WBJEE Mathematics Question Paper 2009 (Eng)

WB-JEE - 2009-Mathematics

1.   If C is the reflecton of A (2, 4) in x-axis and B is the reflection of C in y-axis, then |AB| is

(A) 20       (B) 2√5      (C) 4√5      (D) 4

Ans : (C)

2.  The value of $\cos {15^ \circ }\cos 7{{{1^ \circ }} \over 2}\sin 7{{{1^ \circ }} \over 2}$ is

(A) ${1 \over 2}$     (B) ${1 \over 8}$     (C) ${1 \over 4}$      (D) ${1 \over {16}}$

Ans : (B)

3.  The value of integral $\int\limits_{ - 1}^1 {{{\left| {x + 2} \right|} \over {x + 2}}} dx$ is

(A) 1     (B) 2      (C) 0      (D) –1

Ans : (B)

4.  The line $y = 2{t^2}$ intersects the ellipse ${{{x^2}} \over 9} + {{{y^2}} \over 4} = 1$ in real points if

(A) | t | ≤ 1       (B) | t | < 1       (C) | t | > 1       (D) | t | ≥ 1

Ans : (A)

5.  General solution of $\sin x + \cos x = {\min }\limits_{a \in IR} \left\{ {1,{a^2} - 4a + 6} \right\}$ is

(A) ${{n\pi } \over 2} + {( - 1)^n}{\pi \over 4}$       (B) $2n\pi + {( - 1)^n}{\pi \over 4}$       (C) $n\pi + {( - 1)^{n + 1}}{\pi \over 4}$      (D) $n\pi + {( - 1)^n}{\pi \over 4} - {\pi \over 4}$

Ans : (D)

6.  If A and B square matrices of the same order and AB = 3I, then  A^–1 is equal to

(A) 3B       (B) ${1 \over 3}B$      (C) $3{B^{ - 1}}$       (D) ${1 \over 3}{B^{ - 1}}$

Ans : (B)

7.   The co-ordinates of the focus of the parabola described parametrically by $x = 5{t^2} + 2$ , $y= 10t + 4$ are

(A) (7, 4)        (B) (3, 4)       (C) (3, –4)       (D) (–7, 4)

Ans : (A)

8.   For any two sets A and B, A – (A – B) equals

(A) B     (B) A – B      (C) A ∩ B      (D) A^c ∩ B^c

Ans : (C)

9.  If a = 2√2 , b = 6 , A = 45°, then

(A) no triangle is possible           (B) one triangle is possible

(C) two triangle are possible       (D) either no triangle or two triangles are possible

Ans : (A)

10.   A Mapping from IN to IN is defined as follows :f : IN → INf(n) = (n + 5)² , n ∈ IN(IN is the set of natural numbers). Then

(A)  f is not one-to-one       (B) f is onto      (C) f is both one-to-one and onto      (D) f is one-to-one but not onto

Ans : (D)

11.  In a triangle ABC if $\sin A\sin B = {{ab} \over {{c^2}}}$ then the triangle is

(A) equilateral      (B) isosceles       (C) right angled       (D) obtuse angled

Ans : (C)

12.  $\int {{{dx} \over {\sin x + \sqrt 3 \cos x}}}$ equals

(A) ${1 \over 2}1n\left| {\tan \left( {{x \over 2} - {\pi \over 6}} \right)} \right| + c$      (B) ${1 \over 2}1n\left| {\tan \left( {{x \over 4} - {\pi \over 6}} \right)} \right| + c$

(C) ${1 \over 2}1n\left| {\tan \left( {{x \over 2} + {\pi \over 6}} \right)} \right| + c$      (D) ${1 \over 2}1n\left| {\tan \left( {{x \over 4} + {\pi \over 3}} \right)} \right| + c$

where c is an arbitrary constant

Ans : (C)

