Exercise 4.1 2 Questions – Check Quadratic Equations
Check whether the following are quadratic equations:
(i) (x+1)² = 2(x'3) ' x²+2x+1 = 2x'6 ' x²+7 = 0 ' Yes, quadratic (can write as x²+0·x+7=0)
(ii) x²'2x = ('2)(3'x) ' x²'2x = '6+2x ' x²'4x+6 = 0 ' Yes
(iii) (x'2)(x+1) = (x'1)(x+3) ' x²'x'2 = x²+2x'3 ' '3x+1 = 0 ' No (linear)
(iv) (x'3)(2x+1) = x(x+5) ' 2x²'5x'3 = x²+5x ' x²'10x'3 = 0 ' Yes
(v) (2x'1)(x'3) = (x+5)(x'1) ' 2x²'7x+3 = x²+4x'5 ' x²'11x+8 = 0 ' Yes
(vi) x²+3x+1 = (x'2)² ' x²+3x+1 = x²'4x+4 ' 7x'3 = 0 ' No
(vii–viii) Check similarly ' expand, simplify, check if x² term remains with non-zero coefficient.
Represent the following situations in the form of quadratic equations:
(i) The area of a rectangular plot is 528 m². The length is 1 more than twice its breadth.
Let breadth = x. Length = 2x+1. Area = x(2x+1) = 528 ' 2x² + x ' 528 = 0
(ii) The product of two consecutive positive integers is 306.
Let integers be x, x+1. x(x+1)=306 ' x² + x ' 306 = 0
(iii) John and Jivanti together have 45 marbles. Both lost 5 each, product of their marbles is 124.
John = x. Jivanti = 45'x. (x'5)(40'x) = 124 ' x² ' 45x + 324 = 0
Exercise 4.2 6 Questions – Factorisation Method
(i) x² ' 3x ' 10 = 0
x²'5x+2x'10 = x(x'5)+2(x'5) = (x'5)(x+2) = 0. ∴ x = 5, '2
(ii) 2x² + x ' 6 = 0
2x²+4x'3x'6 = 2x(x+2)'3(x+2) = (x+2)(2x'3) = 0. ∴ x = '2, 3/2
(iii) √2x² + 7x + 5√2 = 0
√2x²+5x+2x+5√2 = x(√2x+5)+√2(√2x+5) = (√2x+5)(x+√2) = 0. ∴ x = '5/√2, '√2
(iv) 2x² ' x + 1/8 = 0
Multiply by 8: 16x²'8x+1 = 0 ' (4x'1)² = 0. ∴ x = 1/4 (repeated)
(v) 100x² ' 20x + 1 = 0
(10x'1)² = 0. ∴ x = 1/10 (repeated)
Q2: Find two numbers whose sum is 27 and product is 182. ' x(27'x)=182 ' x²'27x+182=0 ' 13 and 14
Q3: Find two consecutive positive integers sum of whose squares is 365. ' x²+(x+1)²=365 ' 13 and 14
Q4: Altitude of right triangle is 7 cm less than base. Hypotenuse = 13 cm. Find the other two sides. ' 5 cm, 12 cm
Q5: A cottage industry produces pottery articles daily. Cost of each article is twice the number of articles plus 3. If total cost is ₹90, find articles produced and cost per article. ' 6 articles, ₹15 each
Q6: Find dimensions of a rectangle whose perimeter is 28 m and diagonal is 10 m. ' 6 m — 8 m
Exercise 4.3 11 Questions – Completing the Square & Quadratic Formula
Q1: Find roots by completing square: 2x²'7x+3=0
Divide by 2: x²'(7/2)x+3/2=0. x²'(7/2)x='3/2. Add (7/4)²=49/16 to both sides: (x'7/4)²=49/16'24/16=25/16. x'7/4=±5/4. x=3, 1/2
Quadratic Formula: x = ['b ± √(b²'4ac)] / 2a
Q2: Solve 2x²+x'4=0 using formula: x = ['1±√(1+32)]/4 = ['1±√33]/4
Q3: 4x²+4√3x+3=0 ' (2x+√3)²=0 ' x='√3/2
Q5: In a class test, sum of Shefali's marks = 30. If she got 2 more in Maths and 3 less in English, product would be 210. Find marks. ' Eng=13, Maths=17 OR Eng=12, Maths=18
Q6: Diagonal of rectangular field is 60 m more than shorter side. Longer side is 30 m more than shorter side. Find sides. ' x²+(x+30)²=(x+60)² ' 90 m, 120 m
Q8: A train travels 360 km at uniform speed. If speed is 5 km/h more, time reduces by 1 hr. Find speed. ' 360/x ' 360/(x+5) = 1 ' 40 km/h
Q9: Two water taps fill a tank in 9⅜ hrs. Larger tap takes 10 hrs less than smaller. Find time for each. ' 25 hrs and 15 hrs
Q11: Sum of areas of two squares = 468 m². Perimeter difference = 24 m. Find sides. ' 12 m, 18 m
Exercise 4.4 5 Questions – Nature of Roots
Find the nature of roots: D = b²'4ac. If D>0: two distinct real roots. D=0: two equal real roots. D<0: no real roots.
(i) 2x²'3x+5=0 ' D=9'40='31<0 ' No real roots
(ii) 3x²'4√3x+4=0 ' D=48'48=0 ' Two equal real roots
(iii) 2x²'6x+3=0 ' D=36'24=12>0 ' Two distinct real roots
Q2: kx(x'2)+6=0 has equal roots. Find k. ' kx²'2kx+6=0. D=4k²'24k=0 ' k = 6
Q3: Is it possible to design a rectangular mango grove with length = 2—breadth and area = 800 m²? ' x(2x)=800, x²=400, x=20. Yes, 20—40 m
Q4: Sum of ages of two friends is 20 years. Four years ago, product was 48. Find ages. ' 16 and 4 (not possible both) ' Not possible
Q5: Perimeter 80 m, area 400 m². Possible? ' x+y=40, xy=400. x(40'x)=400, x²'40x+400=0, D=0. Yes, square 20—20 m
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