CBSE Class 10 Mathematics Pairs of Linear Equations in Two Variables MCQs Set 08

Question. A can do a piece of work in 24 days. If \( B \) is \( 60\% \) more efficient than \( A \), then the number of days required by \( B \) to do the twice as large as the earlier work is
(a) 24
(b) 36
(c) 15
(d) 30
Answer: (d)
Work ratio of \( A : B = 100 : 160 \) or \( 5 : 8 \)
Time ratio \( = 8 : 5 \) or \( 24 : 15 \)
If \( A \) takes 24 days, \( B \) takes 15 days, Hence, \( B \) takes 30 days to do double the work.

Question. A motor boat takes 2 hours to travel a distance 9 km. down the current and it takes 6 hours to travel the same distance against the current. The speed of the boat in still water and that of the current (in km/hour) respectively are
(a) 3, 1.5
(b) 3, 2
(c) 3.5, 2.5
(d) 3, 1
Answer: (a)
Downrate \( = 9 \div 2 = 4.5 \) km/hr
Uprate \( = 9 \div 6 = 1.5 \) km/hr
Speed of the boat \( = (4.5 + 1.5) \div 2 = 3 \) km/hr
Speed of the current \( = (4.5 - 1.5) \div 2 = 1.5 \) km/hr

Question. \( X \)’s salary is half that of \( Y \)’s. If \( X \) got a \( 50\% \) rise in his salary and \( Y \) got \( 25\% \) rise in his salary, then the percentage increase in combined salaries of both is
(a) 30
(b) \( 33 \frac{1}{3} \)
(c) \( 37 \frac{1}{2} \)
(d) 75
Answer: (b)
96% of C.P. \( = 240 \)
110% of C.P. \( = \frac{240}{960} \times 1100 = 275 \)

Question. The 2 digit number which becomes \( (5/6) \)th of itself when its digits are reversed. The difference in the digits of the number being 1, then the two digits number is
(a) 45
(b) 54
(c) 36
(d) None of these
Answer: (b)
If the two digits are \( x \) and \( y \), then the number is \( 10x + y \).
Now \( \frac{5}{6}(10x + y) = 10y + x \)
Solving, we get \( 44x + 5y = 60y + 6x \Rightarrow 38x = 55y \Rightarrow \frac{x}{y} = \frac{5}{4} \)
Also \( x - y = 1 \). Solving them, we get \( x = 5 \) and \( y = 4 \). Therefore, number is 54.

Question. The points \( (7, 2) \) and \( (-1, 0) \) lie on a line
(a) \( 7y = 3x - 7 \)
(b) \( 4y = x + 1 \)
(c) \( y = 7x + 7 \)
(d) \( x = 4y + 1 \)
Answer: (b)
The point satisfy the line, \( 4y = x + 1 \).

Question. In a number of two digits, unit’s digit is twice the tens digit. If 36 be added to the number, the digits are reversed. The number is
(a) 36
(b) 63
(c) 48
(d) 84
Answer: (c)
Let unit’s digit : \( x \)
tens digit : \( y \)
Then, \( x = 2y \)
Number \( = 10y + x \)
Also, \( 10y + x + 36 = 10x + y \)
\( 9x - 9y = 36 \)
or \( x - y = 4 \)
Solve, \( x = 2y \)
\( x - y = 4 \)

Question. At present ages of a father and his son are in the ratio \( 7 : 3 \), and they will be in the ratio \( 2 : 1 \) after 10 years. Then the present age of father (in years) is
(a) 42
(b) 56
(c) 70
(d) 77
Answer: (c)
Let the ages of father and son be \( 7x \), \( 3x \)
Hence, \( (7x + 10) : (3x + 10) = 2 : 1 \)
or \( x = 10 \)
Age of the father is 70 years.

