Quiz - 1 For Class X (Quadratic Equation)

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.

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.

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.

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.

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.

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.

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.

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.

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.

Explanation
The sum of the squares of two consecutive odd positive integers, 11 and 13, is 290.