are independent events and $P(A)\;=\;0.3,\; P(B)\;=\;0.6$
and are events such that 
and 
, then 
$\begin{array}{l} P(A)=0.4, P(B)=0.8 \text { and } \\ P(B / A)=0.6 \\ P(B / A)=\frac{P(A \cap B)}{P(A)}=0.6 \\ P(A \cap B)=0.24 \\ \Rightarrow P(A / B)=\frac{P(A \cap B)}{P(B)}=0.3 \end{array}$...
and be two events such that 
and 
, show that and are independent events.As per the given question,
and 
be defined as follows: 
first toss is head, 
second toss is head, and 
exactly two heads are tossed in a row. 
As per the given question,
Answer : Equation of line parallel to y - axis (vertical) is given by x = constant, as x - coordinate is constant for every point lying on line i.e. 6. So, the required equation of line is given as...
and be defined as follows: first toss is head, second toss is head, and exactly two heads are tossed in a row. 
As per the given question,
Answer : Equation of line parallel to x - axis (horizontal) is y = constant, as y - coordinate of every point on the line parallel to x - axis is - 2 i.e. constant. Therefore equation of the line...
Answer: Equation of line parallel to x - axis is given by y = constant, as x - coordinate of every point on the line parallel to y - axis is - 3 i.e. constant. So, the required equation of line is y...
Answer : (i) Equation of line parallel to y - axis is given by x = constant, as the x - coordinate of every point on the line parallel to y - axis is 6 i.e. constant. Now the point lies to the right...
Answer : (i) Equation of line parallel to x - axis is given by y = constant, as the y - coordinate of every point on the line parallel to x - axis is 4,i.e. constant. Now the point lies above x -...
so that each card is equally likely to be selected. State whether events and are independent if, the card drawn is spade, the card drawn in an aceAs per the given question,
and are independent if, the card drawn is a king or queen the card drawn is a queen or jack (ii) A card is drawn from a pack of 52 cards so that each card is equally likely to be selected. State whether events and are independent if, the card drawn is black, the card drawn is a kingAs per the given question, (i) (ii)
on the first die is independent of the occurrence of on the second die.As per the given question,
and are independent if, the number of heads is two, the last throw results in headAs per the given question,
and are independent if, the first throw results in head, the last throw results in tail (ii) A coin is tossed thrice and all the eight outcomes are assumed equally likely. State whether events and are independent if, the number of heads is odd, the number of tails is oddAs per the given question, So, $A\;and\;B$ are independent events. (ii)
As per the given question, (i) Let $'A'$ be the event that both the children born are girls. Let $'B'$ be the event that the youngest is a girl. We have to find conditional probability $P(A/B).$...
through are placed in a box, mixed up thoroughly and then one card is drawn randomly. If it is known that the number on the drawn card is more than 
what is the probability that it is an even number?As per the given question,
out of which 
are girls. It is known that out of 
of the girls study in class 
. What is the probability that a student chosen randomly studies in class given that the chosen student is a girl?As per the given question,
, the probability that he will buy a trouser is , and the probability that he will buy a shirt given that he buys a trouser is . Find the probability that he will buy both a shirt and a trouser. Find also the probability that he will buy a trouser given that he buys a shirt.As per the given question,
, and the probability that he/she passes in Mathematics and Computer Science is . What is the probability that he/she will pass in Computer Science if it is known that he/she has passed in Mathematics?As per the given question,
be the event that the sum is greater than or equal to 10 and 
be the event “5 appears on the first-die”. Find 
. If is the event “5 appears on at least one die”, find .As per the given question,
Find the probability that the sum of the numbers showing on the dice is 
As per the given question,
What is the conditional probability that the number has appeared at least one?As per the given question, A dice is thrown twice,
through 
If the sum is even find the probability that both the numbers are odd.As per the given question,
given that at least one die does not show five.As per the given question,
or more, if appears on the first die.As per the given question,
As per the given question,
as the sum if it is known that the second die always exhibits a prime number.As per the given question,
as the sum, if it is known that the second die always exhibits an odd number.As per the given question,
As per the given question,
What is the conditional probability that the number has appeared at least once?The sample space for the experiment is ${(1,1),(1,2),(1,3)...(6,6)}$ consisting of $36$ outcomes
and are two events qiven by 
There are three person for photograph father (F), mother (M), son (S).
