Show that (x − 1) is a factor of x^3 − 1.

Show that

i)(a − b)(a + b) = a^2 − b^2

ii)(a + b)2 = a^2 + 2ab + b^2

iii)(a − b)2 = a^2 − 2ab + b^2

iv)(a + b)3 = a^3 + 3a^2b + 3ab^2 + b^3

Expand and simplify:

i)(x + y)(a + b)

i)(x + 2y + 3z)(x − 4 − z)

i)(x^2 + x + 3)(x^2 − 3x + 5)

Multiply

i)(−21abc) and (−5a^2b)

ii)i)(−7rs) and 3pq

i)Multiply 3a+7 and 2b-5a

ii)Find (3 + 5x^2)(2a^2 + 3a + 4b)

iii)Simplify (x + 1)(x + 2)(x + 3)

Evaluate (2x + 7)(3y + z).

Multiply :

i)2x^2y and 3xy^3.

ii)3x^2 and 7x^3 + 2x^2y − 3xy^2.

Simplify:

i)(9x + b + c) − (6x + 2b + 3c) + 7x + 10

ii)100x^2 + 7y^2 − {(36x^2 + y^2) − (6x^2 + 10y^2)} + y^2

iii)pqr + rst − (3rst + 7y) + (106 + 6pqr)

Add:

i)50t^3 to 100t^3

ii)−7x to 17x

iii)ab + xy to 4xy + 7ab

iv)t^3 − 7t + 10 to a + bt^2 + 4t

Simplify

i)15pqr – 20pqr

ii)5x^2y + 5yx^2 + x^2y

Add 3ab, 5ab,−7ab and 20ab.

Subtract ab − 6abc from 7abc + 9ab

(3x^2 − 4y^3) + (9x^2 + 7y^3)

Distinguish between a term and a monomial. Give an example for a term that is not a monomial.

If an amount P is invested at the simple annual interest rate r then the amount available at the end of t years is A = P + Prt. What is the amount got after 5 years if you invest Rs.100 at the simple interest rate of 3 per cent per year?

The number of diagonals of a polygon with n sides is (n(n-3))/2. Verify this formula for a square.

The length of the hypotenuse H of right-angled triangle is given by √(A^2+B^2) where A and B are the lengths of the perpendicular sides.

a)Find the hypotenuse if A = 5 metres and B = 12 metres.

b)What is the third side if the hypotenuse is 5 metres and one side is 4 metres?

The power, P watts, used in an electric bulb is i^2R where i is the current in amperes and R is the resistance in ohms. What will be the power consumption if i = 0.9 amps and R is 160 ohms?

Find the value of each of the following expressions for the values of the letters given against each.

i)x^2 + 3x + 5; x = 3

ii)a^2 + 2ab − 7; a = −1, b = 2.

iii)(2x+3y-9z^2)/(7+2y-z^3); x = 2, y = −1, z = −2.

iv)1/(x − 3); x = 3

Identify variables and constants in each of the following expressions:

i)4πr^2, the area of the surface of a sphere with radius r.

ii)S(1 + PT/100), the amount of money obtained if rupees S is invested at simple interest P per cent per year for T years.

iii)1/2gt^2, the distance in centimetres that a freely falling object covers in t seconds, g being equal to 980 cm/sec^2.

iv)mMG/d^2, the force of attraction between two objects of mass m and M situated at a distance d apart, G being the gravitational constant.

An object falling freely under gravity covers a distance 4.9t^2 metres in t seconds. How far would an object fall in 5 seconds?

Find the value of (2x^2+7y+9)/(3z+(y-x)^2) at x = 1, y = 2 and z = 3.

What is the degree of xy − 3x + 5y − 7y^3, where x and y are variables?

Distinguish amongst −2a, a^−2 and (−a)^2, where a ≠ 0.

Given 2x + 3y − 4 = 0, complete the following ordered pairs so as to produce a

solution of this equation.

 (0, _), (1, _), (_, 2), (_, 9)

Formulate and solve the linear equation corresponding to the puzzle involving the age of Diophantus.

Find at least two different solutions of each of the following:

i) x − 5y + 6 = 0

ii) 2x − 5y + 9 = 0

iii) x + 3 = y − 7

The house you bought two years before for 15 lakhs costs 20 lakhs now. What is the percentage of increase in price?

Suppose you are planning to go to the USA and you need $1000 for your expenses there. If the exchange rate is $1 = 43.64Rs, how many rupees should you convert into dollars so that you get $1000?

For a father-son pair, the age of the father is 25 more than that of the son. After 5 years, the father will be twice as old as the son. What the are the ages of the father and son?

Solve the following equations:

a)3x + 1 = 7

b)5x + 1 = 7x − 9

c)3x/2 + 1 = x + 1

d)5/x + 1 = 7/x + 1

Solve the following equations:

a)2x + 1 = 3

b)3x + 1 = 2x + 3

c)3/x + 1 = 5/x + 2

Remove brackets in

 i) (2y^2) × (5y^3)

 ii) −2(−3x + 4y)

i) Show that (−a)^4 = a^4.

 ii) Expand (−5a)(7b + a).

