Numerical Analysis MCQ

NUMERICAL ANALYSIS- MCQ 1. Encircle the correct response. (i) What is the correct transformation of the equation ln x + (a) x =

(b) x =

1 = 5 to the form x = g(x)? x2

1 1 − 5 ln x 1 1 − 5 ln x 1

(c) x = e5 − e x2 1

(d) x = e5− x2 1

(e) x = e5+ x2 (ii) What is the approximate value of the root of the equation x 2 − 3 x + 1 = 0 when the iterative method is applied on the the interval [3,5] with the transformation x2 + 1 x= 3 and the initial guess x0 = 3? (a) 3.333 (b) 4.037 (c) 5.766 (d) 6.845 (e) None of the above (iii) Let α be a root of the equation x 2 + x − 2 = 0. Which of the following statement/s is/are true? I. α ∈ [0,1) II. α ∈ (2,3] III. The bisection algorithm is applicable to [1,2] to approximate α . (a) (b) (c) (d) (e)

I only II only III only I and III only None

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(iv) The bisection method was used to approximate a root of the equation f(x) = 0, starting from the interval [1 , 2]. It was found that the 10th and 11th iterations are x10 = 1.2435 x11 = 1.2436 What can you conclude? I. 10 > ln 2 – ln(0.001) II. 10 < ln 2 − ln(0.001) III. x12 < 1.2436 (a) (b) (c) (d) (e)

I only II only III only I and III only None

(v) Suppose that the Newton-Raphson method was applied to determine an approximate root of the polynomial x3 + 3x2 – 3 ,taking 0.5 as the initial guess. What is the fourth iteration? (a) (b) (c) (d) (e)

0.814814 0.879339 0.870467 0.811976 0.873215

(vi) The Newton-Raphson method was applied to solve the equation e x = x + 1 with the initial guess x0=0. What is the answer, correct up to 4 decimal places? (a) (b) (c) (d) (e)

1.3546 0.8793 3.8704 5.4693 None of the above

(vii) f is the function defined on the set ¡ − {2} as f ( x) = Then f(1)f(3)<0 .What can you conclude from this?

1 . x−2

I. f has a root in (1,3) and it can be approximated by the bisection method. II. f has a root in (1 , 3) but it may or may not be approximated by the bisection method. III. f does not have a root in (1,2). (a) (b) (c) (d) (e)

I only II only III only I and III only I, II and III

2

(viii) For which of the following equations, it is possible to obtain an approximate root by using the bisection method? I. x – e x = 0 II. x2 – 2x + 3 = 0 III. x3 – x – 3 = 0 (a) (b) (c) (d) (e)


(ix) Consider the following transformations of equations to the form x = g(x), illustrated by graphs. Which of them will give monotonic sequences of iterates, when the fixed point iterative method is properly applied? I.

II.

III.

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(a) (b) (c) (d) (e)


(x) let f be a function defined on [0, ∞ ) as follows: −1  x 0≤ x<2  8 f ( x) =  x−5+ 7 − 3 x>2  x x2  Which of the following is/are true? I. It is possible to apply the bisection method on [1 , 4] to obtain an approximate value of a root of the equation f(x) = 0. II. It is possible to apply the bisection method on [2 , 4] and to obtain an approximate value of a root of the equation f(x) = 0. III. It is possible to apply the Newton-Raphson method taking the initial guess x0 =2, to obtain an approximate value of a root of the equation f(x) = 0. (a) (b) (c) (d) (e)

I only I and II only II and III only I, II and III none

2. Encircle the correct response. (i) Consider the series expansion of ex: x 2 x3 x 4 e x = 1 + x + + + + .... 2! 3! 4! What is the least number of terms so that the sum yields the value of e0.1 correct up to three decimal places? (a) (b) (c) (d) (e)

2 3 5 6 10

4

Problem (ii) and (iii) are based on the following forward difference table: ∆

x

y

0

-3

1

3

2

11

3

27

4

57

∆2

∆3

∆4

6 2 8

6 8

16

0 6

14 30

(ii) What is the polynomial that can be fitted into these data by Newton’s forward difference formula? (a) (b) (c) (d) (e)

x3 − 2 x2 − 7 x − 3 x3 − 3x2 − 5 x + 2 x3 − 7 x2 − 4 x − 4 3x3 − 2 x2 − 5 x − 3 5 x3 − 2 x2 − 3x + 5

(iii) What is the approximate value of y for x = 3.5 given by Newton’s interpolation? (a) –8 (b) 48 (c) 53 (d) 40 (e) 32 (iv) Which one of the following expressions is incorrect? (a) (b) (c) (d) (e)

∆( f ( x) + g ( x )) = ∆f ( x) + ∆g ( x) ∆( f ( x).g ( x)) = ∆f ( x ).∆g ( x) ∆(cf ( x )) = c∆f ( x ) where c is a constant. ∆ 2 (e x ) = e(e − 1)2 ∆ m + n f ( x) = ∆ m ∆n f ( x)

