Sunday, September 30, 2012

CBSE Class 10 - Maths - CH4 Quadratic Equations (Ex 4.1)

Quadratic Equations
Quadratic Euqations
NCERT Exercise 4.1

Standard form of equation is 

ax2 + bx + c = 0
where a, b and c are the co-efficients and a ≠ 0

Q1: Check whether the following are quadratic equations :
(i)   (x + 1)2 = 2(x – 3)
(ii)  x2 – 2x = (–2) (3 – x)
(iii) (x – 2)(x + 1) = (x – 1)(x + 3)
(iv) (x – 3)(2x +1) = x(x + 5)
(v)  (2x – 1)(x – 3) = (x + 5)(x – 1)
(vi) x2 + 3x + 1 = (x – 2)2

Answer:


(i)   (x + 1)2 = 2(x – 3)
⇒  x2 + 2x + 1 = 3x - 6
⇒  x2 + (2-3)x + 1 + 6 = 0
⇒  x2 - x + 7 = 0
Since the equation is of the form ax2 + bx + c = 0, it is a quadratic equation.

(ii) x2 – 2x = (–2) (3 – x)
⇒  x2– 2x = (–2 × 3) + ( –2 × –x)
⇒x2– 2x = –6 + 2x
⇒  x2– 2x - 2x + 6 = 0
⇒  x2– 4x + 6 = 0
Since the equation is of the form ax2 + bx + c = 0, it is a quadratic equation.

(iii) (x – 2)(x + 1) = (x – 1)(x + 3)
⇒ x2 + x -2x - 2 = x2 + 3x -x -3
⇒ x2 - x - 2 = x2 + 2x - 3
⇒ x2 - x - 2 - x2 - 2x + 3 = 0
⇒ -3x + 1 = 0
 ⇒ 3x - 1 = 0
Since the equation is NOT of the form ax2 + bx + c = 0, it is NOT a quadratic equation.

(iv) (x – 3)(2x +1) = x(x + 5)
⇒ 2x2 + x - 6x - 3 = x2 + 5x
⇒ 2x2 - 5x - 3 -  x2 - 5x = 0
⇒ x2- 10x - 3 = 0
Since the equation is of the form ax2 + bx + c = 0, it is a quadratic equation.

(v)  (2x – 1)(x – 3) = (x + 5)(x – 1)
⇒ 2x2 -  6x  - 1x + 3 = x2 - 1x + 5x - 5
⇒ 2x2 - 7x + 3 = x2+ 4x - 5
⇒ 2x2 - 7x + 3 - x2 - 4x + 5 = 0
⇒ x2 - 11x + 8 = 0
Since the equation is of the form ax2 + bx + c = 0, it is a quadratic equation.

(vi) x2 + 3x + 1 = (x – 2)2
⇒ x2 + 3x + 1 = x2 + 4 - 4x
⇒ x2 + 3x + 1 - x2 - 4 + 4x =0
⇒ 7x - 3 = 0
Since the equation is NOT of the form ax2 + bx + c = 0, it is NOT a quadratic equation.

Q2: Represent the following situations in the form of quadratic equations:

(i) The area of a rectangular plot is 528 m2. The length of the plot (in metres) is one more than twice its breadth. We need to find the length and breadth of the plot.

Answer: Let the breadth of the rectangular plot = b m
∴ Length of the plot = (2 × b + 1) m = 2b + 1
Area = length × breadth = 528 m2
⇒ (2b+1) × b = 528
⇒ 2b2 + b =528⇒ 2b2 + b  - 528 = 0

(ii) The product of two consecutive positive integers is 306. We need to find the integers.

Answer: Let the first integer number be = p
The next consecutive positive integer =  p+1
Product = p × (p +1) = 306
⇒ p2+ p = 306
⇒ p2+ p - 206 = 0

(iii) Rohan’s mother is 26 years older than him. The product of their ages (in years) 3 years from now will be 360. We would like to find Rohan’s present age. 

Answer: Let Rohan's age = t years
Rohan's mother age = t + 26 years
3 years from now, Rohan's age = t + 3
Age of Rohan's mother after 3 years = t + 26 + 3 = t + 29
Product of their ages after three years = (t + 3)(t + 29) = 360
⇒ t2 + 29t + 3t + 87 = 360
⇒ t2 + 32t + 87 - 360 = 0
⇒ t2 + 32t - 273  = 0

(iv) A train travels a distance of 480 km at a uniform speed. If the speed had been 8 km/h less, then it would have taken 3 hours more to cover the same distance. We need to find the speed of the train.

Answer: Let the speed of the train = v km/h
∵ Distance = speed  × time

Time taken by  train (t) = 480/v  h
If speed decreases by 8 km/h, new speed = v - 8  km/h
Time taken to cover 480 km with new speed  = t + 3 hours = 480/v + 3 hours
∵ Distance = speed × time
⇒  (v - 8) × (480/v + 3) = 480
⇒  480 + 3v - 3840/v - 24 = 480

⇒  480 + 3v - 3840/v - 24 - 480 = 0
3v - 3840/v - 24 = 0
⇒ 3v2 - 3840 - 24v = 0
⇒ v2 - 8v - 1280  = 0

Q3(FA Manual): Which of the following are quadratic equations?
(i) √x - x = 4
(ii) x + 1/x = 5
(iii) 7 v2 = 49

Answer:
(i) √x - x = 4
x1/2- x = 4
Since Since the equation is NOT of the form ax2 + bx + c = 0, it is NOT a quadratic equation.

(ii) x + 1/x = 5
⇒ x + 1/x - 5 = 0
Multiply by x to both sides,
x2 + 1 - 5x = 0
x2 -5x + 1 = 0
Since the equation is of the form ax2 + bx + c = 0, it is a quadratic equation.

(iii) 7 v2 = 49
7v2 - 49 = 0
⇒  7v2+ 0v  - 49 = 0


Since the equation is of the form ax2 + bx + c = 0, it is a quadratic equation.

[Note: Equation is quadratic, when the first term must be  ax2 and a ≠ 0 but coefficients b and c can be zero].







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2 comments:

  1. Thanks for your grateful informations, this blogs will be really help for chemistry notes.

    ReplyDelete
  2. in 1st question there is some mistake. instead of 3x-6 it should be 2x-6.

    ReplyDelete