{"id":13843,"date":"2023-06-19T12:58:08","date_gmt":"2023-06-19T07:28:08","guid":{"rendered":"https:\/\/apboardsolutions.in\/?p=13843"},"modified":"2023-06-20T17:04:00","modified_gmt":"2023-06-20T11:34:00","slug":"ap-10th-class-maths-bits-chapter-5","status":"publish","type":"post","link":"https:\/\/apboardsolutions.in\/ap-10th-class-maths-bits-chapter-5\/","title":{"rendered":"AP 10th Class Maths Bits Chapter 5 Quadratic Equations with Answers"},"content":{"rendered":"

Practice the AP 10th Class Maths Bits with Answers<\/a> Chapter 5 Quadratic Equations on a regular basis so that you can attempt exams with utmost confidence.<\/p>\n

AP SSC 10th Class Maths Bits 5th Lesson Quadratic Equations with Answers<\/h2>\n

Question 1.
\nIf x2<\/sup> – px + q = 0(p,q\u2208R and p \u2260 0, q \u2260 0) has distinct real roots, then write
\nthe condition.
\nAnswer: p2<\/sup> > 4q.<\/p>\n

Question 2.
\nIf one root of 2x2<\/sup> + kx – 6 is 2., then find k.
\nAnswer:
\nk = – 1
\nExplanation:
\n2(2)2<\/sup> + k(2) – 6 = 0
\n\u21d2 8 + 2k – 6 = O
\n\u21d2 2k + 2 = 0 \u21d2 k = -1<\/p>\n

Question 3.
\nIf the equation x2<\/sup> + 5x + k = 0 has real and distinct roots, then find the value of \u2018k\u2019.
\nAnswer:
\nk > 6.25
\nExplanation:
\nReal and distinct roots so,
\nb2<\/sup> – 4ac > 0
\n\u21d2 25 – 4 . 1. k > 0
\n\u21d2 25 > 4k = k > \\(\\frac{25}{4}\\) > 6.25<\/p>\n

\"AP<\/p>\n

Question 4.
\nFrame a quadratic equation, whose roots are 2 + \\(\\sqrt{3}\\) and 2 – \\(\\sqrt{3}\\) ?
\nAnswer:
\nx2<\/sup> – 4x + 1 = 0
\nExplanation:
\nx2<\/sup> – (2 + \\(\\sqrt{3}\\) +2 – \\(\\sqrt{3}\\))x + (2 + \\(\\sqrt{3}\\)) (2 – \\(\\sqrt{3}\\))
\n\u21d2 x2<\/sup> – 4x + 1 = 0<\/p>\n

Question 5.
\nIn a quadratic equation ax2<\/sup> + bx + c = 0, if b2<\/sup> – 4ac > 0, then write the nature of the roots.
\nAnswer:
\nRoots are real and distinct.<\/p>\n

Question 6.
\nCreate the quadratic equation, whose zeroes are \\(\\sqrt{2}\\) and – \\(\\sqrt{2}\\) ?
\nAnswer:
\nx2<\/sup> – 2 = 0.
\nExplanation:
\n\\(x^{2}-(\\sqrt{2}-\\sqrt{2}) x+(\\sqrt{2})(-\\sqrt{2})=0\\)
\n\u21d2 x2<\/sup> – 2 = 0<\/p>\n

Question 7.
\nFor which positive value of x the qua-dratic equation 4x\\(\\sqrt{3}\\) -9 = 0 satisfies ?
\nAnswer:
\n\\(\\frac{3}{2}\\)<\/p>\n

Question 8.
\nIf the roots of x2<\/sup> + 6x + 5 = 0 are a and P, then find the value of sum of the roots.
\nAnswer:
\n-6
\nExplanation:
\n\u03b1 + \u03b2 = \\(\\frac{-b}{a}\\) = \u2014 6<\/p>\n

Question 9.
\nWrite the discriminant of 6x2<\/sup> – 5x + 1 = 0.
\nAnswer:
\nD = 1
\nExplanation:
\nD = b2<\/sup> – 4ac = 25 – 4 . 6 . 1
\n\u21d2 25 – 24 = 1 > 0 D = 1<\/p>\n

Question 10.
\nWrite the quadratic polynomial having \\(\\frac { 1 }{ 3 }\\) and \\(\\frac { 1 }{ 2 }\\) as its zeroes.
\nAnswer:
\nx2<\/sup> – \\(\\frac{5 x-1}{6}\\) = 0 \u21d2 6x2<\/sup> – 5x + 1 = 0
\nExplanation:
\n\"AP<\/p>\n

Question 11.
\nIf a number is 132 smaller than its square, then find the number.
\nAnswer:
\n12
\nExplanation:
\nx + 132 = x2<\/sup>
\n\u21d2 x2<\/sup> – x – 132 = 0
\nBy solve the equation, \u2234 x = 12<\/p>\n

Question 12.
\nWrite the general form of a quadratic equation in variable ‘x’.
\nAnswer:
\nax2<\/sup> + bx + c = 0 (a \u2260 0).<\/p>\n

Question 13.
\nMake the quadratic polynomial, whose zeroes are 2 and 3.
\nAnswer:
\nx2<\/sup> – 5x + 6.<\/p>\n

Question 14.
\nIf \u03b1, \u03b2 are the roots of x2<\/sup> – 10x + 9 = 0, thep find the value of | \u03b1 – \u03b2 |.
\nAnswer:
\n8
\nExplanation:
\nx2<\/sup> – 9x – x + 9 = 0
\n\u21d2 x(x – 9) – 1 (x – 9) = 0
\n\u21d2 (x-9)(x- 1) = 0
\nx = 9 and 1, |\u03b1 – \u03b2| = |9- 1| = 8<\/p>\n

Question 15.
\nWrite the discriminant of adjacent dia-gram indicates.
\n\"AP
\nAnswer:
\nb2<\/sup> – 4ac > 0.<\/p>\n

Question 16.
\nIf the roots of a quadratic equation px2<\/sup> + qx + r = 0 are imaginary, then write the condition of discriminant.
\nAnswer:
\nq2<\/sup> < 4pr (or) q2<\/sup> – 4pr < 0.<\/p>\n

\"AP<\/p>\n

Question 17.
\nTwo angles are complementary. If the large angle is twice the measure of a smaller angle, then find the value of smaller angle.
\nAnswer:
\n30\u00b0
\nExplanation:
\nx + y = 90\u00b0
\n\u21d2 x + 2x = 90\u00b0
\n\u21d2 3x = 90\u00b0 \u21d2 x = 30\u00b0<\/p>\n

Question 18.
\nObserve the following graphs.
\n\"AP
\nWhich as them are the graphs of qua-dratic polynomials ?
\nAnswer:
\n(i) and (iv).<\/p>\n

Question 19.
\nWrite the possible number of roots to a quadratic equation.
\nAnswer:
\nAt a maximum of 2.<\/p>\n

Question 20.
\nIf 1 is a common root of ax2<\/sup> + ax + 2 = 0 and x2<\/sup> + x + 6 = 0, then find a-b.
\nAnswer:
\n2<\/p>\n

