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JEE Main 2019 questions

449 questions, by subject. Each opens with its options and answer.

Mathematics

  1. If z and w are two complex numbers such that |zw| = 1 and arg(z) - arg(w) = (π)/2, then

    Complex Numbers and Quadratic Equations · with solution

  2. Let P(4, -4) and Q(9, 6) be two points on the parabola, y^2 = 4x and let X be any point on the arc POQ of this parabola, where O is the vertex of this parabola, such that the area of triangle PXQ is maximum. Then, this…

    Conic Sections

  3. If the tangent to the curve y = x/(x^2 - 3), x ∈ R, (x ≠ ±√(3)), at a point (α, β) ≠ (0, 0) on it is parallel to the line 2x + 6y - 11 = 0, then

    Application of Derivatives

  4. Let f(x) = 5 - |x - 2| and g(x) = |x + 1|, x ∈ R. If f(x) attains maximum value at α and g(x) attains minimum value of β, then lim_x → -αβ ((x-1)(x^2 - 5x + 6))/(x^2 - 6x + 8) is equal to

    Application of Derivatives · with solution

  5. Let a, b and c be in GP with common ratio r, where a ≠ 0 and 0 < r ≤ 1/2. If 3a, 7b and 15c are the first three terms of an AP, then the 4th term of this AP is

    Sequences and Series · with solution

  6. If a_1, a_2, a_3, …, a_n are in AP and a_1 + a_4 + a_7 + … + a_16 = 114, then a_1 + a_6 + a_11 + a_16 is equal to

    Sequences and Series · with solution

  7. The sum of series 1 + (1^3 + 2^3)/(1 + 2) + (1^3 + 2^3 + 3^3)/(1 + 2 + 3) + … + (1^3 + 2^3 + 3^3 + … + 15^3)/(1 + 2 + 3 + … + 15) - 1/2(1 + 2 + 3 + … + 15) is equal to

    Sequences and Series · with solution

  8. If ∫ x^2 e^-x^2 dx = g(x)e^-x^2 + C, where C is a constant of integration, then g(-1) is equal to

    Indefinite Integration

  9. The integral ∫ sec^2/3 x cosec^4/3 x dx is equal to (here C is a constant of integration)

    Indefinite Integration · with solution

  10. The value of int_-π/2^π/2 dx/([x] + [sin x] + 4), where [t] denotes the greatest integer less than or equal to t, is

    Definite Integrals and Areas

  11. A value of α such that int_α^α+1 dx/((x+α)(x+α+1)) = log_e (9/8 ) is

    Definite Integrals and Areas · with solution

  12. lim_n → ∞ ( frac(n+1)^1/3n^4/3 + frac(n+2)^1/3n^4/3 + ..... + frac(2n)^1/3n^4/3 ) is equal to

    Definite Integrals and Areas · with solution

  13. Let z = (dfrac√(3)2 + i/2 )^5 + (dfrac√(3)2 - i/2 )^5. If R(z) and I(z) respectively denote the real and imaginary parts of z, then

    Complex Numbers and Quadratic Equations

  14. The equation |z - i| = |z - 1|, i = √(-1), represents

    Complex Numbers and Quadratic Equations · with solution

  15. Let z ∈ C with Im(z) = 10 and it satisfies (2z - n)/(2z + n) = 2i - 1 for some natural number n, then

    Complex Numbers and Quadratic Equations

  16. If int_x^x f(t) dt = x^2 + int_x^1 t^2 f(t) dt, then f (1/2 ) is

    Definite Integrals and Areas

  17. If the area (in sq units) bounded by the parabola y^2 = 4λ x and the line y = λ x, λ > 0, is 1/9, then λ is equal to

    Definite Integrals and Areas · with solution

  18. If the area enclosed between the curves y = kx^2 and x = ky^2, (k > 0), is 1 square unit. Then, k is

    Definite Integrals and Areas

  19. Let f be a differentiable function such that f(1) = 2 and f'(x) = f(x) for all x ∈ R. If h(x) = f(f(x)), then h'(1) is equal to

    Differential Equations · with solution

  20. Given that the slope of the tangent to a curve y = y(x) at any point (x, y) is 2y/(x^2). If the curve passes through the centre of the circle x^2 + y^2 - 2x - 2y = 0, then its equation is

