Question 11 Mark
Using suitable identity, evaluate the following:
$999^2$
Answer$(999)^2=(1000-1)^2=(1000)^2-2 \times(1000) \times 1+1^2$
$=1000000-2000+1$
$=998001$
View full question & answer→Question 21 Mark
Classify the following as a constant, linear, quadratic and cubic polynomial:
$2-x^2+x^3$
AnswerPolynomial $2-x^2+x^3$ is a cubic polynomial, because maximum exponent of $x$ is $3$ .
View full question & answer→Question 31 Mark
Classify the following as a constant, linear, quadratic and cubic polynomial: $5\text{t}-\sqrt{7}$
AnswerPolynomial $5\text{t}-\sqrt{7}$ is a linear polynomial, because maximum exponent of $t$ is $1.$
View full question & answer→Question 41 Mark
Which of the following expression are polynomials? $\sqrt{3}\text{x}^2-2\text{x}$
AnswerPolynomial, because the exponent of the variable of $\sqrt{3}\text{x}^2-2\text{x}$ is a whole number.
View full question & answer→Question 51 Mark
Classify the following polynomials as polynomials in one variable, two variables etc.
$x^2-2 x y+y^2+1$
Answer$x^2-2 x y+y^2+1$ is a polynomial in three variable.
View full question & answer→Question 61 Mark
Which of the following expression are polynomials$? 8$
AnswerPolynomial, because the exponent of the variable of $8$ or $8x^\circ $ is $0$ which is a whole number.
View full question & answer→Question 71 Mark
Classify the following as a constant, linear, quadratic and cubic polynomial:
$3 x^3$
AnswerPolynomial $3 x^3$ is a cubic polynomial, because maximum exponent of $x$ is $3 .$
View full question & answer→Question 81 Mark
Verify whether the following are True or False: $-3$ is a zero of $x - 3$
AnswerBecause zero of $x - 3$ is $3. [\therefore x - 3 = 0 ⇒ x = 3]$
View full question & answer→Question 91 Mark
Write the coefficient of $x^2$ in the following:
$\frac{\pi}{6}\text{x}+\text{x}^2-1$
AnswerThe coefficient of $x^2$ in $\frac{\pi}{6}\text{x}+\text{x}^2-1$ is $1$.
View full question & answer→Question 101 Mark
Classify the following as a constant, linear, quadratic and cubic polynomial:
$y^3-y$
AnswerPolynomial $y^3-y$ is a cubic polynomial, because maximum exponent of $y$ is $3 .$
View full question & answer→Question 111 Mark
Classify the following polynomials as polynomials in one variable, two variables etc. $xy + yz + zx$
Answer$xy + yz + zx$ is a polynomial in three variable.
View full question & answer→Question 121 Mark
Which of the following expression are polynomials?
$\frac{1}{7}\text{a}^3-\frac{2}{\sqrt{3}}\text{a}^2+4\text{a}-7$
Answer Polyonimial, because the exponent of the variable of $\frac{1}{7}\text{a}^3-\frac{2}{\sqrt{3}}\text{a}^2+4\text{a}-7$ is a whole number.
View full question & answer→Question 131 Mark
Factorise:
$6 x^2+7 x-3$
Answer$6 x^2+7 x-3=6 x^2+9 x-2 x-3$
$=3 x(2 x+3)-1(2 x+3)=(3 x-1)(2 x+3)[\text { by splitting middle term }]$
View full question & answer→Question 141 Mark
Which of the following expression are polynomials$?$
$\frac{1}{\text{x}+1}$
AnswerNot Polynomial, as the polynomial is expressed as $a_0+a_1 x+a_2 x+\ldots a_n x^n$,
where $a_0, a_1, a_{2, \ldots}, a_n$ are constants.
Now, $\mathrm{f}(\mathrm{x})=\frac{\mathrm{p}(\mathrm{x})}{\mathrm{q}(\mathrm{x})}$ is a rational expression where $\mathrm{q}(\mathrm{x}) \neq 0, \mathrm{p}(\mathrm{x})$ and $\mathrm{q}(\mathrm{x})$ are polyonimials.
