Sample Paper – 2014
Class – XII
Subject – Mathematics
Class – XII
Subject – Mathematics
SECTION A ( 1X10=1M)
(Questions 1 to 10 carry 1 mark each)
1. Let * be the binary operation on N given by a *b = HCF of a and b. Find 20*16
2. What is Sin-1(Sin 7Ï€/6) ?
3. Find x and y if
= ![](file:///C:%5CDOCUME%7E1%5CADMINI%7E1%5CLOCALS%7E1%5CTemp%5Cmsohtml1%5C01%5Cclip_image004.png)
![](file:///C:%5CDOCUME%7E1%5CADMINI%7E1%5CLOCALS%7E1%5CTemp%5Cmsohtml1%5C01%5Cclip_image002.png)
![](file:///C:%5CDOCUME%7E1%5CADMINI%7E1%5CLOCALS%7E1%5CTemp%5Cmsohtml1%5C01%5Cclip_image004.png)
4. If A is a square matrix of order 3 and
= 64 then find
.
![](file:///C:%5CDOCUME%7E1%5CADMINI%7E1%5CLOCALS%7E1%5CTemp%5Cmsohtml1%5C01%5Cclip_image006.png)
![](file:///C:%5CDOCUME%7E1%5CADMINI%7E1%5CLOCALS%7E1%5CTemp%5Cmsohtml1%5C01%5Cclip_image008.png)
6. Find the adj A of
.
![](file:///C:%5CDOCUME%7E1%5CADMINI%7E1%5CLOCALS%7E1%5CTemp%5Cmsohtml1%5C01%5Cclip_image012.png)
7. ![](file:///C:%5CDOCUME%7E1%5CADMINI%7E1%5CLOCALS%7E1%5CTemp%5Cmsohtml1%5C01%5Cclip_image014.png)
![](file:///C:%5CDOCUME%7E1%5CADMINI%7E1%5CLOCALS%7E1%5CTemp%5Cmsohtml1%5C01%5Cclip_image014.png)
8. Find the value of α so that
= αi + 2j + k is perpendicular to
= 4i – 9j + 2k
![](file:///C:%5CDOCUME%7E1%5CADMINI%7E1%5CLOCALS%7E1%5CTemp%5Cmsohtml1%5C01%5Cclip_image016.png)
![](file:///C:%5CDOCUME%7E1%5CADMINI%7E1%5CLOCALS%7E1%5CTemp%5Cmsohtml1%5C01%5Cclip_image018.png)
9. Find the unit vector in the direction of
if![](file:///C:%5CDOCUME%7E1%5CADMINI%7E1%5CLOCALS%7E1%5CTemp%5Cmsohtml1%5C01%5Cclip_image022.png)
![](file:///C:%5CDOCUME%7E1%5CADMINI%7E1%5CLOCALS%7E1%5CTemp%5Cmsohtml1%5C01%5Cclip_image020.png)
![](file:///C:%5CDOCUME%7E1%5CADMINI%7E1%5CLOCALS%7E1%5CTemp%5Cmsohtml1%5C01%5Cclip_image022.png)
10. Find k if the lines
and
are perpendicular.
![](file:///C:%5CDOCUME%7E1%5CADMINI%7E1%5CLOCALS%7E1%5CTemp%5Cmsohtml1%5C01%5Cclip_image024.png)
![](file:///C:%5CDOCUME%7E1%5CADMINI%7E1%5CLOCALS%7E1%5CTemp%5Cmsohtml1%5C01%5Cclip_image026.png)
SECTION B(Q. 11 to 22 carry 4 marks each)
11. Show that the relation R on NXN defined by ( a,b) R (c,d)
a+d= b+c is an equivalence relation. (or)
![](file:///C:%5CDOCUME%7E1%5CADMINI%7E1%5CLOCALS%7E1%5CTemp%5Cmsohtml1%5C01%5Cclip_image028.png)
Let f : R
R be a function defined by f(x) = 4 + 3x . Show that f is invertible and find the inverse of f.
![](file:///C:%5CDOCUME%7E1%5CADMINI%7E1%5CLOCALS%7E1%5CTemp%5Cmsohtml1%5C01%5Cclip_image030.png)
12. Prove that tan-1 (
-
)/
+
) = Ï€/4 – ½ Cos-1x .