Question 21.
\nFind the product of roots of quadratic equation ax2<\/sup> + bx + c = 0.
\nAnswer:
\n\\(\\frac{\\text { c }}{\\text { a }}\\)<\/p>\n

Question 22.
\nWrite the number of diagonals in a polygon, having ‘n’ sides.
\nAnswer:
\n\\(\\frac{n(n-3)}{2}\\)<\/p>\n

Question 23.
\nFind the discriminant of quadratic equation 2x2<\/sup> + x – 4 = 0.
\nAnswer:
\n33<\/p>\n

Question 24.
\nA quadratic equation ax2<\/sup> + bx + c = 0 has two distinct real roots, then write the condition.
\nAnswer:
\nb2<\/sup> – 4ac >0.<\/p>\n

Question 25.
\nDraw the shape of quadratic equation which having distinct roots ?
\nAnswer:
\n\"AP<\/p>\n

Question 26.
\nThe sum of a number and its reciprocal is \\(\\frac { 5 }{ 2 }\\) then find the number.
\nAnswer:
\n2 or \\(\\frac { 1 }{ 2 }\\)
\nExplanation:
\nx + \\(\\frac{1}{x}=\\frac{5}{2}\\)
\n\u21d2 \\(\\frac{x^{2}+1}{x}=\\frac{5}{2}\\)
\n\u21d2 2x2<\/sup> + 2 = 5x
\n\u21d2 2x2<\/sup> – 5x + 2 = 0
\n\u21d2 2x2<\/sup> – 4x r x + 2 = 0
\n\u21d2 2x(x – 2) – 1 (x – 2) – 0 \u21d2 (x – 2) (2x – 1) = 0 1
\n\u2234 x = 2 or 1\/2.<\/p>\n

\"AP<\/p>\n

Question 27.
\nFind the roots of the equation 4x2<\/sup> – 4\\(\\sqrt{3}\\) x + 3 = 0.
\nAnswer:
\n\\(-\\frac{\\sqrt{3}}{2}\\)<\/p>\n

Question 28.
\nFind the positive root of \\(\\sqrt{3 x^{2}+6}=9\\)
\nAnswer:
\n5
\nExplanation:
\n3x2<\/sup> + 6 = 81
\n\u21d2 3x2<\/sup> = 81 – 6 = 75
\n\u21d2 x2<\/sup> = \\(\\) = 25 \u21d2 x = 5<\/p>\n

Question 29.
\nFind the roots of the quadratic equation (7x – 1) (2x + 3) = 0.
\nAnswer:
\n\\(\\frac{1}{7}, \\frac{-3}{2}\\)<\/p>\n

Question 30.
\nIf the sum of the squares of two con-secutive odd numbers is 74, then find the smaller number.
\nAnswer:
\n5 (or)-7
\nExplanation:
\n(2x + 1)2<\/sup> + (2x + 3)2<\/sup> – 74
\n\u21d2 4x2<\/sup> + 4x + 1 + 4x2<\/sup> + 12x + 9 \u2014 74′
\n\u21d2 8x2<\/sup> + 16x + 10 = 74
\n\u21d2 8x2<\/sup> + 16x – 64 = 0
\n\u21d2 8(x2<\/sup> + 2x – 8) = 0
\n\u21d2 x2<\/sup> + 4x – 2x – 8 = 0
\n\u21d2 x(x + 4) – 2 (x + 4) = 0
\n\u21d2 x = – 4, 2
\n\u2234 x = – 4, then smaller number
\n= 2 . (-4) + 1 = -8 + 1 = -7
\n\u2234 x = 2, then smaller number
\n= 2 . (2) + 1 = 4 + 1 = 5<\/p>\n

Question 31.
\nWrite the standard form of a cubic polynomial.
\nAnswer:
\nax3<\/sup> + bx2<\/sup> + cx + d = 0; (a \u2260 0).<\/p>\n

Question 32.
\nWrite the discriminant of 5x2<\/sup>– 3x – 2 = 0.
\nAnswer:
\n49<\/p>\n

Question 33.
\nCreate the quadratic equation whose roots are – 2 and – 3.
\nAnswer:
\nx2<\/sup> + 5x + 6 = 0<\/p>\n

Question 34.
\nFind the roots of the quadratic equation \\(\\frac{x^{2}-8}{x^{2}+20}=\\frac{1}{2}\\)
\nAnswer:
\n\u00b16
\nExplanation:
\n2x2<\/sup> – 16 = x2<\/sup> + 20
\n\u21d2 x2<\/sup> – 36 \u21d2 x = \u00b16.<\/p>\n

Question 35.
\nFind the roots of the equation 3x2<\/sup> – 2\\(\\sqrt{6}\\) x + 2 = 0.
\nAnswer:
\n\\(\\sqrt{\\frac{2}{3}}, \\sqrt{\\frac{2}{3}}\\)<\/p>\n

Question 36.
\nFind the roots of the quadratic equa- tion \\(\\left(x-\\frac{1}{3}\\right)^{2}\\) = 9.
\nAnswer:
\n\\(\\frac { 10 }{ 3 }\\) (or) \\(\\frac { -8 }{ 3 }\\).
\nExplanation:
\n\"AP<\/p>\n

Question 37.
\nOn solving x2<\/sup> + 5 = – 6x, find the value of ‘x’
\nAnswer:
\n– 1 or – 5.<\/p>\n

Question 38.
\nSimplified form of \\(\\frac{\\mathbf{x}}{\\mathbf{x}-\\mathbf{y}}-\\frac{\\mathbf{y}}{\\mathbf{x}+\\mathbf{y}}\\)
\nAnswer:
\n\\(\\frac{x^{2}+y^{2}}{x^{2}-y^{2}}\\)<\/p>\n

Question 39.
\nFind the sum of roots of bx2<\/sup> + ax + c = 0.
\nAnswer:
\n\\(\\frac{-a}{b}\\)<\/p>\n

\"AP<\/p>\n

Question 40.
\nFind the roots of 2x2<\/sup> – x + \\(\\frac { 1 }{ 8 }\\) = 0.
\nAnswer:
\n\\(\\frac{1}{4}, \\frac{1}{4}\\)<\/p>\n

Question 41.
\nIf x + \\(\\frac{1}{x}\\) = 2, then find \\(x^{2}+\\frac{1}{x^{2}}\\).
\nAnswer:
\n2
\nExplanation:
\nx + \\(\\frac { 1 }{ x }\\) = 2
\n\u21d2 \\(x^{2}+\\frac{1}{x^{2}}\\) + 2 = 4 \u21d2 x2<\/sup> + \\(x^{2}+\\frac{1}{x^{2}}\\) = 2<\/p>\n

Question 42.
\nIf 3y2<\/sup> \u2014 192, then find ‘y’.
\nAnswer:
\ny = \u00b1 8<\/p>\n

Question 43.
\nHow many diagonals has a pentagon?
\nAnswer:
\n\u20199′<\/p>\n

Question 44.
\nIf \u03b1 and \u03b2 are the roots of the quadratic equation x2<\/sup> – 3x + 1 = 0, then find \\(\\left(\\frac{1}{\\alpha^{2}}+\\frac{1}{\\beta^{2}}\\right)\\)
\nAnswer:
\n7<\/p>\n

