10th Maths 3.5
NCERT Class 10th solution of Exercise 3.1
NCERT Class 10th solution of Exercise 3.6
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. :
i)
`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. :
i)
`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. :
i)
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.
v)
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.
Comments