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Bangalore University 2010-2nd Year M.C.A Entrance -Year -Maths-Practice -5 - Question Paper

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TEST PAPER 5

Total Questions: 60 Time allotted 75 minutes

Q. 1) A person at the top of a hill observes that the angles of depression of two consecutive kilometer stones on a road leading to the foot of hill are 30⁰ and 60⁰. The height of the hill is

1	(b) *

km	<d>

km

^(c)

4 2 cos^-1 + tan^-1

5 3

is

(b)

(d)

7

16

17

6

Q.2) The value of tan

(a) 17

. N 16

^(c) 7

Q.3) If for a triangle ABC,

1 + cos A + cos B + cos C = 0 , then the triangle must be

(a) equilateral (b) isosceles

(c) right-angled (d) obtuse-angled

Q.4) The lines x + y = 1,8x - 3 y = and y = 0 are

(a) sides of an isosceles triangle (b) concurrent

(c) parallel (d) sides of an equilateral triangle

Q.5) The line 3x + y - = 0 cuts the x-axis at a andy-axis at B. The incentre of the triangle OAB, where

O is the origin, is at

(a) (, ) (b) (3, 3)

(c) (, 3) (d) (3, )

Q.6) The centre of the circle passing through the points (0, 0), (1, 0) and touching the circle x² + y² = 9 is

(b) If

(a) If!

1, -V

(c) I -

(d)

1 1

"V^,

Q.7) Match List I (Equation of circles) with List II (Their centres) and select the correct answer using the

codes given below the lists: List I (Equation of circles)		List II (Their centres)
^A. (x + ) +(y +1) = 1	1.	(1, -)
^B. (x -1) +(y + ) = 1	.	(-1, )
^C. (x +1) +(y - ) =	3.	(, 1)
^D. (x - ) +(y -1) =	.	(-, -1)

Codes:

A B C D

(a) 4 1 2 3

(b) 4 2 1 3

(c) 3 12 4

(d) 3 2 1 4

The eccentricity of hyperbola 25x2 - 9y2 = 144 is >/34 V34

Q.8)

^(a) 12- ^(b) T"

(c) (d) 3

2 2 x y

The distance from the major axis of any point on the ellipse +- = 1 and its corresponding point

Q.9)

a b

on the auxiliary circle are in the ratio

Q.10)

/ \ a ^(a) b	(b) ^b a
_{( )} a² ^(c) V	(d) a
_x² The distance of a point on the ellipse--+
point is
(a) n	(b) n
(c) f	(d) n

2

= 1 from the centre is 2. The eccentric angle of the 4

The focus of the curve y² + 4 x - 6 y +13 = 0 is (a) (2, 3) (b) (-2, 3)

Q.11)

Q.12)

Q.13)

(c) (2, -3) (d) (-2, -3)

If a line makes angles of 60⁰ and 45⁰ with the positive direction of the axes of x and y respectively,

then the angle made by the line with positive direction of the z-axis is equal to

(a) 60⁰ (b) 120⁰

(c) either 60⁰ or 120⁰ (d) neither 60⁰ nor 120⁰

Codes:

Match List I (Equation of curves) with List II (Their types) and select the correct answer using the codes given below the lists:

	List I (Equation of curves)		List II (Their types)
A.	x² + x +1 = y	1.	Circle
B.	x + y¹ + 2 x + 2 y - 6 = 0	2.	Parabola
C.	2 x² + 3 y² + 4 x + 6 y = 0	3.	Ellipse
D.	3x² - 2y² + 6x - 4y = 0	4.	Hyperbola
		5.	Pair of straight lines

A B C D

(a) 2 3 1 4

(b) 4 3 1 5

(c) 2 1 3 4

(d) 4 1 3 5

The direction cosines of the line perpendicular to the line, with direction ratios (1, -2, -2) and (0, 2, 1)

are

(b) I- HI

2 1

3^,3^;

2 _ 1 _ 2

3^, 3^, 3

(d)

(c)

The length of the normal from the origin to the plane x + 2 y _ 2 x = 9 is equal to

Q15)

Q.16)

Q.17)

Q.18)

Q.19)

Codes:

(a) 2 (b) 3

(c) 4 (d) 5

If the angle between the lines joining the end points of minor axis of an ellipse with its foci is -j ,then the eccentricity of the ellipse is

^{(b) Ti}

1

(d)

(c)

(2V2)

2

The lines 2x = 3y = _z and 6x = _y = _4z

(a) area parallel (b) are perpendicular

(c) interest at an angle of 45⁰ (d) interest at an angle of 60⁰

The distance between the parallel planes 4 x _ 2 y + 4 z + 9 = 0 and 8x _ 4 y + 8 z + 21 = 0 is

