**Question 1: ****What is curl of the vector field 2x**^{2}yi + 5z^{2}j - 4yzk? [2019 : 1 Mark, Set-ll]

(a) -14zi-2x^{2}k

(b) 6zi + 4x^{2}j - 2x^{2}k

(c) - 14zi + 6yj + 2x^{2}k

(d) 6zi - 8xyj + 2x^{2}yk

Answer: **(a)**

**Solution: **

**Question 2: Euclidean norm (length) of the vector [4 -2 -6]**^{r} is [2019 : 1 Mark, Set-ll]

**Answer: (b)**

**Solution: **

Euclidean norm length

**Question 3: The following inequality is true for all x close to 0.**

**What is tha value of [2019 : 1 Mark, Set-ll]**

**(a) 1**

**(b) 0**

**(c) 1/2**

**(d) 2**

**Answer: (d)**

**Solution: **

**Question 4: Consider the functions: x = y In φ and y = φ In y. which one of the following the correct expression for [2019 : 2 Mark, Set-l]**

**Answer: (a)**

**Solution: **

** **......(i)

**Question 5: Which one of the following is NOT a correct statement? [2019 : 2 Marks, Set-I]**

**(a) The functionhas the global minima at x = e**

**(b) The function has the global maxima at x = a **

**(c) The function x**^{3} has neither global minima nor global maxima

**(d) The function |x| has the global minima at x = 0**

**Answer: (a) **

**Solution: **Let y = x^{1/x}

log y = logx/x

y maximum (or) minimum when,

is maximum (or) minimum

**Question 6: For a small value of h, the Taylor series expansion for f(x +h) is [2019 : 1 Mark, Set-I]**

**Answer: **(c)

**Solution: **Taylor series of f (x + h) at x.

f(x + h) = f(x) + (x + h - x)

**Question 7: Which of the following is correct? [2019 : 1 Mark, Set-I]**

**Answer: (d)**

**Solution: **

**Question 8: The value (up to two decimal places) of a line along C which is a straight line joining (0, 0) to (1, 1) is _____. [2018 : 2 Marks, Set-II]**

**Solution:**

(0, 0) to (1, 1) line is y = x

**Question 9: The value of the integral [2018 : 2 Marks, Set-I]**

**(d) ∏**^{2}

**Answer:** (b)

**Solution:**

**Question 10: At the point x = 0, the function f(x) = x**^{3} has [2018 : 1 Marks, Set-I]

**(a) local maximum **

**(b) local minimum **

**(c) both local maximum and minimum**

**(d) neither local maximum nor local minimum**

**Answer: **(d)

**Solution: **

**Question 11: Consider the following definite integral:**

**The value of the integral is [2017 : 2 Marks, Set-II]**

**Answer: **(a)

**Solution:**

**Question 12: The tangent to the curve represented by y = x In x is required to have 45° inclination with the x-axis. The coordinates of the tangent point would be [2017 : 2 Marks, Set-II]**

**(a) (1,0) **

**(b) (0,1) **

**(c) (1,1) **

**(d) **

**Answer: (a)**

**Solution: **

tan 45° = In x + 1

1 = lnx + 1

⇒ Inx = 0

∴ x = 1

Putting x = 1 in the eq. of curve, we get y = 0.

**Question 13: The divergence of the vector field V = x**^{2}i + 2y^{3}j + z^{4}k at x = 1, y = 2, z = 3 is _________. [2017 : 1 Mark, Set-II]

**Solution: **

**Question 14: Let w= f(x, y), where x and yare functions of t. Then, according to the chain rule dw/dt [2017 : 1 Mark, Set-II]**

**Answer: (c)**

**Solution: **W = f(x, y)

By Chain rule,

**Question 15: Let x be a continuous variable defined over the interval (-∞, ∞), and ****The integral is equal to [2017 : 1 Mark, Set-I]**

**Answer: **(b)

**Solution: **

Let e^{-x} = t