    Differential Equations · with solution

  21. The general solution of the differential equation (y^2 - x^3)dx - xy dy = 0 (x ≠ 0) is (where, C is a constant of integration)

    Differential Equations

  22. A straight line L at a distance of 4 units from the origin makes positive intercepts on the coordinate axes and the perpendicular from the origin to this line makes an angle of 60° with the line x + y = 0. Then, an…

    Straight Lines and Pair of Lines · with solution

  23. The sum sum_k=1^20 k · 1/(2^k) is equal to

    Sequences and Series

  24. A triangle has a vertex at (1, 2) and the mid-points of the two sides through it are (-1, 1) and (2, 3). Then, the centroid of this triangle is

    Straight Lines and Pair of Lines · with solution

  25. If the circles x^2 + y^2 + 5Kx + 2y + K = 0 and 2(x^2 + y^2) + 2Kx + 3y - 1 = 0, (K ∈ R), intersect at the points P and Q, then the line 4x + 5y - K = 0 passes through P and Q, for

    Circles · with solution

  26. The locus of the centres of the circles, which touch the circle, x^2 + y^2 = 1 externally, also touch the Y-axis and lie in the first quadrant, is

    Circles

  27. If the angle of intersection at a point where the two circles with radii 5 cm and 12 cm intersect is 90°, then the length (in cm) of their common chord is

    Circles · with solution

  28. If the parabolas y^2 = 4b(x - c) and y^2 = 8ax have a common normal, then which one of the following is a valid choice for the ordered triad (a, b, c)?

    Conic Sections

  29. If the area of the triangle whose one vertex is at the vertex of the parabola, y^2 + 4(x - a^2) = 0 and the other two vertices are the points of intersection of the parabola and Y-axis, is 250 sq units, then a value of…

    Conic Sections

  30. If the line ax + y = c, touches both the curves x^2 + y^2 = 1 and y^2 = 4√(2)x, then |c| is equal to

    Conic Sections · with solution

  31. An ellipse, with foci at (0, 2) and (0, -2) and minor axis of length 4, passes through which of the following points?

    Conic Sections · with solution

  32. If the normal to the ellipse 3x^2 + 4y^2 = 12 at a point P on it is parallel to the line, 2x + y = 4 and the tangent to the ellipse at P passes through Q(4, 1) then PQ is equal to

    Conic Sections

  33. The number of ways of choosing 10 objects out of 31 objects of which 10 are identical and the remaining 21 are distinct, is

    Permutations and Combinations · with solution

  34. A group of students comprises of 5 boys and n girls. If the number of ways, in which a team of 3 students can randomly be selected from this group such that there is at least one boy and at least one girl in each team,…

    Permutations and Combinations · with solution

  35. The maximum value of 3cosθ + 5sin (θ - (π)/6 ) for any real value of θ is

    Trigonometry · with solution

  36. The value of sin 10° sin 30° sin 50° sin 70° is

    Trigonometry · with solution

  37. The angle of elevation of the top of a vertical tower standing on a horizontal plane is observed to be 45° from a point A on the plane. Let B be the point 30 m vertically above the point A. If the angle of elevation of…

    Trigonometry

  38. All x satisfying the inequality (cot^-1 x)^2 - 7(cot^-1 x) + 10 > 0, lie in the interval

    Inverse Trigonometric Functions

  39. Let S be the set of all α ∈ R such that the equation, cos 2x + α sin x = 2α - 7 has a solution. Then, S is equal to

    Trigonometry · with solution

  40. The number of four-digit numbers strictly greater than 4321 that can be formed using the digits 0, 1, 2, 3, 4, 5 (repetition of digits is allowed) is

    Permutations and Combinations · with solution

  41. With the usual notation, in triangle ABC, if ∠ A + ∠ B = 120°, a = √(3) + 1 and b = √(3) - 1, then the ratio ∠ A : ∠ B is

    Trigonometry · with solution

  42. Two vertices of a triangle are (0, 2) and (4, 3). If its orthocentre is at the origin, then its third vertex lies in which quadrant?

