Mathematics and Statistics Model set 1 by shaalaa.com 2024-2025 HSC Commerce (English Medium) 12th Standard Board Exam Question Paper Solution

The false statement in the following is ______.

p ˄ (∼ p) is contradiction

(p → q) ↔ (∼ q → ∼ p) is a contradiction

∼ (∼ p) ↔ p is a tautology

p ˅ (∼ p) ↔ p is a tautology

`int_a^b f(x)dx` = ______.

`int_b^a f(x)dx`

`-int_a^b f(x)dx`

`-int_b^a f(x)dx`

`int_b^a f(x)dx`

Choose the correct alternative:

`int (x + 2)/(2x^2 + 6x + 5) "d"x = "p"int (4x + 6)/(2x^2 + 6x + 5) "d"x + 1/2 int 1/(2x^2 + 6x + 5)"d"x`, then p = ?

`1/3`

`1/2`

`1/4`

2

Find `(d^2y)/(dy^2)`, if y = e^4x

8 e^4x

16 e^4x

13 e^4x

22 e^4x

If 0 < η < 1, then the demand is ______.

constant

inelastic

unitary elastic

elastic

If y = e^logx then `dy/dx` = ?

`(e^(logx))/x`

`1/x`

0

`1/2`

The derivative of a^x is a^x log a.

True

False

Conditional of p → q is equivalent to p → ∼ q.

True

False

A^–1 exists if |A| = 0.

True

False

If y = x log x, then `(d^2y)/dx^2`= _____.

`int 1/(4x^2 - 1) dx` = ______.

If x = `y + 1/y`, then `dy/dx` = ____.

Obtain the differential equation by eliminating arbitrary constants from the following equation:

y = Ae^3x + Be^–3x

Find the equation of tangent and normal to the curve y = x² + 5 where the tangent is parallel to the line 4x – y + 1 = 0.

Write the converse, inverse, and contrapositive of the following statement.

If he studies, then he will go to college.

For the demand function D = 100 – `p^2/2`. Find the elasticity of demand at p = 10 and comment on the results.

Solve the following differential equations:

x²ydx – (x³ – y³)dy = 0

Find the area of the region bounded by the following curves, the X-axis and the given lines:

y = x² + 1, x = 0, x = 3

The rate of growth of population is proportional to the number present. If the population doubled in the last 25 years and the present population is 1 lac, when will the city have population 4,00,000?

Evaluate: `int_2^5 sqrt(x)/(sqrt(x) + sqrt(7) - x)dx`

Express the following equations in matrix form and solve them by method of reduction.

3x – y = 1, 4x + y = 6

Evaluate the following integrals:

`int_1^3 (root(3)(x + 5))/(root(3)(x + 5) + root(3)(9 - x))*dx`

Using the truth table, prove the following logical equivalence.

p ↔ q ≡ ~(p ∧ ~q) ∧ ~(q ∧ ~p)

If `int_a^b x^3 dx` = 0, then `(x^4/square)_a^b` = 0

⇒ `1/4 (square - square)` = 0

⇒ b⁴ – `square` = 0

⇒ (b² – a²)(`square` + `square`) = 0

⇒ b² – `square` = 0 as a² + b² ≠ 0

⇒ b = ± `square`

Find the value of x for which the function f(x)= 2x³ – 9x² + 12x + 2 is decreasing.

Given f(x) = 2x³ – 9x² + 12x + 2

∴ f'(x) = `squarex^2 - square + square`

∴ f'(x) = `6(x - 1)(square)`

Now f'(x) < 0

∴ 6(x – 1)(x – 2) < 0

Since ab < 0 ⇔a < 0 and b < 0 or a > 0 and b < 0

Case 1: (x – 1) < 0 and (x – 2) < 0

∴ x < `square` and x > `square`

Which is contradiction

Case 2: x – 1 and x – 2 < 0

∴ x > `square` and x < `square`

1 < `square` < 2

f(x) is decreasing if and only if x ∈ `square`

Feasible region is the set of points which satisfy ______.

the objective function

all of the given constraints

some of the given constraints

only one constraint

If F(x) is distribution function of discrete r.v.x with p.m.f. P(x) = `(x - 1)/(3)`; for x = 0, 1 2, 3, and P(x) = 0 otherwise then F(4) = _______.

