### Introduction to unknowns - class-VI

 Description: introduction to unknowns Number of Questions: 72 Created by: Vijay Palan Tags: numbers and algebra algebraic expression polynomials unchanging relations fundamental concepts real numbers algebra making sense of algebra unknown numbers maths fundamental concepts - algebra variables concept of algebraic variables algebraic expressions and operations on them algebraic expressions introduction to algebra
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Zero degree polynomial is considered as

1. variable

2. coefficent

3. constant

4. zero

Correct Option: C
Explanation:

Zero degree polynomial is considered as constant.
Example: $p(x)= k$
Therefore, $k$ is constant.
The constant term in an expression or equation has a fixed value and does not contain variables.

In ancient times, algebra is used to find

1. polynomial and negative numbers

2. linear equations

4. Both B and C

Correct Option: D
Explanation:

In ancient times, algebra is used to find linear and quadratic equations.

al-Khwarizmi was a ______ scientist.

1. german

2. russian

3. persian

4. italian

Correct Option: C
Explanation:

al-Khwarizmi was a persian scientist.

Determine the constant term in the expression: $4x^2+5x^6-7x^2-7+2x^2-7x^6$.

1. $4$

2. $6$

3. $-7$

4. $7$

Correct Option: C
Explanation:

The constant term in an expression or equation has a fixed value and does not contain variables.
So, $-7$ is the constant term in the expression, $4x^2+5x^6-7x^2-7+2x^2-7x^6$

______ used algebraic equations and notations in presenting problems and solutions in Arithmetica.

1. Rene Descartes

2. Brahmagupta

3. Al-Khwarizmi

4. Diophantus of Alexandria

Correct Option: D
Explanation:

Diophantus of Alexandria used algebraic equations and notations in presenting problems and solutions in Arithmetica.

Find the constant for the given polynomial: $x^3+2x^2-1+x^5-5x(x^2)$

1. $1$

2. $-1$

3. $3$

4. $2$

Correct Option: B
Explanation:

Given polynomial is $x^3+2x^2-1+x^5-5x(x^2)$

The constant term in an expression or equation has a fixed value and does not contain variables.
So, here $-1$ is the constant term.

In a quadratic equation, $3x^2+x-3$, what is the constant term?

1. $3$

2. $2$

3. $-3$

4. $1$

Correct Option: C
Explanation:

A constant term is a value that do not have variable.
So, in this quadratic equation, 3−3 is a  constant term.

$abc=$

1. $Rrs$

2. $4Rr\triangle$

3. $4Rrs$

4. $4Rr\triangle s$

Correct Option: A

State True or False, if the following expression is polynomial in one variable
$4x^2-3x+7$

1. True

2. False

Correct Option: A
Explanation:

Clearly, the expression $4x^2-3x+7$  is a polynomial in one variable x because there is only one variable x in the expression having the highest power 2, which is whole number.

State True or False, if he following expression is polynomial in one variable

$y^2+\sqrt 2$

1. True

2. False

Correct Option: A
Explanation:

Clearly, the expression $y^2+\sqrt 2$  is a polynomial in one variable y because there is only one variable y in the expression having the highest power 2, which is whole number.

State True or False, if the following expression is polynomial in one variable.

$3\sqrt t+t\sqrt 2$

1. True

2. False

Correct Option: B
Explanation:

The expression $3\sqrt t+t\sqrt2$ contain the term $3\sqrt t$, here exponent of $t$ is $\dfrac12$, which is not a whole number.
Therefore, the given expression is not a polynomial in one variable.

State True or False, if the following expression is polynomial in one variable (State reason for your answer):
$y+\dfrac {2}{y}$
1. True

2. False

Correct Option: B
Explanation:

The expression $y+\dfrac2y$ contain the term $\dfrac2y$, here

the exponent of $y$ is $-1$, which is not a whole number.
Therefore, the given expression is not a polynomial in one variable.

State whether true/false:

The following expression is a polynomial in one variable:
$x^{10}+y^3+t^{50}$

1. True

2. False

Correct Option: B
Explanation:

Clearly, the given expression $x^{10}+y^3+t^{50}$ contains three variables $x,y\space and\space t$
Hence, the given expression is not a polynomial in one variable instead in three variables.

