If a_ij = \({1\over2}(3i-2j)\) and A = [a_ij]_2x2 is

(a)

\(\begin{bmatrix} {1\over 2}& 2 \\ -{1\over2} & 1 \end{bmatrix}\)

(b)

\(\begin{bmatrix} {1\over 2}& -{1\over2} \\ 2& 1 \end{bmatrix}\)

(c)

\(\begin{bmatrix} 2& 2\\ {1\over 2}& -{1\over2} \end{bmatrix}\)

(d)

\(\begin{bmatrix} -{1\over 2}& {1\over2} \\ 1& 2 \end{bmatrix}\)

What must be the matrix X, if 2x +\(\begin{bmatrix} 1& 2 \\ 3 & 4 \end{bmatrix}=\begin{bmatrix} 3 & 8 \\ 7 & 2 \end{bmatrix}?\)

(a)

\(\begin{bmatrix} 1& 3 \\ 2 &-1 \end{bmatrix}\)

(b)

\(\begin{bmatrix} 1& -3 \\ 2 &-1 \end{bmatrix}\)

(c)

\(\begin{bmatrix} 2& 6 \\ 4 &-2 \end{bmatrix}\)

(d)

\(\begin{bmatrix} 2& -6 \\ 4 &-2 \end{bmatrix}\)

Which one of the following is not true about the matrix \(\begin{bmatrix} 1 &0 &0 \\ 0 & 0 &0 \\ 0 & 0 & 5 \end{bmatrix}?\)

(a)

a scalar matrix

(b)

a diagonal matrix

(c)

an upper triangular matrix

(d)

a lower triangular matrix

If A and B are two matrices such that A + B and AB are both defined, then

(a)

A and B are two matrices not necessarily of same order

(b)

A and B are square matrices of same order

(c)

Number of columns of A is equal to the number of rows of B

(d)

A = B.

If A = \(\begin{bmatrix}\lambda & 1 \\ -1 & -\lambda \end{bmatrix}\), then for what value of \(\lambda\), A² = O?

(a)

0

(b)

\(\pm 1\)

(c)

-1

(d)

1

If A =\(\begin{bmatrix} 1 & -1 \\ 2 &-1 \end{bmatrix}\), B = \(\begin{bmatrix} a & 1 \\ b &-1 \end{bmatrix}\) and (A + B)² = A² + B², then the values of a and b are

(a)

a = 4, b = 1

(b)

a = 1, b = 4

(c)

a = 0, b = 4

(d)

a = 2, b = 4

If A =\(\begin{bmatrix} 1& 2 &2 \\ 2 & 1 & -2 \\ a & 2 & b \end{bmatrix}\) is a matrix satisfying the equation AA^T = 9I, where I is 3 \(\times\) 3 identity matrix, then the ordered pair (a, b) is equal to

(a)

(2, - 1)

(b)

(- 2, 1)

(c)

(2, 1)

(d)

(- 2, - 1)

If A is a square matrix, then which of the following is not symmetric?

(a)

A + A^T

(b)

AA^T

(c)

A^T A

(d)

A − A^T

If A and B are symmetric matrices of order n, where (A \(\neq\) B), then

(a)

A + B is skew-symmetric

(b)

A + B is symmetric

(c)

A + B is a diagonal matrix

(d)

A + B is a zero matrix

If A = \(\begin{bmatrix}a & x \\ y& a \end{bmatrix}\) and if xy = 1, then det (A A^T ) is equal to

(a)

(a −1)²

(b)

(a² +1)²

(c)

a² −1

(d)

(a² −1)²

The value of x, for which the matrix A = \(\begin{bmatrix} e^{x-2}& e^{7+x} \\ e^{2+x} & e^{2x+3} \end{bmatrix}\) is singular

(a)

9

(b)

8

(c)

7

(d)

6

If the points (x,−2), (5, 2), (8, 8) are collinear, then x is equal to

(a)

-3

(b)

\({1\over 3}\)

(c)

1

(d)

