Download LINEAR EQUATION IN TWO VARIABLE CLASS 10 TEST PAPER

LINEAR EQUATIONS
Cynesure Institute
Class 10
1. Solve the pair or linear equations by elimination method: x – y = 2 and x + y = 2. Then find k
if k = 2x + 3y
2. Solve for x and y by substitution method:
x +1
2
y−1
+
3
= 8 and
x +1
3
+
y−1
2
=8
3. Solve the following pair of linear equations:
21x + 47y = 110; 47x + 21y = 162
𝑦
4. If 2x + y = 23 and 4x – y = 19, find the values of 5y – 2x and 𝑥 – 2.
5. Solve for x and y by elimination method :
x +6
4
= 6:
3y−8
y
=5.
[Ans x=3, y=2]
6. Solve the following system of linear equations using the method of cross-multiplication:
a. ax +by =1: bx +ay =
7. Solve for x and y:
2
2x+y
-
(a+b)2
a 2 +b 2
1
x−2y
+
5
9
=1
=0 :
9
2x+y
-
6
x−2y
+4=0
[Ans x=2 , y=1/2]
8. Solve the following system of equations:
bx
a
-
ay
b
+a +b=0 :
bx –ay +2ab=0 [Ans x= -3a, y= -b]
1. Find k so that the equations x + 2y = – 7 and 2x + ky + 14 = 0 will represent co-incident lines.
2. Find a linear equations which is coincident with 2 x + 3y - 5 = 0
3. What is the value of k for which (3, k) lies on 2x – 3y = 5?
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4. Arman is walking on the line joining (1, 4) and (0, 6). Sudhir is walking on the line joining
(3, 4,) and (1,0). Represent this situation on graph and find the point where they cross each
other.
5. For what value of k the equations are parallel. 2x + ky = 10 and 3x + (k + 3) y = 12
6. Graphically check if the pair of linear equations 3x + 5y = 15, x – y = 5 is consistent or not.
7. For what value of p, the pair of linear equations (P + 2) x – (2 p + 1)y = 3 (2p – 1) , 2x – 3y =
7 has unique solution?
8. How many solutions can every linear equation in two variables?
9. If ax + by = c and lx + my = n has unique solution, then what is the relation between the
coefficients?
10. For what values of p and q, will the following pair of linear equations have infinitely many
solutions?
4x + 5y = 2 ; (2p + 7q) x + (p + 8q) y = 2q – p + 1.
11. Draw the graphs of the pair of linear equations: x – y + 2 = 0 and 4x – y – 4 = 0. Evaluate the
area of the triangle formed by the lines so drawn and the x-axis.
12. If the pair of linear equations 2x+3y=11 and 2px+(p+q)y=p+5q has infinitely many solution,
prove that q=2p.
13. Find the value of k for which the given system of equations has unique solution: 2x+3y-5=0,
kx-6y-8=0
[Ans k≠ -4]
14. Find the value of a if (-3, a) lies on 7x+2y=14.
15. Draw the graph the following equations: 2x+3y-12=0 and 7x-3y-15=0.
16. For what all real values of a, the pair of equations: x – 2y = 8 and 5x – 10y = a have a unique
solution?
17. Is the line represented by x = 7 parallel to the x–axis? Explain.
18. For what value/s of λ can the pair of linear equations λx + y = λ2 and x + λy = 1 have (i) no
solution (ii) infinitely many solutions (iii) a unique solution?
19. Two straight lines are represented by the equations x – 3y = 2 and –2x + 6y = 5. Check if the
paths cross each other or not.
20. By the graphical method, find whether the following pair of equations are consistent or not.
If consistent, solve them.
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i) 3x + y + 4 = 0 6x – 2y + 4 = 0
(ii) x – 2y = 6 : 3x – 6y = 0
(iii) x + y = 3: 3x + 3y = 9
1. The sum of numerator and denominator of a fraction is 3 less than twice the denominator. If
numerator and denominator both are decreased by 1, the numerator becomes half the
denominator. Form a pair of linear equations in two variables.
2. Roohi gives Rs. 9000 to some athletes of a school as scholarship every month. Had there
been 20 more athletes, then each would have got Rs. 160 less. Form a pair of linear equations
in two variables.