If $\displaystyle A=\pi \left ( R^{2}-r^{2} \right )$, then $R$ is equal to

1. $\displaystyle \sqrt{\frac{A-\pi r^{2}}{\pi }}$

2. $\displaystyle \sqrt{\frac{A+\pi r^{2}}{\pi }}$

3. $\displaystyle \sqrt{\frac{r^{2}\pi -A}{\pi }}$

4. $\displaystyle \sqrt{\frac{r^{2}\pi -A}{r}}$

Correct Option: B
Explanation:
Given, $A=\pi(R^2-r^2)$
Therefore, $A =$ $\displaystyle \pi R^{2}-\pi r^{2}$
$\Rightarrow A+\pi r^{2}=\pi R^{2}$
$\displaystyle \Rightarrow R^{2}=\frac{A+\pi r^{2}}{\pi }$
$\displaystyle \Rightarrow$ $\displaystyle R=\sqrt{\frac{A+\pi r^{2}}{\pi }}$

The sum of the reciprocals of $\displaystyle\frac{x+3}{x^2+1}$ and $\displaystyle\frac{x^2-9}{x^2+3}$ is

1. $\displaystyle\frac{x^3+2x^2-x}{x^2-9}$

2. $\displaystyle\frac{x^3-2x^2+x}{x^2-9}$

3. 1

4. 0

Correct Option: B
Explanation:

Reciprocals will be $\frac { { { x }^{ 2 } }+1 }{ x+3 }$,$\frac { { x }^{ 2 }+3 }{ { x }^{ 2 }-9 }$
Their sum will be
$\frac { { { x }^{ 2 } }+1 }{ x+3 } +\frac { { x }^{ 2 }+3 }{ { x }^{ 2 }-9 }$
$=\frac { \left( x-3 \right) \left( { x }^{ 2 }+1 \right) +{ x }^{ 2 }+3 }{ { x }^{ 2 }-9 }$
$=\frac { { x }^{ 3 }+x-3{ x }^{ 2 }-3+{ x }^{ 2 }+3 }{ { x }^{ 2 }-9 }$
$=\frac { { x }^{ 3 }-2{ x }^{ 2 }+x }{ { x }^{ 2 }-9 }$

If $\displaystyle x^{2}-3x+1=0$ then the value of $\displaystyle x-\frac{1}{x}$ is

1. $\displaystyle \sqrt{5}$

2. $\displaystyle \sqrt{3}$

3. $\displaystyle \sqrt{2}$

4. $\displaystyle \sqrt{6}$

Correct Option: A
Explanation:

$x^{2}-3x+1$

$\therefore x^{2}+1=3x\Rightarrow \frac{x^{2}+1}{x}=\frac{3x}{x}\Rightarrow x+\frac{1}{x}=3$
$x^{2}+\frac{1}{x}^{2}=\left ( x+\frac{1}{x} \right )^{2}-2=(3)^{2}-=9-2=7$
We know
$\left ( x-\frac{1}{x} \right )^{2}=x^{2}+\frac{1}{x}^{2}-2\Rightarrow 7-2=5$
$\therefore x-\frac{1}{x}=\sqrt{5}$

If $\displaystyle x-\frac{1}{x}=3$; then the value of $\displaystyle \frac{3x^{2}-3}{x^{2}+2x-1}$ is

1. 9/5

2. 8/5

3. 7/5

4. 6/5

Correct Option: A
Explanation:

Given $x+\frac{1}{x}=3$ multiply by x both sides

Then $x^{2}-1=3x$
$\Rightarrow x^{2}-3x-1=0$
So $\frac{3x^{2}-3}{x^{2}+2x-1}=\frac{3(x^{2}-1)}{x^{2}-3x-1+5x}= \frac{3\times 3x}{0+5x}= \frac{9x}{5x}=\frac{9}{5}$

If $x=2$, $y=3$, then $x^x+y^y$ is equal to

1. $30$

2. $37$

3. $33$

4. $31$

Correct Option: D
Explanation:

Given $x=2, y=3$

${ x }^{ x }+{ y }^{ y }$
$={ 2 }^{ 2 }+{ 3 }^{ 3 }=4+27=31$

If $\displaystyle x^{2}-11x+1=0$ then the value of $\displaystyle x+\frac{1}{x}$ is

