PRACTICE QUESTION PAPER

SECTION A

1. The area (in sq units) bounded by the curve x = √ , the y–axis, y = 1 and y = 4 is
   (a) 11/3
   (b) 1/4
   (c) 14/3
   (d) none of these

2. Sampling which provides for a known-zero equal chance of selection is
   (a) systematic sampling
   (b) convenience sampling
   (c) quota sampling
   (d) purposive sampling

3. The cost function of a firm is C = 3x² + 2x – 3. The marginal cost, when x = 3 is
   (a) 10
   (b) 25
   (c) 5
   (d) 20

4. It is given that at x = 1, the function f(x) = x³ – 12x² + kx + 7 attains maximum value, then the value of k is
   (a) 10
   (b) 12
   (c) 21
   (d) 13

5. For the purpose of t–test of significance, a random sample of size (n) 2025 is drawn from a normal population, then the degree of freedom is
   (a) ______
   (b) ______
   (c) 2025
   (d) 2024

6. The region represented by the inequation x – y
   (a) bounded
   (b) unbounded
   (c) do not exist
   (d) triangular region

7. The shape of normal distribution curve is
   (a) bell shaped
   (b) flat
   (c) circular
   (d) spiked

8. A machine costing Rs 50,000 has a useful life of 4 years. The estimated scrap value is Rs 10,000, then the annual depreciation is
   (a) Rs 20,000
   (b) Rs 10,000
   (c) Rs 5000
   (d) Rs 2500

9. dy/dx of x² + y² – 2x – 3 = 0
   (a) 2x – y
   (b) 2y – x
   (c) x + y
   (d) none of these

10. The probability that a student is not a swimmer is 1/5, then the probability that out of 5 students 4 are swimmers is
    (a) C(5,4)(4/5)⁴(1/5)
    (b) (4/5)⁴(1/5)
    (c) C(5,0)(4/5)⁴(1/5)
    (d) none of these

11. It is currently 8 am time. In next 500 hours time will be
    (a) 4 am
    (b) 5 am
    (c) 6 am
    (d) none of these

12. Integration of 2x³ – 7 w.r.t x is
    (a) x⁻⁵ + c
    (b) x + 5 + c
    (c) 2x + 5
    (d) none of these

13. Trend can be measured using by the following methods
    (a) graphical method
    (b) semi average method
    (c) moving averages method
    (d) All of the above

14. The present value of sequence of payments of Rs 60 made at the end of each 6 months and continuing forever, if money is worth 4% p.a compounded semi-annually
    (a) Rs 3000
    (b) Rs 3500
    (c) Rs 4000
    (d) Rs 4500

15. Y = e⁻ˣ + ax + b is a solution of differential equation
    (a) e⁻ˣ yʺ = 1
    (b) eˣ yʺ = 1
    (c) eˣ (yʹ)² = 1
    (d) e⁻ˣ (yʹ)² = 1

16. If A is a square matrix of order 3 and |A| = 5 then the value of |3 adj A| is
    (a) 25
    (b) 50
    (c) 75
    (d) none of these

17. If A² + 4A – 5I = O then A⁻¹ is equal to
    (a) 1/7 (A – 5I)
    (b) 1/7 (5I – A)
    (c) –1/7 (5I – A)
    (d) none of these

18. The sum of order and degree of the differential equation ________ = eˣ
    (a) 2
    (b) 3
    (c) 5
    (d) none of these

ASSERTION AND REASON QUESTIONS

In Q19 and Q20 Assertion (A) is followed by a statement of Reason (R).
Mark the correct choice:

(a) Both A and R are true and R is the correct explanation of A
(b) Both A and R are true but R is not the correct explanation of A
(c) A is true but R is false
(d) A is false but R is true

19. Assertion (A): If the nominal rate of interest is 12.5% and inflation is 2% then the effective rate of interest is 10.5%
    Reason (R): If the nominal rate is calculated only at the end of a year, then the effective rate of interest is same as the nominal rate

20. Assertion (A): [ ] is a skew symmetric matrix
    Reason (R): Every square matrix A can be expressed as a sum of symmetric and skew symmetric matrix

SECTION B

21. Tea worth Rs 126 per kg and Rs 135 per kg are mixed with a third variety in the ratio
    1 : 1 : 2. If the mixture is worth Rs 153 per kg, find the price of third variety per kg.

OR

Pipes A and B together can fill a tank in 4 hours. Pipe B takes 6 hours more than A to fill the tank. If they are opened separately, find the time taken by A alone.

22. A and B can cover a 200 m race in 22 seconds and 25 seconds respectively. When A finished the race, then B is at what distance from the finishing line?

23. There are 3 points P, Q and R in a straight line such that Q is equidistant from P and R.
    A can swim from P to R downstream in 24 hours and from Q to P upstream in 16 hours.
    Find the ratio of speed of man in still water to speed of stream.

