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Find the roots of the equation $x^3 + 2x - 5 = 0$ by Regula falsi method.
Find the roots of the equation $x^3 + 2x - 5 = 0$ by Regula falsi method.
The roots can be determined through iterative calculations. Provide the approximate roots after 3 iterations.
Find the roots of the equation $2x + 3 ext{sin}(x) - 5 = 0$ by Regula falsi method.
Find the roots of the equation $2x + 3 ext{sin}(x) - 5 = 0$ by Regula falsi method.
The roots can be determined through iterative calculations.
Find the real positive root of the equation $\log_{10} x - 1.2 = 0$ lying between 2 and 3 correct up to four decimal places by Regula falsi method.
Find the real positive root of the equation $\log_{10} x - 1.2 = 0$ lying between 2 and 3 correct up to four decimal places by Regula falsi method.
The real positive root is approximately 15.8498.
Using Regula falsi method, find the root of the equation $3x - ext{cos}(x) - 1 = 0$ lying between 0 and 1.
Using Regula falsi method, find the root of the equation $3x - ext{cos}(x) - 1 = 0$ lying between 0 and 1.
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Find the real root of the equation $e^x - 4x = 0$ correct to three decimal places by Newton-Raphson method.
Find the real root of the equation $e^x - 4x = 0$ correct to three decimal places by Newton-Raphson method.
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Find the real root of the equation $x^3 + x - 1 = 0$ correct to three decimal places by Newton-Raphson method.
Find the real root of the equation $x^3 + x - 1 = 0$ correct to three decimal places by Newton-Raphson method.
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Find the real root of the equation $\cos(x) - x \cos(x) = 0$ correct to three decimal places by Newton-Raphson method.
Find the real root of the equation $\cos(x) - x \cos(x) = 0$ correct to three decimal places by Newton-Raphson method.
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Find the real root of the equation $e^{-x} - ext{sin}(x) = 0$ correct to three decimal places by Newton-Raphson method.
Find the real root of the equation $e^{-x} - ext{sin}(x) = 0$ correct to three decimal places by Newton-Raphson method.
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Solve the following equations by Jacobi's iteration method: $4x + y + 3z = 17$, $x + 5y + z = 14$, $2x - y + 8z = 12$.
Solve the following equations by Jacobi's iteration method: $4x + y + 3z = 17$, $x + 5y + z = 14$, $2x - y + 8z = 12$.
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Solve the following equations by Gauss-Seidel method: $27x + 6y - z = 85$, $6x + 15y + 2z = 72$, $x + y + 54z = 110$ up to 3 iterations.
Solve the following equations by Gauss-Seidel method: $27x + 6y - z = 85$, $6x + 15y + 2z = 72$, $x + y + 54z = 110$ up to 3 iterations.
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If $ ext{tanh}(x) = 2/3$, find the value of x and then $ ext{cosh}(2x)$.
If $ ext{tanh}(x) = 2/3$, find the value of x and then $ ext{cosh}(2x)$.
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Solve the equation for real values of x; $17 ext{cosh}(x) + 18 ext{sinh}(x) = 3/2$.
Solve the equation for real values of x; $17 ext{cosh}(x) + 18 ext{sinh}(x) = 3/2$.
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If $ ext{cosh}(eta) = x/2$ and $ ext{cos}(eta) = ext{sinh}(eta) = y/2$, show that cosec($ ext{α} − ieta$) + cosec($ ext{α} + ieta$) = $4x/(x^2 + y^2)$.
If $ ext{cosh}(eta) = x/2$ and $ ext{cos}(eta) = ext{sinh}(eta) = y/2$, show that cosec($ ext{α} − ieta$) + cosec($ ext{α} + ieta$) = $4x/(x^2 + y^2)$.
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If $ ext{cosh}(x) = ext{sec}( heta)$, prove that i) $x = ext{log}( ext{sec}( heta) + an( heta))$ and ii) $ ext{tanh}(x/2) = an( heta/2)$.
If $ ext{cosh}(x) = ext{sec}( heta)$, prove that i) $x = ext{log}( ext{sec}( heta) + an( heta))$ and ii) $ ext{tanh}(x/2) = an( heta/2)$.
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If $u + iv = ext{cosec}( rac{ ext{π}}{4} + ix)$, prove that $(u^2 + v^2)^2 = 2(u^2 - v^2)$.
If $u + iv = ext{cosec}( rac{ ext{π}}{4} + ix)$, prove that $(u^2 + v^2)^2 = 2(u^2 - v^2)$.
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Prove that $ ext{cosh}^{-1}( ext{√}(1 + x^2)) = ext{tanh}^{-1}(x/ ext{√}(1 + x^2))$.
Prove that $ ext{cosh}^{-1}( ext{√}(1 + x^2)) = ext{tanh}^{-1}(x/ ext{√}(1 + x^2))$.
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Separate into real and imaginary parts of $ ext{tan}^{-1}(x + iy)$.
