8/11/2019 Class 12 Cbse Maths Sample Paper Model 5

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MATHEMATICS

SAMPLE TEST PAPER

CLASS XII

Class:12 Max Mks:100

Time 3hrs No of pages: 4

Ò General Instructions:

Ò All questions are compulsory.

Ò The question paper consists of 29 questions divided into three sections - A, B and C.

Ò Section - A comprises of 10 questions of one mark each, Section B is of 12 questions of four

marks each, Section C comprises of 7 questions of six marks each.

Ò Internal choice has been provided in four marks question and six marks question. You have to

attempt any one of the alternatives in all such questions

Ò use of calculator not permitted..

SECTION A

Question number 1 to 10 carry 1 mark each

1. Show that the functions f:R→R given by f(x) = x3

is injective.

2. Find the value of sin-1x = y when 0<y<π

3. Find the value of x,y an d z in from the following equations.

4. Find the derivative with respect to x sec(tan√x)

5. The radius of an air bubble increasing at the rate of 5cm/s. At what rate is the volume of the

 bubble increasing when the radius is 2cm.

6.Find the area between the curve y=x and y = x

2

7. Find the direction cosine of the ides of the triangle whose vertices are (3,5,-4) ,(-1,1,2) and (-5,-

5,-2)

8. Evaluate sin-1

(sin1000)+cos

-1(cos100

0)

9. f(x) = e2log(sinx) then find f' (π/4)

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10. Find the amplitude of the number (1+cos Ө+isin Ө)

SECTION B

Question numbers 11 to 22 carry 4 marks each.

11. show that the function f:R→{xέ R:-1<X<1} defined by f(x) = x

(1+ ∣ x∣) , x belongs to R is one

and onto function

12. Simplify

13. Write the Minors and Co factors of the determinant

or 

Given A , show that A2-5A+7I = 0. Hence find A-1

14. Without eliminating the parameter finddydx x = cos θ -cos2θ , y= sin θ -sin2θ

or 

x = 5at3, y = 2at

2+6t

15. Find the equation of the tangent and normal to the hyperbola x

2

a2 -

 y2

b2 =1 at the point (x0,y0)

16. Evaluate: limit 0-π/2  √ sinx

 sinx+   cosxdx

or 

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Evaluate: limit a-0  √ x

√ x+√a− xdx

17. Find the area enclosed between the parabola y2

= 4ax and the line y = mx

18. Find the differential solution of the differential equationdy

dx = x+  1

2− ,y# 2

or 

Find the general solution of:dy

dx =(1+x)2

(1+y)2

19. Two adjacent sides of a parallelogram are 2  ⃗ i -4  ⃗  j +4  ⃗k  and ⃗ i -2  ⃗ j -3  ⃗k  , find the unit vector

 parallel to its diagonal. Also find its area.

20. Find the shortest distance between the lines whose vector equation are ⃗r = (1-t)  ⃗ i +(t-2) ⃗ j +(3-

2t)  ⃗k  and re ⃗r = (s+1)  ⃗ i +(2s-1)  ⃗ j -(2s+1)  ⃗k 

21. A company manufactures two types of novelly souvenirs made of plywood. Souvenirs of type

A require 5 mint each for cutting and 10 mint each for assembling. Souvenirs of type B

requires 8 mint each for cutting 8 mint and 8 mint for assembling. The profit is Rs 5 each for

type A and Rs 6 each for type B souvenirs. How many souvenirs of each type should the

company manufactures in order to maximize the profit ?

22. A and b throw a die alternately till one of them gets a '6' and wins the game . Find their

respective probability of winning , if A starts first.

SECTION C

Question numbers 23 to 29 carry 6 marks each.

23. Write the function in simplest form tan-11+ x

2

− 1

 x, x # 0

or 

Write the function in simplest form tan-11− cosx

1+ cosx ,x<π

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24. If A = , prove that A3-6A

2+7A+2I = 0

25. Solve system of linear equations, using matrix method

2x+3y+3z = 5

x-2y+z = - 4

3x-y-2z = 3

26. Differentiate with respect to x tan-1

  sinx

1+ cosx

27. Find (   cotx+   tanx dx)

28. Find the equation of the plane through the line of intersection of the planes x+y+z = 1 and

2x+3y+4z = 5 which is perpendicular to the plane x-y+z = 0.

29. A dietician wishes to mix together two kinds of food X and Y in such a way that the mixture

contains at least 10 units of vitamins A,12 units of vitamin B and 8 units of vitamin C. The

vitamin content of 1 kg food is below. 1 kg of food X costs Rs 16 and 1 kg of food Y costs Rs

20.Find the least cost of mixture which will produce the required diet?