1. POLYNOMIAL
If 𝑝(𝑥) is a polynomial in 𝑥, and if 𝑘 is any real number, then the value obtained by replacing x by k in 𝑝(𝑥), is called the value of 𝑝(𝑥) at 𝑥 =
𝑘 , and is denoted by 𝑝(𝑘).
consider the polynomial p(x) = 𝑥2
– 3𝑥 – 4.
Then, put 𝑥 = 2
𝑝(2) = 22
– 3 × 2 – 4 = – 6.
𝑝(0) is the value of 𝑝(𝑥) at 𝑥 = 0,
𝑝(0) = – 4
• A real number k is said to be a zero of a polynomial p(x), if 𝑝(𝑘) = 0
What is the value of 𝑝(𝑥) = 𝑥2
– 3𝑥 – 4 at 𝑥 = – 1?
𝑝(– 1) = (−1)2
– {3 × (– 1)} – 4 = 0
Also, 𝑝(4) = 42
– (3 × 4) – 4 = 0.
𝑝(– 1) = 0 and 𝑝(4) = 0,
–1 and 4 are called the zeroes of the quadratic polynomial 𝑥2
– 3𝑥 – 4.
2. POLYNOMIAL
How to find the zeroes of a linear polynomial.
In general, if k is a zero of 𝑝(𝑥) = 𝑎𝑥 + 𝑏,
then 𝑝(𝑘) = 𝑎𝑘 + 𝑏 = 0, i.e., 𝑘 =
−𝑏
𝑎
So, the zero of the linear polynomial ax + b is
−𝑏
𝑎
=
− (Constant term)
Coefficient of x
Thus, the zero of a linear polynomial is related to its coefficients.
Does this happen in the case of other polynomials too?
For example, are the zeroes of a quadratic polynomial also related to its coefficients?
For example, if k is a zero of 𝑝(𝑥) = 2𝑥 + 3,
then p(k) = 0 gives us 2𝑘 + 3 = 0, i.e., 𝑘 =
−3
1
3. POLYNOMIAL
For any quadratic polynomial a𝑥2
+ 𝑏𝑥 + 𝑐, 𝑎 ≠ 0,
The graph of the corresponding equation 𝑦 = 𝑎𝑥2
+ 𝑏𝑥 + 𝑐 has one of the two shapes either open upwards like or open downwards like
depending on whether 𝑎 > 0 or 𝑎 < 0. (These curves are called parabolas.)
–1 and 4 are zeroes of the quadratic polynomial.
–1 and 4 are the x-coordinates of the points where the graph of 𝑦 = 𝑥2
– 3𝑥 – 4 intersects the x-axis. Thus, the zeroes of the quadratic
polynomial x 2 – 3𝑥 – 4 are x-coordinates of the points where the graph of
𝑦 = 𝑥2
– 3𝑥 – 4 intersects the x-axis.
• This fact is true for any quadratic polynomial, i.e., the zeroes of a quadratic polynomial a𝑥2
+ 𝑏𝑥 + 𝑐, 𝑎 ≠ 0, are precisely the x-coordinates
of the points where the parabola representing y = a𝑥2
+ 𝑏𝑥 + c intersects the x-axis.
5. PAIR OF LINEAR EQUATION IN TWO VARIABLE
Examples (i) , (ii) and(iii) note down what kind of pair they are geometrically.
(i) x – 2y = 0 and 3x + 4y – 20 = 0
The lines intersect
𝑎1
𝑎2
≠
𝑏1
𝑏2
⟹
1
2
≠
−2
4
(ii) 2x + 3y – 9 = 0 and 4x + 6y – 18 = 0
The lines coincide
𝑎1
𝑎2
=
𝑏1
𝑏2
=
𝑐1
𝑐2
⟹
2
4
=
3
6
=
−9
−18
(iii) x + 2y – 4 = 0 and 2x + 4y – 12 = 0
The lines are parallel
𝑎1
𝑎2
=
𝑏1
𝑏2
≠
𝑐1
𝑐2
⟹
1
2
=
2
4
≠
−4
−12
If the lines represented by the equation
𝑎1𝑥 + 𝑏1𝑦 + 𝑐1 = 0
and 𝑎2𝑥 + 𝑏2𝑦 + 𝑐2 = are
(i) intersecting, then
𝑎1
𝑎2
≠
𝑏1
𝑏2
(ii) coincident, then
𝑎1
𝑎2
=
𝑏1
𝑏2
=
𝑐1
𝑐2
(iii) parallel, then
𝑎1
𝑎2
=
𝑏1
𝑏2
≠
𝑐1
𝑐2
6. PAIR OF LINEAR EQUATION IN TWO VARIABLE
EXERCISE 3.2
1. Form the pair of linear equations in the following problems, and find their solutions graphically.
(i) 10 students of Class X took part in a Mathematics quiz. If the number of girls is 4 more than the number of boys, find the number of boys and girls who took part in the quiz.
