Tag: maths

Questions Related to maths

If $a+b+c=6$ and $ ab+bc+ca = 11 $
Find $\left( { a }^{ 2 }+{ b }^{ 2 }+{ c }^{ 2 } \right)$ ?
maths squares and square roots finding the square of a number finding square of a number patterns in square numbers

  1. $14$

  2. $25$

  3. $36$

  4. $47$

Show answer
Correct Option: A
Explanation:
Given $a+b+c=6$ 

$(a+b+c)^2=36$

$(a^2+b^2+c^2)+2(ab+ba+ca)=36$

$a^2+b^2+c^2+(2\ast 11)=36$

$a^2+b^2+c^2=36-22=14$

Evaluating the following :
$(3+\sqrt{2})^{5}-(3-\sqrt{2})^{5}$

maths squares and square roots finding the square of a number finding square of a number patterns in square numbers

  1. $1718\sqrt 3$

  2. $1718\sqrt 2$

  3. $1178\sqrt 3$

  4. $1178\sqrt 2$

Show answer
Correct Option: D
Explanation:

Given term is $(3+\sqrt{2})^{5}-(3-\sqrt{2})^{5}$


$\Rightarrow 2\left[\ ^{5}C _{1}\times 3^{4}\times (\sqrt{2})^{1}+\ ^{5}C _{3}\times 3^{2}\times (\sqrt{2})^{3}+\ ^{5}C _{5}\times 3^{0}\times (\sqrt{2})^{5}\right]$

$\Rightarrow 2\left[5\times 81\times\sqrt{2}+10\times 9\times 2\sqrt{2}+4\sqrt{2}\right]$

$\Rightarrow 2\sqrt{2}(405+180+4)$

$\Rightarrow 1178\sqrt{2}$

Evaluating the following :
$(1+2\sqrt{x})^{5}+(1-2\sqrt{x})^{5}$

maths squares and square roots finding the square of a number finding square of a number patterns in square numbers

  1. $2(1+40x^2+80x)$

  2. $2(1-40x+81x^2)$

  3. $2(1+40x+80x^2)$

  4. None of these

Show answer
Correct Option: C
Explanation:
Given to evaluate is $(1+2\sqrt x)^5 +(1-2\sqrt x)^5$

$\Rightarrow 2[^{5}C _{0} (2\sqrt x)^0 +^5C _2 (2\sqrt x)^2 +^5C _4 (2\sqrt x)^4]$

$\Rightarrow 2[1+10\times 4x+5\times 16x^2]$

$\Rightarrow 2[1+40x+80x^2]$

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:

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