Question 11 Mark
$103^2- 102^2= \_\_\_\_\_\_\_\_ × (103 - 102) = \_\_\_\_\_\_\_\_\_.$
Answer$103^2- 102^2= (103 + 102) \times (103 - 102) = 205.$
Solution:
We have, $103^2- 102^2= (103 + 102)(103 - 102) = 205 \times 1 = 205$
View full question & answer→Question 21 Mark
On dividing $\frac{\text{p}}{3}$ by $\frac{3}{\text{p}}$, the quotient is $9.$
AnswerFalse.Solution:
We have,$\frac{\text{p}}{3}\div\frac{3}{\text{p}}=\frac{\text{p}}{3}\times\frac{\text{p}}{3}$
$=\frac{1}{9}\text{p}^2$
Hence, the quotient is $\frac{1}{9}\text{p}^2$
View full question & answer→Question 31 Mark
$a(b + c) = ax$ ____ $\times ax$ _____.
Answer$a(b + c) = ax\ b \times ax\ c. $
Solution:
we have, $a(b + c) = a \times b + a \times c [$using left distributive law$]$
View full question & answer→Question 41 Mark
Volume of a rectangular box with $l = b = h = 2x$ is _________.
AnswerVolume of a rectangular box with $l = b = h = 2x$ is $8x^3$.
Solution:
Volume of a rectangular box $= l \times b \times h = 2x \times 2x \times 2x = (2 \times 2 \times 2)x^3= 8x^3$
View full question & answer→Question 51 Mark
Factorise the expressions and divide them as directed:
$\left(2 x^3-12 x^2+16 x\right) \div(x-2)(x-4)$
Answer$\left(2 x^3-12 x^2+16 x\right) \div(x-2)(x-4)$
We have,$=\frac{2\text{x}^3-12\text{x}^2+16\text{x}}{(\text{x}-2)(\text{x}-4)}$
$=\frac{2\text{x}(\text{x}^2-6\text{x}+8)}{(\text{x}-2)(\text{x}-4)}$
$=\frac{2\text{x}(\text{x}^2-4\text{x}-2\text{x}+8)}{(\text{x}-2)(\text{x}-4)}$
$=\frac{2\text{x}[\text{x}(\text{x}-4)-2(\text{x}-4)]}{(\text{x}-2)(\text{x}-4)}$
$=\frac{2\text{x}(\text{x}-4)(\text{x}-2)}{(\text{x}-2)(\text{x}-4)}$
$=2\text{x}$
View full question & answer→Question 61 Mark
Factorise the expressions and divide them as directed:
$\left(x^4-16\right) \div x^3+2 x^2+4 x+8$
Answer$\left(x^4-16\right) \div x^3+2 x^2+4 x+8$
We have,$=\frac{\text{x}^4-16}{\text{x}^3+2\text{x}+4\text{x}+8}$
$=\frac{({\text{x}^2})^2-4^2}{\text{x}^2(\text{x}+2)+4(\text{x}+2)} [\therefore\text{a}^2-\text{b}^2=(\text{a}+\text{b})(\text{a}-\text{b})]$
$=\frac{(\text{x}^2+4)(\text{x}^2-4)}{(\text{x}^2+4)(\text{x}+2)}$
$=\frac{\text{x}^2-2^2}{\text{x}+2}$
$=\frac{(\text{x}+2)(\text{x}-2)}{\text{x}+2}$
$=\text{x}-2$
View full question & answer→Question 71 Mark
The product of one negative and one positive term is a negative term.
AnswerTrue.Solution:
When we multiply a negative term by a positive term, the result will be a negative term, i-e. $(-) \times (+) = (-).$
View full question & answer→Question 81 Mark
Subtract: $5a^2b^2c^2$ from $-7a^2b^2c^2$
AnswerThe required difference is given by
$ -7 a^2 b^2 c^2-5 a^2 b^2 c^2 $
$ =(-7-5) a^2 b^2 c^2 $
$ =-12 a^2 b^2 c^2 $
View full question & answer→Question 91 Mark
Carry out the following divisions:
$76 x^3 y z^3 \div 19 x^2 y^2$
Answer$76 x^3 y z^3 \div 19 x^2 y^2$
$\frac{76\text{x}^3\text{yz}^3}{19\text{x}^2\text{y}}$
$=\frac{4\times9\times\text{x}\times\text{x}\times\text{x}\times\text{y}\times\text{z}\times\text{z}\times\text{z}}{19\times\text{x}\times\text{x}\times\text{y}\times\text{y}}$
$=\frac{4\text{xz}^3}{\text{y}}$
View full question & answer→Question 101 Mark
h is a factor of $2\pi(\text{h}+\text{r})$.
