Students can Download Class 10 Maths Chapter 10 Quadratic Equations Additional Questions, Notes Pdf, KSEEB Solutions for Class 10 Maths helps you to revise the complete Karnataka State Board Syllabus and score more marks in your examinations.

## Karnataka State Syllabus Class 10 Maths Chapter 10 Quadratic Equations Additional Questions

I. Multiple Choice Questions:

Question 1.

The degree of a quadratic equations is

a. 1

b. 2

c. 3

d. 4

Answer:

b. 2

Question 2.

The standard form of quadratic equation is

a. ax^{2} + bx + c = 0

b. ax + c = 0

c. ax^{3} + bx^{2} + c = 0

d. ax^{4} + bx^{3} + cx^{2} + dx + c= 0

Answer:

a. ax^{2} + bx + c = 0

Question 3.

The standard form of pure quadratic equation is

a. ax^{2} + bx + c = 0

b. ax^{2} + c = 0

c. ax^{3} + bx^{2} + cx = 0

d. ax^{4} + bx^{3} + cx^{2} + dx + c= 0

Answer:

b. ax^{2} + c = 0

Question 4.

An quadratic equation has only

a. 1 root

b. 2 roots

c. 3 roots

d. 4 roots

Answer:

b. 2 roots

Question 5.

The roots of the quadratic equation 3x^{2} – 6x = 0 are

a. (0, 2)

b. (3, 6)

c. (0, – 2)

d. (0, 6)

Answer:

a. (0, 2)

Question 6.

The roots of the quadratic equation (2x + 1)( 3x – 5) = 0 are

Answer:

a. \(\left[\frac{-1}{2}, \frac{5}{3}\right]\)

Question 7.

The roots of the quadratic equation 3x^{2} = 36 are

a. ±2\(\sqrt{3}\)

b. ±3\(\sqrt{2}\)

c. ±12

d. ±\(\sqrt{2}\)

Answer:

a. ±2\(\sqrt{3}\)

Question 8.

The consecutive even integers are

a. (2x) (x + 2)

b. (x) (x + 2)

c. (x) (x + 1)

d. (x) (x – 1)

Answer:

b. (x) (x + 2)

Question 9.

Two consecutive positive integers differ by

a. 2

b. 1

c. 3

d. 4

Answer:

b. 1

Question 10.

The sum of the squares of two consecutive natural numbers is 25 represent this statement in the form of a quadratic equation.

a. x^{2} + (x + 1)^{2} = 25

b. x^{2} – (x – 1)^{2} = 25

c. (x + 1) – x^{2} = 25

d. x^{2} + (x + 1)^{2} + 25 = 0

Answer:

a. x^{2} + (x + 1)^{2} = 25

Question 11.

The discriminant of the quadratic equation ax^{2} + bx + c = 0 is

a. ∆ = b^{2} + 4ac

b. ∆ = b^{2} – 4ac

c. ∆ = 4abc

d. ∆ = b^{2} × 4ac

Answer:

b. ∆ = b^{2} – 4ac

Question 12.

The nature of the roots of the quadratic equation 2x^{2} – 4x + 3 = 0 are

a. real and distinct

b. real and equal

c. no real roots

d. imaginary

Answer:

d. imaginary

Question 13.

If k = \(\frac{1}{2}\) mv^{2} then v is equal to

Answer:

a. ±\(\sqrt{\frac{2 \mathrm{k}}{\mathrm{m}}}\)

Question 14.

If ax^{2} + bx + c = 0 then quadratic formula is

Answer:

a. x = \(\frac{-b \pm \sqrt{b^{2}-4 a c}}{2 a}\)

Question 15.

The quadratic equation whose roots are 2 and 3 is

a. x^{2} + 5x + 6 = 0

b. x^{2} – 5x + 6 = 0

c. x^{2} – 5x – 6 = 0

d. x^{2} + 5x – 6 = 0

Answer:

b. x^{2} – 5x + 6 = 0

Question 16.

The sum of the roots of the equation 2x^{2} – 8 = 0 is

a. 2

b. 4

c. – 4

d. 0

Answer:

d. 0

Question 17.

The product of the equation 3x^{2} = 9x is

a. 0

b. – 3

c. 2

d. 9

Answer:

a. 0

Question 18.

If l^{2} = r^{2} + d^{2} then the value of d is equal to

Answer:

c. d = ±\(\sqrt{l^{2}-r^{2}}\)

Question 19.

