Maths M.C.Q. [1 Marks Each]

We solve the given equation as follows.
$2 (a - 3) + 4b - 2 (a - b) + 5$
$= 2a - 6 + 4b - 2a + 2b + 5$
$= 6b - 1$

The perimeter is $= 6a^2 - 4a + 9$
$\Rightarrow $ The sum of three sides give the perimeter
$\therefore$ the third side is $6a^2 - 4a + 9$
$-(a^2 - 2a + 1 + 3a^2 - 5a + 3)$
$\Rightarrow 2a^2 + 3a + 5$

$5\text{x}2 + 6\text{x} - 3\\\underline{2\text{x}2 - 7\text{x} - 9 }\\3\text{x}2 + 13\text{x} + 6$

Since, the coefficient of $x^2$ in $-\frac{5}{3}\text{x}^2\text{y}$ is equal to $-\frac{5}{3}\text{y}$
Hence, the correct alternative is option $(b).$

$(3 + 7x) + 11x = 3 + 18x$ is a binomial.

$(x + y)^{-1}(x^{-1} + y^{-1})$
$\Rightarrow\frac{1}{\text{(x+y)}}\times\Big(\frac{1}{\text{x}}+\frac{1}{\text{y}}\Big)$
$\Rightarrow\frac{1}{\text{(x+y)}}\times\Big(\frac{\text{x+y}}{\text{xy}}\Big)$
$\Rightarrow\frac{1}{\text{x+y}}\Rightarrow({\text{xy}})^{-1}$

$(1):$ Any expression with one or more terms is called a Polynomial.
$(2):$ Terms which have the same algebraic factors are like terms.
$(3):$ The coefficient is the numerical factor in the term.
$(4):$ Algebraic expressions are formed from variables and constants.

The power of x must be a non-negative integer in a polynomial. As we can write option $\text{c as x}^2+3\text{x}^{\frac{3}{2}-\frac{1}{2}}=\text{x}^2+3\text{x}$

$2x^3 + 7x^2 + 6x + 5$ Coefficient $= 7$

Let us substitute $m = 2$ and $n = 1$ in the given polynomial $4m^2n$ as shown below.
$4m^2n = 4(2)^21 = 4 × 4 = 16$
$4m^2n = 16,$ if $m = 2$ and $n = 1.$

Terms in the expression $4x^2​​​​​ - 3xy$ are $4x^2$ and $-3xy$

Since, vertex is formed by joining two sides. Diagonal is line segment joining the two opposite vertex.
So, number of diagonal formed by one vertex $= n - 3.$

A polynomial is a mathematic expression containing several terms where the variables have positive integral exponents. $A$ and $C$ have negative exponents, so they are not polynomials. $B$ and $D$ are correct.

$\sqrt{\text{y}^3​}+\text{y}^2=\text{y}^{\frac{3}{2}}+{\text{y}}^2 $
$\therefore$ the exponent is $\frac{3}{2}$

Given, $3x^2 - y^3 + 5xy - 4$ They are all unlike terms, nothing can be combined, so it cannot be simplified.

Apolynomialis anexpressionconsisting of variables and coefficients, that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponents. the variable is $x$ but in $\frac{7}{\text{x}}$ power of xis not a whole number. this is not a polynomial.

Given,
$\Rightarrow a+b=10 \Rightarrow a b=16 $
$ \Rightarrow(a+b)^2=a^2+b^2+2 a b $
$ \Rightarrow 10^2=a^2+b^2+2(16) $
$ \therefore a^2+b^2=68 $
$ \Rightarrow a^2+b^2+a b=68+16=84 $
$ \Rightarrow a^2+b^2-a b=68-16=52$

The maximum no. of terms in a polynomial of degree $10$ is a polynomial that can have terms with powers of $x$ as $0, 1, 2, 3, 4, 5, 6, 7, 8, 9$ and $10$. there are $11$ such terms that can be possible with these powers of $x$ and $a$ real coefficient.