and , if 
appears on the third toss 
and appear respectively on first two tosses.As per the given question,
in each of the following: (i) Tail appears on one coin, One coin shows head. (ii) No tail appears, No head appears.As per the given question,
in each of the following: At most two tails, At least one tail.As per the given question,
in each of the following: (i) Heads on third toss, Heads on first two tosses (ii) At least two heads, At most two headsAs per the given question,
and 
, find As per the given question,
and , find (i) 
(ii) As per the given question
and are two events such that 
and 
, find As per the given question,
![\[y=\frac{\cos x+\sin x}{\cos x-\sin x},\frac{dy}{dx}={{\sec }^{2}}(\frac{\pi }{4}-x)\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-7436b98af8cd47b2abb43b4a3f7dc219_l3.png "Rendered by QuickLaTeX.com")
![\[y=\frac{\cos x-\sin x}{\cos x+\sin x},\frac{dy}{dx}+{{y}^{2}}+1=0\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-144a2057d1d1974a98c2ebfe8519871c_l3.png "Rendered by QuickLaTeX.com")
![\[\frac{\sin x+{{x}^{2}}}{\cot 2x}\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-8c58976b85869547c9c27aefb8620919_l3.png "Rendered by QuickLaTeX.com")
![\[\sin \left( \frac{1+{{x}^{2}}}{1-{{x}^{2}}} \right)\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-4ac248d7d5c0004794d384b4ceddd0ac_l3.png "Rendered by QuickLaTeX.com")
![\[\cos 4x.\cos 2x\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-48a05688c711aee46d03749c1e0bdc87_l3.png "Rendered by QuickLaTeX.com")
![\[\sin 2x.\sin x\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-a0bb224cdd697fc6c576f75e2148c59f_l3.png "Rendered by QuickLaTeX.com")
![\[\sin 5x.\cos 3x\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-569c60085ef4405b107f96e043a291e8_l3.png "Rendered by QuickLaTeX.com")
![\[\sqrt{\cos ec({{x}^{3}}+1)}\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-d2f55737d6d0d56ab4c78578c7770187_l3.png "Rendered by QuickLaTeX.com")
![\[\cos (\sin \sqrt{ax+b})\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-a40472eeabfe7df2d39ef6f6ced13db1_l3.png "Rendered by QuickLaTeX.com")
$=\frac{1}{2}\int{_{0}^{\frac{\pi }{4}}(cox-\cos 5x)dx}$ $=\frac{1}{2}\left[ \sin x-\frac{\sin 5x}{5} \right]$ $=\frac{1}{2}\left[ \sin \left( \frac{\pi }{4} \right)-\frac{\sin \left( \frac{5\pi...
![\[{{\cot }^{3}}{{x}^{2}}\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-f6b7ac37dc2363d44a27e0dd1121840c_l3.png "Rendered by QuickLaTeX.com")
$\int{_{\frac{\pi }{2}}^{\frac{\pi }{2}}\frac{dx}{2{{\sin }^{2}}x}=\int{_{\frac{\pi }{2}}^{\frac{\pi }{2}}\frac{1}{2}\frac{dx}{2}\cos e{{c}^{2}}xdx}}$ $\int{_{\frac{\pi }{2}}^{\frac{\pi...
![\[\sqrt{\cot \sqrt{x}}\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-7a257d90b638c09c1b8ccf7139357ecb_l3.png "Rendered by QuickLaTeX.com")
![\[\sqrt{x\sin x}\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-4e7ae5831720f4f9c886685c46541cb8_l3.png "Rendered by QuickLaTeX.com")
![\[\sqrt{\sin {{x}^{3}}}\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-3b18fd99216cbeff4882c0c46216032e_l3.png "Rendered by QuickLaTeX.com")
$\int{_{0}^{\frac{\pi }{4}}\frac{dx}{1+\cos 2x}=\int{_{0}^{\frac{\pi }{4}}\frac{1}{2}{{\sec }^{2}}xdx}}$ $\int{_{0}^{\frac{\pi }{4}}\frac{1}{2}{{\sec }^{2}}xdx=\frac{1}{2}[\tan x]}$...
