3 Marks Question

Question 13 Marks

Identify the coefficients of the terms of following expressions.
4x-3y, a + b + 5, 2y + 5, 2xy

Answer

We know that the numerical factor of a term is said to be the numerical coefficient or simply the coefficient of the same term. So, the terms and corresponding coefficients are given below.

S. No.	Expression	Terms	Coefficient
(i)	4x - 3y	4x, -3y	4, -3
(ii)	a + b + 5	a, b, 5	1, 1, 5
(iii)	2y + 5	2y, 5	2, 5
(iv)	2xy	2xy	2

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Question 23 Marks

Classify the following expressions as a monomial, a binomial or a trinomial. a, a + b, ab + a + b, ab + a + b - 5 xy, xy + 5, $5 x^2-x+2$, 4pq - 3q + 5p, 7, 4m - 7n + 10, 4mn + 7.

Answer

Given expressions and their type are given below.

Expression	Type of polynomial
a	Monomial
a+b	Binomial
ab+a+b	Trinomial
ab+a+b-5	Polynomial
xy	Monomial
xy+5	Binomial
5x^(2)-x+2	Trinomial
4pq-3q+5p	Trinomial
7	Monomial
4m-7n+10	Trinomial
4mn+7	Binomial

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Question 33 Marks

Simplify the expression $2\left(a^2+a b\right)+3-a b$ and find Its value, when a = 5 and b =-3.

Answer

We have, $2\left(a^2+a b\right)+3-a b=2 a^2+2 a b+3-a b$
$=2 a^2+2 a b-a b+3$ [rearranging terms]
$=2 a^2+a b(2-1)+3=2 a^2+a b+3$
On putting a = 5 and b = -3 we get
The value of $2 a^2+a b+3=2(5)^2+5 \times(-3)+3$
= 2 × 5 × 5 + 5(-3) + 3
= 50 - 15 + 3 = 38
Hence, the simplified expression is $2 a^2+a b+3$ and its required value is 38.

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Question 43 Marks

What should be the value of a, if the value of $2 x^2+x-a$ equals to 5 , when $x=0$?

Answer

Given, the value of $2 x^2+x-a$ is equal to 5 i.e.
$2 x^2+x-a=5$
On putting x = 0 we get $2(0)^2+(0)-a=5$
$\Rightarrow$ 0 + 0 - a = 5 $\Rightarrow$ -a = 5 $\Rightarrow$ a = -5
Hence, the required value of a is -5.

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Question 53 Marks

If a = 2 b = -2 then find the value of
(i) $a^2+b^2$$\quad$ (ii) $a^2+a b+b^2$$\quad$ (iii) $a^2-b^2$

Answer

(i) On substituting $a=2$ and $b=-2$ in $a^2+b^2$, we get
$a^2+b^2=(2)^2+(-2)^2$
= 4 + 4 = 8
(ii) On substituting $a=2$ and $b=-2$ in $a^2+a b+b^2$, we get
$a^2+a b+b^2=(2)^2+2 \times(-2)+(-2)^2$
= 4 - 4 + 4 = 4
(iii) On substituting $a=2$ and $b=-2$ in $a^2-b^2$, we get
$a^2-b^2=(2)^2-(-2)^2=4-4=0$

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Question 63 Marks

Identify terms which contain $y^2$ and give the coefficient of $y^2$
(a) $8-x y^2$
(b) $5 y^2+7 x$
(c) $2 x^2 y-15 x y^2+7 y^2$

Answer

(a) In the expression $8-x y^2$, the term which contains $y^2$ is $-x y^2$, so the coefficient of $y^2$ is $-x$.
(b) In the expression $5 y^2+7 x$, the term which contains $y^2$ is $5 y^2$, so the coefficient of $y^2$ is 5.
(c) In the expression $2 x^2 y-15 x y^2+7 y^2$, the terms which contain $y^2$ are $-15 x y^2$ and $7 y^2$, so the coefficients of $y^2$ are $-15 x$ and 7 , respectively.

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Question 73 Marks

Classify the following as monomials, binomials and trinomials.
(i) $2 x^2-5$
(ii) $\frac{3}{22} x y z$
(iii) $2 x^2-7 x$
(iv) $3 x^2-2$
(v) $4 x^3+3 x^2+6$
(vi) $6 x^2-3 x+1$