(ii) 5 pencils and 7 pens together cost ₹ 50, whereas 7 pencils and 5 pens together cost ₹ 46. Find the cost of one pencil and that of one pen.
2. On comparing the ratios
𝑎1
𝑎2
,
𝑏1
𝑏2
, and
𝑐1
𝑐2
, find out whether the lines representing the following pairs of linear equations intersect at a point, are parallel or coincident:
(i) 5x – 4y + 8 = 0
7x + 6y – 9 = 0
(ii) 9x + 3y + 12 = 0
18x + 6y + 24 = 0
(iii) 6x – 3y + 10 = 0
2x – y + 9 = 0
3. On comparing the ratios
𝑎1
𝑎2
,
𝑏1
𝑏2
, and
𝑐1
𝑐2
, find out whether the following pair of linear equations are consistent, or inconsistent.
(i) 3x + 2y = 5
2x – 3y = 7
(ii) 2x – 3y = 8
4x – 6y = 9
(iii)
3
2
x +
5
3
y = 7
9x – 10y = 14
(iv) 5x – 3y = 11
– 10x + 6y = –22
(v)
4
3
x + 2y = 8
2x + 3y = 12
7. PAIR OF LINEAR EQUATION IN TWO VARIABLE
4. Which of the following pairs of linear equations are consistent/inconsistent? If consistent, obtain the solution graphically:
(i) x + y = 5,
2x + 2y = 10
(ii) x – y = 8,
3x – 3y = 16
(iii) 2x + y – 6 = 0,
4x – 2y – 4 = 0
(iv) 2x – 2y – 2 = 0,
4x – 4y – 5 = 0
8. PAIR OF LINEAR EQUATION IN TWO VARIABLE
5. Half the perimeter of a rectangular garden, whose length is 4 m more than its width, is 36 m. Find the dimensions of the garden.
6. Given the linear equation 2x + 3y – 8 = 0, write another linear equation in two variables such that the geometrical representation of the pair so
formed is:
(i) intersecting lines
(ii) parallel lines
(iii) coincident lines
7. Draw the graphs of the equations x – y + 1 = 0 and 3x + 2y – 12 = 0.
Determine the coordinates of the vertices of the triangle formed by these lines and the x-axis, and shade the triangular region.
9. PAIR OF LINEAR EQUATION IN TWO VARIABLE
• Cross - Multiplication Method
Example:- The cost of 5 oranges and 3 apples is ₹ 35 and the cost of 2 oranges and 4 apples is ₹ 28.
Let us find the cost of an orange and an apple.
Let us denote the cost of an orange by ₹ x
the cost of an apple by ₹ y.
Then, 5𝑥 + 3𝑦 = 35
5𝑥 + 3𝑦 – 35 = 0 …….(i)
2x + 4y = 28
2𝑥 + 4𝑦 – 28 = 0 …….(ii)
Use the elimination method
Multiply Equation (i) by 4 and Equation (ii) by 3.
(4)(5)𝑥 + (4)(3)𝑦 + (4)(– 35) = 0 ……(iii)
(3)(2)𝑥 + (3)(4)𝑦 + (3)(– 28) = 0 …….(iv)
Subtracting Equation (iv) from Equation (iii),
we get [(5 × 4) – (3 × 2)]𝑥 + [(4 × 3) – (3 × 4)]𝑦 + [4(– 35) – (3) × (– 28)] = 0
10. PAIR OF LINEAR EQUATION IN TWO VARIABLE
Let us now see how this method works for any pair of linear equations in two variables of the form
𝑎1𝑥 + 𝑏1𝑦 + 𝑐1 = 0 ……..(i)
and 𝑎2𝑥 + 𝑏2𝑦 + 𝑐2 = 0 ……..(ii)
To obtain the values of x and y as shown above, we follow the following steps:
Step 1 : Multiply Equation (1) by 𝑏2 and Equation (2) by 𝑏1 , to get 𝑏2
𝑏2 𝑎1𝑥 + 𝑏2 𝑏1𝑦 + 𝑏2𝑐1 = 0 …….(iii)
𝑏1 𝑎2𝑥 + 𝑏1 𝑏2𝑦 + 𝑏1 𝑐2 = 0 ……..(iv)
Step 2 : Subtracting Equation (iv) from (iii),
𝑏2 𝑎1 − 𝑏1 𝑎2 𝑥 + 𝑏2 𝑏1 − 𝑏1 𝑏2 𝑦 + 𝑏2𝑐1 − 𝑏1 𝑐2 = 0
i.e., 𝑏2 𝑎1 − 𝑏1 𝑎2 𝑥 = 𝑏1 𝑐2 − 𝑏2𝑐1
So, x =
𝑏1 𝑐2−𝑏2𝑐1
𝑏2 𝑎1−𝑏1 𝑎2
, provided 𝑏2 𝑎1 − 𝑏1 𝑎2 ≠ 0 ………(v)
Step 3 : Substituting this value of x in (1) or (2), we get y =
𝑎2𝑐1−𝑎1 𝑐2
𝑏2 𝑎1−𝑏1 𝑎2
……..(vi)
11. PAIR OF LINEAR EQUATION IN TWO VARIABLE
Now, two cases arise :
Case 1 : 𝑏2 𝑎1 − 𝑏1 𝑎2 ≠ 0.