13.  The value of $\left( {1 + \cos {\pi \over 6}} \right)\left( {1 + \cos {\pi \over 3}} \right)\left( {1 + \cos {{2\pi} \over 3}} \right)\left( {1 + \cos {{7\pi } \over 6}} \right)$ is

(A) ${3 \over {16}}$      (B) ${3 \over 8}$      (C) ${3 \over 4}$      (D) ${1 \over 2}$

Ans : (A)

14.  If $P = {1 \over 2}{\sin ^2}\theta + {1 \over 3}{\cos ^2}\theta$ then

(A) ${1 \over 3} \le P \le {1 \over 2}$       (B) $P \ge {1 \over 2}$      (C) $2 \le P \le 3$      (D) $- {{\sqrt {13} } \over 6} \le P \le {{\sqrt {13} } \over 6}$

Ans : (A)

15.  A positive acute angle is divided into two parts whose tangents are ${1 \over 2}$ and ${1 \over 3}$. Then the angle is

(A)  ${\pi \over 4}$       (B)  ${\pi \over 5}$        (C) ${\pi \over 3}$       (D)  ${\pi \over6}$

Ans : (A)

16.  If $f(x) = f(a - x)$ then $\int\limits_0^a x f(x)dx$ is equal to

(A) $\int\limits_0^a f (x)dx$       (B) ${{{a^2}} \over 2}\int\limits_0^a f (x)dx$      (C) ${a \over2}\int\limits_0^a f (x)dx$      (D) $- {a \over 2}\int\limits_0^a f (x)dx$

Ans : (C)

17.  The value of $\int\limits_0^\infty {{{dx} \over {({x^2} + 4)({x^2} + 9)}}}$ is

(A) ${\pi \over {60}}$      (B) ${\pi \over {20}}$       (C) ${\pi \over {40}}$      (D) ${\pi \over {80}}$

Ans : (A)

18.  If ${I_1} = \int\limits_0^{\pi /4} {{{\sin }^2}xdx}$ and ${I_1} = \int\limits_0^{\pi /4} {{{\cos }^2}xdx}$ ,  then,

(A) ${I_1} = {I_2}$      (B) ${I_1} < {I_2}$       (C) ${I_1} > {I_2}$       (D) ${I_2} = {I_1}+ \pi /4$

Ans : (B)

19.  The second order derivative of a sin³t with respect to a cos³t at $t = {\pi \over 4}$ is

(A) 2      (B) ${1 \over {12a}}$       (C) ${{4\sqrt 2 } \over {3a}}$      (D) ${{3a} \over {4\sqrt 2}}$

Ans : (C)

20.  The smallest value of 5 cos θ + 12 is

(A) 5      (B) 12      (C) 7      (D) 17

Ans : (C)

21.  The general solution of the differential equation ${{dy} \over {dx}} = {e^{y + x}} + {e^{y - x}}$ is

(A) e^–y = e^x – e^–x + c      (B) e^–y = e^-x – e^x + c       (C) e^–y = e^x + e^–x + c       (D) e^y = e^x + e^–x + c

where c is an arbitrary constant

Ans : (B)

22.  Product of any r consecutive natural numbers is always divisible by

(A) r !      (B) (r + 4) !       (C) (r + 1) !       (D) (r + 2) !

Ans : (A)

23.  The integrating factor of the differential equation $x\log x{{dy} \over {dx}} + y = 2\log x$ is given by

(A) e^x     (B) log x      (C) log (log x)      (D) x

Ans : (B)

24.  If x²  + y² = 1 then

(A) yy′′ − (2y′)² + 1 = 0        (B) yy′′ + ( y′)² +1 = 0       (C) yy′′ − (y′)² −1 = 0       (D) yy′′ + (2y′)² + 1 = 0

Ans : (B)

25.  If c₀,  c₁,  c₂, ..................., c_n denote the co-efficients in the expansion of (1 + x)ⁿ

then the value of c₁ + 2c₂ + 3c₃ + ..... + nc_n is

(A) n.2^n-1      (B) (n + 1)^n-1      (C)  (n + 1)2ⁿ      (D) (n + 2) 2^n-1

Ans. (A)

