CBSE
Class 10 Class 12
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General Instruction:
| 1. | Show that the relation R in the set {1, 2, 3} given by R = {(1, 1), (2, 2), (3, 3), (1, 2), (2, 3)} is reflexive but neither symmetric nor transitive. | [1] |
| 2. |
For what values of k, the system of linear equations | [1] |
| 3. |
| [1] |
| 4. | Show that the function f : N → N given by f(1) = f (2) = 1 and f (x) = x – 1, for every x > 2, is onto but not one-one. | [1] |
| 5. |
| [2] |
| 6. |
Evaluate the following definite integrals as limit of sums. | [2] |
| 7. |
Show that all the diagonal elements of a skew symmetric matrix are zero. | [2] |
| 8. |
Write the order and degree of the differential equation | [2] |
| 9. |
Give a condition that the three vectors  form the three sides of a triangle. What are the other possibilities?
| [2] |
| 10. |
Often it is taken that a truthful person commands, more respect in the society. A man is known to speak the truth 4 out of 5 times. He throws a die and reports that it is a six. | [2] |
| 11. |
| [2] |
| 12. |
PQRS is a parallelogram. If  are the position vectors of the vertices P, Q and R respectively with reference to the origin of reference O ; find the position vector of S with reference to O. | [2] |
| 13. |
Find x such that the four points A(4, 1, 2), B(5, x, 6) , C(5, 1, -1) and D(7, 4, 0) are coplanar. | [4] |
Find the coordinates of the foot of perpendicular drawn from the point A
(-1,8,4) to the line joining the points B(0,-1,3) and C(2,-3,-1). Hence find the image of the point A in the line BC.
| 14. |
A bag X contains 4 white balls and 2 black balls, while another bag Y contains 3 white balls and 3 black balls. Two balls are drawn (without replacement) at random from one of the bags and were found to be one white and one black. Find the probability that the balls were drawn from bag Y. | [4] |
| 15. |
If xm yn = (x + y)m + n, prove that | [4] |
| 16. |
Show that the function | [4] |
| 17. |
In a set of 10 coins, 2 coins are with heads on both the sides. A coin is selected at random from this set and tossed five times. If all the five times, the result was heads, find the probability that the selected coin had heads on both the sides | [4] |
| 18. |
| [4] |
| 19. |
Find the particular solution of the differential equation | [4] |
Prove that x2 – y2 = c(x2 + y2)2 is the general solution of the differential equation (x3 – 3xy2)dx = (y3 – 3x2y) dy, where C is a parameter.
| 20. |
| [4] |
| 21. |
Find all points of discontinuity of f where
| [4] |
| 22. |
Show that the four points A, B, C and D with position vectors | [4] |
| 23. |
Find the value(s) of x for which y = | [4] |
| 24. |
Using properties of determinants, prove that | [6] |
| 25. |
Let A= R × R and * be a binary operation on A defined by | [6] |
| 26. |
By using the properties of definite integrals, evaluate the following: | [6] |
| 27. |
Sketch the region bounded by the curves | [6] |
Using the method of integration, find the area of the triangular region whose vertices are (2, -2), (4, 3) and (1, 2).
| 28. |
Minimum and maximum z = 5x + 2y subject to the following constraints: | [6] |
A retired person wants to invest an amount of Rs. 50, 000. His broker recommends investing in two type of bonds ‘A’ and ‘B’ yielding 10% and 9% return respectively on the invested amount. He decides to invest at least Rs. 20,000 in bond ‘A’ and at least Rs. 10,000 in bond ‘B’. He also wants to invest at least as much in bond ‘A’ as in bond ‘B’. Solve this linear programming problem graphically to maximise his returns.
| 29. |
Show that lines: | [6] |
is continuous but not differentiable at x=3. 
respectively are coplanar.
is an increasing function.
and find its area using integration. 