Question 45.
\nIf \\(\\mathbf{a}^{\\mathbf{x}^{2}-4 \\mathbf{x}+3}\\) = 1, then find x (a # 0).
\nAnswer:
\n1 or 3.<\/p>\n

Question 46.
\nFind discriminant of the quadratic equation x + \\(\\frac { 1 }{ x }\\) = 3.
\nAnswer:
\n5<\/p>\n

Question 47.
\nCreate the quadratic equation with 2 < x < 3.
\nAnswer:
\nx2<\/sup> – 5x + 6 < 0.
\nExplanation:
\nx2<\/sup> – (2 + 3)x + 2 . 3 < 0
\n\u21d2 x2<\/sup> – 5x + 6 < 0<\/p>\n

Question 48.
\np(x) = x2<\/sup> + 2x + 1, then find p(x2<\/sup>).
\nAnswer:
\nx4<\/sup> + 2x2<\/sup> + 1<\/p>\n

Question 49.
\nx2<\/sup> – 7x – 60 = 0, then find ‘x’.
\nAnswer:
\n12 and -5.<\/p>\n

Question 50.
\n\\(\\frac{1}{a+3}+\\frac{1}{a-3}+\\frac{6}{9-a^{2}}\\) is equal to ?
\nAnswer:
\n\\(\\frac{2}{a+3}\\)<\/p>\n

Question 51.
\nFind the roots of \\(\\sqrt{2} x^{2}+7 x+5 \\sqrt{2}\\) = 0.
\nAnswer:
\n\\(\\frac{-5}{\\sqrt{2}} \\text { or }-\\sqrt{2}\\)<\/p>\n

Question 52.
\nFind the roots of a quadratic equation \\((\\sqrt{2} x+3)(5 x-\\sqrt{3})=0\\)
\nAnswer:
\n\\(\\frac{-3}{\\sqrt{2}}, \\frac{\\sqrt{3}}{5}\\)<\/p>\n

Question 53.
\n4x2<\/sup> + ky – 2 = 0 has no real roots, then find ‘k’.
\nAnswer:
\nk < – \\(\\sqrt{32}\\)<\/p>\n

\"AP<\/p>\n

Question 54.
\nThe sum of a number and its reciprocal is \\(\\frac { 50 }{ 7 }\\), then find the number.
\nAnswer:
\n7 (or) \\(\\frac { 1 }{ 7 }\\)
\nExplanation:
\nx + \\(\\frac{1}{x}=\\frac{50}{7}\\)
\n\u21d2 \\(\\frac{x^{2}+1}{x}=\\frac{50}{7}\\)
\n\u21d2 7x2<\/sup> + 7 \u2014 50x
\n\u21d2 7x2<\/sup> – 50x + 7 \u2014 0
\n\u21d2 7x2<\/sup> – 49x – x + 7 = 0
\n\u21d2 7x (x – 7) – 1 (x – 7) – 0
\n\u21d2 (x – 7) (7x – 1) – 0
\n=+ x = 7 (or) 1\/7<\/p>\n

Question 55.
\nFind the roots of the quadratic equation \\(\\frac{9}{x^{2}-27}=\\frac{25}{x^{2}-11}\\)
\nAnswer:
\n\u00b16;<\/p>\n

Question 56.
\nWrite the nature of the roots of a qua-dratic equation 4x2<\/sup> – 12x + 9 = 0.
\nAnswer:
\nReal and equal.<\/p>\n

Question 57.
\n3x2<\/sup> + (- k)x + 8 = 0 has no real roots, then find k\u2019.
\nAnswer:
\nk < 4\\(\\sqrt{6}\\)
\nExplanation:
\nNo real roots. So D < 0,
\n(-k)2<\/sup> – 4 . 3 . 8 < 0
\n\u21d2 k2<\/sup> – 96 < 0
\n\u21d2 k2<\/sup> < 96
\n\u21d2 k < \\(\\sqrt{96}\\)
\n\u21d2 k < \\(4 \\sqrt{6}\\)<\/p>\n

Question 58.
\nFind the discriminant of 3x2<\/sup> – 2x = \\(\\frac{-1}{3}\\).
\nAnswer:
\nD = 0<\/p>\n

Question 59.
\nFind the product of the roots of 1 =x2<\/sup>.
\nAnswer:
\n-1<\/p>\n

Question 60.
\nx(x + 4) = 12, then find ‘x’.
\nAnswer:
\n– 6 or 2.<\/p>\n

Question 61.
\nForm a quadratic equation from, x3<\/sup> – 4x2<\/sup> – x + 1 = (x – 2)3<\/sup>.
\nAnswer:
\n2x2<\/sup> – 13x + 9 = 0.
\nExplanation:
\nx3<\/sup> – 4x2<\/sup> – x + 1 = x3<\/sup> – 3 . x2<\/sup> . 2 + 3 . x . 22<\/sup> – 23<\/sup>
\n\u21d2 x3<\/sup> – 4x2<\/sup> – x + 1 = x3<\/sup> – 6x2<\/sup> + 12x – 8
\n\u21d2 x3<\/sup> – 4x2<\/sup> – x + 1 = x3<\/sup> + 6x2<\/sup> – 12x + 8 = 0
\n\u21d2 2x2<\/sup> – 13x + 9 = 0<\/p>\n

Question 62.
\nFind the product of the roots of x2<\/sup> + 7x = 0.
\nAnswer:
\n0<\/p>\n

Question 63.
\n\\(\\frac{2 a^{2}+a-1}{a+1}+\\frac{3 a^{2}+5 a+2}{3 a+2}+\\frac{4-a^{2}}{a+2}\\) is equal to ?
\nAnswer:
\n2 (a +1)<\/p>\n

Question 64.
\n1 and \\(\\frac { 3 }{ 2 }\\) are the roots of which qua-dratic equation ?
\nAnswer:
\n2x2<\/sup> – 5x + 3 = 0.<\/p>\n

Question 65.
\nIf b2<\/sup> < 4ac, then draw the shape of graph.
\nAnswer:
\n\"AP<\/p>\n

Question 66.
\n\\(\\sqrt{\\mathbf{k}+\\mathbf{1}}\\) = 3, then find ‘k’.
\nAnswer:
\nk = 8<\/p>\n

Question 67.
\n\\(\\sqrt{x}=\\sqrt{2 x-1}\\), then find ‘x’.
\nAnswer:
\nx = 1<\/p>\n

\"AP<\/p>\n

Question 68.
\nIf \\(\\frac{1}{x-2}+\\frac{2}{x-1}=\\frac{6}{x}\\) then find ‘x’
\nAnswer:
\n3 or \\(\\frac { 4 }{ 3 }\\)<\/p>\n

Question 69.
\nFind the coefficient of ‘x’ in a pine qua-dratic equation.
\nAnswer:
\n0<\/p>\n