26.   A polygon has 44 diagonals. The number of its sides is

(A) 10      (B) 11       (C) 12       (D) 13

Ans : (B)

27.   If α, β be the roots of x² – a(x – 1) + b = 0, then the value of ${1 \over {{\alpha ^2} - a\alpha }} + {1 \over{{\beta ^2} - a\beta }} + {2 \over {a + b}}$

(A) ${4 \over {a + b}}$       (B)  ${1 \over {a + b}}$       (C) 0      (D) –1

Ans : (C)

28.  The angle between the lines joining the foci of an ellipse to one particular extremity of the minor axis is 90° .  The eccentricity of the ellipse is

(A) ${1 \over 8}$      (B) ${1 \over {\sqrt 3 }}$      (C) $\sqrt {{2 \over 3}}$      (D) $\sqrt{{1 \over 2}}$

Ans : (D)

29.  The order of the differential equation ${{{d^2}y} \over {d{x^2}}} = \sqrt {1 - {{\left( {{{dy} \over {dx}}}\right)}^2}}$ is

(A) 3      (B) 2      (C) 1      (D) 4

Ans : (B)

30.  The sum of all real roots of the equation |x – 2|² + |x – 2| – 2 = 0

(A) 7      (B) 4      (C) 1      (D) 5

Ans : (B)

31.  If $\int\limits_{ - 1}^4 {f(x)dx = 4}$ and  $\int\limits_2^4 {\left\{ {3 - f(x)} \right\}dx = 7}$ then the value of $\int\limits_{ - 1}^2 {f(x)dx}$

(A) –2      (B) 3       (C) 4       (D) 5

Ans : (D)

32.  For each n∈ N,  2³ⁿ – 1 is divisible by

(A) 7       (B) 8       (C) 6       (D) 16

where N is a set of natural numbers

Ans : (A)

33.  The Rolle’s theorem is applicable in the interval – 1 ≤ x ≤ 1 for the function

(A) ƒ(x) = x      (B) ƒ(x) = x²      (C) ƒ(x) = 2x³ + 3       (D) ƒ(x) = |x|

Ans : (B)

34.  The distance covered by a particle in t seconds is given by x = 3 + 8t – 4t² .  After 1 second velocity will be

(A) 0 unit/second       (B) 3 units/second       (C) 4 units/second       (D) 7 units/second

Ans : (A)

35.  If the co-efficients of x² and x³ in the expansion of (3 + ax)⁹ be same, then the value of ‘a’  is

(A) ${3 \over 7}$       (B) ${7 \over 3}$       (C) ${7 \over 9}$       (D) ${9 \over 7}$

Ans : (D)

36.  The value of $\left( {{1 \over {{{\log }_3}12}} + {1 \over {{{\log }_4}12}}} \right)$ is

(A) 0       (B) ${1 \over 2}$        (C) 1       (D) 2

Ans : (C)

37.  If x = log_a bc,  y = log_b ca,  z = log_c ab,  then the value of ${1\over {1 + x}} + {1 \over {1 + y}} + {1 \over {1 + z}}$ will be

(A) x + y + z       (B) 1       (C) ab + bc + ca        (D) abc

Ans : (B)

38.  Using binomial theorem, the value of (0.999)³ correct to 3 decimal places is

(A) 0.999        (B) 0.998        (C) 0.997       (D) 0.995

Ans : (C)

39.  If the rate of increase of the radius of a circle is 5 cm/.sec., then the rate of increase of its area, when  the radius is 20 cm, will be

(A) 10π        (B) 20π       (C) 200π        (D) 400π

Ans : (C)

40.  The quadratic equation whose roots are three times the roots of 3ax² + 3bx + c = 0 is

(A) ax² + 3bx + 3c = 0      (B) ax² + 3bx + c = 0      (C) 9ax² + 9bx + c = 0       (D) ax² + bx + 3c = 0