3

If \(\begin{vmatrix}2a & x_1 &y_1 \\ 2b & x_2 & y_2 \\ 2c & x_3 &y_3 \end{vmatrix}={abc\over 2}\neq 0,\) then the area of the triangle whose vertices are \(\begin{pmatrix} {x_1\over a}, {y_1\over a} \end{pmatrix}\), \(\begin{pmatrix} {x_2\over b}, {y_2\over b} \end{pmatrix}\), \(\begin{pmatrix} {x_3\over c}, {y_3\over c} \end{pmatrix}\) is

(a)

\({1\over 4}\)

(b)

\({1\over 4} abc\)

(c)

\({1\over 8}\)

(d)

\({1\over 8}abc\)

If the square of the matrix \(\begin{bmatrix} \alpha & \beta \\ \gamma & -\alpha \end{bmatrix}\) is the unit matrix of order 2, then \(\alpha ,\beta \) and \(\gamma\) should satisfy the relation.

(a)

1 + \(\alpha ^2+\beta \gamma=0\)

(b)

1 - \(\alpha ^2-\beta \gamma=0\)

(c)

1 - \(\alpha ^2+\beta \gamma=0\)

(d)

1 + \(\alpha ^2-\beta \gamma=0\)

If \(\triangle\) = \(\begin{vmatrix} a&b &c \\ x & y & z \\ p &q &r \end{vmatrix}\), then \(\begin{vmatrix} ka&kb &kc \\ kx & ky & kz \\k p &kq &kr \end{vmatrix}\) is

(a)

\(\triangle\)

(b)

k\(\triangle\)

(c)

3k\(\triangle\)

(d)

k³\(\triangle\)

A root of the equation \(\begin{vmatrix} 3-x&-6 &3 \\ -6 & 3-x & 3 \\ 3 &3 &-6-x \end{vmatrix}=0 \ is\)

(a)

6

(b)

3

(c)

0

(d)

-6

The value of the determinant of A = \(\begin{bmatrix} 0&a &-b \\ -a & 0 & c \\ b & -c & 0 \end{bmatrix}is\)

(a)

-2abc

(b)

abc

(c)

0

(d)

a² + b² + c²

If x₁, x₂, x₃ as well as y₁, y₂, y₃ are in geometric progression with the same common ratio, then the points (x₁, y₁ ), (x₂, y₂), (x₃, y₃ ) are

(a)

vertices of an equilateral triangle

(b)

vertices of a right angled triangle

(c)

vertices of a right angled isosceles triangle

(d)

collinear

If a \(\neq\) b, b, c satisfy \(\begin{vmatrix} a&2b &2c \\3 & b & c \\ 4 & a & b \end{vmatrix}=0,\) then abc =

(a)

a + b + c

(b)

0

(c)

b³

(d)

ab + bc

If A = \(\begin{vmatrix}-1 & 2 &4 \\ 3 &1 &0 \\ -2& 4 &2 \end{vmatrix}\) and B = \(\begin{vmatrix}-2 & 4 &2 \\ 6 &2 &0 \\ -2& 4 &8 \end{vmatrix}\), then B is given by

(a)

B = 4A

(b)

B = -4A

(c)

B = -A

(d)

B = 6A

If \(\left\lfloor . \right\rfloor \) denotes the greatest integer less than or equal to the real number under consideration and −1\(\le\) x < 0, 0 \(\le\) y < 1, 1 \(\le\) z < 2, then the value of the determinant \(\begin{vmatrix} \left\lfloor x \right\rfloor +1& \left\lfloor y \right\rfloor & \left\lfloor z \right\rfloor \\ \left\lfloor x \right\rfloor & \left\lfloor y \right\rfloor +1& \left\lfloor z \right\rfloor \\ \left\lfloor x \right\rfloor & \left\lfloor y \right\rfloor & \left\lfloor z \right\rfloor +1\end{vmatrix}\) is

(a)

\(\left\lfloor z \right\rfloor \)

