CBSE Class 10 Pair of Linear Equations in Two Variables Sure Shot Questions Set 09

EXERCISE 

A. Very Short Answer Type Questions

In each of the following verify whether the given value of the x is a solution or not :

Question. \( \frac{x}{3} + \frac{x}{4} = 8, x = 12 \)
Answer: No

Question. \( (4x + 7) – 2 = 3x + 1, x = – 4 \)
Answer: Yes

Question. \( \frac{5x + 4}{4} - \frac{3x - 2}{2} = 5, x = \frac{1}{2} \)
Answer: No

Question. \( 2x – 4 + 1 = 3x – 6, x = 3 \)
Answer: Yes

Question. Solve : \( \frac{6}{x} + 11 = \frac{3}{x} + 12 \)
Answer: 3

Question. If \( 2x – 8 = 8 \), then find the value of \( x^2 + x – 70 \).
Answer: 2

Question. For each of the following, state the quadrant in which the point lies.
(i) (3, 3)
(ii) (–3, 2)
(iii) (2, –4)
(iv) (–1, –2)
(v) (–5, –5)
(vi) (5, 3).
Answer: (i) Ist (ii) IInd (iii) IVth (iv) IIIrd (v) IIIrd (vi) Ist

Question. Draw the graph of \( y = x \). Show that point (4, 4) is on the graph.
Answer: The point (4, 4) satisfies the equation \( y = x \), hence it lies on the graph.

Question. Express x in terms of y, given that \( 3x + 4y = 6 \). Check whether the point (3, 2) is on the given line.
Answer: (i) \( x = \frac{6 - 4y}{3} \), (ii) No

Question. Draw the graph of \( y = – 2x \). Show that the point (2, –5) is not on the graph.
Answer: Substituting \( x = 2 \) in \( y = -2x \) gives \( y = -4 \). Since \( -4 \neq -5 \), the point (2, -5) is not on the graph.

B. Short answer type Questions

Question. Indicate the quadrants in which the following points lie and plot them on a graph paper.
(i) (–2, 0)
(ii) (0, 1)
(iii) (–2, –3)
Answer: (i) lies on x-axis on negative side (ii) lies on y-axis on + ve side. (iii) IIIrd quadrant

Question. Draw the graph of (i) \( x = 3 \) (ii) \( y = –2 \).
Answer: (i) The graph of \( x = 3 \) is a straight line parallel to y-axis. (ii) The graph of \( y = –2 \) is a straight line below x-axis.

Question. Find the value of k, if line represented by the equation \( 2x – ky = 9 \) passes through the point (–1, –1).
Answer: k = 11

Question. Express x in terms of y, it is being given that \( 7x – 3y = 15 \). Check if the line represented by the given equation intersects the y-axis at \( y = – 5 \)
Answer: (i) \( x = \frac{15 + 3y}{7} \), (ii) Yes

Question. Draw the graph of \( 6 – 1.5x = 0 \).
Answer: The equation simplifies to \( x = 4 \), which is a line parallel to the y-axis.

Question. The following observed values of x and y are thought to fulfil the law \( y = ax + b \). Find the values of a and b.
x : 1, 2, –3, 0, 5
y : 12, 19, –16, 5, –30
Answer: a = 7, b = 5

Question. Show that the points A (1, 2), B (–1, –16), C(0, –7) are on the graph \( y = 9x – 7 \).
Answer: All points satisfy the equation \( y = 9x - 7 \). For A: \( 2 = 9(1) - 7 \); For B: \( -16 = 9(-1) - 7 \); For C: \( -7 = 9(0) - 7 \).

Question. Find the point of intersection of the line represented by the equation \( 7x + y = –2 \) with x-axis. Check whether the point (2, 1) is a solution set of the given equation.
Answer: (i) (–2/7, 0) (ii) No

Question. Express y in terms of x, given that \( 2x – 5y = 7 \). Check whether the point (–3, –2) is on the given line.
Answer: (i) \( y = \frac{2x - 7}{5} \) (ii) No

Question. Verify whether \( x = 2, y = 1 \) and \( x = 1 \) and \( y = 2 \) are the solutions of the linear equation \( 2x + y = 5 \). Find two more solutions.
Answer: \( x = 2, y = 1 \) is the solution but \( x = 1 \) and \( y = 2 \) is not the solution. Other solutions are \( x = 3, y = – 1 \) and \( x = 1, y = 3 \).

