2. CONTENT
1. What is algebra
2. Variables and constant
3. Equations
4. Coefficient
5. Like and unlike terms
6. Polynomial
3. What is Algebra
• The main feature of the Algebra is to study is the use of letters.
• Use of letters will allow us to write rules and formulas in a general way. By
using letters, we can talk about any number and not just a particular number.
• Secondly, letters may stand for unknown quantities. By learning methods of
determining unknowns, we develop powerful tools for solving puzzles and
many problems from daily life.
• Thirdly, since letters stand for numbers, operations can be performed on
them as on numbers. This leads to the study of algebraic expressions and
their properties.
4. VARIABLES AND CONSTANT
• Variables- x, y, z, a, b c, etc. Its value is not fixed; it can take any value 1, 2,
3, 4, ...
• The word variable means something that can vary, i.e. change.
A variable takes on different numerical values. From variables, we form
expressions. The expressions are formed by performing operations like
addition, subtraction, multiplication and division on the variables. From x, we
formed the expression (4x + 5). For this, first we multiplied x by 4 and then
added 5 to the product.
• Let a and b be two variables which can take any number value.
Then, a + b = b + a
• commutativity of multiplication of two numbers as a × b = b × a
• Distribution of variables, a × (b + c) = a × b + a × c
• Constant has a fixed value. Examples of constants are: 4, 100, –17, etc.
5. EQUATIONS
• any equation, is a condition on a variable. It is satisfied only for a definite value of the
variable. For example, the equation 2n = 10 is satisfied only by the value 5 of the
variable n. Similarly, the equation x – 3 = 11 is satisfied only by the value 14 of the
variable x.
• Note that an equation has an equal sign (=) between its two sides. The equation says
that the value of the left hand side (LHS) is equal to the value of the right hand side
(RHS). If the LHS is not equal to the RHS, we do not get an equation.
• if we add the same number to both sides of an equality, it still holds.
• if we subtract the same number from both sides of an equality, it still holds.
• if we multiply or divide both sides of the equality by the same non-zero number, it still
holds.
• Transposing means moving to the other side. Transposition of a number has the same
effect as adding same number to (or subtracting the same number from) both sides of
the equation. When you transpose a number from one side of the equation to the other
side, you change its sign. For example, transposing +3 from the LHS to the RHS in
equation x + 3 = 8 gives x = 8 – 3 (= 5). We can carry out the transposition of an
expression in the same way as the transposition of a number.
• value of the variable in an equation which satisfies the equation is called a solution to
the equation
6. Coefficient
(a) What are the coefficients of x in the following expressions?
4x – 3y, 8 – x + y, y2x – y, 2z – 5xz
(b) What are the coefficients of y in the following expressions?
4x – 3y, 8 + yz, yz2 + 5, my + m
When terms have the same algebraic factors, they are like terms. When terms
have different algebraic factors, they are unlike terms. For example, in the
expression 2xy – 3x + 5xy – 4, look at the terms 2xy and 5xy. The factors of
2xy are 2, x and y. The factors of 5xy are 5, x and y.
Like and Unlike terms
7. POLYNOMIAL
• An expression with only one term is called a monomial; for example, 7xy, –
5m, 3z2, 4 etc.
• An expression which contains two unlike terms is called a binomial; for
example, x + y, m – 5, mn + 4m, a2 – b2 are binomials
• An expression which contains three terms is called a trinomial; for example,
the expressions x + y + 7, ab + a +b, 3x2 – 5x + 2, m + n + 10 are trinomials
• an expression with one or more terms is called a polynomial. Thus a
monomial, a binomial and a trinomial are all polynomials.
8. Solve the following equations:
5t + 28 = 10
6z + 10 = –2
Solve the following equations:
(a) 2(x + 4) = 12 (b) 3(n – 5) = 21 (c) 3(n – 5) = – 21
(d) – 4(2 + x) = 8 (e) 4(2 – x) = 8
Solve the following equations:
(a) 4 = 5(p – 2) (b) – 4 = 5(p – 2) (c) 16 = 4 + 3(t + 2)
(d) 4 + 5(p – 1) =34 (e) 0 = 16 + 4(m – 6)
The sum of three times a number and 11 is 32. Find the number.