–1

0

1

4

The payment date after adding 3 days of grace period is known as ______.

The legal due date

The nominal due date

Days of grace

Date of drawing

Which component of time series refers to erratic time series movements that follow no recognizable or regular pattern?

Trend

Seasonal

Cyclical

Irregular

Insurance companies collect a fixed amount from their customers at a fixed interval of time. This amount is called ______.

EMI

Installment

Contribution

Premium

The set of feasible solutions of LPP is a ______.

Concave set

Convex set

Null set

None of these

Broker is an agent who gives a guarantee to seller that the buyer will pay the sale price of goods.

True

False

Graphical solution set of x ≤ 0, y ≥ 0 in xy system lies in second quadrant.

True

False

`sqrt((sump_1q_0)/(sump_0q_0)) xx sqrt((sump_1q_1)/(sump_0q_1)) xx 100`

True

False

Fill in the blank :

_______ component of time series is indicated by a smooth line.

Solution which satisfy all constraints is called ______ solution.

Marshall-Edgeworth's Price Index Number is given by ______

For the certain bivariate data on 5 pairs of observations given

∑x = 20, ∑y = 20, ∑x² = 90, ∑y² = 90, ∑xy = 76

Calculate: 

1. cov(x, y)
2. b_yx and b_xy
3. r

Find the sequence that minimizes the total elapsed time to complete the following jobs in the order AB. Find the total elapsed time and idle times for both the machines.

A salesman receives 3% commission on the sales up to ₹ 50,000 and 4% commission on the sales over ₹ 50,000. Find his total income on the sale of ₹ 2,00,000.

The following table shows the production of gasoline in U.S.A. for the years 1962 to 1976.

1. Obtain trend values for the above data using 5-yearly moving averages.
2. Plot the original time series and trend values obtained above on the same graph.

A stock worth ₹ 7,00,000 was insured for ₹ 4,50,000. Fire burnt stock worth ₹ 3,00,000 completely and damaged the remaining stock to the extent of 75 % of its value. What amount can be claimed under the policy?

Find x if Laspeyre’s Price Index Number is same as Paasche’s Price Index Number for the following data

The difference between true discount and banker's discount on a bill due 6 months hence at 4% is ₹ 160. Calculate true discount, banker's discount and amount of bill.

Solve the following problem :

Find the sequence that minimizes the total elapsed time to complete the following jobs. Each job is processed in order AB.

Determine the sequence for the jobs so as to minimize the processing time. Find the total elapsed time and the idle times for both the machines.

From the data of 20 pairs of observations on X and Y, following results are obtained.

`barx` = 199, `bary` = 94,

`sum(x_i - barx)^2` = 1200, `sum(y_i - bary)^2` = 300,

`sum(x_i - bar x)(y_i - bar y)` = –250

Find:

1. The line of regression of Y on X.
2. The line of regression of X on Y.
3. Correlation coefficient between X and Y.

Find the optimal sequence that minimizes total time required to complete the following jobs in the order ABC. The processing times are given in hours.

A department store has four workers to pack goods. The times (in minutes) required for each worker to complete the packings per item sold is given below. How should the manager of the store assign the jobs to the workers, so as to minimize the total time of packing?

Shraddho wants to invest at most ₹ 25,000/- in saving certificates and fixed deposits. She wants to invest at least ₹ 10,000/- in saving certificate and at least ₹ 15,000/- in fixed deposits. The rate of interest on saving certificate is 5% and that on fixed deposits is 7% per annum. Formulate the above problem as LPP to determine maximum income yearly.

If X has Poisson distribution with parameter m and P(X = 2) = P(X = 3), then find P(X ≥ 2). Use e^–3 = 0.0497.

P[X = x] = `square`

Since P[X = 2] = P[X = 3]

`square` = `square`

`m^2/2 = m^3/6` 

∴ m = `square`

Now, P[X ≥ 2] = 1 – P[x < 2]

= 1 – {P[X = 0] + P[X = 1]

= `1 - {square/(0!) + square/(1!)}`

= 1 – e^–3[1 + 3]

= 1 – `square` = `square`