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Expression - Mathematics - Exam, Exams of Mathematics

Babasaheb Bhimrao Ambedkar UniversityMathematics

This is the Past Exam of Mathematics which includes Identify, Natural Domain, Function, Inverse Function, Inequalities, Sketch, Corresponding Inverse Function, Giving Reason etc. Key important points are: Expression, Simplify, Geometric Series, Formula, Common Ratio, Infinite Geometric Series, Equation, Terms, Indefinite Integrals, Evaluate

Typology: Exams

2012/2013

Uploaded on 02/26/2013

deveshwar
deveshwar ๐Ÿ‡ฎ๐Ÿ‡ณ

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SECTION A
1. Simplify:
(a) x6(yโˆ’1z3)4
x3y2z12 (b) 9โˆ’4a2
9โˆ’12a+ 4a2.
[4 marks]
2. Write 9
9xโˆ’7โˆ’7
9x2โˆ’7xas a single fraction, and simplify it as far as
possible.
[4 marks]
3. Solve the following quadratic equations:
(a) x2โˆ’5xโˆ’24 = 0 (b) 20x2โˆ’21xโˆ’27 = 0.
[4 marks]
4. Sketch the graph of each of the functions:
(a) y=โˆ’3x+ 9 (b) y=x2+ 2xโˆ’15 (c) y=|x2+ 2xโˆ’15|.
[7 marks]
5. Given that f(x) = 6x+ 5
9โˆ’7x, obtain an expression for the inverse function
fโˆ’1(x).
[3 marks]
6. (a) Find the sum of the geometric series
5
X
n=1
(โˆ’6)n.
(b) Write down the formula for the sum of the infinite geometric series
โˆž
X
n=1
arnโˆ’1with first term aand common ratio r, when |r|<1.
Hence show that
โˆž
X
n=1 ๎˜11
15 ๎˜‘n=11
4.
[6 marks]
Paper Code MATH 011 Jan-08 Page 2 of 5 CONTINUED
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SECTION A

  1. Simplify: (a) x^6 x( 3 yyโˆ’ (^21) zz 123 ) 4 (b) (^9) โˆ’^9 12 โˆ’a^4 + 4a^2 a 2. [4 marks]
  2. Write (^9) x 9 โˆ’ 7 โˆ’ (^9) x (^2 7) โˆ’ 7 x as a single fraction, and simplify it as far as possible. [4 marks]
  3. Solve the following quadratic equations: (a) x^2 โˆ’ 5 x โˆ’ 24 = 0 (b) 20 x^2 โˆ’ 21 x โˆ’ 27 = 0. [4 marks]
  4. Sketch the graph of each of the functions: (a) y = โˆ’ 3 x + 9 (b) y = x^2 + 2x โˆ’ 15 (c) y = |x^2 + 2x โˆ’ 15 |. [7 marks]
  5. Given that f (x) =^69 x โˆ’^ + 5 7 x, obtain an expression for the inverse function f โˆ’^1 (x). [3 marks]
  6. (^) (a) Find the sum of the geometric seriesโˆ‘^5 n=1^ (โˆ’6)

n (^). โˆ‘^ โˆž (b)^ Write down the formula for the sum of the infinite geometric series n=1^ ar

nโˆ’ (^1) with first term a and common ratio r, when |r| < 1. Hence show thatโˆ‘ n^ โˆž=1^ (^1115 )n = 114. [6 marks]

  1. Evaluate the following limits: (a) (^) nlimโ†’โˆž^12 n 52 โˆ’โˆ’ 37 nn 2 + 5 (b) lim xโ†’ 8 x^2 โˆ’ x (^2 10) โˆ’x 64 + 16. [4 marks]
  2. Differentiate with respect to x (a) (11x + 8)^6 (b) (x^9 โˆ’ 5)^7 /^9 (c) x^5 sin x. [8 marks]

the point where^ 9.^ Write down the equation of the tangent line to the curve x = โˆ’2.^ y^ = 2x^3 โˆ’^ 6 at [3 marks]

  1. Find the indefinite integrals: (a)^ โˆซ (cos x โˆ’ 6 x^7 + 5) dx (b)^ โˆซ eโˆ’^12 x^ dx. [6 marks]
  2. Evaluate the definite integrals: (a)^ โˆซ^0 ฯ€/^18 sin 9x dx (b)^ โˆซ^178 x^16 โˆ’ 7 dx [Substitute u = 8x โˆ’ 7]. [6 marks]
  1. (i) Find the stationary points and the inflection point of the function f (x) = x^3 โˆ’ 4 x^2 โˆ’ 12 x , in each case giving the values of x and f (x) to 2 decimal places. Determine also the nature of the stationary points. the x-axis.(ii)^ Find the three points at which the curve^ y^ =^ x^3 โˆ’^4 x^2 โˆ’^12 x^ crosses 4 x^2 โˆ’(iii) 12 x^ .Using the information from (a) and (b), sketch the curve^ y^ =^ x^3 โˆ’ (iv) Calculate the total area bounded by the curve and the x-axis. [15 marks]
  2. Find the indefinite integrals: (a)^ โˆซ x^8 sin(x^9 โˆ’ 5) dx [Substitute u = x^9 โˆ’ 5] (b)^ โˆซ tan^15 x sec^2 x dx [Substitute t = tan x]. Evaluate the definite integral: (c)^ โˆซ^4

โˆš 73 0 โˆš^ dx 64 โˆ’ 49 x^2 [Substitute^ x^ =

7 sin^ t]. [15 marks]

Paper Code MATH 011 Jan-08 Page 5 of 5 END