The name of the graph of y = 2x + 3 is called

a. Parabola

b. Hyper bola

c. Straight line

d. Bar graph

Answer:

c. Straight line

Question 20.

The name of the graph y = mx + c is called

a. Straight line

b. Parabola

c. Hyper bola

d. Pie chart

Answer:

a. Straight line

II. Short Answer Questions:

Question 1.

Write the standard form of Quadratic equation

Answer:

ax^{2} + bx + c = 0

Question 2.

If b^{2} – 4ac < 0, then find the nature of roots of Quadratic equation ax^{2} + bx + c = 0.

Answer:

Roots are imaginary

Question 3.

Write the standard form of the equation \(\frac{6}{x}+\frac{x}{1}\) = 5

Answer:

Question 4.

If 7y = \(\frac{35}{y}\) then find the value of y.

Answer:

7y = \(\frac{35}{y}\)

7y^{2} = 35

Question 5.

Find the discriminant of x^{2} + 5x + 5 = 0

Answer:

x^{2} + 5x + 5 = 0

It is in the form of ax^{2} + bx + c = 0

a = 1, b = 5 and c = 5

∆ = b^{2} – 4ac

= (5)^{2} – 4 (1) (5) = 25 – 20

∆ = 5

Discriminant = 5

Question 6.

Solve 5x^{2} = 625

Answer:

5x^{2} = 625

Question 7.

Write the formula to find the roots of Quadratic equation ap^{2} + bp + c = 0

Answer:

Question 8.

Is x = – 2 is a solution of 3x^{2} + 13x + 14 = 0

Answer:

LHS 3x^{2} + 13x + 14

= 3 (- 2)^{2} + 13 (- 2) + 14

= 3 × 4 – 26 + 14

= 12 – 26 + 14

= 26 – 26 = 0

∴ x = – 2 is a solution

Question 9.

What is the condition for an equation of the form ax^{2} + bx + c = 0 to become a linear equation.

Answer:

a = 0

Question 10.

Write the formula to find the nature of roots?

Answer:

discriminat = b^{2} – 4ac

Question 11.

Write the formula to form a Quadratic equation if their roots m and n are given.

Answer:

x^{2} – (m + n) x + mn = 0

III. Long Answer Questions:

Question 1.

Factorise \(\sqrt{2}\)x^{2} + 7x + 5\(\sqrt{2}\) = 0

Answer:

Question 2.

Factorize 0.2t^{2} – 0.04t = 0.03

Answer:

0.2t^{2} – 0.04t – 0.03 = 0

Multiply by 100

20t^{2} – 4t – 3 = 0

20t^{2} – 10t + 6t – 3 = 0

10t(2t- 1) + 3(2t – 1) = 0

(2t – 1) (10t + 3) = 0

(2t – 1) = 0 (or) (10t + 3) = 0

2t = 1 (or) 10t = – 3

t = \(\frac{1}{2}\) (or) t = \(\frac{-3}{10}\)

∴ \(\frac{1}{2}\) and \(\frac{-3}{10}\) are the roots of the equation.

Question 3.

Factorize a^{2} b^{2} x^{2} – (a^{2} + b^{2}) x + 1 = 0

Answer:

\(\frac{1}{a^{2}}\) & \(\frac{1}{b^{2}}\) are the roots of the equation.

Question 4.

Solve 2x^{2} + 5x – 3 = 0 by completing squares method.

Answer:

∴ \(\frac{1}{2}\) & – 3 and are the roots of the given equation.

Question 5.

Solve (2x + 3) (3x – 2) + 2 = 0 using formula.

Answer:

(2x + 3) (3x – 2) + 2 = 0

6x^{2} – 4x + 9x – 6 + 2 = 0

6x^{2} + 5x – 4 = 0

It is in the form of ax^{2} + bx + c = 0

a = 6, b = 5 and c = – 4

∴ \(\frac{-4}{3}\) and \(\frac{1}{2}\) are the roots of the given equation.

Question 6.

Solve a (x^{2} + 1) = x (a^{2} + 1) using formula.

Answer:

a(x^{2} + 1) = x (a^{2} + 1)

ax^{2} + a = x (a^{2}) + 1

ax^{2} – (a^{2} + 1)x + a = 0

It is in the form of ax^{2} + bx + c = 0

a = a, b = – (a^{2} + 1) and c = a

x = \(\frac{2 a^{2}}{2 a}\) (or) x = \(\frac{2}{2 \mathrm{a}}\)

x = a (or) x = \(\frac{1}{a}\)

∴ a and \(\frac{1}{a}\) are the roots of the given equation.