In this case
𝑎1
𝑎2
≠
𝑏1
𝑏2
Then the pair of linear equations has a unique solution.
Case 2 : 𝑏2 𝑎1 − 𝑏1 𝑎2 = 0.
If we write
𝑎1
𝑎2
=
𝑏1
𝑏2
= k
Then 𝑎1 = 𝑘𝑎2, 𝑏1 = 𝑘𝑏2.
Substituting the values of 𝑎1 and 𝑏1 in the Equation (i), we get k 𝑎2𝑥 + 𝑏2𝑦 𝑐1 = 0. (vii)
It can be observed that the Equations (vii) and (ii) can both be satisfied only if 𝑐1 = k𝑐2 , i.e.,
𝑐1
𝑐2
= 𝑘
If 𝑐1 = k𝑐2 , any solution of Equation (ii) will satisfy the Equation (1), and vice versa.
If
𝑎1
𝑎2
=
𝑏1
𝑏2
=
𝑐1
𝑐2
= 𝑘 , then there are infinitely many solutions to the pair of linear equations given by (i) and (ii).
If 𝑐1 ≠ k𝑐2 , then any solution of Equation (i) will not satisfy Equation (ii) and vice versa.
Therefore the pair has no solution
12. PAIR OF LINEAR EQUATION IN TWO VARIABLE
𝑎1𝑥 + 𝑏1𝑦 + 𝑐1 = 0 ……..(i)
and 𝑎2𝑥 + 𝑏2𝑦 + 𝑐2 = 0 ……..(ii)
For the pair of linear equations given by (i) and (ii) as follows:
(i) When
𝑎1
𝑎2
≠
𝑏1
𝑏2
, we get a unique solution.
(ii) When
𝑎1
𝑎2
=
𝑏1
𝑏2
=
𝑐1
𝑐2
, there are infinitely many solutions.
(iii) When
𝑎1
𝑎2
=
𝑏1
𝑏2
≠
𝑐1
𝑐2
, there is no solution
Note that you can write the solution given by Equations (v) and (vi) in the following form :
𝑥
𝑏1𝑐2−𝑏2𝑐1
=
𝑦
𝑐1𝑎2−𝑐2𝑎1
=
1
𝑎1𝑏2−𝑎2𝑏1
………(viii)
In remembering the above result, the following diagram may be helpful to you :
The arrows between the two numbers indicate that they are to be multiplied and the second product is to be subtracted from the first
13. PAIR OF LINEAR EQUATION IN TWO VARIABLE
For solving a pair of linear equations by this method, we will follow the following steps :
Step 1 : Write the given equations in the form
𝑎1𝑥 + 𝑏1𝑦 + 𝑐1 = 0 ……..(i)
and 𝑎2𝑥 + 𝑏2𝑦 + 𝑐2 = 0 ……..(ii)
Step 2 : write Equations as
𝑥
𝑏1𝑐2−𝑏2𝑐1
=
𝑦
𝑐1𝑎2−𝑐2𝑎1
=
1
𝑎1𝑏2−𝑎2𝑏1
.
Step 3 : Find x and y, provided 𝑏2 𝑎1 − 𝑏1 𝑎2 ≠ 0 Step 2 above gives you an indication of why this method is called the cross-multiplication
method.
14. PAIR OF LINEAR EQUATION IN TWO VARIABLE
EXERCISE 3.5
1. Which of the following pairs of linear equations has unique solution, no solution, or infinitely many solutions. In case there is a unique solution,
find it by using cross multiplication method.
(i) x – 3y – 3 = 0
3x – 9y – 2 = 0
(ii) 2x + y = 5
3x + 2y = 8
(iii) 3x – 5y = 20
6x – 10y = 40
(iv) x – 3y – 7 = 0
3x – 3y – 15 = 0
2. (i) For which values of a and b does the following pair of linear equations have an infinite number of solutions?
