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Page No 48:
Question 1:
The value of k for which the system of equations has a unique solution, is
kx − y = 2
6x − 2y = 3
(a) =3
(b) ≠3
(c) ≠0
(d) =0
Answer:
The given system of equations are
for unique solution
Here
By cross multiply we get
Hence, the correct choice is.
Page No 48:
Question 2:
The value of k for which the system, of equations has infinite number of solutions, is
2x + 3y = 5
4x + ky = 10
(a) 1
(b) 3
(c) 6
(d) 0
Answer:
The given system of equations are
For the equations to have infinite number of solutions,
Here,
Therefore
By cross multiplication of we get,
And
Therefore the value of k is 6
Hence, the correct choice is .
Page No 48:
Question 3:
The value of k for which the system of equations x + 2y − 3 = 0 and 5x + ky + 7 = 0 has no solution, is
(a) 10
(b) 6
(c) 3
(d) 1
Answer:
The given system of equations are
For the equations to have no solutions,
If we take
Therefore the value of k is10.
Hence, correct choice is.
Page No 49:
Question 4:
The value of k for which the system of equations 3x + 5y = 0 and kx + 10y = 0 has non-zero solution, is
(a) 0
(b) 2
(c) 6
(d) 8
Answer:
The given system of equations are,
Here,
By cross multiply we get
Therefore the value of k is 6,
Hence, the correct choice is.
Page No 49:
Question 5:
The value of k for which the system of equations has no solution is
(a) 6
(b) −6
(c) 3/2
(d) None of these
Answer:
The given system of equation is
If then the equation have no solution.
By cross multiply we get
Hence, the correct choice is.
Page No 49:
Question 6:
If a pair of linear equations in two variables is consistent, then the lines represented by two equations are
(a) intersecting
(b) parallel
(c) always coincident
(d) intersecting or coincident
Answer:
If a pair of linear equations in two variables is consistent, then its solution exists.
∴The lines represented by the equations are either intersecting or coincident.
Hence, correct choice is.
Page No 49:
Question 7:
If the system of equations 2x + 3y = 5, 4x + ky = 10 has infinitely many solutions, then k =
(a) 1
(b) ½
(c) 3
(d) 6
Answer:
The given system of equations
For the equations to have infinite number of solutions
If we take
And
Therefore, the value of k is 6.
Hence, the correct choice is .
Page No 49:
Question 8:
If the system of equations kx − 5y = 2, 6x + 2y = 7 has no solution, then k =
(a) −10
(b) −5
(c) −6
(d) −15
Answer:
The given systems of equations are
If
Here
Hence, the correct choice is .
Page No 49:
Question 9:
If is the solution of the systems of equations and , then the values of and are, respectively
(a) 3 and 1 (b) 3 and 5 (c) 5 and 3 (d) 1 and 3
Answer:
The given equations are
Adding (1) and (2), we get
2x = 6
⇒ x = 3
Putting x = 3 in (1), we get
3 + y = 4
⇒ y = 1
So, x = a = 3 and y = b = 1.
Thus, the values of a and b are 3 and 1, respectively.
Hence, the correct answer is option (a).
Page No 49:
Question 10:
For what value k , do the equations and reperesent coincident lines ?
(a) (b) (c) 2 (d)
Answer:
The given system of equations is
We know that the lines
are coincident iff
Thus, the value of k = 2.
Hence, the correct answer is option (c).
Page No 49:
Question 11:
The pair of linear equations y = 0 and y = –5 has
(a) one solution
(b) two solutions
(c) infinitely many solutions
(d) no solution
Answer:
The given pair of equations are y = 0 and y = 5.
The equation y = 0 represents x-axis and y = 5 is a line parallel to x-axis (no x-intercept).
We can see this graphically, both lines are parallel and thus will have no solution.
Hence, the correct answer is option (d).
Page No 49:
Question 12:
8 chairs and 5 tables cost â¹10,500, while 5 chairs and 3 tables cost â¹6,450. The cost of each chair will be
(a) â¹750
(b) â¹600
(c) â¹850
(d) â¹900
Answer:
Let the cost of each chair be â¹x and for each table be â¹y.
Then,
Multiplying (1) by 3 and (2) by 5, we get
Subtracting (3) from (4), we get
Hence, the correct answer is option (a).