Let the polynomial to be added be $p$
$\therefore x^3+3 x-8+p=3 x^3+x^2+6 $
$ \therefore p=3 x^3+x^2+6-x^3-3 x+8 $
$ \therefore p=2 x^3+x^2-3 x+14$

$(4x^3 + 2x^2 + 4x + 4) × 2$
$= 8x^3 + 4x^2 + 8x + 8$

Since, $(x^2 + 2x - 3) - (-2x^2 + x + 1)$
$= x^2 + 2x - 3 + 2x^2 - x - 1$
$= 3x^2 + x - 4$
So, $-2x^2 + x + 1$ is less than $x^2 + 2x - 3$ by $3x^2 + x - 4.$
Hence, the correct alternative is option $(b).$

Half of x is y or $\frac {1}{2}\text{x = y}$ One-third of y is z or $\frac {1}{3}\text{y = z}$
$\therefore \text{z} = \frac{1}{3}\text{y} = \frac{1}{3} (\frac{1}{2}\text{x})\times\frac{100}{100}\text{x} = \frac{16..66}{100}\text{x} = {16.66}\%\text{ of x}$

We have to just subtract $-8abc$ from $-7abc$
$= (-7abc) - (-8abc)$
$= -7abc + 8abc = abc$

A term consists of numbers and variables combined with the multiplication operation, with the variables optionally having exponents and Numerical Coefficient is often abbreviated to just coefficient. A coefficient is the numerical value in a term. If a term has no coefficient, the coefficient is an unwritten $1$ or in other words it is term without the variables.

Opening the brackets and multiplying the terms, we get $(x^2- 3x + 2) (x - 3)$
$= x^3 - 3x^2 - 3x^2 + 9x + 2x - 6$ So the term not containing $x$ is the independent term $= -6$

Since, $(9x^2 - 7) - (3x^2 + 4) = 9x^2 - 7 - ​3x^2 - 4 = 6x^2 - 11$
So, $6x^2 - 11$ should added to $3x^2 + 4$ to get $9x^2 - 7.$
Hence, the correct alternative is option $(a).$

$\Rightarrow $ Product of $x$ and $a = x \times a = ax$
$\Rightarrow $ Product of $b$ and $y = b \times y = by$
$\Rightarrow $ Product of $x$ and a subtracted from the product of $b$ and $y = by - ax$
$\therefore$ Required algebraic expression is $by - ax.$

given, $(a + b) (c + d) = ac + bc + ad + bd$ In above expression the number of terms are Four $(4)$

Degree of variables in ploynomials $(1), (3)$ and $(4)$ are not whole numbers.
$\therefore$ they are not ploynomials. While in option $(2)$ degrees of variable are whole numbers.
$\therefore$ it is a ploynomial.

$x^2 mx + 3 = (1)^2 - (2) (1) + 3 = 1 - 2 + 3 = 2$

Since, $2x^2 + 1$ has two terms $2x^2$ and $1.$
So, $2x^2 + 1$ is a binomial.
Hence, the correct alternative is option $(a).$

The given expression can be solved as shown below:
$\Rightarrow{60} = \frac{\text{b}}{4}\sqrt{{4}\times{13}^{2}}$
$\Rightarrow{60} = \frac{\text{b}}{4}\sqrt{{4}\times{169}}$
$\Rightarrow{60} = \frac{\text{b}}{4}\times\sqrt{676}$
$\Rightarrow{60} = \frac{\text{b}}{4}\times{26}$
$\Rightarrow{60} \times4 = {26}\text{ b}$
$\text{b} = \frac{240}{26} = \text{b} = 9.23$

According to classification of polynomial based on degree, a polynomial having degree $3$ is known as trinomial (cubic) polynomial.

$6x^2 - 5x^2 - 2x + 3$ has terms and $6x^3, 5x^2, 2x$ and $3,$
$\therefore$ four terms.