Question 7.

For what Positive value of ‘m’ roots x^{2} – mx + 9 = 0 are

i. Equal

ii. distinct

iii. imaginary.

Answer:

x^{2} – mx + 9 = 0

It is in the form of ax^{2} + bx + c = 0

a = 1, b = – m and c = 9

∆ = b^{2} – 4ac

∆ = – (m)^{2} – 4 × 1 × 9

∆ = m^{2} – 36

i) If roots are equal

∆ = 0

m^{2} – 36 = 0

m^{2} = 36

m = \(\sqrt{36}\) = ±6

m = 6

ii) If roots are distinct

∆ > 0

m^{2} – 36 > 0

m^{2} > 36

m > ±6

∴ m > 6

iii) If roots are imaginary

∆ < 0

m^{2} – 36 < 0

m^{2} < 36

∴ m < ±6

∴ m < 6

Question 8.

Answer:

10x^{2} + 12x + 36x – 120 – x^{2} = 0

9x^{2} + 36x – 108 = 0 divide by 9

x^{2} + 4x – 12 = 0

x^{2} + 6x – 2x – 12 = 0

(x + 6) – 2 (x + 6) = 0

(x + 6) (x – 2) = 0

x = – 6 (or) x = 2

Question 9.

Find the value of ‘P’ in a Quadratic equation (3P + 1) c^{2} + 2 (P + 1) c + P = 0 has equal roots.

Answer:

It is in the form of ax^{2} + bx + c = 0

a = (3P + 1) , b = 2(P + 1) and c = P

∆ = b^{2} – 4ac = 0 [has equal roots]

[2(P + 1)]^{2} – 4 × (3P + 1) P = 0

4(P^{2} + 2P + 1) – 4P(3P + 1) = 0

4[P^{2} + 2P + 1 – 3P^{2} – P] = 0

divide by 4

– 2P^{2} + P + 1 = 0 divide by – 1

2P^{2} – P – 1 = 0

2P^{2} – 2P + P – 1 = 0

2P(P – 1) + 1 (P – 1) = 0

(P – 1) (2P + 1) = 0

(P – 1) = 0 (or) 2P + 1 = 0

P = 1 (or) 2P = – 1

P = \(\frac{-1}{2}\)

∴ P = 1 (or) \(\frac{-1}{2}\)

Question 10.

In Rhombus ABCD the diagonals AC and BC intersect at E. If AE = x, BE = x + 7 and AB = x + 8. Find the lengths of the diagonals AC and BC.

Answer:

In Rhombus ABCD, AC and BD are diagonals are intersect at E.

x (x – 5) + 3 ( x – 5) = 0

(x – 5) (x + 3) = 0

x – 5 = 0 & x + 3 = 0

x = 5 & x = – 3

AE = x = 5 cm

∴ AC = 5 + 5 = 10 cm

BE = x + 7 = 5 + 7 = 12 cm

BD= 12 + 12 = 24 cm

∴ two diagonals are 10 cm and 24 cm.

Question 11.

A motor boat whose speed is 15 km/hr in still water goes 30 km downstream and comes back in a total of 4 hours 30 minutes determine the speed of the stream.

Answer:

Let the speed of the stream be x km/hr

Speed of downstream = (15 + x) km/hr.

Time taken by the boat to go 30 km

downstream = \(\frac{30}{15+x}\) hours

Speed upstream = 15 – x hours

Time taken by the boat to go 30 km

Upstream = \(\frac{30}{15 – x}\) hrs

It is given that the boat returns to the same point in 4 hours 30 minutes.

The speed of the stream = 5 km/hr.

Question 12.

A dealer sells an article for ₹ 24 and gains as much percent as the cost price of the article. Find the cost price of the article.

Answer:

Let the C.P of the article = x

S.P = ₹ 24

Gain = x% of C.P

x^{2} = 2400 – 100x

x^{2} + 100x – 2400 = 0

x^{2} + 120x – 20x – 2400 = 0

x(x + 120) – 20(x + 120) = 0

(x + 120) (x – 20) = 0

x + 120 = 0 & x – 20 = 0

x = – 120 & x = 20

∴ The C.P of the article = ₹ 20