^(a) 4 ^(c) i

Match List I (Equations of spheres) with List II (Their centres) and select the correct answer using the codes given below the lists:

List II (Their centres)

List I (Equation of spheres)

A.	x² + y²	+ z²	+ 3x _ 3y + 3z _ 49 = 0	1.	(2, 3, 1)
B.	x² + y²	+ z²	_ 4 x _ 6 y _ 2 z + 9 = 0	2.	(33
					¹ 2^, 2^,
C.	2 x² + 2 y² + 2 z² _ x _ y _ z = 0			3.	(1, 1, 1)
D.	2 2 x + y	+ z²	0 = z 2 _ ^y 2 _ x 2 _	4.	( Hi
					1444.
	A B	C	D
(a)	2 4	1	3
(b)	3 1	4	2
(c)	2 1	4	3
(d)	3 4	1	2

The radius of the sphere 3x² + 3y² + 3z² - 8x + 4y -15 = 0 is equal to (a) 2 (b) 3

(c) 4 (d) 5

The condition for the lines x = az + b, y = cz + d ;and x = a₁ z + b₁ y = c₁ z + d₁ to be perpendicular is

Q.21)

Q22)

Q.23)

(a) ac₁ + a₁c +1 = 0 (b) aa₁ + cc₁ +1 = 0

(c) ac_l + ajc -1 = 0 (d) aa_l + cc_l -1 = 0

If n forcesPA_l,PA₂,...,PA_n diverge from point P and n other forces A₁Q, A₂Q,...,A_nQ converge to a point Q, then the resultant of the 2n forces is represented, in magnitude and direction by (a) nPQ (b) nQP

(c) 2nPQ (d) n² PQ

If (a -

(a - bj, (a + bj = 0, then

(a) a and b are perpendicular (b) a and b are parallel (c) |a| = |b| (d) a = 2b

Q.24)

Q.25)

If A + tB is perpendicular to C, where A = i + 2 j + 3k, B = -i + 2 j + k and C = 3i + j then t is equal to

(a) 4 (b) 5

(c) 6 (d) 7

If a = a_li + a₂j + a₃k where a_l,a₂,a₃ are scalar quantities and i,j,k, unit vectors in three mutually

perpendicular directions, then|a x i | + |a x k-| is equal to

(a) a + a (b) a + a₂ + a3

(c) 2af + a + a₃² (d) af + 2a + a₃²

The magnitude of the projection of the vector A = i - 2 j + k on the vector B = 4i - 4 j + 7k lies between

Q.26)

Q.27)

Q.28)

(a) 1 and 2 (b) 2 and 3

(c) 3 and 4 (d) 4 and 5

The moment about the point A(3, -1, 3) of a force F = 2i + j + 4k through the point B (5, 1, 4) is

(a) 3i + 2j - k (b) 7i - 6j - 2k

(c) 5i + j + 3k (d) i + 2 j - k

If |a x b| + |a.b| = 144 and|a| = 4, thenis equal to

(a) 3 (b) 8

(c) 12 (d) 16

If the work done by a force i + j + 8k along a given vector in the xy-plane is 8 units and the magnitude of the given vector is 473, then the given vector is represented as (a) (4 + 21) +(4 - 22 )j (b) (4i + W2j)

(d) (4 + 2V2 )(i + j)

Consider the following statements:

1. 5^th decile and 50^th percentile are the same.

2. 2^nd quartile and 50^th decile are the same.

3. 2^nd quartile and 50^th percentile are the same.

4. 2^nd quartile, 50^th decile and 50^th percentile are the same.

Which of these is/are correct?

(a) Only 1 (b) 2 and 3

(c) 1, 2 and 3 (d) 1, 2, 3 and 4

The figure formed by joining the mid-points of the upper horizontal sides of each rectangle of a histogram is called

Q.31)

Q32)

Q33)

Q34)

Q.35)

Q.36)

Q.37)

(a) frequency curve (b) frequency polygon

(c) more than ogive (d) less than ogive

Average age of a teacher and three students is 20 years. If all the three boys are of same age and the difference between the ages of a boy and the teacher is 20 years, then the age of the teacher is equal to (a) 25 years (b) 30 years

(c) 35 years (d) 45 years

In an examination, the standard of passing was 40%. Out of 9 students who appeared, 4 failed and the remaining got 80%, 57%, 51%, 68% and 79% marks. The median of the percentage marks is equal to (a) 51% (b) 57%

(c) 68% (d) 79%

The median of 19 observations is 30. Two more observations are made and their values are 8 and 32. The median of the 21 observations taken together is equal to (a) 28 (b) 30

