Important questions

Download Tamil Nadu 11th Standard Maths question papers, model tests, one-mark questions, important questions, and public exam papers in PDF format. Free study materials and answer keys for TN State Board students.

Questions:

Multiple Choice Question

50 x 1 = 50

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)

\(17\over4\)

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)

(b)

(c)

(d)

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)

(b)

(c)

(d)

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 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 \(\overrightarrow{a}\) 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°

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

(a)

(b)

(c)

(d)

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)

\({\pi\over 2}\)

10.

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)

(b)

(c)

(d)

11.

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

(a)

(b)

(c)

(d)

12.

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)

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

13.

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)

\({1\over2}\)

14.

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

15.

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)

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

16.

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)

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

17.

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)

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

18.

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)

\(\overrightarrow{0}\)

19.

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.

20.

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)

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

21.

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)

\(-4\hat{i}\)

22.

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°

23.

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)

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

24.

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

(a)

(b)

\(1\over3\)

(c)

(d)

\(1\over6\)

25.

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

(a)

\(\overrightarrow{AD}\)

(b)

\(\overrightarrow{CA}\)

(c)

\(\overrightarrow{0}\)

(d)

\(-\overrightarrow{AD}\)

26.

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

27.

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)

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

28.

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)

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

29.

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

30.

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

31.

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

32.

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)

(c)

b³

(d)

ab + bc

33.

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

34.

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)

(d)

a²+ b²+ c²

35.

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

(a)

(b)

(c)

(d)

-6

36.

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\(\triangle\)

37.

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 + \(\alpha ^2-\beta \gamma=0\)

38.

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)

\({1\over 8}abc\)

39.

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

(a)

-3

(b)

\({1\over 3}\)

(c)

(d)

40.

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)

(b)

(c)

(d)

41.

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

42.

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

43.

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

44.

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)

45.

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

46.

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

(a)

(b)

\(\pm 1\)

(c)

-1

(d)

47.

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.

48.

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

49.

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)

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

50.

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)

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

Visit QB365-SLMS - https://www.qb365.in/slms/

Answers Key:

Multiple Choice Question

50 x 1 = 50

(b)
\(15\over4\)

\(\vec{b} \times \vec{c}=\left|\begin{array}{ccc} \hat{i} & \hat{j} & \hat{k} \\ 2 & x & 1 \\ 1 & -1 & 4 \end{array}\right|\)
\(=\hat{i}(4 x+1)-\hat{j}(8-1)+\hat{k}(-2-x) \)
\(=\hat{i}(4 x+1)-7 \hat{j}+\hat{k}(-2-x) \)
\(\vec{a} \cdot(\vec{b} \times \vec{c}) =(4 x+1)-7-2-x=70 \)
\(=(\hat{i}+\hat{j}+\hat{k}) \propto(4 x+1) \hat{i}-7 \hat{j}+\hat{k}(-2-x) \)
\(4 x+1-7-2-x =70 \)
\(3 x-8 =70 \)
\(3 x =78 \)
\(x =\frac{78}{3}=26 \)

\(\overrightarrow{O A}=10 \hat{i}+3 \hat{j}, \overrightarrow{O B}=12 \hat{i}-5 \hat{j}, \overrightarrow{O C}=a \hat{i}+11 \hat{j} \)
\(\overrightarrow{A B}=\overrightarrow{O B}-\overrightarrow{O A}=2 \hat{i}-8 \hat{j} \)
\(\overrightarrow{B C} =\overrightarrow{O C}-\overrightarrow{O B} \)
\(=(a-12) \hat{i}+16 \hat{j} \)
Condition:
\(\overrightarrow{B C} =-2 \overrightarrow{A B} \)
\((a-12) \hat{i}+16 \hat{j} =2(2 \hat{i}-8 \hat{j}) \)
\((a-12) \hat{i}+16 \hat{j} =-4 \hat{i}+16 \hat{j} \)
\(a-12 =-4 \)
\(a =-4+12=8 \)
\(a =8 \)