Ans : (A)

41.  Angle between y² = x and x² = y at the origin is

(A) $2{\tan ^{ - 1}}\left( {{3 \over 4}} \right)$      (B) ${\tan ^{ - 1}}\left( {{4 \over 3}} \right)$      (C)${\pi \over 2}$      (D) ${\pi \over 4}$

Ans : (C)

42.  In triangle ABC, a = 2, b = 3 and $\sin A = {2 \over 3}$ , then B is equal to

(A) 30°       (B) 60°       (C) 90°       (D) 120°

Ans : (C)

43.  $\int\limits_0^{1000} {{e^{x - [x]}}}$  is equal to

(A) ${{{e^{1000}} - 1} \over {e - 1}}$       (B) ${{{e^{1000}} - 1} \over {1000}}$       (C) ${{e - 1}\over {1000}}$      (D) 1000 (e – 1)

Ans : (D)

44.  The coefficient of xⁿ,  where n is any positive integer, in the expansion of (1 + 2x + 3x² + ......... ∞)^½  is

(A) 1       (B) ${{n + 1} \over 2}$        (C) 2n + 1        (D) n + 1

Ans : (A)

45.  The circles x² + y² – 10x + 16 = 0 and x² + y² = a² intersect at two distinct points if

(A) a < 2        (B) 2 < a < 8        (C) a > 8         (D) a = 2

Ans. (B)

46.  $\int {{{{{\sin }^{ - 1}}x} \over {\sqrt {1 - {x^2}} }}dx}$ is equal to

(A) $\log ({\sin ^{ - 1}}x) + c$      (B) ${1 \over 2}{({\sin ^{ - 1}}x)^2} + c$      (C) $\log \left({\sqrt {1 - {x^2}} } \right) + c$      (D) $\sin \left( {{{\cos }^{ - 1}}x} \right) + c$

where c is an arbitrary constant

Ans : (B)

47.  The number of points on the line x + y = 4 which are unit distance apart from the line 2x + 2y = 5 is

(A) 0       (B) 1       (C) 2       (D) Infinity

Ans : (A)

48.   Simplest form of ${2 \over {\sqrt {2 + \sqrt {2 + \sqrt {2 + 2\cos 4x} } } }}$ is

(A)  $\sec {x \over 2}$      (B) sec x      (C) cosec x        (D) 1

Ans : (A)

49.  If $y = {\tan ^{- 1}}\sqrt {{{1 - \sin x} \over {1 + \sin x}}}$ , then the value of ${{dy} \over {dx}}$ at $x = {\pi \over 6}$

(A) $- {1 \over 2}$      (B) ${1 \over 2}$       (C) 1      (D) –1

Ans : (A)

50.  If three positive real numbers a , b ,  c are in A.P. and abc = 4 then minimum possible value of b is

(A) 2^3/2       (B) 2^2/3       (C) 2^1/3       (D) 2^5/2

Ans : (B)

51.  If $5\cos 2\theta + 2{\cos ^2} {{\theta} \over 2} + 1 = 0$ , when (0 < θ < π), then the values of θ are :

(A) ${\pi \over 3} \pm \pi$      (B) ${\pi \over 3},{\cos ^{ - 1}}\left ( {{3 \over 5}} \right)$      (C) ${\cos ^{ - 1}}\left ( {{3 \over 5}} \right) \pm \pi$       (D) ${\pi \over 3},\pi - {\cos ^{ - 1}}\left ( {{3 \over5}} \right)$

Ans : (D)

52.   For any complex number z, the minimum value of |z| + |z – 1| is

(A) 0       (B) 1       (C) 2       (D) –1

Ans : (B)

53.  For the two circles x² + y² = 16 and x² + y² – 2y = 0 there is / are

(A) one pair of common tangents         (B) only one common tangent

(C) three common tangents                  (D) no common tangent

Ans : (D)