(b)

\(\left\lfloor y \right\rfloor \)

(c)

\(\left\lfloor x \right\rfloor \)

(d)

\(\left\lfloor x \right\rfloor \)+1

If A is skew-symmetric of order n and C is a column matrix of order n \(\times\) 1, then C^T AC is

(a)

an identity matrix of order n

(b)

an identity matrix of order 1

(c)

a zero matrix of order 1

(d)

an identity matrix of order 2

The matrix A satisfying the equation \(\begin{bmatrix} 1 & 3 \\ 0 & 1 \end{bmatrix}\) A = \(\begin{bmatrix} 1 & 1 \\ 0 & -1 \end{bmatrix}\) is

(a)

\(\begin{bmatrix} 1 & 4 \\ -1 & 0 \end{bmatrix}\)

(b)

\(\begin{bmatrix} 1 & -4 \\ 1 & 0 \end{bmatrix}\)

(c)

\(\begin{bmatrix} 1 & 4 \\ 0 & -1 \end{bmatrix}\)

(d)

\(\begin{bmatrix} 1 & -4 \\ 1 & 1 \end{bmatrix}\)

If A + I =\(\begin{bmatrix} 3& -2 \\ 4 & 1 \end{bmatrix}\), then (A + I )(A - I) is equal to

(a)

\(\begin{bmatrix} -5& -4 \\ 8 & -9 \end{bmatrix}\)

(b)

\(\begin{bmatrix} -5& 4 \\ -8 & 9 \end{bmatrix}\)

(c)

\(\begin{bmatrix} 5& 4 \\ 8 & 9 \end{bmatrix}\)

(d)

\(\begin{bmatrix} -5& -4 \\ -8 & -9 \end{bmatrix}\)

Let A and B be two symmetric matrices of same order. Then which one of the following statement is not true?

(a)

A + B is a symmetric matrix

(b)

AB is a symmetric matrix

(c)

AB = (BA)^T

(d)

A^T B = AB^T

The value of \(\overrightarrow{AB}+\overrightarrow{BC}+\overrightarrow{DA}+\overrightarrow{CD}\) is

(a)

\(\overrightarrow{AD}\)

(b)

\(\overrightarrow{CA}\)

(c)

\(\overrightarrow{0}\)

(d)

\(-\overrightarrow{AD}\)

If \(\overrightarrow{a}+2\overrightarrow{b}\) and \(3\overrightarrow{a}+m\overrightarrow{b}\) are parallel, then the value of m is

(a)

3

(b)

\(1\over3\)

(c)

6

(d)

\(1\over6\)

The unit vector parallel to the resultant of the vectors \(\hat{i}+\hat{j}-\hat{k}\) and \(\hat{i}-2\hat{j}+\hat{k}\) is

(a)

\({\hat{i}-\hat{j}+\hat{k}\over\sqrt{5}}\)

(b)

\({2\hat{i}+\hat{j}\over\sqrt{5}}\)

(c)

\({2\hat{i}-\hat{j}+\hat{k}\over\sqrt{5}}\)

(d)

\({2\hat{i}-\hat{j}\over\sqrt{5}}\)

A vector \(\overrightarrow{OP}\) makes 60° and 45° with the positive direction of the x and y axes respectively.  Then the angle between \(\overrightarrow{OP}\)and the z-axis is

(a)

45°

(b)

60°

(c)

90°

(d)

30°

If \(\overrightarrow{BA}=3\hat{i}+2\hat{j}+\hat{k}\) and the position vector of B is \(\hat{i}+3\hat{j}-\hat{k}\) ,then the position vector of A is

(a)

\(4\hat{i}+2\hat{j}+\hat{k}\)

(b)

\(4\hat{i}+5\hat{j}\)

(c)

\(4\hat{i}\)

(d)

\(-4\hat{i}\)

A vector makes equal angle with the positive direction of the coordinate axes. Then each angle is equal to

(a)

\(cos^{-1}({1\over 3})\)

(b)

\(cos^{-1}({2\over 3})\)

(c)

\(cos^{-1}({1\over\sqrt 3})\)

(d)

\(cos^{-1}({2\over\sqrt 3})\)

The vectors \(\overrightarrow{a}-\overrightarrow{b},\overrightarrow{b}-\overrightarrow{c},\overrightarrow{c}-\overrightarrow{a}\) are

(a)

parallel to each other

(b)

unit vectors

(c)

mutually perpendicular vectors

(d)

coplanar vectors.