Remove brackets in :

i) (3x^2)(2y^3) 

ii) 2a(5b + 7c) 

iii) +(3x + 4y) 

iv) −(7a − 9)

Express (−a)^3 as an exponent of a.

Evaluate the following by using properties vii) and prove your results by direct computation.

 i) x^2x^4, ii) x^6/x^3, iii) (p^2)^3, iv) (pq)^3

In (−7xy^2z^3), find the coefficient of :

i)x

ii)y^2

iii)xz^3

iv)7xy

Find the coefficients of x, y and z and numerical coefficients in each of the following terms:

i)xyz

ii)3x^2yz

iii)-yzx^5

Identify the polynomials from amongst the following and label them as monomial, binomial or trinomial.

i)x + 3a2; where x and a are variables.

ii)3ab2c3d4e5 + 1030; where a, b, c, d, e are all variables.

iii)(1/2)p + (1/3)q + (1/4r); where p, q, r are variables.

iv)(x+964)/(y-964)

v)x/y

vi)1

Solve each of the following explicitly for x (i.e. write as x = . . .)

i) x + y + 6 = 0

ii) x − y + 1 = 0

iii) 2x − y = 3

Solve each of the following explicitly for y (i.e. write as y = . . .)

i) x + y + 6 = 0

ii) x − y + 1 = 0

iii) 2x − y = 3

Suppose Ram encashed his national saving certificates and spent Rs.20,000 out of the money he got on a colour television. If he invests the remaining amount at 8% interest for 6 years, he will get Rs.30,000 for spending on the education of his children. How much money did Ram get by encashing the national saving certificates?

Rewrite the following statements in the language of algebra stating the meaning of each letter you use:

i)The volume of a cube is the third power of the length of its edge.

ii)Two times a number plus three is equal to four.

iii)Three times a number decreased by 10 is 73.

iv)The I.Q. is calculated by multiplying 100 with the quotient of mental age by actual age.

v)The perimeter of a rectangle is twice the sum of its length and breadth.

vi)The temperature on the Fahrenheit scale is equal to 32 plus 1.8 times the temperature on the Celsius scale.

Rewrite the following using numbers, letters and symbols of arithmetic operations:

i)excess of 9 over 10y

ii)product of a raised to the fifth power and 11

iii)x times y times z

iv)2 times a added to b

v)Add 3x to the product of 2y and z

A rectangle has sides of length (√5+√2) metres and (√5-√2) metres. Find the area of the rectangle and the length of its diagonal.

Simplify the following:

a)√3 √3

b)√2 √4 (2 + 2√2)

c)(√5 − √9)(√5 + √9)

d)(√3 − √5)(2√6 + √7)

The lengths of two sides of a rectangle are √2 + 1 metres and √2 − 1 metres. Calculate its area and the length of a diagonal.

Simplify the following:

 4√2 + 3√32

Express each of the following radicals in its simplest form:

a)(32)^1/3

b)5√18 

c)√3000

Express 5√1210 in its simplest form.

Calculate:

a)(100)^−3/2

b)(7^3/2)^−2

c)(5^3/2) × (5^−5/2)

d)(1/8)^1/3(2^−1/3(64^2/3))

Write the numbers as powers of 2:

a)8

b)1/8 

c)1/1024

Evaluate:

a)(100)0

b)5^2 × 5^−2

c)3^−5 × 3^−2 

d)5(2^3)^−2

e)((2^-3))/((2^-5)) 

f)(3^2)^−5 (2^5)^−1 

g)1/y^2 (x^3 y^−5)^2

h)1/a^2  (b/a^2)^(-3)

Evaluate the following:

a)3^−4

b)3^−2 × 3^−4

c)3^3 × (2^−2)^3

d)1/x^2 (x^3y^−5)^2

e)1/p^6 (p^4q^3)^3

Find the value of m and n:

a)3^2 × 3^5 = 3^n

b)3^2 × 5^3 × 3^3 × 5^2 = 3^m × 5^n

Find the missing numbers:

a)(5^4)^3 = 5^n

b)(2^2)^3 × (2^2)^5 = 2^n

Find the value of n:

a)(2^2)^3 = 2^n

b)(3^3)^3 × (3^2)^2 = 3^n

Find the value of m and n:

a)2 × 2^5 = 2^n

b)a^3 × a^13 = a^n

c)2^5 × 3^7 × 2^4 × 3^2 = 2^m × 3^n

Find the missing numbers:

a)(2^2 × 3^3 × 5^2^)2 × (2^3 × 5^3)^3 = 2^x × 3^y × 5^z

b)((2^2 × 5^3)/7^2)^2×((3^2 × 5^3)/7^3)^2 = (2^x × 3^y × 5^z)/7^n

Find the missing numbers:

a)(3^2 × 7^3 × 5^2)^2 × (2^3 × 7^3^)3 = 2^x × 3^y × 5^z × 7^n

b)((2^2 × 7^3)/5^2)^2×((3^2 × 7^3)/5^3)^2 = (2^x × 3^y × 7^z)/5^n

c)[(1/4).(1/3)]^2

Calculate:

a)16^1/2

b)(125)^1/3

c)(23)^1/3

d)(8)^1/2