(c) 32 (d) 34

If X follows binomial distribution with mean 3 and variance 2, then P (X > 8) is equal to

(a) 19 (b) 19

^(c) F ^(d) F

The most probable number of heads in 80 tosses and a biased coin, given that the probability of a head . 3 .

in a single toss is 5, is

(a) 48 (b) 32

(c) 16 (d) 12

In tossing a coin twice, let E and F denote occurrence of head on first toss and second toss respectively, P (E u F) is equal to

<^a> -4 <> 2

31 <0 4 i) 3

The probability of having a king and a queen when the two cards are drawn at random from a pact of 52 cards is

16

663

(d)

663

(c)

663

Q.39) A card is drawn from an ordinary pack and a gambler bets that it is either a spade or an ace. The odds against his winning are (a) 9 : 4 (b) 9 : 5

(c) 9 : 6 (d) 9 : 8

Q.40) Thirteen cards are drawn simultaneously from a deck of 52. If aces count 1, face cards 10 and others according to denomination, then expectation of the total score on 13 cards is

(b)

13

(c) 13

Q.41) The probability that a leap year, selected at random, will contain 53 Sundays is

(b)

^(c) i

Directions: The following 4 (four) items consists of two statements, one labeled as Assertion () and the other labeled as Reason (R). You are to examine these two statements carefully and decide if the Assertion (A) and the Reason (R) are individually true if so, whether the Reason R is the correct explanation for the given Assertion A. Select you answers to these items using the codes given below and mark your answer sheet accordingly.

Codes:

(a) Both A and R are individually true and R is the correct explanation of A.

(b) Both A and R are individually true and R is not the correct explanation of A.

(c) A is true but R is false.

(d) A is false and R is true.

The vector product of a Force F and displacement r is equal to the work done. Work done is not a vector.

If lim f (x) = l,limg (x) = m, then lim {f (x) g (x)} = lm.

tan x , lim-= 1

^x .^a x

Q.44) Assertion (A) : f (x) = x

10, x = 0

2 / \ om v- ~t~ \j

Reason (R) : Both h(x) = x and g (x) = <! x are continuous at x = 0

(a) 85

(d) 4

(a)

Q.42) Assertion (A) Reason (R) :

Q.43) Assertion (A) Reason (R) :

is continuous at x = 0

| x | is not continuous at x = 0.

Q.45)

Assertion (A)

The domain of the function y/x +1 is

Vvx

V3-1

(b) 0 < x < (d) 0 < x <

(a) 0 < x <

(c) {¹ "I < x < 0

The inverse of the function y =¹⁰¹⁰ is equal

Q.47)

10^x +10

^(b) :2^log₁₀ (² x -¹) 1 + x

^(d) T^log1

4 1 - x

(a) ^log10 (² - x)

r 2 x

^(c) T^log10 4 2 - x

Q48)

(b) 0 _ (d) e

1/3

lim | ^{tan x} I is equal to

x

(a) 1 (c) e

a x

If lim-= -1, then a is equal to

Q49)

x^a - a^a

(b) 0 (d) 2

(a) -1 (c) 1

Iff(x) = |x - 3| + |x - 4|, then in the interval [0, 5], the functionfx) is

Q.50)

Q.51)

(a) differentiable at x = 3

(b) differentiable at x = 4

(c) not differentiable at x = 3 and x = 4

(d) not continuous in the interval [0, 5]

If y = 3x² + 2 and if x changes from 10 to 10.1, then the approximate change in y will be (a) 4 (b) 5

(c) 6 (d) 8

V1 - x² -1

and z = tan ¹| ^2x |, then is equal to

1 - x² J dz

<> 2 ^(d) 8

Q.52)

If y = tan

(a) 1

^(c) 4

+ log_e V-x² , then I ^d^y I is equal to

I ^dx J x=0 (b) 1

x sin x

Q.53)

If y =

VT-

(d)

Q.5) A flower-bed in the form of a circular sector has been fenced by a wire of 0 m length. If the flowerbed has the greatest possible surface area, then the radius of the circle is

(a) 5 m (b) 0 m

(c) 10 m (d) 5 m

Q.55) J sin³ xcos xdx is equal to

(b) sin x + c

sinx (d) + c

sin x

(c) + c

Q.56) The area of the region bounded by the parabola y = ax and its latus rectum is