Question. Draw the graph of the equation \( 4x – 5y = 20 \) and check whether the points (3, 1) and (5, 0) lie on the graph.
Answer: Point (3, 1) does not lie on the lines and the point (5, 0) lies on the line.

Question. Draw the graph of the equation \( 3x + 4y = 14 \) and check whether \( x = 1 \) and \( y = 2 \) is a solution or not.
Answer: Not

Question. Draw the graph of the equation \( 2y + x = 7 \) and determine from the graph whether \( x = 3 \) and \( y = 2 \) is a solution
Answer: Yes

Question. Solve the following system of equations graphically. Also, find out the points, where these lines meet the x-axis.
\( x – 2y = 1 \)
\( 2x + y = 7 \)
Answer: \( x = 3, y = 1, (1, 0), (\frac{7}{2}, 0) \)

Question. Solve the following system of equations graphically. Also, find out the points, where these lines meet the y-axis.
(i) \( x + 2y – 7 = 0; 2x – y + 1 = 0 \)
(ii) \( 2x + y = 8; x + 1 = 2y \)
(iii) \( 2x + 3y = 12; 2y – 1 = x \)
Answer: (i) \( x = 1, y = 3, (0, \frac{7}{2}), (0, 1) \)
(ii) \( x = 3, y = 2, (0, 8), (0, \frac{1}{2}) \)
(iii) \( x = 3, y = 2, (0, 4), (0, \frac{1}{2}) \)

Question. Draw the graphs of the following systems of equations, state whether it is consistent (dependent), consistent (independent) or inconsistent :
(i) \( x + y = 7; 2x – 3y = 9 \)
(ii) \( 2x + 4y = 7; 3x + 6y = 10 \)
(iii) \( 2x + 3y – 12 = 0; 2x + 3y – 6 = 0 \)
(iv) \( 3x – 5y + 4 = 0; 9x = 15y – 12 \)
(v) \( x + 3y = 1; 2x + 6y = 2 \)
(vi) \( x + 4y = 7; 2x – y = 5 \)
Answer: (i) Consistent (independent) with unique solution
(ii) Inconsistent
(iii) Inconsistent
(iv) Consistent (dependent) with infinitely many solutions
(v) Consistent (dependent) with infinitely many solutions
(vi) Consistent (independent) with unique solution

Question. Solve the following pair of linear equations by the substitution method :
(i) \( 7x – 15y = 2; x + 2y = 3 \)
(ii) \( 2x + 3y = 9; 4x + 6y = 18 \)
(iii) \( x + 2y = 5; 2x + 3y = 8 \)
(iv) \( 0.2x + 0.3y = 1.3; 0.4x + 0.5y = 2.3 \)
(v) \( x + 2y = – 1; 2x – 3y = 12 \)
(vi) \( 3x – 5y + 1 = 0; x – y + 1 = 0 \)
Answer: (i) \( x = \frac{49}{29}, y = \frac{19}{29} \)
(ii) \( x = 3, y = 1; x = 0, y = 3 .... \)
(iii) \( x = 1, y = 2 \)
(iv) \( x = 2, y = 3 \)
(v) \( x = 3, y = – 2 \)
(vi) \( x = – 2, y = – 1 \)

Question. Solve the following equations by the method of elimination by equating the coefficients.
(i) \( 12x + 5y = 17; 7x – y = 6 \)
(ii) \( 17x + 12y = – 2; 15x + 8y = 6 \)
(iii) \( 23x + 17y = 6; 39x – 19y = 58 \)
(iv) \( 43x – 37y = 31; 13x + 23y = – 59 \)
(v) \( 0.4x + 3y = 1.2, 7x – 2y = \frac{17}{6} \)
(vi) \( (a + 2b) x + (2a – b) y = 2, (a – 2b) x + (2a + b) y = 3 \)
(vii) \( a(x + y) + b(x – y) = a^2 – ab + b^2, a(x + y) – b(x – y) = a^2 + ab + b^2 \)
Answer: (i) \( x = 1, y = 1 \)
(ii) \( x = 2, y = –3 \)
(iii) \( x = 1, y = – 1 \)
(iv) \( x = – 1, y = – 2 \)
(v) \( x = \frac{1}{2}, y = \frac{1}{3} \)
(vi) \( x = \frac{5b - 2a}{10ab}, y = \frac{a + 10b}{10ab} \)
(vii) \( x = \frac{b^2}{2a}, y = \frac{2a^2 + b^2}{2a} \)

