10th Maths 3.5

Exercise 3.5

Q1. Which of the following pairs of linear equations has unique solution, no solution, or infinitely many solutions. In case there is a unique solution, find it by using cross multiplication method.

(i) $x-3y-3=0; 3x-9y-2=0$

(ii) $2x+y=5; 3x+2y=8$

(iii) $3x-5y=20; 6x-10y=40$

(iv) $x-3y-7=0; 3x-3y-15=0$

Sol. :

$x-3y-3=0$ _______(1)

$3x-9y-2=0$ ______(2)

$a_1/a_2=1/3, b_1/b_2=(-3)/-9=1/3,$ and

$c_1/c_2=(-3)/-2=3/2$

$a_1/a_2=b_1/b_2nec_1/c_2$

Answer :

No solution.

ii)

$2x+y=5$ ________(1)

$3x+2y=8$ _______(2)

$a_1/a_2=2/3, b_1/b_2=1/2,$ and

$c_1/c_2=(-5)/-8=5/8$

$a_1/a_2neb_1/b_2nec_1/c_2$

By Cross Multiplication Method :

$x/(b_1c_2-b_2c_1)=y/(c_1a_2-c_2a_1)=1/(a_1b_2-a_2b_1)$

$x/((1)(-8)-(2)(-5))=y/((-5)(3)-(-8)(2))=1/((2)(2)-(3)(1))$

$x/(-8+10)=y/(-15+16)=1/(4-3)$

$x/2=y/1=1/1$

$x=2times1=2,$ and

$y=1times1=1$

Answer :

Unique solution ;

$x=2, y=1$ .

iii)

$3x-5y=20$

$3x-5y-20=0$ __________(1)

$6x-10y=40$

$6x-10y-40=0$ _________(2)

$a_1/a_2=3/6=1/2, b_1/b_2=(-5)/-10=1/2,$ and

$c_1/c_2=(-20)/-40=1/2$

$a_1/a_2=b_1/b_2=c_1/c_2$

Answer :

Infinitely many solutions.

iv)

$x-3y-7=0$ __________(1)

$3x-3y-15=0$ ________(2)

$a_1/a_2=1/3, b_1/b_2=(-3)/-3=1/1,$ and

$c_1/c_2=(-7)/(-15)=7/15$

$a_1/a_2neb_1/b_2nec_1/c_2$

By Cross Multiplication Method :

$x/(b_1c_2-b_2c_1)=y/(c_1a_2-c_2a_1)=1/(a_1b_2-a_2b_1)$

$x/((-3)(-15)-(-3)(-7))=y/((-7)(3)-(-15)(1))=1/((1)(-3)-(3)(-3))$

$x/(45-21)=y/(-21+15)=1/(-3+9)$

$x/24=y/-6=1/6$

$x/4=y/-1=1/1$

$x=1times4=4,$ and

$y=-1times1=-1$

Answer :

Unique solution ;

$x=4, y=-1$ .

Q2.

(i) For which values of $a$ and $b$ does the following pair of linear equations have an infinite number of solutions?

$2x+3y=7$

$(a-b)x+(a+b)y=3a+b-2$

(ii) For which value of $k$ will the following pair of linear equations have no solution?

$3x+y=7$

$(2k-1)x+(k-1)y=2k+1$

Sol. :

$2x+3y=7$ ______________(1)

$(a-b)x+(a+b)y=3a+b-2$ ___(2)

The given pair of linear equations have infinite number of solutions

$a_1/a_2=2/(a-b), b_1/b_2=3/(a+b),$ and

$c_1/c_2=7/(3a+b-2)$

$7(a-b)=2(3a+b-2)$

$7a-7b=6a+2b-4$

$7a-6a-7b-2b+4=0$

$a-9b+4=0$ ____________(3)

and

$7(a+b)=3(3a+b-2)$

$7a+7b=9a+3b-6$

$7a-9a+7b-3b+6=0$

$-2a+4b+6=0$

$2a-4b-6=0$ ___________(4)

By Cross Multiplication Method :

from equation (3) and (4)

$x/(b_1c_2-b_2c_1)=y/(c_1a_2-c_2a_1)=1/(a_1b_2-a_2b_1)$

$a/((-9)(-6)-(-4)(4))=b/((4)(2)-(-6)(1))=1/((1)(-4)-(2)(-9))$

$a/(54+16)=b/(8+6)=1/(-4+18)$

$a/70=b/14=1/14$

$a/5=b/1=1/1$

$a=5times1=5, b=1times1,$

Answer :

$a=5, b=1$ .

ii)

$3x+y=1$ _______________(1)

$(2k-1)x+(k-1)y=2k+1$ ____(2)

The given pair of linear equations have no solution

$3/(2k-1)=1/(k-1)ne1/(2k+1)$

$3k-3=2k-1$

$k=3-1=2$

Answer :

$k=2$ .

Q3. Solve the following pair of linear equations by the substitution and cross-multiplication methods :

$8x+5y=9$

$3x+2y=4$

Sol.

By Substitution Method :

$8x+5y=9$ __________(1)

$3x+2y=4$ __________(2)

from equation (1)

$y=(9-8x)/5$ _________(3)

Substituting this value in equation (2)

$3x+2((9-8x)/5)=4$

$15x+18-16x=20$

$-x=20-18$

$x=-2$

Put in equation (3)

$y=(9-8(-2))/5$

$y=(9+16)/5$

$y=25/5$

$y=5$

By Cross Multiplication Method :

$8x+5y-9=0$ ____________(1)

$3x+2y-4=0$ ____________(2)

$x/(b_1c_2-b_2c_1)=y/(c_1a_2-c_2a_1)=1/(a_1b_2-a_2b_1)$

$x/((2)(-9)-(5)(-4))=y/((-4)(8)-(-9)(3))=1/((3)(5)-(8)(2))$

$x/(-18+20)=y/(-32+27)=1/(15-16)$

$x/2=y/-5=1/-1$

$x=2(-1)=-2$ and

$y=(-5)(-1)=5$

Answer :

$x=-2, y=5$ .