If ABCD is a parallelogram, then \(\overrightarrow{AB}+\overrightarrow{AD}+\overrightarrow{CB}+\overrightarrow{CD}\) is equal to

(a)

\(2(\overrightarrow{AB}+\overrightarrow{AD})\)

(b)

\(4\overrightarrow{AC}\)

(c)

\(4\overrightarrow{BD}\)

(d)

\(\overrightarrow{0}\)

One of the diagonals of parallelogram ABCD with \(\overrightarrow{a}\) and \(\overrightarrow{b}\) as adjacent sides is \(\overrightarrow{a}+\overrightarrow{b}\) The other diagonal \(\overrightarrow{BD}\) is

(a)

\(\overrightarrow{a}-\overrightarrow{b}\)

(b)

\(\overrightarrow{b}-\overrightarrow{a}\)

(c)

\(\overrightarrow{a}+\overrightarrow{b}\)

(d)

\(\overrightarrow{a}+\overrightarrow{b}\over 2\)

If \(\overrightarrow{a},\overrightarrow{b}\) are the position vectors A and B, then which one of the following points whose position vector lies on AB, is

(a)

\(\overrightarrow{a}+\overrightarrow{b}\)

(b)

\({2\overrightarrow{a}-\overrightarrow{b}\over 2}\)

(c)

\({2\overrightarrow{a}+\overrightarrow{b}\over 3}\)

(d)

\({\overrightarrow{a}-\overrightarrow{b}\over 3}\)

If \(\overrightarrow{a}, \overrightarrow{b}, \overrightarrow{c}\) are the position vectors of three collinear points, then which of the following is true?

(a)

\(\overrightarrow{a}=\overrightarrow{b}+\overrightarrow{c}\)

(b)

\(2\overrightarrow{a}=\overrightarrow{b}+\overrightarrow{c}\)

(c)

\(\overrightarrow{b}=\overrightarrow{c}+\overrightarrow{a}\)

(d)

\(4\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c}=0\)

If \(\overrightarrow{r}={9\overrightarrow{a}+7\overrightarrow{b}\over16}\), then the point P whose position vector \(\overrightarrow{r}\) divides the line joining the points with position vectors \(\overrightarrow{a}\) and \(\overrightarrow{b}\) in the ratio

(a)

7: 9 internally

(b)

9: 7 internally

(c)

9: 7 externally

(d)

7: 9 externally

If \(\lambda \hat{i}+2\lambda \hat{j}+2\lambda \hat{k}\) is a unit vector, then the value of \(\lambda\) is

(a)

\({1\over3}\)

(b)

\({1\over4}\)

(c)

\({1\over9}\)

(d)

\({1\over2}\)

Two vertices of a triangle have position vectors \(3\hat{i}+4\hat{j}-4\hat{k}\) and \(2\hat{i}+3\hat{j}+4\hat{k}\) . If the position vector of the centroid is \(\hat{i}+2\hat{j}+3\hat{k}\), then the position vector of the third vertex is

(a)

\(-2\hat{i}-\hat{j}+9\hat{k}\)

(b)

\(-2\hat{i}-\hat{j}-6\hat{k}\)

(c)

\(2\hat{i}-\hat{j}+6\hat{k}\)

(d)

\(-2\hat{i}+\hat{j}+6\hat{k}\)

If \(|\overrightarrow{a}+\overrightarrow{b}|=60,\) \(|\overrightarrow{a} - \overrightarrow{b}|=40\)  and \(|\overrightarrow{b}|=46\) , then \(|\overrightarrow{a}|\) is