Question. Solve the following system of equations by cross-multiplication method :
(i) \( 3x – 4y = 7; 5x + 2y = 3 \)
(ii) \( 3x – 5y = 1; 7x + 2y = 16 \)
(iii) \( 2x + 3y = 8; 3x + 2y = 7 \)
(iv) \( 3x – 4y = 1; 4x – 3y = 6 \)
(v) \( 3x – 4y = 10; 4x + 3y = 5 \)
(vi) \( 2x – 6y + 10 = 0; 3x – 9y + 15 = 0 \)
(vii) \( \frac{2}{x - 1} + \frac{3}{y + 1} = 2, \frac{3}{x - 1} + \frac{2}{y + 1} = \frac{13}{6} \)
(viii) \( \frac{5}{x + y} - \frac{2}{x - y} = -1, \frac{15}{x + y} + \frac{7}{x - y} = 10 \)
Answer: (i) \( x = 1, y = – 1 \)
(ii) \( x = 2, y = 1 \)
(iii) \( x = 1, y = 2 \)
(iv) \( x = 3, y = 2 \)
(v) \( x = 2, y = – 1 \)
(vi) Infinite solutions
(vii) \( x = 3, y = 2 \)
(viii) \( x = 3, y = 2 \)

Question. For what value of k will the following system of equations have a unique solution.
(i) \( 2x + ky = 1 \) and \( 3x – 5y = 7 \)
(ii) \( x – 2y = 3 \) and \( 3x + ky = 1 \)
(iii) \( 2x + 5y = 7 \) and \( 3x – ky = 5 \)
Answer: (i) \( k \neq \frac{-10}{3} \)
(ii) \( k \neq – 6 \)
(iii) \( k \neq \frac{-15}{2} \)

Question. For what value of k will the following system of equations have infinitely many solutions.
(i) \( 7x – y = 5 \) and \( 21x – 3y = k \)
(ii) \( 5x + 2y = k \) and \( 10x + 4y = 3 \)
(iii) \( kx + 4y = k – 4 \) and \( 16x + ky = k \)
Answer: (i) \( k = 15 \)
(ii) \( k = \frac{3}{2} \)
(iii) \( k = 8 \)

Question. Find the conditions so that the following systems of equations have infinitely many solutions.
(i) \( 3x – (a + 1) y = 2b – 1 \) and \( 5x + (1 – 2a) y = 3b \), find a and b.
(ii) \( 2x + 3y = 7 \) and \( (p + q) x + (2p – q) y = 3(p + q + 1) \), find p and q.
(iii) \( 2x – (2a + 5) y = 5 \) and \( (2b + 1) x – 9y = 15 \), find a and b.
Answer: (i) \( a = 8, b = 5 \)
(ii) \( p = 5, q = 1 \)
(iii) \( a = –1, b = \frac{5}{2} \)

Question. Show that the following systems of equation are inconsistent.
(i) \( x – 2y = 6; 3x – 6y = 0 \)
(ii) \( 2y – x = 9; 6y – 3x = 21 \)
(iii) \( 2x – y = 9; 4x – 2y = 15 \)
Answer: The ratios \( \frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2} \) for all cases.

Question. For what value of k the following systems of equations have no solution.
(i) \( 8x + 5y = 9 \) and \( kx + 10y = 8 \)
(ii) \( x – 4y = 6 \) and \( 3x + ky = 5 \)
(iii) \( kx – 5y = 2 \) and \( 6x + 2y = 7 \)
(iv) \( 4x + 6y = 11 \) and \( 2x + ky = 7 \)
(v) \( 2x + ky = 11 \) and \( 5x – 7y = 5 \)
Answer: (i) \( k = 16 \)
(ii) \( k = – 12 \)
(iii) \( k = – 15 \)
(iv) \( k = 3 \)
(v) \( k = \frac{-14}{5} \)

