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Free CA Quantitative Aptitude Practice Questions & Answers
500 exam-style Quantitative Aptitude questions. Pick your answer, hit Check answer, and see the worked solution — free to start, no signup.
100% free · No login to startQuestion 91
The integral of 1/x with respect to x is:
Select an option first.
Correct answer: D — log x plus c
Explanation: The power rule for integration fails when the index is -1, because it would require division by zero. This single case is handled separately: the integral of 1/x is the natural logarithm of x plus a constant. It is the exact reverse of the derivative of log x being 1/x - learn the pair together.
Question 92
The integral of 1 over x squared with respect to x is:
Select an option first.
Correct answer: B — -1/x plus c
Explanation: Write the integrand as x to the power -2. The power rule raises the index to -1 and divides by -1, giving -1/x plus a constant. The rule works for every index except -1 itself, where the logarithm appears instead. Differentiate the answer to check: the derivative of -x to the power -1 is x to the power -2.
Question 93
The integral of (3x^2 + 2x) with respect to x is:
Select an option first.
Correct answer: B — x^3 + x^2 + c
Explanation: Integrate term by term using the power rule, raising the index by one and dividing by the new index: 3x^2 gives x^3 and 2x gives x^2, so the result is x^3 + x^2 + c. Distractor A differentiates instead of integrating. The constant of integration must always be written - omitting it costs a mark in the descriptive papers and marks a careless habit here.
Question 94
The integral of x cubed with respect to x is:
Select an option first.
Correct answer: A — x to the power 4, divided by 4, plus c
Explanation: Raise the index by one and divide by the new index: x to the power 4, over 4, plus the constant. Distractor B differentiates instead of integrating. Differentiating the answer should return the original integrand - the quickest possible check on any integration.
Question 95
If 12 men complete a piece of work in 15 days, the number of days 20 men would take is:
Select an option first.
Correct answer: A — 9
Explanation: Men and days are INVERSELY proportional, so more men means fewer days: 12 x 15 = 20 x d, giving d = 9 days. Distractor B applies direct proportion, which would absurdly make more men take longer. Ask which way the relationship runs before setting up the equation - the total work, 180 man-days, is the constant.
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Question 96
The value of (a raised to the power m) raised to the power n, divided by a raised to the power mn, is:
Select an option first.
Correct answer: B — 1
Explanation: A power raised to a power multiplies the indices, so the numerator is a to the power mn - identical to the denominator. The quotient is therefore a to the power zero, which is 1. Note that the answer is 1, not 0: it is the EXPONENT that vanishes, not the value.
Question 97
The value of (x^a / x^b)^(a+b) multiplied by (x^b / x^c)^(b+c) multiplied by (x^c / x^a)^(c+a) is:
Select an option first.
Correct answer: C — 1
Explanation: Each factor reduces to x raised to a difference of squares: (a - b)(a + b) + (b - c)(b + c) + (c - a)(c + a) = (a^2 - b^2) + (b^2 - c^2) + (c^2 - a^2) = 0. Since x to the power zero is 1, the whole expression is 1. Watch the difference between an answer of 1 and an answer of 0 - the EXPONENT is zero, not the value.
Question 98
The value of x satisfying 3(x - 2) = 2(x + 3) is:
Select an option first.
Correct answer: D — 12
Explanation: Expanding gives 3x - 6 = 2x + 6, so x = 12. Substituting back: 3(10) = 30 and 2(15) = 30, which confirms it. Expand both brackets fully before collecting terms - attempting to cancel across an equals sign before expanding is where sign errors enter.
Question 99
The region satisfying x >= 0, y >= 0 and x + y
Select an option first.
Correct answer: D — A bounded triangle in the first quadrant with vertices (0, 0), (6, 0) and (0, 6)
Explanation: The first two conditions confine the region to the first quadrant; the third cuts it off at the line joining (6, 0) and (0, 6). The feasible region is therefore the closed triangle with those three vertices. In linear programming the optimum of a linear objective always occurs at a CORNER point, which is why identifying the vertices matters more than shading the area.
Question 100
The solution set of 3x - 7 > 5 is:
Select an option first.
Correct answer: B — x > 4
Explanation: Adding 7 to both sides gives 3x > 12, so x > 4. The inequality stays strict because no equality was involved, and the direction is unchanged because both sides were divided by a POSITIVE number. Dividing or multiplying an inequality by a negative number reverses the sign - the single rule that separates this topic from ordinary equations.