    Straight Lines and Pair of Lines · with solution

  43. Let α = (λ - 2)a + b and β = (4λ - 2)a + 3b be two given vectors where vectors a and b are non-collinear. The value of λ for which vectors α and β are collinear, is

    Vector Algebra · with solution

  44. Let a = i - j, b = i + j + k and c be a vector such that a × c + b = 0 and a · c = 4, then |c|^2 is equal to

    Vector Algebra · with solution

  45. Let A(3, 0, -1), B(2, 10, 6) and C(1, 2, 1) be the vertices of a triangle and M be the mid-point of AC. If G divides BM in the ratio 2:1, then cos(∠ GOA) (O being the origin) is equal to

    Vector Algebra · with solution

  46. The sum of the distinct real values of μ, for which the vectors, mui+j+k, i+muj+k, i+j+muk are coplanar, is

    Vector Algebra · with solution

  47. Let a = 3i + 2j + 2k and b = i + 2j - 2k be two vectors. If a vector perpendicular to both of the vectors a + b and a - b has the magnitude 12, then one such vector is

    Vector Algebra

  48. The distance of the point having position vector -i + 2j + 6k from the straight line passing through the point (2, 3, -4) and parallel to the vector, 6i + 3j - 4k is

    Three Dimensional Geometry

  49. The number of 6 digits numbers that can be formed using the digits 0, 1, 2, 5, 7 and 9 which are divisible by 11 and no digit is repeated, is

    Permutations and Combinations

  50. Let (x+10)^50 + (x-10)^50 = a_0 + a_1x + a_2x^2 + … + a_50x^50 for all x ∈ R; then a_2/a_0 is equal to

    Binomial Theorem

  51. The coefficient of x^8 in the product (1+x)^11(1-x)^10(1+x+x^2)^9 is

    Binomial Theorem

  52. A person throws two fair dice. He wins ₹ 15 for throwing a doublet (same numbers on the two dice), wins ₹ 12 when the throw results in the sum of 9, and loses ₹ 6 for any other outcome on the throw. Then, the expected…

    Probability · with solution

  53. For an initial screening of an admission test, a candidate is given fifty problems to solve. If the probability that the candidate can solve any problem is 4/5, then the probability that he is unable to solve less than…

    Probability

  54. Assume that each born child is equally likely to be a boy or a girl. If two families have two children each, then the conditional probability that all children are girls given that at least two are girls; is

    Probability · with solution

  55. A pot contain 5 red and 2 green balls. At random a ball is drawn from this pot. If a drawn ball is green then put a red ball in the pot and if a drawn ball is red, then put a green ball in the pot, while drawn ball is…

    Probability

  56. A value of θ ∈ (0, π/3), for which beginvmatrix 1 + cos^2θ & sin^2θ & 4cos 6θ \ cos^2θ & 1 + sin^2θ & 4cos 6θ \ cos^2θ & sin^2θ & 1 + 4cos 6θ endvmatrix = 0 is

    Matrices and Determinants · with solution

  57. If [x] denotes the greatest integer ≤ x, then the system of liner equations [sinθ]x + [-cosθ]y = 0, [cotθ]x + y = 0

    Matrices and Determinants

  58. If B = beginbmatrix 5 & 2α & 1 \ 0 & 2 & 1 \ α & 3 & -1 endbmatrix is the inverse of a 3 × 3 matrix A, then the sum of all values of α for which det(A) + 1 = 0, is

    Matrices and Determinants

  59. If A is a symmetric matrix and B is a skew-symmetric matrix such that A + B = beginbmatrix 2 & 3 \ 5 & -1 endbmatrix, then AB is equal to

    Matrices and Determinants · with solution

  60. For x ∈ (0, 3/2 ), let f(x) = √(x), g(x) = tan x and h(x) = (1-x^2)/(1+x^2). If φ(x) = ((hof)og)(x), then φ ((π)/3 ) is equal to

    Sets Relations and Functions

  61. The domain of the definition of the function f(x) = 1/(4-x^2) + log_10(x^3 - x) is

    Sets Relations and Functions

  62. If the function f: R - \-1, 1\ → A defined by f(x) = (x^2)/(1 - x^2), is surjective, then A is equal to

    Sets Relations and Functions

  63. lim_x → 0 dfracx + 2sin x√(x^2 + 2sin x + 1) - √(sin^2 x - x + 1) is

    Limits Continuity and Differentiability · with solution

  64. If ^20C_1 + (2^2) ^20C_2 + (3^2) ^20C_3 + … + (20^2) ^20C_20 = A(2^β), then the ordered pair (A, β) is equal to

    Binomial Theorem

  65. If the function f defined on ((π)/6, (π)/3 ) by f(x) = begincases dfrac√(2)cos x - 1cot x - 1, & x ≠ (π)/4 \ k, & x = (π)/4 endcases is continuous, then k is equal to