Hence, $\frac{1}{x+1}$ is a rational expression but not a polyonimial.
View full question & answer→Question 151 Mark
Give an example of a polynomial, which is: Monomial of degree $1$
Answer$3x$ is monomial of degree $1.$
View full question & answer→Question 161 Mark
Find the zeroes of the polynomial in following: $h(y) = 2y$
AnswerGiven, polynomial is $h(y) = 2y$ For zero of polynomial,
put $h(y) = 0 2y = 0$
Hence, zero of polynomial is $0.$
View full question & answer→Question 171 Mark
Factorise:
$2 x^2-7 x-15$
Answer$2 x^2-7 x-15=2 x^2-10 x+3 x-15$
$=2 x(x-5)+3(x-5)=(2 x+3)(x-5)$ [by splitting middle term]
View full question & answer→Question 181 Mark
Find the zeroes of the polynomial in following:
$g(x) = 3 - 6x$
AnswerGiven, polynomial is
$g(x) = 3 - 6x$
For zero of polynomial, put $g(x) = 0$
$3 - 6x = 0 ⇒ 6x = 3$
$\Rightarrow\text{x}=\frac{1}{2}$
Hence, zero of polynomial is $\text{x}=\frac{1}{2}.$
View full question & answer→Question 191 Mark
Determine the degree of the following polynomial:
$x^3-9 x+3 x^5$
AnswerSince the highest power of $x$ is $5$ , the degree of the polynomial $x^3-9 x+3 x^5$ is $5$.
View full question & answer→Question 201 Mark
Verify whether the following are True or False:
$0$ and $2$ are the zeroes of $t^2-2$
AnswerNow，
$\mathrm{t}^2-2 \mathrm{t}=\mathrm{t}(\mathrm{t}-2)$
Hence, zeroes of $\mathrm{t}^2-2 \mathrm{t}$ are $0$ and $2 $.
View full question & answer→Question 211 Mark
Which of the following expression are polynomials? $\frac{1}{2\text{x}}$
AnswerNot polyonimial, because the exponent of the variable of is $\frac{1}{2\text{x}}$ or $\frac{1}{2}\text{x}^{-1}$ is $-1$ which is not a whole number.
View full question & answer→Question 221 Mark
Which of the following expression are polynomials? $\frac{(\text{x}-2)(\text{x}-4)}{\text{x}}$
AnswerNot Polynomial, because the exponent of the variable of $\frac{(\text{x}-2)(\text{x}-4)}{\text{x}}=\frac{\text{x}^2-6\text{x}+8}{\text{x}}=\text{x}-6+8\text{x}^{-1}$ is $-1$ which is not whole number.
View full question & answer→Question 231 Mark
Verify whether the following are True or False:
$-\frac{1}{3}$ is a zero of $3x + 1$
AnswerBecause zero of $3x + 1$ is $-\frac{1}{3}$. $\big[\therefore 3x + 1 = 0 ⇒ x =-\frac{1}{3}\big]$
View full question & answer→Question 241 Mark
Find the zeroes of the polynomial in following:
$q(x) = 2x - 7$
AnswerGiven, polynomial is
$q(x) = 2x - 7$
For zero of polynomial, put $q(x) = 0$
$2x - 7 = 0 ⇒ 2x = 7$
$\Rightarrow\text{x}=\frac{7}{2}$
Hence, zero of polynomial is $\text{x}=\frac{7}{2}.$
View full question & answer→Question 251 Mark
Classify the following as a constant, linear, quadratic and cubic polynomial:
$4-5 y^2$
AnswerPolynomial $4-5 y^2$ is a quadratic polynomial, because maximum exponent of $y$ is $2 .$
View full question & answer→Question 261 Mark
Determine the degree of the following polynomial:
$y^3\left(1-y^4\right)$
Answer$y^3\left(1-y^4\right)=y^3-y^7$ Since the highest power of $x$ is $7 ,$ the degree of the polynomial is $7 .$
View full question & answer→Question 271 Mark
Determine the degree of the following polynomial:
$–10$
Answer$-10$ is a non-zero constant. A non-zero constant term is always regarded as having degree $0.$
View full question & answer→Question 281 Mark
Classify the following as a constant, linear, quadratic and cubic polynomial:
$1+x+x^2$
AnswerPolynomial $1+x+x^2$ is a quadratic polynomial, because maximum exponent of $x$ is $2$.