Find a number, such that one-fourth of the number is 3 more than 7.
Raju’s father’s age is 5 years more than three times Raju’s age. Find Raju’s age, if his father is 44 years
old.
PRACTICE QUESTIONS
9. Solve the following:
(a) The teacher tells the class that the highest marks obtained by a student in her class is twice
the lowest marks plus 7. The highest score is 87. What is the lowest score?
(b) In an isosceles triangle, the base angles are equal. The vertex angle is 40°. What are the
base angles of the triangle? (Remember, the sum of three angles of a triangle is 180°).
(c) Sachin scored twice as many runs as Rahul. Together, their runs fell two short of a double
century. How many runs did each one score?
Solve the following:
(i) Irfan says that he has 7 marbles more than five times the marbles Parmit has. Irfan has 37
marbles. How many marbles does Parmit have?
(ii) Laxmi’s father is 49 years old. He is 4 years older than three times Laxmi’s age. What is
Laxmi’s age?
(iii) People of Sundargram planted trees in the village garden. Some of the trees were fruit trees.
The number of non-fruit trees were two more than three times the number of fruit trees. What
was the number of fruit trees planted if the number of non-fruit trees planted was 77?
10. Identify the numerical coefficients of terms (other than constants) in the
following expressions:
(i) 5 – 3t2 (ii) 1 + t + t2 + t3 (iii) x + 2xy + 3y
(iv) 100m + 1000n (v) – p2q2 + 7pq (vi) 1.2 a + 0.8 b
(vii) 3.14 r2 (viii) 2 (l + b) (ix) 0.1 y + 0.01 y2
Identify terms which contain x and give the coefficient of x.
(i) y2x + y (ii) 13y2 – 8yx (iii) x + y + 2
(iv) 5 + z + zx (v) 1 + x + xy (vi) 12xy2 + 25
(vii) 7x + xy2
Identify terms which contain y2 and give the coefficient of y2.
(i) 8 – xy2 (ii) 5y2 + 7x (iii) 2x2y – 15xy2 + 7y2
Classify into monomials, binomials and trinomials.
(i) 4y – 7z (ii) y2 (iii) x + y – xy (iv) 100
(v) ab – a – b (vi) 5 – 3t (vii) 4p2q – 4pq2 (viii) 7mn
(ix) z2 – 3z + 8 (x) a2 + b2 (xi) z2 + z
(xii) 1 + x + x2
12. Subtract:
(i) –5y2 from y2
(ii) 6xy from –12xy
(iii) (a – b) from (a + b)
(iv) a (b – 5) from b (5 – a)
(v) –m2 + 5mn from 4m2 – 3mn + 8
(vi) – x2 + 10x – 5 from 5x – 10
(vii) 5a2 – 7ab + 5b2 from 3ab – 2a2 – 2b2
(viii) 4pq – 5q2 – 3p2 from 5p2 + 3q2 – pq
(a) What should be added to x2 + xy + y2 to obtain 2x2 + 3xy?
(b) What should be subtracted from 2a + 8b + 10 to get – 3a + 7b + 16?
What should be taken away from 3x2 – 4y2 + 5xy + 20 to obtain
– x2 – y2 + 6xy + 20?
(a) From the sum of 3x – y + 11 and – y – 11, subtract 3x – y – 11.
(b) From the sum of 4 + 3x and 5 – 4x + 2x2, subtract the sum of 3x2 – 5x and
–x2 + 2x + 5.
13. Find the values of the following expressions for x = 2.
(i) x + 4 (ii) 4x – 3 (iii) 19 – 5x2
(iv) 100 – 10x3
Find the value of the following expressions when n = – 2.
(i) 5n – 2 (ii) 5n2 + 5n – 2 (iii) n3 + 5n2 + 5n – 2
Find the value of the following expressions for a = 3, b = 2.
(i) a + b (ii) 7a – 4b (iii) a2 + 2ab + b2
(iv) a3 – b3
If z = 10, find the value of z3 – 3(z – 10).
(ii) If p = – 10, find the value of p2 – 2p – 100
What should be the value of a if the value of 2x2 + x – a equals to 5, when x = 0?
