Cbse 12 Maths Sample Paper 1

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CLASS 12 MATHEMATICS SAMPLE PAPER 1

SECTION A(Questions number 1 to 10 carry 1 marks each)

1. Find the principal value of 1 1 1tan 2cos 2sin2

.

2. Solve 1 1tan (1 ) tan (1 ) ( 1 1).2

x x x

3. Find the order and degree of the differential equation.2

52

log 1ed y

xdx

.

4. Without actual expanding write the value of0 54 98

54 0 67

98 67 0

.

5. If '' 2 3 1 0and , then find 2 .1 2 1 2

A B A B

6. Differentiate w.r.t. x, 2logcos(2 )x .7. Find the distance between the parallel planes .(2 3 ) 4 6 3 9 13 0r i j k and x y z .

8.1 0 1

1 2 , then write 3 and find the value of k, if A is singular matrix.

0 1 4

A k A

9. Find a vector in the direction of given vector i j k having magnitude 8 units.10.Find the sine of the angle between the vectors 2 3 and 3 2a i j k b i j k

.

SECTION-B(Questions number 11 to 22 carry 4 marks each.)

11.

1 0

Let : and ( ) 0 0 and g(x)= , then does and coinside in [-1,0)?

1 0

x

f R R f x x x fog gof

x

OR

Given , let : P(X) P(X) P(X) be defined as A*B = (A-B) (B-A), A, B P(X)X

show that

is the identity element for the operation * and all the elements A of P(X) are

invertible with A-1=A.

12. By using the properties of determinants prove that :1 1 1

1 1 11 1 1 1 .

1 1 1

a

b abca b c

c

13. Show that 1 11

1 1 1tan cos . where 1.

4 21 1 2

x xx x x

x x


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OR

Show that , 1 1 112 4 63

sin cos tan13 5 16

14. If

2cos ( )cos cos( ), prove that .

cos

dy a yy x a y

dx a 15.Verify the applicability of the Rolles Theorem for the function

4 4( ) sin cos on [0, /2].f x x x

16. For what value ofa and b, the function defined as:3 ; 1

( ) 11 ; 1 is continuous at 1.

5 2 ; 1

ax b if x

f x if x x

ax b if x

17. Evaluate8 8

2 2

sin cos.

1 2sin .cos

x xdx

x x

18. Evaluate

2

5 3.

4 10

xdx What should be the ideal role of youth in national integration?

x x

19. Evaluate2

0

log(tan cot ) .x x dx

20.

Let a = i + 4 j + 2k, b = 3i - 2 j + 7k and c = 2i - j + 4k, find a vector d which is perpendicular

.

to both a and b, and c d =15. " .Directionless youth is the burden on nation" comment on it

OR

A girl walks 5 km towards west, and then she walks 3 km in a direction 60 east of north

and stops. Determine the girls displacement from her initial point of departure. Respect

the girl implies respect the nation comment on it.

21. Find the length of the perpendicular drawn from the point (2, -3,1) to the line1 3 2 4

2 3 2

x y z.

22. Consider the experiment of tossing a coin. If the coin shows head, toss it again but if itshows tail, then throw a die. Find the conditional probability of the event that the die

shows a number greater than 4 given that there is at least one tail.

OR

LetEandFbe two independent events. The probability that exactly one of them occurs is11

25and the probability of none of them occurring is

2

25. If P(T) denotes the probability of

the occurrence of the event T, then find the value of P(E) and P(F).


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SECTION-C

(Question number 23 to 29 carry 6marks each)

23. Using elementary transformations, find the inverse of

1 3 2

3 0 5 .

2 5 0

24. The total cost function and demand function of an item are given

by3

2( ) 7 111 50 and 100 respectively.3

xC x x x p x Write the total revenue

function. Find the number of items when the profit will be maximum. Find the

maximum profit also.

OR

Find the equation of tangents to the curve y= cos (x +y), 2 2x that are

parallel to the line x + 2y = 0.

25. Find the area lying above x-axis and included between the circle2 2 8x y x and

inside the parabola 2 4y x .

26. Solve the differential equation 2 1 ; (y 0)ye x dydxy y

and (1) 2y .

OR

Solve the differential equation ( . . ) .sin ( . . ). .cos .y y

x dy y dx y y dx x dy xx x

27. Assume that the chance of a patient having a heart attack is 40%. It is also assumedthat a meditation and yoga course reduce the risk of heart attack by 30% andprescription of certain drug reduces its chances by 25%. At a time a patient can choose

any one of the two options with equal probabilities. It is given that after going through

one of the two options the patient selected at random suffers a heart attack. Find the

probability that the patient followed a course of meditation and yoga? Why Meditation

and Yoga is necessary and sufficient thing for peace in mind and for good health.

28. Find the vector equation of the line passing through (1, 2, 3) and parallel to theplane

.( 2 ) 5 and 3 6.r i j k x y z

29. An aero plane can carry a maximum of 200 passengers. A profit of 1000 is made oneach executive class ticket and a profit of

600 is made on each economy class ticket.

The airline reserves at least 20 seats for executive class. However, at least 4 times as

many passengers prefer to travel by economy class than by the executive class.

Determine how many tickets of each type must be sold in order to maximize the profit

for the airline. What is the maximum profit? How one should respect the hard earn

money of parents/guardians in a best economical way.

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Cbse 12 Maths Sample Paper 1