Important Question paper

11th Standard

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Maths

Use Blue Pen only
Time : 01:00:00 Hrs
Total Marks : 60

    Part-A

    Answer all the questions

    10 x 1 = 10
  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. 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

  4. 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

  5. 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

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

    (a)

    \(\overrightarrow{AD}\)

    (b)

    \(\overrightarrow{CA}\)

    (c)

    \(\overrightarrow{0}\)

    (d)

    \(-\overrightarrow{AD}\)

  7. 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)

    \(1\over6\)

  8. 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

  9. 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

  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)

    \(17\over4\)

  11. Part-B

    Answer all the questions

    10 x 2 = 20
  12. Simplify : \(sec \theta \begin{bmatrix} sec \theta & tan\theta \\ tan\theta & sec\theta \end{bmatrix}\)-\(tan \theta \begin{bmatrix} tan \theta & sec\theta \\ sec\theta & tan\theta \end{bmatrix}\)

  13. If A=\(\begin{bmatrix} 0 &c &b \\ c & 0 &a \\ b & a & 0 \end{bmatrix}\),compute A2

  14. Consider the matrix Aa=\(\begin{bmatrix} cos \alpha & -sin\alpha \\ sin\alpha & cos\alpha \end{bmatrix}\)
    Find all possible real values of α satisfying the condition \(A\alpha +A^T_{\alpha}=I\)

  15. If A=\(\begin{bmatrix} 4 & 2 \\ -1 & x \end{bmatrix}\) and such that (A- 2I)(A-3I)=O, find the value of x.

  16. Can a vector have direction angles 30°,45°,60°?

  17. Find the direction cosines of \(\overrightarrow{AB},\)where A is (2, 3, 1) and B is (3, - 1, 2).

  18. Find the direction cosines of the line joining (2, 3, 1) and (3, - 1, 2).

  19. If \(\overrightarrow{a}\) and \(\overrightarrow{b}\)are two vectors such that | \(\overrightarrow{a}\) |=10,| \(\overrightarrow{b}\) |=15 and \(\overrightarrow{a}\).\(\overrightarrow{b}\) = 75 \(\sqrt{2}\), find the angle between \(\overrightarrow{a}\)and \(\overrightarrow{b}\).

  20. Find the angle between the vectors  \(2\hat{i}+3\hat{j}-6\hat{k}\) and \(6\hat{i}-3\hat{j}+2\hat{k}\)

  21. Five mangoes and 4 apples are in a box. If two fruits are chosen at random, find the probability that (i) one is a mango and the other is an apple (ii) both are of the same variety

  22. Part-C

    Answer all the questions

    5 x 3 = 15
  23. If A =\(\begin{bmatrix} 4 & 6 & 2 \\ 0 & 1 & 5 \\ 0 & 3 & 2 \end{bmatrix}\) and B= \(\begin{bmatrix} 0 & 1 & -1 \\ 3 & -1 & 4 \\ -1 & 2 & 1 \end{bmatrix}\)
    verify(AB)T=BTAT

  24. If A =\(\begin{bmatrix} 4 & 6 & 2 \\ 0 & 1 & 5 \\ 0 & 3 & 2 \end{bmatrix}\) and B= \(\begin{bmatrix} 0 & 1 & -1 \\ 3 & -1 & 4 \\ -1 & 2 & 1 \end{bmatrix}\)
    verify(A+B)T=AT+BT

  25. If A =\(\begin{bmatrix} 4 & 6 & 2 \\ 0 & 1 & 5 \\ 0 & 3 & 2 \end{bmatrix}\) and B= \(\begin{bmatrix} 0 & 1 & -1 \\ 3 & -1 & 4 \\ -1 & 2 & 1 \end{bmatrix}\)
    verify(A-B)T=AT-BT

  26. If A1, B1, C1 are the cofactors of a1, b2,c3, respectively, i = 1 to 3 in
    |A|=\(\begin{vmatrix} a_1 &b_1 &c_1 \\ a_2 & b_2 &c_2 \\ a_3 & b_3 & c_3 \end{vmatrix}\), show that \(\begin{vmatrix} A_1 &B_1 &C_1 \\ A_2 & B_2 &C_2 \\ A_3 & B_3 & C_3 \end{vmatrix}\) =|A|2

  27. Identify the singular and non-singular matrices:\(\begin{bmatrix} 1&2 &3 \\ 4 & 5 &6 \\ 7 & 8 & 9 \end{bmatrix}\)

  28. Part-D

    Answer all the questions

    3 x 5 = 15
  29. If \(\begin{bmatrix} 0 & p& 3 \\ 2 & q^2 & -1 \\ r & 1 & 0 \end{bmatrix}\) is skew-symmetric, find the values of p,q, and r.

  30. Construct the matrix \(A=[a_{ij}]_{3\times 3}\), where \(a_{ij}=i-j.\) State whether A is symmetric or skew-symmetric.

  31. Let A and B be two symmetric matrices. Prove that AB = BA if and only if AB is a symmetric matrix.

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