Page No 49:
Question 13:
If ABCD is a rectangle shown in the given figure then
(a) x = 10, y = 2
(b) x = 12, y = 8
(c) x = 2, y = 10
(d) x = 20, y = 0
Answer:
By the property of rectangle, we know that opposite sides are equal i.e. CD = AB.
So, x + y = 12 .....(1)
Similarly, AD = BC.
So, x − y = 8 .....(2)
On adding (1) and (2), we get
2x = 20
x = 10
On substituting x = 10 in (1), we get
y = 2
Hence, the correct answer is option (a).
Page No 49:
Question 14:
The pair of linear equations 3x + 5y = 3 and 6x + ky = 8 do not have a solution, if k
(a) = 5
(b) = 10
(c) ≠ 10
(d) ≠ 5
Answer:
The pair of linear equations and , the condition for no solution is given by .
Comparing the coefficients of two equations, with general equation of the form ax + by + c = 0, we get:
Hence, the correct answer is option (b).
Page No 49:
Question 15:
If the sum of the ages of a father and his son in years is 65 and twice the difference of their ages in years is 50, then the age of father is
(a) 40 years
(b) 45 years
(c) 55 years
(d) 65 years
Answer:
Let the ages of a father and his son in years be x and y respectively.
According to the question,
Adding (1) and (2), we get
x = 45 and y = 20
Thus, the age of father is 45 years.
Hence, the correct answer is option (b).
Page No 49:
Question 16:
If the system of equations has infinitely many solutions, then
2x + 3y = 7
(a + b)x + (2a − b)y = 21
(a) a = 1, b = 5
(b) a = 5, b = 1
(c) a = −1, b = 5
(d) a = 5, b = −1
Answer:
The given systems of equations are
For the equations to have infinite number of solutions,
Here ,
Let us take
By cross multiplication we get,
Now take
By cross multiplication we get,
Substitute in the above equation
Substitute in equation we get,
Therefore and.
Hence, the correct choice is.
Page No 49:
Question 17:
If the system of equations is inconsistent, then k =
(a) 1
(b) 0
(c) −1
(d) 2
Answer:
The given system of equations is inconsistent,
If the system of equations is in consistent, we have
Therefore, the value of k is2.
Hence, the correct choice is .
Page No 49:
Question 18:
If am ≠ bl, then the system of equations
(a) has a unique solution
(b) has no solution
(c) has infinitely many solutions
(d) may or may not have a solution
Answer:
Given the system of equations has
We know that intersecting lines have unique solution
Here
Therefore intersecting lines, have unique solution
Hence, the correct choice is
Page No 49:
Question 19:
If the system of equations has infinitely many solutions, then
(a) a = 2b
(b) b = 2a
(c) a + 2b = 0
(d) 2a + b = 0
Answer:
Given the system of equations are
For the equations to have infinite number of solutions,
By cross multiplication we have
Divide both sides by 2. we get
Hence, the correct choice is .
Page No 50:
Question 20:
If 2x − 3y = 7 and (a + b)x − (a + b − 3)y = 4a + b represent coincident lines, then a and b satisfy the equation
(a) a + 5b = 0
(b) 5a + b = 0
(c) a − 5b = 0
(d) 5a − b = 0
Answer:
The given system of equations are
For coincident lines , infinite number of solution
Option A.:
Option B:
Option.C:
a - b = 0
-5 - (-1) = -4 0
None of the option satisfies the values.
Page No 50:
Question 21:
The area of the triangle formed by the line with the coordinate axes is
(a) ab
(b) 2ab
(c)
(d)
Answer:
Given the area of the triangle formed by the line
If in the equation either A and B approaches infinity, The line become parallel to either x axis or y axis respectively,
Therefore
Area of triangle
Hence, the correct choice is .
Page No 50:
Question 22:
The area of the triangle formed by the lines y = x, x = 6 and y = 0 is
(a) 36 sq. units
(b) 18 sq. units
(c) 9 sq. units
(d) 72 sq. units
Answer:
Given and
We have plotting points as when
Therefore, area of
Area of triangle is square units
Hence, the correct choice is .
Page No 50:
Question 23:
The area of the triangle formed by the lines x = 3, y = 4 and x = y is
(a) ½ sq. unit
(b) 1 sq. unit
(c) 2 sq. unit
(d) None of these
Answer:
Given and
We have plotting points as when
Therefore, area of
Area of triangle is square units
Hence, the correct choice is
Page No 50:
Question 24:
The sum of the digits of a two digit number is 9 . If 27 is added to it , the digits of the number get reversed . th number is
(a) 25 (b) 72 (c) 63 (d) 36
Answer:
Let the digits at the tens and the ones place be x and y, respectively. So, the two digit number is 10x + y.
