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PADASALAI .NET PRE HALF YEARLY MODEL QUESTION PAPER 2018-2019

PART - III

MATHEMATICS

XII std

( ENGLISH VERSION)

TIME ALLOWED: 2.30 HRS

(2018-2019)

MAXIMUM MARKS: 90

INSTRUCTIONS : 1) Check the question paper for fairness of printing. If there is

any lack of fairness, inform the hall supervisor immaditely.

2 ) Use Blue or black link to write and pencil to draw diagrams

Part-A

NOTE :

1) Answer all the 20 questions

2) Choose the most suitable answer from the given four alternatives

and write the option code and the correspoinding answer.

Padasalai.20xN1=20 et 1) Every homogeneous system a) is always consistent b) has only trivial solution

c) has infinitely many solutions 2) The projection of i j on z ?axis is a) 0 3) The focus of the parabola y 2 8x is

d) need not be consistent

b) 1 c) -1

d) 2

a) (0,-2)

b) (0,2)

x2 y2 4) The centre of the ellipse 1 is

49

c) (-2,0)

d) (2,0)

a) (0,3)

b) (2,3)

c) (0,0)

d) (3,0)

5) The point of intersection of tangents at ' t1 ' and ' t 2 ' to the parabola y 2 4ax is

a) a t1 t2 , at1t2

b) at1t2 , a t1 t2 c) at 2 , 2at d) at1t2 , a t1 t2

6) f is a twice differentiable function on an interval I and if f " ( x ) > 0 for all x in the domain I of f then f is

a) concave upward

b) convex upward

c) increasing

d) decreasing







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7) The curve y 2 x 2 1 x is defined only for

a) x 1

b) x 1

c) x 1

d) x 1

8) If n is a positive integer then x n e ax dx

0

n! a)

an

n 1 ! b)

an

n1 ! c)

a n1

n ! d)

a n1

9) The arc length of the curve y = f (x ) from x = a to x = b is

b

a)

a

1 dy 2

dx

b)

d

dx

c

2

1

dx dy

dx

Identify the incorrect statement 10)

b

c) 2 y

a

1

dy

2

dx

d)

2

b

y

dx

a

2

1

dx dy

dx

a) The order of a differential equation is the order of the highest derivative occurring in it.

b) The degree of the differential equation is the degree of the highest order derivative which occurs in it (the

derivatives

are free from radicals and fractions)

dy c) dx

f1 x , y f2 x , y

is the first order first degree homogeneous differential equation

d) dy xy e x is a linear differential equation in x. dx

11) Which of the following are not statements ?

i. All natural numbers are integers.

ii. A square has five sides

iii. The sky is blue

iv. How are you ?

a) (iv) only

b) (i) and (ii)

c) (i) (ii) and (iii)

d) (iii) and (iv)

12. If p is true and q is false then which of the following statements is not true ?

a) p q is false

b) p q is true

c) p q is false

d) p q is true

13. Which of the following is not true? i) Negation of a negation of a statement is the statement itself ii) If the last column of its truth table contain only T then it is tautology iii) If the last column of its truth table contains only F then it is contradiction







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iv) If p and q are any two statements then p q is a tautology

14. Which of the following are correct?

i) EaX b aE X b

iii) 2 variance

ii) 2 2 ' 1'2 iv) var aX b a 2 var X

a)all

b) (i) , (ii) , (iii)

c) (ii) , (iii)

d) (i) , (iv)

15. Which of the following is not true regarding the normal distribution?

a) skewness is zero

c) the Points of inflection are at X

16. If u x , y x 4 y 3 3x 2 y 2 3x 2 y then u is

y

b) Mean = median = mode

1 d) maximum height of the curve is

2

a) 3y 2 6xy 3x 2

b) 3y 2 6xy 2 3x 2

c) 3y 2 6x 2 y 3x 2

d) 3y 2 6x 2 y 2 3x 2

17. The equation of the latus rectum of y 2 8x is

a) y-2=0

b) y+2 =0

c) x -2 = 0

d) x+2 =0

18. The length of the latus rectum y 2 8x is

a) 8

b) 6

c) 4

d) -8

19. The values of Z Z is

a) 2 Re( Z )

b) Re( Z )

20) If Z Z1 = Z Z 2 then the locus of Z is

c) Im( Z )

d) 2 Im( Z )

a) a circle with centre at the origin b) a circle with centre at Z 1 c) a straight line passing through the origin d) is a perpendicular bisector of the line joining Z1 andz2







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Part -B

7 x 2 =14

NOTE:: (i) Answer 7 questions.

(ii) Q.no 30 is compulsory .select any 6 question from first 9question.