\(\text {Given} \ \overrightarrow{O A}=\hat{i}+2 \hat{j}+4 \hat{k}, \overrightarrow{O B}=2 \hat{i}-3 \lambda \hat{j}-3 \hat{k} \)
\(\overrightarrow{A B}=\overrightarrow{O B}-\overrightarrow{O A}=2 \hat{i}-3 \lambda \hat{j}-3 \hat{k}-\hat{i}-2 \hat{j}-4 \hat{k}\)
\(\hat{i}+5 \hat{j}-7 \hat{k}=\hat{i}+(-3 \lambda-2) \hat{j}-7 \hat{k}\)
\(-3 \lambda-2=+5\) \((\because \text { compare coefft of } \hat{j}\text { on both sides) }\)
\(-3 \lambda=7 \)
\(\lambda=-\frac{7}{3} \)

\(\text { Projection }=\frac{5-3-3 \lambda}{\sqrt{1+9+\lambda^{2}}}=\frac{5-3-3 \lambda}{\sqrt{25+1+9}}\)
\(\sqrt{10+\lambda^{2}} =\sqrt{35} \)
\(\lambda^{2}+10 =35 \)
\(\lambda^{2} =25 \)
\(\lambda =\pm 5 \)

\(A B=B C=2 \text { (ie) } \vec{a}=\vec{b}=2\)
\(\therefore A B C \text { is isosceles triangle }\)
\(\therefore \angle A+\angle B =180^{\circ}-120^{\circ} \)
\(2 \angle A =60^{\circ} \)
\(\angle A =30^{\circ} \)

\((\vec{a}+3 \vec{b}) \times(3 \vec{a}-\vec{b}) =3(\vec{a} \times \vec{a})-\vec{a} \times \vec{b}+9 \vec{b} \times \vec{a}-3 \vec{b} \times \vec{b} \)
\(=\overrightarrow{0}+\vec{b} \times \vec{a}+9(\vec{b} \times \vec{a})-\overrightarrow{0} \)
\(=10 \vec{b} \times \vec{a} \)
\(\frac{\sqrt{3}}{2} \times 2 =|\vec{a} \times \vec{b}| \)
\(|\vec{a} \times \vec{b}| =\sqrt{3} \)
\([(\vec{a}+3 \vec{b}) \times(3 \vec{a}-\vec{b})]^{2} =100(\vec{b} \times \vec{a})^{2} \)
\(=100(\vec{a} \times \vec{b})^{2}=100(\sqrt{3})^{2} \)
\(=100 \times 3=300 \)

\(\text { W.K.T }|\vec{a} \cdot \vec{b}|^{2}+|\vec{a} \times \vec{b}|^{2}=|\vec{a}|^{2} \cdot|\vec{b}|^{2}\)
\(|\vec{a} \times \vec{b}|^{2} =13^{2} \cdot 5^{2}-|\vec{a} \cdot \vec{b}|^{2} \)
\(=(169)(25)-3600=4225-3600 =625 \)
\(|\vec{a} \times \vec{b}| =25 \)

\(\vec{a} \perp \vec{b} \Rightarrow \quad \vec{a} \cdot \vec{b} =0 \)
\((\sin \theta \hat{i}+\cos \theta \hat{j}) \cdot(\hat{i}-\sqrt{3} \hat{j}+2 \hat{k}) =0 \)
\(\sin \theta-\sqrt{3} \cos \theta =0 \)
\(\sin \theta =\sqrt{3} \cos \theta \)
\(\frac{\sin \theta}{\cos \theta} =\sqrt{3} \)
\(\tan \theta =\sqrt{3} \)
\(\theta =60^{\circ} \text { or } \frac{\pi}{3} \)

10.

\(\cos 60^{\circ} =\frac{\vec{a} \cdot \vec{b}}{|\vec{a}||\vec{b}|} \)
\(\frac{1}{2} =\frac{\frac{1}{2}}{|\vec{a} \| \vec{a}|} \)
\(|\vec{a}|^{2} =1 \Rightarrow|a|=1 \)

11.