Question 70.
\nWrite number of distinct line segments that can be formed out of n – points.
\nAnswer:
\n\\(\\frac{n(n-1)}{2}\\)<\/p>\n

Question 71.
\nThe product of two consecutive positive integers is 306, then find the largest number.
\nAnswer:
\n18
\nExplanation:
\nx(x + 1) = 306 \u21d2 x2<\/sup> + x – 306 = 0
\nby solving thik Q.E., x = 17
\n\u2234 Largest number = x + 1
\n= 17 + 1 = 18.<\/p>\n

Question 72.
\nWrite the nature of roots of 3x2<\/sup> + 13x – 2 = 0.
\nAnswer:
\nReal and unequal.<\/p>\n

Question 73.
\nIf \u03b1 and \u03b2 are the roots of x2<\/sup> – 2x + 3 = 0, then find \u03b12<\/sup> + \u03b22<\/sup>
\nAnswer:
\n\u03b12<\/sup> + \u03b22<\/sup> = – 2.<\/p>\n

Question 74.
\nIf (2x – 1) (2x + 3) = 0, then find ‘x’.
\nAnswer:
\n\\(\\frac { 1 }{ 2 }\\) or \\(\\frac { -3 }{ 2 }\\)<\/p>\n

Question 75.
\nWrite the quadratic equation whose one root is 2 – \\(\\sqrt{3}\\) .
\nAnswer:
\nx2<\/sup> – 4x + 1 = 0<\/p>\n

Question 76.
\nIf b2<\/sup> – 4ac > 0, then write nature of the roots of the quadratic equation.
\nAnswer:
\nReal and distinct.<\/p>\n

Question 77.
\nFind product of the roots of ax2<\/sup> + bx + c = 0. c
\nAnswer:
\nc\/a<\/p>\n

Question 78.
\nWrite the nature of the roots of a qua-dratic equation 4x2<\/sup> + 5x + 1 = 0.
\nAnswer:
\nReal and distinct.<\/p>\n

Question 79.
\nWrite the quadratic equation whose roots are – 1,6.
\nAnswer:
\nx2<\/sup> – 5x – 6 = 0.<\/p>\n

Question 80.
\nCreate the quadratic equation whose roots are – 3 and – 4.
\nAnswer:
\nx2<\/sup> + 7x 4- 12 = 0.<\/p>\n

Question 81.
\nFind the roots of the quadratic equation (3x + 4)2<\/sup> – 49 = 0.
\nAnswer:
\n1, \\(\\frac{-11}{3}\\)<\/p>\n

Question 82.
\nIf x2<\/sup> – 2x + 1 = 0, then find x + \\(\\frac{1}{x}\\).
\nAnswer:
\n2<\/p>\n

Question 83.
\nWrite nature of the roots of 5x2<\/sup> – x + 1 = 0.
\nAnswer:
\nImaginary roots.<\/p>\n

\"AP<\/p>\n

Question 84.
\nWrite the nature of the roots of qua-dratic equation 3x2<\/sup> + x + 8 = 0.
\nAnswer:
\nImaginary roots.<\/p>\n

Question 85.
\nFind product of the roots of the qua-dratic equation 3x2<\/sup> – 6x + 11 = 0.
\nAnswer:
\n\\(\\frac{11}{3}\\)<\/p>\n

Question 86.
\nForm a quadratic equation whose roots are k and 1\/k.
\nAnswer:
\nx2<\/sup> – (\\(\\mathrm{k}+\\frac{1}{\\mathrm{k}}\\))x + 1 = 0<\/p>\n

Question 87.
\nIf k2<\/sup> – 8kx + 16 = 0 has equal roots, then find the value of ‘k’.
\nAnswer:
\nk = \u00b1 1.
\nExplanation:
\n(-8k)2<\/sup> – 4(1) (16) = 0
\n\u21d2 64k2<\/sup> = 64 \u21d2 k2<\/sup> = 1 \u21d2 k = \u00b11<\/p>\n

Question 88.
\nIf the roots of a quadratic equation ax2<\/sup> + bx + c = 0 are real and equal, then find ‘b2<\/sup>‘.
\nAnswer:
\n4ac<\/p>\n

Question 89.
\n3(x – 4)2<\/sup> – 5(x – 4) = 12, then find ‘x’.
\nAnswer:
\n7 (or) 8\/3.
\nExplanation:
\n3(x – 4)2<\/sup> – 5 (x – 4) = 12
\n3[x2<\/sup> + 16 – 8x] – 5x + 20 \u2014 12
\n3x2<\/sup> + 48 – 24x – 5x + 20 – 12 \u2014 0
\n\u21d2 3x2<\/sup> – 29x + 56 = 0
\n\u21d2 3x2<\/sup> – 21x – 8x + 56 \u2014 0
\n\u21d2 3x (x – 7) – 8 (x – 7) = 0
\n\u21d2 (x – 7) (3x – 8) = 0
\n\u21d2 x = 7 (or) x = \\(\\frac { 8 }{ 3 }\\)<\/p>\n

Question 90.
\nIf a and pare the roots of x2<\/sup> + x + 1 = 0, then find \u03b12<\/sup> + \u03b22<\/sup>.
\nAnswer:
\n\u03b12<\/sup> + \u03b22<\/sup> = – 1.<\/p>\n

Question 91.
\n\\(\\frac{1-\\frac{1}{1+x}}{\\frac{1}{1+x}}\\) is equal to ?
\nAnswer:
\nx<\/p>\n

Question 92.
\nFind sum of the roots of a pure quadratic equation.
\nAnswer:
\n0<\/p>\n

Question 93.
\n\\(\\frac{\\mathbf{x}}{\\mathbf{a}-\\mathbf{b}}=\\frac{\\mathbf{a}}{\\mathbf{x}-\\mathbf{b}}\\) , then find ‘x’.
\nAnswer:
\nb – a (or) – a<\/p>\n

Question 94.
\n\\(\\frac{1}{x+4}-\\frac{1}{x-7}=\\frac{11}{30}\\) x \u2260 -4 x or 7 find x\u2019.
\nAnswer:
\n2 or 1<\/p>\n

Question 95.
\n(1 – 5x) (9x +1) is equal to ?
\nAnswer:
\n1 + 4x – 5x2<\/sup>.<\/p>\n

Question 96.
\nFrom the figure, find \u2019x’.
\n\"AP
\nAnswer:
\n\u00b1 10
\nExplanation:
\nBy Pythagoras theorem,
\nx2<\/sup> = 62<\/sup> + 82<\/sup> = 64 + 36 = 100
\nx = \\(\\sqrt{100}\\) = \u00b1 10<\/p>\n

\"AP<\/p>\n

Question 97.
\nFind the sum of the roots of the equation 3x2<\/sup> – 7x + 11 = 0.
\nAnswer:
\n7\/3<\/p>\n

Question 98.
\nFind the roots of the quadratic equation \\((\\sqrt{5} x-3)(\\sqrt{5} x-3)\\) – 0.
\nAnswer:
\n\\(\\frac{3}{\\sqrt{5}}, \\frac{3}{\\sqrt{5}}\\)<\/p>\n

