## WBJEE 2014 | Mathematics

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

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Question 1 |

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 2 |

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 3 |

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 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 |

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 6 |

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 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 |

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 9 |

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 10 |

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 11 |

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 12 |

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 13 |

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 14 |

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 15 |

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

A | (-∞,∞) |

B | (-∞,0) union (0,∞) |

C | (0,∞) |

D | [0,∞) |

Question 16 |

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 17 |

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 18 |

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 19 |

The principle value of sin

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

B | -2pi/3 |

C | pi/3 |

D | 4pi/3 |

Question 20 |

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 21 |

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 22 |

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 23 |

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 24 |

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 25 |

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 26 |

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 27 |

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 28 |

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 29 |

The value of (0.16)

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

B | 3 |

C | 4 |

D | none of these |

Question 30 |

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 31 |

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 32 |

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 33 |

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 34 |

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 35 |

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 36 |

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 37 |

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 38 |

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 39 |

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 40 |

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 41 |

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 42 |

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 43 |

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 44 |

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 45 |

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 46 |

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 |

Question 47 |

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 48 |

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 49 |

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 50 |

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) |

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