(a)

42

(b)

12

(c)

22

(d)

32

If \(\overrightarrow{a}\)  and \(\overrightarrow{b}\) having same magnitude and angle between them is 60° and their scalar product is \({1\over2}\) then \(|\overrightarrow{a}|\) is 

(a)

2

(b)

3

(c)

7

(d)

1

The value of  \(\theta \in (0,{\pi\over 2})\) for which the vectors \(\overrightarrow{a}=(sin \theta)\hat{i}+(cos\theta)\hat{j}\) and \(\overrightarrow{b}=\hat{i}-\sqrt{3}\hat{j}+2\hat{k}\) are perpendicular, is equal to

(a)

\({\pi\over 3}\)

(b)

\({\pi\over 6}\)

(c)

\({\pi\over 4}\)

(d)

\({\pi\over 2}\)

If \(|\overrightarrow{a}|=13,|\overrightarrow{b}|=5\)  and \(\overrightarrow{a}.\overrightarrow{b}=60^o\) then \(|\overrightarrow{a}\times\overrightarrow{b}|\) is

(a)

15

(b)

35

(c)

45

(d)

25

Vectors \(\overrightarrow{a}\) and \(\overrightarrow{b}\) are inclined at an angle \(\theta =120^o\). If \(|\overrightarrow{a}|=1,|\overrightarrow{b}|=2,\) then \([(\overrightarrow{a}+3\overrightarrow{b})\times (3\overrightarrow{a}-\overrightarrow{b})]^2\) is equal to

(a)

225

(b)

275

(c)

325

(d)

300

If   \(\overrightarrow{a}\)  and   \(\overrightarrow{b}\) are two vectors of magnitude 2 and inclined at an angle 60°, then the angle between   \(\overrightarrow{a}\)  and \(\overrightarrow{a}+\overrightarrow{b}\) is

(a)

30°

(b)

60°

(c)

45°

(d)

90°

If the projection of \(5\hat{i}-\hat{j}-3\hat{k}\) on the vector \(\hat{i}+3\hat{j}+\lambda\hat{k}\) is same as the projection of \(\hat{i}+3\hat{j}+\lambda\hat{k}\) on \(5\hat{i}-\hat{j}-3\hat{k}\), then \(\lambda\) is equal to

(a)

\(\pm 4\)

(b)

\(\pm 3\)

(c)

\(\pm 5\)

(d)

\(\pm 1\)

If (1, 2, 4) and (2, -3\(\lambda\), -3) are the initial and terminal points of the vector \(\hat{i}+5\hat{j}-7\hat{k}\) , then the value of \(\lambda\) is equal to

(a)

\(7\over 3\)

(b)

-\(7\over 3\)

(c)

-\(5\over 3\)

(d)

\(5\over 3\)

If the points whose position vectors \(10\hat{i}+3\hat{j},12\hat{i}-5\hat{j}\) and \(a\hat{i}+11\hat{j}\) are collinear then a is equal to

(a)

6

(b)

3

(c)

5

(d)

8

If \(\overrightarrow{a}=\hat{i}+\hat{j}+\hat{k},\overrightarrow{b}=2\hat{i}+x\hat{j}+\hat{k},\overrightarrow{c}=\hat{i}-\hat{j}+4\hat{k}\) and \(\overrightarrow{a}.(\overrightarrow{b}\times \overrightarrow{c})=70,\) then x is equal to

(a)

5

(b)

7

(c)

26

(d)

10

If \(\overrightarrow{a}=\hat{i}+2\hat{j}+2\hat{k},|\overrightarrow{b}|=5\) and the angle between \(\overrightarrow{a}\) and \(\overrightarrow{b}\) is \({\pi\over 6},\) then the area of the triangle formed by these two vectors as two sides, is

(a)

\(7\over4\)

(b)

\(15\over4\)

(c)

\(3\over4\)

(d)

\(17\over4\)