Test Paper 1 12th

Guess Paper – 2010 Subject – Maths Class – XII neral Instructions: General Instructions: The question paper consists of three sections A, B and C. (i) Question numbers 1 to 10 in section A are of 1 mark each. (ii) Question numbers 11 to 22 in section B are of 4 marks each. (iii) Question numbers 23 to 29 in section C are of 6 marks each. (iv) All questions are compulsory. (v) No Internal choices have been provided (vi) Use of calculators is not permitted. However, you may ask for logarithmic and statistical tables, if required.

SECTION A 1.

Find the value of cos (sec–1 x + cosec–1 x), | x | ≥ 1

2.

If A =  1

3.

2

3 1 0 2  and I = 0 1 Find x and y Such that A = xA + yI. 2      cos θ sin θ  sin θ − cos θ Simplify : cos θ   + sin θ cos θ sin θ  − sin θ cos θ    

4.

3 Find values of x for which the matrix is Singular  x 

5.

 If Sin   sin 

−1

1 +cos 5

−1

 x  =1 , then 

find

the values

x  1 

of x

6. 7. 8.

If A is a square matrix of order 3 and detA = 5, find the value of det(AdjA). 2 Find the absolute maximum value of f ( x) = x − 4 x +1, [ 0,3] 2 2 Find the point on the curve x + y = 1 , at which the tangent is parallel to Xaxis .

9.

−1 If y = sec 

10.

 

 x +1   + sin −1    x −1  

x −1   , then find dy/dx. x +1  

Check the monotonocity i.e increasing & decreasing of f ( x) = cos 2 x, [π / 2, π ] .

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Pr ove that

11.

tan

−1

 1 + x 2 + 1−x 2  2 2   1+x − 1−x

 π 1  = + cos 4 2  

−1

x2

12.

a − cb + bc 222 P o t rh+vca b ea − ac ( t ++= ) acba b(++ c ) − ba + ab c

13.

1: x ≤ 3   If the function f (x ) = ax + b;3 < x < 5 is continuous at x = 3 and x = 5, then find  7; x ≥ 5 

the

14. 15. 16.

value of a & b

If x m y n =( x + y ) m +n prove that

d2 y dy −x −2 = 0 2 dx dx −1  x − 1  −1  2x − 1  −1  33   + tan   = tan  . Solve for x : tan   x + 1  2x + 1   36  2 If y = (sin-1x)2 , show that (1 − x )

3

17.

dy y = dx x

Evaluate :

∫ (x

2

)

+ 5 x + 1 dx as a limit of a sum

1

18.

19.

Find the equation of tangent to the curve y = 3 x −2 which is parallel to the line 4x – 2y + 5 = 0. α  − tan   0 2 and I is the identity matrix of order 2, show that If A =  α   tan 0   2   cos α

20.

− sin α 

(I +A ) = (I – A)  sin α cos α  .   Water is dripping out from a conical funnel, at a uniform rate of 2cm3/sec through a tiny hole at the vertex at the bottom, when the slant height of the water is 4cm , find the rate of decrease of the slant height of the water, given that the vertical angle of funnel is 1200.

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21.

Find the intervals in which f ( x ) = cos(2 x + π 4 ) is strictly increasing or decreasing function on (0, π)

22.

Verify Rolle’s theorem for the function

[ 2]

f ( x) = sin x + cosx, 0, π

SECTION C

23.

 a −b

θ

b + acos θ

  −1 cos −1   Prove that tan  a + b tan 2  = a + bcos θ 2     24.

1  Find A −1 , where A = 2 3

equations

2 3 −3

− 3 2  . Hence, Solve the system of linear − 4

x + 2y - 3z = -4, 2x + 3y + 2z = 2, and (y + z) 2

25.

1

Show that

xy

xy (x + z)

xz

3x - 3y - 4z = 11

zx 2

yz

yz

= 2xyz (x+y+z)3

(z + y) 2

26.

If the lengths of three sides of trapezium other than base are equal to 10 cm, then find the area of the trapezium when it is maximum.

27.

Prove that the volume of the largest cone that can be inscribed in a sphere of radius R is 8/27 of the volume of the sphere. Make a rough sketch of the region given below and find the area using the method of integration : ( x, y );0 ≤ y ≤ x 2 + 3,0 ≤ y ≤ 2x + 3,0 ≤ x ≤ 3 Sketch the region common to the circle x2 + y2 = 16 and the parabola x2 = 6y. Also find the area of the region using integration

28.

29.

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Paper Submitted by: Pankaj Email : [email protected] Mob No. : 9810217980

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