Solution of Chapter 4. Quadratic Equations (NCERT - Exemplar Mathematics Book)

Which of the following is a quadratic equation?

Which of the following is not a quadratic equation?

Which of the following equation has 2 as a root?

If is a root of the equation , then the value of k is

Which of the following equations has the sum of its roots as 3?

Value(s) of k for which the quadratic equation 2x² – kx + k = 0 has equal roots is/are

Which constant value must be added and subtracted to solve the quadratic equation

by the method of completing the squares?

The quadratic equation has

Which of the following equations has two distinct real roots?

Which of the following equations has no real roots?

(x² + 1)² - x² = 0 has

State whether the following quadratic equations have two distinct real roots. Justify your answer.

x² - 3x + 4 = 0

State whether the following quadratic equations have two distinct real roots. Justify your answer.

2x² + x – 1 = 0

State whether the following quadratic equations have two distinct real roots. Justify your answer.

State whether the following quadratic equations have two distinct real roots. Justify your answer.

3x² – 4x + 1 = 0

State whether the following quadratic equations have two distinct real roots. Justify your answer.

(x + 4)² - 8x = 0

State whether the following quadratic equations have two distinct real roots. Justify your answer.

State whether the following quadratic equations have two distinct real roots. Justify your answer.

State whether the following quadratic equations have two distinct real roots. Justify your answer.

x (1 - x) – 2 = 0

State whether the following quadratic equations have two distinct real roots. Justify your answer.

(x - 1) (x + 2) + 2 = 0

State whether the following quadratic equations have two distinct real roots. Justify your answer.

(x + 1) (x - 2) + x = 0

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 at least one real root.

(iii) Every quadratic equation has at least two roots.

(iv) Every quadratic equation has at most two roots.

(v) If the coefficient of x² and the constant term of a quadratic equation have opposite signs, then the quadratic equation has real roots.

(vi) If the coefficient of x² 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.

A quadratic equation with integral coefficient has integral roots. Justify your answer.

Does there exist a quadratic equation whose coefficients are rational but both of its roots are irrational? Justify your answer.

Does there exist a quadratic equation whose coefficients are all distinct irrationals but both the roots are rational? Justify.

Is 0.2 a root of the equation x² - 0.4 = 0? Justify your answer.

If b = 0, c < 0, is it true the roots of x² + bx + c = 0 are numerically equal and opposite in sign? Justify your answer.

Find the roots of the quadratic equations by using the quadratic formula in each of the following:

2x² - 3x – 5 = 0

Find the roots of the quadratic equations by using the quadratic formula in each of the following:

5x² + 13x + 8 = 0

Find the roots of the quadratic equations by using the quadratic formula in each of the following:

-3x² + 5x + 12 = 0

Find the roots of the quadratic equations by using the quadratic formula in each of the following:

-x² + 7x – 10 = 0

Find the roots of the quadratic equations by using the quadratic formula in each of the following:

Find the roots of the quadratic equations by using the quadratic formula in each of the following:

Find the roots of the quadratic equations by using the quadratic formula in each of the following:

Find the roots of the following quadratic equations by factorisation method.

Find the roots of the following quadratic equations by factorisation method.

Find the roots of the following quadratic equations by factorisation method.

Find the roots of the following quadratic equations by factorisation method.

Find the roots of the following quadratic equations by factorisation method.

Find whether the following equations have real roots. If real roots exist, find them.

8x² + 2x – 3 = 0

Find whether the following equations have real roots. If real roots exist, find them.

-2x² + 3x + 2 = 0

Find whether the following equations have real roots. If real roots exist, find them.

5x² - 2x – 10 = 0

Find whether the following equations have real roots. If real roots exist, find them.

Find whether the following equations have real roots. If real roots exist, find them.

Find a natural number whose square diminished by 84 is equal to thrice of 8 more than the given number.

A natural number, when increased by 12, equals 160 times its reciprocal. Find the number.

A train, travelling at a uniform speed for 360 km, would have taken 48 mins less to travel the same distance, if its peed were 5 km/hr more. Find the original speed of the train.

If Zeba were younger by 5 years than what she really is, then the square of her age (in years) would have been 11 more than five times her actual age, what is her age now?

At present Asha’s age (in years) is 2 more than the square of her daughter Nisha’s age. When Nisha grows to her mother’s present age, Asha’s age would be one year less than 10 times the present age of Nisha. Find the present ages of both Asha and Nisha.

In the centre of a rectangular lawn of dimensions 50 m x 40 m, a rectangular pond has to be constructed, so that the area of the grass surrounding the pond would be 1184 m² [see figure]. Find the length and breadth of the pond.

At t min past 2 pm, the time needed by the minute hand of a clock to show 3pm was found to be 3 min less than min. Find t.