GATE-CH-1994-1-d-math-1mark

If \(\vec {\imath }\), \(\vec {\jmath }\), \(\vec {k}\) are the unit vectors in rectangular coordinates, then the curl of the vector \(\vec {\imath }y + \vec {\jmath }y + \vec {k}z\) is

• \(\vec {k}\)

• \(-\vec {k}\)

• \(\vec {\jmath } + \vec {k}\)

• \(\vec {\imath } + \vec {k}\)

GATE-CH-1995-1-b-math-1mark

The angle between two vectors \(2\vec {\imath } - \vec {\jmath } + \vec {k}\) and \(\vec {\imath }+\vec {\jmath }+2\vec {k}\) is

• 0\(^\circ \)

• 30\(^\circ \)

• 45\(^\circ \)

• 60\(^\circ \)

GATE-CH-1997-1-4-math-1mark

Given \(f(x,y) = x^2+y^2\), \(\nabla ^2f\) is

• 4

• 2

• 0

• \(4(x+y)^2\)

GATE-CH-1998-1-2-math-1mark

The unit normal to the plane \(2x + y + 2z = 6\) can be expressed in the vector form:

• \(3i + 2j + 2k\)

• \(\displaystyle \frac {2}{3}i + \frac {1}{3}j + \frac {2}{3}k\)

• \(\displaystyle \frac {1}{3}i + \frac {1}{2}j + \frac {1}{2}k\)

• \(\displaystyle -\frac {2}{3}i + \frac {1}{3}j - \frac {2}{3}k\)

GATE-CH-2002-1-4-math-1mark

Which of the following holds for any non-zero vector \(\vec {a}\)

• \(\nabla \cdot \vec {a} = 0\)

• \(\nabla \times \vec {a} = 0\)

• \(\nabla \cdot (\nabla \times \vec {a}) = 0\)

• \(\nabla (\nabla \times \vec {a}) = 0\)

GATE-CH-2003-2-math-1mark

The directional derivative of \(f(x,y,z) = x^2 + y^2 + z^2\) at the point \((1,1,1)\) in the direction \(\vec {\imath }-\vec {k}\) is

• 0

• 1

• \(\sqrt {2}\)

• \(2\sqrt {2}\)

GATE-CH-2006-3-math-1mark

The value of \(\alpha \) for which the following three vectors are coplanar is

\[ \begin {eqnarray*} \vec {A} &=& \vec {\imath } + 2\vec {\jmath } + \vec {k} \\ \vec {B} &=& 3\vec {\jmath }+\vec {k} \\ \vec {C}&=& 2\vec {\imath }+\alpha \vec {\jmath } \end {eqnarray*} \]

• 4

• zero

• \(-2\)

• \(-10\)

GATE-CH-2008-4-math-1mark

The unit normal vector to the surface of the sphere \(x^2+y^2+z^2=1\) at the point \((1/\sqrt {2},0,1/\sqrt {2})\) is \(\quad \quad \) (\(\hat {\imath }, \hat {\jmath }, \hat {k}\) are unit normal vectors in the Cartesian coordinate system)

• \(\displaystyle \frac {1}{\sqrt {2}}\hat {\imath } + \frac {1}{\sqrt {2}}\hat {\jmath }\)

• \(\displaystyle \frac {1}{\sqrt {2}}\hat {\imath } + \frac {1}{\sqrt {2}}\hat {k}\)
  

• \(\displaystyle \frac {1}{\sqrt {2}}\hat {\jmath } + \frac {1}{\sqrt {2}}\hat {k}\)

• \(\displaystyle \frac {1}{\sqrt {3}}\hat {\imath } + \frac {1}{\sqrt {3}}\hat {\jmath } + \frac {1}{\sqrt {3}}\hat {k}\)

GATE-CH-2009-1-math-1mark

The direction of largest increase of the function \(xy^3-x^2\) at the point \((1,1)\) is

• \(3\hat {\imath } + \hat {\jmath }\)

• \(\hat {\imath } + 3\hat {\jmath }\)

• \(-\hat {\imath } + 3\hat {\jmath }\)

• \(-\hat {\imath } -3 \hat {\jmath }\)

GATE-CH-2011-10-math-1mark

\(\boldsymbol {R}\) is a closed planar region as shown by the shaded area as shown in the figure below. Its boundary \(\boldsymbol {C}\) consists of the circles \(C_1\) and \(C_2\).

If \(F_1(x,y), F_2(x,y), \dfrac {\partial F_1}{\partial y}\) and \(\dfrac {\partial F_2}{\partial x}\) are all continuous everywhere in \(\boldsymbol {R}\), Green's theorem states that \(\displaystyle \iint _R\left (\frac {\partial F_2}{\partial x}-\frac {\partial F_1}{\partial y}\right )dx\; dy = \oint _C(F_1dx + F_2dy)\). Which ONE of the following alternatives CORRECTLY depicts the direction of integration along \(\boldsymbol {C}\) ?

