**Introduction**

The Graduate Aptitude Test in Engineering is an examination held in India that primarily assesses a student's complete mastery of numerous undergraduate engineering and scientific disciplines in preparation for a master's degree program and work in the public sector. GATE Online Application Processing System is where candidates must register and fill out the application form (GOAPS). You can visit the official site __GATE__ for getting information on registration dates and deadlines.

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

**Questions with Solutions**

1. What is the value of the following limit?

**Solution: **This is 0/0 form, by applying the L'Hospital rule.

2. Which of the following functions is/are increasing everywhere in [0,1]?

**Solution) **III only

3. What is the value of the following limit?

**Solution) **We will get 0/0 form when x = 3, so we can apply the L'Hospital rule.

4. Consider the equation , a quadratic equation with coefficients in a base b coefficients. This equation has two solutions in the same base b: x=5 and x=6. Then b is equal to =

**Solution) **8

5. If , then the constants R and S are respectively.

**Solution) **4/π and 0

**Now, integrate equation (1) w.r.t. x, we get,**

6. Let g(x)=f′(x) be the derivative of f(x), and f(x)=f′(x) be the polynomial. If (f(x)+f(-x)) has a degree of 10, then (g(x)-g(-x)) has a degree of =

**Solution) **9

7. In the plot below, choose the most acceptable equation for the function represented as a thick line.

**Solution)** x = - (y - |y|)

8. The value of the double integral given below is -

**Solution) **½

9. What is the value of the following limit?

**Solution)** ∞

10. If the slope of the tangent is -2x/y at every point of a certain curve, then the curve is

**Solution)** An ellipse

11. In the interval [-1,1] the function y=|x| is

**Solution) **Continuous but not differentiable

The function y = | x | in the interval [-1, 1 ] is

| x | is continuous and differentiable everywhere except at x = 0, where it is continuous but not differentiable.

Since [-1, 1] contains 0, in this interval it is continuous but not differentiable.

12. The formula for computing an approximation for a function f's second derivative at a location x_{0} is

**Solution)**

13. What is the value of the following integral?

**Solution) **(π/8) + (¼)

14. What is the value of the following limit?

**Solution) 0**

15. Take a look at these two statements about the function f(x)=|x|:

**For all real x values, P.f(x) is continuous.**

**For any real x values, Q.f(x) is differentiable.**

Which of the following statements is correct?

**Solution) **P is true and Q is false

f(x)=∣x∣. Here, for all values of x,f(x) exists. Therefore, it is continuous for all real values of x.

At x=0,f(x) is not differentiable. Because if we take the left hand limit here, it is negative while the right hand limit is positive making

LHL≠RHL

16. What is the value of the following limit?

**Solution) **1

17. What is the value of the following double integral?

**Solution) **13.5

18. What is the value of the following limit?

**Solution) **1

19. What is the value of the following limit?

**Solution) **e^{-2}

20. In the interval x∈[π/4,7π/4], consider the function f(x)=sin(x). The number and location(s) of this function's local minima are

**Solution) **One, at (3π/2)

21. The function f is defined at the following points-

Using the trapezoidal rule computes the value of the following integral-

**Solution) **9.045

22. Which of the following functions is continuous at x=3?

**Solution) **(a)

23. Let the function f be

Where θ∈[π/6,π/3] and f(θ) denote the derivative of f with respect to θ. Which of the following statements is/are TRUE?

I) There exists θ∈[π/6,π/3] such that f(θ) =0.

II) There exists θ∈[π/6,π/3] such that f(θ) ≠0.

**Solution) **Both I and II

24. What is the value of k in the following integral?

**Solution) 4**

25. A function f(x) is linear, which has the value of 29 at x=−2 and 39 at x=3. Then find its value at x=5

**Solution) **43

26. What is the value of the following limit?

**Solution) **1

27. What is the value of the following limit?

**Solution) **1