    Limits Continuity and Differentiability

  66. Let f: R → R be a continuously differentiable function such that f(2) = 6 and f'(2) = 1/48. If int_6^f(x) 4t^3 dt = (x-2)g(x), then lim_x → 2 g(x) is equal to

    Limits Continuity and Differentiability

  67. Let f: R → R be a differentiable function satisfying f'(3) + f'(2) = 0. Then lim_x → 0 ((1 + f(3+x) - f(3))/(1 + f(2-x) - f(2)) )^1/x is equal to

    Limits Continuity and Differentiability

  68. If α and β are the roots of the equation 375x^2 - 25x - 2 = 0, then lim_n → ∞ sum_r=1^n α^r + lim_n → ∞ sum_r=1^n β^r is equal to

    Limits Continuity and Differentiability

  69. A spherical iron ball of radius 10 cm is coated with a layer of ice of uniform thickness that melts at a rate of 50 cm³/min. When the thickness of the ice is 5 cm, then the rate at which the thickness (in cm/min) of the…

    Application of Derivatives

  70. Let α ∈ (0, π/2) be fixed. If the integral ∫ (tan x + tan α)/(tan x - tan α) dx = A(x) cos 2α + B(x) sin 2α + C, where C is a constant of integration, then the functions A(x) and B(x) are respectively

    Indefinite Integration

  71. The common tangent to the circles x^2 + y^2 = 4 and x^2 + y^2 + 6x + 8y - 24 = 0 also passes through the point

    Circles

  72. A circle touching the X-axis at (3, 0) and making a intercept of length 8 on the Y-axis passes through the point

    Circles

  73. The number of real roots of the equation 5 + |2^x - 1| = 2^x(2^x - 2) is

    Complex Numbers and Quadratic Equations · with solution

  74. If α, β and γ are three consecutive terms of a non-constant GP such that the equations α x^2 + 2β x + γ = 0 and x^2 + x - 1 = 0 have a common root, then, α(β + γ) is equal to

    Complex Numbers and Quadratic Equations

  75. Let P be the point of intersection of the common tangents to the parabola y^2 = 12x and the hyperbola 8x^2 - y^2 = 8. If S and S' denotes the foci of the hyperbola where S lies on the positive X-axis then P divides SS'…

    Conic Sections · with solution

  76. The value of sin^-1 (12/13 ) - sin^-1 (3/5 ) is equal to

    Inverse Trigonometric Functions

  77. The angles A, B and C of a triangle ABC are in AP and a:b=1:√(3). If c=4 cm, then the area (in sq cm) of this triangle is

    Trigonometry

  78. All the points in the set S = \(α + i)/(α - i) : α ∈ R \ (i = √(-1)) lie on a

    Complex Numbers and Quadratic Equations · with solution

  79. The vertices B and C of a triangle ABC lie on the line, (x+2)/3 = (y-1)/0 = z/4 such that BC = 5 units. Then, the area (in sq units) of this triangle, given that the point A(1, -1, 2) is

    Three Dimensional Geometry · with solution

  80. The length of the perpendicular drawn from the point (2, 1, 4) to the plane containing the lines r = (i + j) + λ(i + 2j - k) and r = (i + j) + μ(-i + j - 2k) is

    Three Dimensional Geometry · with solution

  81. The integral ∫ (2x^3 - 1)/(x^4 + x) dx is equal to (here C is a constant of integration)

    Indefinite Integration

  82. If ∫ e^sec x (sec x tan x f(x) + (sec x tan x + sec^2 x)) dx = e^sec x f(x) + C, then a possible choice of f(x) is

    Indefinite Integration · with solution

  83. lim_n → ∞ ( n/(n^2+1^2) + n/(n^2+2^2) + n/(n^2+3^2) + … + 1/5n ) is equal to

    Definite Integrals and Areas

  84. If int_0^π/2 (cot x)/(cot x + cosec x) dx = m(π + n), then m · n is equal to

    Definite Integrals and Areas

  85. If z = dfrac√(3)2 + i/2 (i = √(-1) ), then (1 + iz + z^5 + iz^8)^9 is equal to

    Complex Numbers and Quadratic Equations

  86. If the area (in sq units) of the region \(x, y): y^2 ≤ 4x, x + y ≤ 1, x ≥ 0, y ≥ 0\ is a√(2) + b, then a - b is equal to

