Solutions: Quadratic Equations - Vardaan Learning Institute

1. Which of the following is a quadratic equation?

Ans: (d) $x^3 - x^2 = (x-1)^3$
$(x-1)^3 = x^3 - 1 - 3x^2 + 3x$.
Eq: $x^3 - x^2 = x^3 - 3x^2 + 3x - 1$.
$2x^2 - 3x + 1 = 0$. Quadratic.

2. If the discriminant of $ax^2 + bx + c = 0$ is zero, then the roots are:

Ans: (b) Real and equal

3. The roots of the quadratic equation $x^2 - 0.04 = 0$ are:

Ans: (a) $\pm 0.2$

4. Values of k for which the quadratic equation $2x^2 - kx + k = 0$ has equal roots is:

Ans: (d) 0, 8
$D = b^2 - 4ac = k^2 - 4(2)(k)$
$= k^2 - 8k = 0$.
$k(k-8) = 0 \Rightarrow k = 0, 8$.

5. If 1/2 is a root of the equation $x^2 + kx - 5/4 = 0$, then value of k is:

Ans: (a) 2
$(1/2)^2 + k(1/2) - 5/4 = 0$.
$1/4 + k/2 - 5/4 = 0$.
$k/2 - 1 = 0 \Rightarrow k = 2$.

6. The quadratic equation $2x^2 - \sqrt{5}x + 1 = 0$ has:

Ans: (c) No real roots
$D = b^2 - 4ac = (-\sqrt{5})^2 - 4(2)(1)$
$= 5 - 8 = -3 < 0$. No real roots.

7. Constant to add/subtract for $9x^2 + \frac{3}{4}x - \sqrt{2} = 0$:

Ans: (b) 1/64
Eq: $(3x)^2 + 2(3x)(1/8) - \sqrt{2} = 0$.
$2ab = 3x/4 \Rightarrow b = 1/8$. Add $b^2 = (1/8)^2 = 1/64$.

8. If $9x^2 + 6kx + 4 = 0$ has equal roots, k is:

Ans: (a) $\pm 2$
$D = (6k)^2 - 4(9)(4) = 0$.
$36k^2 - 144 = 0 \Rightarrow k^2 = 4 \Rightarrow k = \pm 2$.

9. Roots of $x^2 + 7x + 10 = 0$:

Ans: (b) -2, -5
$(x+2)(x+5)=0$. Roots are -2, -5.

10. Assertion: Linear Eq. Reason: Max power 1.

Ans: (a)
$x^2+3x+1 = x^2-4x+4 \Rightarrow 7x-3=0$. Linear.

11. Find the roots of the quadratic equation by factorization: $3x^2 - 5x + 2 = 0$.

Ans: 1, 2/3
$3x^2 - 3x - 2x + 2 = 0$
$\Rightarrow 3x(x-1) - 2(x-1) = 0$.
$(3x-2)(x-1) = 0$.

12. Find the value of p so that the quadratic equation $px(x-3) + 9 = 0$ has two equal roots.

Ans: p = 4
$px^2 - 3px + 9 = 0$.
$D = (-3p)^2 - 4(p)(9)$
$= 9p^2 - 36p = 0$.
$9p(p-4) = 0 \Rightarrow p = 4$.

13. Solve for x: $4x^2 - 4a^2x + (a^4 - b^4) = 0$.

Ans: $\frac{a^2 \pm b^2}{2}$
$x = [4a^2 \pm \sqrt{16a^4 - 4(4)(a^4-b^4)}] / 8$.
$x = [4a^2 \pm \sqrt{16b^4}] / 8$
$= (4a^2 \pm 4b^2)/8$
$= (a^2 \pm b^2)/2$.

14. The product of two consecutive positive integers is 306. Form the quadratic equation to find the integers.

Ans: $x^2 + x - 306 = 0$
$x(x+1) = 306 \Rightarrow x^2+x-306=0$.

15. A motor boat whose speed is 18 km/h... Speed of stream?

Ans: 6 km/h
$24/(18-s) - 24/(18+s) = 1$.
$24[18+s - (18-s)] = 324 - s^2$.
$48s = 324 - s^2 \Rightarrow s^2 + 48s - 324 = 0$.
$(s+54)(s-6) = 0 \Rightarrow s=6$.

16. Find the roots of $5x^2 - 6x - 2 = 0$ by completing the square.

Ans: $(3 \pm \sqrt{19})/5$
$x = [-(-6) \pm \sqrt{36 - 4(5)(-2)}]/10$
$= (6 \pm \sqrt{76})/10$
$= (3 \pm \sqrt{19})/5$.

17. Two water taps together can fill a tank in $9\frac{3}{8}$ hours...

Ans: Larger: 15 hrs, Smaller: 25 hrs
Total time = 75/8 hrs. Let smaller tap take x hrs.
$1/x + 1/(x-10) = 8/75$.
Solving gives $x = 25$ ($x=15/4$ rejected).

18. Is it possible to design a rectangular park of perimeter 80 m and area 400 $m^2$?

Ans: Yes. Length 20m, Breadth 20m.
$2(l+b)=80 \Rightarrow l+b=40 \Rightarrow b=40-l$.
$l(40-l) = 400 \Rightarrow l^2 - 40l + 400 = 0$.
$(l-20)^2 = 0 \Rightarrow l=20$. It's a square.

19. Show $c^2 = a^2(1+m^2)$ if equal roots.

$D = (2mc)^2 - 4(1+m^2)(c^2-a^2) = 0$.
$4m^2c^2 - 4(c^2 - a^2 + m^2c^2 - m^2a^2) = 0$.
$m^2c^2 - c^2 + a^2 - m^2c^2 + m^2a^2 = 0$.
$-c^2 + a^2(1+m^2) = 0 \Rightarrow c^2 = a^2(1+m^2)$.

20. Solve for x: $\frac{1}{a+b+x} = ... $

Ans: -a, -b
$\frac{1}{a+b+x} - \frac{1}{x} = \frac{1}{a} + \frac{1}{b}$.
$\frac{x-a-b-x}{x(a+b+x)} = \frac{a+b}{ab}$.
$\frac{-(a+b)}{x^2+ax+bx} = \frac{a+b}{ab}$.
$-ab = x^2+ax+bx \Rightarrow x^2+ax+bx+ab=0$.
$(x+a)(x+b)=0$.

21. Plane late by 30 mins...

Ans: 750 km/h
$1500/x - 1500/(x+250) = 1/2$.
$3000(250) = x(x+250) \Rightarrow x^2 + 250x - 750000 = 0$.
$(x+1000)(x-750) = 0 \Rightarrow x=750$.

22. Rs 6500 divided...

Ans: 50 persons
Let x be number of persons.
$6500/x - 6500/(x+15) = 30$.
$650[1/x - 1/(x+15)] = 3$.
$650(15) = 3x(x+15) \Rightarrow x^2 + 15x - 3250 = 0$.
$(x+65)(x-50) = 0 \Rightarrow x=50$.

23. Case Study: Basketball Trajectory

Detailed Solution:
(i) At t=0, $h = 6$ feet.
(ii) $h(0.5) = -16(0.25) + 20(0.5) + 6$
$= -4 + 10 + 6 = 12$ feet.
(iii) $-16t^2 + 20t + 6 = 0 \Rightarrow 8t^2 - 10t - 3 = 0$.
$8t^2 - 12t + 2t - 3 = 0 \Rightarrow (4t+1)(2t-3) = 0$.
$t = 1.5$ seconds.