Algebraic Expressions

Q. Question 1 (x)
Find the zeroes of the following polynomials by factorization method and verify the relations between the zeroes and the coefficient of the polynomials

(x) $7y^2–\frac{11}{3}y-\frac{2}{3}$

Q.

Find the HCF of $81$ and $237$ and express it as a linear combination of $81$ and $237$. What does inear combination mean?

Q.

If $k$ is said to be a zero of a polynomial function $P\left(x\right)$, then $P\left(k\right)=$ _____.

Q.

If (x – p) and (x – q) are the factors of the polynomial P(x) = $x^2$ + ax + b =0, then P(x) can be written as:

1. (x + p) (x + q)

2. (x +p ) ( x – q)

3. (x – p) (x – q)

4. (x – p) ( x + q)

Q. If p, q and r are the roots of a cubic polynomial $a{x}^3+b{x}^2+cx+d$, then what will be the value of pq + qr + rp?

1. $-\frac{b}{a}$
2. $-\frac{d}{a}$
3. $\frac{b}{a}$
4. $\frac{c}{a}$

Q. Which of the following is a constant?

1. 7x - 6x - 2x + 6 - 4
2. $\frac{21x}{7x}$
3. 7x - 5x + 2
4. 12x -5 + 16x + 5

Q.

If (x-p) and (x-q) are the factors of the expression $a{x}^2+bx+c$, then the values of $a$ and $b$ are:

1. 1, p + q

2. 1, -(p+q)

3. None of these

4. -1, -(p + q)

Q. Arrange the polynomials in the increasing order of their degree (bottom to top).

1. $2x-1$
2. $-10$
3. $x^3-9x+3x^5$
4. $y^3(1-y^4)$

Q. 5 chairs and 4 tables together cost Rs.5600, while 4 chairs and 3 tables together cost Rs.4340. Find the cost (in rupees) of a chair and that of a table respectively.

1. 700, 560
2. 700, 700
3. 560, 560
4. 560, 700

Q.

Verify that $-5$ and $7$ are the zeroes of the polynomial $p\left(x\right)=\left(x+5\right)\left(x-7\right)$.

Q. The expression $a^3+b^3+c^3-3abc$ can be expressed as a product of two expressions. What is the product?

1. $(a+b+c)(a^2-b^2-c^2-ab-bc-ca)$
2. $(a-b-c)(a^2+b^2+c^2-ab-bc-ca)$
3. $(a+b+c)(a^2+b^2+c^2+ab+bc+ca)$
4. $(a+b+c)(a^2+b^2+c^2-ab-bc-ca)$

Q. Question 1 (ii)
Find the zeroes of the following polynomials by factorization method and verify the relations between the zeroes and the coefficient of the polynomials
(ii) $3x^2+4x–4$

Q. Expansion of $a^3+b^3+c^3-3abc$ is:

1. $\frac{1}{2}(a+b+c)\left[\right(a-b{)}^2+(b-c{)}^2+(c-a{)}^2]$
2. $(a+b+c)(a^2+b^2+c^2-ab-bc-ca$
3. Both (A) & (B)
4. None of the above

Q. solve for real x (x-3) (x-7) (x-12) <0

Q. Question 1 (iii)
Find the zeroes of the following polynomials by factorization method and verify the relations between the zeroes and the coefficient of the polynomials
(iii) $5t^2+12t+7$

Q. On solving $3x-4y+11=0\mathrm{and}7\mathrm{x}+11\mathrm{y}-15=0,$ we get:
(a) x = 1, y = −2
(b) x = −1, y = 2
(c) x = 0, y = $\frac{15}{11}$
(d) x = $\frac{-11}{3}$, y = 0

Q. Simplify:
$(2x-y)+(2y-3x)+(3y-x)$

Q.

If $x+y+4=0,$ find the value of $x^3+y^3-12xy+64.$

Q.

Find the remainder when $27x^3-9x^2+5x-1$ is divided by $3x+1$

1. x – y

2. x² + y

3. x² – y

4. x + y

Q. If $\frac{2}{x}+\frac{3}{y}=6\mathrm{and}\frac{1}{x}+\frac{1}{2y}=2$ then
(a) $x=1,y=\frac{2}{3}$ (b) $x=\frac{2}{3},y=1$ (c) $x=1,y=\frac{3}{2}$ (d) $x=\frac{3}{2},y=1$

Q.

If ‘p’ and ‘q’ are the zeroes of the polynomial P(x) = $x^2$ + ax + b =0, then P(x) can be written as

1. (x + p) (x + q)

2. (x – p)(x – q)

3. (x +p )( x – q)

4. (x – p) ( x + q)

Q. Mark the number of terms of the polynomial $3a-2a^2+3+4a^3$.

1. 4

Q. Evaluate $\frac{(8a^3-27b^3)}{(64x^3+y^3)}$, where $(64x^3+y^3)\ne 0$.

1. $\frac{(2a-3b)(4a^2+6ab+9b^2)}{(4x+y)(16x^2-4xy+y^2)}$
2. $\frac{(2a-3b)(4a^2+6ab+9b^2)}{(4x-y)(16x^2-4xy+y^2)}$
3. $\frac{(2a+3b)(4a^2+6ab+9b^2)}{(4x+y)(16x^2-4xy+y^2)}$
4. $\frac{(2a+3b)(4a^2+6ab+9b^2)}{(4x-y)(16x^2-4xy+y^2)}$

Q. If $p\left(x\right)=4x^2-4x-80$ and $q\left(x\right)=8x^2+24x-32$, find their HCF.

Q.

Verify $\frac{2}{7}$ is a zero of the polynomial $r\left(x\right)=2-7x$.

Q. x^2 + xy + yz + zx = 30
y^2 + xy + yz + zx = 15
z^2 + xy + yz + zx = 18
Find x^2 + y^2 + z^2

Q. Solve:|2/x-4|>1, x not equal to 4

Q. The degree of zero polynomial is .

1. \N
2. 1
3. undefined

Q. $x^2+x^3+3x^2+1$ is a .

1. Monomial
2. Binomial
3. Trinomial

Q. State true or false
$(a-2b+5c)(a-b)-(a-b-c)(2a+3c)+(6a+b)(2c-3a-5b)$=$19ca-37ab-19a^2$

1. True
2. False