    Definite Integrals and Areas

  87. Consider the quadratic equation, (c-5)x^2 - 2cx + (c-4) = 0, c ≠ 5. Let S be the set of all integral values of c for which one root of the equation lies in the interval (0, 2) and its other root lies in the interval (2,…

    Complex Numbers and Quadratic Equations · with solution

  88. All the pairs (x, y) that satisfy the inequality 2^√(sin^2 x - 2sin x + 5) · dfrac14^sin^2 y ≤ 1 also satisfy the equation

    Trigonometry

  89. The solution of the differential equation, dy/dx = (x - y)^2, when y(1) = 1, is

    Differential Equations · with solution

  90. The curve amongst the family of curves represented by the differential equation, (x^2 - y^2)dx + 2xy dy = 0, which passes through (1, 1), is

    Differential Equations · with solution

  91. Consider the differential equation, y^2dx + (x - 1/y )dy = 0. If value of y is 1 when x = 1, then the value of x for which y = 2, is

    Differential Equations

  92. The equation y = sin x sin(x+2) - sin^2(x+1) represents a straight line lying in

    Trigonometry

  93. If 19th term of a non-zero AP is zero, then its (49th term) : (29th term) is

    Sequences and Series · with solution

  94. Let S_n denote the sum of the first n terms of an AP. If S_4 = 16 and S_6 = -48, then S_10 is equal to

    Sequences and Series · with solution

  95. The sum of series (3 × 1^3)/(1^2) + (5 × (1^3 + 2^3))/(1^2 + 2^2) + (7 × (1^3 + 2^3 + 3^3))/(1^2 + 2^2 + 3^2) + … upto 10th term, is

    Sequences and Series

  96. Let S_n = 1 + q + q^2 + ... + q^n and T_n = 1 + ((q+1)/2 ) + ((q+1)/2 )^2 + ... + ((q+1)/2 )^n, where q is a real number and q ≠ 1. If ^101C_1 + ^101C_2 · S_1 + ... + ^101C_101 · S_100 = α T_100, then α is equal to

    Sequences and Series

  97. Consider the set of all lines px + qy + r = 0 such that 3p + 2q + 4r = 0. Which one of the following statements is true?

    Straight Lines and Pair of Lines · with solution

  98. If a tangent to the circle x^2 + y^2 = 1 intersects the coordinate axes at distinct points P and Q, then the locus of the mid-point of PQ is

    Circles · with solution

  99. The sum of the squares of the lengths of the chords intercepted on the circle, x^2 + y^2 = 16, by the lines, x + y = n, n ∈ N, where N is the set of all natural numbers, is

    Circles · with solution

  100. A rectangle is inscribed in a circle with a diameter lying along the line 3y = x + 7. If the two adjacent vertices of the rectangle are (-8, 5) and (6, 5), then the area of the rectangle (in sq units) is

    Circles · with solution

  101. Let O(0, 0) and A(0, 1) be two fixed points, then the locus of a point P such that the perimeter of triangle AOP is 4, is

    Conic Sections · with solution

  102. The line x = y touches a circle at the point (1, 1). If the circle also passes through the point (1, -3), then its radius is

    Circles · with solution

  103. A circle cuts a chord of length 4a on the X-axis and passes through a point on the Y-axis, distant 2b from the origin. Then, the locus of the centre of this circle, is

    Circles · with solution

  104. The tangents to the curve y = (x-2)^2 - 1 at its points of intersection with the line x - y = 3, intersect at the point

    Conic Sections · with solution

  105. In an ellipse, with centre at the origin, if the difference of the lengths of major axis and minor axis is 10 and one of the foci is at (0, 5√(3)), then the length of its latus rectum is

    Conic Sections · with solution

  106. The tangent and normal to the ellipse 3x^2 + 5y^2 = 32 at the point P(2, 2) meets the X-axis at Q and R, respectively. Then, the area (in sq units) of the triangle PQR is

