InterviewSolution
Saved Bookmarks
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.
| 1. |
Find whether the following equations have real roots. If real roots exist, find them `5x^(2)-2x-10=0` |
|
Answer» Given equation is `5x^(2)-2x-10=0` on comparing with `ax^(2)+bx+c=0` we get `a=5,b=-2` and `c=-10` `:.` Discriminant `D=b^(2)-4ac` `=(-2)^(2)-4(5)(-10)` `=4+200=204gt0` Therefore the equation `5x^(2)-2x-10=0` has two distinct real roots. Roots `x=(-b+-sqrt(D))/(2a)` `=(-(-2)+-sqrt(204))/(2xx5)=(2+-2sqrt(51))/10` `=(1+-sqrt(51))/5=(1+sqrt(51))/5,(1-sqrt(51))/5` |
|
| 2. |
Find whether the following equations have real roots. If real roots exist, find them `-2x^(2)+3x+2=0` |
|
Answer» Given equation is `-2x^(2)+3x+2=0` on comparing with `ax^(2)+bx+c=0` we get `a=-2,b=3` and `c=2`. `:.` Discriminant `D=b^(2)-4ac` `=(3)^(2)-4(-2)(2)` `=9+16=25gt0` Therefore, the equation `-2x^(2)+3x+2=0` has two distinct real roots because we know that if the equation `ax^(2)+bx+c=0` has its discriminant greater than zero, then it has two distinct real roots. Roots `x=(-b+-sqrt(D))/(2a)=(-3+-sqrt(25))/(2(-2))` `=(-3+-5)/(-4)=(-3+5)/(-4),(-3-5)/(-4)` `=2/(-4),(-8)/(-4)=-1/2,2` |
|
| 3. |
What is the nature of roots of the quadratic equation `2x^2-sqrt5x+1=0 ?`A. two distinct real rootsB. two equal real rootsC. no real rootsD. more than 2 real roots |
|
Answer» Correct Answer - C Given equation is `2x^(2)-sqrt(5)x+1=0` On comparing with `ax^(2)+bx+c=0` we get `a=2,b=-sqrt(5)` and `c=1` `:.` Discriminant `D=b^(2)-4ac=(-sqrt(5))^(2)-4xx(2)xx(1)=5-8` `=-3lt0` since discriminant is negatice therefore quadratic equation `2x^(2)-sqrt(5)x+1=0` has no real roots i.e. imaginary roots. |
|
| 4. |
A quadratic equation with integral coefficients has integral roots. Justify your answer. |
| Answer» No, consider the quadratic equation `2x^(2)+x-6=0` with integral coefficient. The roots of the given quadratic equation are `-2` and `3/2` which are not integers. | |
| 5. |
State whether the following quadratic equations have two distinct real roots. Justicy your answer: `x(1-x)-2=0` |
|
Answer» Given equation is `x(1-x)-2=0` `implies x-x^(2)-2=0` `implies x^(2)-x=2=0` On comparing with `ax^(2)-bx+c=0` we get `a=1, b=-1` and `c=2` `:.` Discriminant `D=b^(2)-4ac` `=(-1)^(2)-4(1)(2)=1-8=-7lt0` i.e. `Dlt0` Hence the equation `x(1-x)-2=0` has imaginary roots i.e. no real roots. |
|
| 6. |
Write whether the following statements are true or false. Justify your answers. (i) Every quadratic equation has exactly one root. (ii) Every quadratic equation has atleast one real root. (ii) Every quadratic equation has atleast two roots. (iv) Every quadratic equations atmost two roots. (v) If he coefficient of `x^(2)` and the constnat term of a quadratic equation have opposite sigh, then the quadratic equation has real roots. (vi) If the coefficient of `x^(2)` and the constant term have the same sign and if the coefficient of `x` term is zero, then the quadratic equation has no real roots. |