Now,
x + y = 9 .....(i)
Also,
10x + y + 27 = 10y + x
⇒ 9x − 9y = −27
⇒ x − y = −3 .....(ii)
Adding (i) and (ii), we get
2x = 6
⇒ x = 3
Putting x = 3 in (i), we get
3 + y = 9
⇒ y = 6
Thus, the required number is 10 × 3 + 6 = 36.
Hence, the correct answer is option (d).
Page No 50:
Question 25:
Aruna has only â¹1 and â¹2 coins with her . If the total number of coins that she has is 50 and the amount of money with her is â¹75 , then the number of â¹1 and â¹2 coins are , respectively
(a) 35 and 15 (b) 35 and 20 (c) 15 and 35 (d) 25 and 25
Answer:
Let the number of â¹1 coins be x and that of â¹2 coins be y.
Now,
Total number of coins = 50
So, x + y = 50 .....(i)
Also,
â¹1 × x + â¹2 × y = â¹75
∴ x + 2y = 75 .....(ii)
Subtracting (i) from (ii), we get
y = 25
Putting y = 25 in (i), we get
x + 25 = 50
⇒ x = 25
So, the number of â¹1 coins and â¹2 coins are 25 and 25, respectively.
Hence, the correct answer is option (d).
Disclaimer: The answer given in the book does not match with the one obtained.
Page No 50:
Question 26:
If x = a, y = b is the solution of the pair of linear equations 37x + 43y = 123, 43x + 37y = 117, then a3 + b3 is equal to
(a) –7
(6) 7
(c) 9
(d) –9
Answer:
37x + 43y = 123 ....(1)
43x + 37y = 117 ....(2)
Subtracting (1) from (2), we get
6x − 6y = −6
⇒ x = y − 1
Substitute in (1), we get
37(y − 1) + 43y = 123
⇒ 80y = 123 + 37 = 160
⇒ y = 2 = b
⇒ x = 1 = a
∴ a3 + b3 = 13 + 23 = 9
Hence, the correct answer is option (c).
Page No 50:
Question 27:
The value of k for which the lines 5x + 7y = 3 and 15x + 21y = k coincide is
(a) 9
(b) 5
(c) 7
(d) 18
Answer:
Comparing the coefficients of two equations, with general equation of the form ax + by + c = 0, we get:
As , which means that lines are coincident
Solving we get k = 9.
Hence, the correct answer is option (a).
Page No 50:
Question 28:
One equation of a pair of dependent linear equations is –5x + 7y = 2. The second equation is
(a) 10x + 14y + 4 = 0
(b) –10x – 14y + 4 = 0
(c) –10x + 14y + 4 = 0
(d) 10x – 14y = –4
Answer:
The condition for pair of equation to be dependent linear equations is .
Comparing the coefficients of the first equation, with general equation of the form ax + by + c = 0, we get
Let the second equation be .
, where k is any arbitrary constant.
Putting , we get
The required equation becomes .
Hence, the correct answer is option (d).
Page No 50:
Question 29:
If 217x + 131y = 913 and 131x + 217y = 827, then x + y is equal to
(a) 5
(b) 6
(c) 7
(d) 8
Answer:
217x + 131y = 913 .....(1)
131x + 217y = 827 .....(2)
Adding (1) and (2), we get
348x + 348y = 1740
⇒ x + y = 5 ....(3)
Subtracting (2) from (1), we get
86x − 86y = 86
⇒ x − y = 1 .....(4)
Adding (3) and (4), we get
⇒ x = 3 and y = 2
⇒ x + y = 3 + 2 = 5
Hence, the correct answer is option (a).
Page No 50:
Question 30:
The number of solutions of 3x + y = 243 and 243x – y = 3 is
(a) 0
(b) 1
(c) 2
(d) infinite
Answer:
3x + y = 243
⇒ 3x + y = 35
⇒ x + y = 5 .....(1)
And, 243x – y = 3
⇒ (35)x – y = 31
⇒ 5(x – y) = 1
⇒ x – y = ..... (2)
Adding (1) and (2), we get
⇒
Thus, the number of solutions is 1.
Hence, the correct answer is option (b).