(v) If the interval of differencing is one, the value of ∆ 4 ( x3 + 3x + 1) + ∆3 (2 x3 + 1) is (a) x 3 + 3x + 1 + 2 x3 + 1 (b) 12 (c) 8 (d) 0 (e) -8 5

(vi) Which one of the following expressions is incorrect? (a) (b) (c) (d) (e)

∆E = E ∆ 1+ E = ∆ ∆E −1 = E −1∆ (1 + ∆)(1 − ∇) = 1 ∇ = 1 − E −1

(vii) Which of the following statement/s is/are true? I. The linear interpolant is continuous at the given data points x1, x2, …xn. II. The linear interpolant is differentiable at the given data points x1, x2, …xn. III. The linear interpolant passes through all the given data points x1, x2, …xn. (a) (b) (c) (d) (e)


(viii) Which of the following statement/s is/are true? I. The quality of Newton’s interpolation increases with the degree of the polynomial used. II. The quality of Newton’s interpolation decreases with the degree of the polynomial used. III. In Newton’s interpolation, the degree of the polynomial is independent of the number of given data points. (a) (b) (c) (d) (e)

I only II only III only I and III only none

(ix) Let S(x) be the cubic spline function which was obtained based on a given set of data points x1, x2, …xn where n>2.Which of the following statement/s is/are true? I. S(x) is a cubic polynomial. II. S(x) is differentiable on (x1,xn) . III. S ′ is differentiable on (x1,xn).

6

(a) (b) (c) (d) (e)

I only I and II only II and III only I, II and III none

(x) Let Pn denotes the Newton’s interpolation polynomial of degree n. Suppose that we consider n points of some known functions to check the validity of the interpolation formula. Which one of these functions would give a maximum deviation from its interpolation polynomial for sufficiently large n? (a) f(x) = x3 + 2x , x∈ ¡ (b) g(x) = x4 – x3 , x ∈ ¡ (c) h(x) =1/x , x > 0 1 (d) k(x) = 2 , x∈ ¡ x +1 (e) q(x) = c where c is a constant, x ∈ ¡ .

1. A discrete model for Newton's Law of Motion is (a). yk+1 – yk = dt*c*(ysur – yk) (b). yk+1 – yk = dt*(-32 + k*yk*yk)/m (c). y'(t) = c(ysur –y(t)) (d). y'(t) = (-32 + k y2)/m (e). yk+1 – y = c(-32 –y(t)) 2. A continuous model for Newton's Law of Motion is (a). yk+1 – yk = dt*c*(ysur – yk) (b). yk+1 – yk = dt*(-32 + k*y*y)/m (c). y'(t) = c(ysur –y(t)) (d). my'(t) = (-32 + k y2) (e). yk+1 – y = c(-32 –y(t)) 3. A steady state solution of y' = (-32 + y2) is (a). y(t) = 32 (b). y(t) = 321/2 (c). y(t) = -32 (d). y(t) = e-32t (e). all of the above 4. A solution of the differential equation y' = (-32 + y2) with initial condition y(0) = 0 (a). can be found using separation variables (b). can be found using Maple (c). can be approximated using Euler method (d). can be approximated using improved Euler method (e). all of the above 5. In the code file imeuler.m the expression y(1) (a). represents the speed function evaluated at time 1 (b). represents the position function evaluated at time 1 (c). the first entry in the y array whose value is the initial speed 7

(d). the first entry in the y array whose value is the initial position (e). a Matlab command

8

6. The reason for using numerical methods is (a). one cannot always find the exact solution (b). the graphics are nice (c). one can easily experiment with model parameters (d). a modern way of doing engineering (e). (a) and (c) 7. The improved Euler method is better because (a). it is the second variation of the Euler method (b). the error is one half the error of the Euler method (c). the error is often proportional to the step size squared (d). the code is a little longer (e). gives better graphics 8. The imeuler.m code is executed by (a). first entering fmass at the Matlab prompt (b). first entering fmass(1,2) at the Matlab prompt (c). entering imeuler(0) at the Matlab prompt (d). entering imeuler.m at the Matlab prompt (e). entering imeuler at the Matlab prompt 9. The graph of the numerical solution for the speed array, y, versus the time array, t, in Matlab can be generated by (a). plot(t,speed) (b). plot(t,y(10)) (c). plot(t,y) (d). graph(t,speed) (e). graph(t,y) 10. The best reason for using a model for falling mass is (a). good calculus problem (b). good computing problem (c). one wants to predict the speed (d). allows for repeated experimentation with different parameters so as to obtain a desired result (e). none, just another Matlab problem

Question No: 1

- Please choose one

A series of constant, or level, cash flows that occur at the end of each period for some fixed number of periods is called a/an:

► Present Value ► Future Value ► Ordinary Annuity ► Ordinary Share Question No: 2

- Please choose one 9

The Ratios showing the ability of a firm to pay its bills in short-run are called:

► Leverage Ratios ► Liquidity Ratios ► Profitability Ratios ► Market Value Ratios Question No: 3

- Please choose one

GAAP stands for:

► Generally Accepted Accounting Principles ► Generally All-rounder Accounting Principles ► General Accepting Accounts Principles ► None of the given options Question No: 4

- Please choose one

A contract between the bond issuer and bond holder is called:

► Bond Indenture ► Bond Debenture ► Bond Value ► None of the given options Question No: 5

- Please choose one

Suppose you have a portfolio comprised of two securities X and Y. In the portfolio, 60 shares are of stock X valued at Rs.10 per share and 40 shares are of stock Y valued at Rs.3 per share. What is the approximate weight of stock X in the portfolio?

