JEE Questions

Trigonometry

The sum of the radii of inscribed and circumscribed circles for an n sided regular polygon of side a, is

  1. a/2 cot(π/2n)
  2. a/4 cot(π/2n)
  3. a cot(π/n)
  4. a cot(π/2n)

Trigonometry

Let cos(a + b) = 4/5 and let sin(a - b) = 5/13 where 0 ≤ a,b ≤ π/4. Then tan 2a is

  1. 20/17
  2. 25/16
  3. 56/33
  4. 19/12

Trigonometry

In a ∆PQR, if 3 sin P + 4 cos Q = 6 and 4 sin Q + 3 cos P = 1, then the angle R is equal to

  1. π/4
  2. 3π/4
  3. 5π/6
  4. π/6

Trigonometry

If A = sin2 x + cos4 x, then for all real x

  1. 3/4 ≤ A ≤ 1
  2. 1 ≤ A ≤ 2
  3. 3/4 ≤ A ≤ 13/16
  4. 13/16 ≤ A ≤ 1

Trigonometry

Let fk(x) = 1/k(sink x + cosk x) where x ∈ R and k ≥ 1. Then f4(x) - f6(x) equals

  1. 1/12
  2. 1/6
  3. 1/4
  4. 1/3

Statistics Probability

The mean and variance of a random variable X having binomial distribution are 4 and 2 respectively, then P (X = 1) is

  1. 1/4
  2. 1/8
  3. 1/16
  4. 1/32

Probability

Consider 5 independent Bernoulli's trials each with probability of success P. If the probability of at least one failure is greater than or equal to 31/32, then P lies in the interval

  1. (3/4,11/2]
  2. (11/2,1]
  3. (1/2,3/4]
  4. [0,1/2]

Probability

Three numbers are chosen at random without replacement from {1, 2, 3, ...... 8}. The probability that their minimum is 3, given that their maximum is 6, is

  1. 2/5
  2. 1/4
  3. 3/8
  4. 1/5

Probability

Assuming the balls to be identical except for difference in colours, the number of ways in which one or more balls can be selected from 10 white, 9 green and 7 black balls is

  1. 880
  2. 879
  3. 629
  4. 630

Probability

One ticket is selected at random from 50 tickets numbered 00, 01, 02, ... , 49. Then the probability that the sum of the digits on the selected ticket is 8, given that the product of these digits is zero, equals

  1. 1/7
  2. 1/14
  3. 5/14
  4. 1/50

Probability

Five horses are in a race. Mr.A selects two of the horses at random and bets on them. The probability that Mr.A selected the winning horse is

  1. 1/5
  2. 2/5
  3. 3/5
  4. 4/5

Statistics Sequences Series

If the mean deviation about the median of the numbers a, 2a, ..., 50a is 50, then |a| equals

  1. 2
  2. 3
  3. 4
  4. 5

Statistics

The average marks of boys in class is 52 and that of girls is 42. The average marks of boys and girls combined is 50. The percentage of boys in the class is

  1. 20
  2. 40
  3. 60
  4. 80

Statistics Sequences Series

If the mean deviation of the numbers 1, 1 + d, 1 + 2d, ... , 1 + 100d from their mean is 255, then the d is equal to

  1. 10
  2. 10.1
  3. 20
  4. 20.2

Statistics

In a frequency distribution, the mean and median are 21 and 22 respectively, then its mode is approximately

  1. 25.5
  2. 24.0
  3. 22.0
  4. 20.5

3D Geometry

The angle between the lines whose direction cosines satisfy the equations l + m + n = 0 and l2 + m2 + n2 = 0 is

  1. π/2
  2. π/3
  3. π/4
  4. π/6

3D Geometry

If the lines (x - 1)/2 = (y + 1)/2 = (z - 1)/4 and (x - 3)/1 = (y - k)/2 = z/1 intersect, then k is equal to

  1. 9/2
  2. 2/9
  3. -1
  4. 0

3D Geometry

If the angle between the line x = (y - 1)/2 = (z - 3)/λ and the plane x + 2y + 3z = 4 is cos-1(5/14) then λ is equal to

  1. 3/2
  2. 2/5
  3. 2/3
  4. 5/3

3D Geometry

A line AB in three-dimensional space makes angles 45° and 120° with the positive x-axis and the positive y-axis respectively. If AB makes an acute angle θ with the positive z-axis, then θ equals