Separate into real and imaginary parts of $ ext{tan}^{-1}(x + iy)$.
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Separate into real and imaginary parts of $ ext{tan}^{-1}(e^{i heta})$.
Separate into real and imaginary parts of $ ext{tan}^{-1}(e^{i heta})$.
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Prove that $ ext{coth}^{-1}(x) = rac{1}{2} ext{log} rac{(x + 1)}{(x - 1)}$.
Prove that $ ext{coth}^{-1}(x) = rac{1}{2} ext{log} rac{(x + 1)}{(x - 1)}$.
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Prove that $ ext{sin}igg[ rac{i ext{log}(a - ib)}{(a + ib)}igg] = rac{2ab}{a^2 + b^2}$.
Prove that $ ext{sin}igg[ rac{i ext{log}(a - ib)}{(a + ib)}igg] = rac{2ab}{a^2 + b^2}$.
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If $ rac{(a + ib)x + iy}{(a + ib)x + iy} = ext{α} + ieta$, find α and β.
If $ rac{(a + ib)x + iy}{(a + ib)x + iy} = ext{α} + ieta$, find α and β.
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Show that for real values of a and b, $e^{2a} ext{cot}^{-1}igg( rac{b}{b - 1}igg) = 1$.
Show that for real values of a and b, $e^{2a} ext{cot}^{-1}igg( rac{b}{b - 1}igg) = 1$.
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Solve the equation $x^5 - x^4 + x^3 - x^2 + x - 1 = 0$.
Solve the equation $x^5 - x^4 + x^3 - x^2 + x - 1 = 0$.
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Show that all the roots of $(x + 1)^6 + (x - 1)^6 = 0$ are given by $-i ext{cot}igg( rac{(2k + 1) ext{π}}{12}igg)$ where $k = 0, 1, 2, 3, 4, 5$.
Show that all the roots of $(x + 1)^6 + (x - 1)^6 = 0$ are given by $-i ext{cot}igg( rac{(2k + 1) ext{π}}{12}igg)$ where $k = 0, 1, 2, 3, 4, 5$.
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Find the cube root of unity. If $ ext{ω}$ is the complex cube root of unity, then prove that $(1 - ext{ω})^6 = -27$.
Find the cube root of unity. If $ ext{ω}$ is the complex cube root of unity, then prove that $(1 - ext{ω})^6 = -27$.
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Express $ ext{sin}(7 heta)$ and $ ext{cos}(7 heta)$ in terms of $ ext{sin}( heta)$ and $ ext{cos}( heta)$.
Express $ ext{sin}(7 heta)$ and $ ext{cos}(7 heta)$ in terms of $ ext{sin}( heta)$ and $ ext{cos}( heta)$.
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Express $ ext{cos}^5( heta)$ in a series of cosines of multiples of $ heta$.
Express $ ext{cos}^5( heta)$ in a series of cosines of multiples of $ heta$.
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Prove that $ ext{cos}^5( heta) ext{sin}^3( heta) = rac{1}{27}( ext{sin}(8 heta) + 2 ext{sin}(6 heta) - 2 ext{sin}(4 heta) - 6 ext{sin}(2 heta))$.
Prove that $ ext{cos}^5( heta) ext{sin}^3( heta) = rac{1}{27}( ext{sin}(8 heta) + 2 ext{sin}(6 heta) - 2 ext{sin}(4 heta) - 6 ext{sin}(2 heta))$.
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Prove that $ ext{cos}^6( heta) - ext{sin}^6( heta) = rac{1}{16}( ext{cos}(6 heta) + 15 ext{cos}(2 heta))$.
Prove that $ ext{cos}^6( heta) - ext{sin}^6( heta) = rac{1}{16}( ext{cos}(6 heta) + 15 ext{cos}(2 heta))$.
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Prove that $ rac{ ext{sin}(6 heta)}{ ext{sin}(2 heta)} = 16 ext{cos}^4( heta) - 16 ext{cos}^2( heta) + 3$.
Prove that $ rac{ ext{sin}(6 heta)}{ ext{sin}(2 heta)} = 16 ext{cos}^4( heta) - 16 ext{cos}^2( heta) + 3$.
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Find the continued product of the roots of $igg( rac{1}{2} + rac{i ext{√}3}{2}igg)^4$.
Find the continued product of the roots of $igg( rac{1}{2} + rac{i ext{√}3}{2}igg)^4$.
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Show that $ ext{sin}^5( heta) = rac{1}{16}( ext{sin}(5 heta) - 5 ext{sin}(3 heta) + 10 ext{sin}( heta))$.
Show that $ ext{sin}^5( heta) = rac{1}{16}( ext{sin}(5 heta) - 5 ext{sin}(3 heta) + 10 ext{sin}( heta))$.
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If $ ext{sin}(6 heta) = a ext{cos}^5( heta) ext{sin}( heta) + b ext{cos}^3( heta) + c ext{cos}( heta) ext{sin}^5( heta)$, find the values of a, b, and c.
