Quadratic equations practice · Medium

Option D ($\cot \beta$) is correct because the tangent of the sum of angles can be found using Vieta's formulas and trigonometric identities.

Option C ($3 / 2$) is correct because it is the value of x that satisfies the given exponential equation after simplification and comparison of exponents.

Option D ($\frac{{ - 2bc}}{{b + c}}$) is correct because solving the equation and using the condition that the product of roots is zero leads to the given sum of roots.

Option B (GP) is correct because the given condition implies that the ratio of the roots is the same, leading to a relationship between the coefficients that satisfies the condition for a geometric progression.

Option B (1) is correct because the equation simplifies to |x-1|=0, resulting in only one real solution.

Option C $((-\infty, 3))$ is correct because the condition for at least one negative root requires $a < 3$.

Option A (4,4) is the correct solution because it satisfies the given equation when substituted for x and y, derived from completing the square and discriminant analysis.

Option C ($\frac{1}{4}$) is correct because it is the minimum value of m that ensures the given expression is always non-negative, derived from the discriminant condition of a quadratic.

Option B (3) is correct because solving the inequality yields -1 < x < 3, which has three integer solutions: 0, 1, and 2.

Option C (No solution) is correct because the expression ${x^2} + 15|x| + 14$ is always positive for all real values of x, so it can never be equal to zero.

Option A (0) is correct because the expression simplifies to zero after substituting the sum and product of the roots.

Option D (All of these) is correct because all the given expressions are equivalent to the other root $\beta$ of the quadratic equation.

Option A ($4 x^2-12 x-7=0$) is correct because given the sum and difference of the roots, we can find the roots and then construct the quadratic equation.

Option B ($p = 1$ or 0) is correct because these are the possible values of p that satisfy the given conditions for the roots of the equation.

Option A (${a^2},{c^2},{b^2}$ are in A.P.) is correct because it is derived from the condition that the given quadratic equations have a common root, and $a, b, c$ are in A.P.

Option B ($- \frac{1}{2}\left( {{{{a^2} + {b^2}}}} \right)$) is correct because it represents the product of the roots under the given condition that the sum of the roots is zero.

Option B (real and both negative) is correct because the roots of the resulting quadratic equation are -7/3 and -1/2, both real and negative.

Option B ($n\left( {n + 1} \right),n \in Z$) is correct because the discriminant must be a perfect square, which leads to m being of the form n(n+1).

Option B (16) is correct because by finding the value of lambda using the even prime root and substituting it into the second equation, we can find the value of mu using the discriminant condition for equal roots.

Option A ($a^2+b^2$) is correct because the maximum value of the given expression is obtained by completing the square and recognizing that the squared term is always non-negative.

Option B (2) is correct because the function is not differentiable at x=1 and x=4 due to the presence of absolute value functions creating sharp edges.

Option A ($p = 2, q = 32$) is correct because the values satisfy the conditions of the increasing geometric progression formed by the roots of the given quadratic equations.

Option B is correct because the discriminant of the quadratic equation is greater than 0, implying real roots.

Option D (-2/a) is correct because simplifying the given expression using Vieta's formulas leads to -2/a.

Option C ($\frac{{ - 9}}{8}$) is correct because it represents the minimum value of the quadratic function, found by completing the square or using the vertex formula.

The correct answer is A ($p=1, q=-2$) because substituting these values into the equations $p+q=-p$ and $pq=q$ derived from Vieta's formulas satisfies the conditions.

The correct answer is 5 because substituting $\beta$ into the given equation and simplifying the expression leads to the value 5.

Option A is correct because it represents the condition where the discriminant is negative, ensuring no real roots, and also satisfies the condition derived from f(-1) > 0.

Option A (1) is correct because only one integer value of x (x=3) satisfies both inequalities simultaneously.

Option C (8) is correct because the product of all x values satisfying the equation is found by solving for x² and then multiplying the roots.