5 Marks Questions

Question 15 Marks

Identify the numerical coefficients of terms (other than constants) in the following expressions.
(i) $5-3 t^2$
(ii) $1+t+t^2+t^3$
(iii) x + 2xy + 3y
(iv) 100m + 1000n
(v) $-p^2 q^2+7 p q$
(vi) 1.2a + 0.8b
(vii) $3.14 r^2$
(viii) 2(l + b)
(ix) $0.1 y+0.01 y^2$

Answer

(i) In the expression $5-3 t^2$, term with variable is $-3 t^2$. So, its numerical coefficient is -3.
(ii) In the expression $1+t+t^2+t^3$, terms with variables are t, $t^2$ and $t^3$. So, their numerical coefficients are 1, 1 and 1, respectively.
(iii) In the expression x + 2xy + 3y terms with variables are x, 2xy and 3y. So, their numerical coefficients are 1, 2 and 3, respectively.
(iv) In the expression 100m + 1000n terms with variables are 100m and 1000n. So, their numerical coefficients are 100 and 1000, respectively.
(v) In the expression $-p^2 q^2+7 p q$, terms with variables are $-p^2 q^2$ and 7pq. So, their numerical coefficients are-1 and 7, respectively.
(vi) In the expression 1.2a + 0.8b terms with variables are 1.2a and 0.8b. So, their numerical coefficients are 1.2 and 0.8, respectively.
(vii) In the expression $3.14 r^2$, term with variable is $314 r^2$. So, its numerical coefficient is 3.14.
(viii) In the expression $2(l+b)$, terms with variables are 21 and 2b. So, their numerical coefficients are 2 and 2, respectively.
(ix) In the expression $0.1 y+0.01 y^2$, terms with variables are 0.1 y and $0.01 y^2$. So, their numerical coefficients are-1 and 7, respectively.

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Question 25 Marks

A wire is (7x - 3) m long. A length of (3x - 4) m is cut for use. Now, answer the following questions.
(i) How much wire is left?
(ii) If this left out wire is used for making an equilateral triangle, then what is the length of each side of the triangle so formed?

Answer

A wire is (7x - 3) m long.
Wire is cut (3x - 4) m for use.
(i) Wire left = Total wire - Wire cut for use
= (7x - 3) - (3x - 4) = 4x + 1
Thus, wire left is (4x + 1) m.
(ii) Wire of (4x + 1) m long is formed in the shape of equilateral triangle. In equilateral triangle, all sides are equal.
$\therefore$ Length of each side = $\frac{1}{3}(4 x+1)$ [as wire is divided into 3 equal parts]

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Question 35 Marks

(i) Critical Thinking Write two different algebraic expressions for the word phrase
$\left(\frac{1}{4}\right)$ of the sum of $x$ and 7.
(ii) What's the Error? A student wrote an algebraic expression for "5 less than a number n divided by $3^{\prime \prime}$ as $\frac{n}{3}-5$. What error did the student make?
(iii) Write About If Shashi used addition to solve a word problem about the weekly cost of commuting by toll tax for ₹15 each day. Ravi solved the same problem by multiplying. They both got correct answer. How is this possible?

Answer

(i) First expression $=\frac{1}{4}(x+7)$
As we know, the addition is commutative.
So, it can also be written as = $\frac{1}{4}(7+x)$
(ii) Since, the expression of 5 less than a number n
= n - 5
So, 5 less than a number n divided by 3 will be written = $\frac{n-5}{3}$
So, student makes an error of quotient.
(iii) By addition method,
Total weekly cost = 15 + 15 + 15 + 15 + 15 + 15 + 15 = ₹105
By multiplication method,
Total weekly cost = Cost of one day × Seven days
= 15 × 7 = ₹105

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Question 45 Marks

At age of 2 yr, a cat or a dog is considered 24 "human" years old. Each year, after age 2 is equivalent to 4 "human" years. Fill in the expression [24+....(a-2)], so that it represents the age of a cat or dog in human years. Also, you need to determine for what 'a' stands for. Copy the chart and use your expression to complete it.

Age	[24+......(a-2)]	Age (human years)
2
3
4
5
6

Answer

The expression is [24 + 4(a - 2)]
Here, 'a' represents the present age of dog or cat.

Age	[24+......(a-2)]	Age (human years)
2	[24 + 4(2 - 2)]	24
3	[24 + 4(3 - 2)]	28
4	[24 + 4(4 - 2)]	32
5	[24 + 4(5 - 2)]	36
6	[24 + 4(6 - 2)]	40

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Question 55 Marks

Consider an algebraic expression
$8 p^2-2 p q+10 q^2 p$.
Find its terms, factors and the numerical coefficient of $p^2$. Then, find the value of given expression when p = 1 and q = 2.

Answer

Given expression is $8 p^2-2 p q+10 q^2 p$
Terms of given expression are $8 p^2,-2 p q, 10 q^2 p$.
Factors of $8 p^2$ are $8, p$ and $p$.
Factors of-2pq are (-2), p and q.
Factors of $10 q^2 p$ are $10, q, q$ and $p$.
Numerical coefficient of $p^2$ is 8.
The value of given expresssion\
$8 p^2-2 p q+10 q^2 p$ at $p=1$ and $q=2$
$=8 \times 1^2-2 \times 1 \times 2+10 \times 2 \times 2 \times 1$
= 8 - 4 + 40 = 44

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Question 65 Marks

Find the value of $x^2+y^2+2 z^2+2 x y+3 y z+4 z x$ when $x=2, y=-3$ and z = 4.

Answer

On putting the value of x = 2y = -3 and z = 4 in the given expression, we get
$x^2+y^2+2 z^2+2 x y+3 y z+4 z x$
$=(2)^2+(-3)^2+2(4)^2+2(2)(-3)+3(-3)(4)+4(4)(2)$
= 4 + 9 + 32 - 12 - 36 + 32 = 77 - 48 = 29

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Question 75 Marks

Find the values of the following expressions when a = -2 and b = 3.
(i) $a^2+2 a b+b^2$
(ii) $a^2-2 a b+b^2$
(iii) $a^3+3 a^2 b+3 a b^2+b^3$
(iv) $a^3-3 a^2 b+3 a b^2-b^3$
(v) $\frac{a^2+b^2}{3}$

Answer

Given, a = - 2 and b = 3
So, putting a = -2 and b = 3 in the given expressions,
we get
(i) a^2+2 a b+b^2=(-2)^2+2(-2)(3)+(3)^2=4-12+9=1
(ii) $a^2-2 a b+b^2=(-2)^2-2(-2)(3)+(3)^2$
= 4 + 12 + 9 = 25
(iii) $a^3+3 a^2 b+3 a b^2+b^3$
$=(-2)^3+3(-2)^2(3)+3(-2)(3)^2+(3)^3$
= -8 + 36 - 54 + 27 = 1
(iv) $a^3-3 a^2 b+3 a b^2-b^3$
$=(-2)^3-3(-2)^2(3)+3(-2)(3)^2-(3)^3$
-8 - 36 - 54 - 27= -125
(v) $\frac{a^2+b^2}{3}=\frac{(-2)^2+(3)^2}{3}=\frac{4+9}{3}=\frac{13}{3}$

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MATHS 5 Marks Questions

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