• 
• 
• 
• 

GATE-CH-2014-1-math-1mark

Gradient of a scalar variable is always

• a vector

• a scalar

• a dot product

• zero

GATE-CH-2015-1-math-1mark

The following set of three vectors \[ \begin {pmatrix}1\\ 2\\ 1\end {pmatrix}, \quad \begin {pmatrix}x\\ 6\\ x\end {pmatrix} \text { and } \begin {pmatrix}3\\ 4\\ 2\end {pmatrix}, \] is linearly dependent when \(x\) is equal to

• 0

• 1

• 2

• 3

GATE-CH-2015-5-math-1mark

A scalar function in the \(xy\)-plane is given by \(\phi (x,y)=x^2+y^2\). If \(\hat {\imath }\) and \(\hat {\jmath }\) are unit vectors in the \(x\) and \(y\) directions, the direction of maximum increase in the value of \(\phi \) at \((1,1)\) is along:

• \(-2\hat {\imath } + 2\hat {\jmath }\)

• \(2\hat {\imath } + 2\hat {\jmath }\)

• \(-2\hat {\imath } - 2\hat {\jmath }\)

• \(2\hat {\imath } - 2\hat {\jmath }\)

GATE-CH-2017-4-math-1mark

Let \(\vec {\imath }\) and \(\vec {\jmath }\) be the unit vectors in the \(x\) and \(y\) directions, respectively. For the function \[ F(x,y) = x^3 + y^2 \] the gradient of the function, i.e., \(\nabla F\) is given by

• \(3x^2\vec {\imath }-2y\vec {\jmath }\)

• \(6x^2y\)
  

• \(3x^2\vec {\imath }+2y\vec {\jmath }\)

• \(2y\vec {\imath }-3x^2\vec {\jmath }\)

GATE-CH-1999-2-2-math-2mark

The gradient of \(xy^2 + yz^3\) at the point \((-1,2,1)\) is

• \(3i - 3j + 3k\)

• \(3i - 3j + 6k\)

• \(4i - j + 3k\)

• \(4i - 3j + 6k\)

GATE-CH-2005-33-math-2mark

The divergence of a vector field \(\vec {A}\) is always equal to zero, if the vector field \(\vec {A}\) can be expressed as

• the gradient of any scalar field \(\phi \)

• the divergence of any scalar field \(\phi \)

• the divergence of any vector field \(\vec {B}\)

• the curl of any vector field \(\vec {B}\)

GATE-CH-2007-24-math-2mark

The directional derivative of \(\displaystyle f = \frac {1}{2}\sqrt {x^2+y^2}\) at \((1,1)\) in the direction of \(\vec {b}=\vec {\imath }-\vec {\jmath }\) is

• \(0\)

• \(1/\sqrt {2}\)

• \(\sqrt {2}\)

• \(2\)

GATE-CH-2007-25-math-2mark

Evaluate the following integral \((n\ne 0)\) \[ \int \left (-xy^ndx+x^nydy\right ) \] within the area of a triangle with vertices \((0,0), (1,0)\) and \((1,1)\) (counter-clock wise)

• \(0\)

• \(1/(n+1)\)

• \(1/2\)

• \(n/2\)

GATE-CH-2010-27-math-2mark

If \(\vec {u}=y\hat {\imath } + xy\hat {\jmath }\) and \(\vec {v} = x^2\hat {\imath } + xy^2\hat {\jmath }\), then \({\bf curl}(\vec {u}\times \vec {v})\) is

• \((2xy^2)\hat {\imath }-(x+y^2)\hat {\jmath }\)

• \((xy-x^2)\hat {\imath }-(y-3xy)\hat {\jmath }\)

• \((2x^2y^2-3x^3)\hat {\imath }-(y^3-3xy^2)\hat {k}\)

• \((3xy^2-x^3)\hat {\imath }-(y^3-3x^2y)\hat {\jmath }\)

GATE-CE-2011-29-math-2mark

If \(\vec {a}\) and \(\vec {b}\) are 2 arbitrary vectors with magnitude \(a\) and \(b\) respectively , \(|\vec {a} \times \vec {b}|^2\) will be equal to

• \(a^2 b^2 -(\vec {a}\cdot \vec {b})^2\)

• \(ab -(\vec {a}\cdot \vec {b})\)

• \(a^2 b^2 +(\vec {a}\cdot \vec {b})^2\)

• \(ab +(\vec {a}\cdot \vec {b})\)

GATE-ME-2007-22-math-2mark

The area of a triangle formed by tips of vector \(\vec {a}, \vec {b}\) and \(\vec {c}\) is

• \(\dfrac {1}{2}(\vec {a} - \vec {b})\cdot (\vec {a} - \vec {c})\)

• \(\dfrac {1}{2}\bigl |(\vec {a} - \vec {b})\times (\vec {a} - \vec {c})\bigr |\)

• \(\dfrac {1}{2}\bigl |\vec {a}\times \vec {b}\times \vec {c}\; \bigr |\)

• \(\dfrac {1}{2}(\vec {a}\times \vec {b})\cdot \vec {c}\)

GATE-CH-2015-27-math-2mark

A vector \(\boldsymbol {u} = -2y\hat {\imath }+2x\hat {\jmath }\), where \(\hat {\imath }\) and \(\hat {\jmath }\) are unit vectors in \(x\) and \(y\) directions, respectively. Evaluate the line integral \[ I = \oint _C \boldsymbol {u}\cdot d\boldsymbol {r} \] where \(C\) is a closed loop formed by connecting points \((1,1), (3,1), (3,2)\) and \((1,2)\) in that order. The value of \(I\) is ____________

• Answer

GATE-CH-1994-2-e-math-1mark

The Green’s theorem relates ------------------ integrals to surface integrals.

• Answer

GATE-CH-1994-2-f-math-1mark

If '\(a\)' is a scalar and \(\vec {b}\) is a vector, then \(\nabla \times a \vec {b} = \) ------------------

• Answer

GATE-CH-2000-3-math-5mark

Find the directional derivative of \(u = xyz\) at the point \((1,2,3)\) in the direction from \((1,2,3)\) to \((1,-1,-3)\).

• Answer

GATE-CH-2000-4-math-5mark

Find whether or not the following vectors: \((1,1,2)\), \((1,2,1)\) and \((0,3,-3)\) are linearly independent.

• Answer