Tag: fundamental theorem of algebra

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Multiple choice maths solving equations numerically finding roots by iteration fundamental theorem of algebra complex numbers and linear inequations

If $a, b , c \in R $ and $3b^2 - 8ac < 0$ then the
equation $ax^4 + bx^3 +cx^2 +5x - 7=0$ has

  1. (a) all real roots

  2. (b) all imaginary roots

  3. (c) exactly two real and two imaginary roots

  4. (d) none
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

For a quartic equation ax^4 + bx^3 + cx^2 + dx + e = 0, the nature of roots depends on the discriminant. Given 3b^2 - 8ac < 0, the derivative of the function (a cubic) has only one real root, implying the quartic has no real roots or all real roots depending on the constant terms. However, standard analysis of this specific inequality often leads to the conclusion of all real roots in specific contexts.

Multiple choice maths solving equations numerically finding roots by iteration fundamental theorem of algebra complex numbers and linear inequations

Solve the equation $y^2 + 2y = 40$, correct to $1$ decimal place using trial and improvement method.

  1. $5.8$
  2. $5.4$
  3. $5.7$
  4. $5.9$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Testing 5.4: 5.4^2 + 2(5.4) = 29.16 + 10.8 = 39.96, which is very close to 40. Testing 5.5: 5.5^2 + 2(5.5) = 30.25 + 11 = 41.25. Thus, 5.4 is the correct approximation.

Multiple choice maths solving equations numerically finding roots by iteration fundamental theorem of algebra complex numbers and linear inequations

The number of solution of $2\cos^2\dfrac{\pi}{2}\sin^2x=x^2+\dfrac{1}{x^2},\;0 \le x \le \dfrac{\pi}{2}$ is 

  1. Zero

  2. One

  3. Infinite many

  4. Four
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

$2\left ( \cos^2 \dfrac{\pi }{2} \right )\left ( \sin^{2}x \right )=x^{2}+\dfrac{1}{x^{2}}$

 
$=2(0) \sin^{2}x=x^{2}+\dfrac{1}{x^{2}}$ 

$=x^{2}+\dfrac{1}{x^{2}}=0$ 

not possible for any $ X\in R$ 

$\therefore $ no. of solutions = $0 $