|
Answer» (i) False since a quadratic equation has two and onloy two roots. (ii) False, for example `x^(2)+4=0` has no real root. (iii) False, Since a quadratic equations has to only two roots (iv) True, because every quadratic polynomial has atmost two roots. (v) True, since in this cae discriminant is always positive, so it has always real roots i.e. `aclt0` and so `b^(2)-4acgt0` (vi) True, Since in this case discriminant is always negative so it has no real roots i.e. , if `b=0` then `b^(2)-4acimplies-4aclt0` and `acgt0`. |
|
| 7. |
State whether the following quadratic equations have two distinct real roots. Justicy your answer: `(x-1)(x+2)+2=0` |
|
Answer» Given equation is `(x-1)(x+2)+2=0` `implies x^(2)+x-2+2=0` `implies x^(2)+x+0=0` on comparing the equation with `ax^(2)+bx+c=0` we have `a=1,b=1` and `c=0` `:.` Discriminant `D=b^(2)-4ac` `=1-4(1)(0)=1lt0` i.e. `Dlt0` Hence equation has two distinct real roots. |
|
| 8. |
State whether the following quadratic equations have two distinct real roots. Justicy your answer: `2x^(2)+x-1=0` |
|
Answer» Gien equation is `2x^(2)+x-1=0` On comparing with `ax^(2)+bx+c=0` we get `a=2,b=1` and `c=-1` `:.` Discriminant `D=b^(2)-4ac=(1)^(2)-4(2)(-1)` `=1+8=9gt0` i.e. `Dlt0` Hence the equation `2x^(2)+x-1=0` has two distinct real roots. |
|
| 9. |
State whether the following quadratic equations have two distinct real roots. Justicy your answer: `x^(2)-3x+4=0` |
|
Answer» Given equation is `x^(2)-3x+4=0` On comparing with `ax^(2)+bx+c=0` we get `a=1,b=-3` and `c=4` `:.` Discriminant `D=b^(2)-4ac=(-3)^(2)-4(1)(4)` `=9-16=-7 lt 0` i.e. `Dlt0` Hence equation `x^(2)-3x+4=0` has no real roots. |
|
| 10. |
Which of the following equations has no real roots?A. `x^(2)-4x+3sqrt(2)=0`B. `x^(2)+4x-3sqrt(2)=0`C. `x^(2)-4x-3sqrt(2)=0`D. `3x^(2)+4sqrt(3)x+4=0` |
|
Answer» (a) The given equation is `x^(2)-4x+3sqrt(2)=0` On comparing with `ax^(2)+bx+c=0`, we get `a=1,b=-4` and `c=3sqrt(2)` The discriminant of `x^(2)-4x+3sqrt(2)=0` is `D=b^(2)-4ac` `=(-4)^(2)-4(1)(3sqrt(2))=16-sqrt(2)=16-12xx(1.41)` `=16-16.92=-0.92` `implies b^(2)-4aclt0` (b) The given equation is `x^(2)+4x-3sqrt(2)=0` On comparing the equation with `ax^(2)+bx+c=0` we get `a=1,b=4` and `c=-3sqrt(2)` Then `D=b^(2)-4ac=(-4)^(2)-4(1)(-3sqrt(2))` `=16+12sqrt(2)gt0` Hence the equation has real roots. (c) Given equation is `x^(2)-4x-3sqrt(2)=0` On comparing the equation with `ax^(2)+bx+c=0` we get `a=1,b=-4` and `c=-3sqrt(2)` Then `D=b^(2)-4ac=(-4)^(2)-4(1)(-2sqrt(2))` `=16+12sqrt(2)gt0` Hence, the equation has real roots. (d) Given equation is `3x^(2)+4sqrt(3)x+4=0` On comparing the equation with `ax^(2)+bx+c=0` we get `a=3,b=4sqrt(3)` and `c=4` Then `D=b^(2)-4ac=(4sqrt(3))^(2)-4(3)(4)=48-48=0` Hence, the equation has real roots. Hence `x^(2)-4x+3sqrt(2)=0` has no real roots. |
|
| 11. |