\(\text { W.K.T }|\vec{a}+\vec{b}|^{2}+|\vec{a}-\vec{b}|^{2}=2\left[|\vec{a}|^{2}+|\vec{b}|^{2}\right]\)
\(60^{2}+40^{2} =2\left(|\vec{a}|^{2}+46^{2}\right) \)
\(3600+1600 =2\left(|\vec{a}|^{2}+2116\right) \)
\(\frac{5200}{2} =|\vec{a}|^{2}+2116 \)
\(2600-2116 =|\vec{a}|^{2} \)
\(|\vec{a}|^{2} =484\)
\(|\vec{a}| =22 \)

12.

\(\left(\frac{3+2+x_{1}}{3},\right. \left.\frac{4+3+x_{2}}{3}, \frac{-4+4+x_{3}}{3}\right)=(1,2,3) \)
\(3+2+x_{1} =3 ; 4+3+x_{2}=6 ;-4+4+x_{3}=9 \)
\(5+x_{1}=3 7+x_{2}=6 \)
\(x_{1}=-2 \quad x_{2}=-1 \)
\(\therefore \text { Third vertex is }(-2,-1,9)\)

13.

\(\text { Unit vector }=\frac{\lambda \hat{i}+2 \lambda \hat{j}+2 \lambda \hat{k}}{\sqrt{\lambda^{2}+4 \lambda^{2}+4 \lambda^{2}}}\)
\(=\frac{\lambda \hat{k}+2 \lambda \hat{j}+2 \lambda \hat{k}}{\sqrt{9 \lambda^{2}}}=\frac{\lambda \hat{i}+2 \lambda \hat{j}+2 \lambda \hat{k}}{3 \lambda} \)
\(=\frac{\lambda(\hat{i}+2 \hat{j}+2 \hat{k})}{3 \lambda}=\frac{1}{3}(\hat{i}+2 \hat{j}+2 \hat{k}) \)
\(\lambda =\frac{1}{3} \)

14.

\(\vec{r}=\frac{n \vec{a}+m \vec{b}}{m+n} \text { internally }\)

\(\text { Given } \vec{r}=\frac{9 \vec{a}+7 \vec{b}}{16}\)
Comparing (1) & (2),
we get n = 9; m = 7
7 : 9 internally

15.

\(2 \vec{a}=\vec{b}+\vec{c} \Rightarrow \vec{a}+\vec{a}=\vec{b}+\vec{c} \Rightarrow \vec{a}-\vec{b}=\vec{c}-\vec{a} \)
\(\overrightarrow{O A}-\overrightarrow{O B}=\overrightarrow{O C}-\overrightarrow{O A} \Rightarrow \overrightarrow{B A}=\overrightarrow{A C} \)
\(\Rightarrow \vec{a}, \vec{b}, \vec{c} \text { are collinear }\)

16.

\(\vec{m}=\frac{|\vec{b}+2 \vec{a}|}{1+2}=\frac{2 \vec{a}+\vec{b}}{3}\)

17.

\(\text { In } \Delta \mathrm{BCD}, \overrightarrow{\mathrm{BD}}=\overrightarrow{\mathrm{BC}}+\overrightarrow{\mathrm{CD}}\)
\(=\overrightarrow{\mathrm{b}}-\overrightarrow{\mathrm{a}}\)

18.

\(\overrightarrow{A B}=-\overrightarrow{C D}\)
\(\overrightarrow{A D}=\overrightarrow{B C}=-\overrightarrow{C B}\)
\(\therefore \overrightarrow{A B}+\overrightarrow{A D}+\overrightarrow{C B}+\overrightarrow{C D}=-\overrightarrow{C D}-\overrightarrow{C B}+\overrightarrow{C B}+\overrightarrow{C D}=\overrightarrow{0}\)

19.

(d)
coplanar vectors.

20.

All angle are equal
\(\therefore \alpha=\beta=\gamma\)
\(\text { W.K.T } \cos ^{2} \alpha+\cos ^{2} \beta+\cos ^{2} \gamma=1\)
\(\cos ^{2} \alpha+\cos ^{2} \alpha+\cos ^{2} \alpha =1 \)
\(3 \cos ^{2} \alpha =1 \)
\(\cos ^{2} \alpha =\frac{1}{3} \)
\(\cos \alpha =\pm \frac{1}{\sqrt{3}} \)
\(\alpha =\cos ^{-1}\left(\frac{1}{\sqrt{3}}\right) \)

21.

