Problem 1: Find the result of subtracting $3$ from $3 + 2\sqrt{2}$.
Solution:
$(3 + 2\sqrt{2}) – 3 = 2\sqrt{2}$
Problem 2: Find the result of subtracting $-4$ from $-8$.
Solution:
$-8 – (-4) = -8 + 4 = -4$
Problem 3:Find the values of $k$ and $k_1$ for the given radical equations.
Solution:
Part A: $\sqrt{8} = k\sqrt{2}$
$\sqrt{4 \times 2} = k\sqrt{2}$
$2\sqrt{2} = k\sqrt{2}$
$k = 2$
Part B: $\sqrt{18} = k_1\sqrt{2}$
$\sqrt{9 \times 2} = k_1\sqrt{2}$
$3\sqrt{2} = k_1\sqrt{2}$
$k_1 = 3$
Problem 4: Solve for $n$: $3n = 81 + 24$
Solution:
$3n = 105$
$n = \frac{105}{3}$
$n = 35$
Problem 5: Find ‘n’ for an Arithmetic Progression where the first term ($a$) is $5$, the common difference ($d$) is $6$, and the last term is $301$.
Solution:
Formula used: $a_n = a + (n-1)d$
$301 = 5 + (n-1)6$
$301 = 5 + 6n – 6$
$301 = 6n – 1$
$301 + 1 = 6n$
$6n = 302$
$n = \frac{302}{6} = \frac{151}{3}$
$n = \frac{151}{3}$
Solution:
$(n-1)(-2) = -18$
$(n-1) = \frac{-18}{-2}$
$(n-1) = 9$
$n = 9 + 1$
$n = 10$
Problem 7: Calculate the value of $\frac{100 \times 101}{2}$.
$\frac{10100}{2}$
$5050$
Given values from previous problems: $n = 21$, $a = 100$, $d = 50$.
Formula used: $S_n = \frac{n}{2} [2a + (n-1)d]$
$S = \frac{21}{2} [2 \times 100 + (21-1)50]$
$S = \frac{21}{2} [200 + (20)50]$
$S = \frac{21}{2} [200 + 1000]$
$S = \frac{21}{2} [1200]$
$S = 21 \times 600$
Result: $12600$
Problem 9: Solve the quadratic equation by factoring.
Equation: $n^2 – 17n + 52 = 0$
Factor by splitting the middle term (finding factors of $52$ that add to $-17$, which are $-4$ and $-13$):
$n^2 – 4n – 13n + 52 = 0$
$n(n – 4) – 13(n – 4) = 0$
$(n – 4)(n – 13) = 0$
Set each factor to zero:
$n – 4 = 0 \implies n = 4$
$n – 13 = 0 \implies n = 13$
Solutions: $n = 4$, $n = 13$
Solution:
Using the elimination method, subtract equation (1) from equation (2):
$$=> (a + 6d) – (a + 2d) = 700 – 600$$
$$=> 4d = 100$$
$$=> d = \frac{100}{4}$$
$d = 25$
Next, substitute the value of $d$ back into the first equation to solve for $a$:
$$a + 2(25) = 600$$
$$a + 50 = 600$$
$$a = 600 – 50$$
$a = 550$
The final solution is $a = 550$, $d = 25$.