54.   If C is a point on the line segment joining A (–3, 4) and B (2, 1) such that AC = 2BC , then the coordinateof C is

(A) $\left( {{1 \over 3},2} \right)$       (B) $\left( {2,{1 \over 3}} \right)$       (C) (2, 7)      (D) (7, 2)

Ans : (A)

55.   If a , b , c are real, then both the roots of the equation (x – b) (x – c) + (x – c) (x – a) + (x – a) (x – b) = 0 are always

(A) positive       (B) negative          (C) real        (D) imaginary

Ans : (C)

56.  The sum of the infinite series $1 + {1 \over {2!}} + {{1.3} \over {4!}} + {{1.3.5} \over {6!}} + \cdots \cdots$

(A) e       (B) e²     (C) √e     (D) ${1 \over e}$

Ans : (C)

57.  The point (–4, 5) is the vertex of a square and one of its diagonals is 7x – y + 8 = 0.  The equation of the other diagonal is

(A) 7x – y + 23 = 0        (B) 7y + x = 30      (C) 7y + x = 31       (D) x – 7y = 30

Ans : (C)

58.  The domain of definition of the function $f(x) = \sqrt {1 + {{\log }_e}(1 - x)}$  is

(A) $- \infty < x \le 0$       (B) $- \infty < x \le {{e - 1} \over e}$      (C) $- \infty < x \le 1$     (D) $x \ge 1 - e$

Ans : (B)

59.  For what value of m, ${{{a^{m + 1}} + {b^{m + 1}}} \over {{a^m} + {b^m}}}$  is the arithmetic meanof ‘a’ and ‘b’ ?

(A) 1       (B) 0       (C) 2       (D) None

Ans : (B)

60.  The value of the limit ${\lim }\limits_{x \to 1} {{\sin ({e^{x - 1}} - 1)} \over {\log x}}$  is

(A) 0      (B) e       (C) ${1 \over e}$       (D) 1

Ans : (D)

61.   Let $f(x) = {{\sqrt {x + 3} } \over {x + 1}}$  then the value of ${Lt}\limits_{x \to - 3 - 0} f(x)$  is

(A) 0      (B) does not exist      (C) ${1 \over 2}$       (D) $-{1 \over 2}$

Ans : (B)

62.  ƒ(x) = x + | x | is continuous for

(A) x∈(−∞,∞)        (B) x∈(−∞,∞) −{0}        (C) only x > 0        (D) no value of x

Ans : (A)

63.   $\tan \left[ {{\pi \over 4} + {1 \over 2}{{\cos }^{ - 1}}\left( {{a \over b}} \right)} \right] + \tan \left[ {{\pi \over4} - {1 \over 2}{{\cos }^{ - 1}}\left( {{a \over b}} \right)} \right]$ is equal to

(A) ${{2a} \over b}$        (B) ${{2b} \over a}$        (C) ${a \over b}$        (D) ${b \over a}$

Ans : (B)

64.  If $i = \sqrt { - 1}$  and n is a positive integer, then ${i^n} + {i^{n + 1}} + {i^{n + 2}} + {i^{n + 3}}$ is  euqal to

(A) 1       (B) i       (C) iⁿ       (D) 0

Ans : (D)

65.   $\int {{{dx} \over {x(x + 1)}}}$ equals

(A) $ln\left| {{{x + 1} \over x}} \right| + c$       (B) $ln\left| {{x \over {x + 1}}} \right| + c$       (C)$ln\left| {{{x - 1} \over x}} \right| + c$       (D) $ln\left| {{{x - 1} \over {x + 1}}} \right| + c$

where c is an arbitrary constant.

Ans : (B)

66.   If a, b, c are in G.P. (a > 1,  b > 1,  c > 1), then for any real number x (with x > 0,  x ≠ 1), log_a x ,  log_b x, log_c x are in

(A) G..P.     (B) A.P.      (C) H.P.      (D) G..P. but not in H.P.

