Maharashtra Msbshse Class 12 Mathematics and Statistics — Real Previous-Year Papers

State whether the following statement is true or false: If y = 20 + 15x + x² then (dx)/(dy) = (1)/(15 + 2x)

State whether the following statement is true or false: If ∫ (x)/((1 + x)(2 + x))dx = ∫ ((A)/(1 + x) + (B)/(2 + x))dx then A = 1, B = 2.

State whether the following statement is true or false: Order and degree of a differential equation are always positive integers. (a) True (b) False

Statement p rightarrow q is true, when p and q have ___ truth values.

If y = eax, then x · (dy)/(dx) = ___.

If f(x) = x · log x then its minimum value is ___.

Examine whether the following statement pattern is a tautology, a contradiction or a contingency. (p wedge sim q) arrow (sim p wedge sim q)

If ex + ey = e(x + y), then show that (dy)/(dx) = -ey - x.

Evaluate: ∫ (1 + x)/(x) + e-xdx

Find the inverse of 1 -1 1 \2 1 -3 \1 1 1 by adjoint method.

The consumption expenditure Ec of a person with the income x. is given by Ec = 0.0006x² + 0.003x. Find MPC, MPS, APC and APS when the income x = 200.

Evaluate the following integrals: ∫₂⁷ frac√(x)√(x) + √(9 - x)dx

If p : He swims q : Water is warm Give the verbal statement for the following symbolic statement: p rightarrow sim q

If p : He swims q : Water is warm Give the verbal statement for the following symbolic statement. sim (p vee q)

If p : He swims q : Water is warm Give the verbal statement for the following symbolic statement. q arrow p

If the demand function is D = 50 - 3p - p². Find the elasticity of demand at p = 5 comment on the result.

Evaluate the following. ∫ (1)/(4x² - 20x + 17)dx

Solve the following differential equation: (x² - y²)dx + 2xydy = 0

Find the area of the region bounded by the curve x² = 16y and the line y = 4.

Express the following equations in matrix form and solve them by the method of reduction: x - y + z = 1, 2x - y = 1, 3x + 3y - 4z = 2 Solution: The given equations can be written in the matrix form as: 1 -1 1 \2 -1 0 \3 3 -4 x y z = 1 \1 \2 By R₂ arrow R₂ - 2R₁, 1 -1 1 ; square square square \3 3 -4 x y z = 1 \-1 \2 By R₃ arrow R₃ - 3R₁ 1 -1 1 \0 1 -2 ; square square square x y z = 1 \-1 \-1 By R₃ arrow R₃ - 6R₂ 1 -1 1 \0 1 -2 \0 0 5 x y z = 1 \-1 ; square We write equations as x - y + z = 1 ...(I) y - 2z = -1 ...(II) 5z = 5 ...(III) Solving equations (I), (II) and (III) We get x = square, y = square, z = square

Obtain the differential equation from the relation Ax² + By² = 1, where A and B are constants. Solution: The given equation is Ax² + By² = 1 ...[I] Differentiating equation (I) w.r.t. x, we get, squarex + 2By(dy)/(dx) = 0 Ax + By(dy)/(dx) = 0 ...[II] Differentiating equation (II) w.r.t. x, we get, A + Bsquare = 0 ...[III] Since equations (I), (II), and (III) are consistent in A and B. therefore x² y² 1 x y(dy)/(dx) 0 \1 square 0 = 0 therefore \x[y(d² y)/(dx²) + ((dy)/(dx))²] - y(dy)/(dx)\ = 0 therefore square + x((dy)/(dx))² - y(dy)/(dx) = 0

The difference between face value and present worth is called ___. (a) Banker’s discount (b) True discount (c) Banker’s gain (d) Cash value

Insurance companies collect a fixed amount from their customers at a fixed interval of time. This amount is called ___. (a) EMI (b) Installment (c) Contribution (d) Premium

If byx > 1 then bxy is ___. (a) > 1 (b) < 0 (c) = 0 (d) < 1

Which of the following can’t be a component of a time series? (a) Seasonality (b) Cyclical (c) Trend (d) Mean

Price Index Number by Simple Aggregate Method is given by ___. (a) Σ (p₁)/(p₀) × 100 (b) Σ (p₀)/(p₁) × 100 (c) (Σ p₁)/(Σ p₀) × 100 (d) (Σ p₀)/(Σ p₁) × 100

If the corner points of the feasible region are (0, 10), (2, 2) and (4, 0) then the point of minimum z = 3x + 2y is ___. (a) (2, 2) (b) (0, 10) (c) (4, 0) (d) (2, 4)

State whether the following statement is True or False. The banker’s discount is always lower than the true discount. (a) True (b) False

The cost of living Index Number using weighted relative method is given by (Σ IW)/(Σ W).

To convert an assignment problem into a minimization problem, the smallest element in the matrix is deducted from all other elements.

The difference between the banker’s discount and the true discount is called ___.

If P₀₁(L) = 225, P₀₁(P) = 144 then P₀₁(F) = ___.

A dish washing machine holds up to 40 pieces of large crockery (x). This constraint is given by ___.

The following table shows the production of pig-iron and ferroalloys ('000 metric tons) | Years | 1974 | 1975 | 1976 | 1977 | 1978 | 1979 | 1980 | 1981 | 1982 | | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | | Production | 0 | 4 | 9 | 9 | 8 | 5 | 4 | 8 | 10 | Find trend values for the above data by using 5 yearly moving averages.

A shop is valued at ₹3,60,000 for 75% of its value. If the rate of premium is 0.9%, find the premium paid by the owner of the shop. Also, find the agents commission if the agent gets commission at 15% of the premium.