Question 101
In a linear programming problem, the feasible region is always:
Select an option first.
Correct answer: B — A convex set
Explanation: The feasible region is the intersection of half-planes defined by linear constraints, and any intersection of convex sets is itself convex - meaning the segment joining any two feasible points stays inside the region. This is why the optimum of a linear objective always occurs at a CORNER point, which is the basis of the corner point method.
Question 102
A loan of Rs.1,00,000 is to be repaid in 5 equal annual instalments at 12% per annum. Given a PV annuity factor of 3.6048, the annual instalment is:
Select an option first.
Correct answer: A — Rs.27,741
Explanation: The instalment is the loan divided by the annuity factor: 1,00,000 / 3.6048 = Rs.27,740.79, or about Rs.27,741. Distractor B is simple repayment of principal with no interest, which must be too low. Distractor D multiplies instead of dividing - a sanity check is that the instalment must exceed 1,00,000/5 because interest is being paid as well.
Question 103
A loan of Rs.2,00,000 is repayable in 4 equal annual instalments at 10%, the present value annuity factor being 3.1699. Each instalment is:
Select an option first.
Correct answer: A — Rs.63,093.47
Explanation: Instalment = 2,00,000 / 3.1699 = Rs.63,093.47. Distractor B repays only the principal in equal parts and ignores interest altogether, so the true instalment must exceed it. Divide by the annuity factor - multiplying by it is the mirror-image error and gives an absurdly large figure.
Question 104
The value of log 1, to any valid base, is:
Select an option first.
Correct answer: A
Explanation: Any base raised to the power 0 equals 1, so log 1 = 0 whatever the base. Two companion results are worth holding alongside it: log of the base itself is 1, and the log of 0 is undefined because no power of a positive base gives zero. These three cover most one-line logarithm questions.
Question 105
If log x to the base 2 equals 5, then x is:
Select an option first.
Correct answer: C — 32
Explanation: The statement means 2 raised to the power 5 = x, so x = 32. Converting a logarithmic statement into its exponential form is the first move in almost every log question - base raised to the answer gives the number. Distractor D would be the answer for a log of 6.
Question 106
The value of log 32 to the base 4 is:
Select an option first.
Correct answer: D — 2.5
Explanation: Express both as powers of 2: log 32 / log 4 = 5 log 2 / 2 log 2 = 2.5. Sense check by bracketing - 4 squared is 16 and 4 cubed is 64, so the answer must lie between 2 and 3, which rules out two options before any working.
Question 107
If log x = 2, the base being 10, then x is:
Select an option first.
Correct answer: A — 100
Explanation: The statement means 10 squared = x, so x = 100. Distractor B multiplies the base by the logarithm instead of raising it to that power - the single commonest misreading of logarithmic notation. Convert to exponential form as the first step every time.
Question 108
Given that log 2 = 0.3010, the number of digits in 2^64 is:
Select an option first.
Correct answer: C — 20
Explanation: log(2^64) = 64 x 0.3010 = 19.264. The characteristic is 19, and the number of digits equals the characteristic plus one, so 2^64 has 20 digits. The rule follows from the fact that a number with n digits lies between 10^(n-1) and 10^n, giving a characteristic of n - 1. Taking 19 straight from the characteristic is the standard slip.
Question 109
The value of log to the base 8 of 32 is:
Select an option first.
Correct answer: C — 5/3
Explanation: Write both numbers as powers of 2: 32 = 2^5 and 8 = 2^3. Using the change of base rule, the answer is log 32 / log 8 = 5 log 2 / 3 log 2 = 5/3. Reducing to a common base is almost always faster than using a calculator relation, and it leaves an exact fraction rather than a decimal.
Question 110
If the total cost function is C = 500 + 20q, the marginal cost is:
Select an option first.
Correct answer: B — 20
Explanation: Marginal cost is the derivative of total cost with respect to quantity: dC/dq = 20, a constant. The 500 is fixed cost, which disappears on differentiation because it does not vary with output. Note that average cost, (500 + 20q)/q, is NOT constant - distinguishing marginal from average is a standard examiner trap.
