INTERNATIONAL INDIAN SCHOOL, AL-JUABAIL

WORKSHEET
CLASS: X
SUBJECT: MATHEMATICS TOPIC: 3-PAIR OF LINEAR EQUATIONS IN TWO VARIABLES

1 1 3 1 1 −1
1. Solve + = ; − = .
3𝑥+𝑦 3𝑥−𝑦 4 2(3𝑥+𝑦) 2(3𝑥−𝑦) 8
2. Solve152𝑥 − 378𝑦 = −74; −378𝑥 + 152𝑦 = −604.
4 3
3. Solve + 3𝑦 = 14; − 4𝑦 = 23.
𝑥 𝑥
4. Obtain the condition for the following system of linear equations to have a unique solution 𝑎𝑥 + 𝑏𝑦 = 𝑐;
a. 𝑙𝑥 + 𝑚𝑦 = 𝑛.
5. On selling a T.V. at 5% gain and a fridge at 10% gain, a shopkeeper gains ₹.2000. But if he sells the T.V. at
10% gain and the fridge at 5% loss. He gains ₹ 1500 on the transaction. Find the actual prices of T.V. and
fridge.
6. The sum of a two digit number and the number formed by interchanging its digits is 110. If 10 is subtracted
from the first number, the new number is 4 more than 5 times the sum of the digits in the first number.
Find the first number.
7. If three times the larger of the two numbers is divided by the smaller one, we get 4 as quotient and 3 as the
remainder. Also if seven times the smaller number is divided by the larger one, we get 5 as quotient and 1
as remainder. Find the numbers.
8. Father’s age is three times the sum of ages of his two children. After 5 years his age will be twice the sum of
the ages of two children. Find the age of father.
9. Points A and B are 90km apart from each other on a highway. A car starts from A and another from B at the
same time. If they go in the same direction they meet in 9 hours and if they go in opposite directions they
meet in 9/7 hours. Find their speeds.
10. Determine by drawing graphs, whether the following pair of linear equations has a unique solution or not:
2𝑥 − 5 = 0, 𝑦 + 4 = 0.
11. Two numbers are in the ration 4:5. If 30 is subtracted from each numbers, the ratio becomes 1:2. Form the
pair of linear equations for the above situation and represent them graphically.
12. The sum of the digits of a two digit numbers is 15. The number obtained by interchanging digits exceeds
the given number by 9. Find the number (Graphically).
13. Solve the following system of linear equations graphically, shade the region bounded by lines and x-axis.
Also find the area of shaded region: 3𝑥 + 𝑦 − 12 = 0; 𝑥 − 3𝑦 + 6 = 0.
14. Solve for x and y using substitution method.
𝑎𝑥 𝑏𝑦
i) − = 𝑎 + 𝑏; 𝑎𝑥 − 𝑏𝑦 = 2𝑎𝑏. 𝑖𝑖) 𝑎2 𝑥 − 𝑏 2 𝑦 = 𝑎2 − 2𝑏 2 ; 𝑏 2 𝑥 + 𝑎2 𝑦 = 𝑏 2 + 2𝑎2
𝑏 𝑎

15. Solve the following equations for x and y:

a. 2(𝑎𝑥 − 𝑏𝑦) + (𝑎 + 4𝑏) = 0; 2(𝑏𝑥 + 𝑎𝑦) + (𝑏 − 4𝑎) = 0
16. Students of a class are made to stand in rows. If 4 students are extra in a row, there would be two rows
less. If 4 students are less in a row, there would be four more rows. Find the number of students in the
class.
17. The sum of the digits of a two digit number is 9. The number obtained by reversing the order of digits of
the given number exceeds the given number by 27. Find the given number.
18. The sum of a two digit number and the number formed by reversing the order of digits is 154. If the two
digits differ by 4, find the number.
19. Solve for x and y: 𝑎2 𝑥 + 𝑏 2 𝑦 = 𝑐 2 ; 𝑏 2 𝑥 + 𝑎2 𝑦 = 𝑑2
20. Find the value of k so that the following system of equations has no solution. 3𝑥 − 𝑦 − 5 = 0; 6𝑥 − 2𝑦 +
𝑘 = 0.
21. Find the value of k so that the following system of equations has infinite solutions. 3𝑥 − 𝑦 − 5 = 0; 6𝑥 −
2𝑦 + 𝑘 = 0.
22. Find the value of m so that the following system of equations has infinite solutions.
a. 2𝑥 + 3𝑦 − 7 = 0 𝑎𝑛𝑑 (𝑚 − 1)𝑥 + (𝑚 + 1)𝑦 = (3𝑚 − 1)
23. Find the values of a and b so that the following system of equations has infinite solutions.
a. 2𝑥 − 3𝑦 = 7 𝑎𝑛𝑑 (𝑎 + 𝑏)𝑥 − (𝑎 + 𝑏 − 3)𝑦 = (4𝑎 + 𝑏)
24. Solve the system of linear equations 99𝑥 + 101𝑦 = 499; 101𝑥 + 99𝑦 = 501.
 CHAPTER 4 QUADRATIC EQUATIONS.
1 . Find the discriminant of the equation: 5𝑥 2 − √2 𝑥 + 3 = 0
2 . Solve the following quadratic equations:
1 1 1 1
(a) = + + , a+b≠0
𝑎+𝑏+𝑥 𝑎 𝑏 𝑥

