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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.
| 201. |
Write thediscriminant of the following quadratic equations:`sqrt(3)x^2+2sqrt(2)x-2sqrt(3)=0`(ii) `x^2-x+1=0` |
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Answer» Correct Answer - `x=sqrt(6)" or "x=(-sqrt(2))/(sqrt(3))` `sqrt(3)x^(2)-2sqrt(2)x-2sqrt(3)=0impliessqrt(3)x^(2)-3sqrt(2)x+sqrt(2)x-2sqrt(3)=0` `" "impliessqrt(3)x(x-sqrt(6))+sqrt(2)(x-sqrt(6))=0.` |
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| 202. |
Obtain a quadratic equation whose roots are -3 and -7. |
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Answer» Correct Answer - The required quadratic equation is `x^(2) + 10x + 21 = 0` Let `alpha = - 3` and `beta = - 7` Then `alpha + beta = - 3-7 = -10` and `alpha beta = ( -3) xx ( -7) = 21 ` The required quadratic equation is `x^(2) - (alpha + beta) x + alpha beta = 0 ` `:. x^(2) - ( -10) x + 21 = 0` `:. x^(2) + 10x + 21 = 0` ....( Substituting the values ) |
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| 203. |
Determine the nature of the roots of the following quadratic equations from their discriminant `:` (i) `3y^(2) + 9y+4=0` (ii) `2x^(2) + 5 sqrt(3)x + 16=0` (iii) `4x^(2) + 12x + 9=0` (iv) `m^(2) + sqrt(2) m +1=0` (v) `x^(2) - (1)/(2) x + (1)/( 16) = 0 ` (vi)` x^(2) - 4x -4=0` |
| Answer» (i) Real and unequal (ii) Not real (iii) Real and equal (iv) Not real (v) Real and equal (vi) Real and unequal . | |
| 204. |
Determine the nature of the roots of the following quadratic equations from their discriminant `:` (i) `2x^(2) - 3 x -4 =0` (ii) ` x^(2) - 2 x+ ( 9 )/(4) =0`. |
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Answer» Correct Answer - The roots of the given quadratic equation are real and unequal (ii) The roots of the given quadratic equation are not real. (i) `2x^(2) - 3 x - 4 = 0 ` Comparing with `ax^(2) + bx + c = 0` `a = 2, b = - 3, c = - 4 ` `Delta = b^(2) -4ac = ( -3)^(2) - 4(2) ( -4)` `= 9 + 32 = 41 ` `:. Delta gt 0` (ii) `x^(2) - 2x + (9)/(4) = 0 ` Comparing with `ax^(2) + bx + c = 0 ` `a = 1 , b = - 2 , c = ( 9)/(4)` `Delta = b^(2) - 4ac = ( -2)^(2) - 4(1) ((9)/(4))` `= 4-9= - 5 ` `:. Delta lt 5 ` |
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| 205. |
Which of the following is the value of the discriminant for the quadratic equation `2x^(2) + 5sqrt(3)x + 6 = `?A. 27B. 72C. 123D. `25sqrt(3) - 48` |
| Answer» Correct Answer - A | |
| 206. |
If `alpha +beta =3, alpha^3 + beta^3 = 7`, then `alpha` and `beta` are the roots of |
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Answer» Here, `alpha^3+beta^3 = 7` `=>(alpha+beta)^3 - 3alphabeta(alpha+beta) = 7` `=>3^3-3alphabeta(3) = 7` `=>9alphabeta = 20` `=>alphabeta = 20/9.` So, the equation will be, `x^2-3x+20/9 = 0` `=>9x^2-27x+20 = 0` So,option `(b)` is the correct option. |
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| 207. |
Find the discriminant of the equation `3x^2-2x+1/3=0`and hence find the nature of its roots. Find them, if they are real. |
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Answer» Here, equation is , `3x^2-2x+1/3 = 0` Comparing it with, `ax^2+bx+c = 0` `a = 3, b = -2 and c = 1/3` So, discriminant,`(d) = sqrt(b^2-4ac)` `d = sqrt((-2)^2-4(3)(1/3)) = 0` We know, if `d >=0`, roots are real and if `d<0`, roots are unreal.Here, as `d = 0`, roots are real and equal.Now, roots are` = (-b+-sqrt(d))/(2a) = -b/(2a) = 1/3`So, roots are `(1/3,1/3)` |
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| 208. |
If `x=(sqrt(p^6+q^2)+sqrt(p^2-q^2))/(sqrt(p^2+q^2)-sqrt(p^2-q^2))` then `q^2x^2-2p^2x+q^2=?` |
