## WBJEE 2014 | Mathematics

- Total No of Questions - 50
- Time Limit - 50 minutes
- No negative marking

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Question 1 |

If the normal at the one end of the latus rectum of an ellipse x

^{2}/a^{2}+ y^{2}/b^{2}= 1 passes through the one end of the minor axis, then :A | e ^{4} - e^{2} + 1 = 0 |

B | e ^{2} - e + 1 = 0 |

C | e ^{2} + e + 1 = 0 |

D | e ^{2} + e^{2} - 1 = 0 |

Question 2 |

If a circle of radius 3k passes through the origin and meet the axes at A and B, the locus of the centroid of ΔOAB is :

A | xz ^{2} + y^{2} = k^{2} |

B | x ^{2} + y^{2} = 2k^{2} |

C | x ^{2} +y^{2} = 3k^{2} |

D | none of these |

Question 3 |

If tan θ = - 4/3, then sin θ is :

A | -4/5 but not 4/5 |

B | -4/5 or -4/5 |

C | 4/5 but not -4/5 |

D | none of the above |

Question 4 |

If the complex numbers z

_{1}, z_{2}, z_{3}are in AP. Then they lie on aA | circle |

B | parabola |

C | line |

D | ellipse |

Question 5 |

If f(x) = |log|x||, then

A | f(x) is continuous and differentiable for all x in its domain. |

B | f(x) is continuous for all x in its domain but not differentiable at x=1 or x=-1 |

C | f(x) is neither continuous nor differentiable at x = 1 or x = -1 |

D | none of the above |

Question 6 |

The side of a triangle are 3x+4y, 4x+3y and 5x+5y unit where x,y>0. The triangle is :

A | right angled |

B | equilateral |

C | obtuse angle |

D | none of these |

Question 7 |

The two parts of 100 for which the sum double of first and square of second part is minimum, are :

A | 60, 2 |

B | 99, 1 |

C | 98, 3 |

D | none of these |

Question 8 |

The eccentricity of the hyperbola with latus rectum 12 and semi-conjugate axis 2√3 is

A | 2 |

B | 3 |

C | √3/2 |

D | 2√3 |

Question 9 |

If two circles a(x

^{2}+ y^{2}) +bx +cy = 0 and A(x^{2}) + y^{2}) +Bx +Cy = 0 touch each other then :A | aC = cA |

B | bC = cB |

C | aB = bA |

D | aA=bB=cC |

Question 10 |

The eccentricity of the hyperbola 3x

^{2}- 4y^{2}= -12 is equal to :A | √(7/3) |

B | √7/2 |

C | -√7/3 |

D | -√7/2 |

Question 11 |

For the curve x = t

^{2}- 1, y = t^{2}- t, the tangent line is perpendicular to x-axis where :A | t = 0 |

B | t = ∞ |

C | t = 1/√3 |

D | t = - 1/√3 |

Question 12 |

If the function f : R → A given by x²/(x² + 1) is surjection, then A is equal to ;

A | R |

B | [0,1] |

C | (0,1) |

D | [0,1) |

Question 13 |

The point of intersection of the curve whose parametric eqn are x = t

^{2}+ 1, y = 2t and x = 2s, y = 2/s is given by :A | (1,-3) |

B | (2,2) |

C | (-2,4) |

D | (1,2) |

Question 14 |

If [x] denotes the greatest integer less than or equal to x, then lim

_{x→∞ }[1²x] + [2²x] + [3²x] + ... + [n²x] equal :A | x/2 |

B | x/3 |

C | x/6 |

D | 0 |

Question 15 |

The eqn asin x + bcos x + c, where |c|>√(a

^{2}+ b^{2}) has :A | a unique solution |

B | infinite no of solutions |

C | no solution |

D | none of the above |

Question 16 |

the common tangents to the circle x

^{2}+ y^{2}+ 2x = 0 and x^{2}+ y^{2}- 6x = 0 form a triangle which isA | equilateral |

B | isosceles |

C | right angled |

D | none of the above |

Question 17 |

The curve for which the normal at any point (x,y) and line joining origin to the points from the isosceles triangle with x axis as base is :

