CBSE
Class 10 Class 12
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General Instruction:
| 1. | Let A be the set of all students of a boys school. Show that the relation R in A given by R = {(a, b) : a is sister of b} is the empty relation and R’ = {(a, b) : the difference between heights of a and b is less than 3 meters} is the universal relation. | [1] |
| 2. |
Evaluates : | [1] |
| 3. |
If P1 , P2, P3, P4 are points in a plane or space and O, the origin of vectors, show that P4 coincides with O if an only if  | [1] |
| 4. | Show that the function f : N → N given by f(x) = 2x, is one-one but not onto. | [1] |
| 5. |
Using principle value, evaluate the following : | [2] |
| 6. |
Construct a 3 x 4 matrix whose elements are ai j = i –J | [2] |
| 7. | Examine whether the function f given by f(x) = x2 is continuous at x = 0. | [2] |
| 8. |
The cost function C(x), in rupees, of producing x items (x ≥ 15) in a certain factory is given by | [2] |
| 9. |
Evaluate the following integral: | [2] |
| 10. |
Determine the order and degree of the differential equation: | [2] |
| 11. |
Find values of x for which  is a unit vector.
| [2] |
| 12. |
Find the angle between the vector | [2] |
| 13. |
| [4] |
| 14. |
Discuss continuity of the function f given by f(x) = | x – 1| + | x – 2 ] at x = 1 and x = 2. | [4] |
| 15. |
The volume of a cube is increasing at a rate of 9 cubic centimeters per second. How fast is the surface area increasing when the length of an edge is 10 centimeters? | [4] |
| 16. |
Show that the following differential equation is homogeneous and find a primitive of it. Derive the solution wherever possible: | [4] |
Solve 
| 17. |
Find all the points of discontinuity of the function f defined by
| [4] |
| 18. |
Evaluate | [4] |
| 19. |
If P, Q, R, S are the points (– 2, 3, 4), (– 4, 4, 6), (4, 3, 5), (0, 1, 2), prove by projection that PQ is perpendicular to RS. | [4] |
| 20. |
A die is thrown three times. Events A and B are defined as below: | [4] |
| 21. | A point source of light along a straight road is at a height of ‘a’ metres. A boy ‘b’ metres in height is walking along the road. How fast is his shadow increasing if he is walking away from the light at the rate of c metres per minute? | [4] |
| 22. |
ABCDEF is a regular hexagon. Show that (i)  (ii)  where O is centre of the hexagon. | [4] |
| 23. | An unbiased die is thrown twice. Let the event A be ‘odd number on the first throw’ and B the event ‘odd number on the second throw’. Check the independence of the events A and B. | [4] |
| 24. |
Show that each of the relation R in the set A = {x ∈ Z : 0 ≤ x ≤ 12 }, given by | [6] |
| 25. |
| [6] |
| 27. |
Evaluate: | [6] |
Draw a graph of 
and evaluate area bounded by it.
| 28. |
A (– 1, 2, – 3), B (5, 0, – 6), C (0, 4, – 1) are three points. Show that the direction cosines of the bisectors of are proportional to 25, 8, 5 and -11, 
| [6] |
For the cartesian and vector equation of a line which passes through the point (1, 2, 3) and is parallel to the line 
| 29. |
Solve the following linear programming problem graphically: Maximise Z = 4x + y subject to the constraints: x + y ≤ 50, 3x + y ≤ 90, x ≥ 0, y ≥ 0 | [6] |
Determine the marginal cost function and the marginal cost of producing 100 items. 