Question. If \( 3x + 4y : x + 2y = 9 : 4 \), then \( 3x + 5y : 3x - y \) is equal to
(a) 4 : 1
(b) 1 : 4
(c) 7 : 1
(d) 1 : 7
Answer: (c)
\( \frac{3x+4y}{x+2y} = \frac{9}{4} \)
Hence, \( 12x + 16y = 9x + 18y \)
or \( 3x = 2y \)
\( x = \frac{2}{3}y \)
Substitute \( x = \frac{2}{3}y \) in the required expression.
\( \frac{3(\frac{2}{3}y) + 5y}{3(\frac{2}{3}y) - y} = \frac{7y}{y} = \frac{7}{1} = 7 : 1 \)

Question. A fraction becomes 4 when 1 is added to both the numerator and denominator and it becomes 7 when 1 is subtracted from both the numerator and denominator. The numerator of the given fraction is
(a) 2
(b) 3
(c) 5
(d) 15
Answer: (d)
Let the fraction be \( \frac{x}{y} \),
\( \frac{x+1}{y+1} = 4 \) ...(1)
and \( \frac{x-1}{y-1} = 7 \) ...(2)
Solving (1) and (2),
We have \( x = 15 \), \( y = 3 \),
i.e. \( x = 15 \)

Question. \( x \) and \( y \) are 2 different digits. If the sum of the two digit numbers formed by using both the digits is a perfect square, then value of \( x + y \) is
(a) 10
(b) 11
(c) 12
(d) 13
Answer: (b)
The numbers that can be formed are \( xy \) and \( yx \).
Hence, \( (10x + y) + (10y + x) = 11(x + y) \). If this is a perfect square that \( x + y = 11 \).

Question. The pair of equations \( 3^{x+y} = 81 \), \( 81^{x-y} = 3 \) has
(a) no solution
(b) unique solution
(c) infinitely many solutions
(d) \( x = 2\frac{1}{8}, y = 1\frac{7}{8} \)
Answer: (d)
Given, \( 3^{x+y} = 81 \Rightarrow 3^{x+y} = 3^4 \Rightarrow x + y = 4 \) ...(1)
and \( 81^{x-y} = 3 \Rightarrow (3^4)^{x-y} = 3^1 \Rightarrow 4(x - y) = 1 \Rightarrow x - y = \frac{1}{4} \) ...(2)
On adding Eq. (1) and (2), we get
\( 2x = 4 + \frac{1}{4} = \frac{17}{4} \Rightarrow x = \frac{17}{8} = 2\frac{1}{8} \)
From Eq. (1), we get \( y = \frac{15}{8} = 1\frac{7}{8} \)

Question. A man can row a boat in still water at the rate of 6 km per hour. If the stream flows at the rate of 2 km/hour, he takes half the time going downstream than going upstream the same distance. His average speed for upstream and down stream trip is
(a) 6 km/hour
(b) 16/3 km/hour
(c) Insufficient data to arrive at the answer
(d) none of the above
Answer: (b)
Upstream speed \( = 4 \) km/hr
and time \( = x \) hrs
Downstream \( = 8 \) km/hr
and time taken \( = x/2 \) hrs
Hence, average speed \( = \frac{4x + 8 \times x/2}{x + x/2} = \frac{16}{3} \) km/hr.

Question. A boat travels with a speed of 15 km/h in still water. In a river flowing at 5 km/hr. the boat travels some distance downstream and them returns. The ratio of average speed to the speed in still water is
(a) 8 : 3
(b) 3 : 8
(c) 8 : 9
(d) 9 : 8
Answer: (c)
Let distance \( = d \)
Time taken upstream \( = \frac{d}{15-5} = \frac{d}{10} \)
Time taken downstream \( = \frac{d}{15+5} = \frac{d}{20} \)
Hence, average speed \( = \frac{2d}{\frac{d}{10} + \frac{d}{20}} = \frac{2d \times 20}{3d} = \frac{40}{3} \) km/hr
Ratio \( = \frac{40}{3} : 15 = 40 : 45 = 8 : 9 \)

FILL IN THE BLANK

Question. Every solution of a linear equation in two variables is a point on the .......... representing it.
Answer: line

Question. If \( \frac{1}{x} + \frac{1}{y} = k \) and \( \frac{1}{x} - \frac{1}{y} = k \), then the value of \( y \) is ..........
Answer: Does not exist

Question. If a pair of linear equations has infinitely many solutions, then its graph is represented by a pair of .......... lines.
Answer: coincident

