Grade 10 : Mathematics - 4. Quadratic Equations (Set 4)

Question 1

What is the nature of the roots of the quadratic equation $2x^2 - 4x + 3 = 0$?

Accepted Answer

C) No real roots

Answer

A) Two distinct real roots

Answer

B) Two equal real roots

Answer

D) Three real roots

Question 2

A rectangular playground has a length that is 3 m more than twice its breadth. The total area of the playground is $299m^2$.

What is the breadth of the playground?

Accepted Answer

A) 11 m

Answer

B) 13 m

Answer

C) 15 m

Answer

D) 25 m

Question 3

For what non-zero value of $k$ does the quadratic equation $kx(x - 2) + 6 = 0$ have equal real roots?

Accepted Answer

B) $k = 6$

Answer

A) $k = 4$

Answer

C) $k = 8$

Answer

D) $k = 12$

Question 4

If $\alpha$ and $\beta$ are the roots of the quadratic equation $3x^2 - 7x + 2 = 0$, what is the value of $\frac{1}{\alpha} + \frac{1}{\beta}$?

Accepted Answer

C) $\frac{7}{2}$

Answer

A) $\frac{7}{3}$

Answer

B) $\frac{2}{3}$

Answer

D) $\frac{3}{7}$

Question 5

A motor boat whose speed is 18 km/h in still water takes 1 hour more to go 24 km upstream than to return downstream to the same spot.

Find the speed of the stream.

Accepted Answer

C) 6 km/h

Answer

A) 4 km/h

Answer

B) 5 km/h

Answer

D) 8 km/h

Question 6

Find two consecutive odd positive integers, the sum of whose squares is 290.

Accepted Answer

B) 11 and 13

Answer

A) 9 and 11

Answer

C) 13 and 15

Answer

D) 15 and 17

Question 7

Time $t$ (seconds)	Height $h(t)$ (meters)
0	25
1	40
2	45
3	40
4	25

The vertical height $h(t)$ of a model rocket in meters is modeled by $h(t) = -5t^2 + 20t + 25$. After how many seconds will the rocket reach the ground ($h(t) = 0$)?

Accepted Answer

B) 5 seconds

Answer

A) 4 seconds

Answer

C) 6 seconds

Answer

D) 8 seconds

Question 8

In a right-angled triangle, the hypotenuse is 1 cm more than twice the shortest side. The third side is 7 cm more than the shortest side.

Find the length of the shortest side of the triangle.

Accepted Answer

C) 8 cm

Answer

A) 6 cm

Answer

B) 7 cm

Answer

D) 10 cm

Question 9

Two water taps running together can fill a tank in $9\frac{3}{8}$ hours. The tap of larger diameter takes 10 hours less than the smaller one to fill the tank separately.

Find the time taken by the smaller tap to fill the tank separately.

Accepted Answer

C) 25 hours

Answer

A) 15 hours

Answer

B) 20 hours

Answer

D) 30 hours

Question 10

A rectangular photograph measuring 12 cm by 8 cm is surrounded by a border of uniform width $x$ cm. The total area of the photo and border combined is 165 cm$^2$.

What is the uniform width $x$ of the border?

Accepted Answer

B) 1.5 cm

Answer

A) 1 cm

Answer

C) 2 cm

Answer

D) 2.5 cm

Question 11

Which quadratic equation has roots given by $3 + \sqrt{5}$ and $3 - \sqrt{5}$?

Accepted Answer

A) $x^2 - 6x + 4 = 0$

Answer

B) $x^2 + 6x + 4 = 0$

Answer

C) $x^2 - 6x - 4 = 0$

Answer

D) $x^2 + 6x - 4 = 0$

Question 12

Solve for $x$: $x + \frac{1}{x} = \frac{10}{3}$ where $x eq 0$.

Accepted Answer

B) $3, \frac{1}{3}$

Answer

A) $2, \frac{1}{2}$

Answer

C) $4, \frac{1}{4}$

Answer

D) $5, \frac{1}{5}$

Question 13

If $\alpha$ and $\beta$ are roots of $x^2 - 5x + k = 0$ such that $\alpha - \beta = 1$, find the value of $k$.

Accepted Answer

C) 6

Answer

A) 4

Answer

B) 5

Answer

D) 8

Question 14

A group of students organized an excursion with a fixed food budget of Rs 500. Five members could not attend, causing the cost per attending student to increase by Rs 5.

How many students actually attended the excursion?

Accepted Answer

B) 20

Answer

A) 15

Answer

C) 25

Answer

D) 30

Question 15

Which of the following is the correct completed square form for $2x^2 - 8x + 5 = 0$?

Accepted Answer

A) $2(x - 2)^2 - 3 = 0$

Answer

B) $2(x - 4)^2 - 3 = 0$

Answer

C) $2(x - 2)^2 + 3 = 0$

Answer

D) $2(x - 2)^2 - 11 = 0$

Question 16

An express train takes 3 hours less than a freight train for a journey of 600 km. If the average speed of the freight train is 10 km/h less than that of the express train, find the speed of the express train.

What is the speed of the express train?

Accepted Answer

C) 50 km/h

Answer

A) 40 km/h

Answer

B) 45 km/h

Answer

D) 60 km/h

Question 17

Is it possible to design a rectangular park of perimeter 80 m and area $400m^2$? If so, what are its dimensions?

Accepted Answer

A) Yes, length $= 20m$, breadth $= 20m$

Answer

B) Yes, length $= 25m$, breadth $= 15m$

Answer

C) Yes, length $= 30m$, breadth $= 10m$

Answer

D) No, such a park cannot be designed

Question 18

Statement I: The quadratic equation $x^2 + 4x + 5 = 0$ has no real roots.
Statement II: If the discriminant $D = b^2 - 4ac < 0$ for $ax^2 + bx + c = 0$, then the equation has no real roots.

Which of the following statements is correct regarding Statements I and II?

Accepted Answer

A) Both Statement I and Statement II are true, and Statement II is the correct explanation for Statement I.

Answer

B) Both Statement I and Statement II are true, but Statement II is NOT the correct explanation for Statement I.

Answer

C) Statement I is true, but Statement II is false.

Answer

D) Statement I is false, but Statement II is true.

Question 19

The sum of the squares of two consecutive positive even integers is 340. Find the smaller integer.

Accepted Answer

B) 12

Answer

A) 10

Answer

C) 14

Answer

D) 16

Question 20

Solve the quadratic equation $a^2b^2x^2 + b^2x - a^2x - 1 = 0$ for $x$ ($a eq 0, b eq 0$).

Accepted Answer

A) $x = \frac{1}{b^2}$ or $x = -\frac{1}{a^2}$

Answer

B) $x = -\frac{1}{b^2}$ or $x = \frac{1}{a^2}$

Answer

C) $x = \frac{a^2}{b^2}$ or $x = -\frac{b^2}{a^2}$

Answer

D) $x = a^2$ or $x = -b^2$

Grade 10 : Mathematics - 4. Quadratic Equations (Set 4)