Ans : (C)

67.  A line through the point A (2, 0) which makes an angle of 30° with the positive direction of x-axis is rotated about A in clockwise direction through an angle 15°. Then the equation of the straight line in the new position is

(A) (2 - √3)x + y - 4 + 2√3 = 0        (B) (2 - √3)x - y - 4 + 2√3 = 0

(C) (2 - √3)x - y + 4 + 2√3 = 0        (D) (2 - √3)x + y + 4 + 2√3 = 0

Ans : (B)

68.  The equation $\sqrt 3 \sin x + \cos x = 4$ has

(A) only one solution        (B) two solutions        (C) infinitely many solutions        (D) no solution

Ans : (D)

69.  The slope at any point of a curve y = ƒ(x) is given by ${{dy} \over {dx}} = 3{x^2}$ and it passes through (–1 , 1). The equation of the curve is

(A) y = x³ + 2        (B) y = – x³ – 2        (C) y = 3x³ + 4       (D) y = – x³ + 2

Ans : (A)

70.  The modulus of ${{1 - i} \over {3 + i}} + {{4i} \over 5}$ is

(A) $\sqrt 5$ unit       (B) ${{\sqrt {11} } \over 5}$ unit        (C) ${{\sqrt 5 } \over 5}$ unit     (D) ${{\sqrt {12} } \over 5}$ unit

Ans : (C)

71.  The equation of the tangent to the conic x² – y² – 8x + 2y + 11 = 0 at (2, 1) is

(A) x + 2 = 0        (B) 2x + 1 = 0        (C) x + y + 1 = 0         (D) x – 2 = 0

Ans : (D)

72.  A and B are two independent events such that P(A∪B') = 0.8 and P(A) = 0.3. The P(B) is

(A) ${2 \over 7}$       (B)  ${2 \over 3}$       (C)  ${3 \over 8}$        (D) ${1 \over 8}$

Ans : (A)

73.  The total number of tangents through the point (3, 5) that can be drawn to the ellipses 3x² + 5y² = 32 and 25x² + 9y² = 450 is

(A) 0       (B) 2      (C) 3       (D) 4

Ans : (C)

74.  The value of ${\lim }\limits_{n \to \infty } \left[ {{n \over {{n^2} + {1^2}}} + {n \over {{n^2} + {2^2}}} + \cdots \cdots {n \over {{n^2} + {n^2}}}} \right]$ is

(A) ${\pi \over 4}$      (B) log 2      (C) zero      (D)1

Ans : (A)

75.   A particle is moving in a straight line. At time t, the distance between the particle from its starting pointis given by x = t – 6t² + t³.  Its acceleration will be zero at

(A) t = 1 unit time         (B) t = 2 unit time         (C) t = 3 unit time         (D) t = 4 unit time

Ans : (B)

76.  Three numbers are chosen at random from 1 to 20.  The probability that they are consecutive is

(A) ${1 \over {190}}$        (B) ${1 \over {120}}$       (C) ${3 \over {190}}$       (D) ${5\over {190}}$

Ans : (C)

77.  The co-ordinates of the foot of the perpendicular from (0, 0) upon the line x + y = 2 are

(A) (2, –1)        (B) (–2, 1)       (C) (1, 1)        (D) (1, 2)

Ans : (C)

78.  If A is a square matrix then,

(A) A + A^T is symmetric      (B) AA^T is skew - symmetric     (C) A^T +  A is skew-symmetric     (D) A^TA is skew symmetric

Ans : (A)

79.   The equation of the chord of the circle x² + y² – 4x = 0 whose mid point is (1, 0) is

(A) y = 2       (B) y = 1       (C) x = 2       (D) x = 1

Ans : (D)

80.  If A² – A + I = 0, then the inverse of the matrix A is

(A) A – I       (B) I – A       (C) A + I       (D) A

Ans : (B)

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