View full question & answer→Question 291 Mark
Classify the following polynomials as polynomials in one variable, two variables etc.
$y^3-5 y$
Answer$y^3-5 y$ is a polynomial in one variable.
View full question & answer→Question 301 Mark
Factorise the following:
$9 y^2-66 y z+121 z^2$
Answer$9 y^2-66 y z+121 z^2=(3 y)^2-2 \times 3 y \times 11 z+(11 z)^2$
$=(3 y-11 z)^2\left[\text { Using identity, } a^2-2 a b+b^2=(a-b)^2\right]$
View full question & answer→Question 311 Mark
Determine the degree of following polynomial:
$2x - 1$
AnswerSince the highest power of $x$ is $1,$ the degree of the polynomial $2x - 1$ is $1.$
View full question & answer→Question 321 Mark
Which of the following expression are polynomials?
$1-\sqrt{5\text{x}}$
Answer Not Polynomial, because the exponent of the variable of $1-\sqrt{5\text{x}}$ or $1-\sqrt{5\text{x}}^{\frac{1}{2}}$ is $\frac{1}{2}$ which is not a whole number.
View full question & answer→Question 331 Mark
Factorise the following: $1-64 a^3-12 a+48 a^2$
Answer$1-64 a^3-12 a+48 a^2=(1)^3-(4 a)^3-3 \times 1^2 \times 4 a+3 \times 1 \times(4 a)^2$
${\left[\text { Using identity, }(a-b)^3=a^3-b^3+3 a(-b)(a-b)\right]}$
$=(1-4 a)^3=(1-4 a)(1-4 a)(1-4 a)$
View full question & answer→Question 341 Mark
Which of the following expression are polynomials? $\frac{1}{5\text{x}^{-2}}+5\text{x}+7$
AnswerPolynomial, because the exponent of the variable of $\frac{1}{5\text{x}^{-2}}+5\text{x}+7=\frac{1}{5}\text{x}^2+5\text{x}+7,$ is a which is whole number.
View full question & answer→Question 351 Mark
Factorise the following:
$4 x^2+20 x+25$
Answer$4 x^2+20 x+25=(2 x)^2+2 \times 2 x \times 5+(5)^2$
$=(2 x+5)^2\left[\text { Using identity, } a^2+2 a b+b^2=(a+b)^2\right]$
View full question & answer→Question 361 Mark
Write the coefficient of $x^2$ in the following:
$(2 x-5)\left(2 x^2-3 x+1\right)$
Answer$\text { Let } p(x)=(2 x-5)\left(2 x^2-3 x+1\right)$
$=2 x\left(2 x^2-3 x+1\right)-5\left(2 x^2-3 x+1\right)$
$=4 x^3-6 x^2+2 x-10 x^2+15 x-5$
$=4 x^3-16 x^2+17 x-5$
Hence, the cofficient of $x^2$ in $p(x)$ is $-16$
View full question & answer→Question 371 Mark
Classify the following polynomials as polynomials in one variable, two variables etc.
$x^2+x+1$
Answer$x^2+x+1$ is a polynomial in one variable.
View full question & answer→Question 381 Mark
Factorise:
$84-2 r-2 r^2$
Answer$84-2 r-2 r^2$
$=-2\left(r^2+r-42\right)$
$=-2\left(r^2+7 r-6 r-42\right)$
$=-2[r(r+7)-6(r+7)]$
$=-2(r-6)(r+7)=2(6-r)(r+7)$
View full question & answer→Question 391 Mark
Classify the following as a constant, linear, quadratic and cubic polynomial: $2 + x$
AnswerPolynomial $2 + x$ is a linear polynomial, because maximum exponent of $x$ is $1.$
View full question & answer→Question 401 Mark
Verify whether the following are True or False:
$-3$ is a zero of $y^2+y-6$
AnswerNow,
$y^2+y-6=y^2+3 y-2 y-6[$ by splitting middle term]
$=y(y+3)-2(y+3)$
$=(y-2)(y+3)$
Hence, zeroes of $y^2+y-6$ are $2$ and $-3 .$
View full question & answer→Question 411 Mark
Classify the following as a constant, linear, quadratic and cubic polynomial:
$t^2$
AnswerPolynomial $t^2$ is a quadratic polynomial, because maximum exponent of $t$ is $2.$
View full question & answer→Question 421 Mark
Write the coefficient of $x^2$ in the following:
$3 x-5$
AnswerThe coefficient of $x^2$ in $3 x-5$ is $0$.