A polynomial having terms more than 3 is known as multinomial. for eg $-3x^4 + 2x^2+ x - 4$

The value of $(5 - 6i) - (-7i + 16) = 5 - 6i + 7i - 16 = i - 11$

Solve the polynomial as follows:$ -3(x^3 - x^2 - 2x - 5) - (4x^3 - 7x - 1)$
$= -3x^3+ 3x^2 + 6x + 15 - 4x^3 + 7x + 1$
$= -7x^3 + 3x^2 + 13x + 16$

$2x^2 + 5x + 12$ The polynomial in coefficient form is $(2, 5, 12)$

$x^2+ y - 2$ is an example of algebraic expression. An algebraic expression is a collection of real numbers, variables, grouping and operation symbols.

There are $3$ terms in the given expression i.e. $5x^3, 7x^2, 8xy.$

We need to find missing term in $(2a^3 - 3) (5a^3- 2) = 10a^6 + ..... + 6$
$(2a^3- 3) (5a^3- 2)$
$= 2a^3 (5a^3 - 2) -3 (5a^3 - 2)$
$= 10a^6- 4a^3 - 15a^3 + 6$
$= 10a^6 - 19a^3 + 6$ missing term is $-19a^3.$

The Coefficient of $\frac{\text{y}}{3} \text{ is }\frac{1}{3}$

Since, the coefficient in the monomial $3a^2b$ is $3$ and the coefficient in the monomial $-2ab^2$ is $-2.$
So, the sum of the coefficients in the monomials $3a^2b$ and $-2ab^2 = 3 + (-2) = 3 - 2 = 1$
Hence, the correct alternative is option $(c).$

Monthly salary is $Rs. 2p^2 + p - 3$
Annual salary is $= 12 × (2p^2 + p - 3)$
$= 24p^2 + 12p - 36$
Annual expenditure is $= 14p^2 + 6p - 10$
$\therefore$ Annual savings = salary - expenditue
$= 24p^2+ 12p - 36 - (14p^2 + 6p - 10)$
$= 24p^2 + 12p - 36 - 14p^2 - 6p + 10$
$= 10p^2+ 6p - 26$

$-5y + 10$ (or $10 - 5y)$: Do not forget to reverse the signs of every term in a subtracted expression
$(4 - y) -2 (2y - 3) = 4 - y - 4y + 6 = -5y + 10$ (or $10 - 5y)$

A binomial is a polynomial that contains $2$ unlike terms. $3x + 1$ is a binomial.

Let X be subtracted from $3a^2 - 6ab - 3b^2$ Then,
$3a^2 - 6ab - 3b^2 - 1 - X = 4a^2 - 7ab - 4b^2 + 1$
$x = 3a^2 - 6ab - 3b^2 - 1 -(4a^2 - 7ab - 4b^2 + 1)$
$x = 3a^2 - 6ab - 3b^2- 1 -4a^2 + 7ab + 4b^2 - 1$
$x = -a^2 + ab + b^2 - 2$

$(3x + 2y - 9) (2x - 6y + 2) - [(4x - 9y - 1) + (-3x + 8y + 7)]$
$= (6x^2 - 18xy + 6x + 4xy + 4y - 12y^2 - 18x + 54y - 18) -[4x - 9y - 1 - 3x + 8y + 7]$
$=6x^2 - 14xy - 12y^2 - 13x + 59y - 24$

Given, side of the square $= (2x + 3$
Perimeter of square $= 4 x$ (Side)
$= 4 × (2x + 3)$
$= 8x + 1$

Two like terms will add upto a single term. Eg. $5xy + 4xy = 8xy$

$5$ times of $y = 5y$
Now, subtraction of $5$ times of $y$ from $x$ is written as $x - 5y.$

A binomial is a polynomial that contains $2$ unlike terms. $3x + 1$ is a binomial.

