Important questions

11th Standard

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Maths

Part A

Answer all the questions 

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

    Part A

    Answer all the questions

    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 is \(\hat{i}+3\hat{j}-\hat{k}\) ,then the position vector 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)

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

  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)

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

    (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)

    \(17\over4\)

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