TEST PAPERS For ENGINEERING

M. A- R

1) The area bounded by the curve 

y= cos⁻¹(sin2x) - sin⁻¹(cosx) and the lines y=0, x=3π/2, x= 2π is 

a) π²/4 b) π²/2 c) π² d) 2π² 

2) Let f(x)= x³+ ax²+ bx+ c. If f(-2)= f(1)= 0 and f'(1)< 0, then the smallest integer value of c is

a) 1 2 3 4

3) A= {z: z¹²=1} and B={w: w¹⁸ =1} are sets of complex roots of unity. The set C= {zw: z ∈ A, w∈ B} is also set of complex roots of unity. If n is the number of distinct elements of C, then the sum of the digits of n is

a) 7 b) 9 c) 11 d) 12

4) Consider the sequence a₁ , a₂ , a₃ , .... Let bₙ = ∆aₙ = aₙ₊₁ - aₙ , n∈N. If ∆bₙ = 1 and a₁₁ = a₂₁ = 0, then a₁ =0

a) 20 b) 40 c) 80 d) 100

5) If two distinct chords of a parabola y²= 4ax passing through the point (a,2a) are bisected by the line x+ y =1, then the length of the latus rectum can be.

a) 1 b) 2 c) 4 d) 8

COMPREHENSION

Consider 5 digit numbers formed using the digits 0,1,2,3,4,5 without repetition of digits.

6) The number of numbers divisible by 4 is

a) 48 b) 54 c) 66 d) 144

7) The number of numbers divisible by 12 is

a) 54 b) 66 c) 108 d) 144

8) The number of numbers divisible by 15 is

a) 48 b) 66 c) 108 d) 54

9) In triangle ABC, a= 4, b=3, c=2

Column I                             Column II

a) The length of the internal    p) 6

bisector of angle B is               q) √6

                                                    r) √10

b) The length of the external  s) 2√10

bisector of angle B is              t) 3√10

c) The length of the external

bisector of angle C

10) Let N be the number of 5 letter words formed using the letters of the word NAGARJUNA. The sum of the digits of N is

AIEEE - 2012

1) LeuX be the universal set for sets A and B. If n(A)= 200, n(B)= 300 and n(A∩B)= 100, then n(A' ∩ B') is equal to 300 provided n(X) is equal to

a) 600 b) 700 c) 800 d) 900

2) The sum of n terms of the infinite series 1.3²+ 2.5²+ 3.7²+...... ∞ is

a) (n/6)(n+1)(6n²+14n+7) b) (n/6)(2n+1)(3n+1) c) 4n³+ 4n²+ n d) none

3) If x²ᵏ occurs in the expansion of (x + 1/x²)ⁿ⁻³, then

a) n - 2k is a multiple of 2.

b) n - 2k is a multiple of 3.

c) n = 0 d) none

4) If y= 1- x + x²/2! - x³/3! + x⁴/4! - ........, then d²y/dx² is

a) x  b) - x  c) - y  d) y

5) The value of ∫ secx dx/√{sin(2x+θ)+ sinθ} is

a) √{(tanx + tanθ) secθ}+ c

b) √{2(tanx + tanθ) secθ}+ c

c) √{2(sinx + tanθ) secθ}+ c d) none

6) The solution of the equation x² d²y/dx² = log x when x=1, y=0 and dy/dx = -1 is

a) y= (1/2)(logx)²+ logx

b) y= (1/2)(logx)²- logx

c) y= -(1/2)(logx)²+ logx

d) y= - (1/2)(logx)²- logx

7) If C²+ S²= 1 then (1+ C + iS)/(1+ C - iS) is equal to

a) C+ iS b) C- iS c) S+ iC d) S- iC 

8) The number of real roots of (x + 1/x)³+ (x + 1/x)= 0 is

a) 4 b) 6 c) 2 d) 0

9) If Tₚ , Tq , Tᵣ are pᵗʰ , qᵗʰ and rᵗʰ terms of an AP, then

Tₚ      Tq     Tᵣ

 p       q       r

 1        1      1 is equal to

a) p+ q+ r b) 0 c) 1 d) -1

10) If A is a square Matrix of order n x n and k is a scalar then adj.(kA) is equal to

