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InterviewSolution
This section includes InterviewSolutions, each offering curated multiple-choice questions to sharpen your knowledge and support exam preparation. Choose a topic below to get started.
| 51. |
Given that 2 is a root of the equation `3x^2-p(x+1)=0` and that the equation `px^2-qx+9=0` has equal roots, find the values of p and q |
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Answer» 1)`3x^2-p(x+1)=0` x=2 is a root `3(2)^2-p(2+1)=0` `12-3p=0` `p=4` `px^2-qx+9=0` `4x^2-qx+9=0` `D=0` `b^2-4ac=0` `q^2=4*4*9` `q=2*2*3` `q=pm12`. |
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| 52. |
If -4 is a root of the equation `x^2+2x+4p=0` then find the value of `k` for which the equation `x^2+px(1+3k)+7(3+2k)=0` has equal roots |
| Answer» Correct Answer - `k=2" or "k=(-10)/(9)` | |
| 53. |
(i) Find the values of k for which the quadratic equation `(3k+1)x^(2)+2(k+1)x+1=0` has real and equal roots. (ii) Find the value of k for which the equation `x^(2)+k(2x+k-1)+2=0` has real and equal roots. |
| Answer» Correct Answer - (i) `k=0" or "k=1" "(ii)" "k=2` | |
| 54. |
Solve each of the following equatins : `9x^(2)+6x+1=0` |
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Answer» Correct Answer - `x=(-1)/(3)x,=(-1)/(3)` `9x^(2)+6x+1=0implies(3x+1)^(2)=0implies3x+1=0.` |
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| 55. |
If the sum of the roots of the equation `kx^(2)+2x+3k=0` is equal to their product then the value of k isA. `(1)/(3)`B. `(-1)/(3)`C. `(2)/(3)`D. `(-2)/(3)` |
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Answer» Correct Answer - D Sum of roots `=(-2)/(k)` and product of roots `=(3k)/(k)=3.` `(-2)/(k)=3impliesk=(-2)/(3)` |
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| 56. |
Find the roots of the quadratic equations by using the quadratic formula in each of the following `5x^(2)+13x+8=0` |
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Answer» Correct Answer - `x=-1" or "x=(-8)/(5)` `5x^(2)+13x+8=0implies5x^(2)+5x+8x+8=0.` |
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| 57. |
If one root of `5x^(2)+13x+k=0` be the reciprocal of the other root then the value of k is |
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Answer» Correct Answer - D Let the roots be `alpha" and "(1)/(alpha).` Then, product of roots `=(alphaxx(1)/(alpha)=1." So, "(k)/(5)=1impliesk=5.` |
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| 58. |
Solve the following problems `:` (i) The sume of a natural number and its reciprocal is `( 50)/( 7)` . Find the number. (ii) In a two-digit , the digit at the units place is equal to the square of the digit at tens place. If 18 is added to the number, the digits get interchanged. Find the number. (iii) If the speed of a car is decreased by 8 km `//` h , it takes 1 hour more to cover a distance of 240 km. Find the original speed of the car. |
| Answer» (i) The number is 7 (ii) The number is 24 (iii) The original speed of the car `: ` 48 km `//` h | |
| 59. |
The sum of a number and its reciprocal is `2(1)/(30)`. Find the numbers. |