A | an ellipse |

B | a rectangular hyperbola |

C | a circle |

D | none of the above |

Question 18 |

A farmer has to go 500 m due north, 400 m dues east and 200 m due south to reach his field. If he takes 20 m to reach the feild, then a average speed of farmer during the walk is:

A | 50 m/min |

B | 55 m/min |

C | 60 m/min |

D | 57 m/min |

Question 19 |

If x = 2 + 2

^{2/3}+ 2^{1/3}, then the value of x^{3}- 6x^{2}+ 6x is :A | 3 |

B | 2 |

C | 1 |

D | none of these |

Question 20 |

The value of (0.16)

^{log25[1/3 + 1/32 + 1/33 ... + ∞]}is :A | 2 |

B | 3 |

C | 4 |

D | none of these |

Question 21 |

If y

^{1/m}= [x + √(1+x^{2})], then (1+x^{2})y_{2}+ xy_{1}is equal to :A | m ^{2}y |

B | my ^{2} |

C | m ^{2}y^{2} |

D | none of these |

Question 22 |

In a triangle ABC sin A - cos B = cos C, then angle B is :

A | pi/2 |

B | pi/3 |

C | pi/4 |

D | pi/6 |

Question 23 |

An ellipse slide between two perpendicular straight lines. Then the locus of the centre is a/an :

A | parabola |

B | ellipse |

C | hyperbola |

D | circle |

Question 24 |

A perticle is moving with a velocity of v = (3+6t+9t

^{2}) cm/s. The displacement of the perticle in the interval t=5s to t=8s is:A | 1287 m |

B | 1285 m |

C | 1280 m |

D | 1290 m |

Question 25 |

x

_{1}, x_{2}, x_{3}, x_{4}are the root pf the eqn x^{4}- x^{3}sin 2β + x^{2}cos 2β - x cosβ - sinβ = 0 then tan^{-1}x_{1}+ tan^{-1}x_{2}+ tan^{-1}x_{3}+ tan^{-1}x_{4}is equal to :A | β |

B | pi/2 - β |

C | pi - β |

D | - β |

Question 26 |

The image of the point (3,8) in the line x + 3y = 7 is :

A | (1,4) |

B | (4,1) |

C | (-1,-4) |

D | (-4,-1) |

Question 27 |

The eqn of the tangents to 2x

^{2}- 3y^{2}= 36 which are parallel to the straight line x + 2y - 10 = 0 are :A | x+2y=0 |

B | x+2y+√288/15=0 |

C | x+2y+√1(/15)=0 |

D | none of these |

Question 28 |

An equilateral triangle is inscribed in the parabola y

^{2}= 4ax whose one vertex is at the vertex of the parabola, the length of its side is :A | 4a√3 |

B | 2a√3 |

C | 16a√3 |

D | 8a√3 |

Question 29 |

AB is a diameter of x

^{2}+ 9y^{2}= 25. The eccentric angle of A is pi/6, then the eccentric angle of B is :A | 5 pi/6 |

B | - 5pi/6 |

C | - 2pi/3 |

D | none of these |

Question 30 |

The locus of point z satisfying Re(1/z) = k, where k is a non-zero real number, is :

A | a straight line |

B | a circle |

C | an ellipse |

D | a hyperbola |

Question 31 |

If sin A + cos A = m and sin

^{3}A + cos^{3}A = n, thenA | m ^{3} - 3m + n = 0 |

B | m ^{3} - 3n + 2m= 0 |

C | m ^{3} - 3m + 2n = 0 |

D | m ^{3} + 3m + 2n = 0 |

Question 32 |

The smallest angle of the triangle whose sides are 6 + √12, √48, √24 is :

A | pi/3 |

B | pi/4 |

C | pi/6 |

D | none of these |

Question 33 |

The set of points where the function f(x) = x|x| is differentiable is :

A | (-∞,∞) |

B | (-∞,0) union (0,∞) |

C | (0,∞) |

D | [0,∞) |

Question 34 |

The locus of the point z satisfying Re(1/z) = k, where k is a non-zero real number, is :

A | a straight line |

B | a circle |

C | an ellipse |

Question 35 |

AB is a chord of the parabola y

^{2}= 4ax with vertex at ABC is drawn perpendicular to AB meeting the axes at C, the projection of BC on the axis of the parabola is :A | a |

