Q1. The quadratic equation $2x^2 - \sqrt{5}x + 1 = 0$ has:
Q2. If the equation $x^2 + 4x + k = 0$ has real and distinct roots, then:
Q3. The value of $c$ for which the equation $ax^2 + 2bx + c = 0$ has equal roots is:
Q4. Assertion (A): The equation $x^2 + 3x + 4 = 0$ has no real roots. Reason (R): If Discriminant $D < 0$, the roots are imaginary.
Q5. Find the discriminant of the equation $3x^2 - 2x + \frac{1}{3} = 0$ and hence find the nature of its roots.
Solve on white paper.
Q6. Find the value of $k$ for which the quadratic equation $kx(x-2) + 6 = 0$ has two equal roots.
Q7. Determine the nature of roots of the quadratic equation $2x^2 - 6x + 3 = 0$.
Q8. Find the values of $p$ for which the quadratic equation $(p+1)x^2 - 6(p+1)x + 3(p+9) = 0, p \neq -1$ has equal roots.
Q9. If the roots of the equation $(a^2+b^2)x^2 - 2(ac+bd)x + (c^2+d^2) = 0$ are equal, prove that $\frac{a}{b} = \frac{c}{d}$.
Q10. If the equation $(1+m^2)x^2 + 2mcx + c^2 - a^2 = 0$ has equal roots, show that $c^2 = a^2(1+m^2)$.
Q11. Case Study: Garden Design
A rectangular park is to be designed whose breadth is 3 m less than its length. Its area is to be 4 square metres more than the area of a park that has already been made in the shape of an isosceles triangle with its base as the breadth of the rectangular park and of altitude 12 m.
(i) Form the quadratic equation representing the situation.
(ii) Determine the nature of roots of the equation formed.
(iii) Find the length and breadth of the rectangular park.