\(\overrightarrow{B A}=3 \hat{i}+2 \hat{j}+\hat{k} \)
\(\overrightarrow{O A}-\overrightarrow{O B}=3 \hat{i}+2 \hat{j}+\hat{k} \)
\(\overrightarrow{O A}=3 \hat{i}+2 \hat{j}+\hat{k}+\overrightarrow{O B}=3 \hat{i}+2 \hat{j}+\hat{k}+\hat{i}+3 \hat{j}-\hat{k} \)
\(=4 \hat{i}+5 \hat{j} \)

22.

\(\cos ^{2} \alpha+\cos ^{2} \beta+\cos ^{2} \gamma=1 \)
\(\cos ^{2} 60^{\circ}+\cos ^{2} 45^{\circ}+\cos ^{2} \gamma=1 \)
\(\left(\frac{1}{2}\right)^{2}+\left(\frac{1}{\sqrt{2}}\right)^{2}+\cos ^{2} \gamma=1 \Rightarrow \cos ^{2} \gamma=1-\frac{1}{4}-\frac{1}{2} \)
\(\cos ^{2} \gamma=\frac{4-1-2}{4}=\frac{1}{4} \Rightarrow \cos \gamma=\frac{1}{2} \Rightarrow \gamma=60^{\circ} \)
\(\text { Angle between } \overrightarrow{O P} \text { and } z \text { axis is } 60^{\circ}\)

23.

\(\vec{a}+\vec{b} =2 \hat{i}-\hat{j} \)
\(|\vec{a}+\vec{b}| =\sqrt{4+1}=\sqrt{5} \)
\(\text { Unit vector }=\frac{2 \hat{i}-\hat{j}}{\sqrt{5}}\)

24.

25.

\(\underbrace{\overrightarrow{A B}}+ \overrightarrow{B C}+\overrightarrow{C D}+\overrightarrow{D A} \)
\(=\underbrace{\overrightarrow{A C}+\overrightarrow{C D}}+\overrightarrow{D A} \)
\(=\underbrace{\overrightarrow{A D}+\overrightarrow{D A}} \)
\(=\overrightarrow{A A}=\overrightarrow{0} . \)

26.

\(A \& B \text { are symmetric }\)
\(A^{T}=A \text {, and } B^{T}=B\)
\(\text { 1) }(A+B)^{T}=B^{T}+A^{T}=B+A=A+B \Rightarrow A+B\text { is symmetric }\)
\(\text { 2) }(A B)^{T}=B^{T} A^{T}=B A \neq A B\)
\(\therefore A B \text { is not symmetric }\)
\(\text { 3) } A B=A^{T} B^{T}=(B A)^{T}\)
\(\text { 4) } A^{T} B=A B=A B^{T}\)

27.

\(A+I =\left[\begin{array}{cc} 3 & -2 \\ 4 & 1 \end{array}\right] \)
\(A =\left[\begin{array}{cc} 3 & -2 \\ 4 & 1 \end{array}\right]-\left[\begin{array}{ll} 1 & 0 \\ 0 & 1 \end{array}\right]=\left[\begin{array}{cc} 2 & -2 \\ 4 & 0 \end{array}\right] \)
\(A-I =\left[\begin{array}{cc} 2 & -2 \\ 4 & 0 \end{array}\right]-\left[\begin{array}{ll} 1 & 0 \\ 0 & 1 \end{array}\right]=\left[\begin{array}{ll} 1 & -2 \\ 4 & -1 \end{array}\right] \)
\((A+I)(A-I) =\left[\begin{array}{ll} 3 & -2 \\ 4 & 1 \end{array}\right]\left[\begin{array}{ll} 1 & -2 \\ 4 & -1 \end{array}\right] \)
\(=\left[\begin{array}{ll} 3-8 & -6+2 \\ 4+4 & -8-1 \end{array}\right] \)
\(=\left[\begin{array}{cc} -5 & -4 \\ 8 & -9 \end{array}\right] \)

28.