B | 2a |

C | 4a |

D | 8a |

Question 36 |

In a triangle ABC, the value of 1 - tan B/2 tan C/2 = 2a/(a + b + c) is equal to :

A | 2a/(a + b + c) |

B | 2/(a + b + c) |

C | 2a/(a - b + c) |

D | none of these |

Question 37 |

The no of natural nos smaller than 10

^{4}in the decimal notation of which all the digit are different, is :A | 5274 |

B | 5265 |

C | 4676 |

D | none of these |

Question 38 |

If the parametric eqn of a curve is given x = e

^{t}cos t, y = e^{t}sin t, then the tangent to curve at the point t = pi/4 makes with the x axis of the angle :A | 0 |

B | pi/4 |

C | pi/3 |

D | pi/2 |

Question 39 |

Ten different letters of an alphabet are given, words with five letters are formed from these given letters. Then the number of words which have at least one letter repeated, is :

A | 69760 |

B | 30240 |

C | 99748 |

D | none of these |

Question 40 |

If cos α + cos β + cos γ = sin α + sin β + sin γ = 0, then the value of cos 3α + cos 3β + cos 3γ is

A | 0 |

B | cos (α + β + γ) |

C | 3 cos(α + β + γ) |

D | 3 sin(α + β + γ) |

Question 41 |

If f(x) = ax + b and g(x) = cx + d, then f(g(x)) = g(f(x))↔

A | f(a) = g(c) |

B | f(b) = g(b) |

C | f(d) = g(b) |

D | f(c) = g(a) |

Question 42 |

The principle value of sin

^{-1}(sin 2pi/3) is:A | 2pi/3 |

B | -2pi/3 |

C | pi/3 |

D | 4pi/3 |

Question 43 |

The function f(x) = max {(1 - x), (1 + x), 2}, x belongs to (-∞, ∞ ) is :

A | continuous at all points |

B | differentiable at all points |

C | not differentiable at all points except at x = 1 and x = -1 |

D | continuous at all points except at x = 1, and x = -1 where it discontinuous |

Question 44 |

Let a, b, c be real numbers, a ≠ 0, If α is a root of a²x² + bx + c = 0, β is the root of a²x² - bx - c = 0 and 0<α<β, then the eqn a²x² + 2bx + 2c = 0 has root γ that always satisfies :

A | γ = (α + β)/2 |

B | γ = α + β/2 |

C | γ = α/2 + β |

D | α<γ<β |

Question 45 |

If M is the foot of the perpendicular from a point P on a parabola to its directrix and SPM is an equilateral triangle, where S is the locus, then SP is equal to ;

A | a |

B | 2a |

C | 3a |

D | 4a |

Question 46 |

The plane 2x - (1+ρ)y + 3ρz = 0 passes through the intersection of the planes :

A | 2x-y=0 and y+3z=0 |

B | 2x-y=0 and y-3z=0 |

C | 2x+3z=0 and y=0 |

D | none of the above |

Question 47 |

If x, y, z are in AP as well as in GP and x+3, y+3, z+3 are in HP, then :

A | y = 2 |

B | y = 3 |

C | y = 1 |

D | y = 0 |

Question 48 |

If a,b,c are in GP, then the eqn ax

^{2}+ 2bx + c = 0 and dx^{2}+ 2ex + f = 0 have a common root, if d/a,e/b,f/c are in :A | AP |

B | GP |

C | HP |

D | none of these |

Question 49 |

Let a = cos 2pi/7 + i sin 2pi/7, α= a + a

^{2}+a^{4}and β= a^{3}+ a^{5}+ a^{6}.Then the eqn whose roots are α,β is :A | x ^{2} - x + 2 = 0 |

B | x ^{2} + x -2 = 0 |

C | x ^{2} - x -2 = 0 |

D | x ^{2} + x +2 = 0 |

Question 50 |

The ordr of the differentiable wqn associated with the primitive y = c

_{1}+ c_{2}e^{x}+ c_{3}e^{-2x + c4 }, where c_{1}, c_{2}, c_{3}, c_{4}are arbitery constants,A | 3 |

B | 4 |

C | 2 |

D | none of these |

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