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Answer» Correct Answer - `(5)/(6)"or"(6)/(5)` Let the required number be x. Then, `x+(1)/(x)=(61)/(30).` `:." "30x^(2)-61x+30=0implies30x^(2)-36x-25x+30=0.` |
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| 60. |
A two digit number issuch that the product of the digits is 14. When 45 isadded to the number, then the digits are reversed. Find the number. |
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Answer» Correct Answer - 27 Let the tens digit be x and units digit be y. Then, `xy=14" and "10x+y+4=10y+x` `implies" "xy=14" and "9(y-x)=45impliesxy=14" and "y-x=5` `implies" "x(x+5)=14" and "y=(5+x).` |
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| 61. |
The roots of the equation `ax^(2)+bx+c=0` will be reciprocal of each other ifA. a=bB. b=cC. c=aD. None of these |
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Answer» Correct Answer - C Product of roots `=(c)/(a)." Also,"(alphaxx(1)/(alpha))=1.` `:." "(c)/(a)=1impliesc=a.` |
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| 62. |
The denominator of a positive fraction is one more than twice the numerator. If the sum of the fraction and its reciprocal is 2.9, find the fraction, |
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Answer» Correct Answer - `(2)/(5)` Let the required fraction be `(a)/((a+3))`. Then, `(a)/((a+3))+((a+3))/(a)=(29)/(10)implies10[a^(2)+(a+3)^(2)]=29a(a+3)` `implies9a^(2)+27a-90=0impliesa^(2)+3a-10=0.` |
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| 63. |
The numerator of a fraction is 3 less than its denominator.If 11 is added to the denominator,the fraction is decreased by `1/15` Find the fraction |
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Answer» Correct Answer - `(2)/(5)` Let the required fraction in simplest from be `(x-3)/(x)`, Then, `(x-3)/(x)-(x-3)/(x+1)=(1)/(15)implies(x-3)[(1)/(x)-(1)/(x+1)]=(1)/(15)` `:." "15(x-3)=x(x+1)impliesx^(2)-14x+45=0impliesx=5" or "x=9.` So, the required fraction is `(2)/(5)." "["neglecting "(6)/(9)]` |
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| 64. |
The denominator of a fraction is one more than twice the numerator. Ifthe sum of the fraction and its reciprocal is `2(16)/(21),`find the fraction. |
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Answer» Let the numerator of the required fraction be x. Then, its denominator `=(2x+1).` `:." ""fraction "=(x)/((2x+1))" and its reciprocal "=((2x+1))/(x).` `:." "(x)/((2x+1))+((2x+1))/(x)=(58)/(21)` `implies" "21xx[x^(2)+(2x+1)^(2)]=58x(2x+1)` `implies" "21xx[5x^(2)+4x+1]=116x^(2)+58x` `implies" "11x^(2)-26x-21=0` `implies" "11x^(2)-33x+7x-21=0implies11x(x-3)+7(x-3)=0` `implies" "(x-3)(11x+7)=0impliesx-3=0" or "11x+7=0` `implies" "x=3" or "x=(-7)/(11)` `:." "x=3" "[because" numerator cannot be a negative fraction"].` `:." ""required fraction "=(x)/((2x+1))=(3)/(7).` |
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| 65. |
Which of the following is a quadratic equation?A. `x^(2)-3sqrt(x)+2=0`B. `x+(1)/(x)=x^(2)`C. `x^(2)+(1)/(x^(2))=5`D. `2x^(2)-5x=(x-1)^(2)` |
| Answer» Correct Answer - D | |
| 66. |
