2022 Paper 2 (16/01/2025)

Question 1

First Principles Equation?

Answer 1

lim h->0 ( f(x+h) - f(x) ) / h and after expanding h due to it tending to 0 is replaced with 0 leaving resultant of first principles calculation

Question 2

Sigma Notation Sum?

Answer 2

n Σ x = 1 , Find the amount of times n goes into the order. Multiply this whole number by the sum of each order and add the additional terms.

Question 3

Trapezium Rule?

Answer 3

1/2 x h ( y0 + yn + 2 ( y1 + y2 + y3 + ….. ) ). Where h is the difference between the x terms. Y0 is the first y value in the table. Yn is the last y value in the table. Y1 + Y2 … is every term in the table of y values between YO and YN

Question 4

Finding a desired maximum on a trigonometric graph?

Answer 4

Since it’s a maximum it means the differential needs to be found and then set to zero. Since the graph is repeatable means the differentiated function can be repeated until the position of the maximum on the graph has been generated in number of solutions

Question 5

Root in interval?

Answer 5

This is the case when the function is continuous at this interval and there is a sign change between each interval value when subbed into the function and producing between them one negative value and one positive value

Question 6

Newton-Rapherson?

Answer 6

x n+1 = xn - f(xn) / f1(xn). Where “n+1” is either x = n0 or the last term subbed into the iteration. “x n+1” is the iteration that is trying to be found. “f(xn)” is the original function with “x” replaced with the last iteration or x = n0. “f1(xn) is the differentiated function with”x” replaced with the last iteration or x = n0.

Question 7

Overestimate of Binomial Expansion?

Answer 7

For this to be the case, if all the terms in the binomial expansion will produce negative values it will consequently mean the expansion is an overestimate

Question 8

Possible Translations of trigonometric values?

Answer 8

A coefficient in front of the sinx will increase the maximum value from initially one. A coefficient in front of the “x” will stretch the graph in the “x” direction meaning the period, otherwise before one loop of the function is complete, is longer. To find the translation left or right this is taking the function equal to 1 and finding the angle produced by dividing 1 the coefficient in front of the trigonometric function b

Question 9

Domain and Range Relationship?

Answer 9

The relationship between the domain and range is the domain of an inverse function if the range of the standard function or the range of the inverse function is the domain of the standard function.

Question 10

Reasoning for odd and even numbers?

Answer 10

n ∊ ℕ otherwise true for all natural numbers

Question 11

Significance of vector points on a straight line?

Answer 11

It means using a constant one vector can equal another vector. For example, if a,b,c lie on a line then λBA = CA

Question 12

Limit of differential equation?

Answer 12

Find the value of the input typically “x” when the output typically “y” equals 0. Then substitute this into the differential equation but replace the non-input value with this value. Then make the value found equal to zero and the resultant of this will be the limit as whats being found is the value when the function tends to infinity

Question 13

Obtuse Solution?

Answer 13

If angle found isn’t obtuse take the solution away from 180 to get the obtuse solution