Grade 10 : Mathematics - 4. Quadratic Equations (Set 3)

Question 1

Find the positive value of $k$ for which the quadratic equation $2x^2 - kx + 8 = 0$ has real and equal roots.

Accepted Answer

B) 8

Answer

A) 4

Answer

C) 16

Answer

D) 64

Question 2

Determine the nature of the roots of the quadratic equation $3x^2 - 2\sqrt{6}x + 2 = 0$.

Accepted Answer

B) Two equal real roots

Answer

A) Two distinct real roots

Answer

C) No real roots

Answer

D) Three real roots

Question 3

If $\alpha$ and $\beta$ are the roots of the quadratic equation $x^2 - 7x + 12 = 0$, calculate the value of $\alpha^2 + \beta^2$.

Accepted Answer

A) 25

Answer

B) 37

Answer

C) 49

Answer

D) 13

Question 4

Find the value of $k$ if $x = 3$ is a root of the quadratic equation $kx^2 - 11x + 6 = 0$.

Accepted Answer

C) 3

Answer

A) 1

Answer

B) 2

Answer

D) 9

Question 5

A right-angled triangle has a hypotenuse of length 13 cm. One side of the triangle is 7 cm longer than the other side.

Find the length of the shorter side of the triangle.

Accepted Answer

B) 5 cm

Answer

A) 3 cm

Answer

C) 8 cm

Answer

D) 12 cm

Question 6

Solve the algebraic equation for $x$: $\frac{x+1}{x-1} + \frac{x-2}{x+2} = 3$ (where $x eq 1, -2$).

Accepted Answer

A) x = 2, -5

Answer

B) x = -2, 5

Answer

C) x = 3, -4

Answer

D) x = 1, -10

Question 7

Find the quadratic equation whose roots are the reciprocals of the roots of $2x^2 - 5x + 3 = 0$.

Accepted Answer

A) 3x² - 5x + 2 = 0

Answer

B) 2x² + 5x - 3 = 0

Answer

C) 3x² + 5x - 2 = 0

Answer

D) 5x² - 3x + 2 = 0

Question 8

A express train travels 360 km at a uniform speed. If its speed had been 5 km/h faster, it would have taken 1 hour less for the same journey.

Find the original speed of the train.

Accepted Answer

C) 40 km/h

Answer

A) 30 km/h

Answer

B) 35 km/h

Answer

D) 45 km/h

Question 9

Which condition guarantees that the quadratic equation $ax^2 + bx + c = 0$ ($a eq 0$) has no real roots?

Accepted Answer

C) b² - 4ac < 0

Answer

A) b² - 4ac > 0

Answer

B) b² - 4ac = 0

Answer

D) b² - 4ac ≥ 0

Question 10

If one root of $x^2 - px + 8 = 0$ is 2, find the value of $p$ and the other root.

Accepted Answer

A) p = 6, other root = 4

Answer

B) p = 4, other root = 6

Answer

C) p = 10, other root = 8

Answer

D) p = -6, other root = -4

Question 11

The sum of a non-zero number and its reciprocal is $\frac{10}{3}$. Find the number.

Accepted Answer

A) 3 or 1/3

Answer

B) 2 or 1/2

Answer

C) 4 or 1/4

Answer

D) 5 or 1/5

Question 12

If $\alpha$ and $\beta$ are roots of $x^2 + x - 2 = 0$, evaluate $\frac{1}{\alpha} + \frac{1}{\beta}$.

Accepted Answer

B) 1/2

Answer

A) -1/2

Answer

C) -2

Answer

D) 2

Question 13

Two water taps together can fill a tank in $1\frac{7}{8}$ hours (which is $\frac{15}{8}$ hours). The tap of larger diameter takes 2 hours less than the smaller one to fill the tank separately.

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

Accepted Answer

C) 5 hours

Answer

A) 3 hours

Answer

B) 4 hours

Answer

D) 6 hours

Question 14

If the quadratic equation $px^2 - 2\sqrt{5}px + 15 = 0$ ($p eq 0$) has two equal roots, find the value of $p$.

Accepted Answer

C) 3

Answer

A) 1

Answer

B) 2

Answer

D) 5

Question 15

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

Accepted Answer

B) 12 and 14

Answer

A) 10 and 12

Answer

C) 14 and 16

Answer

D) 8 and 10

Question 16

Find the values of $m$ for which the quadratic equation $x^2 - (m+1)x + 4 = 0$ has real and equal roots.

Accepted Answer

A) m = 3 or -5

Answer

B) m = 4 or -4

Answer

C) m = 2 or -3

Answer

D) m = 5 or -3

Question 17

Determine whether the equation $x(x+1) + 8 = (x+2)(x-2)$ is a quadratic equation.

Accepted Answer

B) No, because x² terms cancel out

Answer

A) Yes, because it has degree 2

Answer

C) Yes, because it simplifies to x² + 12 = 0

Answer

D) Cannot be determined

Question 18

A rectangular garden has length 3 m more than its width. If the total area of the garden is $70m^2$, calculate its perimeter.

What is the perimeter of the garden?

Accepted Answer

B) 34 m

Answer

A) 28 m

Answer

C) 40 m

Answer

D) 70 m

Question 19

If $\alpha$ and $\beta$ are roots of $2x^2 + 3x - 5 = 0$, evaluate $\alpha^3 + \beta^3$.

Accepted Answer

A) -117/8

Answer

B) 117/8

Answer

C) -27/8

Answer

D) -63/8

Question 20

Form the quadratic equation whose roots are $2 + \sqrt{3}$ and $2 - \sqrt{3}$.

Accepted Answer

A) x² - 4x + 1 = 0

Answer

B) x² + 4x + 1 = 0

Answer

C) x² - 4x - 1 = 0

Answer

D) x² - 2x + 3 = 0

Grade 10 : Mathematics - 4. Quadratic Equations (Set 3)