AnswerFalse. Solution: h is not a factor of $2\pi(\text{h}+\text{r})$ This expression has only two factors $2\pi$ and $(h + r).$
View full question & answer→Question 111 Mark
The difference of the squares of two consecutive numbers is their sum.
AnswerTrueSolution:
Let $n$ and $n + 1$ be two consecutive numbers, then their sum $= n + n + 1 = 2n + 1$
Now, the difference of their squares,
$(n+1) 2-n^2=n^2+1+2 n-n^2$
View full question & answer→Question 121 Mark
Factorise the expressions and divide them as directed:
$\left(3 x^4-1875\right) \div\left(3 x^2-75\right)$
Answer$\left(3 x^4-1875\right) \div\left(3 x^2-75\right)$
We have,$=\frac{3\text{x}^4-1875}{3\text{x}^2-75}$
$=\frac{\text{x}^4-625}{\text{x}^2-25}$
$=\frac{(\text{x}^2)^2-(25)^2}{\text{x}^2-25}$
$=\frac{(\text{x}^2+25)(\text{x}^2-25)}{(\text{x}^2-25)}$
$=\text{x}^2+25$
View full question & answer→Question 131 Mark
$(a + b)^2- 2ab =$___________ $+$ ____________
Answer$(a + b)^2- 2ab =$ $ a^2+ b^2$
Solution:
We have,$(a+b)^2-2 a b=a^2+b^2+2 a b-2 a b=a^2+b^2$
View full question & answer→Question 141 Mark
$p^2 q+q^2 r+r^2 q$ is a binomial.
AnswerFalse. Solution: Since, the given expression contains three unlike terms, so it is a trinomial.
View full question & answer→Question 151 Mark
$(9x - 51) ÷ 9$ is $x - 51$
AnswerWe have, $(9x - 51) ÷ 9$
$=\frac{9\text{x}-51}{9}$
$=\frac{9\text{x}}{9}-\frac{51}{9}$
$=\text{x}-\frac{51}{9}$
View full question & answer→Question 161 Mark
$67^2-37^2=(67-37) ×$ ________ $=$ _________.
Answer$67^2-37^2=(67-37) \times 67 + 37 = 3120.$
Solution:
We have, $67^2-37^2=(67-37)(67 + 37) = 30 \times 104 = 3120$
View full question & answer→MCQ 171 Mark
$a^2- b^2$ is equal to:
- A
$(a - b)^2$
- B
$(a - b) (a - b)$
- ✓
$(a + b) (a - b)$
- D
$(a + b) (a + b)$
AnswerCorrect option: C. $(a + b) (a - b)$
$(a + b) (a - b)$
View full question & answer→Question 181 Mark
$38 x^3 y^2 z \div 19 x y^2$ is equal to _________.
Answer$38 x^3 y^2 z \div 19 x y^2$ is equal to $2x^2z.$
Solution:
We have,$38\text{x}^3\text{y}^2\text{z}\div19\text{xyz}^2$
$\frac{38\text{x}^3\text{y}^2\text{z}}{19\text{xy}^2}$
$=\frac{38\times\text{x}\times\text{x}\times\text{x}\times\text{y}\times\text{y}\times\text{z}}{19\times\text{x}\times\text{y}\times\text{z}}$
$=\frac{38}{19}\text{x}^2\text{z}$
$=2\text{x}^2\text{z}$
View full question & answer→Question 191 Mark
Carry out the following divisions:
$-121 p^3 q^3 r^3 \div\left(-11 x y^2 z^3\right)$
Answer$-121 p^3 q^3 r^3 \div\left(-11 x y^2 z^3\right)$
$\frac{-121\text{p}^3\text{q}^3\text{r}^3}{-11\text{xy}^2\text{z}^3}$
$=\frac{-11\times11\times\text{p}\times\text{p}\times\text{p}\times\text{q}\times\text{q}\times\text{q}\times\text{r}\times\text{r}\times\text{r}}{-11\times\text{x}\times\text{y}\times\text{y}\times\text{z}\times\text{z}\times\text{z}}$
View full question & answer→Question 201 Mark
The product of two terms with like signs is _________ a term.
Answer The product of two terms with like signs is Positive a term. Solution:
If both the like terms are either positive or negative, then the resultant term will always be positive. View full question & answer→Question 211 Mark
$a^2- b^2= (a + b )$ __________.