1. 10

2. 11

3. 12

4. 13

Correct Option: B
Explanation:

$\therefore x^{2}+1=3x$

$\frac{x^{2}+1}{x}= \frac{11x}{x}$
$\Rightarrow x+\frac{1}{x}=3$

If $\displaystyle x+\frac{a}{x}=b$ then the value of $\displaystyle \frac{x^{2}+bx+a}{bx^{2}-x^{3}}$ is

1. $\displaystyle \frac{6b}{a}$

2. $\displaystyle \frac{5b}{a}$

3. $\displaystyle \frac{2b}{a}$

4. $\displaystyle \frac{4b}{a}$

Correct Option: C
Explanation:

Given $x+\frac{a}{x}=b$ Multiply by x both sides

$x^{2}+a=bx$
$\Rightarrow x^{2}-bx+a=0$
So $\frac{x^{2}+bx+a}{bx^{2}-x^{3}}=\frac{x^{2}-bx+a+2bx}{-x(x^{2}-bx)}=\frac{0+2bx}{-x(-a)}=\frac{2bx}{ax}=\frac{2b}{a}$

If $x<-1$, then $x^2$

1. $=1$

2. $< 1$

3. $>1$

4. none of these

Correct Option: C
Explanation:

$x<-1$
${ x }^{ 2 }<{ \left( -1 \right) }^{ 2 }\quad squaring\quad both\quad side$
${ x }^{ 2 }>1$

If $a+b+c=0$ then $a^3+b^3+c^3$ is equal to

1. 3abc

2. $\displaystyle\frac{3}{abc}$

3. $3a^3b^3c^3$

4. zero

Correct Option: A
Explanation:

$Using\quad { a }^{ 3 }+\quad { b }^{ 3 }+\quad { c }^{ 3 }-3abc=\left( a+b+c \right) \left( { a }^{ 2 }+{ b }^{ 2 }+{ c }^{ 2 }-ab-bc-ca \right)$

$Using\quad { a }^{ 3 }+\quad { b }^{ 3 }+\quad { c }^{ 3 }-3abc=0\times \left( { a }^{ 2 }+{ b }^{ 2 }+{ c }^{ 2 }-ab-bc-ca \right)$
$Using\quad { a }^{ 3 }+\quad { b }^{ 3 }+\quad { c }^{ 3 }=3abc$
Here if a+b+c is 0 then answer will be 3abc

If $\displaystyle a-\frac{1}{3}=\frac{1}{a}$ then the value of $\displaystyle a^{3}-\frac{1}{a^{3}}$ is

1. $\displaystyle 1\frac{1}{27}$

2. $\displaystyle 1\frac{2}{27}$

3. $\displaystyle 1\frac{3}{27}$

4. $\displaystyle 1\frac{4}{27}$

Correct Option: A
Explanation:

Given $a-\frac{1}{3}=\frac{1}{a}$

=$a-\frac{1}{a}=\frac{1}{3}$
We know that
$x^{3}-y^{3}=(x-y)^{3}+3xy(x+y)$
Then $a^{3}-\left ( \frac{1}{a} \right )^{3}=(a-\left ( \frac{1}{a} \right ))^{3}+3a\left ( \frac{1}{a} \right )(a+\left ( \frac{1}{a} \right ))$
=$\left ( \frac{1}{3} \right )^{3}+3\times\left ( \frac{1}{3} \right )$
=$\frac{1}{27}+1=\frac{28}{27}=1\frac{1}{27}$

Which of the following terms contain maximum number of variables ?

1. $3xy$

2. $5x^3$

3. $8yz$

4. $xyz$

Correct Option: D
Explanation:

Option d) contains the variables $x, y$ and $z$

Determine the constant in the equation $3x^2+5y^2=7$?

1. $5$

2. $8$

3. $7$

4. Cannot be determined

Correct Option: C
Explanation:

Here, the constant in the given equation is $7$ as it contains no variable.

How many variables are there in the expression $5x^3+25xy$ ?

1. $1$

2. $2$

3. $3$

4. cannot be determined

Correct Option: B
Explanation:

The variables are $x$ and $y$. Number of variables $=2$

What is a constant?

1. A symbol having a fixed numerical value

2. A variable that takes a fixed value

3. A symbol that can takes different values

4. can"t be determined

Correct Option: A
Explanation:

A constant is a symbol having a fixed numerical value.

