Free Topic-Wise General Studies MCQs
Evaluate your algebraic skills with Linear and Quadratic Equations MCQs. Standard practice questions and step-by-step solutions tailored for the UPSC Civil Services Prelims.
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Explanation: Let the tens digit be x and units digit be y. Express the original and reversed numbers algebraically, set up two equations based on the digit sum and the numerical difference, and solve the system.
Explanation: Transpose the constant term (a-b) to the right side, factor the resulting numerator as (a+b), then divide both sides by (a+b) to isolate x.
Explanation: Let the common root be r. Subtract one equation from the other to eliminate r², factor out (a-b), and deduce that r = 1. Substitute r = 1 back into either original equation.
Explanation: Let the original number of pens be x and price per pen be p. Set up two equations from the given conditions (xp = 160 and (x+5)(p-2) = 150). Substitute p = 160/x into the second equation, clear fractions, and solve the quadratic.
Explanation: First find the original roots by factorization. Then calculate the sum and product of their squares. Use the identity α² + β² = (α+β)² - 2αβ to find the sum, and α²β² = (αβ)² for the product.
Explanation: Calculate the combined work rate per day, determine the fraction of work completed in 3 days, subtract from 1 to find the remaining work, then divide by B's individual work rate.
Explanation: For a quadratic f(x) = ax² + bx + c with a < 0, the maximum occurs at x = -b/(2a). Substitute this x-value back into the function to find the maximum value.
Explanation: Identify the feasible region bounded by the constraint lines. Find the corner points by solving the constraint equations simultaneously. Evaluate the objective function at each corner point to find the maximum.
Explanation: For a quadratic to be positive for all x, the leading coefficient must be positive and the discriminant must be negative. Set up these two inequalities and find their intersection.
Explanation: Factor the quadratic to find the critical points where the expression equals zero. Test the sign of the expression in each interval determined by these critical points, and select where the expression is positive.
Explanation: Factor the denominator on the right as (x-2)(x+2), multiply every term by this common denominator to clear fractions, then solve the resulting linear equation and verify the solution does not make any original denominator zero.
Explanation: Substitute y = x² to reduce the quartic to a quadratic in y. Solve for y, then for each positive y-value, solve x² = y. The product of all real roots can be found by multiplying ±√y values.
Explanation: For a quadratic ax² + bx + c with a > 0, the minimum value occurs at x = -b/(2a). Calculate this x-coordinate and substitute it back into the expression to find the minimum value.
Explanation: Calculate the sum of new roots as (α+2)+(β+2) = (α+β)+4, and product as (α+2)(β+2) = αβ + 2(α+β) + 4. Use the original sum and product formulas to compute these values.
Explanation: Calculate the discriminant D = b² - 4ac. For real and distinct roots, D > 0. Solve the inequality, and ensure the leading coefficient is non-zero to maintain the quadratic nature.
Explanation: For no solution, the lines must be parallel but distinct, so a₁/a₂ = b₁/b₂ ≠ c₁/c₂. Set the ratio of x-coefficients equal to the ratio of y-coefficients, solve for k, and verify it does not equal the ratio of constants.
Explanation: Let the cost of a pen be p and a pencil be c. Set up two linear equations from the given information. Add the equations to find p + c, subtract to find p - c, then solve the resulting system.
Explanation: Substitute y = 2^x to convert the exponential equation into a quadratic in y. Solve for y, then solve 2^x = y for each valid y-value and sum the solutions.
Explanation: Let the roots be r and 2r. Apply the sum of roots formula to find r, then use the product of roots formula to calculate k = r · 2r.
Explanation: Use the replacement formula: final quantity = initial × (1 - replaced/volume) for each replacement step. Multiply the successive retention factors to find the remaining pure milk.
Explanation: Use the identity (x+y)² = x² + y² + 2xy to find xy. Then apply the sum of cubes formula x³ + y³ = (x+y)³ - 3xy(x+y) to calculate the result.
Explanation: Calculate the total score for each class by multiplying the average by the number of students, add the totals, and divide by the combined number of students.
Explanation: Clear fractions by multiplying both sides by the LCM of denominators (12). Expand each term, collect like terms, and isolate x to solve the linear equation.
Explanation: Apply the quadratic formula x = (-b ± √D)/(2a). Calculate the discriminant D = b² - 4ac, find its square root, then evaluate both roots and select the positive one.
Explanation: Let the roots be r and 2r. Use the sum of roots formula to express r in terms of the coefficients, then use the product of roots formula to establish an equation in k and solve.
Explanation: A system has a unique solution when the determinant ad - bc ≠ 0. Calculate the determinant in terms of a and b, set it equal to zero to find the condition for non-unique solutions, and identify the excluded case.
Explanation: Let the distance be d and scheduled time be t. Form two equations using time = distance/speed, equate the expressions for t from both scenarios, and solve for d.
Explanation: Let the cost of a pen be p and a pencil be c. Set up two linear equations from the given information. Add the equations to find p + c, subtract to find p - c, then solve the resulting system.