  1. 30°
  2. 45°
  3. 60°
  4. 75°

3D Geometry

Let the line (x - 2)/3 =  (y - 1)/-5 =  (z - 2)/2 lie in the plane x + 3y - αz + β = 0. Then (α, β) equals

  1. (-5, 5)
  2. (5, -15)
  3. (-6, 7)
  4. (6, -17)

3D Geometry

The distance of the point (1,0,2) from the point of intersection of the line (x - 2)/3 = (y + 1)/4 = (z - 2)/12 and the plane x - y + z = 16 is 

  1. 13
  2. 8
  3. 3√21
  4. 2√14

Coordinate Geometry

Let P be the point on the parabola, y2 = 8x which is at a minimum distance from the centre C of the circle, x2 + (y + 6)2 = 1. Then the equation of the circle, passing through C and having its centre at P is:

  1. x2 + y2 – x/4 + 2y – 24 = 0
  2. x2 + y2 – 4x + 8y + 12 = 0
  3. x2 + y2 – 4x + 9y + 18 = 0
  4. x2 + y2 – x + 4y – 12 = 0

Coordinate Geometry

The locus of the foot of perpendicular drawn from the center of the ellipse x2 + 3y2 = 6 on any tangent to it is

  1. (x2 - y2)2 = 6x2 + 2y2
  2. (x2 - y2)2 = 6x2 - 2y2
  3. (x2 + y2)2 = 6x2 - 2y2
  4. (x2 + y2)2 = 6x2 + 2y2

Coordinate Geometry

An ellipse is drawn by taking a diameter of the circle (x - 1) + y = 1 as its semi-minor axis and a diameter of the circle x2 + (y - 2)2 = 4 as its semi-major axis. If the centre of the ellipse is the origin and its axes are the coordinate axes, then the equation of the ellipse is

  1. x2 + 4y2 = 8
  2. 4x2 + y2 = 4
  3. x2 + 4y2 = 16
  4. 4x2 + y2 = 8

Coordinate Geometry

The slope of the line touching both the parabolas y2 = 4x and x2 = -32y is

  1. 1/2
  2. 2/3
  3. 3/2
  4. 1/8

Coordinate Geometry

The eccentricity of the hyperbola whose length of the latus rectum is equal to 8 and the length of its conjugate axis is equal half of the distance between its foci, is:

  1. 4/√3
  2. √3
  3. 4/3
  4. 2/√3

Coordinate Geometry

Let O be the vertex and Q be any point on the parabola, x2 = 8y. If the point P divides the line segment OQ internally in the ratio 1 : 3, then the locus of P is

  1. x2 = y
  2. y2 = x
  3. x2 = 2y
  4. y2 = 2x

Coordinate Geometry

Let a, b, c and d be non-zero numbers. If the point of intersection of the lines 4ax + 2ay + c = 0 and 5bx + 2by + d = 0 lies in the fourth quadrant and is equidistant from the two axes then

  1. 3bc - 2ad = 0
  2. 3bc + 2ad = 0
  3. 2bc + 3ad = 0
  4. 2bc - 3ad = 0

Coordinate Geometry

If the pair of straight lines x2 - 2pxy - y2 = 0 and x2 - 2qxy - y2 = 0 be such that each pair bisects the angle between the other pair, then

  1. pq = -1
  2. p = -q
  3. pq = 1
  4. p = q

Coordinate Geometry

The shortest distance between the line y - x = 1 and curve y = x2 is

  1. √3/4
  2. 4/√3
  3. 8/3√2
  4. 3√2/8

Coordinate Geometry

If the line 2x + y = k passes through the point which divides the line segment joining the points (1, 1) and (2, 4) in the ratio 3:2, then k equals

  1. 5
  2. 6
  3. 29/5
  4. 11/5

Coordinate Geometry

The lines 2x - 3y = 5 and 3x - 4y = 7 are diameters of a circle having area as 154 sq. units. Then the equation of the circle is

  1. x2 + y2 + 2x + 2y = 47
  2. x2 + y2 + 2x - 2y = 47
  3. x2 + y2 - 2x + 2y = 62
  4. x2 + y2 - 2x + 2y = 47