Which of the following contains minimum number of variables?

1. $15$

2. $5x^3y^2$

3. $yz$

4. $y^3$

Correct Option: A
Explanation:

The number of variables in $(A)$ is zero.

Which expression has more variables ?
(1) $x^3+3x^2+5x^2y^2+7y$
(2) $5x+3y+z$

1. $1$

2. $2$

3. Both have same

4. cannot be determined

Correct Option: B
Explanation:

variables in $(1)$ are $x$ and $y$ where as variables in $(2$) are $x, y, z$.

How many constants are there in the expression $3x^2+y$ ?

1. $1$

2. $2$

3. $3$

4. $0$

Correct Option: D
Explanation:

Both the terms of the given expression have either $x$ or $y$ as the variable.

Constants are terms without variables.

Hence, there are no constants.

What is a variable?

1. A symbol that takes a fixed numerical value.

2. A symbol that takes various numerical value.

3. A symbol some time fixed and some time variable.

4. cannot be determined.

Correct Option: B
Explanation:

A variable is a symbol or letter, such as $x$ or $y$, that represents a value. In algebraic equations, the value of one variable is often dependent on the value of another. Hence it's a symbol that takes various numerical values.

e.g. in the polynomial $x+5, \quad x$ is a variable.

Find the constant in the polynomial $x + 5$

1. $x$

2. $5$

3. $2$

4. All of the above

Correct Option: B
Explanation:

The constant in the polynomial $x + 5$ is $5$.

Identify the number of constants in the expression $5x^3-8xy$.

1. $0$

2. $1$

3. $2$

4. $5$

Correct Option: A
Explanation:

Both the terms in the given expression contain atleast one variable, $x$ and $xy$..

Hence, there is no constant term in the given expression.

How many variables are there in the algebraic expression $ax^2+bxy+cy^2$  where $a, b, c$ are constants ?

1. $1$

2. $2$

3. $3$

4. $5$

Correct Option: B
Explanation:

The variables used are $x$ and $y$.

Which of the following is correct?

1. Constant can vary in a polynomial

2. Constant may or may not vary in polynomial

3. Constant cannot change in a polynomial

4. All of the above

Correct Option: C
Explanation:

For a particular polynomial, its constant cannot change otherwise polynomial will change.

Find the constant in the polynomial $y^{3} + y^{2} + y$

1. $y^{3}$

2. $y^{2}$

3. $y$

4. None of the above

Correct Option: D
Explanation:

The terms $y^{3}, y^{2}$ and $y$ are not constants.

Therefore, there are no constants in the given polynomial.

The variable in the polynomial $z^3+2z^2+5z+1$ is

1. $1$

2. $2$

3. $z$

4. All of the above

Correct Option: C
Explanation:

The value of the polynomial changes as the variable changes.

Hence, $z$ is the variable in the polynomial.

The variable in the polynomial $x^2+3x+5$ is:

1. $x$

2. $3$

3. $5$

4. All of the above

Correct Option: A
Explanation:

The value of the polynomial changes as the variable changes.

Hence, $x$ is the variable in the polynomial.

Who is the father of algebra?

1. Rene Descartes

2. Brahmagupta

3. Al-Khwarizmi

4. Albert Einstein

Correct Option: C
Explanation:

al-Khwarizmi is the father of Algebra.

An important development in algebra in the $16^{th}$ century was the

1. introduction of unknown symbols

2. arithmetic mean

3. introduction to exponents

4. introduction to powers

Correct Option: A
Explanation:

An important development in algebra in the $16^{th}$ century was the introduction to unknown symbols.

What is the literal meaning of algebra?

1. science of restoration

2. unknown quantities

3. powers and equations

4. re-union of broken parts

Correct Option: D
Explanation:

The word Algebra literally means the re-union of broken parts.

$-6$ is the ______ in $q(y)=y^3-3y^2-6+y$

1. constant term

2. coefficient of $y$

3. variable

4. degree of the polynomial

Correct Option: A
Explanation:

The constant term in an expression or equation has a fixed value and does not contain variables.
So, $-6$ is the constant term in $q(y)=y^3-3y^2-6+y$

Who used the symbol heap for the unknown in algebra?