![](file:///C:%5CDOCUME%7E1%5CADMINI%7E1%5CLOCALS%7E1%5CTemp%5Cmsohtml1%5C01%5Cclip_image032.png)
![](file:///C:%5CDOCUME%7E1%5CADMINI%7E1%5CLOCALS%7E1%5CTemp%5Cmsohtml1%5C01%5Cclip_image034.png)
![](file:///C:%5CDOCUME%7E1%5CADMINI%7E1%5CLOCALS%7E1%5CTemp%5Cmsohtml1%5C01%5Cclip_image032.png)
![](file:///C:%5CDOCUME%7E1%5CADMINI%7E1%5CLOCALS%7E1%5CTemp%5Cmsohtml1%5C01%5Cclip_image036.png)
13. Using properties of determinants Prove that
= 4
.
![](file:///C:%5CDOCUME%7E1%5CADMINI%7E1%5CLOCALS%7E1%5CTemp%5Cmsohtml1%5C01%5Cclip_image038.png)
![](file:///C:%5CDOCUME%7E1%5CADMINI%7E1%5CLOCALS%7E1%5CTemp%5Cmsohtml1%5C01%5Cclip_image040.png)
14. Test the continuity of the following function at x = 0 ,
![](file:///C:%5CDOCUME%7E1%5CADMINI%7E1%5CLOCALS%7E1%5CTemp%5Cmsohtml1%5C01%5Cclip_image042.png)
If x = a ( t + Sint ) , y = a ( 1 – Cost ) , show that y’’ = 1/a, at t=
( or ) If xp y q =
,Prove that y’ = y/x.
![](file:///C:%5CDOCUME%7E1%5CADMINI%7E1%5CLOCALS%7E1%5CTemp%5Cmsohtml1%5C01%5Cclip_image044.png)
![](file:///C:%5CDOCUME%7E1%5CADMINI%7E1%5CLOCALS%7E1%5CTemp%5Cmsohtml1%5C01%5Cclip_image046.png)
15. Find the intervals where the function f (x) =2x3 – 9x2 + 12x + 30 is a) increasing b) decreasing.
16. Evaluate: ![](file:///C:%5CDOCUME%7E1%5CADMINI%7E1%5CLOCALS%7E1%5CTemp%5Cmsohtml1%5C01%5Cclip_image048.png)
![](file:///C:%5CDOCUME%7E1%5CADMINI%7E1%5CLOCALS%7E1%5CTemp%5Cmsohtml1%5C01%5Cclip_image048.png)
(or)
Evaluate as sum of limits
17. Solve the differential equation x2y’ = x2-2
+xy
![](file:///C:%5CDOCUME%7E1%5CADMINI%7E1%5CLOCALS%7E1%5CTemp%5Cmsohtml1%5C01%5Cclip_image052.png)
( or)
Form the differential equation representing the family of ellipses having foci on x-axis and centre at the origin.
Solve the differential equation Cos2x y’ + y = tanx.
i.
18. Three vectors
,
satisfying the condition
+
+
= 0 . Evaluate the quantity
+
+
if
= 1 ,
= 4
= 2.
![](file:///C:%5CDOCUME%7E1%5CADMINI%7E1%5CLOCALS%7E1%5CTemp%5Cmsohtml1%5C01%5Cclip_image054.png)
![](file:///C:%5CDOCUME%7E1%5CADMINI%7E1%5CLOCALS%7E1%5CTemp%5Cmsohtml1%5C01%5Cclip_image056.png)
![](file:///C:%5CDOCUME%7E1%5CADMINI%7E1%5CLOCALS%7E1%5CTemp%5Cmsohtml1%5C01%5Cclip_image016.png)
![](file:///C:%5CDOCUME%7E1%5CADMINI%7E1%5CLOCALS%7E1%5CTemp%5Cmsohtml1%5C01%5Cclip_image018.png)
![](file:///C:%5CDOCUME%7E1%5CADMINI%7E1%5CLOCALS%7E1%5CTemp%5Cmsohtml1%5C01%5Cclip_image058.png)
![](file:///C:%5CDOCUME%7E1%5CADMINI%7E1%5CLOCALS%7E1%5CTemp%5Cmsohtml1%5C01%5Cclip_image060.png)
![](file:///C:%5CDOCUME%7E1%5CADMINI%7E1%5CLOCALS%7E1%5CTemp%5Cmsohtml1%5C01%5Cclip_image062.png)
![](file:///C:%5CDOCUME%7E1%5CADMINI%7E1%5CLOCALS%7E1%5CTemp%5Cmsohtml1%5C01%5Cclip_image064.png)
![](file:///C:%5CDOCUME%7E1%5CADMINI%7E1%5CLOCALS%7E1%5CTemp%5Cmsohtml1%5C01%5Cclip_image066.png)
![](file:///C:%5CDOCUME%7E1%5CADMINI%7E1%5CLOCALS%7E1%5CTemp%5Cmsohtml1%5C01%5Cclip_image068.png)
![](file:///C:%5CDOCUME%7E1%5CADMINI%7E1%5CLOCALS%7E1%5CTemp%5Cmsohtml1%5C01%5Cclip_image070.png)
19. Find the shortest distance between the lines
= I + j + K ( 2i – j + k ) and
= ( 2i + j - k ) + p( 3i -5j+2k).
