engg mathematics

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– 1 –

SECTION - I

Single Correct Answer Type

This section contains 25 multiple choice questions.

Each question has 4 choices (1), (2), (3) and (4) for its

answer, out of which ONLY ONE is correct.

1. The number of solut ions of the equat ion

43232

sin 2

x x

x

(1) Is zero

(2) Is only one

(3) Is only two

(4) Is greater than 2

2. Number of solution of equation2

sinx

x

is/are

(1) 3 (2) 2

(3) 1 (4) Infinite

3. If sin20° – cos20° = x , then cos40° =

(1) 22 x x (2) 2 2 x x

(3) 22 x x (4) 2

2 x x

4. { x } and [ x ] represent fractional and integral part of x ,

then

2007

1 2007

}{][

k

k x x is equal to

(1) [ x ] (2) x

(3) 2007{ x } (4)2

}{ x x

5. If ,11

)(

x x

x f then f (2 x ) is

(1)3)(

1)(

x f

x f (2)

3)(

1)(3

x f

x f

(3) 1)(

3)(

x f

x f (4) 1)(3

3)(3

x f

x f

Instructions:

(i) This question paper is divided into two sections - Section-I (Single Correct Answer Type) and Section-II

(Multiple Correct Answer Type).

(i) Use ball point pen only to darken the appropriate circle.

(ii) Mark should be dark and should completely fill the circle.

(iv) Dark the circle in the space provided only.

(v) Rough work must not be done on the Answer sheet and do not use white-fluid or any other rubbing

material on Answer sheet.

(vi) Each question carries 3 marks. For every wrong response 1 mark shall be deducted from total score.

Topics covered : Trigonometry, Algebra (Sets, Relations, Functions, Quadratic Equation, Complex Numbers,

Permutations and Combination, Binomial Theorem, Sequence and Series, Probability), Calculus (Limits and

Derivatives, Application of Derivatives, Indefinite Integration, Definite Integration, Differential Equation)

Regd. Off ice : Aakash Tower, Plot No.-4, Sec-11, MLU, Dwarka, New Delhi-110075

Ph.: 011-47623456, Fax : 011-47623472

Time: 1½ hours Campus Recruitment Test MM: 105

Sample Paper MATHEMATICS (Engineering)

Code -



– 2 –

6. If are the roots of the equation

x 3 + 2 x 2 + 3 x + 3 = 0, then the va lue of

333

111

is

(1) 14 (2) 44

(3) 45 (4) 15

7. The set of al l values of a R for which the

equation 2 x 2 – 2(2a + 1) x + a(a – 1) = 0 has roots

and satisfying < a < is

(1) (–, –3) (0, )

(2) (–3, 0)

(3) (–, 0) (3, )

(4)3 7 3 7

, ,2 2

8. The area of the triangle whose vertices are the roots

of z 3 + iz 2 + 2i = 0 is

(1)4

3(2)

7

3

(3)4

73(4) 2

9. If a dict ionary is formed using al l possible

arrangements of the letters of the word MOTHER,

then the rank of the word MOTHER in the

dictionary will be

(1) 240 (2) 261

(3) 308 (4) 309

10. There are n married couples at a party. Each

person shakes hand with every person other than

her or his spouse. The total number of hand shakes

must be

(1) 2nC 2 – 2n (2) 2nC

2 – (n – 1)

(3) 2n (n – 1) (4) 2nC 2

11. If 1,

2,

3,........,

nare the nth roots of unity,

then nC 1

1+ nC

2

2+ nC

3

3+.......+ nC

n

n

equals

(1)2

1

(2) }1){( 2

212

1

n

(3) }1)1{( 22

1

n (4) }1){( 212

1

n

12. If 12)12( nR and f = R – [R ] where [ ]

denotes the greatest integer function, then [R ]

equals

(1)

f

f 1

(2)

f

f 1

(3) f f

1

(4)f

f 1

13. The sum to infinity of the series

......343

16

49

9

7

41 is

(1)271

481(2)

27

49

(3)

344

518(4)

34

53

14. Let A, B, C be three coins in a bag. Suppose A is

a fair coin, B is two headed and C is weighted so

that the probability of heads is1

3. A coin is selected

at random and tossed. Which of following is true?

(1) The probability that tail appears is5

18

(2) If tail appears then the probability that it is the

coin C is4

7(3) If head appears, the probability that it is the

fair coin A is2

11

(4) The probability that head appears is13

18

15. equals 3214

lim4

53

x x

x x x

x

(1)

2

1(2)

3

1

(3)5

1(4)

4

1

16. x x

x

5lim

5

is equal to

(1) ln 5

(2)5ln

!5

(3) 5)5(ln

!5

(4) 0



– 3 –

17. I f the funct ion

0,

0,)(cos)(

1

x K

x x x f x is

continuous at x = 0, then the value of K is

(1) 0 (2) 1

(3) –1 (4) e

18. Tangent of acute angle between the curves

2 21 and 7y x y x at the points of

intersection is

(1)5 3

2(2)

3 5

2

(3)5 3

4(4)

3 5

4

19. The greatest value of the function

4sin

2sin)(

x

x x

on the interval

2

,0 is

(1) 2 (2) 2

(3)2

1(4) 1

20. dx x x

1

1

4is equal to

(1) tan –1 x 2 + K (2)2

1sec –1 x + K

(3) sec –1 x 2 + K (4)2

1sec –1 x 2 + K

21. dx x x x

2)1(

1 is equal to

(1) K x

x

112 (2) K

x x

11

(3) K x

x

1

)1(2(4) K

x

x

1

12

22.