Q4. Form the pair of linear equations in the following problems and find their solutions ( if they exist ) by any algebraic method :

(i) A part of monthly hostel charges is fixed and the remaining depends on the number of days one has taken food in the mess. When a student $A$ takes food for $20$ days she has to pay $₹ 1000$ as hostel charges whereas a student $B$ , who takes food for $26$ days, pays $₹ 1180$ as hostel charges. Find the fixed charges and the cost of food per day.

(ii) A fraction becomes $1/3$ when $1$ is subtracted from the numerator and it becomes $1/4$ when $8$ is added to its denominator. Find the fraction.

(iii) Yash scored $40$ marks in a test, getting $3$ marks for each right answer and losing $1$ mark for each wrong answer. Had $4$ marks been awarded for each correct answer and $2$ marks been deducted for each incorrect answer, then Yash would have scored $50$ marks. How many questions were there in the test?

(iv) Places $A$ and $B$ are $100 km$ apart on a highway. One car starts from $A$ and another from $B$ at the same time. If the cars travel in the same direction at different speeds, they meet in $5$ hours. If they travel towards each other, they meet in $1$ hour. What are the speeds of the two cars?

(v) The area of a rectangle gets reduced by $9$ square units, if its length is reduced by $5$ units and breadth is increased by $3$ units. If we increase the length by $3$ units and the breadth by $2$ units, the area increases the length by $67$ square units. Find the dimensions of the rectangle.

Sol. :

Let the fixed charges

$₹ x$ ,

and the cost of food per day

$₹ y$ .

According to question

$x+20y=1000$ ___________(1)

$x+26y=1180$ ___________(2)

By equation (2) - (1)

$x+26y-(x+20y)=1180-1000$

$x+26y-x-20y=1180-1000$

$6y=180$

$y=180/6$

$y=30$

Put in equation (1)

$x+20(30)=1000$

$x+600=1000$

$x=1000-600$

$x=400$

Answer :

$x=400, y=30$ .

ii)

Let numerator is

$x$ ,

and denominator is

$y$ .

then fraction will be

$x/y$

According to question

$(x-1)/y=1/3$

$3x-3=y$

$3x-y=3$ ______________(1)

$x/(y+8)=1/4$

$4x=y+8$

$4x-y=8$ ______________(2)

By equation (2) - (1)

$4x-y-(3x-y)=8-3$

$4x-y-3x+y=5$

$x=5$

Put in equation (1)

$3(5)-y=3$

$15-y=3$

$y=15-3$

$y=12$

Answer :

$x=5, y=12$ and fraction

$=5/12$ .

iii)

Let correct answers is

$x$ ,

and incorrect answer is

$y$ .

According to question

$3x-y=40$ ___________(1)

$4x-2y=50$ __________(2)

Multiply equation (1) by

$2$ and equation (2) by

$1$

$6x-2y=80$ __________(3)

$4x-2y=50$ __________(4)

By equation (3) - (4)

$6x-2y-(4x-2y) = 80 - 50$

$6x-2y-4x+2y=30$

$2x=30$

$x=30/2$

$x=15$

Put in equation (1)

$3(15)-y=40$

$45-y=40$

$y=45-40$

$y=5$

Total number of questions in the test

$x+y=15+5=20$

Answer :

Total number of questions in the test is

$20$ .

iv)

Let the speed of first car is

$x$ km/h,

and the speed of second car is

$y$ km/h.

Speed, when they travel in the same direction

$=(x-y)$ km/h,

Speed, when they travel in the opposite direction

$=(x+y)$ km/h`.

According to question

$5(x-y)=100$

$x-y=20$ ____________(1)

$1(x+y)=100$

$x+y=100$ ___________(2)

By equation (1) + (2)

$x-y+x+y=20+100$

$2x=120$

$x=120/2$

$x=60$

Put in equation (1)

$60-y=20$

$y=60-20$

$y=40$

Answer :

$x=60$ km/h,

$y=40$ km/h.

Let the length is

$x$ units,

and the breadth is

$y$ units.

Area of the rectangle

$=xtimesy$ square units

According to question

$(x-5)times(y+3)=xy-9$

$xy+3x-5y-15=xy-9$

$3x-5y=15-9$

$3x-5y=6$ __________(1)

$(x+3)(y+2)=xy+67$

$xy+2x+3y+6=xy+67$

$2x+3y+6=67$

$2x+3y=67-6$

$2x+3y=61$ _________(2)

from equation (1)

$x=(6+5y)/3$ _________(3)

Substituting this value in equation (2)

$2((6+5y)/3)+3y=61$

$12+10y+9y=183$

$19y=183-12$

$y=171/19$

$y=9$

put in equation (3)

$x=(6+5(9))/3$

$x=(6+45)/3$

$x=51/3$

$x=17$

Answer :

$x=17$ units,

$y=9$ units.

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10th Maths 3.5

NCERT Class 10th solution of Exercise 3.1

NCERT Class 10th solution of Exercise 3.2

NCERT Class 10th solution of Exercise 3.3

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Exercise 3.5

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