KEE Model Question Paper

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1

MATHEMATICS (Common to all candidates) 1. If I is the unit matrix of order n, where k 0 is a constant, then adj(KI) =____________. a) Kn(adj I)2x The equation x + 2

b) Kn-1(adj I)0 x +1 0 02

c) K2(adj I)

d) K (adj I)

2.

= 0 has the solution

x + 3 x + 4 x +1

a) x = -1, -2, -3 c) x = -2, -3, -4 3.

b) x = 0, -1, i d) x = 0, 0, 0

If (A) = (A, B) then the system is a) b) c) d) consistent and has infinitely many solution consistent and has a unique solution. consistent inconsistentcos

12

+ i sin 0 0

12 cos

20 0

4.

The value of

6

+ i sin 0

6 cos

0

is ____________.8

8

+ i sin

a) 5.

1 i 2

b)

1+ i 2

c)

1+ i 2

d)

1 i 2

If A is a square matrix then AA + AA is a a) unit matrix c) symmetric matrix b) null matrix d) skew symmetric matrix

6.

If a and b are unit vectors having opposite directions, which one of the following is true? a) a . b = 1 b) a . b = 0 c) a x b = 0 d) | a | | b | = 2

2

7.

If a and b are two unit vectors and is the angle between them, then ( a - b ) is a unit vector if a) =

4

b) =

2

c) =

3

d) =

2 3

8.

The angle between the planes x + y + z = 10 and z-axis is ________.

2 a) sin-1 3 9.

1 b) sin-1 3

c) sin-1(2)

d) sin-1( 3 )

If a is any vector, the value of | a x i |2 + | a x j |2 + | a x k |2 is ________. a) a2 b) 2a2 c) 3a2 d) 0

10.

If |z z1| = |z z2| then the locus of z is a) b) c) d) a circle with centre at the origin a circle with centre at z1 a straight line passing through the origin is a perpendicular bisector of the line joining z1 and z2. 1+ x = cos 2 + i sin 2 , then x is equal to 1 x b) i tan 2 c) i cot d) i cot 2

11.

If

a) i tan

12.

Which of the following is incorrect? a) |z1 + z2| |z1| + |z2| c) | z1 - z2| |z1| + |z2| b) |z1 + z2| |z1| + |z2| d) | z1 - z2| |z1| - |z2| 2 2 + i sin then n n

13.

If n is a positive integer greater than one and a = cos 1 + a + a2 + + an-1 = _____________. a) 0 b) 1 c) -1

d) n

3

14.

The point of contact of the tangent y = mx + c and the parabola y2 = 4ax is

a 2a a) 2 , m m 15.

2a a b) 2 , m m

a 2a c) , 2 m m

a 2a d) 2 , m m

The curve with parametric equation x = 1 + 4 cos, y = 2 + 3 sin is _____________. a) a circle b) a parabola c) an ellipse d) a hyperbola

16.

The intercepts cut off by the plane 2x + y z = 5 with the axes is 5 1 2 1 1 c) 2, 1, -1 b) , , 5 a) , , 5 5 5 2 5

d) -2, -1, 1x2 y2 =1 a2 b2

17.

The condition that the line lx + my + n = 0 may be a normal to the hyperbola is a) al3 + 2alm2 + m2n = 0a 2 b 2 (a 2 b 2 ) 2 c) 2 + 2 = l m n2 a 2 b 2 (a 2 + b 2 ) 2 + = l 2 m2 n2 a 2 b 2 (a 2 + b 2 ) 2 d) 2 2 = l m n2

b)

18.

The hyperbola with foci at (0, -1), (0, 3) and one vertex at the origin is _____________. a) 3y2 x2 6y = 0 c) 3x2 y2 + 6y = 0 b) 3x2 y2 + 6x = 0 d) 3x2 y2 6x = 0

19.

x = x0 is a root of even order for the equation f (x) = 0 then x = x0 is a a) maximum point c) inflexion point b) minimum point d) critical point

20.

x2 y2 The area of the largest rectangle that can be inscribed in the ellipse 2 + 2 = 1 is a b ___________.

a) ab

b) a2b2

c) 2ab

d)

2ab

4

21.

If the length of the diagonal of a square is increasing at the rate of 0.2 m / sec, what is the 30 rate of increase of its area when the side is cm? 2 a) 3 cm2 / sec b) 6 2 cm2 / sec c) 3 2 cm2 / sec d) 6 cm2 / sec

22.

In the law of mean, the value of satisfies the condition a) > 0 b) < 0 c) < 1 d) 0 < < 1

23.

The curve y2 = (x a) (x b)2, a, b > 0 and a > b does not exist for a) x a b) x = a c) b < x < a d) x = a

24.

If there is an error of 0.01 cm in the diameter of a sphere when its radius is 5 cm, then the percentage error in its surface area is a) 0.1 % b) 0.2 % c) 0.02 % d) 2.0 %

25.

In which region the curve y2 (a + x) = x2(3a x) does not lie? a) x > 0 c) -a < x < 3a b) x -a and x > 3a d) 0 < x < 3a

26.

The curve x3 + y3 = 3axy is symmetrical about ___________. a) x = 0

b) y = 0

c) both axis

d) y = x

27.

x e0

x 6 2

dx = _____________

a)

6 27a

b)

6 26

c) 26 6

d) 27 6

28.