Question 111
If the total cost function is C = q cubed - 6q squared + 15q, the marginal cost at q = 2 is:
Select an option first.
Correct answer: B — 3
Explanation: Differentiating, MC = 3q squared - 12q + 15. At q = 2 this is 12 - 24 + 15 = 3. Distractor A is the TOTAL cost at q = 2, namely 8 - 24 + 30 = 14, and the average cost would be 7 - total, average and marginal cost are three separate quantities, and the question asks for the rate of change.
Question 112
If the demand function is p = 100 - 2q, the marginal revenue is:
Select an option first.
Correct answer: A — 100 - 4q
Explanation: Total revenue is price times quantity: R = (100 - 2q)q = 100q - 2q squared. Differentiating, MR = 100 - 4q. Note that MR falls twice as fast as the demand curve - the standard result for a linear demand function, and a check worth applying to any answer here.
Question 113
If y = 3x squared + 5x, the marginal value of y at x = 2 is:
Select an option first.
Correct answer: A — 17
Explanation: The marginal value is the derivative: dy/dx = 6x + 5, which at x = 2 equals 17. Distractor B is the VALUE of the function at x = 2, that is 12 + 10 = 22, rather than its rate of change. In economics questions, marginal always means the derivative - total, average and marginal are three different quantities.
Question 114
The function y = x^2 - 8x + 5 has a minimum at x equal to:
Select an option first.
Correct answer: D — 4
Explanation: Setting dy/dx = 2x - 8 = 0 gives x = 4. The second derivative is 2, which is positive, confirming a MINIMUM. Because the coefficient of x squared is positive the parabola opens upwards, so a minimum is expected - a shape check that confirms the calculus without further work.
Question 115
A firm's profit function is P = -2x squared + 40x - 50. Profit is maximum when x equals:
Select an option first.
Correct answer: A — 10
Explanation: Differentiate and set to zero: dP/dx = -4x + 40 = 0, so x = 10. The second derivative is -4, which is negative, confirming a maximum. The negative coefficient of x squared means the parabola opens downwards, so a maximum was expected - always confirm the nature of the turning point rather than assuming it.
Question 116
The mean proportional between 8 and 32 is:
Select an option first.
Correct answer: C — 16
Explanation: The mean proportional between a and b is the square root of their product: the square root of 256 = 16. Distractor A is the arithmetic mean, which is always larger unless the two numbers are equal. The mean proportional is simply the geometric mean, so 8, 16 and 32 form a geometric progression with ratio 2.
Question 117
The equation 2x^2 - 6x + 5 = 0 has:
Select an option first.
Correct answer: D — Complex (imaginary) roots
Explanation: The discriminant is b^2 - 4ac = 36 - 40 = -4, which is negative, so the roots are complex conjugates. The rule: a positive discriminant gives two distinct real roots, zero gives coincident real roots, and a negative value gives a conjugate pair. Distractor C is impossible for a quadratic with real coefficients - complex roots always occur in pairs.
Question 118
A machine costing Rs.5,00,000 yields a net cash inflow of Rs.1,50,000 at the end of each year for 5 years. At a cost of capital of 10%, the PV annuity factor being 3.7908, the NPV is:
Select an option first.
Correct answer: A — Rs.68,620
Explanation: Present value of inflows = 1,50,000 x 3.7908 = Rs.5,68,620; deducting the outlay of Rs.5,00,000 gives an NPV of Rs.68,620. Distractor C is the gross present value rather than the NET figure. A positive NPV means the project earns more than the cost of capital and should be accepted.
Question 119
If the effective annual rate is 10.25% with half-yearly compounding, the nominal rate is:
Select an option first.
Correct answer: B — 10%
Explanation: Solve (1 + r/2) squared = 1.1025, so 1 + r/2 = 1.05, giving r = 10% per annum. Distractor C is the rate per half year rather than the nominal ANNUAL rate. The nominal rate is always the smaller of the two whenever compounding is more frequent than annual.
Question 120
The number of terms in the progression 7, 11, 15, ..., 95 is:
Select an option first.
Correct answer: C — 23
Explanation: Using the nth term formula, 7 + (n - 1)4 = 95 gives (n - 1) = 22 and n = 23. The plus one restores the first term, which the subtraction of 7 removed. Distractor A stops at 22, which is the number of STEPS rather than the number of terms - the classic fence-post error.
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