\(\left[\begin{array}{ll} 1 & 3 \\ 0 & 1 \end{array}\right] A=\left[\begin{array}{cc} 1 & 1 \\ 0 & -1 \end{array}\right]\)
\(\text { Let } A=\left[\begin{array}{ll} a & b \\ c & d \end{array}\right]\)
\(\left[\begin{array}{ll} 1 & 3 \\ 0 & 1 \end{array}\right]\left[\begin{array}{ll} a & b \\ c & d \end{array}\right]=\left[\begin{array}{cc} 1 & 1 \\ 0 & -1 \end{array}\right]\)
\(\left[\begin{array}{cc} a+3 c & b+3 d \\ 0+c & 0+d \end{array}\right]=\left[\begin{array}{cc} 1 & 1 \\ 0 & -1 \end{array}\right]\)
\(c=0,d=-l\)
\(a+3 c=1 \quad b+3 d=1\)
\(a+0=1 \quad b-3=1\)
\(a=1 \quad b=4\)
\(\therefore A=\left[\begin{array}{cc} 1 & 4 \\ 0 & -1 \end{array}\right]\)

29.

30.

\(-1 \leq x<0 \Rightarrow\lfloor x\rfloor=-1 ;\)
\(0 \leq y<1 \Rightarrow\lfloor y\rfloor=0 ; 1 \leq z<2 \Rightarrow\lfloor z\rfloor=1 \)
\(\therefore|\cdot A|=\left|\begin{array}{ccc} -1+1 & 0 & 1 \\ -1 & 0+1 & 1 \\ -1 & 0 & 1+1 \end{array}\right|=\left|\begin{array}{ccc} 0 & 0 & 1 \\ -1 & 1 & 1 \\ -1 & 0 & 2 \end{array}\right|=1(0+1)=1 \)
\(|A|=1=\lfloor z\rfloor \)

31.

\(B =(-)\left|\begin{array}{ccc} -2 & 4 & 8 \\ 6 & 2 & 0 \\ -2 & 4 & 2 \end{array}\right| \quad R_{1} \leftrightarrow R_{3} \)
\(=-(2 \times 2)\left|\begin{array}{ccc} -1 & 2 & 4 \\ 3 & 1 & 0 \\ -2 & 4 & 2 \end{array}\right| \)
\(\text { Take } 2 \text { from } R_{1} \& R_{2}\)
\(B=-4 A\)

32.

\(\left|\begin{array}{ccc} a & 2 b & 2 c \\ 3 & b & c \\ 4 & a & b \end{array}\right| =0 \)
\(\Rightarrow \frac{1}{2}\left|\begin{array}{lll} a & 2 b & 2 c \\ 6 & 2 b & 2 c \\ 4 & a & b \end{array}\right| =0 \quad R_{2} \rightarrow 2 R_{2} \)
\(\frac{1}{2}\left|\begin{array}{ccc} a-6 & 0 & 0 \\ 6 & 2 b & 2 c \\ 4 & a & b \end{array}\right| =0 \quad R_{1} \rightarrow R_{1}-R_{2} \)
\(\Rightarrow \frac{1}{2}\left[(a-6)\left(2 b^{2}-2 a c\right)\right] =0 \)
\((a-6)\left(2 b^{2}-2 a c\right) =0 \)
\(a=6,2 b^{2} =2 a c \)
\(b^{2} =a c \quad \therefore b^{3}=a b c \)

33.

(d)
collinear

34.

\(|A|=\left|\begin{array}{ccc} 0 & a & -b \\ -a & 0 & c \\ b & -c & 0 \end{array}\right|\)
\(=0-a(-b c)-b(a c)=a b c-a b c=0\)

35.

Put x = 0
\(|A|=\left|\begin{array}{ccc} 3 & -6 & 3 \\ -6 & 3 & 3 \\ 3 & 3 & -6 \end{array}\right|=\left|\begin{array}{ccc} 0 & -6 & 3 \\ 0 & 3 & 3 \\ 0 & 3 & -6 \end{array}\right|\)
\(C_{1} \rightarrow C_{1}+C_{2}+C_{3}\)
= 0
\(\therefore\) x = 0 is a root.

36.