► 23 % ► 40 % ► 60 % ► 83 % Question No: 6

- Please choose one 10

In which market, previously issued securities are traded among investors?

► Primary Market ► Secondary Market ► Tertiary Market ► None of the given options Question No: 7 - Please choose one Which of the following is the present value of a series of future net cash flows that will result from an investment, minus the amount of the original investment?

► Present Value ► Future Value ► Net Present Value ► Terminal Value Question No: 8

- Please choose one

You earn a 5 percent real return. If the inflation rate is 4 percent, what is your nominal return?

► 8.96 % ► 9.05 % ► 9.20 % ► 9.92 % Question No: 9

- Please choose one

Fee paid to the consultant for evaluating the option of launching a new product will be considered as:

► Sunk Cost ► Opportunity Cost ► Financing Cost ► Operating Cost Question No: 10 - Please choose one 11

A risk that affects a single or at most a small number of assets is called:

► Unsystematic Risk ► Unique Risk ► Diversifiable Risk ► All of the given options Question No: 11 - Please choose one What will be the payback period of a Rs.70,000 investment with the following cash inflows? Years 1 2 3 4 5

Cash flows Rs. 15,000 Rs. 20,000 Rs. 25,000 Rs. 15,000 Rs. 5,000

► 3.57 years ► 3.67 years ► 4.57 years ► 4.67 years Question No: 12

- Please choose one

Which of the following is the required return on a firm's debt by its creditors?

► Cost of Equity ► Cost of Debt ► Cost of Preferred Stock ► Cost of Capital Question No: 13

- Please choose one

Which one of the followings is the overall required return the firm must earn on its existing assets to maintain the value of the stock?

► AAR (Average Accounting Return) ► IRR (Internal Rate of Return) 12

► MIRR (Modified Internal Rate of Return) ► WACC (Weighted Average Cost of Capital) Question No: 14

- Please choose one

The costs to store and finance the assets are known as:

► Carrying Costs ► Shortage Costs ► Manufacturing Costs ► None of the given options Question No: 15

- Please choose one

The minimum level of inventory that a firm keeps on hand is called:

► Common stock ► Safety Stock ► Preferred Stock ► Dangerous Stock Question No: 16

- Please choose one

Realization Principle is one of the basic principles of GAAP.

► True ► False Question No: 17

- Please choose one

Whenever the word Dividend is used, it always refers to a long-term loan.


- Please choose one

A preferred dividend is exactly like interest on bond. 13


- Please choose one

By IRR rule, take a project when its IRR exceeds the required return.


- Please choose one

Diversification is the group of assets such as stocks and bonds held by investor.

► True ► False Question No: 21 ___________________ is a special case of Annuity, where the stream of cash flows continues forever. Question No: 22 ________________ is the value of a present amount at a certain date in the future based on a determined rate of return. Question No: 23 The amount of time required for an investment to generate cash flows sufficient to recover its initial cost is called its ____________________. Question No: 24 __________________ refers to the extent to which a firm relies on its debt. Question No: 25 The difference between the return on a risky investment and that on a risk free investment is called ____________________. Question No: 26 What is the difference between Flexible Policy and Restrictive Policy regarding size of investment in current assets while making short-term financial policy? Question No: 27 14

Differentiate between Systematic Risk and Unsystematic Risk. Which of them can be eliminated by diversification? Question No: 28 Suppose common stocks of a company are currently selling for Rs.30 per share. Stock market analysts estimated a dividend of Rs.2 per share for the next year and it is expected that the dividend will grow by 10% more or less indefinitely. What return does this stock offer? Question No: 29 A bank is offering 12% interest rate compounded quarterly on its saving account. What would be the Effective Annual Rate (EAR) ? Question No: 30 “An investment is acceptable if the IRR exceeds the required return. It should be rejected otherwise.” Explain. Question No: Sumi Inc. has outstanding Rs.1, 000- face –value bond with a 16 percent coupon rate and 6 years remaining until final maturity. Interest payments are made quarterly. What would be the value of this bond if your nominal annual required rate of return is : (i) 13 %, (ii) 19 %. Question No: 32 S&T Company just paid a dividend of Rs.2 per share and has a share price of Rs.30. The dividends are expected to grow @ 10% forever. S&T Company has Rs.75 million in equity and Rs.75 million in debt in its total capital. The tax rate for the firm is 35% and the Cost of debt is 8%. What will be the Weighted Average Cost of Capital (WACC) for S&T Company ?

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Numerical Analysis MCQ

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