    Conic Sections · with solution

  107. If the line y = mx + 7√(3) is normal to the hyperbola (x^2)/24 - (y^2)/18 = 1, then a value of m is

    Conic Sections

  108. If 5x + 9 = 0 is the directrix of the hyperbola 16x^2 - 9y^2 = 144, then its corresponding focus is

    Conic Sections

  109. The value of cos^2 10° - cos 10° cos 50° + cos^2 50° is

    Trigonometry · with solution

  110. Let S be the set of all triangles in the xy-plane, each having one vertex at the origin and the other two vertices lie on coordinate axes with integral coordinates. If each triangle in S has area 50 sq. units, then the…

    Permutations and Combinations · with solution

  111. Suppose that 20 pillars of the same height have been erected along the boundary of a circular stadium. If the top of each pillar has been connected by beams with the top of all its non-adjacent pillars, then the total…

    Permutations and Combinations

  112. If cos^-1x - cos^-1y/2 = α, where -1 ≤ x ≤ 1, -2 ≤ y ≤ 2, x ≤ y/2, then for all x, y, 4x^2 - 4xycosα + y^2 is equal to

    Inverse Trigonometric Functions · with solution

  113. Given, (b+c)/11 = (c+a)/12 = (a+b)/13 for a triangle ABC with usual notation. If (cos A)/(α) = (cos B)/(β) = (cos C)/(γ), then the ordered triad (α, β, γ) has a value

    Trigonometry · with solution

  114. Let the equations of two sides of a triangle be 3x - 2y + 6 = 0 and 4x + 5y - 20 = 0. If the orthocentre of this triangle is at (1, 1) then the equation of its third side is

    Straight Lines and Pair of Lines

  115. Let α ∈ R and the three vectors a = αi + j + 3k, b = 2i + j - αk and c = αi - 2j + 3k. Then, the set S = \α : a, b and c are coplanar\

    Vector Algebra · with solution

  116. A committee of 11 members is to be formed from 8 males and 5 females. If m is the number of ways the committee is formed with at least 6 males and n is the number of ways the committee is formed with atleast 3 females,…

    Permutations and Combinations

  117. Let a , b and c be three unit vectors, out of which vectors b and c are non-parallel. If α and β are the angles which vector a makes with vectors b and c respectively and a × (b × c) = 1/2b, then |α - β| is equal to

    Vector Algebra

  118. Let a = 2i + λ_1j + 3k, b = 4i + (3-λ_2)j + 6k and c = 3i + 6j + (λ_3 - 1)k be three vectors such that b = 2a and a is perpendicular to c. Then a possible value of (λ_1, λ_2, λ_3) is

    Vector Algebra · with solution

  119. If a unit vector a makes angles (π)/3 with i, (π)/4 with j and θ ∈ (0, π) with k, then a value of θ is

    Vector Algebra

  120. Let a = i + 2j + 4k, b = i + λj + 4k and c = 2i + 4j + (λ^2 - 1)k be coplanar vectors. Then, the non-zero vector a × c is

    Vector Algebra

  121. If the fractional part of the number frac2^40315 is k/15, then k is equal to

    Binomial Theorem · with solution

  122. If the coefficients of x^2 and x^3 are both zero, in the expansion of the expression (1 + ax + bx^2)(1 - 3x)^15 in powers of x, then the ordered pair (a, b) is equal to

    Binomial Theorem

  123. In a random experiment, a fair die is rolled until two fours are obtained in succession. The probability that the experiment will end in the fifth throw of the die is equal to

    Probability

  124. Let a random variable X have a binomial distribution with mean 8 and variance 4. If P(X ≤ 2) = dfrack2^16, then k is equal to

    Probability · with solution

  125. Four persons can hit a target correctly with probabilities 1/2, 1/3, 1/4 and 1/8 respectively. If all hit at the target independently, then the probability that the target would be hit, is

    Probability · with solution

  126. The sum of the real roots of the equation beginvmatrix x & -6 & -1 \ 2 & -3x & x-3 \ -3 & 2x & x+2 endvmatrix = 0, is equal to

    Matrices and Determinants

  127. The total number of matrices A = beginbmatrix 0 & 2y & 1 \ 2x & y & -1 \ 2x & -y & 1 endbmatrix, (x, y ∈ R, x ≠ y) for which A^T A = 3I_3 is

    Matrices and Determinants

  128. Let λ be a real number for which the system of linear equations x + y + z = 6, 4x + λ y - λ z = λ - 2 and 3x + 2y - 4z = -5 has infinitely many solutions. Then λ is a root of the quadratic equation

    Matrices and Determinants · with solution

  129. Let f(x) = a^x (a > 0) be written as f(x) = f_1(x) + f_2(x), where f_1(x) is an even function and f_2(x) is an odd function. Then f_1(x + y) + f_1(x - y) equals

    Sets Relations and Functions · with solution

  130. Let f(x) = x^2, x ∈ R. For any A ⊂eq R, define g(A) = \x ∈ R : f(x) ∈ A\. If S = [0, 4], then which one of the following statements is not true?

