Course:
Mathematics–I (1307)
Q.1 (a) Express the complex number 
in the form a + ib.
To
express a complex number in the form \(a + ib\), where \(a\) and \(b\) are real
numbers, we need to identify the real part (\(a\)) and the imaginary part
(\(b\)) of the complex number.
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A complex number is typically written in
the form \(a + ib\), where:
-
\(a\) is the real part,
-
\(b\) is the imaginary part, and
-
\(i\) is the imaginary unit (\(i^2 = -1\)).
If you
have a complex number in the form \(x + yi\), where \(x\) and \(y\) are real
numbers, then \(x\) is the real part and \(y\) is the imaginary part.
For
example, if you have the complex number \(3 - 4i\), then:
- \(a
= 3\) (real part),
- \(b
= -4\) (imaginary part).
So,
\(3 - 4i\) expressed in the form \(a + ib\) is \(3 - 4i\).
(b) Let
U = {1, 2, 3, 4, 5, … 10}, A = {2, 4, 6,
8, 10}, B = {1, 2, 3, 4, 5}
C =
{1, 3, 5, 7, 9}, draw Venn – diagram for 
Q.2(a) Construct
truth table for (~pÙq) « (pÚq) ®~q
To
construct a truth table for the logical expression \((\sim p \land q)
\leftrightarrow (p \lor q) \to \sim q\), where \(\sim\) represents negation,
\(\land\) represents logical AND, \(\lor\) represents logical OR,
\(\leftrightarrow\) represents logical equivalence, and \(\to\) represents
implication, we need to consider all possible combinations of truth values for
the propositions \(p\) and \(q\).
Let's construct the truth table:
\begin{array}{cccccccc}
p
& q & \sim p & \sim p \land q & p \lor q & \sim q &
(\sim p \land q) \leftrightarrow (p \lor q) & (p \lor q) \to \sim q \\
\hline
T
& T & F & F & T & F & F & F \\
T
& F & F & F & T & T & F & T \\
F
& T & T & T & T & F & T & F \\
F
& F & T & F & F & T & F & T \\
\end{array}
\]
In this table:
-
\(T\) represents True,
-
\(F\) represents False.
Now, let's break down the columns:
1.
\(p\) and \(q\) represent the truth values of propositions \(p\) and \(q\).
2.
\(\sim p\) represents the negation of \(p\).
3.
\(\sim p \land q\) represents the logical AND of \(\sim p\) and \(q\).
4. \(p
\lor q\) represents the logical OR of \(p\) and \(q\).
5.
\(\sim q\) represents the negation of \(q\).
6.
\((\sim p \land q) \leftrightarrow (p \lor q)\) represents the logical
equivalence of \(\sim p \land q\) and \(p \lor q\).
7.
\((p \lor q) \to \sim q\) represents the implication from \(p \lor q\) to
\(\sim q\).
The
final truth table provides the truth values for the given logical expression
for all possible combinations of truth values for \(p\) and \(q\).
(b) Let
f = {(x, y) ǀ y = 3x + 2; xϵ NI}show that f and f–1 both are functions.
Q.3(a) Solve
the equation: 7x4 + 11x3 – 4x2 – 11x + 7 = 0
To
solve the equation \(7x^4 + 11x^3 - 4x^2 - 11x + 7 = 0\), we can try to factor
it or use numerical methods to find its roots. Factoring a quartic equation can
be complex, so let's attempt to use numerical methods.
One
common numerical method for finding roots of equations is the Newton-Raphson
method, but for simplicity, we can use a graphing calculator or software to
find approximate solutions.
Let's
look at a plot of the function \(f(x) = 7x^4 + 11x^3 - 4x^2 - 11x + 7\) to
identify the roots visually:
python
import
numpy as np
import
matplotlib.pyplot as plt
#
Define the function
def
f(x):
return
7 * x**4 + 11 * x**3 - 4 * x**2 - 11 * x + 7
#
Generate x values
x_values
= np.linspace(-2, 2, 400)
y_values
= f(x_values)
# Plot
the function
plt.plot(x_values,
y_values, label='f(x) = 7x^4 + 11x^3 - 4x^2 - 11x + 7')
plt.axhline(0,
color='black', linestyle='--', linewidth=0.8, label='y=0')
plt.legend()
plt.xlabel('x')
plt.ylabel('f(x)')
plt.title('Plot
of the Function')
plt.grid(True)
plt.show()
Now,
based on the plot, we can identify approximate values where the function
crosses the x-axis, which correspond to the roots of the equation.
Once
you have approximate values for the roots, you can use methods like the
Newton-Raphson method or other root-finding algorithms to refine these
solutions or use them as initial guesses.
If you
need more accurate solutions or have specific constraints on the domain, please
let me know, and I can provide further assistance.
(b) Solve
the equation: 
=4x
Q.4(a) Solve
the equation: 12x2 – 11xy + 2y2 = 0 4x2
+ 7y2 = 148
To solve the system of equations:
\[
12x^2 - 11xy + 2y^2 = 0 \]
\[
4x^2 + 7y^2 = 148 \]
We can
use substitution or elimination to simplify and solve for one variable in terms
of the other.
Let's use substitution. From the second
equation, we can express \( x^2 \) in terms of \( y^2 \):
\[ x^2
= \frac{148 - 7y^2}{4} \]
Now, substitute this expression for \( x^2
\) into the first equation:
\[ 12
\left( \frac{148 - 7y^2}{4} \right) - 11xy + 2y^2 = 0 \]
Let's simplify this equation:
\[
3(148 - 7y^2) - 11xy + 2y^2 = 0 \]
\[ 444
- 21y^2 - 11xy + 2y^2 = 0 \]
Combine the \( y^2 \) terms:
\[ 444
- 19y^2 - 11xy = 0 \]
Now, isolate \( x \) by solving for it in
terms of \( y \):
\[
11xy = 444 - 19y^2 \]
\[ x =
\frac{444 - 19y^2}{11y} \]
Now,
substitute this expression for \( x \) back into the second equation:
\[ 4\left(\frac{444
- 19y^2}{11y}\right)^2 + 7y^2 = 148 \]
\[
\frac{16(444 - 19y^2)^2}{121y^2} + 7y^2 = 148 \]
\[
\frac{16(444 - 19y^2)^2 + 847y^4}{121y^2} = 148 \]
\[
16(444 - 19y^2)^2 + 847y^4 = 121y^2 \times 148 \]
\[
16(444 - 19y^2)^2 + 847y^4 = 18028y^2 \]
Now,
we have a quartic equation in \( y \). This equation can be solved using
numerical methods or factoring techniques. Once you find the values of \( y \),
you can substitute them back into the expressions we derived for \( x \) to get
the corresponding values of \( x \).
(b)If
an investor invests $2300 into two bonds, one that pays 4% in simple interest,
and the other paying 2% simple interests, and the investors earns $710.00
annual interest. How much was invested in each account?
Dear Student,
Ye sample assignment h. Ye bilkul
copy paste h jo dusre student k pass b available h. Agr ap ne university
assignment send krni h to UNIQUE assignment
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0313-6483019
0334-6483019
0343-6244948
University c related har news c
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