Question 99.
\nWrite the nature of the roots of the quadratic equation \\(\\sqrt{3} x^{2}-2 x-\\sqrt{3}\\).
\nAnswer:
\nReal and distinct.<\/p>\n

Question 100.
\nIf 5x2<\/sup> – kx + 11 = 0 has root x = 3, then find \u2019k’.
\nAnswer:
\nk = \\(\\frac{56}{3}\\)
\nExplanation:
\n5(3)2<\/sup> – k(3) + 11 = 0
\n\u21d2 45 + 11 – 3k = 0
\n\u21d2 56 – 3k = 0
\n\u21d2 3k = 56 \u21d2 k = \\(\\frac { 56 }{ 3 }\\)<\/p>\n

Question 101.
\nFind the value of ‘p’ for which 4x2<\/sup> – 2px + 7 = 0 has a real roots.
\nAnswer:
\np > 2\\(\\sqrt{7}\\)<\/p>\n

Question 102.
\nIf one root of a quadratic equation is 7 – 7\\(\\sqrt{3}\\) , then find the quadratic equation.
\nAnswer:
\nx2<\/sup> – 14x + 46 = 0.<\/p>\n

\"AP<\/p>\n

Question 103.
\nIf b2<\/sup> – 4ac = 0, then write nature of the roots of the quadratic equation.
\nAnswer:
\nReal and equal.<\/p>\n

Question 104.
\nFind sum of the roots of ax2<\/sup> + bx + c = 0.
\nAnswer:
\n\\(-\\frac{b}{a}\\)<\/p>\n

Question 105.
\nIf the equation x2<\/sup> – kx + 1 = 0 has equal roots, then find the value of ‘k’.
\nAnswer:
\nk = \u00b1 2
\nExplanation:
\nb2<\/sup> – 4ac = (- k)2<\/sup> – 4 . 1 . 1 = 0
\n\u21d2 k2<\/sup> – 4 = 0
\n\u21d2 k2<\/sup> = 4 \u21d2 k = \\(\\sqrt{4}\\) = \u00b1 2.<\/p>\n

Question 106.
\nFind (he product of the roots of the qua-dratic equation \\(\\sqrt{2} \\mathrm{x}^{2}-3 \\mathrm{x}+5 \\sqrt{2}\\) = 0.
\nAnswer:
\n5<\/p>\n

Question 107.
\nWrite the nature of roots of 3x2<\/sup> + 6x – 2 = 0.
\nAnswer:
\nReal and distinct.<\/p>\n

Question 108.
\nIf the sum of the roots of the quadratic equation 3x2<\/sup> + (2k + 1)x – (k + 5) = 0 is equal to the product of the roots, then find the value of k.
\nAnswer:
\n4
\nExplanation:
\nSum of the roots = product of the roots
\n\u21d2 \\(\\frac{-(2 k+1)}{3}=\\frac{-(k+5)}{3}\\)
\n\u21d2 – 2k- 1 = -k – 5 \u21d2 k = 4<\/p>\n

\"AP<\/p>\n

Question 109.
\nFind the product of zeroes in the above equation.
\nAnswer:
\n\\(\\frac{-11}{5}\\)<\/p>\n

Question 110.
\nFind the degree of any quadratic equation.
\nAnswer:
\n2<\/p>\n

Question 111.
\nIn the quadratic equation
\nx2<\/sup> + x – 2 = 0, find the value of a + b + c.
\nAnswer:
\na + b + c = 0.<\/p>\n

Question 112.
\nFind the value of \\(\\left(x+\\frac{1}{x}\\right)^{2}-\\left(y+\\frac{1}{y}\\right)^{2}-\\left(x y-\\frac{1}{x y}\\right) \\cdot\\left(\\frac{x}{y}-\\frac{y}{x}\\right)\\)
\nAnswer:
\n0<\/p>\n

Question 113.
\nForm a quadratic equation from x(2x + 3) = x2<\/sup> + 1.
\nAnswer:
\nx2<\/sup> + 3x – 1 = 0.
\nExplanation:
\n2x2<\/sup> + 3x = x2<\/sup> + 1
\n\u21d2 x2<\/sup> + 3x – 1 = 0<\/p>\n

Question 114.
\n(x – \u03b1) (x – \u03b2) = 0, then find the product.
\nAnswer:
\nx2<\/sup> – (\u03b1 + \u03b2)x + \u03b1\u03b2 = 0.<\/p>\n

Question 115.
\nIf \u03b1 and \u03b2 are die roots of x2<\/sup> – 5x + 6 = 0, then find the value of \u03b1 – \u03b2.
\nAnswer:
\n\u00b1 1.<\/p>\n

Question 116.
\nFor what values of m’ are the roots of the equation mx2<\/sup> + (m + 3)x + 4 = 0 are equal ?
\nAnswer:
\n9 or 1.
\nExplanation:
\n(m + 3)2<\/sup> – 4 . m . 4 = 0
\n\u21d2 (m + 3)2<\/sup> – 16m = 0
\n\u21d2 m2<\/sup> + 9 + 6m- 16m = 0
\n\u21d2 m2<\/sup> – 10m + 9 = 0
\n\u21d2 m2<\/sup> – 9m – m + 9 = 0
\n\u21d2 m(m – 9) – 1 (m – 9) = 0
\n\u2234 m = 9 or 1<\/p>\n

Question 117.
\nFind the roots of 2x2<\/sup> + x – 4 = 0.
\nAnswer:
\nx = \\(\\frac{-1 \\pm \\sqrt{33}}{4}\\)<\/p>\n

Question 118.
\nIf kx (x – 2) + 6 = 0 has equal roots, then find k’.
\nAnswer:
\nk = 6.
\nExplanation:
\nkx2<\/sup> – 2kx + 6 = 0
\n\u21d2 (2k)2<\/sup> – 4 . k . 6 = 0
\n\u21d2 4k2<\/sup> – 24k = 0
\n\u21d2 4k (k – 6) = 0 \u21d2 k = 6<\/p>\n

Question 119.
\nIf ‘2’ is a root of x2<\/sup> + 5x + r = 0, then find ‘r’.
\nAnswer:
\nr = -14<\/p>\n

Question 120.
\n(\u03b1 + \u03b2)2<\/sup> – 2\u03b1\u03b2 is sequal to ……………
\nAnswer:
\n\u03b12<\/sup> + \u03b22<\/sup><\/p>\n

Question 121.
\nFind the value of \\(\\sqrt{\\mathbf{a}+\\sqrt{\\mathbf{a}+\\sqrt{\\mathbf{a + \\ldots \\ldots \\infty}}}}\\)
\nAnswer:
\n\\(\\frac{1+\\sqrt{1+4 a}}{2}\\)<\/p>\n

Question 122.
\nIf the sum of the roots of kx2<\/sup> – 3x + 1 = 0 is \\(\\frac{-4}{3}\\) then find ‘k’.
\nAnswer:
\n\\(\\frac{-9}{4}\\)
\nExplanation:
\n\\(\\frac{3}{\\mathrm{k}}=\\frac{-4}{3} \\Rightarrow \\frac{3 \\times 3}{-4}=\\mathrm{k} \\Rightarrow \\mathrm{k}=\\frac{-9}{4}\\)<\/p>\n

