Place For Ample of Sample Papers: Maths

Sample Paper – 2011
Class – XII
Subject – Mathematics

Time 3 Hours Max Marks 100

General Instructions:

1. All questions are compulsory

2. Q 1 – 10 carries 1 marks, Q 11 – 22 carries 4 marks Q-23to 29 carries 6 marks

1. Construct a 2 x 2 matrix A = [ a_ij] whose elements are given by a_{ij =} i² + j²

2. If A_α= cosα sin α then prove that A_α . A_β = A_{α + β}

-sin α cosα

3. Verify whether * defined on the set A = {1,2,3,4,5} defined by

a*b = l.c.m(a,b) is a binary operation.

4. Check the continuity of the function f(x) = x³ - 8 x ≠ 2

x - 2

= 12 if x= 2 at x = 2

5. Find the derivative of sin^-1x with respect to tan^-1x

6. For the function y = x³+21, find the values of x when y increases 75 times as fast as x

7. Find ₀∫ ^π/2 cos³x dx

sin³x + cos³x

8. Solve log(dy/dx) = 3x -4y

9. Find the rate of change of area of a circle with respect to its radius when r = 3 cm

10. Evaluate ∫ 1 + tanx dx

x + logsecx

11. Using properties prove that x+4 2x 2x

2x x+4 2x = (5x+4) (4-x)²

2x 2x x+4

12. Solve cos^-1[ x² – 1] + 1 tan ^-1 2x = 2π

[ x² +1] 2 1-x² 3

13. Find dy/dx at t = 3π/4when x = a { cost + log (tan(t/2))}; y = asint

14. õ sec³ x dx or Evaluate ò e^x (x² +1) dx

(x +1)²

15. Show that the curves y = 4x² and 4xy = k cut at right angles if k² = 1/128

16. ∫ 3x + 5 dx (or) ( x³ – 1 ) ^1/3 x⁵ dx

x³- x² –x + 1

17. Find ₀∫^π xtanxdx

secx + tanx

18. If f(x) = 1 - cos4x , x < 0

x²

a , x = 0

√x , x > 0

( √16 + √x) – 4

Determine the value if ‘a’ if possible, so that the function f is continuous

19. Prove that ₀∫^π/2 √ tanx + √cotx dx = π √2

20 Find the point(s) on the curve y = 5x² – 2x³ at which the tangent is parallel to the line y – 4x = 5 (or)

Find the interval in which the function f(x) = cos(2x + (p/4)) , 0 £ x £ 2 p is increasing or decreasing

21. Evaluate _-2∫² │x +1│dx ( Limits -2 to 2)

22. Solve (3xy + y²) dx₊ ( x² + xy) dy = 0 (or) Solve x² dy/dx = 2xy+y²

23. Consider f :R – {7/5}→ R – {3/5)f(x) = (3x + 4)/(5x-7), show that f is invertible. Find f^-1

24. Solve the following equations by matrix method.

4x+2y+3z = 2; x+y+z = 1; 3x+y-2z = 5

25. Using only elementary column operations find the inverse of

3 4 2

0 2 -3

1 -2 6

26. Prove that a conical tent of given capacity will require the least amount

of canvas when the height is Ö2 times the radius of the base.

27. Evaluate a) ∫ dx ii) dx

Sin⁴x + cos²x e^x + e^-x

28. Find the area of the circle x²+y² = 16 which is exterior to the parabola

y² = 6x

29. Evaluate as limit of sum ₁∫⁵x²-2x+5 dx

(or) Find the area bounded by the curve y = x² – 3x and the line y = 2x

***********************

Sample Paper – 2011
Class – XII
Subject – Mathematics

Note –Atempt all questions .Q. no 1-2one mark ,3-8 four mark ,9-12 six marks each

1. Evaluate : Cos

3. If

4. If

7. Show that function y = be^x + ce^2xis a solution of the differential equation

8. Using differentials, find the approximate value of

9. A wire of length 28 m is to be cut into two pieces. One of the pieces is to be made into a square and the other into a circle. What should be the length of the two pieces so that the combined area of the square and the circle is minimum

10. If

show that

11. Find the inverse of the following matrix using elementary transformation

and hence solve the following system of linear equations:

12. Two tailors A and B earn Rs. 15 and Rs. 20 per day respectively. A can stitch 6 shirts and 4 pants while B can stitch 10 shirts and 4 pants per day. How many days shall each work if it is desired to produce at least 60 shirts and 32 pants at a minimum labour cost? Solve it by graphical method.

Submitted By :- Rakesh Kumar

PH.9350556294, 9312963297

Email – sracademy3311@yahoo.in

Sample Paper – 2011
Class – XII
Subject – Mathematics

SECTION-A

1. If A is a square matrix of order 3 such that

2. If A =

show that A -

is a skew symmetric matrix.

3. Write the value of

4. Find

if y=

5. At what point on the curve

,the tangents are parallel to y-axis.

6. Evaluate

dx (where

7. Write I.F. for

= (1+x)

8. Find the projection of the vector i-j to the vector i+j

9. If P(A) =

, P(B)=0, then find P

10. If X has a Binomial distribution B=

write P(X=1).
SECTION-B

11. Show that the relation R onR defined as R =

is reflexive and transitive but not symmetric. OR On R-

a binary operation * defined by a*b =a+b-ab. Prove that * is commutative and associative . find identity element for * on R-

12. Prove that

=2abc

13. For what value of K the following is continous at x=0.

f (x)=

OR If y=

(1-

)

14. Verify the applicability of LMV Theorem for the following

15. Determine the interval ,where f(x)=

, 0

, is strictly increasing or strictly decreasing.