Question. A pair of linear equations is .......... if it has no solution.
Answer: inconsistent

Question. If \( p + q = k, p - q = n \) and \( k > n \), then \( q \) is ..........
Answer: \( \frac{k - n}{2} \)

Question. A pair of .......... lines represent the pair of linear equations having no solution.
Answer: parallel

Question. If a pair of linear equations has solution, either a unique or infinitely many, then it is said to be ..........
Answer: consistent

TRUE/FALSE

Question. The pair of equations \( 4x - 5y = 8 \) and \( 8x - 10y = 3 \) has a unique solution.
Answer: False

Question. A pair of intersecting lines representing a pair of linear equations in two variables has a unique solution.
Answer: True

Question. \( \sqrt{2}x + \sqrt{3}y = 0 \), \( \sqrt{3}x - \sqrt{8}y = 0 \) has no solution.
Answer: False

Question. A pair of linear equations cannot have exactly two solutions.
Answer: True

Question. If a pair of linear equations is given by \( a_1x + b_1y + c_1 = 0 \) and \( a_2x + b_2y + c_2 = 0 \) and \( \frac{a_1}{a_2} \neq \frac{b_1}{b_2} \). In this case, the pair of linear equations is consistent.
Answer: True

Question. A pair of linear equations in two variables is said to be consistent if it has no solution.
Answer: False

Question. If a pair of linear equations is given by \( a_1x + b_1y + c_1 = 0 \) and \( a_2x + b_2y + c_2 = 0 \) and \( \frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2} \). In this case, the pair of linear equations is consistent.
Answer: True

Question. A pair of linear equations in two variables may not have infinitely many solutions.
Answer: True

MATCHING QUESTIONS

DIRECTION : Each question contains statements given in two columns which have to be matched. Statements (A, B, C, D) in column I have to be matched with statements (p, q, r, s) in column II.

Question. Column-II give value of \( x \) and \( y \) for pair of equation given in Column-I.
Column-I:
(A) \( 2x + y = 8, x + 6y = 15 \)
(B) \( 5x + 3y = 35, 2x + 4y = 28 \)
(C) \( \frac{1}{7x} + \frac{1}{6y} = 3, \frac{1}{2x} - \frac{1}{3y} = 5 \)
(D) \( 15x + 4y = 61, 4x + 15y = 72 \)
Column-II:
(p) \( (3, 4) \)
(q) \( (1/14, 1/6) \)
(r) \( (4, 5) \)
(s) \( (3, 2) \)
Answer: (A) - s, (B) - r, (C) - q, (D) - p.

Question. Column-I:
(A) \( 5y - 4 = 14, y - 2x = 1 \)
(B) \( 6x - 3y + 10 = 0, 2x - y + 9 = 0 \)
(C) \( 3x - 2y = 4, 9x - 6y = 12 \)
(D) \( 2x - 3y = 8, 4x - 6y = 9 \)
Column-II:
(p) Infinite solutions
(q) Consistent
(r) No solution
(s) Inconsistent
Answer: (A) - q, (B) - s, (C) - p, (D) - r

Question. Column-I:
(A) No solution
(B) Infinitely many solutions
(C) Unique solution
(D) System is consistent
Column-II:
(p) \( 5x - 15y = 8, 3x - 9y = \frac{24}{5} \)
(q) \( 2x + 4y = 10, 3x + 6y = 12 \)
(r) \( 3x - 2y = 4, 6x - 4y = 8 \)
(s) \( 2x + 6y = 2, 4x - 2y - 40 = 0 \)
(t) \( 3x - y = 8, x - \frac{y}{3} = 3 \)
(u) \( x - y = 8, 3x - 3y = 16 \)
Answer: (A) - (q, t, u), (B) - (p, r), (C) - s, (D) - (p, r, s)

ASSERTION AND REASON

DIRECTION : In the following questions, a statement of assertion (A) is followed by a statement of reason (R). Mark the correct choice as:
(a) Both assertion (A) and reason (R) are true and reason (R) is the correct explanation of assertion (A).
(b) Both assertion (A) and reason (R) are true but reason (R) is not the correct explanation of assertion (A).
(c) Assertion (A) is true but reason (R) is false.
(d) Assertion (A) is false but reason (R) is true.

