ADVANCED MATHEMATICS - CALCULATING DEVICES (With Marking Scheme)

MITIHANI POPOTE

FORM SIX TOPIC TEST

ADVANCED MATHEMATICS - CALCULATING DEVICES

1. (a) Use non-programmable scientific calculator, find the value of the following to correct four decimal places:

i. e^{√(log 3)} + 4 log(5/√(1 + log₂ 3))

ii. ∫_π^√2 (x cos² x)/√(1 + x²) dx

iii. ((1 + i)/(2 - 3i))¹⁰

(b) Given that:

a^b = ³√((log h)^½ + (4/3) log_e(w/b)) × (h/(πk²))

and b = lim_n→∞ (cosh y/e^y)

Evaluate w if h = 1600, k = 0.56, a = 5060 correct to four (4) decimal places.

2. By using your non-programmable scientific calculator, evaluate each of the following:

Q. [7 ln(3.42)][42.8425]^4/5 / [0.03483]^3/2[11 sin^-1(0.8543)]

D. tan θ = (14.32 tan 16°34′)/76.9

C. Given that:

A =

1	1	0
1	0	-1
1	1	2

B =

1	-2	-1
-3	2	1
1	0	-1

and C =

3

2

5

d. [log₈7.2 - ln(5√3)]²

e. (e^{log 3} + √(log √5))/ln 23

f. √(π² × cos^-1(0)/100)

g. log₈17 - ln(ln(4 × 10²/π)/log √2)

3. (a) By using your non-programmable scientific calculator, evaluate:

i. ∑_x=2⁶ log ³√(1 + tan^-1 x) correct to 6 decimal places

ii. ⁷C₄ + ⁶P₄ -

7	4	11
9	3	6
2	1	2

iii. √(3 cos 88°/(sin³(π/3))² + ⁴√(sin^-1(0.542))) correct to 7 significant figures

(b) Using statistical mode of your non-programmable calculator, compute:

i. Mean

ii. Standard deviation

iii. Variance

Class interval	10-14	15-19	20-24	25-29
Cumulative frequency	3	4	12	16

MITIHANI POPOTE

FORM SIX TOPIC TEST

ADVANCED MATHEMATICS - CALCULATING DEVICES SOLUTIONS

Question 1(a)(i)

Evaluate: e^{√(log 3)} + 4 log(5/√(1 + log₂ 3))

Step 1: Calculate log₂ 3 ≈ 1.58496

Step 2: Calculate denominator: √(1 + 1.58496) ≈ √2.58496 ≈ 1.6078

Step 3: Calculate fraction: 5/1.6078 ≈ 3.1098

Step 4: Calculate log(3.1098) ≈ 0.4928

Step 5: Multiply by 4: 4 × 0.4928 ≈ 1.9712

Step 6: Calculate log 3 ≈ 0.4771

Step 7: Calculate √0.4771 ≈ 0.6907

Step 8: Calculate e^0.6907 ≈ 1.9950

Step 9: Add results: 1.9950 + 1.9712 ≈ 3.9662

Final answer: 3.9662 (to 4 decimal places)

Question 1(a)(ii)

Evaluate: ∫_π^√2 (x cos² x)/√(1 + x²) dx

Step 1: Recognize this requires numerical integration

Step 2: On calculator, use integration function

Step 3: Input function: (X*cos(X)^2)/√(1+X^2)

Step 4: Set lower limit: π ≈ 3.1416

Step 5: Set upper limit: √2 ≈ 1.4142

Note: Upper limit < lower limit → result will be negative

Step 6: Calculate absolute value ≈ 0.0285

Final answer: -0.0285 (to 4 decimal places)

Question 1(a)(iii)

Evaluate: ((1 + i)/(2 - 3i))¹⁰

Step 1: Convert to polar form using calculator

Step 2: Calculate numerator: 1 + i → magnitude √2, angle 45°

Step 3: Calculate denominator: 2 - 3i → magnitude √13, angle -56.31°

Step 4: Divide: (√2/√13) ≈ 0.3922, angle 45° - (-56.31°) = 101.31°

Step 5: Raise to 10th power: (0.3922)¹⁰ ≈ 8.08×10^-5, angle 101.31°×10 = 1013.1°

Step 6: Reduce angle modulo 360°: 1013.1° - 2×360° = 293.1°

Step 7: Convert back to rectangular form ≈ 0.0329 - 0.0739i

Final answer: 0.0329 - 0.0739i (to 4 decimal places)

Question 2(d)

Evaluate: [log₈7.2 - ln(5√3)]²

Step 1: Calculate log₈7.2 using change of base formula: log7.2/log8 ≈ 0.9573

Step 2: Calculate 5√3 ≈ 8.6603

Step 3: Calculate ln(8.6603) ≈ 2.1589

Step 4: Subtract: 0.9573 - 2.1589 ≈ -1.2016

Step 5: Square result: (-1.2016)² ≈ 1.4438

Final answer: 1.4438 (to 4 decimal places)

Question 3(a)(i)

Evaluate: ∑_x=2⁶ log ³√(1 + tan^-1 x) correct to 6 decimal places

Step 1: Calculate each term from x=2 to 6:

x	tan-1x	1 + tan-1x	3√(1 + tan-1x)	log(3√)
2	1.107149	2.107149	1.283382	0.108488
3	1.249046	2.249046	1.307660	0.116479
4	1.325818	2.325818	1.324268	0.121899
5	1.373401	2.373401	1.334839	0.125457
6	1.405648	2.405648	1.341641	0.127693

Step 2: Sum the log terms: 0.108488 + 0.116479 + 0.121899 + 0.125457 + 0.127693 ≈ 0.600016

Final answer: 0.600016 (to 6 decimal places)

Question 3(b)

Using statistical mode, compute mean, standard deviation and variance for:

Class interval	10-14	15-19	20-24	25-29
Cumulative frequency	3	4	12	16

Step 1: Convert cumulative to individual frequencies:

Class	10-14	15-19	20-24	25-29
Frequency (f)	3	1 (4-3)	8 (12-4)	4 (16-12)
Midpoint (x)	12	17	22	27

Step 2: Calculate mean (μ):

μ = (3×12 + 1×17 + 8×22 + 4×27)/16 = 21.0625

Step 3: Calculate variance (σ²):

σ² = (3×(12-21.0625)² + 1×(17-21.0625)² + 8×(22-21.0625)² + 4×(27-21.0625)²)/16 ≈ 28.8086

Step 4: Calculate standard deviation (σ):

σ = √28.8086 ≈ 5.3674

Final answers:

• Mean: 21.0625
• Standard deviation: 5.3674
• Variance: 28.8086