0

2sin

x x dx

(1) 22

(2)

32

2

(3) 2 – (4)2

23. The value of

x x

x x

x

dx e

dx e

0

2

2

0

2

lim is

(1) 1 (2) 2

(3) 3 (4) 0

24. A curve passes through the point 1,4

and its

slope at any point is given by2cos

y y

x x

. Then

the curve has the equation

(1) y = tan –1(lne

x )

(2) y = x tan –1(ln2)

(3) y =1

x tan –1(ln

e

x )

(4) y = x tan –1(lne

x )

25. The differential equation x y x dx

dy y ln1

1 23

has its solution as

(1) c x x y

x

ln

3

2

3

2 3

2

2

(2) c x x x

y

ln

3

2

3

2 3

2

2

(3) c y y x

y

ln

2

3

2

3 3

2

2

(4) c x x

y

x

ln

3

2

3

2 3

2

2

SECTION - II

Multiple Correct Answer Type

This section contains 10 multiple choice questions.

Each question has 4 choices (1), (2), (3) and (4) for its

answer, out of which ONE OR MORE is/are correct.

26. The number of solutions of the equation

sin x + cos x = e x + e – x is/are

(1) 0 (2) Infinite

(3) Finite (4) 1



– 4 –

27. In ABC , if cos A cos B + sin A sin B sin C = 1,

then

(1) a : b : c = 1 : 2 : 1

(2) a : b : c = 1 : 1 : 2

(3) cos A =2

1

(4) sin A =2

1

28. Let cos A + cosB + cosC = 0 and sin A + sinB +

sinC = 0, then which of the following statements is

correct?

(1) cos(2 A – B – C ) = 3

(2) cos(2 A – B – C ) = 0

(3) sin(2 A – B – C ) = 0

(4) sin(2 A – B – C ) = 3

29. If x is the number of five digit numbers, sum of

whose digits is even and y is the number of

5 digit numbers, sum of whose digits is odd, then

(1) x = y (2) x + y = 90000

(3) x = 45000 (4) x < y

30. Let a and b be two positive real numbers. Suppose

A1, A

2are two arithmetic means; G

1, G

2are two

geometric means and H 1, H 2 are two harmonicmeans between a and b, then

(1)1 2 1 2

1 2 1 2

G G A A

H H H H

(2)1 2

1 2

5 2

9 9

G G a b

H H b a

(3)1 2

1 2

9

(2 )( 2 )

H H ab

A A a b a b

(4)1 2 1 2

1 2 1 2

G G H H H H A A

31. Let A and B be two events such that

P ( A Bc ) = 0.25, P ( Ac B) = 0.20, and P ( A B)

= 0.10. Then which of the following is/are true?

(1) P ( A/B) = 0.40

(2) P ( A) = 0.35

(3) P ( A B) = 0.45

(4) P (B/A) = 0.285

32. Let f ( x ) = 1 + 2 sin x + 2cos2 x , 0 x 2

. Then

(1) f ( x ) is greatest at6

(2) f ( x ) is least at 0, 2

(3) f ( x ) is increasing in

6

,0 and decreasing in

2

,6

(4) f ( x ) is continuous in

2

,0

33. dx

x b x a

)sincos(

12222

is equal to

(1) K x a

b

ab

cottan1 1

(2) K x a

b

ab

tantan1 1

(3) K x a

b

ab

tancot

1 1

(4) K x b

a

ab

cotcot1 1

34. Let dx I dx I dx I x x x 2

1

3

1

0

2

1

0

1

232

)2007(,)2007(,)2007(

and dx I x 1

0

4

4

)2007( . Then

(1) I 1

> I 2

(2) I 2

> I 1

(3) I 3

> I 4

(4) I 2

> I 4

35. The differential equation

01221

dy

y

x edx e y

x

y

x

(1) Is homogeneous of degree 0

(2) Is homogeneous of degree 1

(3) Can be solved by the substitution x = vy

(4) Has its solution as k ye x y

x

2



– 5 –

ANSWERS

1. (2)2. (1)

3. (4)

4. (2)

5. (2)

6. (2)

7. (1)

8. (4)

9. (4)

10. (3)

11. (3)

12. (3)

13. (2)

14. (2)

15. (1)

16. (4)

17. (2)

18. (3)

19. (4)

20. (4)

21. (4)

22. (4)

23. (4)

24. (4)

25. (1)

26. (1, 3)

27. (2, 3, 4)

28. (1, 3)

29. (1, 2, 3)

30. (1, 2, 3)

31. (2, 4)

32. (1, 2, 3, 4)

33. (2, 3, 4)34. (1, 3, 4)

35. (2, 3, 4)

Regd. Off ice : Aakash Tower, Plot No.-4, Sec-11, MLU, Dwarka, New Delhi-110075

Ph.: 011-47623456, Fax : 011-47623472

Time: 1½ hours Campus Recruitment Test MM: 105

Sample Paper MATHEMATICS (Engineering)

Code -

engg mathematics

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