Answer$a^2- b^2= (a + b ) (a - b).$
Solution:
We have, $a^2- b^2= (a + b)(a - b)$
Alternative Answer:
Let $(\text{a}^2-\text{b}^2)=(\text{a}+\text{b})\text{x}$$\text{x}=\frac{\text{a}^2-\text{b}^2}{\text{a}+\text{b}}$
$=\frac{(\text{a}+\text{b})(\text{a}-\text{b})}{\text{a}+\text{b}}=\text{a}-\text{b}$
View full question & answer→Question 221 Mark
(a - b) _________$= a^2- 2ab + b^2$
Answer$(a - b) (a - b) = a^2- 2ab + b^2$
Solution:
We know that, $(a - b)(a - b) = (a - b)^2= a^2- 2ab + b^2$
View full question & answer→Question 231 Mark
$(x + a)(x + b) = x^2+ (a + b)x + $________.
Answer$(x + a)(x + b) = x^2+ (a + b)x + ab.$
Solution:
We have, $(x + a)(x + b) = x^2+ bx + ax + ab = x^2+ (a + b)x + ab$
View full question & answer→Question 241 Mark
The product of two terms with unlike signs is ____________ a term.
AnswerThe product of two terms with unlike signs is Negative a term.Solution:
As the product of a positive term and a negative term is always negative.
View full question & answer→Question 251 Mark
The factors of $a^2- 2ab + b^2$ are $(a + b)$ and $(a + b).$
AnswerFalse.
Solution:
We have,
$a^2-2 a b+b^2=(a-b)^2=(a-b)(a-b)$
View full question & answer→Question 261 Mark
The coefficient of the term $-6x^2y^2$ is $-6.$
AnswerTrue.
Solution:
Since, the constant term (i.e. a number) present in the expression $-6x^2y^2$ is $-6.$
View full question & answer→Question 271 Mark
The factorisation of $2x + 4y$ is __________.
AnswerThe factorisation of $2x + 4y$ is $2(x + 2y).$
Soluiton:
We have,$ 2x + 4y = 2x + 2 × 2y = 2(x + 2y)$
View full question & answer→Question 281 Mark
Factorise the expressions and divide them as directed: $(3x^2- 48) ÷ (x - 4)$
Answer$(3x^2- 48) ÷ (x - 4)$
We have,$=\frac{3\text{x}^2-48}{\text{x}-4}$
$=\frac{3(\text{x}^2-16)}{\text{x}-4}$
$=\frac{3(\text{x}^2-4^2)}{\text{x}-4}$
$=\frac{3(\text{x}+4)(\text{x}-4)}{\text{x}-4} [\therefore\text{a}^2-\text{b}^2=(\text{a}+\text{b})(\text{a}-\text{b})$
$=3(\text{x}+4)$
View full question & answer→Question 291 Mark
Common factor of $ax^2+ bx$ is __________.
AnswerCommon factor of $ax^2+ bx$ is x.
Solution:
We have, $ax^2+ bx = x(ax + b)$
View full question & answer→Question 301 Mark
Common factor of $ 12 a^2 b^2+4 a b^2-32 $ is 4.
AnswerTrue.
Solution:
As we have,
$ 12 a^2 b^2+4 a b^2-32 $
$ =2 \times 2 \times 3 \times a \times a \times b \times b+2 \times 2 \times a \times b \times b-2^2 \times 2^3 $
$ =4\left(3 a^2 b^2+a b^2-8\right) $
Thus, the common factor is 4.
View full question & answer→Question 311 Mark
Factorise the expressions and divide them as directed:
$(9x^2- 4) ÷ (3x + 2)$
Answer$(9x^2- 4) ÷ (3x + 2)$
We have,$=\frac{9\text{x}^2-4}{3\text{x}+2}$
$=\frac{(3\text{x})^2-(2)^2}{3\text{x}+2}$
$=\frac{(3\text{x}+2)(3\text{x}-2)}{3\text{x}+2} [\therefore\text{a}^2-\text{b}^2=(\text{a}+\text{b})(\text{a}-\text{b})]$
$=3\text{x}-2$
View full question & answer→Question 321 Mark
Some of the factors of $\frac{\text{n}^2}{2}+\frac{\text{n}}{2}$ are $\frac{1}{2} n,$ and $(n + 1).$
Answer We have,$\frac{\text{n}^2}{2}+\frac{\text{n}}{2}$ $=\frac{\text{n}^2+\text{n}}{2}$
$=\frac{1}{2}\text{n}(\text{n}+1)$
The factors are $\frac{1}{2}\text{n}$ and $(n + 1).$
View full question & answer→Question 331 Mark
Area of a rectangular plot with sides $4x^2$ and $3y^2$ is __________.
