Maharashtra Board Class 12 Mathematics and Statistics (Commerce) Guess Paper 2027

integral from 2 to 7 of sqrt(x)/(sqrt(x) + sqrt(9 - x)) dx = (a) 7/2 (b) 5/2 (c) 7 (d) 2

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

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

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

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

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

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

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

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

Fill in the blank: If the average revenue is 45 and elasticity of demand is 3 then marginal revenue is ____.

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

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): Solve the following L.P.P. by graphical method. Maximize: z = 4x + 6y Subject to: 3x + 2y <= 12, x + y >= 4, x, y >= 0

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): Determine whether the following statement pattern is tautology, contradiction or contingency. [p ^ (~p v q)]

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

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 (4 marks each): Following table shows the all India Infant Mortality Rates (per 000) for years 1980 to 1986: Years 1980 1981 1982 1983 1984 1985 1986; IMR 10 6 5 3 3 1 0. Fit a trend line to the above data by the method of least squares.

Attempt any ONE (4 marks each): 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

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

Price Index Number by Simple Aggregate Method is given by (a) sum(p1/p0) x 100 (b) sum(p0/p1) x 100 (c) (sum p1/sum p0) x 100 (d) (sum p0/sum p1) x 100

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

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

integral from 0 to 2 of e^x dx = _____. (a) e - 1 (b) 1 - e (c) e² - 1 (d) 1 - e²

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

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

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

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

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.

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.

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.'

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 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): 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 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 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.

Attempt any TWO (4 marks each): Find MPC, MPS, APC and APS, if the expenditure Ec of a person with income I is given as Ec = (0.0003) I² + (0.075) I when I = 1000.

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