A train travels a distance of 300km at constant speed. If the speed of the train is increased by 5 km/kr, the journey would have taken 2 hours less. Find the original speed of the train. |
| Answer» Correct Answer - 25 km/hr | |
| 67. |
A train travels a distance of 300km at constant speed . If the speed of the train is increased by 5km/h; the journey would have taken 2 hr less. Find the original speed of the train. |
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Answer» Let the usual (original) speed of train x=km/hr and the increased speed =(x+5)km/hr Usual distance covered =300 km `because` Time taken in original speed `=(300)/(x)` and time taken in increased speed `=(300)/(x+5)` But the difference in time in both cases = 2hrs `because(300)/(x)=(300)/(x+5)=2implies(300(x+5-x))/(x(x+5))=2` `impliesx^(2)+5x=750impliesx^(2)+5x-750=0` `implies(x+30)(x-25)=0` `impliesx=-30orx=25` `becausex=25km//hr` (rejecting x=30, because speed cannot be neagtive) `because` The original speed =25 km/hr |
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| 68. |
A train travels 180 km at a uniform speed. If the speed had been 9 km/hour more, it would have taken 1 hour less for the same journey. Find the speed of the train. |
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Answer» Correct Answer - 36 km/hr Let the speed be x km/hr. Then, `(180)/(x)-(180)/((x+9))=1` `impliesx^(2)+9x-1620=0impliesx^(2)+45x-36x-1620=0.` |
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| 69. |
300 apples are distributed equally among a certain number of students. Had three been 10 more students, each would have received one apple less. Find the number of students. |
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Answer» Correct Answer - 50 Let the total number of students be x. Then, `(300)/(x)-(300)/((x+10))=1impliesx^(2)+10x-3000=0` `implies(x-50)(x+60)=0impliesx=50.` |
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| 70. |
A bookseller buys a number of books for Rs. 1760. If he had bought 4 more books for the same amount, each book would have cost Rs. 22 less. How many books did he buy? |
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Answer» Let the bookseller buy x books for Rs. 1760. Then, cost of each book `= Rs. (1760)/(x).` Again, cost of `(x+4)` books `= Rs. 1760` `:." ""cost of each book, now "= Rs. (1760)/((x+4)).` `:." "(1760)/(x)-(1760)/((x+4))=22` `implies" "(1)/(x)-(1)/((x+4))=(22)/(1760)implies((x+4)-x)/(x(x+4))=(1)/(80)` `implies" "(4)/((x^(2)+4x))=(1)/(80)impliesx^(2)+4x=320` `implies" "x^(2)+4x-320=0impliesx^(2)+20x-16x-320=0` `implies" "x(x+20)-16(x+20)=0implies(x+20)(x-16)=0` `implies" "x+20=0" or "x-16=0impliesx=-20" or "x=16` `implies" "x=16" "[because" number of books cannot be negative"]` Hence, the bookseller bought 16 books. |
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| 71. |
The age of a father is twice the square of the age of his son. Eight years hence, the age of his father will be 4 years more than 3 times the age of the son. Find their present ages. |
