Q: The length of a rectangular plot is 8 m greater than its breadth. If the area of the plot is 308 m2, find the length of the plot.

Let, the breadth of the rectangular plot be x m. Then the length of rectangular plot = (x + 8) m ∴ Area = Length × Breadth = x(x + 8)m2 But the area of the plot is given to be 308 m2 ∴ x( x + 8 ) = 308 ⇒ x2 + 8x - 308 = 0 ⇒ x2 + 22x - 14x - 308 = 0 ⇒ x( x + 22 ) - 14 ( x - 22 ) = 0 ⇒ ( x + 22 ) ( x - 14 ) = 0 ⇒ x = 14, -22 But, x = −22 is not possible, since breath cannot be negative ∴ x = 14 Hence the breadth of the rectangular plot = 14 m Length of the rectangular plot = (14 + 8) m = 22 m.

Q: If x 2 - 3x + 1 = 0, find the value of x + 1/x,

Given equation is x2 - 3x + 1 = 0 ⇒ x2 + 1 = 3x ⇒ x2 + 1/x = 3 ⇒ x2/x + 1/x = 3 ∴ x + 1/x = 3

Q: If one of the roots of quadratic equation 7y 2 - 50y + k = 0 is 7, then what is the value of k?

Given quadratic equation is 7y2 - 50y + k = 0 If one root is 7, then it will satisfy the equation i.e putting y = 7 in equation 7 x (7)2 - 50 x 7 + k = 0 ⇒ 7 x 49 - 350 + k = 0 ⇒ 343 - 350 + k = 0 ∴ k = 7

Q: For what value of k, the equation x 2 + 2(k - 4) x + 2k = 0 has equal roots?

Given equation is x2 + 2(k - 4)x + 2k = 0 On comparing with ax2 + bx + c = 0 Here, a = 1, b = 2(k -4), c = 2k Since, the root are equal, we have D = 0. b2 - 4ac = 0 ∴ 4(k - 4)2 - 8k = 0 4(k2 + 16 - 8k) - 8k = 0 ⇒ 4k2 + 64 - 32k - 8k = 0 ⇒ 4k2 - 40k + 64 = 0 ⇒ k2 - 10k + 16 = 0 ⇒ k2 - 8k - 2k + 16 = 0 ⇒ k(k - 8) -2 (k - 8) = 0 ⇒ (k - 8) (k - 2) = 0 Hence, the value of k 8 or 2.

Q: The quadrictic equation whose roots are 3 and -1, is

Given that, the roots of the quadrictic equation are 3 and -1. Let α = 3 and β = -1 Sum of roots = α + β = 3 - 1 = 2 Products of roots = α . β = (3) (-1) = -3 ∴ Required quadric equation is x2 - (α + β)x + α β = 0 ⇒ x2 - (2)x + (-3) = 0 ⇒ x2 - 2x - 3 = 0

Q: If α and β are the roots of the equation 4x 2 - 19x + 12 = 0, find the equation having the roots 1/α and 1/β

Given equation is 4x2 - 19x + 12 = 0 Let given equation having the roots 1/α and 1/β, Then required equuation is 12x2 - 19x + 4 = 0

Q: The difference in the roots of the equation 2x 2 - 11x + 5 = 0 is

Let α and β be the roots of the quadratic equation 2x2 - 11x + 5 = 0 ∴ α + β = -(-11)/2 = 11/2 ...(i) and αβ = 5/2 Now, (α - β)2 = (α + β)2 - 4αβ = (11/2)2 - 4(5/2) = 121/4 - 20/2 = 121 - 40/4 = 81/4 = (9/2)2 ∴ Difference of roots = (α - β) = 9/2 = 4.5

Q: If {2 1/ 4} is the solution set of a quadric equation, find that equation .

Let α = 2, β = 1/4 Then, α + β = 2 + 1/4 = 9/4 αβ = 2.1/4 = 1/2 Equation having the roots α and β is x2 - (α + β)x + α β = 0 ⇒ x2 - 9/4x + 1/2 = 0 ⇒ 4x2 - 9x + 2 = 0

Q: If p and q are the roots of the equation x 2 + px + q = 0, then

Since, p and q are the roots of the equation x2 + px + q = 0 Then, p + q = - p and pq = q Now, pq = q ⇒ p = 1 Putting the value of p in p + q = - p, we get 1 + q = - 1 ⇒ q = - 2

Question 1

The expression x2 − x + 1 has :

Accepted Answer

According to question , we have Comparing x2 - x + 1 with ax2 + bx + c , we get a = 1, b = −1, c = 1 Here D = b2 - 4ac = ( - 1 )2 - 4 (1)(1) = 1 - 4 = - 3 Since D < 0, so the given expression has no proper linear factor.