Answer

(i) Given, $2 x^2-5$
The expression $2 x^2-5$ is having 2 terms,
i.e. $2 x^2$ and -5 . So, it is a binomial.
(ii) Given, $\frac{3}{22} x y z$
The expression $\frac{3}{22} x y z$ is having only one term,
i.e. $\frac{3 x y z}{22}$. So, it is a monomial.
(iii) Given, $2 x^2-7 x$
The expression $2 x^2-7 x$ is having 2 terms,
i.e. $2 x^2$ and $-7 x$. So, it is a binomial.
(iv) Given, $3 x^2-2$
The expression $3 x^2-2$ is having only two terms,
i.e. $3 x^2$ and -2 . So, it is a binomial.
(v) Given, $4 x^3+3 x^2+6$
The expression $4 x^3+3 x^2+6$ is having three terms,
i.e. $4 x^3, 3 x^2$ and 6 . So, it is a trinomial.
(vi) Given, $6 x^2-3 x+1$
The expression $6 x^2-3 x+1$ is having three terms,
i.e. $4 x^3, 3 x^2$ and 6 . So, it is a trinomial.

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Question 83 Marks

Cross Number Puzzle
Rohit has to solve the given cross number puzzle to qualify for the next round of Mathematics quiz competition. Help him by evaluating the values of given expression at x = 0, y = 1, z = 2. Also help him to fill the cross number along across and downward with the help of given clues. (numbers to be written in words)
Across
(a) xy + yz + zx
(b) $x^2 y^2+z^2-2 x y z$
(c) 8 - (x + y)
(d) $x^2 y^3+y^2 z^3+z^2 x^3$
Down
(e) $x^2-2 x y(y-z)$
(f) $\frac{x^3+y^3+z^3}{3}$
(g) $x^3+y^3+z^3-2 y z^2$
(h) 2x+2y+2z

Answer

Across	Down
(a) Two	(e) Zero
(b) Four	(f) Three
(c) Seven	(g) One
(d) Eight	(h) Six

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Question 93 Marks

Write the factors of the following binomials.
(i) $4 x^2 y+5 x y$
(ii) $5 x^2 y^2+15 x y^2$

Answer

(i) Given, binomial is $4 x^2 y+5 x y$.
So, terms are $4 x^2 y$ and $5 x y$.
Hence, factors of $4 x^2 y$ is $4 \times x \times x \times y$
and factors of $5 x y$ is $5 \times x \times y$.
(ii) Given, binomial is $5 x^2 y^2+15 x y^2$.
So, terms are $5 x^2 y^2$ and $15 x y^2$.
Hence, factors of $5 x^2 y^2$ is $5 \times x \times x \times y \times y$
and factors of $15 x y^2$ is $15 \times x \times y \times y$
or $3 \times 5 \times x \times y \times y$.

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Question 103 Marks

Write the factors of the following monomials.
(i) $2 x^2 y z$
(ii) $6 x y^2 z^2$

Answer

(i) Given, $2 x^2 y z$
Factors of $2 x^2 y z$ will be $2 \times x \times x \times y \times z$.
(ii) Given, $6 x y^2 z^2$
Factors of $6 x y^2 z^2$ will be $6 \times x \times y \times y \times z \times z$.

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Question 113 Marks

Identify the like terms in the following algebraic expressions.
(i) $5 x^2-6 x y+6 x^2+1$
(ii) $2 x^2+3 y^2+5 x^2-9 y^2-3 x y+5 x y$

Answer

(i) Given, algebraic expression $5 x^2-6 x y+6 x^2+1$ Here, $5 x^2$ and $6 x^2$ are like terms.
(ii) Given, algebraic expression
$2 x^2+3 y^2+5 x^2-9 y^2-3 x y+5 x y$
Here, $2 x^2, 5 x^2$ are like terms and $3 y^2,-9 y^2$ are like terms.
Also, 3xy and 5xy are like terms.

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Question 123 Marks

State whether each pair of terms given below is like or unlike.
(i) $3 x^2 y$ and $5 x y^2$
(ii) $4 x^2 y^2 z^2$ and $5 x y z^2$
(iii) $6 x y$ and $3 x y$

Answer

(i) Given, $3 x^2 y$ and $5 x y^2$
Here, $3 x^2 y$ and $5 x y^2$ is unlike pair.
(ii) Given, $4 x^2 y^2 z^2$ and $5 x y z^2$
Here, $4 x^2 y^2 z^2$ and $5 x y z^2$ is unlike pair.
(iii) Given, 6xy and 3xy
Here, 6xy and 3xy is like pair.

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Question 133 Marks

Identify the terms and represent in the form of tree diagram
(i) $2 x^2 y+3 x y^2$
(ii) $5 a b c+7 p^3 q^3$

Answer

(i) Given, $2 x^2 y+3 x y^2$
Total terms $=2$, i.e. $2 x^2 y$ and $3 x y^2$
Tree diagram

(ii) Given, $5 a b c+7 p^3 q^3$
Total terms $=2$ i.e. $5 a b c$ and $7 p^3 q^3$
Tree diagram

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