10th Maths 4.2
NCERT Class 10th solution of Exercise 4.1
NCERT Class 10th solution of Exercise 4.3
NCERT Class 10th solution of Exercise 4.4
Exercise 4.2
Q1. Find the roots of the following quadratic equation by factorisation :
i) `x^2-3x-10=0,`
ii) `2x^2+x-6=0,`
iii) `sqrt2x^2+7x+5sqrt2=0,`
iv) `2x^2-x+1/8=0,`
v) `100x^2-20x+1=0.`
Sol. :
i)
`x^2-3x-10=0`
`x^2-5x+2x-10=0`
`x(x-5)+2(x-5)=0`
`(x-5)(x+2)=0`
`x-5=0`
`x=5`
`x+2=0`
`x=-2`
Answer :
`x=5, x=-2`.
ii)
`2x^2+x-6=0`
`2x^2+4x-3x-6=0`
`2x(x+2)-3(x+2)=0`
`(x+2)(2x-3)=0`
`x+2=0`
`x=-2`,
`2x-3=0`
`x=3/2`
Answer :
`x=-2, x=3/2`.
iii)
`sqrt2x^2+7x+5sqrt2=0`
`sqrt2x^2+5x+2x+5sqrt2=0`
`x(sqrt2x+5)+sqrt2(sqrt2x+5)=0`
`(sqrt2x+5)(x+sqrt2)=0`
`sqrt2x+5=0`
`x=-5/sqrt2`
`x+sqrt2=0`
`x=-sqrt2`
Answer :
`x=-5/sqrt2, x=-sqrt2`.
iv)
`2x^2-x+1/8=0`
`16x^2-8x+1=0`
`16x^2-4x-4x+1=0`
`4x(4x-1)-1(4x-1)=0`
`(4x-1)(4x-1)=0`
`(4x-1)^2=0`
`4x-1=0`
`x=1/4`
Answer :
each `x=1/4`.
v)
`100x^2-20x+1=0`
`100x^2-10x-10x+1=0`
`10x(10x-1)-1(10x-1)=0`
`(10x-1)(10x-1)=0`
`(10x-1)^2=0`
`10x-1=0`
`x=1/10`
Answer :
`x=1/10, x=1/10`.
Q2. i) John and Jivanti together have `45` marbles. Both of them lost `5` marbles each and the product of the number of marbles they now have is `124.` Find out how many marbles they had to start with.
ii) A cottage industry produces a certain number of toys in a day. The cost of production of each toy (in`₹`) was found to be `55` minus the number of toys produced in a day. On a particular day, the total cost of production was `₹750`. Find out the number of toys produced on that day.
Sol. :
i)
Let the John had `x` marbles,
and the Jivanti had `(45-x)` marbles.
According to question
`(x-5)(45-x-5)=124`
`(x-5)(40-x)=124`
`x(40-x)-5(40-x)=124`
`40x-x^2-200+5x=124`
`-x^2+45x-200-124=0`
`x^2-45x-324=0`
`x^2-9x-36x-324=0`
`x(x-9)-36(x-9)=0`
`(x-9)(x-36)=0`
`x-9=0`
`x=9`
`x-36=0`
`x=36`
Answer :
John had `9` marbles and Jivanti had `36` marbles.
ii)
Let the number of toys produced on a day is `x`,
and the cost of each toys `₹(55-x)`
According to question
`xtimes(55-x)=750`
`55x-x^2=750`
`x^2-55x+750=0`
`x^2-25x-30x+750=0`
`x(x-25)-30(x+25)=0`
`(x-25)(x-30)=0`
`x-25=0`
`x=25`
`x-30=0`
`x=30`
Answer :
The number of toys produced a day was `25` or `30`.
Q3. Find two numbers whoes sum is `27` and product is 182`.
Sol. :
Let the first number is `x`,
and the second number `(27-x)`
According to question
`xtimes(27-x)=182`
`27x-x^2=182`
`x^2-27x+182=0`
`x^2-13x-14x+182=0`
`x(x-13)-14(x-13)=0`
`(x-13)(x-14)=0`
`x-13=0`
`x=13`
`x-14=0`
`x=14`
Answer :
Numbers are `13` and `14`.
Q4. Find two consecutive positive integers, sum of whose square is `365`.
Sol.
Let the first number is `x`,
and the second number is `(x+1)`
According to question
`(x)^2+(x+1)^2=365`
`x^2+x^2+2x+1=365`
`2x^2+2x+1-365=0`
`2x^2+2x-364=0`
`x^2+x-182=0`
`x^2+14x-13x-182=0`
`x(x+14)-13(x+14)=0`
`(x+14)(x-13)=0`
`x+14=0`
`x=-14`
`x-13=0`
`x=13`
`x+1=13+1=14`
Answer :
Positive integers are `13` and `14`.
Q5. Altitude of a right triangle is `7` cm less than its base. If the hypotenuse is `13` cm, find the other two sides.
Sol. :
Let the base of a triangle is `x`cm,
the Altitude of a triangle is `(x-7)`cm, and
the hypotenuse of a triangle is `13`cm.
According to question
`(x)^2+(x-7)^2=(13)^2`
`x^2+x^2-14x+49=169`
`2x^2-14x+49-169=0`
`2x^2-14x-120=0`
`x^2-7x-60=0`
`x^2-12x+5x-60=0`
`x(x-12)+5(x-12)=0`
`(x-12)(x+5)=0`
`x-12=0`
`x=12`
`x+5=0`
`x=-5`
`x-7=12-7=5`
Answer :
The base is `12`cm and the altitude is `5`cm.
Q6. A cottage industry produces a certain number of pottery articles in a day. It was observed on a particular day that the cost of production of each article (in`₹`) was `3` more than twice the number of articles produced in that day was `₹90`. Find the number of articles produced and the cost of each article.
Sol. :
Let the number of articles produced a days was `x`,
and the cost of each article was `₹(2x+3)`,
According to question
`xtimes(2x+3)=90`
`2x^2+3x=90`
`2x^2+3x-90=0`
`2x^2+15x-12x-90=0`
`x(2x+15)-6(2x+15)=0`
`(2x+15)(x-6)=0`
`2x+15=0`
`x=-15/2` ( it is not possible)
`x-6=0`
`x=6`
`2x+3=2times6+3=15`
Answer :
The number of articles `=6`, cost of each article `=₹15`.
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