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Quadratic Equations - Mathematics - Exam, Exams of Mathematics

Alagappa UniversityMathematics

This is the Past Exam of Mathematics which includes Rational Functions, Definite Integrals, Indefinite Integrals, Partial Fractions, Constants, Evaluate, Integration, Parts, First Order Differential etc. Key important points are: Quadratic Equations, Single Fraction, Simplify, Graph, Expression, Rational Functions, Constants, Integration, Angle, Rational Multiple

Typology: Exams

2012/2013

Uploaded on 02/26/2013

devkinandan
devkinandan 🇮🇳

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SECTION A
1. Simplify:
(a) x5(y2z3)2
x−2y4z5(b) b2−9
b2−6b+ 9.
[4 marks]
2. Write 3
x2+ 3x+1
x+ 3 as a single fraction, and simplify it as far as possible.
[4 marks]
3. Solve the following quadratic equations:
(a) x2−x−20 = 0 (b) 3x2−13x−10 = 0.
[4 marks]
4. Sketch the graph of each of the functions:
(a) y= 3x−6 (b) y=x2+ 4x−5 (c) y=|x2+ 4x−5|.
[7 marks]
5. Given that f(x) = 1 + 3x
2x−5, obtain an expression for the inverse function
f−1(x).
[3 marks]
6. (a) Find the sum of the geometric series
6
X
n=1
3n.
(b) Write down the formula for the sum of the infinite geometric series
∞
X
n=1
arn−1with first term aand common ratio r, when |r|<1.
Hence show that
∞
X
n=1 5
8n=5
3.
[6 marks]
Paper Code MATH 011 Sept-07 Page 2 of 7 CONTINUED
pf3
pf4
pf5

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SECTION A

  1. Simplify: (a) x x^5 −(y (^22) yz (^43) z) 52 (b) (^) b 2 b−^2 − 6 b^9 + 9. [4 marks]
  2. Write (^) x (^2) + 3^3 x + (^) x + 3^1 as a single fraction, and simplify it as far as possible. [4 marks]
  3. Solve the following quadratic equations: (a) x^2 − x − 20 = 0 (b) 3 x^2 − 13 x − 10 = 0. [4 marks]
  4. Sketch the graph of each of the functions: (a) y = 3x − 6 (b) y = x^2 + 4x − 5 (c) y = |x^2 + 4x − 5 |. [7 marks]
  5. Given that f (x) = 1 + 3 2 x − x 5 , obtain an expression for the inverse function f −^1 (x). [3 marks]
  6. (^) (a) Find the sum of the geometric series∑^6 n=1^3

n (^). ∑^ ∞ (b)^ Write down the formula for the sum of the infinite geometric series n=1^ ar

n− (^1) with first term a and common ratio r, when |r| < 1. Hence show that∑ n^ ∞=1^ (^58 )n = 53. [6 marks]

  1. Evaluate the following limits: (a) (^) nlim→∞^2 − 5 3 −n 2 + 5n 2 n^2 (b) (^) xlim→− 2 x^2 x− (^2) −x^ − 4 6. [4 marks]
  2. Differentiate with respect to x (a) (3x − 5)^5 (b) (x^3 + 2)^43 (c) x^6 sin x. [8 marks]

the point where^ 9.^ Write down the equation of the tangent line to the curve x = 1.^ y^ = 2x^3 −^ 5 at [3 marks]

  1. Find the indefinite integrals: (a)^ ∫ (cos x + 3x^4 − 2) dx (b)^ ∫ e^6 x^ dx. [6 marks]
  2. Evaluate the definite integrals: (a)^ ∫^0 π/^6 sin 6x dx (b)^ ∫^124 x 4 − 3 dx [Substitute u = 4x − 3]. [6 marks]

f (14.x) =^ (i) x^3 +Find the stationary points and the inflection point of the function x^2 − 2 x, in each case giving the corresponding values of x and f (x) to 2 decimal places. Determine also the nature of the stationary points. x-axis.(ii)^ Find the three points at which the curve^ y^ =^ x^3 +^ x^2 −^2 x^ crosses the (iii) Using the information from (i) and (ii), sketch the curve y = x^3 + x^2 − 2 x. (iv) Calculate the total area bounded by the curve and the x-axis. [15 marks]

  1. Find the indefinite integrals: (a)^ ∫ x^3 sin(x^4 + 3) dx [Substitute u = x^4 + 3] (b)^ ∫ tan^5 x sec^2 x dx [Substitute t = tan x]. Evaluate the definite integral: (c)^ ∫^01 √ 4 dx − x 2 [Substitute x = 2 sin t]. [15 marks]

Formulae Handbook

This handbook is designed for examination purposes, to be used in the SemesterExamination. The information contained below is not exhaustive of all the for- mulae given in the lectures which you may need.

  1. Solutions to the quadratic equation ax^2 + bx + c = 0 are x = −b^ ±

√b (^2) − 4 ac 2 a.

  1. Power Laws (a) b^0 = 1 (b) b^1 = b (c) b−n^ = b^1 n (d) b 1 n^ = √nb (e) bmbn^ = bm+n^ (f) b bmn = bmb−n^ = bm−n (g) (bm)n^ = bmn^ = (bn)m (h) (ab)n^ = anbn^ (i)^ (^ a b^ )n = a bnn = anb−n
  2. Logarithms (a) M = bx^ ⇐⇒ x = logb M (b) logb(xy) = logb x + logb y (c) logb^ x y = logb x − logb y (d) logb(xn) = n logb x
  3. Series difference^ (a)^ dThe is an + (th term of an arithmetic series with first termn − 1)d. The sum of its first n terms is na +^1 a^ and common (b) The nth term of a geometric series with first term a and common ratio^2 n(n^ −^ 1)d. r is arn−^1. The sum of its first n terms is a × r rn −− 11. The sum to infinity, when |r| < 1, is (^1) −a r.