Explanation: Let the width be w. Express the length as 2w + 5, set up the area equation w(2w+5) = 150, solve the quadratic for the positive width, then calculate the perimeter using P = 2(length + width).
Explanation: Split the absolute value equation into two cases: the expression inside equals the positive value and the negative value. Solve each linear equation separately, then add the two solutions.
Explanation: Use the substitution or elimination method. Multiply the equations to align coefficients of one variable, subtract to eliminate that variable, solve for the remaining variable, and back-substitute.
Explanation: Cross-multiply to eliminate the fractions, then expand both sides and isolate x by collecting like terms.
Explanation: For infinitely many solutions, the ratios of corresponding coefficients must be equal: a₁/a₂ = b₁/b₂ = c₁/c₂. Set up the proportion using the coefficients of x, y, and constants, then solve for k.
Explanation: Substitute y = |x| to convert the equation into a standard quadratic in y. Solve for y, then for each positive y-value, solve |x| = y to get pairs of opposite roots. The sum of all such roots is zero due to symmetry.
Explanation: Let the roots be a and 2a. Use the sum of roots formula to find a, then use the product of roots formula to calculate q = a · 2a = 2a².
Explanation: For reciprocal roots, their product must equal 1. Using the formula product = c/a = r/p, set r/p = 1 and derive the relationship between p and r.
Explanation: Let the roots be α and α+3. Use the sum of roots formula to find α, then use the product of roots formula to calculate q = α(α+3).
Explanation: Calculate downstream and upstream speeds from distance and time. Let the boat speed be b and stream speed be s. Set up two linear equations (b+s = downstream, b-s = upstream) and solve by adding the equations.
Explanation: Express individual work rates as 1/x and 1/(x+5). The combined rate equals 1/6. Set up the equation, find a common denominator, clear fractions, and solve the resulting quadratic for the positive root.
Explanation: Use the elimination method on the three equations. Add equations to eliminate z, creating a system of two equations in x and y. Solve that system, then substitute back to find z.
Explanation: Since the coefficient of x² is positive, the parabola opens upward and has a minimum value. Find the vertex y-coordinate using f(-b/2a) or completing the square to determine the lower bound of the range.
Explanation: Multiply both sides by the common denominator (x-1)(x-3)(x-2) to clear all fractions. Expand each term carefully, collect like terms, and solve the resulting linear equation.
Explanation: If α and β are roots, the quadratic equation is x² - (α+β)x + αβ = 0. Calculate the sum and product of the given roots and substitute into this standard form.
Explanation: Substitute u = 1/x and v = 1/y to convert the system into linear equations in u and v. Solve for u and v using elimination, then take reciprocals to find x and y.
Explanation: Expand all brackets using the distributive property, collect variable terms on one side and constants on the other, then divide by the coefficient of x.
Explanation: Let the son's present age be x, making the father's age 4x. Express both ages after 20 years, set up the equation based on the future ratio, and solve for x.
Explanation: Use the identity (x+y)² = x² + y² + 2xy to find the product xy. Then apply (x-y)² = x² + y² - 2xy to find the square of the difference, and take the square root.
Explanation: For equal roots, the discriminant must be zero: b² - 4ac = 0. Substitute the coefficients, set the discriminant equal to zero, and solve for k.
Explanation: For a system ax + by = c and dx + ey = f, the determinant D is calculated as ae - bd. Substitute the coefficients from the given equations into this formula.
Explanation: Let the smaller even integer be 2n. The next consecutive even integer is 2n+2. Set up the product equation, solve the quadratic for the positive integer n, then find the sum of the two numbers.
Explanation: Calculate the discriminant D = b² - 4ac. If D > 0, roots are real and distinct; if D = 0, real and equal; if D < 0, complex conjugate pair.
Explanation: Let the original speed be s. Express the original and new times as 300/s and 300/(s+10). Set up the equation based on the time difference, clear fractions by multiplying by s(s+10), and solve the resulting quadratic.
Explanation: Use the elimination method. Multiply the second equation by 2 to make the y-coefficients equal in magnitude, add the equations to eliminate y, solve for x, then substitute back to find y.
Explanation: For rational roots, the discriminant must be a perfect square. Set D = k² - 4ac = k² - 24 equal to m², rearrange as k² - m² = 24, factor as (k-m)(k+m) = 24, and find integer solutions for k.
Explanation: If one root is zero, the constant term must be zero. Set the constant term equal to zero to find k, then use the sum of roots formula to find the other root.
Explanation: For a quadratic equation ax² + bx + c = 0, the product of roots equals c/a. Identify the coefficients and apply this formula directly.
Explanation: The height function is a downward-opening parabola. The maximum occurs at the vertex t = -b/(2a). Calculate this time and substitute back into h(t) to find the maximum height.
Explanation: Eliminate decimals by multiplying both sides by 10, or directly transpose variable terms to one side and constants to the other, then divide by the coefficient.
Explanation: For a quadratic equation ax² + bx + c = 0, the sum of roots equals -b/a. Identify the coefficients and apply this formula directly.
Explanation: Let the roots be α and α+1. Use the sum of roots formula to find α, then use the product of roots formula to calculate k.