If log10(2) = 0.3010, what is log10(8)? A. 0.9030 B. 0.3010 C. 0.6020 D. 0.9040 7. Find the value of x if 2x…

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If log10(2) = 0.3010, what is log10(8)? A. 0.9030 B. 0.3010 C. 0.6020 D. 0.9040 7. Find the value of x if 2x + 3y = 11 and x + y = 5 A. 1 B. 2 C. 3 D. 4 8. What is the sum of the first 10 terms of the arithmetic progression 2, 5, 8, ...? A. 155 B. 160 C. 165 D. 170 9. If P = x: x is a prime number < 10, Q = x: x is an odd number < 10, find P ∩ Q A. 1, 3, 5, 7 B. 2, 3, 5, 7 C. 3, 5, 7 D. 2, 3, 5, 7, 9 10. Find the equation of the circle with centre (2, 3) and radius 4 A. (x − 2)² + (y − 3)² = 16 B. (x + 2)² + (y + 3)² = 16 C. (x − 2)² + (y − 3)² = 4 D. (x + 2)² + (y + 3)² = 4

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let's tackle these math problems! 6. If log₁₀(2) = 0.3010, what is log₁₀(8)? Step 1: Express 8 as a power of 2. 8 = 2³; Step 2: Use the logarithm property log_b(M^p) = p log_b(M). log₁₀(8) = log₁₀(2³) = 3 log₁₀(2) Step 3: Substitute the given value of log₁₀(2). 3 × 0.3010 = 0.9030 The correct option is A. A. 0.9030 7. Find the value of x if 2x + 3y = 11 and x + y = 5. Step 1: From the second equation, express y in terms of x. x + y = 5 ⇒ y = 5 − x Step 2: Substitute this expression for y into the first equation. 2x + 3(5 − x) = 11; 2x + 15 − 3x = 11 Step 3: Simplify and solve for x. −x + 15 = 11; −x = 11 − 15; −x = −4; x = 4 The correct option is D. D. 4 8. What is the sum of the first 10 terms of the arithmetic progression 2, 5, 8, ...? Step 1: Identify the first term (a), common difference (d), and number of terms (n). The first term a = 2. The common difference d = 5 − 2 = 3. The number of terms n = 10. Step 2: Use the formula for the sum of the first n terms of an arithmetic progression, S_n = n/2[2a + (n−1)d]. S₁₀ = 10/2[2(2) + (10−1)3]; S₁₀ = 5[4 + (9)3]; S₁₀ = 5[4 + 27]; S₁₀ = 5[31]; S₁₀ = 155 The correct option is A. A. 155 9. If P = x: x is a prime number < 10, Q = x: x is an odd number < 10, find P ∩ Q. Step 1: List the elements of set P. Prime numbers less than 10 are numbers greater than 1 that have no positive divisors other than 1 and themselves. P = 2, 3, 5, 7 Step 2: List the elements of set Q. Odd numbers less than 10 are integers not divisible by 2. Q = 1, 3, 5, 7, 9 Step 3: Find the intersection P ∩ Q, which consists of elements common to both sets P and Q. P ∩ Q = 3, 5, 7 The correct option is C. C. 3, 5, 7 10. Find the equation of the circle with centre (2, 3) and radius 4. Step 1: Recall the standard equation of a circle with centre (h, k) and radius r. (x − h)² + (y − k)² = r²; Step 2: Substitute the given centre (h, k) = (2, 3) and radius r = 4 into the equation. (x − 2)² + (y − 3)² = 4²; (x − 2)² + (y − 3)² = 16 The correct option is A. A. (x − 2)² + (y − 3)² = 16

Answer

Step-by-step solution

Express 8 as a power of 2.

Use the logarithm property $\log_b(M^p) = p \log_b(M)$ .

Substitute the given value of $\log_{10}(2)$ .

From the second equation, express $y$ in terms of $x$ .

Substitute this expression for $y$ into the first equation.

Simplify and solve for $x$ .

Identify the first term ( $a$ ), common difference ( $d$ ), and number of terms ( $n$ ).

Use the formula for the sum of the first $n$ terms of an arithmetic progression, $S_n = \frac{n}{2}[2a + (n-1)d]$ .

List the elements of set $P$ .

List the elements of set $Q$ .

Find the intersection $P \cap Q$ , which consists of elements common to both sets $P$ and $Q$ .

Recall the standard equation of a circle with centre $(h, k)$ and radius $r$ .

Substitute the given centre $(h, k) = (2, 3)$ and radius $r = 4$ into the equation.

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Answer

Step-by-step solution

Express 8 as a power of 2.

Use the logarithm property log⁡b(Mp)=plog⁡b(M)\log_b(M^p) = p \log_b(M)logb​(Mp)=plogb​(M).

Substitute the given value of log⁡10(2)\log_{10}(2)log10​(2).

From the second equation, express yyy in terms of xxx.

Substitute this expression for yyy into the first equation.

Simplify and solve for xxx.

Identify the first term (aaa), common difference (ddd), and number of terms (nnn).

Use the formula for the sum of the first nnn terms of an arithmetic progression, Sn=n2[2a+(n−1)d]S_n = \frac{n}{2}[2a + (n-1)d]Sn​=2n​[2a+(n−1)d].

List the elements of set PPP.

List the elements of set QQQ.

Find the intersection P∩QP \cap QP∩Q, which consists of elements common to both sets PPP and QQQ.

Recall the standard equation of a circle with centre (h,k)(h, k)(h,k) and radius rrr.

Substitute the given centre (h,k)=(2,3)(h, k) = (2, 3)(h,k)=(2,3) and radius r=4r = 4r=4 into the equation.