0

a

f ( x)dx + f ( 2a x)dx = ___________.0

a)

f ( x)dx0

a

b) 2 f ( x ) dx0

a

2a

c)

f ( x)dx0

2a

d)

f (a x)dx0

5

0

29.

1

| x + 1 | dx1 2

is b) 1 2 c) 2 d) 2

a) 30.

The volume of the solid obtained when the area between the line joining the points (0, 0) and (2, 3) and x axis is rotated about x axis is ___________. a) 2 b) 4 c) 8 d) 6

31.

The area between the parabola y2 = 16x and the line y = x is ___________. a) 442 3 b) 421 3 c) 128 3 d) 256 3

32.

The differential equation formed by eliminating A and B from the relation y = ex (A cos 3x + B sin 3x) is a) y" 2 y 10 y = 0 c) y" + 2 y + 10 y = 0 b) y" 2 y + 10 y = 0 d) y" + 2 y 10 y = 0

33.

If y = e

- 4x

(A cos 3x + B sin 3x) then b) (D2 + 8D + 25 ) y = cos 3x + sin 3x d) (D2 8D + 25 ) y = e- 4x

a) (D2 D 12) y = 0 c) (D2 + 8D + 25) y = 0 34.

The differential equation satisfied by all the straight lines in xy plane is ___________. a) dy = a constant dx b)d2y =0 dx 2

c) y +

dy =0 dx

d)

d2y + y=0 dx 2

35.

The particular integral of a) x 1 cos 3 x + 6 9

d2y + 9 y = 1 + sin 3 x is ___________. dx 2 1 x x x b) sin 3x c) cos 3x + d) cos 3 x + 9 6 10 6 6

36.

If x

dy = y (log y log x + 1) then the solution of the equation is dx

y a) x log = cy x

x b) y log = cx y

x c) log = cy y

y d) log = cx x

6

37.

An element of order 2 in the group (C {0}, ) is ___________. a) 1 i b) 2 + i c) ei

d)

2i 3

38.

The set G = { 1, , 2 } of all cube roots of unity forms an abelian group with respect to multiplication. Then the inverse of a) (1 + 2 ) 1+ + 2 7 is ___________. 1+

b) (1 + )

c)

d) -

39.

If a, b, c are any three elements of the group (G, *) and (a * b) * x = c then x = ___________. a) c * (a-1 * b-1 ) b) c * (b-1 * a-1 ) c) (a-1 * b-1 ) * c d) (b-1 * a-1 ) * c

40.

In congruence modulo 5, { x Z / x = 5 k + 4, k z} represents a) [ 0 ] b) [ 5 ] c) [ 4 ] If f(x) = k sin a) 2 5

d) [ 2 ]

41.

x5

, 0 x 5 is a p.d.f. then the value of k = ___________. b) 3 10 c)

10

d)

5

42.

In a Poisson distribution if standard deviation is 2 then P ( X 1) is ___________. a) 1 e-2

b) 1 + e

c) 1 e2

d) 1 e

-1

43.

A die is thrown 100 tones. If getting an odd number is a success, then the variance of the number of successes is ___________. a. 50 b) 40 c) 25 d) 20

44.

If 2 cards are drawn from a well shuffled pack of 52 cards, the probability that they are of the same colours with replacement is a) 1 2 b) 25 51 c) 26 51 d) 25 102

45.

The binomial distribution have the mean a) n2p b) np c) npq d) np2

7

PHYSICS (Common to all candidates)

46.

Three capacitors each of capacity 4 F are to be connected in such a way that, the effective capacitance is 6 F. This can be done by a) b) c) d) connecting all of them in series connecting them in parallel connecting two in series and one in parallel connecting two in parallel and one in series

47.

Relation between charge density and radius is _____________ a) directly proportional c) equal b) inversely proportional d) no relation

48.

The value of electrical resistance for a given conductor is _____________ a) mL nAe 2 b) mA ne 2L c) m ne 2 d)ne 2 m

49.

Two electric bulbs whose resistances are in the ratio of 1 : 2 are connected in parallel to a constant voltage source. The powers dissipated in them have the ratio a) 1 : 2 b) 1 : 1 c) 2 : 1 d) 1 : 4

50.

To send 10% of the main current through a moving coil galvanometer of resistance 99 ohm the shunt required is a) 10 ohm b) 9.9 ohm c) 9 ohm d) 11 ohm

51.

1A of current produces a deflection of 30o in a tangent galvanometer then its reduction factor is 1 a) 1.414 b) 0.707 c) 1.732 d) 3 In a LCR series circuit, the AC voltage across R, L and C come out as 10V, 10V and 20V respectively. The voltage across the entire combination will be a) 30 V b) 10 3 V c) 20V d) 10 2 V8

52.

53.

Keeping all other factors same, the number of turns of a coil is doubled. The magnetic flux linked with the coil is a) doubled b) halved c) trebled d) quadrupled

54.

A monochromatic light of frequency 3 x 108 MHz propagates through a medium of refractive index 1.55. Its wavelength in the medium is a) 645 nm b) 6.45 x 10-8 m c) 6450 nm d) 6.45 x 10-9 m

55.

Two small angled prisms of refractive indices 1.6 and 1.8 respectively produce same deviation for an incident ray of light. The ratio of the angles of prism is a) 0.75 b