Page No 50:
Question 31:
The area of the triangle formed by the lines 2x + 3y = 12, x − y − 1 = 0 and x = 0 (as shown in Fig. 3.23), is
(a) 7 sq. units
(b) 7.5 sq. units
(c) 6.5 sq. units
(d) 6 sq. units
Answer:
Given and
If We have plotting points as
Therefore, area of
Area of triangle is square units
Hence, the correct choice is
Page No 51:
Question 32:
In the given figure the graph representing two linear equations by lines AB and CD respectively. The area of the triangle formed by these two lines and the line x = 0 is
(a) 3 sq. units
(b) 4 sq. units
(c) 6 sq. units
(d) 8 sq. units
Answer:
Let the intersection point of line CD and x-axis be E.
Thus, EB = 3 units and the perpendicular distance from point C on EB is 2 units.
Area of triangle ECB =
Hence, the correct answer is option (a).
Page No 51:
Question 33:
Teachers and students of class X of a school had gone to Nandan Kannan for study tour. After visiting different places of Nandan Kannan, lastly, they visited bird's sanctuary and deer park. Rohan is a clever boy and keen observer. He put the question to his friends" How many birds are there and how many deer are there (at particular time) in Nandan Kannan?" Rahul's friend, Nishith gave the correct answer as follows:
'Nishith answered that total animals have 1000 eyes and 1400 legs.'
(i) If x and y be the number of birds and deer respectively, what is the equation of total number of eyes?
(b) x + y = 500
(c) x – y = 1000
(d) x – y = 500
(b) x + 2y = 500
(c) x + 2y = 700
(d) 2x – y = 500
(b) 5000
(c) 300
(d) 200
(b) 200
(c) 300
(d) 700
(b) 700
(c) 500
(d) 300
Answer:
(i) Consider x and y be the number of birds and deer respectively.
Since each bird and deer has 2 eyes.
And, total animals have 1000 eyes.
Thus, the equation will be
2x + 2y = 1000
⇒ x + y = 500 .....(1)
Hence, the correct answer is option (b).
(ii) Since, each bird has 2 legs and each deer has 4 legs.
And, total animals have 1400 legs.
Thus, equation will be
2x + 4y = 1400
⇒ x + 2y = 700 .....(2)
Hence, the correct answer is option (c).
(iii) From (1) and (2),
x + y = 500 .....(1)
x + 2y = 700 .....(2)
Multiplying (1), by 2 we get
2x + 2y = 1000 ....(3)
Subtracting (2) from (3), we get
x = 300
Hence, the correct answer is option (c).
(iv) Substituting x = 300 in (1), we get
300 + y = 500
y = 200
Hence, the correct answer is option (b).
(v)
Total number of animals in the zoo are total number of birds and deer, i.e., 300 + 200 = 500.
Hence, the correct answer is option (c).
Page No 52:
Question 34:
Mathematics teacher of a school took the standard 10 students to see the painting exhibition which was held at ART COLLEGE OF EDUCATION, Bangalore. It is the part of art integration of Mathematics. The teacher and students had interest in painting as well. Students were eager to see the above paintings. The teacher explained that the above paintings are based on concept of a pair of linear equations of two variables.
(i) If the speed of boat is 5 km/hr and speed of stream is 2 km/hr. What is the speed of the boat in downstream?
(b) 2 km/hr
(c) 7 km/hr
(d) 3 km/hr
(b) 2 km/hr
(c) 7 km/hr
(d) 3 km/hr
(b) 2 hr
(c) 7 hr
(d) 3 hr
(b) 2 hr
(c) 6 hr
(d) 3 hr
(b) t(x + y) km
(e) 2t(x – y) km
(d) 2t(x + y) km
Answer:
(i) Speed of boat is 5 km/hr and speed of stream is 2 km/hr.
Speed of the boat in downstream = Speed of boat + speed of stream
= 5 + 2 km/hr
= 7 km/hr
Hence, the correct answer is option (c).
(ii) Speed of boat is 5 km/hr and speed of stream is 2 km/hr.
Speed of the boat in upstream = Speed of boat − speed of stream
= 5 − 2 km/hr
= 3 km/hr
Hence, the correct answer is option (d).
(iii) Distance travelled by boat along downstream is 21 km.
Speed of the boat in downstream =
Hence, the correct answer is option (d).
(iv) Distance travelled by boat along upstream is 12 km.
Speed of the boat in upstream =
Hence, the correct answer is option (a).
(v) If speed of boat and stream be x km/hr and y km/hr respectively.