Coordinate Geometry

The two circles x2 + y2 = ax and x2 + y2 = c2 (c > 0) touch each other if

  1. a = 2c
  2. |a| = 2c
  3. |a| = c
  4. 2|a| = c

Differential Equation

The differential equation which represents the family of curves y = c1ec2x where c1 and c2 are arbitrary constants, is

  1. yy'' = y'
  2. y'' = y'y
  3. y' = y2
  4. yy'' = (y')2

Differential Equation

Let y(x) be the solution of the differential equation (x log x)(dy/dx) + y = 2x log(x), (x≥1). Then y(e) is equal to

  1. 0
  2. 2e
  3. e
  4. 2

Differentiation

If dy/dx = y + 3 > 0 and y(0) = 2, then y(ln2) is equal to

  1. -13
  2. 5
  3. 7
  4. 2

Differential Equation

The solution of the equation d2y/dx2 = e-2x is

  1. e-2x/4
  2. 1/4 * e-2x + cx2 + d
  3. e-2x/4 + cx + d
  4. 1/4 * e-4x + cx + d

Integration

The integral 0π √(1 + 4 sin2x/2 - 4 sinx/2) dx equals

  1. 4√3 - 4
  2. 4√3 - 4 - π/3
  3. 2π/3 - 4 - 4√3
  4. π - 4

Integration

The integral ∫(1 + x - 1/x)ex + 1/x dx  is equal to

  1. xex + 1/x + c
  2. (x + 1)ex + 1/x + c
  3. (x - 1)ex + 1/x + c
  4. -xex + 1/x + c

Integration

If the integral ∫ (5 tan x / tan x − 2)dx = x + a ln |sin x - 2 cos x| + k, then a is equal to

  1. -1
  2. 2
  3. -2
  4. 1

Integration

If g(x) = 0x cos4t dt, then g(x + π) equals

  1. g(x) - g(π)
  2. g(x) / g(π)
  3. g(x) + g(π)
  4. g(x).g(π)

Integration

0π [cot x] dx, where [.] denotes the greatest integer function, is equal to

  1. -π/2
  2. 1
  3. -1
  4. π/2

Limits Continuity

Let f(x) = 4 and f′(x) = 4. Then Limx→2 (xf(2) - 2f(x))/(x-2) is given by

  1. -4
  2. 2
  3. -2
  4. 3

Limits Continuity

If f: R → R is a function defined by f(x) = [x] cos((2x - 1)/2)π, where [x] denotes the greatest integer function, then f is

  1. discontinuous only at x = 0
  2. discontinuous only at non-zero integral values of x
  3. continuous only at x = 0
  4. continuous for every real x

Differentiation

Let y be an implicit function of x defined by x2x - 2xcot y - 1 = 0. Then y′(1) equals

  1. -1
  2. 1
  3. -log 2
  4. log 2

Limits Continuity

The value of p and q for which the function

f(x) = (sin(p + 1)x + sinx)/x, x < 0
f(x) = q, x = 0
f(x) = (√(x + x2) - √x)/x3/2, x > 0

is continuous for all x in R, are:

  1. p = 1/2 , q = 3/2
  2. p = -3/2 , q = 1/2
  3. p = 1/2 , q = -3/2
  4. p = 5/2 , q = 1/2

Sequences Series

Let f(x) be a polynomial function of second degree. If f(1) = f(–1) and a, b, c are in A.P., then f'(a), f'(b) and f'(c) are in

  1. G.P.
  2. A.P.
  3. A.P. - G.P.
  4. H.P.

Sequences Series

Let the sum of the first three terms of an A.P. be 39 and the sum of its last four terms be 178. If the first term of this A.P. is 10, then the median of the A.P. is

  1. 31
  2. 29.5
  3. 28
  4. 26.5

Sequences Series

The sum to infinity of the series 1 + 2/3 + 6/32 + 10/33 + ... is

  1. 2
  2. 3
  3. 4
  4. 6

Sequences Series

Three positive numbers form an increasing G.P. If the middle term in this G.P. is doubled, the new numbers are in A.P. Then the common ratio of the G.P. is

  1. √2 + √3
  2. 2 + √3
  3. 2 - √3
  4. 3 + √2