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Core Mathematics Unit - 1 Test No. 1

Total Marks= 145 Total Time= 2 hrs

Q1. Find the set of values of kfor which the line y= kx4 intersects the curve y= x22xat two

distinct points. [4]

Q2.

Q3.

Q4.

The diagram shows pointsA,B and Clying on the line

2y =x + 4. The pointA lies on they-axis andAB =BC. The

line fromD (10, 3) toB is perpendicular toAC. Calculate the

coordinates ofB and C. [6]

The diagram shows part of the curve

y = 6/(3x 2).

Find the gradient of the curve at the point where

x = 2. [2]

The diagram shows the curve y= x36x2+ 9xfor

x0. The curve has a maximum point at A and a

minimum point on the x-axis at B. The normal to the curve

at C(2, 2) meets the normal to the curve at Bat the point

D.

(i) Find the coordinates of A and B. [4](ii) Find the equation of the normal to the curve at C. [3]

(iii) Find the coordinates of point D. [4]

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Q5. The ninth term of an arithmetic progression is 22 and the sum of the first 4 terms is 49.(i) Find the first term of the progression and the common difference.The nth term of the progression is 46. [4](ii) Find the value of n. [3]

Q6. A curve is such that

and the point (9, 2) lies on the curve.(i) Find the equation of the curve. [3](ii) Find the x-coordinate of the stationary point on the curve. [2]

Q7.

Q8. The function f is defined by f (x)=2x212x+ 7.

(i) Express f(x) in the form a(xb)2c. [2]

(ii) Find the set of values of xfor which f(x) < 21. [3]

(iii) The function g is defined by g (x)=2f (x).+ k

Find the value of the constant kfor which the equation g(x) = 0 has two equal roots. [3]

Q9.

The diagram shows part of the curve y= x2+ 5. The point A has coordinates (1, 6). The point Bhas

coordinates (a, a2+ 5), where ais a constant greater than 1. The point Cis on the curve between A

and B.(i) Find by differentiation the value of the gradient of the curve at the point A. [2](ii) The line segment joining the points A and Bhas gradient 2.3. Find the value of a. [2](iii) State a possible value for the gradient of the line segment joining the points A and C. [1]

The diagram shows part of the curve whichcrosses the x-axis at A and the y-axis at B.The normal to the curve at A crosses the y-axis at C.

(i) Show that the equation of the line ACis 9x+ 4y= 27. [4]

(ii) Find the length of BC. [4]

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Q10. Find the equation of the normal to the curve y= x34x2+ 7 at the point (2, 1), giving your

answer in the form ax+ by+ c= 0, where a, band care integers. [4]

Q11. Solve the equation x8x+ 13 = 0, giving your answers in the form p qr, where p, qand r

are integers. [4]

Q12. Given that

(i) find f (x), [2]

(ii) find f (4). [2]

Q13. The quadratic equation kx230x+ 25k= 0 has equal roots. Find the possible values of k. [3]

Q14.

[5]

Q15.

[3]

Q16.

Q17.

[3]

[2]

[2]

Q20.

[3]

Q18.

[2]Q19.

[2]

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Q21.

[4]

Q22.

[4]

Q24.

Q25. The tenth term of an arithmetic progression is equal to twice the fourth term. The twentieth term ofthe progression is 44.(i) Find the first term and the common difference. [4](ii) Find the sum of the first 50 terms. [2]

Q26. The gradient of a curve is given by where ais a constant. The curve passes through

the points (1, 2) and (2, 17). Find the equation of the curve. [8]

Q27.

[3]

Q28.

[4]

Q29.

Q23.

[5]

[2]

[2]

[4]

[2]