Maharashtra Board Class 12 Mathematics and Statistics (Commerce) Important Questions 2027 (Repeated in Board Exams)

Attempt any TWO (3 marks each): Determine whether the following statement pattern is tautology, contradiction or contingency. [p ^ (~p v q)]

Attempt any TWO (3 marks each): Write the converse, inverse and contrapositive of the following statement. 'If a man is rich, then he is happy.'

Fill in the blank: The dual of (p ^ ~q) v t is _____.

Attempt any TWO (3 marks each): Find the inverse of matrix [1 2 3; 1 1 5; 2 4 7] by adjoint method.

Attempt any ONE (4 marks each): Solve the following equation by method of reduction: x - y + z = 4, 2x + y - 3z = 0, x + y + z = 2

Attempt any TWO (4 marks each): Solve the following equations by the method of inversion: 2x - y + z = 1, x + 2y + 3z = 8, 3x + y - 4z = 1

If A = [4 x; 6 3] is a singular matrix then x is (a) 2 (b) -2 (c) 3 (d) -3

Attempt any TWO (3 marks each): Find x, y, z if {5[0 1; 1 0; 1 1] - [2 1; 3 -2; 1 3]}[2; 1] = [x - 1; y + 1; 2z]

Attempt any TWO (3 marks each): Find dy/dx if y = (log x)^x + x⁵

Attempt any TWO (3 marks each): If e^x + e^y = e^(x + y) then show that dy/dx = -e^(y - x).

If x = 2at², y = 4at then dy/dx = ____. (a) -1/(2at²) (b) 1/(2at³) (c) 1/t (d) 1/(4at³)

If y = e^(log x) then dy/dx is (a) e^(log x)/x (b) 1/2 (c) 1 (d) both (a) and (c)

If y = log(e^x / x²) then dy/dx = _____. (a) (2 - x)/x (b) (x - 2)/x (c) (e - x)/(e^x) (d) (x - e)/(ex)

If y = 2x² + log 2 + 5 then dy/dx = ____. (a) x (b) 4x (c) 2x + log 2 (d) -4x

If 0 < eta < 1, then the demand is (a) elastic (b) unitary elastic (c) relatively elastic (d) relatively inelastic

Attempt any TWO (3 marks each): A metal wire of 36 cm length is bent to form a rectangle. Find its dimensions when its area is maximum.

integral x² (3)^(x³) dx = _____ (a) (3)^(x³)/(3 log 3) + c (b) (3)^(x³) + c (c) (log 3)(3)^(x³) + c (d) x² (3)^(x³) + c

integral (1 - x)^(-3) dx = _____. (a) (1/2)(1 - x)^(-2) + c (b) (1/2)(1 + x)^(-2) + c (c) (1/2)(1 - x)^(-2) + x/2 + c (d) (1/2)(1 - x)^(-2) - x/2 + c

Fill in the blank: integral e^x (1/x - 1/x²) dx = _____.

State whether true or false: integral 1/(2x + 3) dx = log|2x + 3| + c

Attempt any TWO (3 marks each): Evaluate: integral 1/(4x² - 20x + 17) dx

State whether true or false: For integral ((x - 1)/(x + 1)³) e^x dx = e^x . f(x) + c, where f(x) = (x + 1)²

Attempt any TWO (4 marks each): Evaluate: integral e^x/sqrt(e^(2x) + 4e^x + 13) dx

Attempt any ONE (4 marks each): Evaluate: integral x² . e^(3x) dx

Attempt any TWO (4 marks each): Evaluate: integral from 2 to 5 of sqrt(x)/(sqrt(x) + sqrt(7 - x)) dx

Attempt any TWO (4 marks each): Evaluate: integral from 1 to 3 of log x dx

If integral from 0 to a of 3x² dx = 8 then a = _____. (a) 2 (b) 0 (c) 8/3 (d) 1

Attempt any ONE (4 marks each): Find the area of the region bounded by the curve x² = 16y and the line y = 4.

Area of the region bounded by the curve y = x², the X-axis and the lines x = 1 and x = 3 is _____. (a) 3/26 sq. units (b) 3 sq. units (c) 26 sq. units (d) 26/3 sq. units

The order and degree of (d²y/dx²)² + (dy/dx)² = a^x are respectively ____ (a) 1, 1 (b) 1, 2 (c) 2, 1 (d) 2, 2

State whether true or false: Order and degree of a differential equation are always positive integers.

Fill in the blank: The integrating factor of the differential equation dy/dx - y = x is _____.

Attempt any ONE (4 marks each): Solve the following differential equation (x² - y²) dx + 2xy dy = 0

State whether true or false: The integrating factor (I.F.) of dy/dx + y = e^(-x) is e^x

Attempt any ONE (4 marks each): In a certain culture of bacteria, the rate of increase is proportional to the number present. If it is found that the number doubles in 4 hours, find the number of times the bacteria increased in 12 hours by using the concept of application of differential equations.

Attempt any ONE (Activity, 4 marks each): Solve the following differential equation (x² - yx²) dy + (y² + xy²) dx = 0. Solution: separating the variables, ((1/y²) - 1/y) dy + ((1/x) + 1) dx = 0; integrate to get -1/y - log y + log x + x = c is the required solution.

The differential equation of y = k1 e^x + k2 e^(-x) is : (a) d²y/dx² - y = 0 (b) d²y/dx² + dy/dx = 0 (c) d²y/dx² + y (dy/dx) = 0 (d) d²y/dx² + y = 0

Attempt any ONE (Activity, 4 marks each): In a certain culture of bacteria, the rate of increase is proportional to the number present. If it is found that the number doubles in 4 hours, find the number of times the bacteria are increased in 12 hours. Solution: dN/dt = KN, integrate to get log N = Kt + C, use conditions to find bacteria increase 8 times in 12 hours.