\(\left|\begin{array}{lll} k a & k b & k c \\ k x & k y & k z \\ k p & k q & k r \end{array}\right|=k^{3}\left|\begin{array}{lll} a & b & c \\ x & y & z \\ p & q & r \end{array}\right|=k^{3} \Delta\)
\(\text { Take } k \text { from } R_{1}, R_{2}, R_{3}\)

37.

\(\text { Given } \text {matrix } A=\left[\begin{array}{cc} \alpha & \beta \\ \gamma & -\alpha \end{array}\right] \text { is unit matrix }\)
\(|A|=1 \)
\(-\alpha^{2}-\beta \gamma=1 \)
\(1+\alpha^{2}+\beta \gamma=0 \)

38.

\(\text { Area of triangle }=\frac{1}{2}\left|\begin{array}{lll} \frac{x_{1}}{a} & \frac{y_{1}}{a} & 1 \\ \frac{x_{2}}{b} & \frac{y_{2}}{b} & 1 \\ \frac{x^{3}}{c} & \frac{y_{3}}{c} & 1 \end{array}\right|\)
\(=\frac{1}{2 a b c}\left|\begin{array}{lll} x_{1} & y_{1} & a \\ x_{2} & y_{2} & b \\ x_{3} & y_{3} & c \end{array}\right| \text { Multiply }R_{1}, R_{2} \& R_{3} \text { by } a, b, c\text { respectively }\)
\(=\frac{1}{2 a b c} \frac{1}{(2)}\left|\begin{array}{ccc} 2 a & x_{1} & y_{1} \\ 2 b & x_{2} & y_{2} \\ 2 c & x_{3} & y_{3} \end{array}\right|=\frac{1}{4 a b c}\left(\frac{a b c}{2}\right)\)
\(=\frac{1}{8}\)

39.

(d)
3

40.

\(|A|=0 \quad \text { [since } A \text { is singular] }\)
\(\left|\begin{array}{ll} e^{x-2} & e^{7+x} \\ e^{2+x} & e^{2 x+3} \end{array}\right| =0 \)
\(e^{x-2+2 x+3}-e^{2+x+7+x} =0 \)
\(e^{3 x+1} =e^{9+2 x} \)
\(3 x+1 =9+2 x \)
\(x =8 \)

41.

\(A^{T}=\left[\begin{array}{ll} a & y \\ x & a \end{array}\right] \quad \therefore A A^{T}=\left[\begin{array}{ll} a & x \\ y & a \end{array}\right]\left[\begin{array}{ll} a & y \\ x & a \end{array}\right] \)
\(=\left[\begin{array}{ll} a^{2}+x^{2} & a y+a x \\ a y+a x & y^{2}+a^{2} \end{array}\right] \)
\(\operatorname{det}\left(A A^{T}\right)=\left|\begin{array}{ll} a^{2}+x^{2} & a y+a x \\ a y+a x & y^{2}+a^{2} \end{array}\right| \)
\(=\left(a^{2}+x^{2}\right)\left(a^{2}+y^{2}\right)-(a y+a x)(a y+a x) \)
\(=a^{4}+a^{2} y^{2}+a^{2} x^{2}+x^{2} y^{2}-a^{2} y^{2}-a^{2} x y -a^{2} x y-a^{2} x^{2}\)
\(=a^{4}+1-2 a^{2} x y=a^{4}-2 a^{2}+1(\because x y=1) \)
\(=\left(a^{2}-1\right)^{2} \)

42.

\(A \& B \text { are symmetric } \Rightarrow A+B \text { is symmetric }\)

43.

(d)
A − A^T

44.