Putting $x = 1$ in given equation we get $3x^2- 5x + 3= 3(1)^2- 5(1) + 3 =3 - 5 + 3 = 1$

Additive inverse of any number is simply the negative of that number. For example Additive inverse of $x$ will be $-x.$
so Additive inverse of $ = \frac{\text{-x}^5-7{\text{x}}^2+18}{\text{x}^3-2}$
will be $ = \frac{\text{-x}^5+7{\text{x}}^2-18}{\text{x}^3-2}$

Given that, $\frac{\text{x}}{\text{y}} = \frac{6}{5}​ $
$\Rightarrow \text{x} = \frac{6\text{y}}{5}$
To find, $ \frac{\text{x}^{2}+\text{y}^{2}}{\text{x}^{2}-\text{y}^{2}}$ .
Substituting value of $x$ in this,
​​​​​​​we get $\therefore \frac {\frac{36\text{y}^{2}}{25} + \text{y}^{2}}{\frac{36\text{y}^{2}}{25} - \text{y}^{2}} = \frac{61\text{y}^{2}}{11\text{y}^{2}} = \frac{61}{11}$

The coefficient of $x$ is $-6y$, not $6y.$
the statement is false.

$x^2 + 6x + 9 = (x)^2 + 2(x) (3) + 3^2 = (x + 3)^2$

$3x - 4y + 5z$ is an example of unlike terms. Because different variables $(x, y, z)$ are used.

Given: $x = a + b 2^x + 2^a⋅2^b = ?$
So. $2^{a + b}+ 2^a+ 2^b⇒ 2^{a + b} + 2^{a + b}$ (according to exponent product rule)
$⇒ 2(2^{a + b})$

Given $ \text{a}\times \text{b}=\frac {\text{ a} }{\text{b}} -\frac{\text{b}}{\text{a}}​$
$\Rightarrow \frac{\text{a}\times\text{b}}{\text{b}\times \text{a}}=\frac{\frac{\text{a}}{\text{b}}-\frac{\text{b}}{\text{a}}}{\frac{\text{b}}{\text{a}}-\frac{\text{a}}{\text{b}}}=-1$

Since, $(-xy - yz - zx) - (xy + yz + zx)$
$= -xy - yz - zx - xy - yz - zx$
$= -2xy - 2yz - 2zx$
$So, -2xy - 2yz - 2zx$ should be added to $xy + yz + zx$ to get $-xy - yz - zx.$
Hence, the correct alternative is option $(a).$

Let $X = -3a + 2b + 3$ and $Y = 2a + 6b - 5$
Let $Z$ be the required expression
Now, $X = Y - Z = > Z = Y - X$
$2a + 6b - 5 - (-3a + 2b + 3)$
$= 2a + 6b - 5 + 3a - 2b - 3$
$= 5a + 4b - 8$

Given, $0.39$ multiple and divide by $100$
$ = {0.39}\times\frac{100}{100}$
$ = \frac{39}{100}$

As, the coefficient of the term $-\frac43\text{ab}^2$ is $-\frac43,$ the coefficient of the term $\frac14\text{bc}^2$ is $\frac14$ and the coefficient of the term $3ca^2$ is $3.$
So, the product of the coefficients of the terms $=-\frac43\times14\times3=-1$
Hence, the correct alternative is option $(c).$

The numerical factor of a term is called coeffiicient. As for example $3x + 6$ Coefficient of $x$ is $3.$

Given, $4x - (-2y + 5x)$
$= 4x + 2y - 5x = -x + 2y$
simplified form is $-x + 2y$

Three cubes of metal whose edges are $6\ cm, 8\ cm$ and $10\ cm$ respectively are melted Then volume of first cube $= (6)^3= 216\ cm^3$And volume of second cube $= (8)^3 = 512\ cm^3$ And volume of first cube $= (10)^3= 1000\ cm^3$Then volume of cube $= (10)^3 = 1000\ cm^3$ Then volume of cube made by melted three cube $= 216 + 512 + 1000 = 1728$ Then side of cube = $\sqrt{1728}=12\text{ cm}$ Then diagonal of cube $=\sqrt{3\times (12)^{2}}=12\sqrt{3}\text{ cm}$

We have,
The expression $x^2 + y^2 + z^2 - xy - yz - zx,$

$(x - 1)(1 - 2x) = -2x^2 + 3x - 1$ as $3$ is multiplied with $x,$ hence, coefficient of $x$ is $3.$

$x^4 + x^3 - 1$ is an example of algebraic expression. An algebraic expression is a collection of real numbers, variables, grouping and arithmetic operational symbols.