Question 123.
\n\\(\\frac{n(n+1)}{2}\\) = 55, then find ‘n’
\nAnswer:
\n10
\nExplanation:
\n\u21d2 n2<\/sup> + n = 110 = 0
\n\u21d2 n2<\/sup> + n – 110 = 0
\n\u21d2 n = 10<\/p>\n

\"AP<\/p>\n

Question 124.
\nIf ‘\u03b1’ is \u03b2 root of ax2<\/sup> + bx + c = 0, then find a\u03b12<\/sup> + b\u03b1 + c.
\nAnswer:
\n0<\/p>\n

Question 125.
\nIf \u03b1 and \u03b2 are the roots of the quadratic equation 2x2<\/sup> + 3x – 7 = 0, then find \\(\\frac{\\alpha^{2}+\\beta^{2}}{\\alpha \\beta}\\)
\nAnswer:
\n\\(\\frac{-37}{14}\\)<\/p>\n

Question 126.
\nFind the sum of the roots of -7x + 3x2<\/sup> – 1 = 0.
\nAnswer:
\n\\(\\frac{7}{3}\\)<\/p>\n

Question 127.
\nFind the roots of a quadratic equation \\(\\frac{\\mathbf{x}}{\\mathbf{p}}=\\frac{\\mathbf{p}}{\\mathbf{x}}\\)
\nAnswer:
\nx = p
\nExplanation:
\nx2<\/sup> = p2<\/sup> \u21d2 x = p<\/p>\n

Question 128.
\nIf (x – 3) (x + 3) = 16, then find the value of ‘x’.
\nAnswer:
\n\u00b1 5.<\/p>\n

Question 129.
\nWrite the roots of a quadratic equation ax2<\/sup> + bx + c = 0.
\nAnswer:
\nx = \\(\\frac{-b \\pm \\sqrt{b^{2}-4 a c}}{2 a}\\)<\/p>\n

Question 130.
\nFind the sum of the roots of the quadratic equation 5x2<\/sup> + 4\\(\\sqrt{3}\\)x – 11 = 0.
\nAnswer:
\n\\(\\frac{-4 \\sqrt{3}}{5}\\)<\/p>\n

Question 131.
\nIf one root of x2<\/sup> – (p – 1)x + 10 = 0 is 5, then find ‘p’.
\nAnswer:
\n7
\nExplanation:
\n52<\/sup> – (p – 1) 5 + 10 = 0
\n\u21d2 25 + 10 – 5p + 5 = 0
\n\u21d2 35 = 5p \u21d2 p = 7<\/p>\n

Question 132.
\nIf one root of x2 – x + k = 0 is square of other, then find ‘k’.
\nAnswer:
\nk = cube of one root
\nExplanation:
\n\u03b1 = x, \u03b2 = x2<\/sup>
\nProduct of roots = \u03b1\u03b2 = \\(\\frac{\\mathrm{c}}{\\mathrm{a}}\\)
\n\u21d2 x.x2<\/sup> = k \u21d2 k = x3<\/sup>
\nk is cube of the first root.<\/p>\n

Question 133.
\nIf \u03b1, \u03b2 are the roots of x2 – px + q = 0, then find \u03b13<\/sup> + \u03b23<\/sup>.
\nAnswer:
\np3<\/sup> – 3pq<\/p>\n

Question 134.
\nx2<\/sup> + (x + 2)2<\/sup> = 290, then find ‘x’.,
\nAnswer:
\n11 or – 13<\/p>\n

Question 135.
\nFind the value of \\(\\sqrt{\\mathbf{a} \\sqrt{\\mathbf{a} \\sqrt{\\mathbf{a}} \\ldots \\ldots \\infty}}\\)
\nAnswer:
\na<\/p>\n

Question 136.
\nIf \\(\\frac{-7}{3}\\) is a root of 6x2<\/sup> – 13x – 63 = 0, then find other root.
\nAnswer:
\n\\(\\frac{9}{2}\\)<\/p>\n

Question 137.
\nIf b2<\/sup>2 – 4ac < 0, then write nature of the roots of the quadratic equation.
\nAnswer:
\nImaginary roots.<\/p>\n

Choose the correct answer satistying the following statements.<\/p>\n

Question 138.
\nStatement (A) : The equation x2<\/sup> + 3x + 1 = (x – 2)2<\/sup> is a quadratic equation.
\nStatement (B) : Any equation of the form ax2 + bx + c = 0 where a \u00b1 0, is called a quadratic equation.
\ni) Both A and B are true.
\nii) A is true, B is false.
\niii) A is false, B is true.
\niv) Both A and B are false.
\nAnswer:
\n(iii)
\nExplanation:
\nWe have, x2<\/sup> + 3x + 1 = (x – 2)2<\/sup>
\n\u21d2 x2<\/sup> + 3x + 1 = x2<\/sup> – 4x + 4
\n\u21d2 7x – 3 = 0, it is not of the form ax2<\/sup> + bx + c = 0
\nSo, A is incorrect but B is correct.
\nHence (iii) is the correct option.<\/p>\n

\"AP<\/p>\n

Question 139.
\nStatement (A) : The roots of the qua-dratic equation x2<\/sup> + 2x + 2 = 0 are imaginary.
\nStatement (B) : If discriminant D = b2<\/sup> – 4ac < 0, then the roots of quadratic equation ax2<\/sup> + bx + c = 0 are imaginary.
\ni) Both A and B are true.
\nii) A is true, B is false.
\niii) A is false, B is true.
\niv) Both A and B are false.
\nAnswer:
\n(i)
\nExplanation:
\nx2<\/sup> + 2x + 2 = 0
\n\u2234 Discriminant, D = b2<\/sup> – 4ac
\n= (2)2<\/sup> – 4 x 1 x 2
\n= 4 – 8 = -4 < 0
\n\u2234 Roots are imaginary.
\nSo, both A and B are correct and B explains Answer: Hence, (i) is the correct option.<\/p>\n

Question 140.
\nStatement (A) : The value of k = 2, if one root of the quadratic equation
\n6x2<\/sup> – x – k = 0 is \\(\\frac{2}{3}\\)
\nStatement (B) : The quadratic equation ax2<\/sup> + bx + c = 0, a \u2260 0 has two roots.
\ni) Both A and B are true.
\nii) A is true, B is false.
\niii) A is false, B is true.
\niv) Both A and B are false.
\nAnswer:
\n(i)
\nExplanation:
\nAs one root is \\(\\frac{2}{3}\\) \u21d2 x = \\(\\frac{2}{3}\\)
\n\"AP
\nSo, both A and B are correct but B does not explain Answer:
\n\u2234 Hence, (i) is the correct option.<\/p>\n

