Diagnostic and Statistical Mathematics Test DSM–IUnited Departments of Mathematics DBS

March 24, 2011

Instructions

This is a diagnostic test for which no marks will be awarded. Thepurpose of this test is to determine your mathematical ability to helpus better plan the courses we offer.The test is divided into parts. Each part has three questions. Thequestions are arranged from easy to hard. It is enough to do one ques-tion from each part–the hardest one you can manage.

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Factorisation

Factorise the following expressions.

Question 1

(a) 12m3np5 − 32m2 p2

(b) 49x2 − 4y2

(c) x2 − 5x− 24

Question 2

(a) xy− xz + wz− wy

(b) 4m2 + 4m + 1

(c) 3x2 + 7x + 4

diagnostic and statistical mathematics test dsm–i 2

Question 3

(a) x4 − 81

(b) a2 + 12ab + 36b2 − 25c2 + 10c− 1

(c) 8x2 − 22xy + 15y2

Quadratic Equations

Solve the following equations.

Question 1

x2 + 11x + 28 = 0

Question 2

x2 − 6x + 11 = 0

Question 3

2x− 1x = 3

Algebraic Fractions

Simplify the following algebraic fractions.

Question 1

(a) 4x−82x+4

(b) 2x3 + 4x

7

(c) 3x4 ×

2p5

Question 2

(a) x2+7x+122x2+6x

(b) xx+2 + 3

x−4

(c) p2−q2

2p−q ÷p+q

4p−2q

diagnostic and statistical mathematics test dsm–i 3

Question 3

(a) x2+3x+2x+5 ÷ 2x2−x−1

x2+4x−5

(b) 185x2−12xy−9y2 − 13

5x2−7xy−6y2

Coordinate Geometry

Question 1

(a) What is the distance between the points A(1, 3) andB(2, 5)?

(b) What is the gradient of the line segment displayed below?

Question 2

(a) M is the midpoint of AB. Find the coordinates of B, if Ais (1, 3) and M is (4,−2).

(b) Given the points A(1, 4), B(−1, 0), C(6, 3) and D(t,−1),find t if

(i) AB is parallel to CD

(ii) AD is perpendicular to BC

(c) Find the equation of the line which passes through points:A(2, 7) and B(−3,−2).

Question 3

(a) What is the equation of the line shown on the graph be-low?

diagnostic and statistical mathematics test dsm–i 4

(b) Find the distance from P(7,−4) to the line with equation2x + y− 5 = 0.

Exponents

Question 1

(a) Express 3x × 34−x in the simplest form.

(b) Express (xy)3y−2 in the simplest form.

(c) Write 3√52 in index form.

Question 2

(a) Simplify 3a×9a

27a+2

(b) Simplify 2n+1 + 3(2n)− 2n−1

(c) Solve 31−2x = 127

Question 3

(a) Factorise 25x + 4(5x)− 12

(b) Solve 9x = 7(3x) + 18

(c) Solve 3x+1 × 9−x = ( 13 )

x+1

Formula Rearrangement

Question 1

Make x the subject of the following equation: 3x + a = bx + c.

Question 2

Make x the subject of the following equation: y = −3− 6x−2 .

Question 3

Make v the subject of the following equation: m = m0√1−( v

c )2. You may

assume that all quantities are greater than 0.