KSEEB Solutions for Class 10 Maths Chapter 10 Quadratic Equations Additional Questions

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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. ax² + bx + c = 0
b. ax + c = 0
c. ax³ + bx² + c = 0
d. ax⁴ + bx³ + cx² + dx + c= 0
Answer:
a. ax² + bx + c = 0

Question 3.
The standard form of pure quadratic equation is
a. ax² + bx + c = 0
b. ax² + c = 0
c. ax³ + bx² + cx = 0
d. ax⁴ + bx³ + cx² + dx + c= 0
Answer:
b. ax² + 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 3x² – 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 3x² = 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. x² + (x + 1)² = 25
b. x² – (x – 1)² = 25
c. (x + 1) – x² = 25
d. x² + (x + 1)² + 25 = 0
Answer:
a. x² + (x + 1)² = 25

Question 11.
The discriminant of the quadratic equation ax² + bx + c = 0 is
a. ∆ = b² + 4ac
b. ∆ = b² – 4ac
c. ∆ = 4abc
d. ∆ = b² × 4ac
Answer:
b. ∆ = b² – 4ac

Question 12.
The nature of the roots of the quadratic equation 2x² – 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 = \(\frac{1}{2}\) mv² then v is equal to

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

Question 14.
If ax² + 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. x² + 5x + 6 = 0
b. x² – 5x + 6 = 0
c. x² – 5x – 6 = 0
d. x² + 5x – 6 = 0
Answer:
b. x² – 5x + 6 = 0

Question 16.
The sum of the roots of the equation 2x² – 8 = 0 is
a. 2
b. 4
c. – 4
d. 0
Answer:
d. 0

Question 17.
The product of the equation 3x² = 9x is
a. 0
b. – 3
c. 2
d. 9
Answer:
a. 0

Question 18.
If l² = r² + d² 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:
ax² + bx + c = 0

Question 2.
If b² – 4ac < 0, then find the nature of roots of Quadratic equation ax² + 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 = \(\frac{35}{y}\)
7y² = 35

Question 5.
Find the discriminant of x² + 5x + 5 = 0
Answer:
x² + 5x + 5 = 0
It is in the form of ax² + bx + c = 0
a = 1, b = 5 and c = 5
∆ = b² – 4ac
= (5)² – 4 (1) (5) = 25 – 20
∆ = 5
Discriminant = 5

Question 6.
Solve 5x² = 625
Answer:
5x² = 625

Question 7.
Write the formula to find the roots of Quadratic equation ap² + bp + c = 0
Answer:

Question 8.
Is x = – 2 is a solution of 3x² + 13x + 14 = 0
Answer:
LHS 3x² + 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 ax² + bx + c = 0 to become a linear equation.
Answer:
a = 0

Question 10.
Write the formula to find the nature of roots?
Answer:
discriminat = b² – 4ac

Question 11.
Write the formula to form a Quadratic equation if their roots m and n are given.
Answer:
x² – (m + n) x + mn = 0

III. Long Answer Questions:

Question 1.
Factorise \(\sqrt{2}\)x² + 7x + 5\(\sqrt{2}\) = 0
Answer:

Question 2.
Factorize 0.2t² – 0.04t = 0.03
Answer:
0.2t² – 0.04t – 0.03 = 0
Multiply by 100
20t² – 4t – 3 = 0
20t² – 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 a² b² x² – (a² + b²) x + 1 = 0
Answer:

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

Question 4.
Solve 2x² + 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
6x² – 4x + 9x – 6 + 2 = 0
6x² + 5x – 4 = 0
It is in the form of ax² + 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 (x² + 1) = x (a² + 1) using formula.
Answer:
a(x² + 1) = x (a² + 1)
ax² + a = x (a²) + 1
ax² – (a² + 1)x + a = 0
It is in the form of ax² + bx + c = 0
a = a, b = – (a² + 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 x² – mx + 9 = 0 are
i. Equal
ii. distinct
iii. imaginary.
Answer:
x² – mx + 9 = 0
It is in the form of ax² + bx + c = 0
a = 1, b = – m and c = 9
∆ = b² – 4ac
∆ = – (m)² – 4 × 1 × 9
∆ = m² – 36
i) If roots are equal
∆ = 0
m² – 36 = 0
m² = 36
m = \(\sqrt{36}\) = ±6
m = 6

ii) If roots are distinct
∆ > 0
m² – 36 > 0
m² > 36
m > ±6
∴ m > 6

iii) If roots are imaginary
∆ < 0
m² – 36 < 0
m² < 36
∴ m < ±6
∴ m < 6

Question 8.

Answer:
10x² + 12x + 36x – 120 – x² = 0
9x² + 36x – 108 = 0 divide by 9
x² + 4x – 12 = 0
x² + 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) c² + 2 (P + 1) c + P = 0 has equal roots.
Answer:
It is in the form of ax² + bx + c = 0
a = (3P + 1) , b = 2(P + 1) and c = P
∆ = b² – 4ac = 0 [has equal roots]
[2(P + 1)]² – 4 × (3P + 1) P = 0
4(P² + 2P + 1) – 4P(3P + 1) = 0
4[P² + 2P + 1 – 3P² – P] = 0
divide by 4
– 2P² + P + 1 = 0 divide by – 1
2P² – P – 1 = 0
2P² – 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

x² = 2400 – 100x
x² + 100x – 2400 = 0
x² + 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