Question 141.
\nStatement (A) : The equation 8x2<\/sup> + 3kx + 2 = 0 has equal roots, then the value of k is \u00b1 \\(\\frac{8}{3}\\).
\nStatement (B) : The equation ax2<\/sup> + bx + c = 0 has equal roots if D = b2<\/sup> – 4ac = 0.
\ni) Both A and B are true.
\nii) A is true, B is false.
\niii) A is false, B is true.
\niv) Both A and B are false.
\nAnswer:
\n(i)
\nExplanation:
\n8x2<\/sup> + 3kx + 2 = 0
\n\u2234 Discriminant, D = b2<\/sup> – 4ac
\n= (3k)2<\/sup> – 4 x 8 x 2
\n= 9k2<\/sup> – 64
\nFor equal roots, D = 0
\n\u21d2 9k2<\/sup> – 64 = 0
\n\u21d2 9k2<\/sup> = 64
\n\u21d2 k2<\/sup> = \\(\\frac { 64 }{ 9 }\\)
\n\u21d2 9k2<\/sup> = \u00b1\\(\\frac { 8 }{ 3 }\\)
\nSo, A and B both correct and B explains Answer: Hence, (i) is the correct option.<\/p>\n

Question 142.
\nStatement (A) : The values of x are \\(\\frac{-a}{2}\\), a for a quadratic equation 2x2<\/sup> + ax – a2<\/sup> = 0.
\nStatement (B) : For quadratic equation ax2<\/sup> + bx + c = 0.
\nx = \\(\\frac{-b \\pm \\sqrt{b^{2}-4 a c}}{2 a}\\)
\ni) Both A and B are true.
\nii) A is true, B is false.
\niii) A is false, B is true.
\niv) Both A and B are false.
\nAnswer:
\n(iii)
\nExplanation:
\n2x2<\/sup> + ax – a2<\/sup> =0
\n\"AP
\nSo, A is incorrect but B is correct. Hence, (iii) is the correct option.<\/p>\n

Question 143.
\nStatement (A) : The equation (x – p) (x – r) + \u03bb(x – q) (x – s) = 0, p < q < r < s, has non-real roots if \u03bb > 0.
\nStatement (B) : The equation ax2<\/sup> + bx + c = 0, a, b,c \u2208 R, has non-real roots if b2<\/sup> – 4ac < 0.
\ni) Both A and B are true.
\nii) A is true, B is false.
\niii) A is false, B is true.
\niv) Both A and B are false.
\nAnswer:
\n(iii)
\nExplanation:
\nStatement (A):
\nLet f(x) = (x – p) (x – r) + \u03bb(x – q) (x- s)
\nf(p) = \u03bb(p – q) (p – s)
\nf(q) = (q – p) (q – r)
\nf(s) = (s – p) (s – r)
\nf(r) = \u03bb(r – q) (r – s)
\nIf \u03bb > 0, then f(p) > 0, f(q) < 0, f(r) < 0 and f(s) > 0.
\n\u21d2 f(x) = 0 has one real root between p and q and other real root between r and s.
\nStatement – B is obviously true. Option (iii) is correct.<\/p>\n

Question 144.
\nStatement (A) : If roots of the equation x2<\/sup> – bx + c = 0 are two consecutive integers, then b2<\/sup> – 4c = 1.
\nStatement (B) : If a, b, c are odd integer, then the roots of the equation 4abc x2<\/sup> + (b2<\/sup> – 4ac)x – b = 0 are real and distinct.
\ni) Both A and B are true.
\nii) A is true, B is false.
\niii) A is false, B is true.
\niv) Both A and B are false.
\nAnswer:
\n(ii)
\nExplanation:
\nStatement (A) : Given equation , x2<\/sup> – bx + c = 0.
\nLet \u03b1, \u03b2 be two roots such that |\u03b1 – \u03b2| = 1. .
\n\u21d2 (\u03b1 + \u03b2)2<\/sup> – 4\u03b1\u03b2 = 1.
\n\u21d2 b2<\/sup> – 4c = 1
\nStatement (B): Given equation
\n4abc x2<\/sup> + (b2<\/sup> – 4ac) x – b = 0
\nD = (b2<\/sup> – 4ac)2<\/sup> + 16 ab2<\/sup> c
\nD = (b2<\/sup> – 4ac)2<\/sup> > 0
\nHence roots are real and unequal. Option (ii) is correct.<\/p>\n

Question 145.
\nStatement (A) : If 1 \u2264 a \u2264 2, then \\(\\sqrt{a+2 \\sqrt{a-1}}+\\sqrt{a-2 \\sqrt{a-1}}=2\\)
\nStatement (B) : If 1 \u2264 a \u2264 2, then (a – 1) > 1.
\ni) Both A and B are true.
\nii) A is true, B is false.
\niii) A is false, B is true.
\niv) Both A and B are false.
\nAnswer:
\n(ii)
\nExplanation:
\nIf 1 \u2264 a \u2264 2 \u21d2 0 \u2264 a- 1 \u2264 1
\n\"AP
\nStatement – A is true.
\nStatement – B is false.
\nOption – (ii) is correct.<\/p>\n

\"AP<\/p>\n

Question 146.
\nStatement (A): If one root is \\(\\sqrt{3}-\\sqrt{2}\\), then the equation of lowest degree with rational coefficients x4<\/sup> – 10x2<\/sup> + 1 = 0.
\nStatement (B): For a polynomial equa-tion with rational coefficient irrational roots occurs in pairs.
\ni) Both A and B are true.
\nii) A is true, B is false.
\niii) A is false, B is true.
\niv) Both A and B are false.
\nAnswer:
\n(i)
\nExplanation:
\nx = \\(\\sqrt{3}-\\sqrt{2}\\), x2<\/sup> = 5 – 2\\(\\sqrt{6}\\)
\n(x2<\/sup> – 5)2<\/sup> = 24
\nx4<\/sup> – 10x2<\/sup> + 25 = 24
\nx4<\/sup> – 10x2<\/sup> + 1 = 0
\nFor polynomial equation with rational coefficients irrational roots occurs in pairs.
\nOption (i) is correct.<\/p>\n

Question 147.
\nStatement (A): Degree of the polynomial 5x2<\/sup> + 3x + 4 is 2.
\nStatement (B) : The degree of a poly-nomial of one variable is the highest value of the exponent of the variable.
\ni) Both A and B are true.
\nii) A is true, B is false.
\niii) A is false, B is true.
\niv) Both A and B are false.
\nAnswer:
\n(i)<\/p>\n

Read the below passages and answer to the following questions.<\/p>\n

Let us consider a quadratic equation x2<\/sup> + 3ax + 2a2<\/sup> = 0.
\nIf the above equation has roots \u03b1,\u03b2 and it is given that \u03b12<\/sup> + \u03b22<\/sup> = 5.<\/p>\n

Question 148.
\nFind value of ‘a’.
\nAnswer:
\n\u00b11.
\nExplanation:
\n\u03b1 + \u03b2 = – 3a; \u03b1\u03b2 = 2a2<\/sup>
\na2<\/sup> + p2<\/sup> = 5
\n\u21d2 (\u03b1 + \u03b2)2<\/sup> – 2\u03b1\u03b2 = 5
\n\u21d2 (- 3a)2<\/sup> – 2(2a2<\/sup>) = 5
\n\u21d2 9a2<\/sup> – 4a2<\/sup> = 5
\n\u21d2 5a2<\/sup> = 5 \u21d2 a = \u00b1 1<\/p>\n