Simplify the expression and find its value when a = 5 and b = – 3.
2(a2 + ab) + 3 – ab
14. If m = 2, find the value of:
(i) m – 2 (ii) 3m – 5 (iii) 9 – 5m
(iv) 3m2 – 2m – 7 (v)
If p = – 2, find the value of:
(i) 4p + 7 (ii) – 3p2 + 4p + 7 (iii) – 2p3 – 3p2 + 4p + 7
Find the value of the following expressions, when x = –1:
(i) 2x – 7 (ii) – x + 2 (iii) x2 + 2x + 1
(iv) 2x2 – x – 2
If a = 2, b = – 2, find the value of:
(i) a2 + b2 (ii) a2 + ab + b2 (iii) a2 – b2
When a = 0, b = – 1, find the value of the given expressions:
(i) 2a + 2b (ii) 2a2 + b2 + 1 (iii) 2a2b + 2ab2 + ab
(iv) a2 + ab + 2
Simplify the expressions and find the value if x is equal to 2
(i) x + 7 + 4 (x – 5) (ii) 3 (x + 2) + 5x – 7
(iii) 6x + 5 (x – 2) (iv) 4(2x – 1) + 3x + 11
Simplify these expressions and find their values if x = 3, a = – 1, b = – 2.
(i) 3x – 5 – x + 9 (ii) 2 – 8x + 4x + 4
(iii) 3a + 5 – 8a + 1 (iv) 10 – 3b – 4 – 5b
(v) 2a – 2b – 4 – 5 + a
15. AMU Class XI Entrance Test ALGEBRA QUESTIONS
If (x + 2) and (x - 1) are factors of (x³ + 10x² + mx + n), then
(a) m = 3, n = -3 (b) m=17, n = -8 (C) m = 23, n = -19 (d) m = 7, n = -18
If x= a + 1/a and y = a - 1/a, then the value of x⁴+ y⁴ - 2x²y² is
(a) 24
(b) 18
(c) 16
(d) 12
If b is a real numbers such that b² = b + 1. Then which of the following is not true?
(a) b³= b² + b
(b) b⁴ = b³+b+1 (c) )b³ =2b+1
(d) b³+b² = b+1
If √(5/3) and - √(5/3) are two zeros of the polynomial 3x⁴ + 6x³ - 2x² - 10x - 5 then
other two zeros are :
(a) -1,-1 (b) 1,-1
(c) 1,1
(d) 3,-3
If x = 3 - 2 √2 , then (√x – 1/ √x)
(a)1
(b) 2
(c) 4
(d) 6
16. If 4x² + 9y² + z² = 6xy + 3yz + 2xz then 8x³ + 27y³ + z³ will be equal to
(a) 0 (b) 2x + 3y + z - 18xyz (c) 18xyz (d) (2x + 3y + z)³
The equation x² + 3x + k = 0, has real root. Then
a) k ≥ 9/4
b) k ≤ 9/4
c) k ≥ 0
d) k ≤ 0
For what value of k does the pair of equation 5x + 2y = 2k and 2 (k + 1) x + ky =
(3k + 4 ) have an infinite number of solutions?
(a) k = 5
(b) k = 4
(c) k = 2/3
(d) k = - 2/3
If (x3+ax2+bx+6) has ( x-2) as a factor and leaves a remainder 3 when divided by ( x-3), then the values of a and b are:
(a) a= -2, b= -1
(b) a= -1, b= -3
(c) a= -3
(d) a= -3, b= 1
17. If one of the zeros of the polynomial (a^2+9)x^2+13x+6a is reciprocal of the other,
then the value of a is
(a) 2
(b) 3
(c) 4
(d) 5
Which of the following is a false statement?
(a) Every positive odd integer is of the form 2q+1, where q is some integer.
(b) Every positive odd integer is of the form 4q+1 or 4q+3, where q is some positive
integer
(c) Every positive odd integer is of the form 6q+ or 6q+3 or 6q+5, where q is some
integer.
(d) -5 and -9 are coprime integers.
If one of the zeros of the cubic polynomial x^3+ax^2+bx+c is 0, then the product of the other two zeros is
(a) 0
(b) a
(c) b
(d) c