![\[\cos e{{c}^{3}}\frac{1}{{{x}^{2}}}\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-c933cdeb92722edb8ddc1d37ea8d97b8_l3.png "Rendered by QuickLaTeX.com")
Let $u=\left( \tan \left( \frac{x}{2} \right)+1 \right)$ $dx=\frac{2}{{{\sec }^{2}}\left( \frac{x}{2} \right)}du$ $=-\frac{2}{u}$ $=-\frac{2}{\tan \left( \frac{x}{2} \right)+1}$ $=2$
and are two events such that 
(ii)If and are two events such that 
and 
, find 
and 
(i) As per the given question, (ii) As per the given question,
$\int{_{0}^{\frac{\pi }{4}}(\cos x-\sin x)dx}$ $=[\sin x+\cos x]$ $=\left[ \cos \left( \frac{\pi }{4} \right)+\sin \left( \frac{\pi }{4} \right)-\cos 0-\sin 0 \right]$ $=\left[...
$=\sqrt{2}\left| \sin x \right|$ $\sqrt{2}\left| \sin \left( \frac{\pi }{4} \right)-\sin 0 \right|$ $\sqrt{2}\left[ \frac{1}{\sqrt{2}} \right]$ $=1$
$\frac{1}{4}\int{_{0}^{\frac{\pi }{2}}(3\sin x-\sin 3x)dx=\frac{1}{4}\left[ -3\cos x+\frac{\cos 3x}{3} \right]}$ $=\frac{1}{4}\left[ -3\cos \left( \frac{\pi }{2} \right)+\frac{\cos \left( \frac{3\pi...
$\frac{1}{4}\int{_{0}^{\frac{\pi }{3}}(3\cos x-\cos 3x)dx=\frac{1}{4}\left[ 3\sin x+\frac{\sin 3x}{3} \right]}$ $\frac{1}{4}\left[ 3\sin \left( \frac{\pi }{3} \right)+\frac{\sin \pi }{3}...
$=-\log \left| \cos ec\left( \frac{\pi }{4} \right)+\cot \left( \frac{\pi }{4} \right) \right|+\log \left| \cos ec\left( \frac{\pi }{6} \right)+\cot \left( \frac{\pi }{6} \right) \right|$ $=-\log...
and are two events such that 
(ii)If and are two events such that 
(i) As per the given question, (ii) As per the given question,
$=2\log (2)-(2)-1\log (1)+(1)$ $=2\log (2)-1$
$=\int{_{0}^{\frac{\pi }{2}}\frac{{{x}^{2}}}{2}(\cos (2x)+\frac{{{x}^{2}}}{2})dx}$ $\int{_{0}^{\frac{\pi }{2}}\frac{{{x}^{2}}}{2}(\cos (2x)+\frac{{{x}^{2}}}{2})dx}=\frac{{{x}^{2}}\sin...
$=-\frac{{{x}^{3}}\cos (3x)}{3}+\frac{{{x}^{2}}\sin (3x)}{3}-\int{\frac{2x\sin x(3x)}{3}dx}$ $=-\frac{{{x}^{3}}\cos (3x)}{3}+\frac{{{x}^{2}}\sin (3x)}{3}+\frac{2x\cos (3x)}{9}-\frac{2\sin (3x)}{27}$...
Find and .As per the given question,
$\int{_{0}^{\frac{\pi }{2}}{{x}^{2}}\sin (2x)dx=\left[ \frac{{{x}^{2}}\sin (2x)}{2}-\frac{\sin (2x)}{2}+\frac{x\cos (2x)}{2} \right]}$ $\left[ \frac{{{\left( \frac{\pi }{2} \right)}^{2}}\sin (\pi...
and are two events such that 
and 
, find .As per the given question,
From integrate by parts: $\int{_{0}^{\frac{\pi }{4}}{{x}^{2}}\sin (x)dx=[-{{x}^{2}}\cos (x)+2\sin (x)+2\cos (x)]}$ $=[2x\sin (x)+(2-{{x}^{2}})\cos (x)]$ $=\left[ \frac{\pi }{2}\sin \left( \frac{\pi...
and are events such that 
and 
, find and 
As per the given question,
$$$\int{_{0}^{\frac{\pi }{2}}{{x}^{2}}\cos (x)dx=[{{x}^{2}}\sin (x)-2x\cos (x)]}$ $=\left[ {{\left( \frac{\pi }{4} \right)}^{2}}\sin \left( \frac{\pi }{2} \right)-2\sin \left( \frac{\pi }{2}...