If $ ext{sin}(6 heta) = a ext{cos}^5( heta) ext{sin}( heta) + b ext{cos}^3( heta) + c ext{cos}( heta) ext{sin}^5( heta)$, find the values of a, b, and c.
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If $ ext{sin}^4( heta) ext{cos}^3( heta) = a ext{cos}( heta) + b ext{cos}(3 heta) + c ext{cos}(5 heta) + d ext{cos}(7 heta)$, then find a, b, c, and d.
If $ ext{sin}^4( heta) ext{cos}^3( heta) = a ext{cos}( heta) + b ext{cos}(3 heta) + c ext{cos}(5 heta) + d ext{cos}(7 heta)$, then find a, b, c, and d.
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Study Notes
Regula Falsi Method
- The Regula Falsi method is used to find the roots of an equation.
- It first requires an initial interval where the root is expected to lie.
- The root is then approximated using the formula below:
$ x_n = x_{n-1} - \frac{f(x_{n-1})(x{n-1} - x_{n - 2})}{f(x_{n-1}) - f(x_{n-2})}$
- This is the weighted average of two points from the previous iteration, giving an approximation of the next point.
Newton-Raphson Method
- The Newton-Raphson method is an iterative root-finding algorithm.
- This method approximates the root of a real-valued function using the formula below:
$ x_{n+1} = x_n - \frac{f(x_n)}{ f'(x_n)} $
- The method requires the derivative of the function.
- The root is then found through successive iterations.
Jacobi's Iteration Method
- Jacobi's method is employed to solve a system of linear equations.
- It is an iterative method that uses the previous iteration's solution to approximate the next solution.
- For example, the system of equations $Ax = b$ can be rewritten as $x = Cx + d$, where $C$ and $d$ are derived from the original system.
- The iterative formula is:
$ x^{(k+1)} = Cx^{(k)} + d $
- The method continues until successive iterations converge to a solution.
- This method may not converge in some cases.
Gauss-Seidel Method
- Gauss-Seidel is another iterative method for solving a system of linear equations.
- It uses the updated values of the variables calculated in the same iteration to improve the solution in the next step, providing better convergence in general.
- This method may not converge in some cases.
Hyperbolic Functions
- Hyperbolic functions are defined in terms of exponential functions.
- Some important hyperbolic functions are:
- sinh(x) = (ex - e-x)/2
- cosh(x) = (ex + e-x)/2
- tanh(x) = sinh(x)/cosh(x)
- coth(x) = cosh(x)/sinh(x)
- sech(x) = 1/cosh(x)
- csch(x) = 1/sinh(x)
- These functions share properties with trigonometric functions, such as identities and derivatives.
Complex Numbers
- Complex numbers are of the form $a + bi$, where $a$ and $b$ are real numbers and $i$ is the imaginary unit (√-1).
- The real part of a complex number is denoted by 'Re(z)' and the imaginary part by 'Im(z)'.
- Complex numbers can be represented in various forms, such as rectangular form, polar form, and exponential form.
- Rectangular form: a + bi
- Polar form: r(cos θ + isin θ)
- Exponential form: reiθ
De Moivre's Theorem
- De Moivre's Theorem relates the powers of complex numbers expressed in polar form to trigonometric functions.
- It states that:
(cos θ + i sin θ)n = cos(nθ) + i sin(nθ)
- This theorem is useful for calculating powers and roots of complex numbers.
Solving Cubic Equations
- Cubic equations are equations of the form ax3 + bx2 + cx + d = 0, where a, b, c, and d are real numbers.
- It can be solved using various methods, including:
- Factoring: Using the Rational Root Theorem to find factors of the polynomial.
- Cardano's Method: Using the cubic formula to find roots.
Trigonometric Identities
- Trigonometric identities relate trigonometric functions to each other.
- They can be used to simplify expressions, solve equations, and prove other trigonometric identities.
- Some fundamental trigonometric identities include:
- sin2θ + cos2θ = 1
- tanθ = sinθ/cosθ
- cotθ = cosθ/sinθ
- secθ = 1/cosθ
- cscθ = 1/sinθ
Continued Products
- The continued product of a finite number of terms is the product of all the terms.
- Example: The continued product of the roots of the equation $x^2 + 3x + 2 = 0$ is -2.
Finding Roots of Equations
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To find the roots of an equation, we need to find the values of $x$ that satisfy the equation.
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This can be done using various methods, including:
- Graphing the equation: Finding the points where the graph intersects the x-axis.
- Using factoring: Finding the values of $x$ that make each factor equal to zero.
- Applying the quadratic formula: A general formula to solve quadratic equations.
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These topics are important in various fields of mathematics, physics, and engineering.
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Description
This quiz explores three important methods used for finding roots of equations: the Regula Falsi method, the Newton-Raphson method, and Jacobi's iteration method. Each method has its unique approach and application in solving mathematical problems. Test your understanding of these techniques and their formulas!