a) kⁿ⁻¹ adj. A b) kⁿ⁺¹ adj. A c) kⁿ adj. A d) k adj. A

11) 6 persons A, B, C, D, E and F are to be seated at a circular table. The number of ways this can be done if A must have either B or C on is right and B must have either C or D on his right is 

a) 24  b) 12 c) 18 d) 36

12) For what values of α

lim ₓ→∞ √(2α²x²+ αx+7) - √(2α²x²+7) will be 1/2√2

a) any value of α  b) α ≠ 0 c)α = 1 d) α = -1

13) A window is in the form of a rectangle with the semicircular bend on the top. If the perimeter of the window is 10m, the radius in metres of the semicircular bend that maximize the amount of light admitted is

a) 20/(4+π) b) 10/(4+π) c) (10 -π) d) none

14) There is line with positive slop λ through origin which cuts off a segment of length √10 between the parallel line 2x - y + 5 = 0 and 2 x - y + 10 = 0. Then λ should be

a)  1/2  b) 1/3  c) 1/5 d) none

15) Let C₁ and C₂ be the circles given by equations x²+ y²- 4x -5= 0 and x²+ y² + 8y +7 = 0. Then the circle having the common chord of C₁ and C₂ as its diameter has

a) centre at (- 1, - 1) and radius 2

b) centre at (1, - 2) and radius √5

c) centre at ( 1, - 2) and radius 2

16) The equation of common tangent to the parabola y²= 16x and the circle x²+ y²= 8 are

a) y= x+ 4; y= - x -4 b) y= 2x+ 4; 2y= - x +9 c) y= x+ 9; y= - x -4 d) none

17) An ellipse has OB as semi minor axis. F and F' its foci and the angle FBF' is a right angle. Then the eccentricity of the ellipse is 

a) 1/√2  b) 1/2 c) 1/4  d) 1/√3

18) The angle between the lines 2 x= 3y= - z and 6x = - y = -4z is 

a) 0° b) 30° c) 45° d) 90°

19) A vector parapendicular to the plane containing the vectors i - 2j - k and 3i - 2j - k is inclined to the vector i + j + k at an angle

a) tan⁻¹√14 b) sec⁻¹√14 c) tan⁻¹√15 d) none

20) Three integers are chosen at random from the first 20 integers. The probability that their product is even, is 

a) 2/19 b) 3/29  c) 17/19 d) 4/29

21) If cotθ + tanθ = m and secθ - cosθ = n, then which of the following is correct?

a) m(mn²)¹⁾³ - n(nm²)¹⁾³ = 1

b) m(m²n)¹⁾³ - n(mn²)¹⁾³ = 1

c) n(mn²)¹⁾³ - m(nm²)¹⁾³ = 1

d) n(m²n)¹⁾³ - m(mn²)¹⁾³ = 1

22) If 3 sin⁻¹{2x/(1+ x²)} - 4 cos⁻¹{(1- x²)/(1+ x²)} + 2 tan⁻¹{2x/(1- x²)} =π/3, then value of x is

a) √3 b) 1/√3 c) 1 d) none

23) The A. M of ²ⁿ⁺¹C₀ , ²ⁿ⁺¹C₁ , ²ⁿ⁺¹C₂ , .... ²ⁿ⁺¹Cₙ is

a) 2ⁿ/n b) 2ⁿ/(n +1) c) 2²ⁿ/n d) 2²ⁿ/(n +1)

24) Let p and q be two statements, then (p∪q) ∪ - p is 

a) tautology b)  contradiction c) Both a and b d) none

25) Let f(x)= x - [x], for every real number of x, where [x] is the integral parts of x. Then  ¹₋₁∫ f(x) dx is equal to

a) 1 b) 2 c) 0 d) -1/2

Directions (26-30): This section contains 5 questions numbered 26 to 30. Each question contains statement-1 (Assertion ) and statement-2 (Reason). Each question has four choices (a), (b), (c) and (d) out of which ONLY ONE is correct.

a) Statement-1 is True , Statement-2 is True , Statement-2 is a correct explanation for Statement -1.

b) statement-1 is true , statement-2 is true;  statement-2 is not a correct explanation for Statement -1.

c) Statement -1 is True , Statement-2 is false .

d) statement-1 is false, statement -2 is true.

