Physics HL Test Topic

Q1

Easy No Calculator

Find the $15^{th}$ term of the arithmetic sequence $7, 11, 15, dots$

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Question Q1 ×

Question Q1

Find the $15^{th}$ term of the arithmetic sequence $7, 11, 15, dots$

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Step 1: Identify $u_1=7$ and $d=4$.

Step 2: Use $u_n = u_1 + (n-1)d$.

$u_{15} = 7 + (14)(4) = 7 + 56 = 63$.

Q2

Easy No Calculator

Solve for $x$: $3^{2x-1} = 27$

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Question Q2 ×

Question Q2

Solve for $x$: $3^{2x-1} = 27$

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$3^{2x-1} = 3^3$

$2x-1 = 3$

$2x = 4 Rightarrow x = 2$.

Q3

Medium Calculator

Find the sum of the first 10 terms of the geometric sequence $2, 6, 18, dots$

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Question Q3 ×

Question Q3

Find the sum of the first 10 terms of the geometric sequence $2, 6, 18, dots$

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$u_1=2, r=3$

$S_{10} = frac{2(3^{10}-1)}{3-1} = 3^{10}-1 = 59048$.

Q4

Medium No Calculator

Given $log_a 2 = p$ and $log_a 5 = q$, express $log_a 20$ in terms of $p$ and $q$.

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Question Q4 ×

Question Q4

Given $log_a 2 = p$ and $log_a 5 = q$, express $log_a 20$ in terms of $p$ and $q$.

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$log_a 20 = log_a (2^2 times 5)$

$= 2log_a 2 + log_a 5 = 2p + q$.

Q5

Medium No Calculator

The $3^{rd}$ term of an arithmetic sequence is 12 and the $7^{th}$ term is 32. Find the common difference.

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Question Q5 ×

Question Q5

The $3^{rd}$ term of an arithmetic sequence is 12 and the $7^{th}$ term is 32. Find the common difference.

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$u_7 = u_3 + 4d$

$32 = 12 + 4d Rightarrow 20 = 4d Rightarrow d = 5$.

Q6

Hard No Calculator

Expand $(x+3)^4$ using the binomial theorem.

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Question Q6 ×

Question Q6

Expand $(x+3)^4$ using the binomial theorem.

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$x^4 + 4(x^3)(3) + 6(x^2)(3^2) + 4(x)(3^3) + 3^4$

$= x^4 + 12x^3 + 54x^2 + 108x + 81$.

Q7

Hard No Calculator

Find the value of $x$ if $sum_{r=1}^{x} (2r-1) = 144$.

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Question Q7 ×

Question Q7

Find the value of $x$ if $sum_{r=1}^{x} (2r-1) = 144$.

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This is a sum of odd numbers: $S_x = x^2$.

$x^2 = 144 Rightarrow x = 12$.

Q8

Easy Calculator

Calculate the amount in a bank account after 5 years if $1000 is invested at 4% interest compounded annually.

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Question Q8 ×

Question Q8

Calculate the amount in a bank account after 5 years if $1000 is invested at 4% interest compounded annually.

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$A = 1000(1.04)^5$

$A approx 1216.65$.

Q9

Medium No Calculator

Solve the inequality $|2x – 3| < 5$.

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Question Q9 ×

Question Q9

Solve the inequality $|2x – 3| < 5$.

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$-5 < 2x - 3 < 5$

$-2 < 2x < 8$

$-1 < x < 4$.

Q10

Hard No Calculator

Find the coefficient of $x^2$ in the expansion of $(2x-1)^6$.

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Question Q10 ×

Question Q10

Find the coefficient of $x^2$ in the expansion of $(2x-1)^6$.

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Term is $binom{6}{4}(2x)^2(-1)^4$

$= 15 times 4x^2 times 1 = 60x^2$.

Coefficient is 60.