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Answer» Let, the present age of the son be x years. Hence, the age of father `=2x^(2)` 8 years hence, the age of son =(x+8) years Accoding to given statement `2x^(2)+8=3(x+8)+4` `implies2x^(2)+8=3x+24+4` `implies2x^(2)-3x-20=0` `implies2x^(2)-8x+5x-20=0` `implies2x(x-4)+5(x-4)=0` `implies(2x+5)(x-4)=0` `impliesx=(-5)/(2)and x=4` `impliesx=4" "(because"age cannot be negative")` Therefore, present age of son is 4 years and present age of father is `2xx4^(2)=32` years. |
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| 72. |
A takes 6 hours less than B to complete a work. If together they complete the work in 13 hours 20 minutes, find how much time will B alone take to complete the work. |
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Answer» Let, B alone completes the work in x hours, then A alone will complete the work in (x-6) hours. `because(1)/(x-6)+(1)/(x)=(3)/(40)" "(13hr+20min=(40)/(3)hrs.)` `implies(x+x-6)/((x-6)x)=(3)/(40)` `implies3x^(2)-18x=80x-240` `implies3x^(2)-98x+240=0` `implies3x^(2)-90x-8x+240=0` `3x(x-30)-8(x-30)=0` `implies(x-30)(3x-8)=0` `impliesx=30orx=(8)/(3)` `impliesx=30" "(ifx=(8)/(3),"then x-6 in negative")` `because` B alone will take 30 hours to complete the work. |
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| 73. |
Two pipes runningtogether can fill a tank in `11 1/9`minutes. If one pipe takes 5minutes more than the other to fill the tank separately, find the time inwhich each pipe would fill the tank separately. |
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Answer» Correct Answer - 20 min, 25 min Let the faster pipe take x min to fill it. Then, the other takes `(x+5)` min. `:." "(1)/(x)+(1)/((x+5))=(9)/(100)implies(100(2x+5)=9(x^(2)+5x)` `implies" "9x^(2)-155x-500=0implies9x^(2)-180x+25x-500=0.` HINT Quadratic formula may be used. |
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| 74. |
Two taps running together can fill a tank in `3(1/13)` hours. If one tap takes3 hours more than the other to fill the tank, then how much time willeach tap take to fill the tank ? |
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Answer» Correct Answer - 5 hours, 8 hours Let the faster pipe take x hr to fill it. Then, the other takes `(x+3)` hr. `:." "(1)/(x)+(1)/((x+3))=(13)/(40)implies((x+3)+x)/(x(x+3))=(13)/(40)` `implies13x^(2)-41x-120=0` `impliesx=(41+-sqrt(1681+6240))/(26)=(41+-sqrt(7921))/(26)=((41+-89))/(26)=x=5.` |
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| 75. |
Find the roots of each of the following equations, if they exist, by applying the quadratic formula: `x^(2)-6x+4=0` |
| Answer» Correct Answer - `x=(3+sqrt(5))" or "x=(3-sqrt(5))` | |
| 76. |
Solve each of the following quadratic equations: `4sqrt(6)x^(2)-13x-2sqrt(6)=0` |
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Answer» Correct Answer - `x=(2sqrt(2))/(sqrt(3))" or "x=(-sqrt(3))/(4sqrt(2))` `4sqrt(6)x^(2)-13x-2sqrt(6)=0implies4sqrt(6)x^(2)-16x+3x-2sqrt(6)=0` `" "implies4sqrt(2)x(sqrt(3)x-2sqrt(2))+sqrt(3)(sqrt(3)x-2sqrt(2))=0.` |
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| 77. |
Solve each of the following quadratic equations: `(1)/(x-2)+(2)/(x-1)=(6)/(x),xne0,1,2` |
| Answer» Correct Answer - `x=3" or "x=(4)/(3)` | |
| 78. |
Solve the following quadratic equation `3q^(2) = 2q+ 8 ` using formula method `:` |
