Q: If $$\tan x = 1.56$$ for $$0 \u003c x \u003c 360^\circ$$, what are the possible values of $$x to $$the nearest degree?

$$x = 57^\circ$$ \, \text{or} \, 237^\circ$

Q: Which of the following is a necessary condition for a point $$x = a to be a $$point of inflection for a function $$f(x)$$?

$$f''(a) = 0$$ \, \text{and} \, f''(x)$$ changes sign at x = a$

Q: If $$\cos x = 0.473$$ for $$0^\circ \leq x \leq 360^\circ$$, what are the possible values of $$x to $$the nearest degree?

$$x = 61^\circ$$ \, \text{or} \, 299^\circ$

Question 1

Which of the following is the correct formula for the sum $$S_n of $$the first $$n$$ terms of an arithmetic series with first term $$a$$ and common difference $$d$$?

Accepted Answer

$$S_n = \frac{n}{2}[2a + (n-1)d]$$

Question 2

Given the quadratic equation $$2x^2 - x - 3 = 0$, which of the following is NOT a valid method to find its roots?

Accepted Answer

Using the trapezium rule

Question 3

A geometric series has first term $$a = 5$$ and common ratio $$r = 0.6. $$What is the sum to infinity $$S_{\infty} of $$this series?

Accepted Answer

$$S_{\infty} = \frac{5}{1 - 0.6} = 12.5$$

Question 4

Which of the following statements about the binomial expansion $$(1 + x)^n$$ for positive integer $$n is $$correct?

Accepted Answer

The coefficients are given by $$\binom{n}{r}$$ for $$r = 0 to$$ $$n.$$

Question 5

A function $$f(x)$$ has a stationary point at $$x = 2. $$Which of the following must be true at $$x = 2$$?

Accepted Answer

$$f'(2) = 0$$

Question 6

The remainder when $$P(x) = 2x^2$ + ax + b is $$divided by $$(x - 2) is$$ $$-6$$, and when divided by $$(x + 2) is$$ $$1. $$What is the value of $$a + b$$?

Accepted Answer

$$-10$$

Question 7

Which of the following is the correct formula for the area under the curve $$y = f(x)$$ from $$x = a to$$ $$x = b$$ using the trapezium rule with $$n$$ intervals of width $$h$$?

Accepted Answer

$$\int_a^b f(x) dx \approx \frac{h}{2}[f(a) + 2f(x_1$) + 2f(x_2$) + \dots + 2f(x_{n-1}) + f(b)]$$

Question 8

If $$\sin x = 0.453$$ for $$0 \leq x \leq 360^\circ$$, what are the possible values of $$x to $$the nearest degree?

Accepted Answer

Question 9

Which of the following is a correct application of the factor theorem?

Accepted Answer

If $$P(a) = 0$$, then $$(x - a) is a $$factor of $$P(x).$$

Question 10

A sequence is defined by $$u_n = 3u_{n-1} - 2$$ with $$u_1 = 4$. What is the value of u_3$?

Accepted Answer

$$u_3 = 28$

Question 11

Which of the following is the correct expression for the $$rth $$term of a geometric sequence with first term $$a$$ and common ratio $$r$$?

Accepted Answer

$$a r$$^{n-1}$$

Question 12

If $$	an x = 1.56$$ for $$0 \u003c x \u003c 360^\circ$$, what are the possible values of $$x to $$the nearest degree?

Accepted Answer

$$x = 57^\circ$$ \, 	ext{or} \, 237^\circ$

Question 13

Which of the following is the correct formula for changing the base of a logarithm $$\log_a b$$?

Accepted Answer

$$\log_a b = \frac{\log_c b}{\log_c a}$$

Question 14

Which of the following is a necessary condition for a point $$x = a to be a $$point of inflection for a function $$f(x)$$?

Accepted Answer

$$f''(a) = 0$$ \, 	ext{and} \, f''(x)$$ changes sign at x = a$

Question 15

Given the polynomial $$P(x) = x^3$ - 2x^2$ - 5x + 6$$, which of the following is a factor of $$P(x)$$?

Accepted Answer

$$(x - 2)$$

Question 16

Which of the following is the correct formula for the $$rth $$binomial coefficient in the expansion of $$(a + b)^n$$?

Accepted Answer

$$\binom{n}{r} = \frac{n!}{r!(n-r)!}$$

Question 17

If $$\cos x = 0.473$$ for $$0^\circ \leq x \leq 360^\circ$$, what are the possible values of $$x to $$the nearest degree?

Accepted Answer

$$x = 61^\circ$$ \, 	ext{or} \, 299^\circ$

Question 18

Which of the following is the correct expression for the difference of two squares?

Accepted Answer

$$x^2$$ - $$y^2 = (x + y)(x - y$$)$$

Edexcel IAL Pure 2 Calculus Revision Notes (May 2022)

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