For which value(s) of \(\lambda\) do the pair of linear equations \(\lambda x + y = \lambda^2\) and \(x + \lambda y = 1\) have (i) no solution, (ii) infinitely many solutions, (iii) a unique solution?
(i) \(\lambda = -1\); (ii) \(\lambda = 1\); (iii) \(\lambda \neq \pm 1\).
For which value(s) of \(k\) will the pair of equations \(kx + 3y = k - 3\) and \(12x + ky = k\) have no solution?
\(k = -6\).
For which values of \(a\) and \(b\) will the pair of equations \(x + 2y = 1\) and \((a-b)x + (a+b)y = a + b - 2\) have infinitely many solutions?
\(a = 3\) and \(b = 1\).
Find the value(s) of p in (i) to (iv) and p and q in (v) for the following pair of equations:
(i) \(3x – y – 5 = 0\) and \(6x – 2y – p = 0\),
if the lines represented by these equations are parallel.
(ii) \(–x + py = 1\) and \(px – y = 1\),
if the pair of equations has no solution.
(iii) \(– 3x + 5y = 7\) and \(2px – 3y = 1\),
if the lines represented by these equations are intersecting at a unique point.
(iv) \(2x + 3y – 5 = 0\) and \(px – 6y – 8 = 0\), if the pair of equations has a unique solution.
(v) \(2x + 3y = 7\) and \(2px + py = 28 – qy\), if the pair of equations have infinitely many solutions.
(i) Any \(p\neq 10\).
(ii) \(p = 1\).
(iii) All \(p \ne \dfrac{9}{10}\).
(iv) All \(p \ne -4\).
(v) \(p = 4\) and \(q = 8\).
The paths \(x - 3y = 2\) and \(-2x + 6y = 5\) represent straight lines. Do the paths cross each other?
No. They are parallel and distinct.
Write a pair of linear equations whose unique solution is \(x = -1\), \(y = 3\). How many such pairs can you write?
Infinitely many. One example: \(x + y = 2\) and \(2x - y = -5\).
If \(2x + y = 23\) and \(4x - y = 19\), find the values of \(5y - 2x\) and \(\dfrac{y}{x} - 2\).
\(5y - 2x = 31\) and \(\dfrac{y}{x} - 2 = -\dfrac{5}{7}\).
In the rectangle, opposite sides are equal. Given the labels in Fig. 3.2, find \(x\) and \(y\): top \(= x + 3y\), bottom \(= 13\), left \(= 3x + y\), right \(= 7\).
\(x = 1\) and \(y = 4\).
(i). \(x + y = 3.3\) and \(\dfrac{0.6}{3x - 2y} = -1\), \(3x - 2y \ne 0\).
(ii). Solve: \(\dfrac{x}{3} + \dfrac{y}{4} = 4\) and \(\dfrac{5x}{6} - \dfrac{y}{8} = 4\).
(iii). Solve: \(4x + \dfrac{6}{y} = 15\) and \(6x - \dfrac{8}{y} = 14\), \(y \ne 0\).
(iv). Solve: \(\dfrac{1}{2x} - \dfrac{1}{y} = -1\) and \(\dfrac{1}{x} + \dfrac{1}{2y} = 8\), \(x,y \ne 0\).
(v). Solve: \(43x + 67y = -24\) and \(67x + 43y = 24\).
(vi). Solve: \(\dfrac{x}{a} + \dfrac{y}{b} = a + b\) and \(\dfrac{x}{a^2} + \dfrac{y}{b^2} = 2\), \(a,b \ne 0\).
(vii). Solve: \(\dfrac{2xy}{x + y} = \dfrac{3}{2}\) and \(\dfrac{xy}{2x - y} = -\dfrac{3}{10}\), with \(x + y \ne 0\) and \(2x - y \ne 0\).
(i). \(x = 1.2\), \(y = 2.1\).
(ii). \(x = 6\), \(y = 8\).
(iii). \(x = 3\), \(y = 2\).
(iv). \(x = \dfrac{1}{6}\), \(y = \dfrac{1}{4}\).
(v). \(x = 1\), \(y = -1\).
(vi). \(x = a^2\), \(y = b^2\).
(vii). \(x = \dfrac{1}{2}\), \(y = -\dfrac{3}{2}\).
Solve the pair \(\dfrac{x}{10} + \dfrac{y}{5} - 1 = 0\) and \(\dfrac{x}{8} + \dfrac{y}{6} = 15\). Hence, if \(y = \lambda x + 5\), find \(\lambda\).
Solution: \(x = 340\), \(y = -165\); hence \(\lambda = -\dfrac{1}{2}\).
By the graphical method, decide consistency and solve:
(i). \(3x + y + 4 = 0\) and \(6x - 2y + 4 = 0\)
(ii). \(x - 2y = 6\) and \(3x - 6y = 0\)
(iii). \(x + y = 3\) and \(3x + 3y = 9\)
(i). Consistent with a unique solution: \(x = -1\), \(y = -1\).
(ii). Inconsistent (parallel). No solution.
(iii). Consistent and dependent: infinitely many solutions (the line \(x + y = 3\)).
Draw the graphs of \(2x + y = 4\) and \(2x - y = 4\). Find the vertices of the triangle formed by these two lines and the y-axis, and its area.
Vertices: \((0,4)\), \((0,-4)\), \((2,0)\). Area \(= 8\) square units.
Find an equation of a line passing through the point that is the solution of \(x + y = 2\) and \(2x - y = 1\). How many such lines are there?
Solution point is \((1,1)\). Infinitely many lines pass through it; e.g., \(y - 1 = m(x - 1)\).
If \(x + 1\) is a factor of \(2x^3 + ax^2 + 2bx + 1\) and \(2a - 3b = 4\), find \(a\) and \(b\).
\(a = 5\), \(b = 2\).
The angles of a triangle are \(x\), \(y\), and \(40^\circ\). Their difference \(|x - y|\) is \(30^\circ\). Find \(x\) and \(y\).
\(x = 85^\circ\), \(y = 55^\circ\) (or vice versa).
Two years ago, Salim was thrice his daughter's age. Six years later, he will be four years older than twice her age. Find their present ages.
Salim: \(38\) years; Daughter: \(14\) years.
A father's present age is twice the sum of the ages of his two children. After 20 years, his age will equal the sum of their ages then. Find the father's present age.
\(40\) years.
Two numbers are in the ratio \(5:6\). If 8 is subtracted from each, the ratio becomes \(4:5\). Find the numbers.
\(40\) and \(48\).
Students in halls A and B: if 10 go from A to B, they become equal. If 20 go from B to A, A becomes double B. Find the original numbers.
A: \(100\) students; B: \(80\) students.
A shopkeeper charges a fixed amount for the first two days and a daily charge thereafter. Latika paid Rs 22 for 6 days; Anand paid Rs 16 for 4 days. Find the fixed charge and the daily charge after two days.
Fixed charge = Rs 10; additional per day = Rs 3.
In a test, +1 mark for a correct answer and \(\dfrac{1}{2}\) mark deducted for a wrong answer. Jayanti answered 120 questions and scored 90 marks. How many did she answer correctly?
\(100\) correct answers.
In cyclic quadrilateral \(ABCD\), \(\angle A = (6x + 10)^\circ\), \(\angle B = (5x)^\circ\), \(\angle C = (x + y)^\circ\), \(\angle D = (3y - 10)^\circ\). Find \(x\) and \(y\), then all four angles.
\(x = 20\), \(y = 30\); angles: \(A=130^\circ\), \(B=100^\circ\), \(C=50^\circ\), \(D=80^\circ\).