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Core Mathematics Unit 2 Test No. 1

Total Marks= 137 Total Time= 2hrs

Q1. The polynomial x4 − 9x2 − 6x − 1 is denoted by f(x).

(i) Find the value of the constant a for which f(x) ≡ (x2 + ax + 1)(x2 − ax − 1). [3]

(ii) Hence solve the equation f(x) = 0, giving your answers in an exact form. [3]

Q2.

[3]

Q3. Find all the values of x in the interval 0° ≤ x ≤ 180° which satisfy the equation sin 3x + 2cos3x = 0. [4]

Q4.

Q5. The equation of a curve is y = √(5x + 4).

(i) Calculate the gradient of the curve at the point where x = 1. [3]

(ii) Find the area enclosed by the curve, the x-axis, the y-axis and the line x = 1. [5]

Q6. The polynomial x4 − 2x3 − 2x2 + a is denoted by f(x). It is given that f(x) is divisible by x2 − 4x + 4.

(i) Find the value of a. [3]

(ii) When a has this value, show that f(x) is never negative. [4]

Q7. A geometric progression has first term 64 and sum to infinity 256. Find

(i) the common ratio, [2]

(ii) the sum of the first ten terms. [2]

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Q8.

Q9.

Q10.

[3+3=6]

Q11.

Q12.

In the diagram, OCD is an isosceles triangle with OC = OD

= 10 cm and angle COD = 0.8 radians.

The points A and B, on OC and OD respectively, are joined

by an arc of a circle with centre O and

radius 6 cm. Find

(i) the area of the shaded region, [3]

(ii) the perimeter of the shaded region. [4]

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Q13. The cubic polynomial 2x3 + ax2 − 13x − 6 is denoted by f(x). It is given that (x − 3) is a factor of f(x).

(i) Find the value of a. [2]

(ii) When a has this value, solve the equation f(x) = 0. [4]

Q14.

The diagram shows the part of the curve y = xe−x for 0 ≤ x ≤ 2

Use the trapezium rule with two intervals to estimate the value of

giving your answer correct to 2 decimal places. [3]

Q15.

Q16. Sketch the graph of y = sec x, for 0 ≤ x ≤ 2π . [3]

Q17. (i) Show that if y = 2x, then the equation 2x − 2−x = 1 can be written as a quadratic equation in y. [2]

(ii) Hence solve the equation 2x − 2−x = 1. [4]

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Q18.

Q19. (i)Find the first 3 terms in the expansion of (2 − x)6 in ascending powers of x. [3]

(ii) Find the value of k for which there is no term in x2 in the expansion of (1 + kx)(2 − x) 6. [2]

Q20.

Q21. A geometric progression has 6 terms. The first term is 192 and the common ratio is 1.5. An arithmetic

progression has 21 terms and common difference 1.5. Given that the sum of all the terms in the geometric

progression is equal to the sum of all the terms in the arithmetic progression, find the first term and the last term of

the arithmetic progression. [6]

Q22.

In the diagram, ABC is a semicircle, centre O and radius 9 cm. The line BD is perpendicular to the diameter AC and

angle AOB = 2.4 radians.

(i) Show that BD = 6.08 cm, correct to 3 significant figures. [2]

(ii) Find the perimeter of the shaded region. [3]

(iii) Find the area of the shaded region. [3]

Q23. A curve has equation y = 4 / √x.

(i) The normal to the curve at the point (4, 2) meets the x-axis at P and the y-axis at Q. Find the length of PQ,

correct to 3 significant figures. [6]

(ii) Find the area of the region enclosed by the curve, the x-axis and the lines x = 1 and x = 4. [4]

The diagram shows a rhombus ABCD. The

points B and D have coordinates (2, 10) and

(6, 2) respectively, and A lies on the x-axis. The

mid-point of BD isM. Find, by calculation, the

coordinates of each of M, A and C. [6]