\(A A^{T}=9 I \)
\({\left[\begin{array}{ccc} 1 & 2 & 2 \\ 2 & 1 & -2 \\ a & 2 & b \end{array}\right]\left[\begin{array}{ccc} 1 & 2 & a \\ 2 & 1 & 2 \\ 2 & -2 & b \end{array}\right]=\left[\begin{array}{ccc} 9 & 0 & 0 \\ 0 & 9 & 0 \\ 0 & 0 & 9 \end{array}\right]} \)
\({\left[\begin{array}{ccc} 1+4+4 & 2+2-4 & a+4+2 b \\ 2+2-4 & 4+1+4 & 2 a+2-2 b \\ a+4+2 b & 2 a+2-2 b & a^{2}+4+b^{2} \end{array}\right]} \)
\(=\left[\begin{array}{ccc} 9 & 0 & 0 \\ 0 & 9 & 0 \\ 0 & 0 & 9 \end{array}\right] \)
\(a+4+2 b =0 \)
\(a+2 b =-4 \)
\(2 a-2 b+2 =0 \)
\(2 a-2 b =-2 \)
\(\text { Solve (1) } \&(2) \text { we get } a=-2 b=-1\)
\(\therefore(a, b)=(-2,1)\)

45.

\(A+B =\left[\begin{array}{ll} 1+a & 0 \\ 2+b & -2 \end{array}\right] \)
\((A+B)^{2} =\left[\begin{array}{ll} 1+a & 0 \\ 2+b & -2 \end{array}\right]\left[\begin{array}{ll} 1+a & 0 \\ 2+b & -2 \end{array}\right] \)
\(=\left[\begin{array}{ll} (1+a)^{2} & 0 \\ (2+b)(1+a)-2(2+b) & 4 \end{array}\right] \)
\(A^{2} =\left[\begin{array}{ll} 1 & -1 \\ 2 & -1 \end{array}\right]\left[\begin{array}{ll} 1 & -1 \\ 2 & -1 \end{array}\right]=\left[\begin{array}{cc} -1 & 0 \\ 0 & -1 \end{array}\right] \)
\(B^{2} =\left[\begin{array}{cc} a & 1 \\ b & -1 \end{array}\right]\left[\begin{array}{cc} a & 1 \\ b & -1 \end{array}\right]=\left[\begin{array}{ll} a^{2}+b & a-1 \\ a b-b & b+1 \end{array}\right] \)
\(A^{2}+B^{2} =\left[\begin{array}{cc} -1 & 0 \\ 0 & -1 \end{array}\right]+\left[\begin{array}{cc} a^{2}+b & a-1 \\ a b-b & b+1 \end{array}\right] \)
\(=\left[\begin{array}{cc} a^{2}+b-1 & a-1 \\ a b-b & b \end{array}\right]\)
\((A+B)^{2} =A^{2}+B^{2} \)
\(\therefore a-1 =0 \)
\(a =1 \)
\(b =4\)

46.

\(A^{2}=A \times A=\left[\begin{array}{cc} \lambda & 1 \\ -1 & -\lambda \end{array}\right]\left[\begin{array}{cc} \lambda & 1 \\ -1 & -\lambda \end{array}\right]\)
\(=\left[\begin{array}{cc} \lambda^{2}-1 & \lambda-\lambda \\ -\lambda+\lambda & -1+\lambda^{2} \end{array}\right]=0\)
\(\therefore \lambda^{2} =1 \)
\(\lambda =\pm 1\)

47.

If A + B and AB are both defined means A & B are square matrices of same order

48.

(a)
a scalar matrix

49.

\(2 X=\left[\begin{array}{ll} 3 & 8 \\ 7 & 2 \end{array}\right]-\left[\begin{array}{ll} 1 & 2 \\ 3 & 4 \end{array}\right]=\left[\begin{array}{cc} 2 & 6 \\ 4 & -2 \end{array}\right]\)
\(X=\left[\begin{array}{cc} 1 & 3 \\ 2 & -1 \end{array}\right]\)

50.

\(A=\left[\begin{array}{ll} a_{11} & a_{12} \\ a_{21} & a_{22} \end{array}\right] \)
\(a_{11}=\frac{1}{2}(3-2)=\frac{1}{2} ; a_{12}=\frac{1}{2}(3-4)=\frac{-1}{2} \)
\(a_{21}=\frac{1}{2}(3(2)-2)=\frac{4}{2}=2 ; a_{22}=\frac{1}{2}(6-4)=\frac{2}{2}=1 \)
\(\therefore A=\left[\begin{array}{ll} \frac{1}{2} & -\frac{1}{2} \\ 2 & 1 \end{array}\right] \)