Which of the following equations has two distinct real roots?A. `2x^(2)-3sqrt(2)x+9/4=0`B. `x^(2)+x-5=0`C. `x^(2)+3x+2sqrt(2)=0`D. `5x^(2)-3x+1=0` |
|
Answer» The given equation is `x^(2)+x-5=0` On comparing with `ax^(2)+bx+c=0` we get `a=1, b=1` and `c=-5` The discriminant of `x^(2)+x-5=0` is `D=b^(2)-4ax=(1)^(2)-4(1)(-5)` `=1+20=21` `implies b^(2)-4acgt0` So `x^(2)+x-5=0` has two distinct real roots. (a) Given equation is `2x^(2)-3sqrt(2)x+9//4=0` On comparing with `ax^(2)+bx+c=0` `a=2,b=-3sqrt(2)` and `c=9//4` Now `D=b^(2)-4ac=(-3sqrt(2))^(2)-4(2)(9//4)=18-18=0` Thus, the equation has real and equal roots. (c) Given equation is `x^(2)+3x+2sqrt(2)=0` On comparing with `ax^(2)+bx+c=0` `a=1,b=3` and `c=2sqrt(2)` Now `D=b^(2)-4ac=(3)^(2)-4(1)(2sqrt(2))=9-8sqrt(2)lt0` `:.` Roots of the equation are not real. (d) Given equation is `5x^(2)+3x+1=0` On comparing with `ax^(2)+bx+c=0` `a=5,b=-3, c=1` Now `D=b^(2)-4ac=(-3)^(2)-4(5)(1)=9-20lt0` Hence roots of the equation are not real. |
|
| 12. |
Find whether the following equations have real roots. If real roots exist, find them `x^(2)+5sqrt(5)x-70=0` |
|
Answer» Given equation is `x^(2)+5sqrt(5)x-70=0` on comparing with `ax^(2)+bx+c=0` we get `a=1,b=5sqrt(5)` and `c=-70` `:.` Discriminant `D=b^(2)-4ac=(5sqrt(5))^(2)-4(1)(-70)` `=125+280=405gt0` Therefore the equation `x^(2)+5sqrt(5)x-70=0` has two distinct real roots. Rots `x=(-b+-sqrt(D))/(2a)` `=(5sqrt(5)+-sqrt(405))/(2(1))=(-5sqrt(5)-9sqrt(5))/2` `=(-5sqrt(5)+9sqrt(5))/2,(-5sqrt(5)-9sqrt(5))/2` `(4sqrt(5))/2,-(14sqrt(5))/2=2sqrt(5),-7sqrt(5)` |
|
| 13. |
Find the roots of the quadratic equations by using the quadratic formula in each of the following `-3x^(2)+5x+12=0` |
|
Answer» Given equation is `-3x^(2)+5x+12=0` On comparing with `ax^(2)+bx+c=0` we get `a=-3, b=5` and `c=12` By quadratic formula `x=(-b+-sqrt(b^(2)-4ac))/(2a)` `(-(5)+-sqrt((5)^(2)-4(-3)(12)))/(2(-3))` `=(5+-sqrt(25+144))/(-6)=(-5+-sqrt(169))/(-6)` `=(-5-13)/(-6)=8/(-6), (-18)/(-6)=-4/3,3` So `-4/3` and 3 are two roots of the given equation. |
|
| 14. |
State whether the following quadratic equations have two distinct real roots. Justicy your answer: `3x^(2)-4x+1=0` |
|
Answer» Given equation is `3x^(2)-4x+1=0` On comparing with `ax^(2)+bx+c=0` we get `a=3,b=-4` and `c=1` `:.` Discriminant `D=b^(2)-4ac=(-4)^(2)-4(3)(1)` `=16-12=4gt0` i.e. `Dgt0` Hence the equation `3x^(2)-4x+1=0` has two distinct real rots. |
|
| 15. |
State whether the following quadratic equations have two distinct real roots. Justicy your answer: `2x^(2)-6x+9/2=0` |
|
Answer» Given equation is `2x^(2)-6x+9/2=0` On comparing with `ax^(2)+bx+c=0` we get `a=2, b=-6` and `c=9/2` `:.` Discriminant `D=b^(2)-4ac` `=(-6)^(2)-4(2)(9/2)=36-36=0` i.e. `D=0` Hence the equation `2x^(2)=6x+9/2=0` has equal to real roots. |
|
| 16. |
Find the roots of the quadratic equations by using the quadratic formula in each of the following `2x^(2)-3x-5=0` |
|
