Numerical methods and calculus- Part 2

Introduction

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Now we will see some Numerical methods and calculus questions with their solutions and proper explanation.

Also check Numerical methods and calculus-1 here.

Questions With Solutions

1. What is the value of the following integral upto three decimal places? (GATE CSE 2018)

Solution) Between 0.27 and 0.30

2. What is the value of the following limit?  (GATE CSE 2017 Set 1)

Solution) 1

3. What is the value of the following integral? (GATE CSE 2015 Set 1)

Solution) -1

4. A denotes the area of the region bounded by f(x)=x^−(1/3) and the X−axis when x varies from −1 to 1. Which of the following statements is/are TRUE?

I. f is continuous in [−1,1]

II. f is not bounded in [−1,1]

III. A is nonzero and finite (GATE CSE 2015 Set 2)

Solution) II and III only

5. If for non-zero x, af(x)+bf(1x)=1x−25

where a≠b, then what is the value of the following integral?  (GATE CSE 2015 Set 3)

Solution)

6. Assume you wish to go from zero to one hundred on the number line. You either travel straight a unit distance or take a shortcut at each step. A shortcut is merely a pair of numbers I j with the prefix I j. If you have the shortcut I j and are on the number line at the point I, you can go straight to j. Assume that T(k) is the fewest number of steps required to get from k to 100. Assume that there is only one shortcut involving any number and that the shortcut from 9 to 15 is in particular. Let T(9) = 1+ min(T(y),T(z)) be the case for y and z. Then the product yz has a value of. (GATE CSE 2014 Set 3)

Solution) 150

7. In the interval [0,2], a function f(x) is continuous. It is well known that f(0) = f(2) = 1 and f(1) = 1. Which of the following statements is required to be correct? (GATE CSE 2014 Set 1)

Solution) in the interval (0,1), there exist a y such that f(y)=f(y+1)

8. What is the value of the following integral? (GATE CSE 2014 Set 3)

Solution) -2π

9. What is the value of t is if the function f(x)= x sinx satisfies the following equation: f"(x)+f(x)+t cosx=0. (GATE CSE 2014 Set 1)

Solution) -2

f′(x)=xcos⁡(x)+sin⁡(x)

f″(x)=x(−sin⁡x)+cos⁡x+cos⁡x

Now f″(x)+f(x)+tcos⁡x=0

⇒x(−sin⁡x)+cos⁡x+cos⁡x+xsin⁡x+tcos⁡x=0

⇒2cos⁡x+tcos⁡x=0

⇒cos⁡x(t+2)=0

⇒t+2=0,t=−2

10. What is the value of the given definite integral? (GATE CS 2011)

Solution) i

11. What is the value of the given definite integral? (GATE CS 2009)

Solution) ½ in 2

12. If a point on a curve is a local minimum or maximum, it is called an extremum. The curve 3x⁴16x³+24x²+37 has a total of how much different extrema. (GATE CS 2008)

Solution) 1

 

13. What is the value of the given definite integral? (GATE CS 2005)

Solution) 1

14. The function f(x,y)=2x²+2xy−y³ has (GATE CS 2002)

Solution) two stationary points at (0,0) and (⅙ , -⅓ )

The function f(x,y)=2x2+2xy−y3 has. only one stationary point at (0,0) two stationary points at (0,0) and (1/6,−1/3) two stationary points at (0,0) and (1,−1)
 

15. Find the points of local maxima and minima if the following function defined in 0≤x≤6

Solution) maximum at x = 1, minimum at x = 2

Here,

 

16. In the interval [0,2] what is the maximum value of the function

f(x)=2x²−2x+6? (GATE CS 2002)

Solution) 10

17. What is the minimal value of f(x) for x(0,2) if f(x) is defined as follows? (GATE CS 2008)

Solution) 2112

18. What is the value of the following integral? (GATE CS 2005)

Solution) 0

19. What is the maximum value of the function

f(x)=2x²−2x+6 in the interval [0,2]? (GATE CS 1997)

Solution) 10

20. Find the points of local maxima and minima, if any, of the following function defined 0≤x≤6. x3−6x2+9x+15 (GATE CS 1998)

Solution) Maximum at x=1

Minimum at x=3

21. What is the value of the following integral? (GATE CS 1998)

Solution) 0

22. With 3.5 as the initial value, the Newton-Raphson method is used to get the root of the equation x²-13=0. After one cycle, the approximation is (GATE CS 2010)

Solution) 3.607

In Newton-Raphson’s method, We use the following formula to get the next value of f(x). f'(x) is the derivative of f(x).

f(x)  =  x2-13

f'(x) =  2x

Applying the above formula, we get

Next x = 3.5 - (3.5*3.5 - 13)/2*3.5

Next x = 3.607

23. If g(x)=1−x & h(x)=x/x−1 then g(h(x))/h(g(x)) is

Solution) h(x)/g(x)

24. What is the value of the following limit?

Solution) 1

25. What is the value of the following limit up to two decimal places? (GATE CS 2021 Set 1)

Solution) 0.25

26.  What is the value of the following limit? (GATE CS 2017 Set 1)

Solution) 1