Question 149.
\nFind value of ‘D’ for the above qua-dratic equation.
\nAnswer:
\nD > 0.
\nExplanation:
\nD = (3a)2<\/sup> – 4(2a2<\/sup>)
\n= 9a2<\/sup> – 8a2<\/sup> = a2<\/sup> = 1 > 0<\/p>\n

Question 150.
\nFind the product of roots.
\nAnswer:
\n2
\nExplanation:
\n\u03b1\u03b2 = 2a2<\/sup> = 2(1) = 2<\/p>\n

Let us consider a quadratic equation x2<\/sup> + \u03bbx + \u03bb + 1.25 = O, where \u03bb is a constant. The value of A such that the above quadratic equation has<\/p>\n

Question 151.
\nTwo distinct roots.
\nAnswer:
\n\u03bb > 5 or \u03bb < – 1.
\nExplanation:
\nThe equation has two distinct roots if b2<\/sup> – 4ac > 0.
\n\u2234 (\u03bb – 5)(\u03bb + 1) > 0
\n\u21d2 Either \u03bb – 5 > 0 (or) \u03bb + 1 > 0
\n\u21d2 \u03bb > 5 (or) \u03bb > -1
\n\u2234 \u03bb > 5
\n\u21d2 \u03bb – 5 <0 (or) \u03bb + 1 < 0
\n\u21d2 \u03bb < 5 (or) \u03bb < – 1
\n\u2234 \u03bb < -1 Hence the given equation has two dis-tinct roots for \u03bb > 5 (or) \u03bb < – 1<\/p>\n

Question 152.
\nTwo coincident roots.
\nAnswer:
\n\u03bb = 5 or \u03bb = -1.
\nExplanation:
\nThe equation has two coincident roots if b2<\/sup> – 4ac = 0
\n\u21d2 (\u03bb – 5) (\u03bb + 1) = 0
\n\u21d2 Either \u03bb – 5 = 0 (or) \u03bb = 5
\n\u21d2 \u03bb + 1 = 0
\n\u21d2 \u03bb = – 1
\n\u21d2 \u03bb = 5 or – 1
\nHence the given equation has coincident roots for \u03bb = 5 or – 1.<\/p>\n

The area of a rectangular plot is 528 m2<\/sup>. The length of the plot is one more than twice its breadth.<\/p>\n

Question 153.
\nWhich mathematical concept is used to find area of above plot ?
\nAnswer:
\nQuadratic equation.<\/p>\n

Question 154.
\nWrite the breadth and length of above given plot.
\nAnswer:
\nLet breadth = x m, length = 2x + 1 m.<\/p>\n

Question 155.
\nWrite the equation of area of above given plot.
\nAnswer:
\nArea = length x breadth
\n= x(2x + 1) – 2x2<\/sup> + x = 528 m2<\/sup>.<\/p>\n

The hypotenuse of a right triangle is 25 cm. We know that the difference in lengthof the other two sides is 5 cm.<\/p>\n

Question 156.
\nWrite the lengths of smaller and larger sides.
\nAnswer:
\nSmaller side = x m
\nLarger side = (x + 5) cm.<\/p>\n

Question 157.
\nWrite the hypotenuse of the triangle.
\nAnswer:
\nx2<\/sup> + (x + 5)2<\/sup> = (25)2<\/sup>
\ni.e., x2<\/sup> + 5x – 300 = 0<\/p>\n

\"AP<\/p>\n

Question 158.
\nWhich mathematical concept is used to find out the values of dimensions ?
\nAnswer:
\nQuadratic equations.<\/p>\n

Question 159.
\nColumn -II give roots of quadratic equations given in column – I, match them correctly.
\n\"AP
\nAnswer:
\nA – (iv), B – (ii).<\/p>\n

Question 160.
\nColumn – II give roots of quadratic equations given in column -1, match them correctly.
\n\"AP
\nAnswer:
\nA – (i), B – (iii).<\/p>\n

Question 161.
\nWrite the correct matching.
\n\"AP
\nAnswer:
\nA – (ii), B – (iv).<\/p>\n

Question 162.
\nWrite die correct matching.
\n\"AP
\nAnswer:
\nA – (iii), B – (i).<\/p>\n

Question 163.
\nColumn – II give pair at two numbers for solution to problems given in column -I. Match them correctly.
\n\"AP
\nAnswer:
\nA – (iv), B – (ii).<\/p>\n

Question 164.
\nColumn – II give pair at two numbers for solution to problems given in column -I.
\nMatch them correctly.
\n\"AP
\nAnswer:
\nA – (i), B – (iii).<\/p>\n

Question 165.
\nD is the discriminait of the quadratic equation ax2<\/sup> + bx + e = 0.
\n\"AP
\nAnswer:
\nA – (ii), B – (i).<\/p>\n

Question 166.
\nD Is the discriminant of the quadratic equation ax2<\/sup> + bx + c = O.
\n\"AP
\nAnswer:
\nA – (ii), B – (i).<\/p>\n

Question 167.
\nWrite a quadratic equation with roots 3 and 4.
\nAnswer:
\nx2<\/sup> – 7x + 12 = 0<\/p>\n

Question 168.
\nDraw the rough graph of the quadratic equation ax2<\/sup> + bx + c = 0, when b2<\/sup> – 4ac < 0.
\nAnswer:
\n\"AP<\/p>\n","protected":false},"excerpt":{"rendered":"

Practice the AP 10th Class Maths Bits with Answers Chapter 5 Quadratic Equations on a regular basis so that you can attempt exams with utmost confidence. AP SSC 10th Class Maths Bits 5th Lesson Quadratic Equations with Answers Question 1. If x2 – px + q = 0(p,q\u2208R and p \u2260 0, q \u2260 0) … Read more<\/a><\/p>\n","protected":false},"author":5,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"spay_email":""},"categories":[2],"tags":[],"jetpack_featured_media_url":"","_links":{"self":[{"href":"https:\/\/apboardsolutions.in\/wp-json\/wp\/v2\/posts\/13843"}],"collection":[{"href":"https:\/\/apboardsolutions.in\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/apboardsolutions.in\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/apboardsolutions.in\/wp-json\/wp\/v2\/users\/5"}],"replies":[{"embeddable":true,"href":"https:\/\/apboardsolutions.in\/wp-json\/wp\/v2\/comments?post=13843"}],"version-history":[{"count":1,"href":"https:\/\/apboardsolutions.in\/wp-json\/wp\/v2\/posts\/13843\/revisions"}],"predecessor-version":[{"id":41313,"href":"https:\/\/apboardsolutions.in\/wp-json\/wp\/v2\/posts\/13843\/revisions\/41313"}],"wp:attachment":[{"href":"https:\/\/apboardsolutions.in\/wp-json\/wp\/v2\/media?parent=13843"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/apboardsolutions.in\/wp-json\/wp\/v2\/categories?post=13843"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/apboardsolutions.in\/wp-json\/wp\/v2\/tags?post=13843"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}