2x + 3y = 7
(a – b) x + (a + b) y = 3a + b – 2
(ii) For which value of k will the following pair of linear equations have no solution?
3x + y = 1
(2k – 1) x + (k – 1) y = 2k + 1
3. Solve the following pair of linear equations by the substitution and cross-multiplication methods :
8x + 5y = 9
3x + 2y = 4
15. PAIR OF LINEAR EQUATION IN TWO VARIABLE
Equations Reducible to a Pair of Linear Equations in Two Variables
The solution of such pairs of equations which are not linear but can be reduced to linear form by making some suitable substitutions.
Example 17 : Solve the pair of equations:
2
𝑥
+
3
𝑦
= 13
5
𝑥
−
4
𝑦
= −2
Solution :
17. PAIR OF LINEAR EQUATION IN TWO VARIABLE
2. Formulate the following problems as a pair of equations, and hence find their solutions:
(i) Ritu can row downstream 20 km in 2 hours, and upstream 4 km in 2 hours. Find her speed of rowing in still water and the speed of the current.
(ii) 2 women and 5 men can together finish an embroidery work in 4 days, while 3 women and 6 men can finish it in 3 days. Find the time taken by
1 woman alone to finish the work, and also that taken by 1 man alone.
(iii) Roohi travels 300 km to her home partly by train and partly by bus. She takes 4 hours if she travels 60 km by train and the remaining by bus. If
she travels 100 km by train and the remaining by bus, she takes 10 minutes longer. Find the speed of the train and the bus separately.
18. PAIR OF LINEAR EQUATION IN TWO VARIABLE
EXERCISE 3.7 (Optional)
1. The ages of two friends Ani and Biju differ by 3 years. Ani’s father Dharam is twice as old as Ani and Biju is twice as old as his sister Cathy. The
ages of Cathy and Dharam differ by 30 years. Find the ages of Ani and Biju.
2. One says, “Give me a hundred, friend! I shall then become twice as rich as you”. The other replies, “If you give me ten, I shall be six times as
rich as you”. Tell me what is the amount of their (respective) capital?
Sol:-
x + 100 = 2(y – 100),
y + 10 = 6(x – 10)
3. A train covered a certain distance at a uniform speed. If the train would have been 10 km/h faster, it would have taken 2 hours less than the
scheduled time. And, if the train were slower by 10 km/h; it would have taken 3 hours more than the scheduled time. Find the distance covered
by the train.
4. The students of a class are made to stand in rows. If 3 students are extra in a row, there would be 1 row less. If 3 students are less in a row,
there would be 2 rows more. Find the number of students in the class.
5. In a ∆ABC, ∠C = 3 ∠B = 2 (∠A + ∠B). Find the three angles.
6. Draw the graphs of the equations 5x – y = 5 and 3x – y = 3.
Determine the co-ordinates of the vertices of the triangle formed by these lines and the y axis.
19. EXERCISE 3.7 (Optional)
7. Solve the following pair of linear equations:
(i) px + qy = p – q
qx – py = p + q
(ii) ax + by = c
bx + ay = 1 + c
(iii)
𝑥
𝑎
−
𝑦
𝑏
= 0
ax + by = 𝑎2
+ 𝑏2
(iv) (a – b)x + (a + b) y = 𝑎2
− 2𝑎𝑏 − 𝑏2
(a + b)(x + y) = 𝑎2
+ 𝑏2
(v) 152x – 378y = – 74
–378x + 152y = – 604
8. ABCD is a cyclic quadrilateral. Find the angles of the cyclic quadrilateral.
∠ 𝐴 = 4𝑦 + 20
∠𝐵 = 3𝑦 − 5
∠𝐶 = −4𝑥
∠𝐷 = −7𝑥 + 5
20. QUADRATIC EQUATION
If 𝑏2 − 4𝑎𝑐 > 0, we get two distinct real roots
−𝑏 + 𝑏2−4𝑎𝑐
2𝑎
and
−𝑏− 𝑏2−4𝑎𝑐
2𝑎
If 𝑏2 − 4𝑎𝑐 = 0, then 𝑥 =
−𝑏
2𝑎
± 0 i.e., 𝑥 =
−𝑏
2𝑎
or
−𝑏
2𝑎
⋅
So, the roots of the equation a𝑥2 + 𝑏𝑥 + 𝑐 = 0 are both
−𝑏
2𝑎
⋅
Therefore, we say that the quadratic equation a𝑥2 + bx + c = 0 has two equal real roots in this case.
If 𝑏2
– 4𝑎𝑐 < 0, then there is no real number whose square is 𝑏2
– 4𝑎𝑐.
Therefore, there are no real roots for the given quadratic equation in this case.