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Answer» Correct Answer - `2, - ( 4)/(3)` are the roots of the given quadratic equation. `3q^(2) = 2q+8` `:. 3q^(2) - 2q-8 = 0 ` Comparing with the standard for `ax^(2) + bx + c = 0 , a = 3, b= -2, c - 8` `:. b^(2) - 4ac = ( -2)^(2) - 4(3)( -8) = 4 + 96 = 100` `:. Sqrt( b^(2) - 4ac ) = 10 ` Now, `q = ( -b+- sqrt(b^(2 ) - 4ac))/(2a) = (-(-2a) +- (10))/(2 xx 3) = ( 2+ 10)/(6)` `:. q = ( 2+ 10)/( 6) ` or `q = ( 2-10)/( 6)` `:. q = ( 12)/( 6)` or ` q = - ( 8)/(6)` `:. q = 2 ` or `q = - ( 4)/( 3) ` |
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| 79. |
Let `f(x)=3x^(2)-5x-1`. Then solve f(x)=0 by (i) factroing the quadratic (ii) using th quadratic formula (iii) completing the square and then rewrite f(x) in the form `A(x+-B)^(2)+-C`. |
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Answer» `f(x)=3x^(2)-5x-1` :.f(x)=0 `implies3x^(2)-5x-1=0` (i) The given quadratic equation cannot be fully factorised using real integers. So it is better to solve this equation by any other method. (ii) `3x^(2)-5x-1=0` Compare it with `ax^(2)+bx+c=0`, we get a=3, b=-5, c=-1 :. Let two roots of this equation are `alpha=(-b+sqrt(b^(2)-4ac))/(2a)andbeta=(-b-sqrt(b^(2)-4ac))/(2a)` `(-(5)+sqrt((-5)^(2)-4(3)(-1)))/(2xx3)` `=(5+sqrt37)/(6)` `=(-(-5)-sqrt((-5)^(2)-4(3)(-1)))/(2xx3)` `beta=(5-sqrt(37))/(6)` :. Two values of x are `(5+sqrt(37))/(6)and(5-sqrt37)/(6)` (iii) `3x^(2)-5x-1=0` `impliesx^(2)-(5)/(3)x(1)/(3)=0" "("dividing both sides by 3")` `impliesx^(2)-(5)/(3)x+.....=(1)/(3)+.....` `impliesx^(2)-(5)/(3)x+((5)/(6))^(2)=(1)/(3)+((5)/(6))^(2)" "["adding"(("coeff. of x")^(2)/(2))"on both sides"]` `implies(x-(5)/(6))^(2)=(1)/(3)+(25)/(36)` `implies(x-(5)/(6))^(2)=(12+25)/(36)` `implies(x-(5)/(6))^(2)=((sqrt37)/(6))^(2)` `impliesx-(5)/(6)=+-(sqrt37)/(6)` `:.x=(5)/(6)+(sqrt37)/(6)=(5+sqrt37)/(6),x=(5)/(6)-(sqrt37)/(6)=(5-sqrt37)/(6)` :. Two values of x are `(5+sqrt37)/(6)and(5-sqrt(37))/(6)`. Now, `f(x)=3x^(2)-5x-1=3(x^(2)-(5)/(3)x)-1` `=3(x^(2)-(5)/(3)x+(25)/(36)-(25)/(36))-1` `=3(x-(5)/(6))^(2)-(25)/(12)-1=3(x-(5)/(6))^(2)-(37)/(12)` which is of the form `A(x-B)^(2)-C`, where A=3, `B=(5)/(6),C=(37)/(12)`. |
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| 80. |
Solve each of the following quadratic equations: `x^(2)+6x+5=0` |
| Answer» Correct Answer - `x=-5" or "x=-1` | |
| 81. |
Solve each of the following quadratic equations: `sqrt(7)x^(2)-6x-13sqrt(7)=0` |
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Answer» Correct Answer - `x=(13)/(sqrt(7))" or "x=-sqrt(7)` `sqrt(7)x^(2)-6x-13sqrt(7)=0impliessqrt(7)x^(2)-13x+7x-13sqrt(7)=0` `" "impliesx(sqrt(7)x-13)+sqrt(7)(sqrt(7)x-13)=0.` |
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| 82. |
Find the roots of each of the following equations, if they exist, by applying the quadratic formula: `2x^(2)+5sqrt(3)x+6=0` |
| Answer» Correct Answer - `x=(-sqrt(3))/(2)" or "x=-2sqrt(3)` | |
| 83. |
Solve each of the following quadratic equations: `(x-1)/(x-2)+(x-3)/(x-4)=3(1)/(3),xne2,4` |
| Answer» Correct Answer - `x=5" or "x=(5)/(2)` | |
| 84. |
Find the roots of each of the following equations, if they exist, by applying the quadratic formula: `2x^(2)+ax-a^(2)=0` |
