Chapter 3 · Pair of Linear Equations in Two Variables

NCERT Exemplar Class 10 Maths Exercise 3.3

All 22 questions of NCERT Exemplar Class 10 Maths Exercise 3.3 (Pair of Linear Equations in Two Variables), in the book's order, each with its answer and a step-by-step solution.

1. Q1
   For which value(s) ​of $\lambda$, ​do the pair of ​linear equations $\lambda x+y={\lambda }^2$ ​and $x+\lambda y=1$ have

   1. (i)no solution?
   2. (ii)​infinitely ​many solutions?
   3. (iii)a ​unique solution?
   Show answer

   (i) No solution: $\lambda =-1$. (ii) Infinitely many solutions: $\lambda =1$. (iii) Unique solution: $\lambda \ne \pm 1$.

   Step-by-step solution →
2. Q2
   For ​which ​value(s) ​of $k$ ​will the pair of equations $kx+3y=k-3$ and $12x+ky=k$ ​have ​no ​solution?
   Show answer

   $k=-6$ is the only value giving no solution; $k=6$ gives infinitely many solutions and is rejected.

   Step-by-step solution →
3. Q3
   For which ​values of $a$ and $b$, will the ​following pair ​of linear equations ​have ​infinitely ​many solutions? $x+2y=1$ ​and $(a-b)x+(a+b)y=a+b-2$.
   Show answer

   $a=3$ and $b=1$.

   Step-by-step solution →
4. Q4
   Find the ​value(s) of $p$ in (i) to (iv) and $p$ ​and $q$ in (v) for the following pair of equations:

   1. (i)$3x-y-5=0$ ​and $6x-2y-p=0$, if the lines represented ​by these equations ​are ​parallel.
   2. (ii)$-x+py=1$ and $px-y=1$, if the pair of equations has ​no solution.
   3. (iii)$-3x+5y=7$ and $2px-3y=1$, if the lines represented by these equations are intersecting at a unique point.
   4. (iv)$2x+3y-5=0$ and $px-6y-8=0$, if the pair of ​equations has a unique solution.
   5. (v)$2x+3y=7$ and $2px+py=28-qy$, if the pair of equations have infinitely many solutions.
   Show answer

   (i) $p\ne 10$; (ii) $p=1$; (iii) $p\ne 9/10$; (iv) $p\ne -4$; (v) $p=4$, $q=8$

   Step-by-step solution →
5. Q5
   Two ​straight paths ​are ​represented ​by ​the equations $x-3y=2$ and $-2x+6y=5$. Check whether the ​paths cross each ​other ​or not.
   Show answer

   The paths do not cross - the lines are parallel since $\frac{a_1}{a_2}=\frac{b_1}{b_2}=-\frac{1}{2}\ne \frac{2}{5}=\frac{c_1}{c_2}$.

   Step-by-step solution →
6. Q6
   Write ​a ​pair of linear ​equations which has the unique ​solution $x=-1$, $y=3$. ​How ​many such pairs can you write?
   Show answer

   One valid pair: $x+y=2$ and $2x-y=-5$, with unique solution $(-1,3)$. Infinitely many such pairs can be written.

   Step-by-step solution →
7. Q7
   If $2x+y=23$ ​and $4x-y=19$, ​find the ​values ​of $5y-2x$ and $\frac{y}{x}-2$.
   Show answer

   $x=7,y=9$; hence $5y-2x=31$ and $\frac{y}{x}-2=-\frac{5}{7}$.

   Step-by-step solution →
8. Q8
   Find ​the ​values ​of $x$ ​and $y$ in ​the following ​rectangle ​[see Fig. 3.2].

   Show answer

   $x=1$ and $y=4$.

   Step-by-step solution →
9. Q9
   Solve ​the ​following pairs ​of ​equations: ​

   1. (i)$x+y=3.3$, $\frac{0.6}{3x-2y}=-1$, $3x-2y\ne 0$.
   2. (ii)$\frac{x}{3}+\frac{y}{4}=4$, $\frac{5x}{6}-\frac{y}{8}=4$.
   3. (iii)$4x+\frac{6}{y}=15$, $6x-\frac{8}{y}=14$, $y\ne 0$. ​
   4. (iv)$\frac{1}{2x}-\frac{1}{y}=-1$, $\frac{1}{x}+\frac{1}{2y}=8$, $x,y\ne 0$.
   5. (v)$43x+67y=-24$, $67x+43y=24$.
   6. (vi)$\frac{x}{a}+\frac{y}{b}=a+b$, $\frac{x}{a^2}+\frac{y}{b^2}=2$, $a,b\ne 0$.
   7. (vii)$\frac{2xy}{x+y}=\frac{3}{2}$, $\frac{xy}{2x-y}=\frac{-3}{10}$, $x+y\ne 0,2x-y\ne 0$.
   Show answer

   (i) $x=1.2,y=2.1$; (ii) $x=6,y=8$; (iii) $x=3,y=2$; (iv) $x=\frac{1}{6},y=\frac{1}{4}$; (v) $x=1,y=-1$; (vi) $x=a^2,y=b^2$; (vii) $x=\frac{1}{2},y=-\frac{3}{2}$.