Question 2

The length of a rectangular plot is 8 m greater than its breadth. If the area of the plot is 308 m2, find the length of the plot.

Accepted Answer

Let, the breadth of the rectangular plot be x m. Then the length of rectangular plot = (x + 8) m ∴ Area = Length × Breadth = x(x + 8)m2 But the area of the plot is given to be 308 m2 ∴ x( x + 8 ) = 308 ⇒ x2 + 8x - 308 = 0 ⇒ x2 + 22x - 14x - 308 = 0 ⇒ x( x + 22 ) - 14 ( x - 22 ) = 0 ⇒ ( x + 22 ) ( x - 14 ) = 0 ⇒ x = 14, -22 But, x = −22 is not possible, since breath cannot be negative ∴ x = 14 Hence the breadth of the rectangular plot = 14 m Length of the rectangular plot = (14 + 8) m = 22 m.

Question 3

If x 2 - 3x + 1 = 0, find the value of x + 1/x,

Accepted Answer

Given equation is x2 - 3x + 1 = 0 ⇒ x2 + 1 = 3x ⇒ x2 + 1/x = 3 ⇒ x2/x + 1/x = 3 ∴ x + 1/x = 3

Question 4

If one of the roots of quadratic equation 7y 2 - 50y + k = 0 is 7, then what is the value of k?

Accepted Answer

Given quadratic equation is 7y2 - 50y + k = 0 If one root is 7, then it will satisfy the equation i.e putting y = 7 in equation 7 x (7)2 - 50 x 7 + k = 0 ⇒ 7 x 49 - 350 + k = 0 ⇒ 343 - 350 + k = 0 ∴ k = 7

Question 5

For what value of k, the equation x 2 + 2(k - 4) x + 2k = 0 has equal roots?

Accepted Answer

Given equation is x2 + 2(k - 4)x + 2k = 0 On comparing with ax2 + bx + c = 0 Here, a = 1, b = 2(k -4), c = 2k Since, the root are equal, we have D = 0. b2 - 4ac = 0 ∴ 4(k - 4)2 - 8k = 0 4(k2 + 16 - 8k) - 8k = 0 ⇒ 4k2 + 64 - 32k - 8k = 0 ⇒ 4k2 - 40k + 64 = 0 ⇒ k2 - 10k + 16 = 0 ⇒ k2 - 8k - 2k + 16 = 0 ⇒ k(k - 8) -2 (k - 8) = 0 ⇒ (k - 8) (k - 2) = 0 Hence, the value of k 8 or 2.

Question 6

The quadrictic equation whose roots are 3 and -1, is

Accepted Answer

Given that, the roots of the quadrictic equation are 3 and -1. Let α = 3 and β = -1 Sum of roots = α + β = 3 - 1 = 2 Products of roots = α . β = (3) (-1) = -3 ∴ Required quadric equation is x2 - (α + β)x + α β = 0 ⇒ x2 - (2)x + (-3) = 0 ⇒ x2 - 2x - 3 = 0

Question 7

If α and β are the roots of the equation 4x 2 - 19x + 12 = 0, find the equation having the roots 1/α and 1/β

Accepted Answer

Given equation is 4x2 - 19x + 12 = 0 Let given equation having the roots 1/α and 1/β, Then required equuation is 12x2 - 19x + 4 = 0

Question 8

The difference in the roots of the equation 2x 2 - 11x + 5 = 0 is

Accepted Answer

Let α and β be the roots of the quadratic equation 2x2 - 11x + 5 = 0 ∴ α + β = -(-11)/2 = 11/2 ...(i) and αβ = 5/2 Now, (α - β)2 = (α + β)2 - 4αβ = (11/2)2 - 4(5/2) = 121/4 - 20/2 = 121 - 40/4 = 81/4 = (9/2)2 ∴ Difference of roots = (α - β) = 9/2 = 4.5

Question 9

If {2 1/ 4} is the solution set of a quadric equation, find that equation .