Question. Assertion : \( x + y - 4 = 0 \) and \( 2x + ky - 3 = 0 \) has no solution if \( k = 2 \).
Reason : \( a_1x + b_1y + c_1 = 0 \) and \( a_2x + b_2y + c_2 = 0 \) are consistent if \( \frac{a_1}{a_2} \neq \frac{b_1}{b_2} \).
(a) Both assertion (A) and reason (R) are true and reason (R) is the correct explanation of assertion (A).
(b) Both assertion (A) and reason (R) are true but reason (R) is not the correct explanation of assertion (A).
(c) Assertion (A) is true but reason (R) is false.
(d) Assertion (A) is false but reason (R) is true.
Answer: (b)

Question. Assertion : Pair of linear equations : \( 9x + 3y + 12 = 0 \), \( 18x + 6y + 24 = 0 \) have infinitely many solutions.
Reason : Pair of linear equations \( a_1x + b_1y + c_1 = 0 \) and \( a_2x + b_2y + c_2 = 0 \) have infinitely many solutions, if \( \frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2} \).
(a) Both assertion (A) and reason (R) are true and reason (R) is the correct explanation of assertion (A).
(b) Both assertion (A) and reason (R) are true but reason (R) is not the correct explanation of assertion (A).
(c) Assertion (A) is true but reason (R) is false.
(d) Assertion (A) is false but reason (R) is true.
Answer: (a)

Question. Assertion : If \( kx - y - 2 = 0 \) and \( 6x - 2y - 3 = 0 \) are inconsistent, then \( k = 3 \).
Reason : \( a_1x + b_1y + c_1 = 0 \) and \( a_2x + b_2y + c_2 = 0 \) are inconsistent of \( \frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2} \).
(a) Both assertion (A) and reason (R) are true and reason (R) is the correct explanation of assertion (A).
(b) Both assertion (A) and reason (R) are true but reason (R) is not the correct explanation of assertion (A).
(c) Assertion (A) is true but reason (R) is false.
(d) Assertion (A) is false but reason (R) is true.
Answer: (a)

Question. Assertion : The lines \( 2x - 5y = 7 \) and \( 6x - 15y = 8 \) are parallel lines.
Reason : The system of linear equations \( a_1x + b_1y + c_1 = 0 \) and \( a_2x + b_2y + c_2 = 0 \) have infinitely many solutions if \( \frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2} \).
(a) Both assertion (A) and reason (R) are true and reason (R) is the correct explanation of assertion (A).
(b) Both assertion (A) and reason (R) are true but reason (R) is not the correct explanation of assertion (A).
(c) Assertion (A) is true but reason (R) is false.
(d) Assertion (A) is false but reason (R) is true.
Answer: (b)

Question. Assertion : \( 3x - 4y = 7 \) and \( 6x - 8y = k \) have infinite number of solution if \( k = 14 \).
Reason : \( a_1x + b_1y + c_1 = 0 \) and \( a_2x + b_2y + c_2 = 0 \) have a unique solution if \( \frac{a_1}{a_2} \neq \frac{b_1}{b_2} \).
(a) Both assertion (A) and reason (R) are true and reason (R) is the correct explanation of assertion (A).
(b) Both assertion (A) and reason (R) are true but reason (R) is not the correct explanation of assertion (A).
(c) Assertion (A) is true but reason (R) is false.
(d) Assertion (A) is false but reason (R) is true.
Answer: (b)

Question. Assertion : The linear equations \( x - 2y - 3 = 0 \) and \( 3x + 4y - 20 = 0 \) have exactly one solution.
Reason : The linear equations \( 2x + 3y - 9 = 0 \) and \( 4x + 6y - 18 = 0 \) have a unique solution.
(a) Both assertion (A) and reason (R) are true and reason (R) is the correct explanation of assertion (A).
(b) Both assertion (A) and reason (R) are true but reason (R) is not the correct explanation of assertion (A).
(c) Assertion (A) is true but reason (R) is false.
(d) Assertion (A) is false but reason (R) is true.
Answer: (c)