CBSE Class 10 Mathematics Pair of Linear Equations in Two Variables Worksheet Set 09

SECTION A

Question. Choose and write the correct option in the following questions.
(i) The cost of production per unit for two products, \( A \) and \( B \), are Rs. 100 and Rs. 80 respectively. In a week, the total production cost is Rs. 32000. In the next week, the production cost reduces by 20%, and the total cost of producing the same number of units of each product is Rs. 25600. Which of these are the equations that can be used to find the number of units of \( A \), \( x \) and the number of units of \( B \), \( y \)?
(a) \( 100x + 80y = 32000 \) and \( 80x + 64y = 25600 \)
(b) \( 100x + 64y = 32000 \) and \( 80x + 100y = 25600 \)
(c) \( 80x + 64y = 25600 \) and \( 80x + 64y = 32000 \)
(d) \( 80x + 80y = 32000 \) and \( 100x + 64y = 25600 \)
Answer: (a) \( 100x + 80y = 32000 \) and \( 80x + 64y = 25600 \)

Question. (ii) Consider the equations shown. \( p + q = 5 \) and \( p - q = 2 \). Which of these are the values of \( p \) and \( q \)?
(a) \( p = 1.5, q = 3.5 \)
(b) \( p = 3.5, q = 1.5 \)
(c) \( p = 2, q = 3 \)
(d) \( p = 3, q = 2 \)
Answer: (b) \( p = 3.5, q = 1.5 \)

Question. (iii) The value of \( k \) for which the lines \( (k + 1)x + 3ky + 15 = 0 \) and \( 5x + ky + 5 = 0 \) are coincident is
(a) 14
(b) 2
(c) -14
(d) -2
Answer: (a) 14

Question. Solve the following questions.
(i) If \( \sqrt{a}x - \sqrt{b}y = b - a \) and \( \sqrt{b}x - \sqrt{a}y = 0 \) then what is the value of \( x - y \)?
Answer: \( \sqrt{b} - \sqrt{a} \)

Question. (ii) For the pair of equations \( \lambda x + 3y = -7 \), \( 2x - 6y = 14 \) to have infinitely many solutions, find the value of \( \lambda \).
Answer: -1

SECTION B

Question. Solve the following questions.
For what value of \( k \), will the following pair of equations have infinitely many solutions?
\( 2x + 3y = 7 \) and \( (k + 2)x - 3(1 - k)y = 5k + 1 \)
Answer: \( k = 4 \)

Question. Find the solution of the pair of equations:
\( \frac{3}{x} + \frac{8}{y} = -1 \), \( \frac{1}{x} - \frac{2}{y} = 2, x, y \neq 0 \)
Answer: \( x = 1, y = -2 \)

Question. Find the value of \( k \) for which the system of equations \( x + 3y - 4 = 0 \) and \( 2x + ky = 7 \) is inconsistent.
Answer: 6

Question. The larger of two supplementary angles exceeds thrice the smaller by 20 degrees. Find them.
Answer: \( 140^\circ, 40^\circ \)

Question. Solve the following questions.
Solve graphically the system of linear equations:
\( 4x - 3y + 4 = 0 \) and \( 4x + 3y - 20 = 0 \)
Find the area bounded by these lines and x-axis.
Answer: \( x = 2, y = 4 \); 12 sq. units

Question. Solve the following pair of linear equations:
\( 3x + 4y = 10 \)
\( 2x - 2y = 2 \)
Answer: \( x = 2, y = 1 \)

Question. Determine graphically the coordinates of the vertices of a triangle, the equations of whose sides are given by \( 2y - x = 8, 5y - x = 14 \) and \( y - 2x = 1 \).
Answer: \( A(-4, 2), B(1, 3), C(2, 5) \)

Question. A person, rowing at the rate of 5 km/h in still water, takes thrice as much time in going 40 km upstream as in going 40 km downstream. Find the speed of the stream.
Answer: 2.5 km/h

Question. Solve the following questions.
Determine graphically, the vertices of the triangle formed by the lines \( y = x, 3y = x, x + y = 8 \).
Answer: (0, 0), (4, 4), (6, 2)

Question. Susan invested certain amount of money in two schemes \( A \) and \( B \), which offer interest at the rate of 8% per annum and 9% per annum respectively. She received Rs. 1860 as annual interest. However, had she interchanged the amount of investment in the two schemes, she would have received Rs. 20 more as annual interest. How much money did she invest in each scheme?
Answer: Scheme \( A \implies \) Rs. 12,000, Scheme \( B \implies \) Rs. 10,000

Question. The sum of a two digit number and the number obtained by reversing the order of its digits is 165. If the digits differ by 3, find the number.
Answer: 69 or 96

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