Algebra Answers

The cost of a dvd in dollars can be represented by the expression 18-d where d is the discount amount.use the distributive property to write and simplify and expression for the cost of 3 dvds.

What is the place value of 9 in 19 ones

julie walked 27 yards . then she walked another 45 yards. how many yards did she walk altogether? Slove the problem 2 way by hundreds chat by number line

Prove or disprove that if 4 divides the product mn of positive integers m and n, then 3 divides m or 3 divides n.

Suppose a is rational and b is irrational. Show that a+b is irrational.

Suppose a is an irrational number. Prove that 1/a is also an irrational number.

3. (a) A coach buys 3 cricket bats and 6 balls for
3,900. Later, he buys 1 bat and 3 balls for
another team for 1,450. Write two linear
equations to represent the purchases. Why
can the linear system so obtained be solved
by Cramer's rule ? Also, use the rule to
solve the system, and interpret the solution
in the given context. 4
(b) Give an example of a 4 x 1 matrix with
entries from Z \ N.

(a) Solve the following linear systems by the
process of elimination : 3
x + 2y + 5z = 9
x — y + $z = 2
3x — 6y z = 26
(b) Give an example, with justification, of each of
the following : 2
(i) An element of (C x Q) \(Q x C);
(ii) An infinite set whose complement in R
is infinite.
2. (a) Find the roots of the polynomial equation
x4— 9x3+ 17x2+ 33x — 90 = 0,
given that two of its roots are equal to 3. 2
(b) Use the principle of mathematical induction
to show that for any positive integer n,
(311— 2n) > O.
(b) Use Cardano's method to obtain the roots of
x3— 3x + 2 = 0.

3. (a) If y = a + 13 is a solution of the cubic equation 3
y3= 4y + B = 0 such that 34 = — A, find the
quadratic equation having a 3and 133as
roots.
(b) Without Expanding the determinant 2
1 2 3
4 5 6
7 8 9
find its value.
4. (a) Prove that A u (A n B) = A for any two
subsets A and B of a set U.
(b) Check if the following system of equations
can be solved by cramer's rule. If it can be
solved, then apply the rule for solving the
system ; otherwise solve it by the Gaussian
elimination method.
x+y+z=6
x + 2y + 3z = 10
x -F 2y + 3z = 6.
5. (a) Show that 22n — 3n —1 is divisible by 9 using
the principle of mathematical induction.
(b) Find the greatest value of (3 — x)5(2 + x)6,
for — 2 < x < 3.

1. Decide which of the following are true or false. 10
If true, furnish the proof, and if false, give a
counter example.
(a) A U (B fl C) = (A U B) n C for any subset
A, B, C of a set U.
(b) 1 .2 1I z2 , = Iz 1 Z21 for any two complex
numbers z1 and z2, where Izi denotes the
absolute value (or modulus) of z.
(c) A linear equation with coefficients in
C\ R in one unknown can have a root which
is real.
(d) The system of equations
4x + 6y = 9
6x + 9y = 1 3
is inconsistent.
(e) For a 3 x 3 matrix A
det (a A) = a (det A)
holds for every scalar a where det A denotes
the determinant of A.
2. (a) Prove that (ax + by) (ay + bx) 4abxy where 2
a, b, x, y are positive real numbers.
(b) Find all the four complex fourth roots of 3
( — 1). Show that these roots are in G.P.