Answer» Given equation is `2x^(2)-3x-5=0` On comparing with `ax^(2)+bx+c=0` we get `a=2,b=-3` and `c=-5` By quadratic formula `x=(-b+1sqrt(b^(2)-4ac))/(2a)` `=(-(-3)+-sqrt((-3)^(2)-4(2)(-5)))/(2(2))=(3+-sqrt(9+40))/4` `=(3+-sqrt(49))/4=(3+-7)/4=10/4 . (-4)/4=5/2 . -1` So `5/2` and `-1` are the roots of the given equation. |
|
| 17. |
State whether the following quadratic equations have two distinct real roots. Justicy your answer: `(x+1)(x-2)+x=0` |
|
Answer» Given equation is `(x+1)(x-2)+x=0` `implies x^(2)+x-2x-2+x=0` `implies x^(2)-2=0` `implies x^(2)+0.x-2=0` On comparing with `ax^(2)+bx+c=0` we get `a=1, b=0` and `c=-2` `:.` Discriminant ` D=b^(2)-4ac=(0)^(2)-4(1)(-2)=0+8=8gt0` Hence the equation `(x+1)(x-2)+x=0` has two distinct real roots. |
|
| 18. |
Is 0.2 a root of the equation `x^(2)-0.4=0`? Justify your answer. |
| Answer» No, since 0.2 does not satisfy the quadratic equation i.e. `(0.2)^(2)-0.4=0.04-0.4!=0` | |
| 19. |
Find whether the following equations have real roots. If real roots exist, find them `8x^(2)+2x-3=0` |
|
Answer» Given equation is `8x^(2)+2x-3=0` on comparing with `ax^(2)+bx+c=0` we get `a=8,b=2` and `c=-3` `:.` Discriminant `D=b^(2)-4ac` `=(2)^(2)-4(8)(-3)` `=4+96=100gt0` Therefore the equation `8x^(2)+2x-3=0` has two distinct real roots because we know that if the equation `ax^(2)+bx+c=0` has discriminant greater than zero, then it has to distinct real roots. Roots `x=(-b+-sqrt(D))/(2a)=(-2+-sqrt(100))/16=(-2+-10)/16` `=(-2+10)/16,(-1-10)/16` `=8/16,-12/16=1/2,-3/4` |
|
| 20. |
Find whether the following equations have real roots. If real roots exist, find them`1/(2x-3)+1/(x-5)=1,x!=3/2,5` |
|
Answer» Given equation is `1/(2x-3)+1/(x-5)=1,x!=3/2,5` `implies (x-5+2x-3)/((2x-4)(x-5))=1` `implies (3x-8)/(2x^(2)-5x-10x+25)=1` `implies (3x-8)/(2x^(2)-15x+25)=1` `implies 3x-8=2x^(2)-15x+25` `implies 2x^(2)-15x-3x+25+8=0` `implies 2x^(2)-18x+33=0` On comparing with `ax^(2)+bx+c=0` we get `a=2,b=-18` and `c=33` `:.` Discriminant `D=b^(2)-4ac` `=(-18)^(2)-4xx2(33)` `=324-264=60gt0` Therefore, the equation `2x^(2)-18x+33=0` has two distinct real roots. Roots `x=(-b+-sqrt(D))/(2a)=(-(-18)+-sqrt(60))/(2(2))` `=(18+-2sqrt(15))/4=(9+-sqrt(15))/2` `=(9+sqrt(15))/2,(9-sqrt(15))/2` |
|
| 21. |
At t minutes past 2 pm, the time needed by the minutes hand of a clock to show 3pm was found to be 3 minutes less than `(t^2)/4 minutes. Find t. |
|
Answer» We know that the time between 2 pm to 3pm `1h=60` min Given that at `t` min past 2pm the time needed by the min hand of a clock to show 3 pm was found to be 3 minless than `(t^(2))/4` min i.e. `t+((t^(2))/4-3)=60` `implies 4t+t^(2)-12=240` `impliest^(2)+4t-252=0` `impliest^(2)+18t-14t-252=0` `implies t^(2)+18t-14t-252=0` [by splitting the middle term] `implies t(t+18)-14(t+18)=0` [since, time cannot be neative so `t!=-18`] `implies (t+18)(t-14)=0` `:. t=14` min Hence the required value to `t` is 14 min. |
|