26) statement -1: 5/3 and 5/4 are the eccentricity of two conjugate hyperbolas.

      Statement -2: If e and e₁ are the eccentricities of two conjugate hyperbolas, then ee₁ > 1.

27) Statement -1: The maximum area of triangle formed by the point (0,0), (a cosθ , b sinθ), (a cosθ , - b sinθ) is (1/2) |ab|.

       Statement -2: Maximum value of sinθ is 2.

28) Statement -1: The Coefficient of xⁿ in the binomial expansion of (1- x)⁻² is (n +1).

       Statement-2: The Coefficient of xʳ in (1- x)⁻ⁿ when n ∈N is ⁿ⁺ʳ⁻¹C ᵣ.

29) Statement -1: 20 persons are sitting in a row. Two of these persons are selected at random, The probability that two selected person are not together is 0.7.

      Statement-2: if A is an event, then

P(not A)= 1- P(A).

30) Statement -1:  A flagstaff of length 100m stands on tower of height h. if at a point on the ground the angle of elevation of the tower and top of the flagstaff be 30°, 45°, then h= 50(√3 +1)m.

     Statement -2: A flagstaff of length 'd' stands on tower of height h. If at a point on the ground the angle of elevation of the tower and top of the flagstaff be α, β then h= d cotβ/(cot α - cotβ).

Answer

1b 2a 3b 4d 5b 6d 7 a 8 d 9 b 10 a 11 c 12 b 13 b 14b 15 c 16 a 17 a 18 d 19a 20c 21a 22b 23d 24a 25a 26b 27c 28a 29d 30a

CHALLENGING PROBLEMS

                    SECTION - I

     (Single Option Correct)

1) ∫ (1+ 2 tanx)dx/{1+ sin2x- cos2x)√(tan²x(sec²x + 2 tanx) -1}=

a) (1/2) tan⁻¹(1+ 2 tanx)+ c

b) sin⁻¹(sec²x + 2 tanx)+ c

c) (1/2) sec⁻¹(2 tanx + tan²x)+ c d) none

2) If f(x)= cosπ √(x -6) then number of solutions of f(x²). f(x²+4)= 1 is

a) 0 b) 2 c) 4 d) infinite 

3) If (2011a + 2012b)(2012c - 2011b)= (2011b + 2012c)(2012b - 2011a)( 0 <a <b< c) and √c. aˣ + √b. cˣ < (√c+ √b)bˣ , then which is a sub-interval of solution of x?

a) (0, 3/2) b) (-1/2, 1/2) c) (0, 1/2) d) none

4) ∫ |cos x| dx/(1+ x⁴) dx at (4π,π) is

a) < 1/π² b) > 1/π² c) = 1/π² d) none

5) If ω be an imaginary cube root of unity and 1, α₁ , α₂ , α₃ ,.....α²⁰¹⁰ be the roots of x²⁰¹¹ = ω²⁰¹³ then ²⁰¹⁰ᵣ₌₁Π{ (αᵣ - ω)/(αᵣ - ω²)}= 

a) 1 b) ω c) ω² d) none

6) ∫ (1+ x)² ˣ√(ₑx²+ x+1) dx =

a) ₑ{(x²+ x+1)/x} + c b) ₓₑ{(x²+ x+1)} + c d) ₓₑ{(x²+ x+1)/x} + c  d) none

7) The number of solution of (cos 20°+ 2 sin 55° + √2sin 65°)= (sin20° -  cos 100°)ˢᶦⁿᶿ ⁻ ᶜᵒˢ²ᶿ  in [0,2π] is

a) 0 b) 1 c) 2 d) 4

8) Find the number of integral values of h if one of the lines represented by 3x²sinθ+ 2hxy + 4y cos θ= 0  bisects the angle between the co-ordinate axes [θ≠ (2n +1)(π/2), where n ∈I]

a) 5 b) 2 c) 4 d) none 

9) In any AP, if sum of first 6 terms is 5 times the sum of next 6 terms then which term is zero ?