   Step-by-step solution →
10. Q10
    Find the ​solution ​of the ​pair of ​equations $\frac{x}{10}+\frac{y}{5}-1=0$ and $\frac{x}{8}+\frac{y}{6}=15$. Hence, ​find $\lambda$, ​if $y=\lambda x+5$.
    Show answer

    $x=340,y=-165$; hence $\lambda =-\frac{1}{2}$.

    Step-by-step solution →
11. Q11
    By the graphical ​method, find whether ​the ​following pair of equations ​are consistent or not. If consistent, ​solve them.

    1. (i)$3x+y+4=0$ ​and $6x-2y+4=0$ ​
    2. (ii)$x-2y=6$ ​and $3x-6y=0$
    3. (iii)$x+y=3$ and $3x+3y=9$.

    1. (i)$3x+y+4=0$; $6x-2y+4=0$
    2. (ii)x - 2y = 6; 3x - 6y = 0
    3. (iii)x + y = 3; 3x + 3y = 9
    Show answer

    (i) Consistent, intersecting, solution $x=-1,y=-1$. (ii) Inconsistent (parallel lines). (iii) Consistent, coincident - infinitely many solutions (every point on $x+y=3$).

    Step-by-step solution →
12. Q12
    Draw the graph of the pair of equations $2x+y=4$ and $2x-y=4$. Write ​the vertices of the triangle formed by ​these ​lines ​and the $y$-axis. Also find the area of this ​triangle.
    Show answer

    Vertices: $(0,4)$, $(0,-4)$, $(2,0)$; Area $=8$ square units.

    Step-by-step solution →
13. Q13
    Write ​an equation ​of ​a line passing through ​the point representing solution ​of the pair ​of linear ​equations $x+y=2$ and $2x-y=1$. How many such lines can we find?
    Show answer

    The solution point is $(1,1)$; one line through it is $y=x$. Infinitely many such lines can be found.

    Step-by-step solution →
14. Q14
    If $x+1$ is ​a ​factor of $2x^3+a{x}^2+2bx+1$, ​then find ​the values of $a$ and $b$ ​given ​that $2a-3b=4$.
    Show answer

    $a=5$ and $b=2$.

    Step-by-step solution →
15. Q15
    The ​angles ​of a ​triangle are $x$, $y$ and $40^{\circ }$. The difference between ​the two ​angles $x$ and $y$ is $30^{\circ }$. ​Find $x$ and $y$.
    Show answer

    $x=85^{\circ }$ and $y=55^{\circ }$.

    Step-by-step solution →
16. Q16
    Two years ​ago, Salim was thrice as ​old ​as ​his daughter ​and six years later, he will be four years older ​than twice her age. ​How old are they ​now?
    Show answer

    Salim is 38 years old and his daughter is 14 years old now.

    Step-by-step solution →
17. Q17
    The ​age of ​the ​father ​is twice the sum of ​the ages of his two children. After ​20 years, his age will be equal to the sum of the ​ages of his children. ​Find the age of the father.
    Show answer

    The father's present age is $40$ years.

    Step-by-step solution →
18. Q18
    Two ​numbers ​are ​in ​the ratio $5:6$. ​If $8$ is subtracted from each of ​the ​numbers, ​the ratio becomes $4:5$. Find the numbers.
    Show answer

    The numbers are $40$ and $48$.

    Step-by-step solution →
19. Q19
    There ​are some students ​in the two examination halls A and B. To make the number of students equal in each hall, 10 students are sent from A to B. But if 20 students ​are ​sent from B to A, the number ​of students in A becomes double ​the number of students in B. Find the number of ​students in the two halls.
    Show answer

    Hall A has $100$ students and hall B has $80$ students.

    Step-by-step solution →
20. Q20
    A shopkeeper gives books ​on rent for reading. She ​takes a fixed charge for the ​first two days, ​and ​an additional charge ​for each day thereafter. Latika paid ​Rs 22 for a book ​kept for six days, while Anand paid Rs 16 for the book kept for four days. Find the fixed charges and the charge for each extra day.
    Show answer

    Fixed charge $=$ Rs $10$ (for the first two days); charge per extra day $=$ Rs $3$.

    Step-by-step solution →
21. Q21
    In a competitive ​examination, one mark is awarded for each correct answer while $\frac{1}{2}$ mark is deducted for ​every wrong ​answer. ​Jayanti answered 120 questions and got ​90 marks. How many questions ​did she answer ​correctly?
    Show answer

    Jayanti answered $100$ questions correctly.

    Step-by-step solution →
22. Q22
    The angles of a cyclic ​quadrilateral ABCD ​are $\angle A=(6x+10{)}^{\circ }$, $\angle B=(5x{)}^{\circ }$, $\angle C=(x+y{)}^{\circ }$, $\angle D=(3y-10{)}^{\circ }$. ​Find $x$ and $y$, ​and hence the values of the ​four ​angles.
    Show answer

    $x=20$, $y=30$; $\angle A=130^{\circ }$, $\angle B=100^{\circ }$, $\angle C=50^{\circ }$, $\angle D=80^{\circ }$.

    Step-by-step solution →