SOLUTION:- According to question, the solution should be:
Express each of the complex number given in the Exercises 1 to 10 in the form a + ib. (1 – i) – (–1 + i6)
\[\begin{array}{*{35}{l}} \left( 1\text{ }-\text{ }i \right)-\text{ }\text{ }\left( \text{ }1\text{ }+\text{ }i6 \right)\text{ }=\text{ }1\text{ }-\text{ }i\text{ }+\text{ }1\text{ }-\text{ }i6 \\...
Express each of the complex number given in the Exercises 1 to 10 in the form a + ib. 3(7 + i7) + i(7 + i7)
\[3(7\text{ }+~i7)\text{ }+~i(7\text{ }+~i7)~\] \[=\text{ }21\text{ }+~i21\text{ }+~i7\text{ }+~{{i}^{2~}}7\] \[=\text{ }21\text{ }+~i28\text{ }\text{ }7~\] \[[{{i}^{2}}~=\text{ }-1]\] \[=\text{...
Expand each of the expressions in Exercises 1 to 5.
From binomial hypothesis extension we can compose as
Express each of the complex number given in the Exercises 1 to 10 in the form a + ib. i-39
\[{{i}^{-39}}~=\text{ }1/\text{ }{{i}^{39}}~=\text{ }1/\text{ }{{i}^{4\text{ }x\text{ }9\text{ }+\text{ }3}}~\] \[=\text{ }1/\text{ }({{1}^{9}}~x\text{ }{{i}^{3}})\text{ }=\text{ }1/\text{...
Expand each of the expressions in Exercises 1 to 5.
(1 – 2x)5 Solution: From binomial theorem expansion we can write as (1 – 2x)5 = 5Co (1)5 – 5C1 (1)4 (2x) + 5C2 (1)3 (2x)2 – 5C3 (1)2 (2x)3 + 5C4 (1)1 (2x)4 – 5C5 (2x)5 = 1 – 5 (2x) + 10 (4x)2 – 10...
Express each of the complex number given in the Exercises 1 to 10 in the form a + ib. i9 + i19
\[{{i}^{9}}~+\text{ }{{i}^{19}}~=\text{ }{{({{i}^{2}})}^{4}}.\text{ }i\text{ }+\text{ }{{({{i}^{2}})}^{9}}.\text{ }i\] \[=\text{ }{{\left( -1 \right)}^{4}}~.\text{ }i\text{ }+\text{ }{{\left( -1...
Express each of the complex number given in the Exercises 1 to 10 in the form a + ib. 1. (5i) (-3/5i)
$(5i)(-3/5i)=5x(-3/5)x{{i}^{2}}$ $=-3x-1[{{i}^{2}}=-1]$ \[=\text{ }3\] Consequently, \[\left( 5i \right)\text{ }\left( -\text{ }3/5i \right)\text{ }=\text{ }3\text{ }+\text{ }i0\]
Prove the following by using the principle of mathematical induction for all n ∈ N: (2n +7) < (n + 3)2
We can compose the given assertion as \[P\left( n \right):\text{ }\left( 2n\text{ }+7 \right)\text{ }<\text{ }\left( n\text{ }+\text{ }3 \right)2\] In the event that \[n\text{ }=\text{ }1\]we get...
Prove the following by using the principle of mathematical induction for all n ∈ N: 41n – 14n is a multiple of 27
We can compose the given assertion as $P(n):{{41}^{n}}-{{14}^{n}}$is a various of \[27\] In the event that \[n\text{ }=\text{ }1\]we get $P(1)={{41}^{1}}-{{14}^{1}}=27$, which is a various by...
Prove the following by using the principle of mathematical induction for all n ∈ N: 32n + 2 – 8n – 9 is divisible by 8
We can compose the given assertion as $P(n):{{3}^{2}}n+2-8n-9$is separable by \[8\] In the event that \[n\text{ }=\text{ }1\]we get $P(1)={{3}^{(2\text{ }\!\!\times\!\!\text{ }1)}}+2-8\text{ ...