24. A radar unit is installed to measure the speed of cars on a highway. The speeds are normally distributed with mean 80 km/hr and standard deviation 10 km/hr. Find the probability of a car running at less than 60 km/hr.
    [Given F(2) = 0.9772]

OR

For a certain type of laptops the charging time of batteries is normally distributed with mean 50 hours and standard deviation 15 hours. Arun has one of the laptops. Find the probability that the charging time of the battery will be between 50 to 70 hours.
[Given F(1.33) = 0.9082 and F(0) = 0.5]

25. Find the integral value of x if [ ][ ][ ] = 0

SECTION C

26. What is the remainder when 17113 is divided by 3?

27. Two batches of the same product are tested for their mean life. Assuming that the lives of the product follow a normal distribution with an unknown variance, test the hypothesis that the mean life for both the batches is the same, given the following information:

Batch I
Sample size: 10
Mean life: 750 hours
Standard deviation: 12 hours

Batch II
Sample size: 8
Mean life: 820 hours
Standard deviation: 14 hours

OR

The manufacturer of electrical items makes bulbs and claims that these bulbs have a mean life of 25 months. The life in months of a random sample of 6 bulbs is:
24, 26, 30, 20, 20 and 18.
Test the validity of the manufacturer’s claim at 1% level of significance.
[Given t₅(0.01) = 4.032]

28. A factory produces bulbs of which 6% are defective. Find the probability that in a sample of 100 bulbs not more than one defective bulb is included.
    (e⁻⁶ = 0.0024)

29. Mrs Anuradha takes a loan of Rs 5,00,000 from a bank at the rate of 6% p.a for 2 years. Calculate her EMI using flat rate method.
    Also, she purchased a car worth Rs 5,00,000 and paid Rs 1,00,000 as cash down payment and the balance in equal monthly installments in 2 years. If bank charges 6% p.a compounded monthly, calculate EMI.
    [Given (1.005)²⁴ = 1.123]

30. Maximum value of z = 2x + 5y subject to constraints
    2x + 3y ≤ 6
    4x – 5y ≤ 20
    y ≤ 3
    x and y are positive.

31. Find the probability distribution of the number of successes of two tosses of a die, where success is defined as “the number greater than 4”.

OR

A die is thrown 5 times. Find the probability of getting an odd number at least 4 times.

SECTION D

32. Given below are the consumer price index of the industrial workers:

YEAR: 2014, 2016, 2017
INDEX NUMBER: 145, 150, 190

Find the best fitted line by method of least square and tabulate the trend values.

OR

The average number, in lakhs of working days lost in strikes during each year of the period 2001 to 2010 was:

2001 – 1.5
2002 – 1.8
2003 – 1.9
2004 – 2.2
2005 – 2.6
2006 – 3.7
2007 – 2.2
2008 – 6.4
2009 – 3.6
2010 – 5.4

Calculate the 3–yearly moving averages and draw the moving averages graph.

33. If
    A = [ ] and B = [ ] find AB.
    Hence solve
    x – y = 3
    2x + 3y + 4z = 17
    y + 2z = 7

OR

Solve the following equations by Cramer’s rule:
x – y = 3
2x + 3y + 4z = 17
y + 2z = 7

34. The demand and supply function for a commodity under pure market competition are
    Pd = 16 – x² and Ps = 4 + 2x²
    where p is the price and x is the quantity.
    Find the consumer surplus and producer’s surplus.

35. A company sets aside a sum of Rs 10,000 at the end of each year in a sinking fund so that at the end of 20 years it would amount to a balance sufficient to repay the machinery. Assuming that the cost of machinery remains the same and money earns 10% p.a compound interest, find the cost of the machinery.
    If the number of years is 10 instead of 20, find the cost of the machinery.
    [Given (1.1)²⁰ = 6.730, (1.1)¹⁰ = 2.594]

SECTION E

CASE STUDY QUESTIONS

36. The relation between the height of the plant (y cm) with respect to exposure to sunlight is governed by the equation
    y = 4x – ½x²,
    where x is the number of days exposed to light.

Based on the above information answer the following questions:

(a) Find the rate of growth of the plant with respect to number of days exposed to sunlight. (1 mark)
(b) What will be the height of the plant after 2 days? (1 mark)
(c) What is the maximum number of days it will take for the plant to grow to the maximum height? What is the maximum height? (2 marks)

OR

(c) If the height of the plant is 7/2 cm, find the number of days it has been exposed to sunlight. (2 marks)

37. Suppose Mr. X invested Rs 1,00,000 in a mutual fund and the value of the investment at the time of redemption was Rs 1,50,000. If CAGR for the investment is 8%.

Based on the above information answer the following questions:

(a) Calculate the number of years for which he has invested the amount. (2 marks)
(b) If CAGR is 4%, what is the number of years? (2 marks)
(You can use log table for calculations)

38. A dietician has to develop a special diet using two foods P and Q.

Each packet (30 g) of food P contains
12 units calcium, 4 units iron, 6 units cholesterol and 6 units vitamin A.

Each packet (30 g) of food Q contains
3 units calcium, 20 units iron, 4 units cholesterol and 3 units vitamin A.

The diet requires at least 240 units of calcium, at least 460 units of iron and at most 300 units of cholesterol.

Based on the above information answer the following:

(a) What is the objective function if the amount of Vitamin A in the diet is to be minimised? (1 mark)
(b) How many packets of each food should be used to minimise the amount of vitamin A using LPP? (3 marks)