Question. Solve the following pair of linear equations
(i) \( \frac{1}{2x} - \frac{1}{y} = - 1 \), \( \frac{1}{x} + \frac{1}{2y} = 8 \)
(ii) \( \frac{2}{x} + \frac{2}{3y} = \frac{1}{6} \), \( \frac{3}{x} + \frac{2}{y} = 0 \); and hence, find a for which \( y = ax – 4 \).
(iii) \( \frac{1}{7x} + \frac{1}{6y} = 3 \), \( \frac{1}{2x} - \frac{1}{3y} = 5 \)
(iv) \( \frac{m}{x} - \frac{n}{y} = a \), \( px – qy = 0 \)
(v) \( \frac{2}{y} + \frac{3}{x} = \frac{7}{xy} \), \( \frac{1}{y} + \frac{9}{x} = \frac{11}{xy} \)
(vi) \( \frac{xy}{x + y} = \frac{6}{5} \), \( \frac{xy}{y - x} = 6 \)
(vii) \( x + y = 5xy, 3x + 2y = 13 xy \)
Answer: (i) \( x = \frac{1}{6}, y = \frac{1}{4} \)
(ii) \( x = 6, y = – 4, a = 0 \)
(iii) \( x = \frac{1}{14}, y = \frac{1}{6} \)
(iv) \( x = \frac{mp - nq}{ap}, y = \frac{mp - nq}{aq} \)
(v) \( x = 2, y = 1 \)
(vi) \( x = 2, y = 3 \)
(vii) \( x = \frac{1}{2}, y = \frac{1}{3} \)

Question. Solve the following pair of linear equations.
(i) \( 3(a + 3b) = 11 ab; 3(2a + b) = 7ab \)
(ii) \( 5x + \frac{4}{y} = 9; 7x – \frac{2}{y} = 5 \)
(iii) \( \frac{3}{x} + 4y = 7; \frac{-2}{x} + 7y = \frac{19}{3} \)
(iv) \( \frac{5}{x + 1} - \frac{2}{y - 1} = \frac{1}{2}; \frac{10}{x + 1} + \frac{2}{y - 1} = \frac{5}{2} \)
(v) \( \frac{6}{x + y} = \frac{7}{x - y} + 3; \frac{1}{2(x + y)} = \frac{1}{3(x - y)} \)
(vi) \( ax + by = c; bx + ay = 1 + c \)
(vii) \( ax + by = 1; bx + ay = \frac{(a + b)^2}{a^2 + b^2} - 1 \)
(viii) \( \frac{148}{x} + \frac{231}{y} = \frac{527}{xy}; \frac{231}{x} + \frac{148}{y} = \frac{610}{xy} \)
Answer: (i) \( a = 1, b = \frac{3}{2} \)
(ii) \( x = 1, y = 1 \)
(iii) \( x = \frac{87}{71}, y = \frac{33}{29} \)
(iv) \( x = 4, y = 5 \)
(v) \( x = \frac{-5}{4}, y = -\frac{1}{4} \)
(vi) \( x = \frac{c}{a + b} - \frac{b}{a^2 - b^2}, y = \frac{c}{a + b} + \frac{a}{a^2 - b^2} \)
(vii) \( x = \frac{a}{a^2 + b^2}, y = \frac{b}{a^2 + b^2} \)
(viii) \( x = 1, y = 2 \)

Question. 2 tables and 3 chairs together cost ₹ 2000 whereas 3 tables and 2 chairs together cost ₹ 2500. Find the total cost of 1 table and 5 chairs.
Answer: ₹ 1700

Question. 3 bags and 4 pens together cost ₹ 257 whereas 4 bags and 3 pens together cost ₹ 324. Find the total cost of 1 bag and 10 pens.
Answer: ₹ 155

Question. Two numbers differ by 4 and their product is 192. Find the numbers.
Answer: 12 and 16

Question. Five years hence, father’s age will be three times the age of his son. Five years ago, father was seven times as old as his son Find their present ages.
Answer: Son’s age = 10 years, father’s age = 40 years

Question. The age of father is 4 times the age of his son. 5 years hence, the age of father will be three times the age of his son. Find their present ages.
Answer: Son’s age = 10 years, father’s age = 40 years