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Answer» `x = (sqrt(p^2+ q^2) + sqrt(p^2 - q^2))/(sqrt (p^2 + q^2) - sqrt(p^2 - q^2))` multiply by `sqrt(p^2 + q^2) + sqrt(p^2 - q^2)` `x = ((sqrt(p^2 + q^2) + sqrt(p^2 - q^2))^2)/(sqrt(p^2 + q^2) - sqrt(p^2 - q^2))` `= (p^2 + q^2 + p^2 - q^2 + 2 sqrt((p^2+ q^2)(p^2- q^2)))/(p^2 + q^2 - p^2 + q^2)` `x = (p^2 + sqrt(p^4 - q^4))/q^2` `x^2 = (p^4 + p^4 - q^4 + 2 p^2 sqrt(p^4 - q^4))/q^4` `= (2p^4 - q^4 + 2p^2 sqrt(p^4 - q^4))/q^2 - (2p^4 - 2p^2 sqrt(p^4 - q^4))/q^2 + q^2` `= -q^4/q^2 + q^2 = 0` here, `p=1, q=1` `x = (sqrt2 - 0)/(sqrt2 - 0) = 1` `q^2x^2 - 2p^2x + q^2` `1 xx 1 - 2 + 1 = 0` answer |
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| 209. |
Find the values of `p`for whch the equadratic equation `(2p+1)x^2-(7p+2)x+(7p-3)=0` |
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Answer» Correct Answer - `p=4" or "p=(-4)/(7)` `D=(7p+2)^(2)-4(2p+1)(7p-3)=-7p^(2)+24p+16.` `:." "D=0implies7p^(2)-24p-16=0implies7p^(2)-28p+4p-16=0` `implies7p(p-4)+4(p-4)=0implies(p-4)(7p+4)=0.` |
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| 210. |
Find the value of p for which the quadratic equation `4x^2+px+3=0` has equal roots |
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Answer» Correct Answer - `p=4sqrt(3)" or "p=-4sqrt(3)` For real and equal roots, we must have D=0. `:." "D=0impliesp^(2)-48=0impliesp^(2)=48impliesp=+-sqrt(48)impliesp=4sqrt(3)" or "-4sqrt(3).` |
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| 211. |
The productof two consecutive positive integers is 306. Form the quadratic equation tofind the integers, if `x`denotes the smaller integer. |
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Answer» Correct Answer - 17 and 18 `x(x+1)=306impliesx^(2)+x-306=0impliesx^(2)+18x-17x-306=0.` |
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| 212. |
A train covers a distance of 480 km at a uniform speed. If the speed had been 8 km/hr less then it would have taken 3 hours more to cover the same distance. Find the usual speed of the train. |
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Answer» Correct Answer - 40 km/hr Let the usual speed of the train be x km/hr. Then, `(480)/((x-8))-(480)/(x)=3implies(1)/(x-8)-(1)/(x)=(3)/(480)=(1)/(160).` `implies160[x-(x-8)]=x(x-8)impliesx^(2)-8x-1280=0` `impliesx^(2)-40x+32x-1280=0.` |
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| 213. |
A truck covers a distance of 150 km at a certain average speed and then covers another 200 km at average speed which is 20 km per hour more than the first speed. If the truck covers the total distance in 5 hours, find the first speed of the truck. |
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Answer» Correct Answer - 60 km/hr Let the first speed be x km/hr. Then, `(150)/(x)+(200)/((x+20))=5implies50[(3)/(x)+(4)/((x+20))]=5` `implies10[3(x+20)+4x]=x(x+20)impliesx^(2)-50x-600=0.` |
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| 214. |
While boarding an aeroplane, a passenger got hurt. The pilot showing promptness and concern, made arrangements to hospitalise the injured and so the plane started late 30 minutes to reach the destination, 1500 km away in time, the pilot increased the speed by 100 km/hr. Find the original speed/hour of the plane. |
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Answer» Correct Answer - 500 km/hr Let the original speed of the plane be x km/hr. Then, `(1500)/(x)-(1500)/((x+100))=(30)/(60)implies(1)/(x)-(1)/((x+100))=(1)/(3000)impliesx^(2)+100x-300000=0` `impliesx^(2)+600x-500x-300000=0.` His promptness is appreciable. |
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| 215. |
The difference of two numbers is 5 and the difference of their reciprocals is `1/10` Find the numbers. |
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Answer» Let the required natural numbers be x and `(x-5)`. Then, `xgtx-5implies(1)/(x)lt(1)/(x-5)implies(1)/(x-5)gt(1)/(x).` `:." "(1)/(x-5)-(1)/(x)=(1)/(10)` `implies" "(x-(x-5))/((x-5)x)=(1)/(10)implies(5)/((x-5)x)=(1)/(10)" "["by cross multiplication"]` `implies" "(x-5)x=50impliesx^(2)-5x-50=0` `implies" "x^(2)-10x+5x-50=0impliesx(x-10)+5(x-10)=0` `implies" "(x-10)(x+5)=0` `implies" "x-10=0" or "x+5=0` `implies" "x=10" or "x=-5` `implies" "x=10" "[:." "-5" is not a natural number"]` Hence, the required natural numbers are 10 and 5. |