AnswerArea of a rectangular plot with sides $4x^2$ and $3y^2$ is $12x^2y^2$.
Solution:
We know that,
Area of rectangle $=$ length $x$ breadth
Area of rectangular plot $= 4x^2× 3y^2$
$= (4 × 3)x^2y^2= 12x^2y^2$
View full question & answer→Question 341 Mark
$x^2+ (a + b)x + ab = (a + b)(x + ab)$
AnswerFalse.Solution:
As we know that,
$ x^2+ (a + b)x + ab = (x + a)(x + b)$
View full question & answer→Question 351 Mark
$(a + b)(a - b) = a^2- b^2$
AnswerTrue.
Solution:
We know that,
$(a + b)(a - b) = a \times a - a \times b + b \times a - b \times b$
$= a^2- b^2$
$= a^2- ab + ba - b^2$
View full question & answer→Question 361 Mark
Volume of a rectangular box with length $2x,$ breadth $3y$ and height $4z$ is _________.
AnswerVolume of a rectangular box with length $2x,$ breadth $3y$ and height $4z$ is $24xyz.$
Solution:
We know that, the volume of a rectangular box $V =$ Length $ x$ Breadth $x$ Height $= 2x × 3y × 4z = (2 × 3 × 4)xyz = 24xyz$
View full question & answer→Question 371 Mark
Common factor of $11 \mathrm{pq}^2, 121 \mathrm{p}^2 \mathrm{q}^3, 1331 \mathrm{p}^2 \mathrm{q}$ is $11 \mathrm{p}^2 \mathrm{q}^2$.
AnswerFalseSolution:
We have,
$ 11 p q^2=11 \times p \times q \times q$
$121 p^2 q^3=11 \times 11 \times p \times p \times q \times q \times q $
$ 1331 p^2 q=11 \times 11 \times 11 \times p \times p \times q$
View full question & answer→Question 381 Mark
The common factor method of factorisation for a polynomial is based on ___________ property.
AnswerThe common factor method of factorisation for a polynomial is based on Distributive property.Solution:
In this method, we regroup the terms in such a way, so that each term in the group contains a common literal or number or both.
View full question & answer→Question 391 Mark
Carry out the following divisions:
$17 a b^2 c^3 \div\left(-a b c^2\right)$
Answer$17 a b^2 c^3 \div\left(-a b c^2\right)$
$\frac{17\text{ab}^2\text{c}^3}{-\text{abc}^2}$
$=\frac{17\times\text{a}\times\text{b}\times\text{b}\times\text{c}\times\text{c}\times\text{c}}{-\text{a}\times\text{b}\times\text{c}\times\text{c}}$
$=-17\text{bc}$
View full question & answer→Question 401 Mark
$(a - b)^2+ \_\_\_\_\_\_\_\_\_\_\_\_ = a^2 - b^2$
Answer$(a - b)^2+ $ $2ab - 2b^2$ $= a^2- b^2$
Solution:
Let $(a-b)^2+x=a^2-b^2 $
$a^2+b^2-2 a b+x=a^2-b^2 $
$ x=a^2-b^2-\left(a^2+b^2-2 a b\right) $
$ =a^2-b^2-a^2-b^2+2 a b $
$ =2 a b-2 b^2 $
View full question & answer→Question 411 Mark
The side of the square of area $9y^2$ is __________.
AnswerThe side of the square of area $9y^2$ is 3y.
Solution:
Given,
Area of a square $= 9y^2$
We know that,
The area of a square with side $a = a^2$
$ a^2=9 y^2 $
$ a^2=(3 y)^2 $
$a = 3y$
View full question & answer→Question 421 Mark
Number of terms in the expression $a^2+ bc × d$ is ________.
AnswerNumber of terms in the expression $a^2+ bc × d$ is $a^2+ bcd$.
Solution:
We have,
$a^2+ bc × d = a^2+ bcd$
The number of terms in this expression is $2$ as $bcd$ is treated as a single term.
View full question & answer→Question 431 Mark
The product of two polynomials is a ________.
AnswerThe product of two polynomials is a polynomial.Solution:
As the product of two polynomials is again a polynomial.
View full question & answer→Question 441 Mark
$(a+b)^2=a^2+b^2$
AnswerFalse.Solution:
We have,
$(a+b)^2=a^2+b^2+ 2ab$
View full question & answer→Question 451 Mark
The product of two negative terms is a negative term.
