# Sample Paper Prepared by M.Srinivasan, Pgt(Maths), Ziet, Mumbai

SAMPLE PAPER PREPARED BY M.SRINIVASAN, PGT(MATHS), ZIET, MUMBAI

SAMPLE PAPER

CLASS XII : MATHEMATICS

Time:03hours Max Marks: 100

General Instructions
All questions are compulsory
The question paper consists of 29 questions divided into three sections A, B and C
Section A contains 10 questions of 1 mark each, Section B is of 12 questions of four marks each and Section C comprises of 7 questions of six marks each
There is no overall choice. However, internal choice has been provided in 4 questions of four marks and 2 questions of six marks each. You have to attempt only one of the alternatives in all such questions.
Use of calculators is not permitted

SECTION – A

1. If a  b = a + 2b, find the value of (9  4)  2
2. Evaluate:
3. If A and B are symmetric matrices of same order, check whether the matrix AB – BA is symmetric of skew-symmetric.
4. For what value of the matrix has no inverse?
5. Evaluate: dx
6. Evaluate: dx where [x] is the greatest integer function.
7. Find the value of for which is a unit vector.
8. For what value of ‘k’ for which the vectors and are parallel.
9. Find the distance between the parallel planes: 2x + 3y + 4z = 4 and 4x + 6y +8z = 12
10. For the matrices A = and B = , find a and b if A = B’

SECTION C

1. For the function f(x) = , x  -1, show that f-1 exist. Hence show that ff-1 is an identity function.
2. Solve for ‘x’: =

(OR)

Evaluate:

1. Using properties of determinants prove that :

= 2 (x + y + z)2

1. Find all points of discontinuity of the function f(x) defined by:
2. Find for the function: (cos x)y = (cos y)x

(OR)

For , y = a sin t , find

1. Find the intervals in which the function f(x) = 20 – 9x + 6x2 – x3is strictlyincreasing or strictly decreasing.

(OR)

Find the equation of the normals to the curve y = x3 + 2x + 6 which are parallel to the line x + 14y + 4 = 0

1. Evaluate: dx
2. Solve the differential equation:
3. Find a particular solution of the differential equation (x – y)(dx + dy) = dx – dy given that y = - 1 when x = 0
4. Show that the points A(1, -2, -8), B(5, 0, -2) and C(11, 3, 7) are collinear and find the ratio in which B divides AC.
5. Three balls are drawn one by one without replacement from a bag containing 5 white and 4 green balls. Find the probability distribution of the number of green balls drawn.

(OR)

A die is thrown 6 times. If getting an odd number is a success, what is the probability of getting a) at least 5 success b) at most 4 success?

1. Find the length and the foot of the perpendicular drawn from the point (2, -1, 5) to the line

SECTION – C

1. Let A = and B = . Verify that AB = BA = 6 I3. Hence solve the system of equations: x – y = 3 ; 2x + 3y + 4z = 17 ; y + 2z = 7
1. Find the largest possible area of a right angled triangle whose hypotenuse is 5 cm long.

(OR)

Show that the volume of the greatest cylinder which can be inscribed in a cone of height ‘h’ and semi-vertical angle  is

1. Using integration, find the area of the region: {(x , y) : 9x2 + y2 ≤ 36, 3x + y  6}

(OR

Using integration, find the area of the triangular region whose vertices are (1 , 0), (2 , 2) and (3 , 1)

1. Evaluate:
2. Find the equation of the plane containing the lines:and . Find the distance of this plane from the origin and also from the point (1, 1, 1)
3. A medicine company has factories at two places A and B. From these places, supply is made to each of the three agencies situated at P, Q and R. The monthly requirement of the agencies are respectively 40, 40 and 50 packets of medicines, while the production capacity of the factories at A and B are 60 and 70 packets respectively. The transportation cost per packet from the factories to the agencies are given below:-

To
From / Transportation cost per packet
P / Q / R
A / 5 / 4 / 3
B / 4 / 2 / 5

How may packets from each factory be transported to each agency so that the cost of transportation is minimum? Also find the minimum cost.

1. Three urns are given, each containing red and black balls as indicated below:

URN / NUMBER OF RED BALLS / NUMEBR OF BLACK BALLS
I / 6 / 4
II / 2 / 6
III / 1 / 8

An urn is chosen at random and a ball is drawn from that urn. The ball drawn is red. Find the probability that the ball is drawn from urn II or from urn III.