View full question & answer→Question 431 Mark
Write the coefficient of $x^2$ in the following:
$(x-1)(3 x-4)$
Answer$\text { Let } p(x)=(x-1)(3 x-4)$
$=3 x^2-7 x+4$
$=3 x^2-4 x-3 x+4$
Hence, the cofficient of $x^2$ in $p(x)$ is $3 .$
View full question & answer→Question 441 Mark
Find $p(0), p(1), p(-2)$ for the following polynomial: $p(x)=10 x-4 x^2-3$
AnswerGiven, polynomial is $p(x)=10 x-4 x^2-3$
On putting $x=0,1$ and $-2$ , respectively in $Eq. (i),$
we get $p(0)=10(0)-4(0)^2$ -$3=0-0-3=-3 p(1)=10(1)-4(1)^2-3$
$=10-4-3=10-7=3$ and $p(-2)$
$=10(-2)-4(-2)^2-3$
$=-20-4 \times 4-3$
$=-20$ $-16-3=-39$
Hence, the values of $p(0), p(1)$ and $p(-2)$ are respectively, $-3,3$ and $-39$ .
View full question & answer→Question 451 Mark
Give an example of a polynomial, which is:
Trinomial of degree $2$
Answer$5 x^2+3 x-1$ is a trinomial of degree $2$.
View full question & answer→Question 461 Mark
Find the zeroes of the polynomial in following: $p(x) = x - 4$
AnswerGiven, polynomial is $p(x) = x - 4$ For zero of polynomial, put $p(x) = x - 4 = 0 ⇒ x = 4$ Hence, zero of polynomial is $4.$
View full question & answer→Question 471 Mark
Factorise:
$x^2+9 x+18$
Answer$x^2+9 x+18=x^2+6 x+3 x+18$
$=x(x+6)+3(x+6)=(x+3)(x+6)[\text { by splitting middle term }]$
View full question & answer→Question 481 Mark
Classify the following as a constant, linear, quadratic and cubic polynomial: $3$
AnswerPolynomial $3$ is a constant polynomial, because the exponent of variable is $0.$
View full question & answer→Question 491 Mark
Find $p(0), p(1), p(-2)$ for the following polynomial: $p(y) = (y + 2)(y - 2)$
AnswerGiven, polynomial is $p(y) = (y + 2)(y - 2)$
On putting $y = 0, 1$ and $-2$, respectively in Eq. $(i),$
We get $p(0) = (0 + 2)(0 - 2)= -4 $
$p(1) = (1 + 2)(1 - 2) = 3 × (-1) = -3$ and
$p(-2) = (-2 + 2)(-2 - 2) = 0(-4) = 0$
Hence, the values of $p(0), p(1)$ and $p(-2)$ are respectively, $-4, -3$ and $0.$
View full question & answer→Question 501 Mark
Classify the following as a constant, linear, quadratic and cubic polynomial: $\sqrt{2}\text{x}-1$
AnswerPolynomial $\sqrt{2}\text{x}-1$ is a linear polynomial, because maximum exponent of $x$ is $1.$
View full question & answer→Question 511 Mark
Verify whether the following are True or False: $\frac{-4}{5}$ is a zero of $4 - 5y$
AnswerBecause zero of $4 - 5y$ is $\frac{4}{5}.$
$\big[\therefore 4 - 5y = 0 ⇒ y =\frac{4}{5}\big]$
View full question & answer→Question 521 Mark
Give an example of a polynomial, which is:
Binomial of degree $20$
Answer$x^{20}-7$ is a binomial of degree $20$.
View full question & answer→