**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

**Question 16: [2017 : 1 Mark, Set-I]**

**Solution: **(Applying L'Hospital rule)

=

**Question 17: The quadratic approximation of **

**f(x) = x**^{3} - 3x^{2} - 5 a the point x = 0 is [2016 : 2 Marks, Set-II]

**(a) 3x**^{2} - 6x - 5

**(b) -3x**^{2} - 5

**(c) -3x**^{2} + 6x - 5

**(d) 3x**^{2} - 5

**Answer: **(b)

**Solution: **The quadratic approximation of f{x) at the point x = 0 is,

**Question 18: The area between the parabola x**^{2} = 8y and the straight line y = 8 is______. [2016 : 2 Marks, Set-II]

**Solution: **Parabola is x^{2} = 8y

and straight is y = 0

At the point of intersection, we have,

⇒

**Question 19: The angle of intersection of the curves x**^{2} = 4y and y^{2} = 4x at point (0, 0) is [2016 : 2 Marks, Set-II]

**(a) 0° **

**(b) 30° **

**(c) 45° **

**(d) 90° **

**Answer: **(d)

**Solution: **Given curve,

x^{2} = 4y .......(i)

and y^{2} = 4x ........(ii)

**Question 20: The area of the region bounded by the parabola y = x**^{2} + 1 and the straight line x + y = 3 is

**(a) 59/6**

**(b) 9/2**

**(c) 10/3**

**(d) 7/6**

**Answer: **(b)

**Solution: **At the point of intersection of the curves,

y = x^{2} + 1 and x + y = 3 i.e., y = 3 - x , we have,

x^{2} + 1 - 3 - x ⇒ x^{2} + x - 2 = 0

⇒ x = -2, 1 and 3 - x __>__ x^{2} + 1

**Question 21: The value of [2016 : 2 Marks, Set-I]**

**(a) π/2**

**(b) π**

**(c) 3π/2 **

**(d) 1**

**Answer: **(b)

**Solution: **

**(Using "division by x")**

(Using definition of Laplace transform)

Put s - 0, we get

**Question 22: What is the value of [2016 : 1 Mark, Set-II]**

**(a) 1**

**(b) -1**

**(c) 0**

**(d) Limit does not exit**

**Answer:** (d)

**Solution: **

(i.e., put x = 0 and then y = 0)

which depends on m.

**Question 23: The optimum value of the function f(x) = x**^{2} - 4x + 2 is [2016 : 1 Mark, Set-II]

**(a) 2 (maximum) **

**(b) 2 (minimum) **

**(c) -2 (maximum) **

**(d) -2 (minimum)**

**Answer: **(d)

**Solution: **

f'(x) = 0

⇒ 2x — 4 = 0

⇒ x = 2 (stationary point)

f"(x) = 2 > 0

⇒ f(x) is minimum at x = ?

i.e., (2)^{2} - 4(2) + 2 = -2

∴ The optimum value of f(x) is -2 (minimum)

**Question 24: While minimizing the function f(x), necessary and sufficient conditions for a point x**_{0} to be a minima are [2015 : 1 Mark, Set-II]

**(a) f' (x**_{0}) > 0 and f" (x_{0}) = 0

**(b) f'(x**_{0})< 0 an d f"(x_{0}) = 0

**(c) f' (x**_{0}) = 0 and f" (x_{0}) < 0

**(d) f' (x**_{0}) = 0 and f" (x_{0}) > 0

**Answer: (d)**

**Solution:** f(x) has a local minimum at x = x_{0}

if f'(x_{0}) = 0 and f''(x_{0}) > 0

**Question 25: is equal to [2015 : 1 Mark, Set-II]**

**(a) e**^{-2}

**(b) e **

**(c) 1 **

**(d) e**^{2}

**Answer: **(d)

**Solution: **

**⇒ **

Which is in the form of

To convert this into 0/0 form, we rewrite as,

**⇒ **

Now it is in 0/0 form.

Using L’Hospital’s rule,

∴ y = e^{2}

**Question 26: The directional derivative of the field u(x, y, z) = x**^{2} - 3yz in the direction of the vector at point (2, - 1, 4) is _________. [2015 : 2 Marks, Set-I]

**Solution: **

Directional derivative

**Question 27: The expression is equal to [2014 : 2 Marks, Set-II]**

**(a) ln x **

**(b) 0 **

**(c) x ln x **

**(d) ∞**

**Answer: **(a)

**Solution: **

**Question 28: [2014 : 1 Marks, Set-I]**

**(a) -****∞**

**(b) 0 **

**(c) 1**

**(d) ∞**

**Answer: (c)**

**Solution: Put **

**Question 29: The value of [2013 : 2 Marks]**

**(a) 0**

**(b) 1/15**

**(c) 1**

**(d) 8/3**

**Answer: **(b)

**Solution: **

**⇒ **

**Alternative Method:**

**Question 30: For the parallelogram OPQR shown in the sketch, The area of the parallelogram is [2011 : 2 Marks]**

**(a) ad - bc **

**(b) ac + bd **

**(c) ad + bc **

**(d) ab - cd**

**Answer: **(a)

**Solution: **

The area of parallelogram OPQR in figure shown above, is the magnitude of the vector product