3. The sum of two natural numbers is 25 and their difference is 7. Find the two numbers.
4. A boat covers 32 km upstream and 36 km downstream in 7 hours. Then it covers 40 km
upstream and 48 km downstream in 9 hours. Find the speed of boat and stream.
5. The three angles of a triangle are x, y and 40°. The difference between x and y is 30°. Find x
and y.
6. Two years ago, Salim was thrice as old as his daughter and six years later, he will be four
years older than twice her age. Find their present ages.
7. The age of the father is twice the sum of the ages of his two children. After 20 years, his age
will be equal to the sum of the ages of his children. Find their present ages.
8. Two numbers are in the ratio 5:6. If 8 is subtracted from each of the numbers, the ratio
becomes 4:5. Find the two numbers.
9. There are two examination halls A and B with students. In order to make the number of
students equal in each hall, 10 students are sent from B to A. But if 20 students are sent from
A to B, the number of students in B becomes double the number of students in A. Find how
many students are in each hall.
10. A shopkeeper gives books on rent for reading. She takes a fixed charge for the first two days,
and an additional charge for each day extra. Lalit paid Rs 22 for a book kept for six days,
while Sumit paid Rs 16 for the book kept for four days. Find the fixed charges and the charge
for each extra day.
11. In an examination, one mark is given for each correct answer while one half mark is
deducted for every wrong answer. Surbhi answered 120 questions and got 90 marks. How
many questions did she answer correctly?
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12. ABCD is a cyclic quadrilateral. Its angles are A = (6x + 10)°, B = (5x)° C = (x + y)°, D = (3y
– 10)° Find x and y, and the four angles.
13. Ramlal sold a table and a chair for Rs 1050, thereby making a profit of 10% on the table and
25% on the chair. If he had taken a profit of 25% on the table and 10% on the chair he would
have got Rsn1065. Find the cost price of each.[ 500, 400]
14. Using two pipes, it takes 12 hours to fill a swimming pool. If the pipe of larger diameter is
used for 4 hours and the pipe of smaller diameter for 9 hours, only half the pool can be filled.
How long would it take for each pipe to fill the pool separately?[ 20, 30]
15. Akshita travels 14 km to her home partly by train and partly by bus. She takes half an hour if
she travels 2 km by train, and the remaining distance by bus. On the other hand, if she travels
4 km by train and the remaining distance by bus, she takes 9 minutes longer. Find the speed
of the train and of the bus.
16. The sum of the numerator and the denominator of a fraction is 20. If we subtract 5 from the
numerator and 5 from denominator, then the ratio of the numerator and the denominator will
be 1:4 .Find the fraction.
[ Ans : 7/13
17. If two digit number is four times the sum of its digits and twice the product of digits. Find the
number. [Ans 36
18. The sum of two natural numbers is 8 and sum of their reciprocals is 8/15. Find the numbers.
[Ans 5 and 3]
19. The sum of the digits of a two- digit number is 12. The number obtained by interchanging the
two digits exceed the given number by 18. Find the number.
[Ans 57]
20. Find the values of x and y in the following rectangle:
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21. Find the values of x and y in the following rectangle:
22. The cost of 4 pens and 4 pencils is Rs 100. Three times the cost of a pen is Rs 15 more than
the cost of a pencil. Find the cost of a pen and a pencil box.
23. A two-digit number is obtained first by multiplying the sum of the digits by 8 and then
subtracting 5. Later the same number is obtained by multiplying the difference of the digits
by 16 and then adding 3. Find the number by forming linear equations in two variable.
24. A half ticket of railway costs half of the full fare. The reservation charges are same on a half
ticket as well as full ticket. A reserved first class ticket from Surat to Mumbai costs Rs 2530.
A reserved first class ticket and one reserved first class half ticket from the same places costs
Rs 3810. Find the full first class fare and the reservation charges for a ticket.
25. Sohan had a number of bananas. He divided them into two parts A and B. He sold the first
part at the rate of Rs 2 for 3 bananas and the second lot at the rate of Re 1 for one banana,
and got a total of Rs 400. If he had sold the first part at the rate of Re 1 per banana, and the
second part at the rate of Rs 4 for 5 bananas, his total collection would have been Rs 460.
Find the total number of bananas he had.
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