Prove the following by using the principle of mathematical induction for all n ∈ N: x^2n – y^2n is divisible by x + y
. We can compose the given assertion as $P(n):{{x}^{2}}n-{{y}^{2}}n$is distinguishable by \[x\text{ }+\text{ }y\] In the event that \[n\text{ }=\text{ }1\]we get $P(1)={{x}^{2}}\text{ ...
Prove the following by using the principle of mathematical induction for all n ∈ N: 10^(2n – 1) + 1 is divisible by 11
We can compose the given assertion as $P(n):{{10}^{2}}n-1+1$is distinct by \[11\] On the off chance that \[n\text{ }=\text{ }1\]we get $P(1)={{10}^{2}}.1-1+1=11$, which is distinct by \[11\] Which...
Prove the following by using the principle of mathematical induction for all n ∈ N: n (n + 1) (n + 5) is a multiple of 3
We can compose the given assertion as \[P\text{ }\left( n \right):\text{ }\left( n\text{ }+\text{ }1 \right)\text{ }\left( n\text{ }+\text{ }5 \right)\], which is a different of \[3\] In the...
Prove the following by using the principle of mathematical induction for all n ∈ N:
SOLUTION:- We can compose the given assertion as \[P\text{ }\left( k\text{ }+\text{ }1 \right)\]is valid at whatever point \[P\text{ }\left( k \right)\]is valid. Thus, by the rule of numerical...
Prove the following by using the principle of mathematical induction for all n ∈ N:
SOLUTION:- \[P\text{ }\left( k\text{ }+\text{ }1 \right)\]is valid at whatever point \[P\text{ }\left( k \right)\]is valid. Thus, by the rule of numerical enlistment, articulation \[P\text{ }\left(...
Prove the following by using the principle of mathematical induction for all n ∈ N:
SOLUTION:- \[P\text{ }\left( k\text{ }+\text{ }1 \right)\]is valid at whatever point \[P\text{ }\left( k \right)\]is valid. Thus, by the rule of numerical enlistment, articulation \[P\text{ }\left(...
Prove the following by using the principle of mathematical induction for all n ∈ N:
SOLUTION:- \[P\text{ }\left( k\text{ }+\text{ }1 \right)\]is valid at whatever point \[P\text{ }\left( k \right)\]is valid. Thus, by the rule of numerical enlistment, articulation \[P\text{ }\left(...
Prove the following by using the principle of mathematical induction for all n ∈ N:
SOLUTION:- By additional improvement \[=\text{ }\left( \mathbf{k}\text{ }+\text{ }\mathbf{1} \right)\text{ }+\text{ }\mathbf{1}\] \[P\text{ }\left( k\text{ }+\text{ }1 \right)\]is valid at whatever...
Prove the following by using the principle of mathematical induction for all n ∈ N:
SOLUTION:- \[P\text{ }\left( k\text{ }+\text{ }1 \right)\]is valid at whatever point \[P\text{ }\left( k \right)\]is valid. Thus, by the rule of numerical enlistment, articulation \[P\text{ }\left(...
Prove the following by using the principle of mathematical induction for all n ∈ N:
SOLUTION:- \[P\text{ }\left( k\text{ }+\text{ }1 \right)\]is valid at whatever point \[P\text{ }\left( k \right)\]is valid. Thus, by the rule of numerical enlistment, articulation \[P\text{ }\left(...
Prove the following by using the principle of mathematical induction for all n ∈ N:
SOLUTION:- \[P\text{ }\left( k\text{ }+\text{ }1 \right)\]is valid at whatever point \[P\text{ }\left( k \right)\]is valid. Thus, by the rule of numerical enlistment, articulation \[P\text{ }\left(...
Prove the following by using the principle of mathematical induction for all n ∈ N:
SOLUTION:- \[P\text{ }\left( k\text{ }+\text{ }1 \right)\]is valid at whatever point \[P\text{ }\left( k \right)\]is valid. Thus, by the rule of numerical enlistment, articulation \[P\text{ }\left(...