Question. 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.
Answer: 64

Question. The sum of a two-digit number and the number formed by interchanging the digits is 132. If 12 is added to the number, the new number becomes 5 times the sum of the digits. Find the number.
Answer: 48

Question. If 2 be added to the numerator of a fraction, it reduces to 1/2 and if 1 be subtracted from the denominator, it reduces to 1/3. Find the fraction.
Answer: \( \frac{3}{10} \)

Question. The sum of the numerator and denominator of a fraction is 18. If the denominator is increased by 2, the fraction reduces to 1/3. Find the fraction.
Answer: \( \frac{5}{13} \)

Question. The length of a rectangle exceeds its width by 8 cm and the area of the rectangle is 240 sq. cm. Find the dimensions of the rectangle.
Answer: Length = 20 cm, Width = 12cm

Question. The side of a square exceeds the side of another square by 4 cm and the sum of the area of the two squares is 400 sq. cm. Find the dimensions of the squares.
Answer: 12 cm and 16 cm

Question. The area of a rectangle gets reduced by 8 sq. metres, if its length is reduced by 5 metres and width is increased by 3 metres. If we increase the length by 3 metres and breadth by 2 metres, the area is increased by 74 sq. metres. Find the length and breadth of the rectangle.
Answer: Length = 19 m, Breadth = 10 m

Question. In a triangle, the sum of two angles is equal to the third. If the difference between them is 50º, find the angles.
Answer: 70º, 20º

Question. Find the four angles of the following cyclic quadrilateral ABCD in which
(i) \( \angle A = 5xº, \angle B = 9xº + 2yº, \angle C = xº + 8yº \) and \( \angle D = xº + 4yº \).
(ii) \( \angle A = (2x + y)º, \angle B = 2(x + y)º, \angle C = (3x + 2y)º, \angle D = (4x – 2y)º \).
Answer: (i) \( \angle A = 50º, \angle B = 120º, \angle C = 130º, \angle D = 70º \)
(ii) \( \angle A = 70º, \angle B = 80º, \angle C = 110º, \angle D = 100º \)

C. Long answer type Questions

Question. The ages of Ram and Mohan are in ratio 2 : 3. If sum of their ages is 65, find the difference of their ages.
Answer: 13

Question. The difference between two numbers is 1365. When larger is divided by the smaller one, the quotient is 6 and remainder is 15. Find the numbers.
Answer: 1635, 270

Question. The denominator of a fraction is 1 more than its numerator. If 1 is subtracted from both the numerator and denominator, the fraction becomes 1/2. Find the fraction.
Answer: 2/3

Question. The measures of angles of a triangle in degrees are x, x + 12 and x + 27. Find the measure of angles.
Answer: 47º, 59º, 74º

Question. Solve for x : \( \frac{4x + 17}{18} - \frac{13x - 2}{17x - 32} + \frac{x}{3} = \frac{7x}{12} - \frac{x + 16}{36} \)
Answer: 4

Question. Solve the following system of equations by cross-multiplication method :
(i) \( ax + by = a^2; bx + ay = b^2 \)
(ii) \( \frac{2x}{a} + \frac{y}{b} = 2; \frac{x}{a} - \frac{y}{b} = 4; a \neq 0, b \neq 0 \)
(iii) \( x – y = a + b; ax + by = a^2 – b^2 \)
(iv) \( \frac{x}{a} + \frac{y}{b} = 2; ax – by = a^2 – b^2; a \neq 0, b \neq 0 \)
(v) \( x + y = a + b; ax – by = a^2 – b^2 \)
Answer: (i) \( x = \frac{a^2 + ab + b^2}{a + b}, y = \frac{-ab}{a + b} \)
(ii) \( x = 2a, y = – 2b \)
(iii) \( x = a, y = – b \)
(iv) \( x = a, y = b \)
(v) \( x = a, y = b \)

Question. Two numbers differ by 4 and their product is 96. Find the numbers.
Answer: 8 and 12

Question. Two numbers are in the ratio of 3 : 5, If 5 is subtracted from each of the number, they become in ratio of 1 : 2. Find the numbers.
Answer: 15 and 25

Question. Two numbers are in the ratio of 3 : 4. If 8 is added to each number, they become in the ratio of 4 : 5. Find the numbers.
Answer: 24 and 32