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Page 1 of 5, Mathematics, (Chapter-2) (Polynomials), (Class X), Exercise 2.2, Question 1:, Find the zeroes of the following quadratic polynomials and verify the, relationship between the zeroes and the coefficients., (i).x' - 2r-8, (ii)4s – 4s +1, (iii)6x - 3-7x, (iv)4u +8u, (v)r -15, (vi)3r -x-4, Answer 1:, (i) x*-2r-8 = (x-4)(x+2), The value of x - 2.x-8 is zero when x - 4 = 0 or x + 2 = 0, i.e., when x, = 4 or x = -2, Therefore, the zeroes of r - 2.x-8 are 4 and -2., -(-2)_ -(Coefficient of x), Coefficient of r, Sum of zeroes =, 4-2 2D., %3D, (-8), Coefficient of r, Constant term, Product of zeroes = 4x(-2)=-8=, %3D, (ii) 4s -4s +1= (2s-1), %3D, The value of 4s? - 4s + 1 is zero when 2s - 1 = 0, i.e., s=-Therefore,, the zeroes of 4s? - 4s + 1 are ½ and ½ ., -(-4) -(Coefficient of s), (Coefficient of s'), 1 1, Sum of zeroes =, =13=, 4, 2 2, 1 1, Constant term, Product of zeroes =-, -X-, =-=, 2 2 4 Coefficient of s, (iї) бх*-3-7х-6r - 7x-3-(3х+1)(2х- 3), The value of 6x? - 3 - 7x is zero when 3x + 1 = 0 or 2x - 3 = 0, i.e.,, 1| Page
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Page 2 of 5, or, 3, x =, -1, Therefore, the zeroes of 6 x? - 3 - 7x are, and, 3, -1,3 7-(-7) -(Coefficient of x), Coefficient of r?, Sum of zeroes =, %3D, %3D, 3 2 6, 6, -1 3, -1, -3, Constant term, Product of zeroes =, %3D, 3 2, Coefficient of xr, (iv) 4u + 8u =4u +8u+0, = 41 (1u1+2), The value of 4u2 + 8u is zero when 4u = 0 or u + 2 = 0, i.e.,, u = 0 or u = -2, Therefore, the zeroes of 4u2 + 8u are 0 and -2., 0+(-2) = -2=3_(Coefficient of u), Coefficient of u?, Sum of zeroes =, %3D, 4, Constant term, Product of zeroes =, 0x(-2)= 0=-, 4, %3D, Coefficient ofu?, (v), -15, =-01-15, The value of t – 15 is zero when 1-V15 = 0 or +V15 = 0, i.e., when, 1= 15 or ( =-, 2 | Page
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Page 3 of 5, Therefore, the zeroes of t2, 15 are 15 and -15., -(Coefficient of ?), (Coefficient of r), -0, Sum of zeroes =, ViS +(-J15) = 0 =, %3D, %3D, Constant term, (Vī5)(-V15) -, -15, =-15 =, 1, Product of zeroes =, Coefficient of r?, (vi) 3x* -х-4, = (3x - 4)(x +1), The value of 3x? - x - 4 is zero when 3x - 4 = 0 or x + 1 = 0, i.e.,, when x=, or x = -1, 3, Therefore, the zeroes of 3x2 - x - 4 are 4/3 and -1., -(Coefficient of x), Coefficient of x, -(-1), Sum of zeroes =, 3, -4, Constant term, Product of zeroes, %3D, 3, Coefficient of x, Question 2:, Find a quadratic polynomial each with the given numbers as the sum and, product of its zeroes respectively., (1) -, (ii), (iii) 0, V5, 4, I 1, 4 4, (vi) 4,1, (iv) 1,1, Answer 2:, (1) -, 3 | Page
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Page 4 of 5, Let the polynomial be ax2 + bx + c and its zeroes be a and ß., 1, -b, a +B =, 4, %3D, -4, - =, 4, aß =, a, If a = 4, then b=-1, c = -4, Therefore, the quadratic polynomial is 4x2 – x- 4., (ii), Let the polynomial be ax2 + bx + c and its zeroes be a and ß., 3/2 -b, a + B =-, VE =, 3, %3D, %3D, a, 1, aß ==, 3, с, %3!, a, If a = 3, then b = -3/2, c =1, Therefore, the quadratic polynomial is 3x2 -, 3/2x + 1., (iii) 0, V5, Let the polynomial be ax2 + bx + c and its zeroes be a and ß., a +B = 0 =-, a, V5, axß= 5 =-, a, If a = 1, then b= 0, c =, Therefore, the quadratic polynomial is x'+ 5., (iv) 1, 1, Let the polynomial be ax? + bx + c and its zeroes be a and ß., 4 | Page
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Page 5 of 5, -b, a +B =1==, ax ß =1==, a, If a = 1, then b=-I, c =1, Therefore, the quadratic polynomial is x-x+1., (v) -, 4'4, Let the polynomial be ax2 + bx + c and its zeroes be a and, -b, -1, a +B =, 4, a, axB%3D, -=-, 4 a, If a = 4, then b= 1, c =1, Therefore, the quadratic polynomial is 4x +.x+1., (vi) 4, 1, Let the polynomial be ax? + bx +c and its zeroes be a and ß., a+B=4=1=-b, -=., a, ax B = 1==, I a, Ifa = 1, then b = -4, c =1, Therefore, the quadratic polynomial isr-4x+1., 5 | Page