21) The area of the region bounded by the curve xy 1, x - axis, x 1 and x . Find the volume of the solid

generated by revolving the area mentioned about x - axis.

1

22)

Solve:

dy dx

1 1

y2 x2

2

0

1

23) Solve:

2D 2

5D 2

ye

2

x

24) Show that the set of all non-zero complex numbers is an abelian group under the usual multiplication of complex numbers.

25) State and prove reversal law on inverse of a group. 26) The probability of success of an event is p and that of failure is q. Find the expected number of trials to get a first success.

27) Find the equation of the asymptotes to the hyperbola 8x 2 10 xy 3y 2 2x 4 y 2 0

28) Find the magnitude and direction cosines of the moment about the point (1,-2,3) of a force 2 i 3 j 6 k whose line of action passes through the origin

29) State and prove the triangle inequality of complex numbers.

30) For any two complex numbers Z1 , Z 2 , show that

(i) Z1 Z 2 Z1 Z 2

(ii) arg Z1 Z 2 arg Z1 arg Z 2







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Part ? c

Note:: (i) Answer any 7 question

7x3=21

(ii) Q.no 40 is compulsory. select any 6 question from first 9 question.

31)

Solve by matrix inversion method x y 3 , 2x 3y 8

32) Prove that a b . c d b c . a d c a . b d 0

33) Find the vector and Cartesian equation of the line through the point (3,-4,2) and parallel to the vector 9 i 6 j 2 k.

34) Solve: (i) x 4 4 0

35) Find the equations of the tangents(i) to the parabola y 2 6x , parallel to 3x 2 y 5 0

36) Verify Lagrange's law of mean for the following functions : f x1 x 2 , 0,3

37) Prove that the inequality 1 xn 1nx is true whenever x 0 and n1.

38) Find

w u

and w if w sin 1 xy v

where x u v , y u v.

39) Find the area of the region bounded by the line y 2x 1, y 3 , y 5 and y ? axis

40) Find the differential equation of the family of straight lines y mx a when (i) m is the parameter ; (ii) a is m

the parameter ; (iii) a , m both are parameters.

Part ? D

7 x 5 = 35

NOTE :: Answer both the question choosing either of the alternatives.

41) a)

If P represents the variable complex number z. Find the locus of P, if

arg

z

1

z3 2

A satellite is traveling around the earth in an elliptical orbit having the earth at a focus and of eccentricity 1/2. Or) b) The shortest distance that the satellite gets to the earth is 400 kms. Find the longest distance that the satellite gets from the earth.







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42) a) On lighting a rocket cracker it gets projected in a parabolic path and reaches a maximum height of 4mts when it is 6 mts away from the point of projection. Finally it reaches the ground 12 mts away from the starting point. Find the angle of projection (or) b) A man is at a point P on a bank of a straight river, 3 km wide, and wants to reach point Q, 8 km downstream on the opposite bank, as quickly as possible. He could row his boat directly across the river to point R and then run to Q, or he could row directly to Q, or he could row to some point to between Q and R and then run to Q. If he can row at 6 km/h and run at 8 km/h where should he land to reach Q as soon as possible?

43) a) Find the cubic polynomial in x which attains its maximum value 4 and minimum value 0 at x = -1 and 1

respectively. (or) b) Solve : 2 xy x dy y dx 0

44) a) Let G be the set of all rational numbers except 1 and * be defined on G by a * b a b ab for all a , b G. Show that ( G , * ) is an infinite abelian group. (or) For what value of the equations x y 3z 0 , 4x 3y z 0 , 2x y 2z 0 have a (i) trivial solution, (ii) non-trivial solution.

45)a) Find the vector and Cartesian equation to the plane through the point (-1,3,2) and perpendicular to the planes x 2 y 2z 5 and 3x y 2z 8. (or) b) The probability density function of a random variable X is

f x

k x 1

e x

0

, x, , 0 , elsewhere

Find (i) k (ii) P ( X > 10 )

46) The mean weight of 500 male students in a certain college in 151 pounds and the standard deviation is 15 pounds. Assuming the weights are normally distributed, find how many students weigh (i) between 120 and 155 pounds (ii) more than 185 pounds. (or) b) If the curve y 2 x and xy k are orthogonal then prove that 8 k 2 1.

47) a) If w u 2 ev where

x u

and

v y log x , find

w

and

w

Find the length of the curve

y

x

y (or) b)

4 y 2 x 3 between x 0 and x 1

******** ALL THE BEST ********

PREPARED BY

S. BALAKRISHNAN , M.SC, B.ED , (PG ASST IN MATHS)

SHRI MARUTHI HIGHER SECONDARY SCHOOL

V. MADEPALLI ?X-ROAD , VEPPANAPALLI

KRISHNAGIRI ?(DT) -635121



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