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Important questions

11th Standard

Reg.No. :
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Maths

Part A

Time : 00:25:00 Hrs
Total Marks : 50

Part A

50 x 1 = 50
1. If aij =${1\over2}(3i-2j)$ and A=[aij]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)

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

2. 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)

$\begin{bmatrix} 2& -6 \\ 4 &-2 \end{bmatrix}$

3. 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

4. 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.

5. 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

6. 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

7. If A=$\begin{bmatrix} 1& 2 &2 \\ 2 & 1 & -2 \\ a & 2 & b \end{bmatrix}$ is a matrix satisfying the equation AAT = 9I, where I is 3 × 3 identity matrix, then the ordered pair (a, b) is equal to

(a)

(2, - 1)

(b)

(- 2, 1)

(c)

(2, 1)

(d)

(- 2, - 1)

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

(a)

A+ AT

(b)

AAT

(c)

AT A

(d)

A− AT

9. 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

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

(a)

(a −1)2

(b)

(a2 +1)2

(c)

a2 −1

(d)

(a2 −1)2

11. 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 is

(a)

9

(b)

8

(c)

7

(d)

6

12. 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

13. 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)

${1\over 8}abc$

14. 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+$\alpha ^2-\beta \gamma=0$

15. 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$\triangle$

16. 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

17. 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

18. 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

19. 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

20. 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

21. 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

22. If A is skew-symmetric of order n and C is a column matrix of order n × 1, then CT 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

23. 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)

$\begin{bmatrix} 1 & -4 \\ 1 & 1 \end{bmatrix}$

24. 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)

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

25. 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)

AT B = ABT

26. The value of $\overrightarrow{AB}+\overrightarrow{BC}+\overrightarrow{DA}+\overrightarrow{CD}$ is

(a)

$\overrightarrow{AD}$

(b)

$\overrightarrow{CA}$

(c)

$\overrightarrow{0}$

(d)

$-\overrightarrow{AD}$

27. 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)

$1\over6$

28. 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)

${2\hat{i}-\hat{j}\over\sqrt{5}}$

29. 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°

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)

$-4\hat{i}$

31. 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)

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

32. 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.

33. 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)

$\overrightarrow{0}$

34. 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)

$\overrightarrow{a}+\overrightarrow{b}\over 2$

35. 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)

${\overrightarrow{a}-\overrightarrow{b}\over 3}$

36. 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)

$4\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c}=0$

37. 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

38. 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)

${1\over2}$

39. 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)

$-2\hat{i}+\hat{j}+6\hat{k}$

40. 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

41. 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

42. 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)

${\pi\over 2}$

43. 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

44. 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

45. 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°

46. 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)

$\pm 1$

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

(a)

$7\over 3$

(b)

-$7\over 3$

(c)

-$5\over 3$

(d)

$5\over 3$

48. 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

49. 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

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)

$17\over4$