a) t₁₀ b) t₁₂ c) t₁₁  d) none

10) If I= π₀∫ sin(sin⁻¹{x}) dx (where {.} denotes fractional part functional) then I ∈

a) [0,π) b) (0,3π/2) c) (-π,π) d) none

11) For function f it is given that f(xy)= f(x) f(y) but x, y ∈ R --{0}, f(x +2)= 6x + 7 +8(1+ x) g(1+ x) and g(0)= 0 then ¹₀∫ {6(1- x)²⁰¹¹ f(x)dx}/f'(x) = 

a) 1/2012 B) 1/2011  c) 1/(2012. 2013) d) none 

             SECTION - II

   (More Than One Option Correct)

12) If aₙ be the co-efficient of xⁿ in the expansion of (1/2) e²ˣ then

a) a₁ , a₂ , (a₃ + a₄) are in AP and GP both

b) a₂ , a₃ , a₄ are in AP

c) a₃ , a₅ = 2a₆

d) a₃ , a₄/4, a₆ are in HP

13) if A and B are events of a random experiment so that P(A)= 1/5, P(A U B)= 3/4 then P(B/A's) is greater than 

a) 1/2 b) 3/4  c) 2/5 d) 5/9 

14) Which of the following is necessary a root of p(x²- 4x +3)+ q(3x²+ x -4)= r(4x²- 3x -1) ?

a) lim ₙ→₀ ⁿᵣ₌₂Π(1- 1/r²)/ ᵣ₌₂ⁿΠ(1-  1/r)

b) e - lim ₙ→ₙ  ⁿᵣ₌₁ ∑eʳ⁾ⁿ 

c) (5/2) lim ₙ→∞  ⁿᵣ₌₁ ∑ r √r/√n⁵

d) lim ₙ→∞ ⁿᵣ₌₁ ∑ 1/(r²+r)

15) If α !(ab)ᵟ /(β!)ʳ be the term independent of x in the expansion of (ax + b/x)¹⁴ then

a) α = 2β b) α = βγ c) α : β : δ = 1:2:2 d) α = δγ

16) If f(x,y)= x⁴+ x²y²+ y² then

a) f(sinα, cosα): f(cosα, sinα)= 1:2

b) f(sinα, cosα) = f(cosα, sinα)

c) f(√e, √e)= f(1,1) will imply that e can be eccentricity of a parabola.

d) f(ω, i²)= 0

17) If z₁ , z₂ , z₃ be the vertices (in anticlockwise sense) of an equilateral triangle inscribed in |z| = k (> 0) and z₁ = 1+ √-3 then

a) z₂ is purely real

b) z₃ = bar z₂

c) z₃ = bar z₁

d) z₁ + z₂ + z₃ = 0

18) If E₁ : sinx + sin2x + sin3x = 3; 

          E₂ : sin2x + cos4x = 2; 

          E₃ : cosx + cos2x + cos3x = 3;

          E₄: tanx = 2 + |tanx| be equation and N(E₁) denotes the number of solution of Eᵢ then in [0,2π]

a) N(E₁ )= 0

b) N(E₂) = N(E₁)

c) N(E₃)=1

d) N(E₄)= 0

19) ²⁰¹²ᵣ₌₁₀₀₇∑ 1/r= 

a)   ²⁰¹¹ᵣ₌₁∑1/(r(r+1))

b) ²⁰¹¹ᵣ₌₁∑ (-1)ʳ

c) ²⁰¹²ᵣ₌₁∑ (-1)ʳ⁺¹/r

d) ¹⁰⁰⁶ᵣ₌₁∑1/((2r-1)2r)

20) Value of (1+ tan6°)(1+ tan66°)(1+ tan42°)(1+ tan78°) can't be less than

a) 64 b) 9 c) 16  d) 27

Answer

1d 2b 3c 4a 5a 6d 7b 8a 9c 10a 11c 12 a,b,c,d 13 a,c,d 14 b,c,d 15 a,b,d 16 b.p,c,d 17 a,c,d 18 a,b,c,d 19 c,d 20 b,c