![](file:///C:%5CDOCUME%7E1%5CADMINI%7E1%5CLOCALS%7E1%5CTemp%5Cmsohtml1%5C01%5Cclip_image072.png)
![](file:///C:%5CDOCUME%7E1%5CADMINI%7E1%5CLOCALS%7E1%5CTemp%5Cmsohtml1%5C01%5Cclip_image074.png)
20. In a factory which manufactures bolts, machine A, B and C respectively 25%, 35% and 40% of the bolts, Of their output s 5,4,and 2 percent are respectively defective bolts. A nolt is drawn random from the product and is found to be defective. What is the probability that it is manufactured from machine A? SECTION C ( Each question carries 6 marks)
21. Find the inverse of
using elementary transformation. ( or) if A =
find A-1 and hence solve the equations 2x+3y+z= 11, -3x+2y+z=4, 5x-4y-2z = -9
![](file:///C:%5CDOCUME%7E1%5CADMINI%7E1%5CLOCALS%7E1%5CTemp%5Cmsohtml1%5C01%5Cclip_image076.png)
![](file:///C:%5CDOCUME%7E1%5CADMINI%7E1%5CLOCALS%7E1%5CTemp%5Cmsohtml1%5C01%5Cclip_image078.png)
24 .Find the maximum area of the isosceles triangle inscribed in an ellipse x2/a2 + y2/b2= 1, whose vertex lies along the major axis. (or) Show that the maximum value of the cylinder which can be inscribed in a sphere of radius 5
is 500Ï€ cm3.
![](file:///C:%5CDOCUME%7E1%5CADMINI%7E1%5CLOCALS%7E1%5CTemp%5Cmsohtml1%5C01%5Cclip_image080.png)
25.Prove that ![](file:///C:%5CDOCUME%7E1%5CADMINI%7E1%5CLOCALS%7E1%5CTemp%5Cmsohtml1%5C01%5Cclip_image082.png)
![](file:///C:%5CDOCUME%7E1%5CADMINI%7E1%5CLOCALS%7E1%5CTemp%5Cmsohtml1%5C01%5Cclip_image082.png)
26. Make a rough sketch of the region given below and find its area using integration. { (x,y) : 0≤y≤2x+3,
}.
![](file:///C:%5CDOCUME%7E1%5CADMINI%7E1%5CLOCALS%7E1%5CTemp%5Cmsohtml1%5C01%5Cclip_image084.png)
27. Find the foot of the perpendicular and the perpendicular distance of the point (3,2,1) from the plane 2x-y+z +1=0. Find the image of the point in the plane.
28. From a lot of 30 bulbs which includes 6 defective, a sample of 4 bulbs is drawn at random with replacement. Find the mean and variance of the number of defective bulbs.
29. A furniture firm manufactures chairs and tables each requiring the use of three machines A,B and C . Production of the chair requires 2 hrs on machine A, 1 hr on machine B, and 1 hr on machine C.Each table requires 1 hr on machine A, 1 hr on machine B and 3 hrs on machine C. The profit obtained by selling one chair is Rs. 30 while by selling one table Rs. 60. The total time available per week on machine A is 70 hrs, machine B 40 hrs, and on machine C 90 hrs. How many chairs and tables should be made per week so as to maximize profit? Formulate the problem as LPP and solve it graphically.