16. Evaluate

17. Evaluate

18. Evaluate

dx OR Find

19. Solve

hat y

20. For any vectors

. OR if the vector

bisects the angle between the vector

and the vector 3i+4j ,then find the unit vector in the direction of

21. A speaks truth in 60% of cases and B in 90% of cases .In what percentage of cases are they likely to contradict to each other in stating the same fact ? . OR Two cards are drawn without replacement from a well- shuffled pack of 52 cards . find the probability that one is spade and other is a queen of red colour

22. Suppose X has a binomial distribution B

Show that X=3 is the most likely outcome. SECTION-C

23.

24. If A=

and B=

are two square matrices , find AB and hence solve the system of linear equation : x –y=3, 2x+ 3y +4z =17, y+2z =7. OR Obtain the inverse of the following matrix using elementary operation : A=

25. Show that the height of the cone of maximum volume that can be inscribed in a sphere of radius 12 cm is 16cm.

26. Find the area of the region

OR Using the integration , find the area of ∆PQR where P is

,Q is

and R is

27. Solve :

28. Find the image of the point having position vector

in the plane

29. A producer has 30 and 17 units of labour and capital respectively which he can use to produce two types of goods Xand Y .to produce one unit of X ,2 units of labour and 3 units of capital is required . similarly ,3 units of labour and 1 unit of capital is ree produquired to produce one unit of Y. If X and Y are priced at Rs 100 and Rs 120 per unit respectively ,how should the producer use his resources to maximize the total revenue ? solve the problem graphically.

29.Let X denote the number of hours you study during a randomly selected school days . the probability that X can take the values x ,has the following form ,where k is some unknown constant .

PAPER SUBMITTED BY:

Name : POONAM SINGH

Email : yashmragendra@gmail.com

Phone No. 07597453169

Sample Paper- 2011

Class- XII

Subject – Mathematics

SECTION – A

Q1. Three coins are tossed once. Find the probability of getting at least one head.

Q2. A problem in mathematics is given to three students A, B and C; their chances of solving it are

respectively. Find the probability that the problem will be solved.

Q3. Given that

= 0 and

= 0. What can you conclude about the vectors

and

Q4. Find the Integrating Factor of the differential equation:(1 – y²)

+ yx = ay, (– 1 < y < 1).

Q5. Write the value of

dx.

Q6. Write the value of

dx.

Q7. State whether the function given by f(x) = 3x + 17 is strictly increasing or strictly decreasing on R.

Q8. If AB = A and BA = B, then find A².

Q9. Give an example of two matrices A and B such that A

0, B

0, AB = 0 and BA

Q10. Let A be a non-singular square matrix of order 3 x 3, then write the relation between | adj A | and | A |.

SECTION - B

Q11. For a positive constant a find

, where

and x =

If y = (tan^{– 1} x)², show that (x² + 1)²y₂ + 2x (x² + 1)y₁ = 2.

Q12. Show that:

Obtain the inverse of the following matrix using elementary operations A =

Q13. Prove that:

(i) tan^–1

(ii)

Q14. Evaluate:

dx.

Q15. Evaluate:

Q16. Prove that:

Q17. Show that the differential equation (x – y)

= x + 2y is homogeneous and solve it.

Solve the differential equation (tan^–1 y – x) dy = (1 + y²) dx

Q18. Let

. Find a vector

which is perpendicular to both

and

= 15

Q19. A coin is tossed 4 times. Find the mean and variance of the probability distribution of the number of heads.

Q20. Suppose a girl throws a die. If she gets a 5 or 6, she tosses a coin three times and notes the number of heads. If she gets 1, 2, 3 or 4, she tosses a coin once and notes whether a head or tail is obtained. If she obtained exactly one head, what is the probability that she threw 1, 2, 3 or 4 with the die?

Q21. Using differentials, find the approximate value of (0.0037)^1/2.

Find the equation of all lines having slope – 1 that are tangents to the curve =

Q22. Examine the continuity of the function: f(x) =

at x = 0

SECTION – C

Q23. (a) Prove that the given function

is one-one and onto theoretical.

(b) Find the image of the point (1, 6, 3) in the line

Q24. (a) Let A = {-1, 0, 1, 2}, B= {-4,-2, 0, 2} and f,g : A

B be the functions defined by f(x) = x²-x, x

A and g(x) = 2

- 1 ,x

A. Are f and g equal? Justify your answer.

(b) Using matrix method, solve the following system of equations: x + 2y + z = 1

2x – y + z = 5

3x + y – z = 0

Q25. A dietician has to develop a special diet using two foods P and Q. Each packet (containing 30 g) of food P contains 12 units of calcium, 4 units of iron, 6 units of cholesterol and 6 units of vitamin A. Each packet of the same quantity of food Q contains 3 units of calcium, 20 units of iron, 4 units of cholesterol and 3 units of vitamin A. The diet requires at least 240 units of calcium, at least 460 units of iron and at most 300 units of cholesterol. How many packets of each food should be used to minimise the amount of vitamin A in the diet? What is the minimum amount of vitamin A?

Q26. A line makes angles α, β, γ and δ with the diagonals of a cube, prove that: cos² α + cos² β + cos² γ + cos² δ =

Q27. Find the shortest distance between the lines

and

Q28. Prove that the curves y² = 4x and x² = 4y divide the area of the square bounded by x = 0, x = 4,

y = 4 and y = 0 into three equal parts.

Find the area bounded by curves {(x, y): y ≥ x² and y = | x |}.

Q29. A point on the hypotenuse of a triangle is at distance a and b from the sides of the triangle. Show that the minimum length of the hypotenuse is

Paper Submitted by

Name: Amit Gupta

Email: competitionfocus11@yahoo.in

Phone No. 09815266446,094

Pages

Maths