	, _x 8a²	_x a²
	(a)	(b)
	3	3
	, _x a²	a²
	(c)	(d)
	3	3
Q.57)	j entail)dx is equal to
	(a) log tan x + c	(b) log sec x + c
	(c) tan x + c	(d) e^tanx + c
Q.58)	J a^xdx is equal to a^x+¹
	(a)-- + c x +1	(b) a^x log x + c a^x
	(c) a^x log a + c n /	(d) --+ c log a
Q.59)	If A_n = J tanⁿ xdx(n > ), than A_n + A_n- is equal to n
	, 1 1	1
	^(a) +	^(b) i
	n n -1	n +1
	(c) -	(d)
	n	n-1
Q.60)	1 The value of J x\|x\| dx is equal to
	-1 (a) 0	(b)
	(c) -	(d)

ANSWER KEYS

1.	(a)	13.	(c)	25.	(d)	37.	(c)	49.	(a)
2.	(d)	14.	(a)	26.	(b)	38.	(b)	50.	(d)
3.	(c)	15.	(b)	27.	(b)	39.	(a)	51.	(c)
4.	(b)	16.	(b)	28.	(a)	40.	(a)	52.	(c)
5.	(a)	17.	(b)	29.	(a)	41.	(b)	53.	(b)
6.	(d)	18.	(a)	30.	(d)	42.	(b)	54.	(c)
7.	(a)	19.	(c)	31.	(b)	43.	(b)	55.	(c)
8.	(b)	20.	*	32.	(c)	44.	(d)	56.	(a)
9.	(b)	21.	(b)	33.	(a)	45.	(b)	57.	(b)
10.	(b)	22.	(a)	34.	(b)	46.	(a)	58.	(d)
11.	(b)	23.	(c)	35.	(c)	47.	(d)	59.	(d)
12.	(c)	24.	(b)	36.	(a)	48.	(d)	60.	(a)

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1.	(a)	13.	(c)	25.	(d)	37.	(c)	49.	(a)
2.	(d)	14.	(a)	26.	(b)	38.	(b)	50.	(d)
3.	(c)	15.	(b)	27.	(b)	39.	(a)	51.	(c)
4.	(b)	16.	(b)	28.	(a)	40.	(a)	52.	(c)
5.	(a)	17.	(b)	29.	(a)	41.	(b)	53.	(b)
6.	(d)	18.	(a)	30.	(d)	42.	(b)	54.	(c)
7.	(a)	19.	(c)	31.	(b)	43.	(b)	55.	(c)
8.	(b)	20.	*	32.	(c)	44.	(d)	56.	(a)
9.	(b)	21.	(b)	33.	(a)	45.	(b)	57.	(b)
10.	(b)	22.	(a)	34.	(b)	46.	(a)	58.	(d)
11.	(b)	23.	(c)	35.	(c)	47.	(d)	59.	(d)
12.	(c)	24.	(b)	36.	(a)	48.	(d)	60.	(a)

1.	(a)	13.	(c)	25.	(d)	37.	(c)	49.	(a)
2.	(d)	14.	(a)	26.	(b)	38.	(b)	50.	(d)
3.	(c)	15.	(b)	27.	(b)	39.	(a)	51.	(c)
4.	(b)	16.	(b)	28.	(a)	40.	(a)	52.	(c)
5.	(a)	17.	(b)	29.	(a)	41.	(b)	53.	(b)
6.	(d)	18.	(a)	30.	(d)	42.	(b)	54.	(c)
7.	(a)	19.	(c)	31.	(b)	43.	(b)	55.	(c)
8.	(b)	20.	*	32.	(c)	44.	(d)	56.	(a)
9.	(b)	21.	(b)	33.	(a)	45.	(b)	57.	(b)
10.	(b)	22.	(a)	34.	(b)	46.	(a)	58.	(d)
11.	(b)	23.	(c)	35.	(c)	47.	(d)	59.	(d)
12.	(c)	24.	(b)	36.	(a)	48.	(d)	60.	(a)

1.	(a)	13.	(c)	25.	(d)	37.	(c)	49.	(a)
2.	(d)	14.	(a)	26.	(b)	38.	(b)	50.	(d)
3.	(c)	15.	(b)	27.	(b)	39.	(a)	51.	(c)
4.	(b)	16.	(b)	28.	(a)	40.	(a)	52.	(c)
5.	(a)	17.	(b)	29.	(a)	41.	(b)	53.	(b)
6.	(d)	18.	(a)	30.	(d)	42.	(b)	54.	(c)
7.	(a)	19.	(c)	31.	(b)	43.	(b)	55.	(c)
8.	(b)	20.	*	32.	(c)	44.	(d)	56.	(a)
9.	(b)	21.	(b)	33.	(a)	45.	(b)	57.	(b)
10.	(b)	22.	(a)	34.	(b)	46.	(a)	58.	(d)
11.	(b)	23.	(c)	35.	(c)	47.	(d)	59.	(d)
12.	(c)	24.	(b)	36.	(a)	48.	(d)	60.	(a)