Given the following information about the production (X) and demand (Y) of a commodity, obtain the regression line of X on Y. | | Production (X) | Demand (Y) | | --- | --- | --- | | Mean | 85 | 90 | | S.D. | 5 | 6 | Coefficient of correlation between X and Y is 0.6. Also, estimate the production when demand is 100.

Solve the following L.P.P. by graphical method: Minimize: z = 8x + 10y Subject to: 2x + y ≥ 7, 2x + 3y ≥ 15, y ≥ 2, x ≥ 0, y ≥ 0.

Solve the following assignment problem for minimization: | Men | Tasks (in hours) | | | | | --- | --- | --- | --- | --- | | I | II | III | IV | | | A | 7 | 25 | 26 | 10 | | B | 12 | 27 | 3 | 25 | | C | 37 | 18 | 17 | 14 | | D | 18 | 25 | 23 | 9 |

The probability distribution of a discrete r.v. X is as follows: | x | 1 | 2 | 3 | 4 | 5 | 6 | | --- | --- | --- | --- | --- | --- | --- | | P(X = x) | k | 2k | 3k | 4k | 5k | 6k | Determine the value of k. Find P(X ≤ 4) P(2 < X < 4) P(X ≥ 3)

The following is the p.d.f. of a r.v. X. f(x) = (x)/(8), for 0 < x < 4 \0, otherwise. Find P(x < 1.5)

The following is the p.d.f. of a r.v. X. f(x) = (x)/(8), for 0 < x < 4 \0, otherwise. Find P(1 < x < 2)

The following is the p.d.f. of a r.v. X. f(x) = (x)/(8), for 0 < x < 4 \0, otherwise. Find P(x > 2)

The true discount on a sum is (3)/(8) of the sum due at 12% p.a. Find the period of the bill.

Following data shows the number of boxes of cereal sold in years 1977 to 1984. | Year | 1977 | 1978 | 1979 | 1980 | 1981 | 1982 | 1983 | 1984 | | --- | --- | --- | --- | --- | --- | --- | --- | --- | | No. of boxes in ten thousand | 1 | 0 | 3 | 8 | 10 | 4 | 5 | 8 | Fit a trend line to the above data by graphical method.

For the following bivariate data obtain the equation of regression line of Y on X. | X | 1 | 2 | 3 | 4 | 5 | | --- | --- | --- | --- | --- | --- | | Y | 5 | 7 | 9 | 11 | 13 |

Calculate (a) Laspeyre’s and (b) Paasche's Price Index Numbers for the following data: | Commodity | Base year | Current year | | | | --- | --- | --- | --- | --- | | Price | Quantity | Price | Quantity | | | P | 12 | 20 | 18 | 24 | | Q | 14 | 12 | 21 | 16 | | R | 8 | 10 | 12 | 18 | | S | 16 | 15 | 20 | 25 |

Determine the optimal sequence of jobs that minimizes the total elapsed time for the data given below (processing time on machines is given in hours). Also find the total elapsed time and the idle time for three machines. | Jobs | I | II | III | IV | V | VI | VII | | --- | --- | --- | --- | --- | --- | --- | --- | | Machine A | 3 | 8 | 7 | 4 | 9 | 8 | 7 | | Machine B | 4 | 3 | 2 | 5 | 1 | 4 | 3 | | Machine C | 6 | 7 | 5 | 11 | 5 | 6 | 12 | Solution: Here min A = 3, min C = 5, Max B = 5. Since Min C ≥ max B is satisfied, the problem can be converted into a two-machine problem. Let G and H be two fictitious machines ∴ G = A + B, H = B + C The above problem can be written as: | Jobs | I | II | III | IV | V | VI | VII | | --- | --- | --- | --- | --- | --- | --- | --- | | Machine G | 7 | 11 | 9 | 9 | 10 | 12 | 10 | | Machine H | 10 | 10 | 7 | 16 | 6 | 10 | 15 | Using the optimal sequence algorithm, the following sequence can be obtained. | | | | VI | II | | | | --- | --- | --- | --- | --- | --- | --- | Work table: | Jobs | Machine A | Machine B | Machine C | | | | | --- | --- | --- | --- | --- | --- | --- | | In | Out | In | Out | In | Out | | | I | 0 | 3 | 3 | 7 | 7 | 13 | | IV | 3 | 7 | 7 | 12 | 13 | 24 | | square | 7 | 14 | 14 | 17 | 24 | 36 | | VI | 14 | 22 | 22 | 26 | 36 | 42 | | II | 22 | 30 | 30 | 33 | square | 49 | | square | 30 | 37 | 37 | 39 | 49 | 54 | | V | 37 | 46 | 46 | 47 | 54 | 59 | ∴ Total elapsed time is = 59 hrs. Idle time for machine A = square hrs Idle time for machine B = square hrs Idle time for machine C = 7 hrs.

In a town, 10 accidents take place in the span of 50 days. Assuming that the number of accidents follows Poisson distribution, find the probability that there will be 3 or more accidents on a day. (Given that e-0.2 = 0.8187) Solution: Here, m = square and X - P(m) with parameter m. The p.m.f. X is: P(X = x) = frace-m · mxx!, x = 0, 1, 2, … P(X ≥ 3) = 1 - P(X < 3) = 1 - [square + square + square] = 1 - [frace-0.2(0.2)⁰0! + frace-0.2(0.2)¹1! + frace-0.2(0.2)²2!] = 1 - [0.8187(1 + 0.2 + 0.02)] = 1 - square = square