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| 216. |
The sum of the squares of two consecutive positive odd numbers is 514. Find the numbers. |
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Answer» Correct Answer - 15 and 17 `x^(2)+(x+2)^(2)=514implies2x^(2)+4x-510=0impliesx^(2)+2x-255=0` `impliesx^(2)+17x-15x-255=0.` |
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| 217. |
The sum of the squaresof two consecutive odd numbers is 394. Find the numbers. |
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Answer» Let the requaired consecutive odd numbers be x and `(x+2).` Then, `x^(2)+(x+2)^(2)=394` `implies" "2x^(2)+4x-390=0impliesx^(2)+2x-195=0` `implies" "x^(2)+15x-13x-195=0` `implies" "x(x+15)-13(x+15)=0` `implies" "x+15=0" or "x-13=0` `implies" "x=-15" or "x=13` `implies" "x=13." "["rejecting "x=-15]` Hence, the required numbers are 13 and 15. |
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| 218. |
Represent the following situations in the form of quadratic equations :(i) The area of a rectangular plot is 528 `m^2`. 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.(ii) The product of two consecutive positive integers is 306. We need to find the integers.(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.(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. |
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Answer» (i) Let breadth of the rectangular plot is `x` metre. Then, length of the plot will be `2x+1` metres. `:.` Area `= x**(2x+1)` `=>x(2x+1) = 528` `=>2x^2+x - 528 =0`, which is the required quadratic equation. (ii) Let the numbers are `x` and `x+1`. Then, `x(x+1) = 306` `=>x^2+x - 306 = 0`, which is the required quadratic equation. (iii) Let age of Rohan is `x` years. Then, age of his mother is `x+26` years. Then, after `3` years, `(x+26+3)(x+3) = 360` `=>(x+29)(x+3) = 360` `=>x^2+29x+3x+87-360 = 0` `=>x^2+32x-273 = 0`, which is the required quadratic equation. (iv) Let speed of the train is `x` km/h. Then, `480/(x-8) - 480/x = 3` `=>480x - 480(x-8) = 3x(x-8)` `=>480x-480x+ 3840 = 3x^2-24x` `=>3x^2-24x-3840 = 0` `=>x^2-8x-1280 = 0`, which is the required quadratic equation. |
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| 219. |
Find two consecutive positiveintegers, sum of whose squares is 365. |
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Answer» Correct Answer - 13 and 14 `x^(2)+(x+1)^(2)=365implies2x^(2)+2x-364=0impliesx^(2)+x-182=0` `:." "x^(2)+14x-13x-182=0.` |
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| 220. |
The length of a rectangle is twice its breadth and its area is 288 `cm^(2)`. Find the dimensions of the rectangle. |
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Answer» Correct Answer - length = 24 cm, breadth = 12 cm `2x xx x=288impliesx^(2)=144impliesx=12 cm.` |
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| 221. |
The product of two consecutive odd numbers in 483. Find the numbers. |
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Answer» Correct Answer - 21 and 23 `x(x+2)=483impliesx^(2)+2x-483=0impliesx^(2)+23x-21x-483=0.` |
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| 222. |
The difference of twonatural numbers is 3 and the difference of their reciprocals is `3/(28)`. Find the numbers. |
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Answer» Correct Answer - 7 and 4 Let the required numbers be x and `(x-3)`. Then, `(1)/((x-3))-(1)/(x)=(3)/(28).` |
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| 223. |
Three consecutive positive integers are such that the sum of the squareof the first and the product of other two is 46, fond the integers. |
