7 Geometry and Elliptic Curves - UCI Mathematics
2 The homogenization of the hyperbola x2 y2 = 3 is the projective curve X2 Y2 = 3Z2 3 The homogenization of x2 + xy2 +2x +1 = 0 is X2Z + XY2 +2XZ2 + Z3 = 0 Definition 7 5 The real projective plane RP2 is defined via an equivalence relation on the set of non-zero points in R3: RP2 = R 3 ˘ where (X,Y, Z) ˘(A, B,C) X Y Z k A B C
180B Lecture Notes, W2011
2 Covariance and Correlation Suppose that (;P) is a probability space We say that X: R is in-tegrable if EjXj
C2 Trigonometr y: Trigonometric Equations PhysicsAndMathsTutor
(b) Solve, for 0 ≤x < 360°, 5sin 2x = 2cos 2x, giving your answers to 1 decimal place (5) (Total 6 marks) 2 (a) Show that the equation 5 sin x = 1 + 2 cos2 x can be written in the form 2 sin2 x + 5 sin x – 3 = 0 (2) (b) Solve, for 0 ≤x < 360°, 2 sin2 x + 5 sin x – 3 = 0 (4) (Total 6 marks) 3 (i) Solve, for –180° ≤ θ < 180°,
6 Gaussian Integers and other Rings
2 S = fmd: m 2Z[i]g, where S is defined as in Definition 6 4 3 Every common divisor of a and b divides d , whence d has the largest norm among all common divisors We leave the proof to the homework: as a hint, part 1 requires the use of the Division Algorithm
GCOC9: Lines and Angles 3 - JMAP
2) 70° 3) 110° 4) 180° 11 Angle A and angle B are complementary angles If m∠A =x, which expression represents the number of degrees in angle B? 1) x −180 2)180−x 3) x −90 4)90−x 12 If the measure of an angle is represented by 2 x, which expression represents the measure of its complement? 1) 180−2x 2) 90 −2x 3) 90 +2x 4) 88x
UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONS
The diagram shows accurate graphs of y = sinx and y = cosx for 0° Y x Y 180° Use the graph to solve the equations (a) sinx – cosx = 0, Answer(a) x = [1] (b) sinx – cosx = 0 5 Answer(b) x = [2] 9 A fence is made from 32 identical pieces of wood, each of length 2 metres correct to the nearest centimetre
Math 180,Exam 2, Spring 2011 Problem 1Solution
Math 180,Exam 2, Spring 2011 Problem 2Solution 2 Sketch the graph of the function f(x) = x−2 −x2 following the steps below (a) Determine the domain of f and find all asymptotes (b) Find the intervals where the graph of f is increasing, decreasing, concave up and concave down
NAME: PAPER B
A teacher asks one of her students to solve the equation 2 cos 2x + 3 = 0 for 0 ⩽ x ⩽ 180° The attempt is shown below 2cos2 = 3x 3 cos2 = 2 x 2 =cos1 3 2 x 2 =150x o x=75o w x o o oor =360 75 =295 so reject as out of range (a) Identify the mistake made by the student (1) (b) Write down the correct solutions to the equation (1)
Math 180, Exam 2, Spring 2008 Problem 1 Solution
Math 180, Exam 2, Spring 2008 Problem 2 Solution 2 Let f(x) = x3 +3x2 − 9x− 11 (a) Find the critical points of f (b) Find the intervals on which f is increasing and the intervals on which f is decreasing
CBSE NCERT Solutions for Class 9 Mathematics Chapter 6
Let the common ratio between a and b be x, ∴a = 2x and b = 3x XY is a straight line, OM and OP rays stands on it Therefore, XOM + MOP + POY = 180 o ⇒b + a + POY + 180o ⇒3x + 2x + 90o= 180o ⇒5x = 90o ⇒x = 18o a = 2x ⇒= 2 × 18 ⇒= 36o b = 3x ⇒= 3 × 18 ⇒= 54o Now, MN is a straight line OX ray stands on it ∠b + ∠c = 180o
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