Mathematics and Statistics Set 1 2016-2017 HSC Commerce (English Medium) 12th Standard Board Exam Question Paper Solution

Write the negation of the following statements :
(a) Radha likes tea or coffee.
(b) `∃x cc` R such that x + 3 ≥ 10.

If A = `[(1,2), (1,3)]`, find A² - 3A

Discuss the continuity of the function f at x = 0

If f(x) = `(2^(3x) - 1)/tanx`, for x ≠ 0

         = 1,   for x = 0

Evaluate : `int ("x" + 1)/("x"("x" + log"x")`dx

Solve the equations x + y = 4 and 2x - y = 5 using the method of reduction.

If the function f is continuous at = 2, then find f(2) where f(x) = `(x^5 - 32)/(x - 2)`, for ≠ 2.

Find the elasticity of demand, if the marginal revenue is 50 and price is Rs 75.

Differentiate  
log (1 + x²) w.r.t. tan^-1 (x)

If   p : It is raining
     q : It is humid

Write the following statements in symbolic form:

(a) It is raining or humid.
(b) If it is raining then it is humid.
(c) It is raining but not humid. 

Using truth table, examine whether the following statement pattern is tautology, contradiction or contingency: p ∨ [∼(p ∧ q)]

Evaluate ; `int x^2/(x^6 - 4x^3 + 13)` dx

If the function f is continuous at x = 0

Where f(x) = 2`sqrt(x^3 + 1)` + a,  for x < 0,
= `x^3 + a + b,  for x > 0
and f (1) = 2, then find a and b.

For manufacturing x units, labour cost is 150 – 54x and processing cost is x². Price of each unit is p = 10800 – 4x². Find the value of x for which Total cost is decreasing.

For manufacturing x units, labour cost is 150 – 54x and processing cost is x². Price of each unit is p = 10800 – 4x². Find the values of x for which Revenue is increasing.

The total cost C for producing x units is Rs (x² + 60x + 50) and the price is Rs (180 - x) per unit. For how many units the profit is maximum.

If y = sin ^-1 `((8x)/(1 + 16x^2))`, find `(dy)/(dx)`

If y = 5^x + x^x, Find `(dy)/(dx)`.

`int (2x + 1)/((x + 1) ( x - 2))` dx

Find the inverse of matrix A by using adjoint method; where A = `[(1, 0, 1), (0, 2, 3), (1, 2, 1)]`

Find the area of the region bounded by the curve (parabola) y² = 4x and the line

Evaluate `int_0^1 (x(sin^-1 x)^2)/sqrt(1 - x^2)` dx

In a class 60% students arc boys and 40% are girls. By admitting 16 boys and 8 girls, the ratio of boys to girls becomes 8 : 5. Find the original number of boys and girls in the class.

An agent charges 12% commission on the sales. How much does he earn if the total sales amounts to ~ 36,000? How muuch does the seller get from this sale?

Find the Age-Specific Death Rate (SDR) for the following data :

If the Crude Death Rate (CDR) for the following data is 13.4 per thousand, find x:

If a fair die is rolled four times,. what is the probability that two of the rolls will show '1'.

Solve the inequation -8 ≤ 3x - 5 < 7 and represent it on the real number line.

Sketch the graph of | y | ≤ 3 in a coordinate plane.

Find the sequence that minimizes the total elapsed time (in hours) required to
complete the following jobs on the machine M₁ , M₂ and M₃ in the order M₁M₂M₃ :

Find accumulated value after 1 year of annuity immediate in which Rs 20,000 is invested every quarter, at .16% p.a. compounded quarterly.
[Given : (1.04)⁴ = 1.1699]

If a random variable X follows Poisson distribution sucli that P(X = 1) = P(X = 2), find:
(a) the mean and
(b) P(X = 0). [Given: e^-2 = 0.1353)

The probability of defective bolts in a workshop is 40%. Find the mean and variance of defective bolts out of 10 bolts.

Given the following table which relates to the number of animals of a certain species at age x . Complete the life table with dx, qx, Lx and Tx.

Co-efficient of correlation between the variables X and Y is 0.3 and their covariance is 12. The variance of X is 9, find the standard deviation of Y.

Compute correlation coefficient for the following data :

Surabhi, Simona and Shruti started a business in a partnership by investing ₹ 60,000, ₹ 40,000 and ₹ 75,000 respectively. At the end of the year they found that they have incurred a loss of ₹ 24,500. Find that how much loss each one had to bear?

Three cars were sold through an agent for Rs 1,60,000, Rs1,48,000 and Rs 1,50,000 respectively. The rates of commission were 17 .5% on the first, and 12.5% on the second. If on the whole, the agent received 14% commission on the total sales, find the rate of commission paid on the third car.

The equations given of the two regression lines are 2x + 3y - 6 = 0 and 5x + 7y - 12 = 0.

Find:

(a) Correlation coefficient

(b) `sigma_x/sigma_y`

If Σ_x1 = 56 Σ_y1 = 56, Σ`x_1^2` = 478,
Σ`y_1^2` = 476, Σ_x1y1 = 469 and n = 7, Find
(a) the regression equation of y on x.
(b) y, if x = 12.

Solve the following minimal assignment problem : 

Solve the following L.P.P.:

Maximize Z = 4x + 5y
subject to       2x + y ≥ 4
                       x + y ≤ 5,
                       0 ≤ x ≤ 3,
                       0 ≤ y ≤ 3