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Answer» Correct Answer - 4,5,6 Let the required numbers be `x,(x+1)" and "(x+2).` Then, `x^(2)+(x+1)(x+2)=46implies2x^(2)+3x-44=0implies2x^(2)+11x-8x-44=0` `implies(2x+11)(x-4).` `xne(-11)/(2)`, since x is an integer. |
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| 224. |
Find two consecutive positive even integers whose product is 288. |
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Answer» Correct Answer - 16 and 18 `x(x+2)=288impliesx^(2)+2x-288=0impliesx^(2)+18x-16x-288=0.` |
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| 225. |
The altitude of a right triangle is 7 cm less than its base. If the hypotenuse is 13 cm, find the other two sides. |
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Answer» let base be`x`cm altitude`= (x-7)`cm hypotenuse`= 13`cm acc to pythagoras theorem `x^2 + (x-7)^2 = 13^2` `x^2 + x^2 - 14x + 49 = 169` `2x^2 - 14x - 120=0` `x^2 - 7x - 60=0` `x^2 - 12x + 5x - 60=0` `x(x-12)+5(x-12)=0` `(x-12)(x+5)=0` `x-12=0 or x+5=0` `x=12,-5` base cant be negative so `x=12` answer |
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| 226. |
A cottageindustry produces a certain number of pottery articles in a day. It wasobserved on a particular day that the cost of production of each article (inrupees) was 3 more than twice the number of articles produced on that day. Ifthe to |
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Answer» Let number of articles produced on a particular day =x `because` Cost of production of each article `=RS.(2x+3)` According to given condition `x(2x+3)=90` `implies2x^(2)+3x=90` `implies2x^(2)+(15-12)x-90=0` `implies2x^(2)+15-12x-90=0` `impliesx(2x+15)-6(2x+15)=0` `implies(x-6)(2x+15)=0` when `x-6=0impliesx=6` and `2x+15=0impliesx=(-15)/(2)` `"Hence", x=6" "(because"no . of articles cannot be negative")` So, the number of articles produced is 6 and the cost of each article is `(2xx6+3)=RS.15`. |
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| 227. |
The area of a right angled triangle is `165m^2`. Determine its base and altitude if the latter exceeds the former by7m. |
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Answer» Correct Answer - `" base"=15m," altitude"=22m` Let the base be x metres. Then, altitude `=(x+7)m.` `:." "(1)/(2)xx x xx(x+7)=165impliesx^(2)+7x-330=0` `implies" "x^(2)+22x-15x-330=0implies(x-15)(x+22)=0impliesx=15. |
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| 228. |
The area of a right angled triangle is `600c m^2dot`If the base of the triangle exceeds the altitude by 10cm, find thedimensions of the triangle. |
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Answer» Correct Answer - `30 cm,40 cm,50 cm` Let the altitude be x cm. Then, base `=(x+10)cm.` `:." "(1)/(2)(x+10)x=600impliesx^(2)+10x-1200=0` `implies" "(x+40)(x-30)=0impliesx=30.` `:." ""base = 40 cm, altitude = 30 cm."` `:." "" hypotenuse "=sqrt((40)^(2)+(30)^(2)))=sqrt(1600+900)=sqrt(2500)=50cm.` |
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| 229. |
The area of a right-angled triangle is 96 sq metres. If the base is three times the altitude, find the base. |
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Answer» Correct Answer - 24 m Let the altitude be x metres. Then, base = 3x metres. `:." "(1)/(2)xx3x xx x=96impliesx^(2)=64impliesx=8m. Hence, base = 24 m. |
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| 230. |
The length of a hall is 3 metres more than its breadth. If the area of the hall is 238 sq metres, calculate its length and breadth. |
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Answer» Correct Answer - breadth = 14 m, length = 17 m `x(x+3)=238impliesx^(2)+3x-238=0impliesx^(2)+17x-14x-238=0.` `:." ""breadth = 14 m and length = 17 m."` |
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