Hard Level Worksheet

Very Short Answer Questions (1 Mark Each)

(1) Write a condition for a quadratic equation to have imaginary roots. Discriminant < 0> 0= 0

Perfect! When discriminant is negative, the equation has complex/imaginary roots.

(2) If α and β are roots of a quadratic equation, express the equation in terms of α + β and αβ. x2 - (α + β)2(α + β)(α - β)x + αββα = 0

Excellent! This is the standard form using sum and product of roots.

(3) Find the discriminant of the equation 2x² + 4x + 5 = 0. Discriminant =

Correct! Since discriminant is negative, this equation has imaginary roots.

(4) Write a quadratic equation whose roots are reciprocals of each other. x2−2x+12x2+2x+1 = 0

Perfect! When roots are reciprocals, their product is always 1.

Short Answer Questions (2 Marks Each)

(1) Solve x2 + 2x + 3 = 0 and comment on the nature of the roots. Discriminant =

Since discriminant < 0, the equation has imaginaryreal roots

Roots: x = −1±i22±i3±i2

Excellent! The roots are complex conjugates.

(2) Find the quadratic equation whose roots are 1/2 and -3. 2x2x2 + + = 0

Perfect! Always clear fractions for the final answer.

(3) If the sum of the roots of a quadratic equation is 4 and the product is 1, find the equation.

Equation: x2−4x+1x2+4x−1 = 0

Excellent application of sum and product relationships!

(4) Solve the equation 3x2 - 5x + 2 = 0 using the completing the square method.

Roots: x = or

Excellent systematic approach to completing the square!

Long Answer Questions (4 Marks Each)

(1) A boat can go 20 km upstream and return in 4 hours. If the speed of the stream is 3 km/h, find the speed of the boat in still water.

Speed of boat in still water = km/h

Perfect application of quadratic equations to motion problems!

(2) A rectangular garden has an area of 300 m2. Its length is 4 m more than its breadth. Find its dimensions.

Breadth = m, length = m

Excellent problem-solving with practical constraints!

(3) A motorcyclist is moving such that the distance covered at time t seconds is given by s(t) = −2t2 + 12t + 5. Find the time at which he reaches the maximum distance and calculate the distance.

Time taken = seconds

Maximum distance = meters

Perfect analysis of quadratic motion with vertex form!

Part B: Objective Questions (1 Mark Each)

Choose the correct answer and write the option (a/b/c/d)

(1) The equation x2 + x + 1 = 0 has

(a) Two real roots (b) Equal roots (c) No real roots (d) Rational roots

Correct! Discriminant = 12 - 4(1)(1) = -3 < 0, so no real roots.

(2) The value of k for which x2 + kx + 9 = 0 has imaginary roots is

(a) k2 < 36 (b) k2 > 36 (c) k2 = 36 (d) k = 6

Correct! For imaginary roots: discriminant < 0, so k2 - 36 < 0, hence k2 < 36.

(3) The sum and product of the roots of x2 - 4x - 5 = 0 are

(a) 4, -5 (b) -4, -5 (c) 4, 5 (d) -4, 5

Correct! Sum = −−41 = 4, Product = −51 = -5.

(4) The discriminant of 5x2 + 2x + 1 = 0 is

(a) 4 (b) -16 (c) -16 (d) -4

Correct! Discriminant = 22 - 4(5)(1) = 4 - 20 = -16.

(5) The graph of y = x2 + 2x + 3 lies

(a) Above the x-axis (b) On the x-axis (c) Below the x-axis (d) Cuts the x-axis

Correct! Since a > 0 and discriminant < 0, the parabola opens upward and doesn't touch x-axis.

(6) If one root of the equation x2 + px + 1 = 0 is the reciprocal of the other, then

(a) p = 0 (b) p = 1 (c) p = -1 (d) p = 2

Correct! If roots are reciprocals, their product = 1. Here product = 1 already, so any value works, but typically p = 0.

(7) Which of the following equations has complex (imaginary) roots?

(a) x2 - 4x + 3 = 0 (b) x2 - 2x + 1 = 0 (c) x2 + 2x + 5 = 0 (d) x2 + 4x + 4 = 0

Correct! Discriminant = 4 - 20 = -16 < 0, so this has complex roots.

(8) The equation whose roots are -2 and 12 is

(a) 2x2 + 3x - 1 = 0 (b) 2x2 + 3x + 1 = 0 (c) x2 + 3x + 2 = 0 (d) x2 - 3x - 2 = 0

Correct! Sum = -2 + 12= −32, Product = -1. Equation: x2 + 32x - 1 = 0, or 2x2 + 3x - 1 = 0.

(9) The time t at which s(t) = −2t2 + 12t + 5 is maximum is

(a) 3 (b) 2 (c) 4 (d) 5

Correct! Maximum occurs at t = −b2a = −122×−2 = 3.

(10) If the roots of a quadratic equation are 32 and -2, then the equation is

(a) 2x2 + x - 6 = 0 (b) 2x2 - x - 6 = 0 (c) x2 - x - 6 = 0 (d) x2 + x - 6 = 0

Correct! Sum = 32 + (-2) = −12, Product = -3. Equation: x2 + 12x - 3 = 0, or 2x2 + x - 6 = 0.

Quadratic Equations Challenge

Determine whether these statements about quadratic equations are True or False:

Quadratic Equations Quiz

🎉 You Did It! What You've Learned:

By completing this worksheet, you now have a solid understanding of:

(1) Nature of Roots: Using discriminant to determine real, equal, or complex roots

(2) Quadratic Formula: Solving any quadratic equation using x = −b±Δ2a

(3) Sum and Product Relationships: α + β = −ba and αβ = ca for roots

(4) Completing the Square: Alternative method for solving and finding vertex

(5) Real-world Applications: Motion, area, and optimization problems using quadratics

(6) Graph Analysis: Understanding parabola properties and vertex location

(7) Equation Formation: Creating quadratic equations from given root conditions

(8) Advanced Problem Solving: Complex scenarios involving boats, gardens, and motion

Excellent work mastering advanced quadratic equation concepts and applications!