1. Spanish

2. Egyptians

3. Babylonian

4. German

Correct Option: B
Explanation:

Egyptians used the symbol heap for the unknown in algebra.

What is the value of the constant term in the expression, $23x^3+12x^2-6x-12$?

1. $12$

2. $6$

3. $-6$

4. $-12$

Correct Option: D
Explanation:

The constant term in an expression or equation has a fixed value and does not contain variables.
So, $-12$ is the constant term in the expression, $23x^3+12x^2-6x-12$.

How many degree of polynomials are there in constant term?

1. one

2. zero

3. two

4. three

Correct Option: B
Explanation:

Constant term has zero degree of polynomial.
Because the constant term in an expression or equation has a fixed value and does not contain variables.
Example: $p(x)=k$
Where $k$ is a constant.

The constant term of $0.4x^{7} - 75y^{2} - 0.75$ is ___

1. $0.4$

2. $0.75$

3. $-0.75$

4. $-75$

Correct Option: C
Explanation:

Given equation is $0.4x^{7}-75y^{2}-075$

A constant term is a term in an algebraic expression that has a value that is constant or cannot change, because it does not contain any modifiable variables.
Then constant term of equation $0.4x^{7}-75y^{2}-075$ is $- 0.75$

Which one is the constant term of $4x^{3} - 3x^{2} + 2x - 5$.

1. $4$

2. $-5$

3. $2$

4. $-3$

Correct Option: B
Explanation:

General equation is $ax^3+bx^2+cx+d$

where d is constant term

so in given equation,

$4x^{3} - 3x^{2} + 2x - 5$

Constant term is $-5$.

Classify the following polynomial as polynomial in one variable, two variables etc.

$x^{2}+x+1$

1. Polynomial in one variable

2. Polynomial in two variables

3. Invalid question

4. None of the above

Correct Option: A
Explanation:

Polynomial has only 1 variable $x$ in the equation.

Classify the following polynomial as polynomial in one variable, two variables etc.
$x^{2}-2xy+y^{2}+1$
1. One Variable

2. Two Variables

3. Three Variables

4. Four Variables

Correct Option: B
Explanation:

There are two variables in the given equation, i.e. $x$ and $y$ while $1$ is a constant.

Classify the following polynomial as polynomial in one variable, two variables etc.

$y^{3}-5y$

1. Polynomial in one variable

2. Polynomial in two variables

3. Polynomial in three variables

4. Polynomial in four variables

Correct Option: A
Explanation:

Polynomial has only 1 variable $y$ in the equation.

Classify the following polynomial as a polynomial in one variable, two variables, etc.

$xy + yz + zx$

1. One Variable

2. Two Variables

3. Three Variables

4. Four Variables

Correct Option: C
Explanation:

The given polynomial has three variables i.e.  $x, y$ and $z$ .

Consider the polynomial $\dfrac{x^{3}+2x+1}{5}-\dfrac{7}{2}x^{2}-x^{6}$.

The constant term is:

1. $\dfrac{1}{7}$

2. $\dfrac{1}{5}$

3. $\dfrac{1}{2}$

4. $\dfrac{1}{3}$

Correct Option: B
Explanation:

$\cfrac { { x^{ 3 }+2x+1 } }{ 5 } -\cfrac { 7 }{ 2 } x^ 2-x^ 6$
$=\cfrac { { x^{ 3 }+2x+1 } }{ 5 } -\cfrac { 7 }{ 2 } x^ 2-x^ 6$
$=-x^ 6+\cfrac { x^{ 3 } }{ 5 } -\cfrac { 7x^ 2 }{ 2 } +\cfrac { 2x }{ 5 } +\cfrac { 1 }{ 5 }$
So, the constant term= $\cfrac { 1 }{ 5 }$

A real variable is a variable whose values are real numbers.

1. True

2. False

Correct Option: A
Explanation:

A variable which can take real numbers only as its value is called as a real variable.

Given $x^2 + \dfrac{1}{N^4} - 142$ Based on the above date answer the following questions. The value $\left(x^2, \dfrac{1}{x^a}\right)$ is

1. $27\sqrt{14}$

2. $17\sqrt{14}$

3. $12\sqrt{14}$

4. $11\sqrt{14}$

Correct Option: A

Which of the following expressions is a polynomial in one variable?