Speed of the boat in downstream = Speed of boat + speed of stream
= (x + y) km/hr
Speed of the boat in downstream =
Hence, the correct answer is option (b).
Page No 53:
Question 35:
Each of the following questions contains STATEMENT-1 (Assertion) and STATEMENT-2 (Reason) and has following four choices (a), (b), (c) and (d), only one of which is the correct answer. Mark correct choice.
(a) Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1.
(b) Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statement-1.
(c) Statement-1 is true, Statement-2 is false.
(d) Statement-1 is false, Statement-2 is true.
Statement-1 (Assertion): The system of linear equations 3x + 5y – 4 = 0 and 15x + 25y – 25 = 0 is inconsistent.
Statement-2 (Reason): The pair of linear equations a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0 represents parallel lines, if
Answer:
The pair of linear equations a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0 represents parallel lines, if
Thus, statement-2 is true.
3x + 5y – 4 = 0 and 15x + 25y – 25 = 0
Comparing the coefficients of two equations, with the general equation of the form ax + by + c = 0, we get
As which means that lines are parallel, so the system of linear equations is inconsistent.
Statement-1 is true, and Statement-2 is a correct explanation for Statement-1.
Hence, the correct answer is option (a).
Page No 53:
Question 36:
Each of the following questions contains STATEMENT-1 (Assertion) and STATEMENT-2 (Reason) and has following four choices (a), (b), (c) and (d), only one of which is the correct answer. Mark correct choice.
(a) Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1.
(b) Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statement-1.
(c) Statement-1 is true, Statement-2 is false.
(d) Statement-1 is false, Statement-2 is true.
Statement-1 (Assertion): The area of the rectangle formed by the lines representing x = 8, y = 6 with the coordinate axes is 24 sq. units.
Statement-2 (Reason): The system of equations x = 8, y = 6 is consistent with a unique solution.
Answer:
The graph of x = 8 is a line parallel to y-axis at a distance of 8 units to the right of it.
So, the line l is the graph of x = 8.
The graph of y = 6 is a line parallel to the x-axis at a distance of 6 units above it.
So, the line m is the graph of y = 6.
The figure enclosed by the lines x = 8, y = 6, the x-axis and the y-axis is OABC, which is a rectangle.
The vertices of the rectangle OABC are O (0, 0), A (8, 0), B (8, 6), C (0, 6).
The length and breadth of this rectangle are 8 units and 6 units, respectively.
As the area of a rectangle = length × breadth,
the area of rectangle OABC = 8 × 6 = 48 sq. units
Thus, statement-1 is false.
From the graph, x = 8, y = 6 has a unique solution (8, 6) and thus a consistent system of equations.
Thus, statement-2 is true.
Hence, the correct answer is option (d).
Page No 53:
Question 37:
Each of the following questions contains STATEMENT-1 (Assertion) and STATEMENT-2 (Reason) and has following four choices (a), (b), (c) and (d), only one of which is the correct answer. Mark correct choice.
(a) Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1.
(b) Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statement-1.
(c) Statement-1 is true, Statement-2 is false.
(d) Statement-1 is false, Statement-2 is true.
Statement-1 (Assertion): If a pair of linear equations represent coincident lines, then the equations are consistent and have a unique solution.
Statement-2 (Reason): A pair of linear equations a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0 represents coincident lines iff
Answer:
A pair of linear equations a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0 represents coincident lines if
Thus, statement-2 is true.
If a pair of linear equations represent coincident lines, then the equations are consistent and have infinitely many solutions.
Thus, statement-1 is false.
Hence, the correct answer is option (d).
Page No 53:
Question 38:
Each of the following questions contains STATEMENT-1 (Assertion) and STATEMENT-2 (Reason) and has following four choices (a), (b), (c) and (d), only one of which is the correct answer. Mark correct choice.
(a) Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1.
(b) Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statement-1.
(c) Statement-1 is true, Statement-2 is false.
(d) Statement-1 is false, Statement-2 is true.
Statement-1 (Assertion): If the system of equations 3x + 6y = 10 and 2x – ky + 5 = 0 is inconsistent, then k = –4.
Statement-2 (Reason): The system of equations a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0 is inconsistent iff
Answer:
The system of equations a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0 is inconsistent if
Thus, statement-2 is false.
The system of equations 3x + 6y = 10 and 2x – ky + 5 = 0 is inconsistent, then
Thus, statement-1 is true.
Hence, the correct answer is option (c).
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