AnswerFalse.Solution:
Since, the product of two negative terms is always a positive term, i.e. $(-) \times (-) = (+).$
View full question & answer→Question 461 Mark
Factorised form of $4y^2- 12y + 9$ is ________.
AnswerFactorised form of $4y^2- 12y + 9$ is $(2y - 3)(2y - 3).$
Solution:
Let $4y^2- 12y + 9 = (2y)^2- 2 × 2y × 3 + 3^2$
$= (2? − 3)^2$
$= (2y - 3)(2y - 3)$
View full question & answer→Question 471 Mark
On Simplification $\frac{3\text{x}+3}{3}$ = ________.
AnswerOn Simplification $\frac{3\text{x}+3}{3} = x + 1.$
Solution:
We have,$\frac{3\text{x}+3}{3}$
$=\frac{3\text{x}}{3}+\frac{3}{3}$
$=\text{x}+1$
View full question & answer→Question 481 Mark
$abc + bca + cab$ is a monomial.
AnswerTrue. Solution: The given expression seems to be a trinomial but it is not as it contains three like terms which can be added to form a monomial, i.e. $abc + abc + abc = 3abc$
View full question & answer→Question 491 Mark
The value of $(a + 1) (a - 1) (a^2+ 1)$ is $a^4- 1.$
AnswerTrue.
Solution:
We have,
$ (a+1)(a-1)\left(a^2+1\right) $
$ =\left(a^2-1\right)\left(a^2+1\right) $
$ =\left(a^2\right)^2-1^2 $
$ =a^4-1 $
View full question & answer→Question 501 Mark
$(a-b)^2=a^2+b^2$
AnswerFalse.Solution:
We have,
$(a-b)^2=a^2+b^2-2 a b$
View full question & answer→Question 511 Mark
Factorised form of $18mn + 10mnp$ is ________.
AnswerFactorised form of $18mn + 10mnp$ is $2mn(9 + 5p).$
Solution:
We have, $18mn + 10mnp = 2 \times 9 \times m \times n + 2 \times 5 \times m \times n \times p = 2mn(9 + 5p)$
View full question & answer→Question 521 Mark
Factorisation of $-3a2 + 3ab + 3ac$ is $3a(-a - b - c).$
AnswerFalse.
Solution:
We have,
$-3a^2+ 3ab + 3ac = 3a(-a + b + c)$
View full question & answer→Question 531 Mark
An equation is true for all values of its variables.
AnswerFalse.Solution:
As equation is true only for some values of its variables, e.g. $2x - 4 = 0$ is true, only for $x = 2.$
View full question & answer→Question 541 Mark
The value of $p$ for $51^2-49^2=100 p$ is $2.$
AnswerWe have,
$51^2-49^2=100 p$
$(51 + 49)(51 - 49)= 100p$
$100 × 2= 100p$
$p = 2$
View full question & answer→Question 551 Mark
The sum of areas of two squares with sides $4a$ and $4b$ is _______.
AnswerThe sum of areas of two squares with sides $4a$ and $4b$ is $16(a^2+ b^2)$.
Solution:
Area of square $= (side)^2$
Area of the square whose one side is $4a = (4a)^2= 16a^2$
Area of the square with side $4b = (4b)^2= 16b^2$
Sum of the areas $= 16a^2+ 16b^2$
$= 16(a^2+ b^2)$
View full question & answer→Question 561 Mark
Carry out the following divisions:
$51 x^3 y^2 z \div 17 x y z$
Answer$51 x^3 y^2 z \div 17 x y z$
$=\frac{51\text{y}^3\text{y}^2\text{z}}{17\text{x}\text{yz}}$
$=\frac{17\times3\times\text{x}\times\text{x}\times\text{x}\times\text{y}\times\text{y}\times\text{z}}{17\times\text{x}\times\text{y}\times\text{z}}$
View full question & answer→Question 571 Mark
Factorised form of $ p^2+ 30p + 216$ is $(p + 18) (p - 12).$
AnswerFalse.
Solution:
We have,
$ p^2+30 p+216=p^2+(12+8) p+216 $
$ =p^2+12 p+18 p+216 $
$= p(p + 12) + 18(p + 12)$
$= (p + 18)(p + 12)$
View full question & answer→Question 581 Mark
The coefficient in $-37abc$ is __________.
AnswerThe coefficient in $-37abc$ is $-37.$
Solution:
The constant term (with their sign) involved in term of an algebraic expression is called the numerical coefficient of that term.
View full question & answer→