Prove the following by using the principle of mathematical induction for all n ∈ N:
SOLUTION:- We can compose the given assertion as \[P\text{ }\left( k\text{ }+\text{ }1 \right)\]is valid at whatever point \[P\text{ }\left( k \right)\]is valid. Thus, by the rule of numerical...
Prove the following by using the principle of mathematical induction for all n ∈ N: 1.2 + 2.22 + 3.22 + … + n.2n = (n – 1) 2n+1 + 2
We can compose the given assertion as \[\mathbf{P}\text{ }\left( \mathbf{n} \right):\text{ }\mathbf{1}.\mathbf{2}\text{ }+\text{ }\mathbf{2}.\mathbf{22}\text{ }+\text{ }\mathbf{3}.\mathbf{22}\text{...
Prove the following by using the principle of mathematical induction for all n ∈ N:
SOLUTION:- \[P\text{ }\left( k\text{ }+\text{ }1 \right)\]is valid at whatever point \[P\text{ }\left( k \right)\]is valid. Thus, by the rule of numerical enlistment, articulation \[P\text{ }\left(...
Prove the following by using the principle of mathematical induction for all n ∈ N:
SOLUTION:- \[P\text{ }\left( k\text{ }+\text{ }1 \right)\]is valid at whatever point \[P\text{ }\left( k \right)\]is valid. Thus, by the rule of numerical enlistment, articulation \[P\text{ }\left(...
Prove the following by using the principle of mathematical induction for all n ∈ N:
SOLUTION:- \[P\text{ }\left( k\text{ }+\text{ }1 \right)\]is valid at whatever point \[P\text{ }\left( k \right)\]is valid. Thus, by the rule of numerical enlistment, articulation \[P\text{ }\left(...
Prove the following by using the principle of mathematical induction for all n ∈ N:
SOLUTION:- \[P\text{ }\left( k\text{ }+\text{ }1 \right)\]is valid at whatever point \[P\text{ }\left( k \right)\]is valid. Thus, by the rule of numerical enlistment, articulation \[P\text{ }\left(...
Prove the following by using the principle of mathematical induction for all n ∈ N:
SOLUTION:- \[P\text{ }\left( k\text{ }+\text{ }1 \right)\]is valid at whatever point \[P\text{ }\left( k \right)\]is valid. Thus, by the rule of numerical enlistment, articulation \[P\text{ }\left(...
Prove the following by using the principle of mathematical induction for all n ∈ N:
SOLUTION:- \[P\text{ }\left( k\text{ }+\text{ }1 \right)\]is valid at whatever point \[P\text{ }\left( k \right)\]is valid. Thus, by the rule of numerical enlistment, articulation \[P\text{ }\left(...
Prove the following by using the principle of mathematical induction for all n ∈ N:
Solution: \[P\text{ }\left( k\text{ }+\text{ }1 \right)\]is valid at whatever point \[P\text{ }\left( k \right)\]is valid. Thus, by the rule of numerical enlistment, articulation \[P\text{ }\left( n...
Find the degree measures corresponding to the following radian measures (Use π = 22/7) (i) 11/16 (ii) -4
\[\left( i \right)\]$$ \[11/16\] Here \[\pi \text{ }radian\text{ }=\text{ }180{}^\circ \] \[\left( ii \right)\text{ }-4\] Here \[\pi \text{ }radian\text{ }=\text{ }180{}^\circ \]
Find the radian measures corresponding to the following degree measures: (iii) \[\mathbf{240}{}^\circ \] (iv) \[\mathbf{520}{}^\circ \]
(iv) $$ \[520{}^\circ \]
Find the radian measures corresponding to the following degree measures: (i) \[\mathbf{25}{}^\circ \] (ii) \[\text{ }\mathbf{47}{}^\circ \text{ }\mathbf{30}\prime \]
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2. $1000$ families with $2$ children were selected randomly, and the following data were recorded:
Number of girls in a family$0$$1$$2$Number of families$333$$392$$275$ Find the probability of a family, having (iii) no girl Solution:- The extent to which an event is likely to occur and measured...