| Answer» Correct Answer - `x=(a)/(2)" or "x=-a` | |
| 85. |
Solve each of the following quadratic equations: ` (3x-4)/(7)+(7)/(3x-4)=(5)/(2),xne(4)/(3)` |
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Answer» Correct Answer - `x=6" or "x=2(1)/(2)` Put `(3x-4)/(7)=y.` Then, `y+(1)/(y)=(5)/(2)implies2y^(2)-5y+2=0.` |
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| 86. |
Solve the x by quadratic formula `p^2x^2+(p^2-q^2)x-q^2=0` |
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Answer» The given equation is `p^(2)x^(2)+(p^(2)-q^(2))x-q^(2)=0.` Comparing in with `ax^(2)+bx+c=0,` we get `a=p^(2),b=(p^(2)-q^(2))" and "c=-q^(2).` `:." "D=(b^(2)-4ac)=(p^(2)-q^(2))^(2)-4xxp^(2)xx(-q^(2))` `=(p^(2)-q^(2))^(2)+4p^(2)q^(2)=(p^(2)+q^(2))^(2)gt0.` So, the given equation has real roots. Now, `sqrt(D)=(p^(2)+q^(2)).` `:." "alpha=(-b+sqrt(D))/(2a)=(-(p^(2)-q^(2))+(p^(2)+q^(2)))/(2p^(2))=(2p^(2))/(2p^(2))=(q^(2))/(p^(2)),` `beta=(-b-sqrt(D))/(2a)=(-(p^(2)-q^(2))-(p^(2)-q^(2)))/(2p^(2))=(-2p^(2))/(2p^(2))=-1.` Hence, `(q^(2))/(p^(2))` and -1 are the roots of the given equation. |
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| 87. |
Solve the following quations by using qardratic formula: `abx^(2)+(b^(2)-ac)x-bc=0` |
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Answer» The given equation is `abx^(2)+(b^(2)-ac)x-bc=0.` This is of the form `Ax^(2)+Bx+C=0,` where `A=ab,B=(b^(2)-ac)" and "C=-bc.` `:." "D=(B^(2)-4AC)=(b^(2)-ac)^(2)+4ab^(2)c` `=b^(4)+a^(2)c^(2)-2ab^(2)c+4ab^(2)c` `=b^(4)+a^(2)c^(2)+2ab^(2)c=(b^(2)+ac)^(2)gt0..` So, the given equation has real roots. Now, `sqrt(D)=(b^(2)+ac).` `:." "alpha=(-B+sqrt(D))/(2A)=(-(b^(2)-ac)+(b^(2)+ac))/(2ab)=(2ac)/(2ab)=(c)/(b),` `" "beta=(-B-sqrt(D))/(2A)=(-(b^(2)-ac)+(b^(2)+ac))/(2ab)=(-2b^(2))/(2ab)=(-b)/(a).` Hence,`(c)/(b)` and `(-b)/(a)` are the roots of the given equation. |
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| 88. |
Solve each of the following quadratic equations: `x^(2)+6x-(a^(2)+2a-8)=0` |
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Answer» Correct Answer - `x=-(a+4)" or "x=(a-2)` `x^(2)+6x-(a+4)(a-2)=0impliesx^(2)+(a+4)x-(a-2)x-(a+4)(a-2)=0.` |
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| 89. |
Solve each of the following quadratic equations: `((x)/(x+1))^(2)-5((x)/(x+1))+6=0,xne-1` |
| Answer» Correct Answer - `x=(-3)/(2)" or "x=-2` | |
| 90. |
Solve each of the following quadratic equations: `sqrt(2)x^(2)+7x+5sqrt(2)=0` |
| Answer» Correct Answer - `x=-sqrt(2)" or "x=(-5)/(sqrt(2))` | |
| 91. |
Solve each of the following quadratic equations: `15x^(2)-28=x` |
| Answer» Correct Answer - `x=(7)/(5)" or "x=(-4)/(3)` | |
| 92. |
Solve each of the following quadratic equations: `(x+3)/(x-2)-(1-x)/(x)=4(1)/(4),xne2,0` |
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Answer» Correct Answer - `x=4" or "x=(-2)/(9)` `(x(+3)-(1-x)(x-2))/(x(x-2))=(17)/(4)implies((x^(2)+3x)(3x-x^(2)-2))/((x^(2)-2x))=(17)/(4)` `implies" "4(2x^(2)+2)=17(x^(2)-2x)implies9x^(2)-34x-8=0implies9x^(2)-36x+2x-8=0.` |
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| 93. |
