Fundamental Concepts of Algebra — Competency Focused Questions

Perimeter of an equilateral triangle, whose each side is $x$ unit is:

1. $4x$

2. $2x$

3. $3 + x$

4. $3x$

Answer

An equilateral triangle has 3 equal sides, each of length $x$ units.

Perimeter = sum of all sides $= x + x + x = 3x$ units.

Hence, option 4 is the correct option.

Which of the following does $2n - 1$ represent?

1. 1 subtracted from the product of $n$ and 2.

2. The difference of two times $n$ and 2.

3. $2n$ added to 1.

4. $n$ subtracted from 2.

Answer

In $2n - 1$, the product of $n$ and 2 is $2n$, and 1 is subtracted from it.

So, $2n - 1$ means "1 subtracted from the product of $n$ and 2".

Hence, option 1 is the correct option.

The quotient of $x$ by 3 is multiplied by $y$, is written as:

1. $\dfrac{x}{3y}$

2. $\dfrac{3x}{y}$

3. $\dfrac{3y}{x}$

4. $\dfrac{xy}{3}$

Answer

Quotient of $x$ by $3 = \dfrac{x}{3}$.

Multiplying it by $y$, we get $\dfrac{x}{3} \times y = \dfrac{xy}{3}$.

Hence, option 4 is the correct option.

The value of $p - q + pq$ for $p = -1, q = -2$ is:

1. 0

2. −1

3. −5

4. 3

Answer

Substituting $p = -1$ and $q = -2$, we get:

$\Rightarrow p - q + pq\\[1em] \Rightarrow (-1) - (-2) + (-1) \times (-2)\\[1em] \Rightarrow -1 + 2 + 2\\[1em] \Rightarrow 3$

Hence, option 4 is the correct option.

Sum of co-efficients of $x^3$ and $x^2$ in $(6x^7 - 7x^3 + 3x^2 + 2x - 1)$ is:

1. 4

2. −4

3. 3

4. −3

Answer

In the given polynomial:

Coefficient of $x^3 = -7$

Coefficient of $x^2 = 3$

Sum of the coefficients = −7 + 3 = −4

Hence, option 2 is the correct option.

If $P = 3a - 5b, Q = 4a + 7b$ and $R = -2a + 9b$, then at $a = -1, b = -2, P - Q + R$ is equal to:

1. 9

2. 8

3. 7

4. 6

Answer

$P - Q + R = (3a - 5b) - (4a + 7b) + (-2a + 9b)$

$\Rightarrow 3a - 5b - 4a - 7b - 2a + 9b\\[1em] \Rightarrow (3a - 4a - 2a) + (-5b - 7b + 9b)\\[1em] \Rightarrow -3a - 3b$

Substituting $a = -1$ and $b = -2$, we get:

$\Rightarrow -3 \times (-1) - 3 \times (-2)\\[1em] \Rightarrow 3 + 6\\[1em] \Rightarrow 9$

Hence, option 1 is the correct option.

Study the following statements and select the correct option:

(i) The number of unlike terms in $4pq^2 - 2q^2r + q^2p + q(3pr + qr) - 8$ is 4.

(ii) An expression having three or more terms is called a trinomial.

1. Both (i) and (ii) are true

2. Both (i) and (ii) are false

3. (i) is true but (ii) is false

4. (i) is false but (ii) is true

Answer

Expanding the given expression, we get:

$\Rightarrow 4pq^2 - 2q^2r + q^2p + q(3pr + qr) - 8\\[1em] \Rightarrow 4pq^2 - 2q^2r + pq^2 + 3pqr + q^2r - 8$

Collecting the like terms:

$\Rightarrow (4pq^2 + pq^2) + (-2q^2r + q^2r) + 3pqr - 8\\[1em] \Rightarrow 5pq^2 - q^2r + 3pqr - 8$

This has 4 unlike terms. Hence, statement (i) is true.

An expression having exactly three terms is called a trinomial; an expression having two or more than two terms is called a multinomial. Hence, statement (ii) is false.

Hence, option 3 is the correct option.

Which of the following options is correct for $p = 2, q = 1, r = 3$ and $s = 4$?

1. $\dfrac{pq + r}{q} = p + s$

2. $\dfrac{p + r}{q + r} = \dfrac{q}{r}$

3. $\dfrac{pq + pr}{ps} = \dfrac{q + r}{s}$

4. $\dfrac{p(q + r)}{p + s} = \dfrac{q + r}{s}$

Answer

Substituting $p = 2, q = 1, r = 3$ and $s = 4$ in each option:

Option 1: L.H.S. $= \dfrac{2 \times 1 + 3}{1} = \dfrac{5}{1} = 5$ and R.H.S. = 2 + 4 = 6. Since 5 ≠ 6, it is not correct.

Option 2: L.H.S. $= \dfrac{2 + 3}{1 + 3} = \dfrac{5}{4} \text{ and R.H.S. }= \dfrac{1}{3}. \text{ Since }\dfrac{5}{4} \neq \dfrac{1}{3}$, it is not correct.

Option 3: L.H.S. $= \dfrac{2 \times 1 + 2 \times 3}{2 \times 4} = \dfrac{2 + 6}{8} = \dfrac{8}{8} = 1 \text{ and R.H.S. }= \dfrac{1 + 3}{4} = \dfrac{4}{4} = 1$. Since L.H.S. = R.H.S., it is correct.

Option 4: L.H.S. $= \dfrac{2(1 + 3)}{2 + 4} = \dfrac{8}{6} = \dfrac{4}{3} \text{ and R.H.S. }= \dfrac{1 + 3}{4} = 1. \text{ Since }\dfrac{4}{3} \neq 1$, it is not correct.

Hence, option 3 is the correct option.

The number of girls in $a$ class is 20 more than 4 times the number of boys. If the number of boys in the class is $m$, then what is number of girls in the class?

1. $4m - 20$

2. $4m$

3. $20m + 4$

4. $4m + 20$

Answer

Number of boys in the class = $m$

4 times the number of boys = $4m$

Number of girls = 20 more than $4m = 4m + 20$

Hence, option 4 is the correct option.