# 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

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

SECTION C

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

(OR)

Evaluate:

- Using properties of determinants prove that :

** = 2 (x + y + z)2**

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

(OR)

For ** , y = a sin t ,** find

- 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 **

- Evaluate: dx
- Solve the differential equation:
- Find a particular solution of the differential equation
**(x – y)(dx + dy) = dx – dy given that y = - 1 when x = 0** - 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.
- 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?**

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

SECTION – C

- 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

- 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

- 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)**

- Evaluate:
- 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)
- 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.

- 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 / TOTAL1 / 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 allotted01 / 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 ff-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