Model question paper 6

11th Standard

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Maths

Use blue pen Only

Time : 00:50:00 Hrs
Total Marks : 55

    Part A

    Answer all the questions

    5 x 1 = 5
  1. The value of \(\overrightarrow{AB}+\overrightarrow{BC}+\overrightarrow{DA}+\overrightarrow{CD}\) is

    (a)

    \(\overrightarrow{AD}\)

    (b)

    \(\overrightarrow{CA}\)

    (c)

    \(\overrightarrow{0}\)

    (d)

    \(-\overrightarrow{AD}\)

  2. 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)

    \(1\over6\)

  3. 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

  4. 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)

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

  5. 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

  6. Part B

    Answer all the questions

    5 x 2 = 10
  7. Represent graphically the displacement of 30 km 60° west of north

  8. Prove that the relation R defined on the set V of all vectors by ‘ \(\overrightarrow{a}R \overrightarrow{b} \ if \overrightarrow{a}=\overrightarrow{b}\) is an equivalence relation on V.

  9. Find a direction ratio and direction cosines of the following vectors \(3\hat{i}+4\hat{j}-6\hat{k}\)
     

  10. Find the direction cosines of a vector whose direction ratios are 3,-1,3

  11. Find the direction cosines and direction ratios for the following vectors.5\(\hat{i}\)-3\(\hat{j}\)-48\(\hat{k}\)

  12. Part C

    Answer all the questions

    5 x 3 = 15
  13. Let A and B be two points with position vectors 2\(\overrightarrow{a}\)+ 4\(\overrightarrow{b}\) and 2\(\overrightarrow{a}\) −8\(\overrightarrow{b}\). Find the position vectors of the points which divide the line segment joining A and B in the ratio 1:3 internally and externally.

  14. Show that the vectors \(5\hat{i}+6\hat{j}+7\hat{k},7\hat{i}-8\hat{j}+9\hat{k},3\hat{i}+20\hat{j}+5\hat{k}\) are coplanar.

  15. Show that the following vectors are coplanar \(\hat{i}\) −2\(\hat{j}\) +3\(\hat{k}\),-2 \(\hat{i}\) +3\(\hat{j}\) - 4 \(\hat{k}\) ,-\(\hat{j}\) +2 \(\hat{k}\) .

  16. Show that the points (4, - 3, 1), (2, - 4, 5) and (1, - 1, 0) form a right angled triangle.

  17. If \(\overrightarrow{a},\overrightarrow{b},\overrightarrow{c}\)are position vectors of the vertices A, B, C of a triangle ABC, show that the area of the triangle ABC is \({1\over 2}|\overrightarrow{a}\times \overrightarrow{b}+ \overrightarrow{b}+\overrightarrow{c}+\overrightarrow{c}\times \overrightarrow{a}|\) .Also deduce the condition for collinearity of the points A, B, and C.

  18. Part D

    Answer all the questions

    4 x 5 = 20
  19. If ABCD is a quadrilateral and E and F are the midpoints of AC and BD respectively, then prove that \(\overrightarrow{AB}\) + \(\overrightarrow{AD}\) +\(\overrightarrow{CB}\)+\(\overrightarrow{CD}\) = 4\(\overrightarrow{EF}\).

  20. The position vectors of the points P, Q, R, S are \(\hat{i}\) + \(\hat{j}\) + \(\hat{k}\),2 \(\hat{i}\)+5\(\hat{j}\),3\(\hat{i}\)+2\(\hat{j}\)-3\(\hat{k}\), and \(\hat{i}\)-6\(\hat{j}\)-\(\hat{k}\)respectively. Prove that the line PQ and RS are parallel.

  21. If \(\overrightarrow{a},\overrightarrow{b}\)are unit vectors and \(\theta\) is the angle between them, show that \(sin {\theta \over 2}={1\over2}|\overrightarrow{a}-\overrightarrow{b}|\)

  22. Find the projection of the vector \(\hat{i}+3\hat{j}+7\hat{k}\) on the vector\(2\hat{i}+6\hat{j}+3\hat{k}\).

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