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

The date on which period of the bill expires is called ____. (a) Legal due date (b) Date of discounting (c) Nominal due date (d) Date of drawing

State whether true or false: The banker's discount is also called as commercial discount.

Attempt any TWO (4 marks each): A bill was drawn on 14th April for Rs 7,000 and was discounted on 6th July at 5% p.a. The banker paid Rs 6,930 for the bills. Find the period of the bill.

Attempt any TWO (3 marks each): A shop is valued at Rs 3,60,000 and is insured 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 agent's commission if the agent gets commission at 15% of the premium.

In an ordinary annuity, payments or receipts occur at (a) Beginning of each period (b) End of each period (c) Mid of each period (d) Quarterly basis

Attempt any TWO (3 marks each): For the following bivariate data obtain equation of regression line Y on X: X 1 2 3 4 5; Y 5 7 9 11 13

In regression analysis b_yx . b_xy = _____. (a) V(x) (b) sigma_x (c) (sigma_y)² (d) r²

Attempt any TWO (3 marks each): Given the following information about the production (X) and demand (Y) of a commodity, obtain the regression line of X on Y. Production (X): Mean 85, S.D. 5; Demand (Y): Mean 90, S.D. 6. Coefficient of correlation between X and Y is 0.6. Also estimate the production when demand is 100.

Attempt any TWO (3 marks each): Obtain the 4-yearly centered moving averages for the following data: Years 1971 1972 1973 1974 1975 1976 1977 1978 1979 1980 1981 1982; Production 1 0 1 2 3 2 3 6 5 1 4 10

Attempt any TWO (3 marks each): Following table shows the number of traffic fatalities (in a state) resulting from drunken driving from years 1975 to 1983: Years 1975 1976 1977 1978 1979 1980 1981 1982 1983; No. of deaths 0 6 3 8 2 9 4 5 10. Fit a trend line by least square method.

Moving averages are useful in identifying ____. (a) Trend component (b) Irregular component (c) Seasonal component (d) Cyclical component

Attempt any TWO (3 marks each): Following data shows the number of bags of cereals sold in years 1977 to 1984. Years 1977 1978 1979 1980 1981 1982 1983 1984; No. of bags (in ten thousands) 1 0 3 8 10 4 5 8. Fit a trend line to the above data by graphical method.

Fill in the blank: If P01(L) = 225, P01(P) = 144 then P01(F) = _____.

Paasche's Price Index Number is given by ____. (a) (sum p0 q0/sum p1 q0) x 100 (b) (sum p0 q1/sum p1 q1) x 100 (c) (sum p1 q0/sum p0 q0) x 100 (d) (sum p1 q1/sum p0 q1) x 100

Attempt any TWO (4 marks each): Minimize: z = 4x + 2y Subject to: 3x + y >= 27, x + y >= 21, x + 2y >= 30, x >= 0, y >= 0. Solve the above L.P.P. by graphical method.

Objective function of LPP is (a) a constraint (b) a function to be maximized or minimized (c) a relation between the decision variables (d) a feasible region

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)

Fill in the blank: Graphical solution set of the inequations x >= 0, y >= 0 lies in _____ quadrant.

State whether true or false: The optimum value of the objective function of L.P.P. occurs at the centre of the feasible region.

Attempt any TWO (3 marks each): Find the sequence that minimizes the total elapsed time to complete the following jobs. Each job is processed in order AB. Also find the idle time for machine B. Jobs (processing times in minutes) I II III IV V VI VII; Machine A 12 6 5 11 5 7 6; Machine B 7 8 9 4 7 8 3

Attempt any TWO (4 marks each): Four new machines M1, M2, M3 and M4 are to be installed in a machine shop. There are five vacant places A, B, C, D and E available. Because of limited space, machine M2 cannot be placed at C and M3 cannot be placed at A. The cost matrix is: M1: A=4 B=6 C=10 D=5 E=6; M2: A=7 B=4 C=- D=5 E=4; M3: A=- B=6 C=9 D=6 E=2; M4: A=9 B=3 C=7 D=2 E=3. Find the optimal assignment schedule.

State whether true or false: To convert an assignment problem into a minimization problem, the smallest element in the matrix is deducted from all other elements.

State whether true or false: In an assignment problem, if the number of columns are greater than number of rows, then a dummy column is added.

Attempt any TWO (4 marks each): A marketing manager has list of salesmen and territories. The total cost per month (in thousand rupees) for each salesman in each territory is: Salesman A: I=11 II=16 III=18 IV=15 V=15; B: 7 19 11 13 17; C: 9 6 14 14 7; D: 13 12 17 11 13. Find the assignment of salesman to territories that will result in minimum cost.

Attempt any TWO (4 marks each): A random variable X has the following probability distribution: x 1 2 3 4 5 6 7; P[X = x] k 2k 2k 3k k² 2k² 7k²⁺k. Determine (a) k (b) P(X < 3) (c) P(X > 6) (d) P(0 < X < 1)

Attempt any TWO (3 marks each): A pair of dice is thrown 3 times. If getting a doublet is considered a success, find the probability of two successes.

Attempt any ONE (4 marks each): A die is thrown 4 times. If 'getting an odd number' is a success, find the probability of (a) 2 successes (b) at least 3 successes (c) at the most 2 successes.

The expected value of the sum of two numbers obtained when two fair dice are rolled is _____. (a) 5 (b) 6 (c) 7 (d) 8