Quiz - 1 For Class X (Quadratic Equation) | Attempts: 414

1.

If the equation x²+ 4x + k = 0 has a real and distinct roots, then
- A.
  
  K < 4
- B.
  
  K > 4
- C.
  
  K ≥ 4
- D.
  
  K ≤4
Correct Answer
A. K < 4

Explanation
If the equation x^2 + 4x + k = 0 has real and distinct roots, it means that the discriminant (b^2 - 4ac) is greater than zero. In this case, the discriminant is 4^2 - 4(1)(k) = 16 - 4k. For the roots to be real and distinct, the discriminant must be greater than zero, so 16 - 4k > 0. Solving this inequality gives k < 4. Therefore, the correct answer is k < 4.

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2.

If the equation x² - ax + 1 = 0 has two distinct roots, then
- A.
  
  |a|= 2
- B.
  
  |a| < 2
- C.
  
  |a| > 2
- D.
  
  None of these
Correct Answer
C. |a| > 2

Explanation
If the equation x^2 - ax + 1 = 0 has two distinct roots, it means that the discriminant (b^2 - 4ac) is greater than 0. In this case, the discriminant is a^2 - 4(1)(1) = a^2 - 4. For the equation to have two distinct roots, the discriminant must be positive, so a^2 - 4 > 0. Solving this inequality, we get a^2 > 4, which implies |a| > 2. Therefore, the correct answer is |a| > 2.

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3.

If ax² + bx + c = 0 has equal roots, then c= ?
- A.
  
  -b/2a
- B.
  
  B/2a
- C.
  
  - b²/2a
- D.
  
  B²/4a
Correct Answer
D. B²/4a

Explanation
If the quadratic equation ax^2 + bx + c = 0 has equal roots, it means that the discriminant (b^2 - 4ac) is equal to zero. The discriminant determines the nature of the roots of a quadratic equation. When it is zero, it indicates that the equation has equal roots. In this case, the discriminant is b^2 - 4ac = 0. Rearranging the equation gives c = b^2/4a, which is the correct answer.

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4.

If x = 1 is a common root of ax² + ax + 2 = 0 and x² + x + b = 0 then ab = ?
- A.
  
  1
- B.
  
  2
- C.
  
  4
- D.
  
  3
Correct Answer
B. 2

Explanation
If x = 1 is a common root of both equations, then substituting x = 1 into the first equation gives a + a + 2 = 0, which simplifies to 2a + 2 = 0. Solving this equation, we find that a = -1. Substituting x = 1 into the second equation gives 1 + 1 + b = 0, which simplifies to b + 2 = 0. Solving this equation, we find that b = -2. Therefore, ab = (-1)(-2) = 2.

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5.

If a and b are roots of the equation x²+ ax + b = 0, then a + b = ?
- A.
  
  1
- B.
  
  2
- C.
  
  - 2
- D.
  
  - 1
Correct Answer
D. - 1

Explanation
If a and b are the roots of the equation x^2 + ax + b = 0, then we can use Vieta's formulas to find the sum of the roots. According to Vieta's formulas, the sum of the roots is equal to the negation of the coefficient of x, divided by the coefficient of x^2. In this equation, the coefficient of x is a and the coefficient of x^2 is 1. Therefore, the sum of the roots is -a/1, which simplifies to -a. So, a + b = -a + b = -1.

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6.

If one root of the equation whose one root is 2 and sum of whose roots is zero, is
- A.
  
  X² + 4 = 0
- B.
  
  X²- 4 = 0
- C.
  
  4x² - 1 = 0
- D.
  
  X² - 2 = 0
Correct Answer
B. X²- 4 = 0

Explanation
The given question states that one root of the equation is 2 and the sum of the roots is zero. The equation that satisfies these conditions is x^2 - 4 = 0. This equation has roots 2 and -2, and their sum is indeed zero. Therefore, the correct answer is x^2 - 4 = 0.

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7.

If y = 1 is a common root of the equations ay² + ay + 3 = 0 and y² + y + b = 0 then ab equals ?
- A.
  
  3
- B.
  
  -7/2
- C.
  
  6
- D.
  
  -3
Correct Answer
A. 3

Explanation
Since y = 1 is a common root of both equations, we can substitute y = 1 into both equations.

For the first equation, we have a(1)^2 + a(1) + 3 = 0, which simplifies to a + a + 3 = 0. This further simplifies to 2a + 3 = 0.

For the second equation, we have (1)^2 + (1) + b = 0, which simplifies to 1 + 1 + b = 0. This further simplifies to 2 + b = 0.

From the first equation, we can solve for a by subtracting 3 from both sides, resulting in 2a = -3. Dividing both sides by 2, we find that a = -3/2.

Substituting this value of a into the second equation, we have 2 + b = 0. Solving for b, we subtract 2 from both sides, resulting in b = -2.

Finally, multiplying a and b together, we have (-3/2)(-2) = 3. Therefore, ab equals 3.

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8.

The values of k for which the quadratic equation 16x² + 4 kx + 9 = 0 has real and equal roots are
- A.
  
  6, -1/6
- B.
  
  36, -36
- C.
  
  6, -6
- D.
  
  3/4, - 3/4
Correct Answer
C. 6, -6

Explanation
The quadratic equation 16x^2 + 4kx + 9 = 0 has real and equal roots when the discriminant is equal to zero. The discriminant is given by b^2 - 4ac, where a = 16, b = 4k, and c = 9. Substituting these values into the discriminant formula, we get (4k)^2 - 4(16)(9) = 16k^2 - 576. Setting this equal to zero and solving for k, we find k = ±6. Therefore, the values of k for which the quadratic equation has real and equal roots are 6 and -6.

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9.

Find two numbers whose sum is 24 and product is 143
- A.
  
  13, 11
- B.
  
  12, 12
- C.
  
  8, 16
- D.
  
  None of these
Correct Answer
A. 13, 11

Explanation
The two numbers that satisfy the given conditions are 13 and 11. When these two numbers are added together, the sum is 24, and when they are multiplied, the product is 143.

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10.

Find two consecutive odd positive integers sum of whose squares is 290
- A.
  
  13, 15
- B.
  
  11, 13
- C.
  
  7, 9
- D.
  
  5, 7
Correct Answer
B. 11, 13

Quiz - 1 For Class X (Quadratic Equation)

If the equation x2 + 4x + k = 0 has a real and distinct roots, then

If the equation x2 - ax + 1 = 0 has two distinct roots, then

If ax2 + bx + c = 0 has equal roots, then c= ?

If x = 1 is a common root of ax2 + ax + 2 = 0 and x2 + x + b = 0 then ab = ?

If a and b are roots of the equation x2 + ax + b = 0, then a + b = ?

If one root of the equation whose one root is 2 and sum of whose roots is zero, is

If y = 1 is a common root of the equations ay2 + ay + 3 = 0 and y2 + y + b = 0 then ab equals ?

The values of k for which the quadratic equation 16x2 + 4 kx + 9 = 0 has real and equal roots are

Find two numbers whose sum is 24 and product is 143

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

If the equation x²+ 4x + k = 0 has a real and distinct roots, then

If the equation x² - ax + 1 = 0 has two distinct roots, then

If ax² + bx + c = 0 has equal roots, then c= ?

If x = 1 is a common root of ax² + ax + 2 = 0 and x² + x + b = 0 then ab = ?

If a and b are roots of the equation x²+ ax + b = 0, then a + b = ?

If y = 1 is a common root of the equations ay² + ay + 3 = 0 and y² + y + b = 0 then ab equals ?

The values of k for which the quadratic equation 16x² + 4 kx + 9 = 0 has real and equal roots are