Tag: complex numbers and linear inequations

Questions Related to complex numbers and linear inequations

What is the value of the sum
$\displaystyle \sum _{ n=2 }^{ 11 }{ \left( { i }^{ n }+{ i }^{ n+1 } \right)  } $ where $i=\sqrt { -1 } $?

maths complex numbers and linear inequations identities of complex numbers powers of imaginary unit i algebra of complex numbers

  1. $i$

  2. $2i$

  3. $-2i$

  4. $1+i$

Show answer
Correct Option: C
Explanation:

$\displaystyle \sum _{n = 1}^{11} (i^{n} + i^{n + 1}) i = \sqrt {-1}$

Now $i^{2} = -1, i^{3} = -i, i^{4} = 1, i^{5} = i$ after this values repeats
$i^{2} + i^{3} + i^{4} + i^{5} = 0$
$\therefore \displaystyle \sum _{n = 2}^{11} i^{n} = i^{11} + i^{12} = -i + 1$
$\therefore \displaystyle \sum _{n = 2}^{11} i^{n} + i^{n + 1} = -2i$.

What are the solutions of the equation $x^2+8x+15=0$? 

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

  1. $3,-5$

  2. $3,5$

  3. $5,-3$

  4. $-5,-3$

Show answer
Correct Option: D
Explanation:

${ x }^{ 2 }+8x+15=0\ { x }^{ 2 }+5x+3x+15=0\ (x+5)(x+3)=0\ x=-5,-3$

If $3$ times the third term of an A.P. is equal to $5$ times the fifth term. Then its $8$ term is

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

  1. $0$

  2. $1$

  3. $2$

  4. $3$

Show answer
Correct Option: A
Explanation:

$3(a+2d)=5(a+4d)\ 3a+6d=5a+20d\ -2a=14d\ a=-7d\ a=+7d=-7d+7d={ 0 }$

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

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

  1. (a) all real roots

  2. (b) all imaginary roots

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

  4. (d) none

Show answer
Correct Option: A

The solution to the equation ${7}^{1+x}+{7}^{1-x}=50$ is

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

  1. $0$

  2. $\pm 1$

  3. $2$

  4. none of these

Show answer
Correct Option: B
Explanation:

$7^{1+x}+7^{1-x}=50$

$7(7^{x}+7^{-x})=50$
$7(7^{2{x}})+7=50(7^{x})$
$\implies (7^{x}-7)(7^{x+1}-1)=0$
$\implies 7^{x}=7^{1}$ or $7^{x+1}=7^{0}\implies x=\pm 1$

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

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

  1. $5.8$

  2. $5.4$

  3. $5.7$

  4. $5.9$

Show answer
Correct Option: B

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 

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

  1. Zero

  2. One

  3. Infinite many

  4. Four

Show answer
Correct Option: A
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 $

find the value of $f(2)$ if $f(x)=x^3+x^2+x+1$

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

  1. 15

  2. 12

  3. 10

  4. None of these

Show answer
Correct Option: A
Explanation:

$f(x)=x^3+x^2+x+1\f(2)=2^3+2^2+2+1\f(2)=8+4+2+1=15$

The value of $p$ if $\dfrac 3p+\dfrac 4p=1$

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

  1. 7

  2. 3

  3. 4

  4. 1

Show answer
Correct Option: A
Explanation:

$\dfrac 3p+\dfrac 4p=1\3+4=p\p=7$

Simplify $(3x-11y) -(17x+13y)$ and choose the right answer. 

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

  1. $7x - 12y$

  2. $14x - 54y$

  3. $-3 (5x-4y)$

  4. $-2 (7x+12y)$

Show answer
Correct Option: D
Explanation:
$\left(3x-11y\right)-\left(17x+13y\right)$

$=3x-11y-17x-13y$

$=-14x-24y$

$=-2\left(7x+12y\right)$