1. $x+\dfrac {2}{3}+3$

2. $3\sqrt {x}+\dfrac {2}{\sqrt {x}}+5$

3. $\sqrt {2x^{2}}-\sqrt {3x}+6$

4. $x^{10}+y^{5}+8$

Correct Option: A

The output of $z^3+2z^2+5z+1$, where $z= 1$, is

1. $7$

2. $2$

3. $9$

4. None of the above

Correct Option: C
Explanation:

Given equation is $z^3+2z^2+5z+1$

Put $z=1$, we get
$z^3+2z^2+5z+1=1^3+2\times 1^2+5\times 1+1=1+2+5+1=9$
Hence, option C is correct.

What is the output of $x^2+3x+5$, where $x$(variable) = $2$?

1. $11$

2. $12$

3. $13$

4. $15$

Correct Option: D
Explanation:

$x^2+3x+5$
$=(2)^2+3\times 2+5$
$=4+6+5$
$=15$

What is the output of $x^2+3x+5$, where $x$(variable) = $-1?$

1. $1$

2. $2$

3. $3$

4. $4$

Correct Option: C
Explanation:

$x^2+3x+5$
$=(-1)^2+3\times (-1)+5$
$=1-3+5$
$=6-3$
$=3$

The output of $z^3+2z^2+5z+1$, where $z= -1$

1. $-1$

2. $-2$

3. $-3$

4. None of the above

Correct Option: C
Explanation:

$z^3+2z^2+5z+1$
$=(-1)^3+2\times (-1)^2+5\times (-1)+1$
$=-1+2-5+1$
$=-3$

The output of $z^3+2z^2+5z+1$, where $z= 0$

1. $1$

2. $2$

3. $3$

4. None of the above

Correct Option: A
Explanation:

$z^3+2z^2+5z+1$
$=(0)+2\times (0)^2+5\times (0)+1$
$=1$

If $\dfrac{2+3}{x}=\dfrac{2+x}{3}$
What one value for $x$ can be correctly entered into the answer grid?

1. -5

2. 3

3. -3

4. 2

Correct Option: B
Explanation:

Given, $\dfrac{2+3}{x}=\dfrac{2+x}{3}$

$\Rightarrow 2x+x^2=15$
$\Rightarrow x^2+2x-15=0$
$\Rightarrow x^2+5x-3x-15=0$
$\Rightarrow x(x+5)-3(x+5)=0$
$\Rightarrow (x+5)(x-3)=0$
$\Rightarrow x=-5,3$
Value of $x$ is not negative, so $x=3$.

Some situations are given below. State true or false:
The temperature of a day is variable.

1. True

2. False

Correct Option: A
Explanation:

Since the temperature of a day depends on the weather. For example, in summers, the temperature will be higher in Celsius whereas in winters, the temperature is low in Celsius. Therefore, the temperature always varies.

Hence, the given statement "The temperature of a day is variable." is true.

Some situations are given below. State true or false:
Length of your classroom is constant.

1. True

2. False

Correct Option: A
Explanation:

Since the length, breadth and height of any area (Room, park, hall, etc) is measured and decided before its construction and then it gets constructed. Therefore, the length of any area will always remain the same.

Hence, the given statement "Length of your classroom is constant." is true.

Some situations are given below. State true or false:
Height of growing plant is constant.

1. True

2. False

Correct Option: B
Explanation:

Since plants have the unique ability to grow indefinitely throughout their life due to the presence of ‘meristems’ in their body. Meristems in the roots and shoots of plants are responsible for ‘primary growth of the plant’. These increase the height of the plant. Therefore, the height of any growing plant varies.

Hence, the given statement "Height of growing plant is constant." is false.

Some situations are given below.State true or false.
The number of days in the month of January are varying.

1. True

2. False

Correct Option: B
Explanation:

Since there are $31$ days in January in every year, therefore the number of days in the month of January doesn't vary.

Hence, the given statement "The number of days in the month of January are varying." is False.