KENDRIYA VIDYALAYA, 1 STC, JABALPUR

I PRE-BOARD EXAMINAIONS: DECEMBER 2011

CLASS XII : MATHEMATICS

BLUE PRINT

S.No. / Topics / VSA / SA / LA / TOTAL
1 / a)RELATIONS AND FUNCTIONS / 1 (1) / 4 (1) / 10 (4)
b)INVERSE TRIGONOMETRIC FUNCTIONS / 1 (1) / 4 (1)
2 / a)MATRICES / 2 (2) / 6(1) / 13 (5)
b)DETERMINANTS / 1 (1) / 4 (1)
3 / a)CONTINUITY & DIFFERENTIABILITY / 8 (2) / 44 (11)
b)APPLICATION OF DERIVATIVES / 4(1) / 6(1)
c)INTEGRATION / 2(2) / 4(1) / 6(1)
d)APPLICATION OF INTEGRALS / 6 (1)
e)DIFFERENTIAL EQUATIONS / 8(2)
4 / a)VECTORS / 2 (2) / 4 (1) / 17 (6)
b)3-DIMENTIONAL GEOMETRY / 1 (1) / 4 (1) / 6 (1)
5 / LINEAR PROGRMMING / 6 (1) / 6 (1)
6 / PROBABILITY / 4 (1) / 6 (1) / 10 (2)
TOTAL / 10 (10) / 48 (12) / 42(7) / 100(29)

KENDRIYA VIDYALAYA, 1 STC, JABALPUR

I PRE-BOARD EXAMINAIONS: DECEMBER 2011

CLASS XII : MATHEMATICS

MARKING SCHEME

S.No. / Answer / Marks allotted
01 / 21 / 1
02 / / 1
3 / AB – BA is skew-symmetric / 1
4 / x = 43 / 1
5 / / 1
6 / 0.5 / 1
7 / x = / 1
8 / k = / 1
9 / Distance = / 1
10 / a = 4 ; b = 2 / 1
11 / Proving one-to-one / 1
Proving ONTO / 1
Finding / 1
Proving ff-1 (x) = x / 1
12 / Using the inverse trigonometric identities / 1 ½ + 1 ½ = 3
Getting x = / 1
12 / Using the inverse trigonometric identities / 1 ½ + 1 ½ = 3
Getting the value as 1 / 1
13 / Using properties and taking factors common / 1 ½ + 1 ½ = 3
Getting the RHS / 1
14 / Proving Continuous at x = 3 / 2
Proving discontinuous at x = -3 / 2
15 / Taking log on both sides / 1
Finding the derivatives / 2
Getting the final answer / 2
15 / Getting the derivatives of x and y with respect to ‘t’ / 2
Getting the value of / 1
Getting the value of y’ = / 1
16 / Getting the critical values / 1
Checking the nature at the intervals / 1 + 1 + 1
16 / Getting the point / 2
Getting the equations of the normals / 2
17 / Applying the integration by parts / 3
Getting the value by applying the limits / 1
S.No. / Answer / Marks allotted
18 / Getting the values of P and Q / 1
Getting I.F / 1
Getting the solution / 2
19 / Expressing in homogenous form / 1
Getting the general solution / 2
Getting the particular solution / 2
20 / Proving the points as collinear / 3
Finding the ratio / 1
21 / Getting the possible values / 1
Getting the probability values / 2
Expressing in the form of table / 1
21 / Getting the value of p and q / 1
Getting the values of probability / 3
22 / Getting the coordinates of foot of the perpendicular / 3
Getting the perpendicular distance / 1
23 / Finding AB , BA / 1 ½ + 1 ½ = 3
Proving AB = BA / 1
Getting the value of x, y , z / 2
24 / Getting the equation connecting height and base / 1
Finding the first derivative / 2
Getting the value of x / 1
Getting the second derivative / 1
Finding the area / 1
24 / Getting the volume function
Finding the first derivative / 2
Getting the value of x / 1
Getting the second derivative / 1
Finding the volume / 1
25 / Identifying the area / 1
Getting the points of intersection / 1
Evaluating the area by integration / 4
25 / Identifying the area / 1
Getting the points of intersection / 1
Evaluating the area by integration / 4
S.No. / Answer / Marks allotted
26 / Applying the properties of integral / 1
Getting the value of the integral / 4
Applying the limits / 1
27 / Finding the equation of the plane / 4
Finding the distances / 1 + 1
28 / Getting the objective function / 1
Getting the constraints / 1
Drawing the graph / 3
Getting the optimal value / 1
29 / Getting the values of the probabilities / 3
Getting the value using baye’s theorem / 3