7𝑥+1 5𝑥−3
(b) 3 ( ) -4( ) = 11
5𝑥−3 7𝑥+1

2𝑥 1 3𝑥+9
(c) + + ( 𝑥−3 )( 2𝑥+3 )
=0
𝑥−3 2𝑥+3

(d) 9𝑥 2 − 9 (𝑎 + 𝑏 )𝑥 + ( 2𝑎2 + 5𝑎𝑏 + 2 𝑏2 ) = 0

(e) √3𝑥 2 + 10 𝑥 + 7√3 = 0

𝑥−1 𝑥−3 10
(f) + =
𝑥+2 𝑥−4 3

(g) √2𝑥 + 9 + 𝑥 = 0

3 . Find 𝑝 for which ( 𝑝 + 1 )𝑥 2 − 6( 𝑝 + 1 )𝑥 + 3 ( 𝑝 + 𝑞 ) = 0, 𝑞 ≠ −1, ℎ𝑎𝑠 𝑒𝑞𝑢𝑎𝑙

roots. Hence find the roots of the equation.
4. If the equation ( 1 + 𝑚2 )𝑥 2 − 2𝑚𝑐𝑥 + ( 𝑐 2 − 𝑎2 ) = 0 has equal roots , prove
𝑐 2 = 𝑎2 ( 1 + 𝑚2 )
5. For what value of 𝑘 the given equation ( 4 − 𝑘 )𝑥 2 + (2𝑘 + 4)𝑥 + ( 8𝑘 + 1 ) = 0
is a perfect square.
6 . Find the positive value of , for which the equation 𝑥 2 + 𝑘𝑥 + 64 = 0 and
𝑥 2 − 8𝑥 + k = 0 will both have real roots.
7 . If − 4 is a root of the quadratic 𝑥 2 + 𝑝𝑥 − 4 = 0 and the quadratic equation
𝑥 2 + 𝑝𝑥 + 𝑘 = 0 has equal roots , find the value of 𝑘.
8 . If 𝛼 , 𝛽 are the roots of the equation 2𝑥 2 − 6𝑥 + 𝑎 = 0 and 2𝛼 + 5𝛽 = 12 , find
the value of 𝑎.

9 . The difference of two numbers is 5 and the difference of their reciprocals is 1⁄10 ..

Find the numbers.

10. A two digit number is seven times the sum of its digits and is also equal to 12 less
than 3 times the product of its digits. Find the number.
11. A piece of cloth costs Rs. 200 . If the piece was 5 m longer and each metre of cloth
Costs Rs. 2 less, the cost of the piece would have remained unchanged. How long is
the piece and what is the original rate per metre?
12. A motor boat whose speed in still water is 18 km/hr takes one hour more to go
24 km upstream than to return down stream to the same spot. Find the speed of
the stream.

13. Two pipes running together can fill a cistern in 3 1⁄13 minutes, If onepipe takes 3

minutes more than the other to fill it, find the time in which each pipe would fill
the cistern.
14. A plane left 30 minutes late than the scheduled time and in order to reach the
destination 1500 km away in time ,it has to increase its speed by 250 km/hr from
its usual speed. Find its usual speed.
15. A numerator of a fraction is 3 less than the denominator. If 2 is added to both the
numerator and the denominator, then the sum of the new fraction and the original
29
fraction is . Find the original fraction.
20

16. A chess board contains 64 equal squares and the area of each square is 6.25 cm². A
border round the board is 2 cm wide. Find the length of the side of the chess board
 ARITHMETIC PROGRESSION
1. Find the common difference of an AP in which a25-a12=52.
2. Find the common difference of an AP whose nth term is 6n+2.
3. For what values of k are 2k, k+10 and 3k+2 in AP?
4. If the sum of the first p terms of an AP is ap2+bp, find it’s common difference.
5. Which term of the AP 115, 110, 105,……………….. is it’s first negative term ?
6. The sum of three numbers in AP is 21.If the product of the first and the third numbers exceeds the
second number by 6, find the numbers.
7. How many terms of the AP 54,51,48…. Be taken so that the sum is 513?
8. For what values of p, are 2p+1, 13, 5p-3 three consecutive terms of an AP?
9. If the 9th term of an AP is zero then prove that the 29 th term of the AP is double its 19th term.
10. Find the middle term of the AP -6,-2,2,….,58.

11. Which term of the AP 3,15,27,39…………will be 120 more than its 21st term?
12. Find the number of three digit natural numbers divisible by 11.
13. The 19th term of an AP is equal to three times its sixth term. If the 9th term is 19,find the AP.

14. Find the common difference of an AP whose first term is 5 and the sum of its first four terms is half
the sum of the next four terms.
15. In an AP, the sum of first ten terms is -150 and sum of its next ten terms is -550.Find the AP.
16. If the sum of first 6 terms of an AP is 36 and that of first 16 terms is 256, find the sum of first 10
terms.
17. The sum of the first five terms of an AP and the sum of first eleven terms of the same AP is 167.If the
sum of the first ten terms of this AP is 235, find the sum of its first twenty terms.
18. If the sum of first 7 terms of an AP is 49 and that of first 17 terms is 289 ,find the sum of its first n
terms.
1 1 1
19. If the pth term is and the qth term is 𝑝, show that the sum of pq terms is 2 (𝑝𝑞 + 1) .
𝑞
20. Find the ninth term from the end of the AP 5,9,13,……..185.
21. Find the sum of all multiples of 7 lying between 500 and 900.
22. If the seventh term of the AP is 1/9 and its ninth term is 1/7, find its 63 rd term.
23. If the sum of first seven terms of an AP is 63 and the sum of next seven terms is 161, find the 28th
term of this AP.
24. Divide 56 in four parts in AP, such that the ratio of the product of their extremes to the product of
their means is 5:6.
25. The digits of a positive number of three digits are in AP and their sum is 15.The number obtained on
reversing the digit is 594 less than the original number. Find the number.
26. Find the 20th term of the AP whose 7th term is 24 less than the 11th term, first term being 12.

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