Solve: $(3x-5)^2 +(3x+5)^2$ = $(18x+10)(x-2)$

1. $\dfrac{-35}{13}$

2. $\dfrac{-25}{13}$

3. $\dfrac{-15}{13}$

4. $\dfrac{-45}{13}$

Correct Option: A
Explanation:

We will solve the given expression $(3x-5)^2+(3x+5)^2=(18x+10)(x-2)$ as shown below:

$(3x-5)^{ 2 }+(3x+5)^{ 2 }=(18x+10)(x-2)\ \Rightarrow [(3x)^{ 2 }+(5)^{ 2 }-(2\times 3x\times 5)]+[(3x)^{ 2 }+(5)^{ 2 }+(2\times 3x\times 5)]=18x(x-2)+10(x-2)\ (\because \quad (a+b)^{ 2 }=a^{ 2 }+b^{ 2 }+2ab,\quad (a-b)^{ 2 }=a^{ 2 }+b^{ 2 }-2ab)\ \Rightarrow 9x^{ 2 }+25-30x+9x^{ 2 }+25+30x=18x^{ 2 }-36x+10x-20\ \Rightarrow 18x^{ 2 }+50=18x^{ 2 }-26x-20\ \Rightarrow 18x^{ 2 }+50-18x^{ 2 }+26x+20=0$
$\Rightarrow 26x+70=0\ \Rightarrow 26x=-70\ \Rightarrow x=-\dfrac { 70 }{ 26 } \ \Rightarrow x=-\dfrac { 35 }{ 13 }$

Hence, $x=-\dfrac { 35 }{ 13 }$.

For $|x| < 1$ the constant terms in the expressions of $\dfrac {1}{x-1(^{2})(x-2)}$ is

1. $1$

2. $2$

3. $0$

4. $-1/2$

Correct Option: B

If ${x}^{3}+m{x}^{2}+nx+6$ has $(x-2)$ as factor and leaves a remainder $3$ when divided by $(x-3)$ find the values of $m,\,n$

1. $m=2,n=2$

2. $m=2,n=-2$

3. $m=-2,n=1$

4. $m=-3,n=-1$

Correct Option: D
Explanation:

$x-2$ is factor

$\Rightarrow x=2$
$f\left( 2 \right) =14+4m+2n$
Remainder is zero

$\Rightarrow 7+2m+n=0\quad \longrightarrow \left( i \right)$
Now, $x-3=0$
gives remainder $3$

$\Rightarrow f\left( 3 \right) =3$
$\Rightarrow 33+9m+3n=3$
$\Rightarrow 10+3m+n=0\quad \longrightarrow \left( ii \right)$

From $(i)$ & $(ii)$
$m=-3$
$n=-1$

The constant term in expression $5xy-4x+8$ is

1. $3$

2. $-5$

3. $8$

4. $1$

Correct Option: C
Explanation:
In $5xy-4x+8$

$\therefore$ The constant term is $=+8$.

If the point (2, -3) lies on $\displaystyle kx^{2}-3y^{2}+2x+y-2=0$ then k is equal to

1. $\displaystyle \frac{1}{7}$

2. 16

3. 7

4. 12

Correct Option: C
Explanation:

As the point lies on the given line, it should satisfy the equation of the line, if we substitute $x = 2$ and $y = -3$ in it.

So, $k({ 2) }^{ 2 }3{ (-3) }^{ 2 }+2(2)-32=0$
$=> 4k -27 + 4 - 3 -2 = 0$
$4k = 28$
$k = 7$

$n^2-n+1$ is an odd number for all

1. $n>1$

2. $n>2$

3. $n\ge1$

4. $n\ge5$

Correct Option: C
Explanation:

For $n = 1$, we have $n^2 - n + 1 = 1^2 -1 + 1 = 1$ which is an odd number

For $n =2$, we have $n^2 - n + 1 = 2^2 -2 + 1 = 3$ which is an odd number

For $n = 3$, we have $n^2 - n + 1 = 3^2 -3 + 1 = 7$ which is an odd number

For $n = 4$, we have $n^2 - n + 1 = 4^2 -4 + 1 = 13$ which is an odd number

For $n = 5$, we have $n^2 - n + 1 = 5^2 -5 + 1 = 19$ which is an odd number

Hence, $n^2 - n + 1 = 1^2 -1 + 1 = 1$ is an odd number for all $n \ge 1$

Find the number of variables in the expression: $3x^2+25xy+7x2+5y^2+z^2$

1. $4$

2. $2$

3. $3$

4. $5$

Correct Option: C
Explanation:

The variables are $x$,$y$ and $z$

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