Solve the following quations by using qardratic formula: `12abx^(2)-(9a^(2)-8^(2))x-6ab=0` |
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Answer» Correct Answer - `x=(3a)/(4b)" or "x=(-2b)/(3a)` `12abx^(2)-9a^(2)x+8b^(2)x-6ab=0implies3ax(4bx-3a)+2b(4bx-3a)=0.` |
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| 94. |
Solve each of the following quadratic equations: `a^(2)b^(2)x^(2)+b^(2)x-a^(2)x-1=0` |
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Answer» Correct Answer - `x=(-1)/(a^(2))" or "x=(1)/(b^(2))` `b^(2)x(a^(2)x+1)-(a^(2)x+1)=0.` |
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| 95. |
Solve for:`1/(2a+b+2x)=1/(2a)+1/b+1/(2x)` |
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Answer» Correct Answer - `x=-a" or "x=(-b)/(2)` `(1)/(2a+b+2x)-(1)/(2x)=(1)/(2a)+(1)/(b).` |
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| 96. |
Solve for `x`: `9x^2-9(a+b)x+(2a^2+5ab+2b^2)=0` |
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Answer» Correct Answer - `x=((a+2b))/(3)" or "x=((2a+b))/(3)` `9x^(2)-9(a+b)x+(2a+b(a+2b)=0` `implies" "9x^(2)-3{(2a+b)+(a+2b)}x+(2a+b)(a+2b)=0` `implies" "9x^(2)-3(2a+b)x-3(a+2b)x+(2a+b)(a+2b)=0.` |
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| 97. |
Find the roots of each of the following equations, if they exist, by applying the quadratic formula: `4x^(2)+4bx-(a^(2)-b^(2))=0` |
| Answer» Correct Answer - `x=(1)/(2)(a-b)" or "x=-(1)/(2)(a+b)` | |
| 98. |
Solve: `6x^(2)-x-2=0` by the factorisation method. |
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Answer» We write, `-x=-4x+3x` as `(-4x)xx3x=-12x^(2)=6x^(2)xx(-2).` `:." "6x^(2)-x-2=0` `implies" "6x^(2)-4x+3x-2=0implies2x(3x-2)+(3x-2)=0` `implies" "(3x-2)(2x+1)=0implies3x-2=0" or "2x+1=0` `implies" "x=(2)/(3)" or "x=(-1)/(2).` Hence, `(2)/(3)` and `(-1)/(2)` are the roots of the given equation. |
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| 99. |
Solve the following equation by using factorisation method: `9x^(2)-6b^(2)x-(a^(4)-b^(4))=0.` |
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Answer» We may write, `-6b^(2)=3(a^(2)-b^(2))-3(a^(2)+b^(2)).` Also, `{3(a^(2)-b^(2))}xx{-3(a^(2)+b^(2)}=-(a^(4)-b^(4))` `:." "9x^(2)-6b^(2)x-(a^(4)-b^(4))=0` `implies" "9x^(2)+3(a^(2)-b^(2))x-3(a^(2)+b^(2))x-(a^(4)-b^(4))=0` `implies" "3x{3x+(a^(2)-b^(2))}-(a^(2)+b^(2)){3x+(a^(2)-b^(2))}=0` `implies" "{3x+(a^(2)-b^(2))}xx{3x-(a^(2)+b^(2))}=0` `implies" "3x+(a^(2)-b^(2))=0" or "3x-(a^(2)+b^(2))=0` `implies" "x=((b^(2)-a^(2)))/(3)" or "x=((a^(2)+b^(2)))/(3).` Hence, `((b^(2)-a^(2)))/(3)` and `((a^(2)+b^(2)))/(3)` are the required roots of the given equation. |
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A person on tour has Rs. 10800 for his expencses. If he extends his tour by 4 days, he has to cut down his daily expenses by Rs. 90. Find the original duration of the tour. |
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Answer» Correct Answer - 20 days Let the original duration of the tour be x days. Then, `(10800)/(x)-(10800)/((x+4))=90implies(1)/(x)-(1)/(x+4)=(1)/(120)impliesx^(2)+4x-480=0` `:." "x^(2)+24x-20x-480=0.` |
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