Accepted Answer

Let α = 2, β = 1/4 Then, α + β = 2 + 1/4 = 9/4 αβ = 2.1/4 = 1/2 Equation having the roots α and β is x2 - (α + β)x + α β = 0 ⇒ x2 - 9/4x + 1/2 = 0 ⇒ 4x2 - 9x + 2 = 0

Question 10

If p and q are the roots of the equation x 2 + px + q = 0, then

Accepted Answer

Since, p and q are the roots of the equation x2 + px + q = 0 Then, p + q = - p and pq = q Now, pq = q ⇒ p = 1 Putting the value of p in p + q = - p, we get 1 + q = - 1 ⇒ q = - 2

Question 11

Number of solution of the equation √ x2 - x + 1 + 1/√ x2 - x + 1 = 2 - x 2 is

Accepted Answer

We know that, AM ≥ GM ∴ √a + 1/√a ≥ 2 Here, √x2 - x + 1 + 1/√x2 - x + 1 ≥ 2 ⇒ 2 - x2 ≥ 2 ⇒ x2 ≤ 0 ⇒ x = 0 Hence, the given equation has only one solution.

Question 12

Two number whose sum is 8 and difference is 4, are the roots of the equation

Accepted Answer

Let the roots be α and β . Then, α + β = 8 ......(i) α - β = 4 ..(ii) On solving Eqs. (i) and (ii), we get α = 6, β = 2 ∴ Required equation is x2 - (α + β) x + (α β) = 0 ⇒ x2 - (6 + 2)x + 6 x 2 = 0 ⇒ x2 - 8x + 12 = 0

Question 13

The sum of a number and its reciprocal is 10/3, then the numbers are

Accepted Answer

Let the number be x and 1/x, Then, x + 1/x = 10/3 ⇒ (x2 + 1)/x = 10/3 ⇒ 3x2 - 10x + 3 = 0 ⇒ 3x2 - 9x - x + 3 = 0 ⇒ 3x(x - 3) -1 (x - 3) = 0 ⇒ (3x - 1) (x - 3) = 0 ∴ x = 1/3, x = 3

Question 14

The values of p for which the difference between the roots of the equation x 2 + px + 8 = 0 is 2 , are

Accepted Answer

α + β = - p, α β = 8 Given, α - β = 2 On squaring both sides, we get (α - β)2 = 4 ⇒ (α + β)2 - 4αβ = 4 ⇒ p2 - 32 = 4 ⇒ p2 = 36 ⇒ p = ± 6

Question 15

x 2 + x - 20 = 0 y 2 - y - 30 = 0

Accepted Answer

x2 + x - 20 = 0 [by factorisation method] ⇒ x2 + 5x - 4x - 20 = 0 ⇒ x(x + 5) - 4(x + 5) = 0 (x + 5) (x - 4) = 0 ∴ x = -5 or 4 and y2 - y - 30 = 0 ⇒ y2 - 6y + 5y - 30 = 0 ⇒ y(y - 6) + 5(y - 6) = 0 ⇒ (y - 6) (y + 5) = 0 ∴ y = 6 or - 5 Hence, y ≥ x or x ≤ y

Question 16

225x 2 - 4 = 0; √ 225y + 2 = 0

Accepted Answer

225x2 - 4 = 0; ⇒ 225x2 = 4 ⇒ x2 = 4/225 ∴ x = √4/225 = ± 2/15, i.e., 2/15 and -2/15 and √225y + 2 = 0 or √225y = -2 On squaring both sides, we get (√225y)2 = (-2)2 ⇒ 225y = 4 ∴ y = 4/225 So, relation cannot be established because 4/225 lies between 2/15 and -2/15.

Question 17

x - √ 121 = 0 y 2 - 121 = 0

Accepted Answer

x - √121 = 0 ⇒ x = √121 ⇒ x = ± 11 and y2 - 121 = 0 ⇒ y2 = 121 ⇒ y = √121 = ± 11 ∴ x = y

Question 18

x 2 - 16 = 0 y 2 - 9y + 20 = 0

Accepted Answer

x2 - 16 = 0 ⇒ x2 = 16 ⇒ x = √16 = ± 4 and y2 - 9y + 20 = 0 y2 - 4y - 5y + 20 = 0 ⇒ y(y - 4) - 5(y - 4) = 0 ⇒ y = 5, 4 ⇒ y ≥ x or x ≤ y

Question 19

x 2 - 32 = 112; y - √ 169 = 0

Accepted Answer

x2 - 32 = 112 ⇒ x2 = 112 + 32 ⇒ x2 = 144 ∴ x = ± 12 and y - √169 = 0 ⇒ y = √169 ⇒ y = ± 13 ∴ Relation cannot be established.

Quadratic Equation Questions