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1. Show that there do not exist integers x and y for which 110x + 315y = 12.
2. If a and b are odd integers, prove that a2 +b2 is divisible by 2 but is NOT divisible by 4. Hint:
Any odd number can be written 2k + 1 for some integer k.
3. Prove that for any prime number p, √ p is irrational. Hint: Follow the same method that was used to prove √ 2 is irrational, by first supposing √ p can be expressed as a=b.
Use Newton's Method to find out an approximation to the solution to cos x = x which lies in the interval [0,2]. Determine the approximation to six decimal places. Solution
On dividing the polynomial 4x 4 - 5x 3 - 39x 2 - 46x - 2 by the polynomial g(x) the quotient is x 2 - 3x - 5 and the remainder is -5x + 8.Find the polynomial g(x). (Ans:4 x 2 +
without a calculator give the exact value of each of the following logarithms. (a) (b) log1000 (c) log 16 16 (d) log 23 1 (e) Solution (b) log10
1.)3 3/8 divided by 4 7/8 plus 3 2.)4 1/2 minus 3/4 divided by 2 3/8
Find the sum of (1 - 1/n ) + (1 - 2/n ) + (1 - 3/n ) ....... upto n terms. Ans: (1 - 1/n ) + (1 - 2/n ) - upto n terms ⇒[1+1+.......+n terms] - [ 1/n + 2/n +....+
There is a number. If the sum of digits is 14, and if 29 is subtracted from the number, the digits become equal. Find the number.
Chain Rule : If f(x) and g(x) are both differentiable functions and we describe F(x) = (f. g)(x) so the derivative of F(x) is F′(x) = f ′(g(x)) g′(x). Proof We will s
In a parallelogram ABCD AB=20cm and AD=12cm.The bisector of angle A meets DC at E and BC produced at F.Find the length of CF.
The scale of a map is 0.5 in 25mi the actual distance between two cities is 725mi write a proportion that represents the relationship how far apart will the cities be on the map
Definition: An equation is considered as function if for any x in the domain of the equation (the domain is the entire x's which can be plugged into the equation) the equation wil
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