    Sets Relations and Functions

  131. Let a function f:(0,∞) → (0,∞) be defined by f(x) = |1 - 1/x |. Then, f is

    Sets Relations and Functions

  132. If f: R → R is a differentiable function and f(2) = 6, then lim_x → 2 int_6^f(x) (2t dt)/((x-2)) is

    Limits Continuity and Differentiability · with solution

  133. The derivative of tan^-1 ((sin x - cos x)/(sin x + cos x) ), with respect to x/2, where (x ∈ (0, (π)/2 ) ) is

    Limits Continuity and Differentiability · with solution

  134. If f(x) = begincases (sin(p+1)x + sin x)/x, & x 0 endcases is continuous at x = 0, then the ordered pair (p, q) is equal to

    Limits Continuity and Differentiability

  135. If lim_x → 1 (x^2 - ax + b)/(x - 1) = 5, then a + b is equal to

    Limits Continuity and Differentiability · with solution

  136. lim_x → 1^- dfrac√(π) - sqrt2sin^-1x√(1-x) is equal to

    Limits Continuity and Differentiability · with solution

  137. If a > 0 and z = ((1+i)^2)/(a-i), has magnitude √(2/5), then z is equal to

    Complex Numbers and Quadratic Equations

  138. Let S be the set of all values of x for which the tangent to the curve y = f(x) = x^3 - x^2 - 2x at (x, y) is parallel to the line segment joining the points (1, f(1)) and (-1, f(-1)), then S is equal to

    Application of Derivatives · with solution

  139. If the volume of parallelopiped formed by the vectors i + λj + k, j + λk and λi + k is minimum, then λ is equal to

    Vector Algebra

  140. The number of solutions of the equation 1 + sin^4 x = cos^2 3x, x ∈ [-(5π)/2, (5π)/2 ] is

    Trigonometry · with solution

  141. Let z_1 and z_2 be two complex numbers satisfying |z_1| = 9 and |z_2 - 3 - 4i| = 4. Then, the minimum value of |z_1 - z_2| is

    Complex Numbers and Quadratic Equations · with solution

  142. The sum of the solutions of the equation |√(x) - 2| + √(x)(√(x) - 4) + 2 = 0 (x > 0) is equal to

    Complex Numbers and Quadratic Equations · with solution

  143. If m is chosen in the quadratic equation (m^2+1)x^2-3x+(m^2+1)^2=0 such that the sum of its roots is greatest, then the absolute difference of the cubes of its roots is

    Complex Numbers and Quadratic Equations

  144. Let √(3)i + j, i + √(3)j and βi + (1-β)j respectively be the position vectors of the points A, B and C with respect to the origin O. If the distance of C from the bisector of the acute angle between OA and OB is…

    Three Dimensional Geometry · with solution

  145. If the tangent to the curve, y = x^3 + ax - b at the point (1, -5) is perpendicular to the line, -x + y + 4 = 0, then which one of the following points lies on the curve?

    Application of Derivatives · with solution

  146. If m is the minimum value of k for which the function f(x) = x√(kx - x^2) is increasing in the interval [0, 3] and M is the maximum value of f in the interval [0, 3] when k = m, then the ordered pair (m, M) is equal to

    Application of Derivatives · with solution

  147. The integral ∫ cos(log_e x) dx is equal to (where C is a constant of integration)

    Indefinite Integration · with solution

  148. The integral int_π/6^π/3 sec^2/3 x csc^4/3 x dx is equal to

    Definite Integrals and Areas

  149. Let z ∈ C be such that |z| < 1. If ω = (5 + 3z)/(5(1-z)), then

    Complex Numbers and Quadratic Equations · with solution

  150. Let z_0 be a root of the quadratic equation, x^2 + x + 1 = 0. If z = 3 + 6iz_0^81 - 3iz_0^93, then arg z is equal to

    Complex Numbers and Quadratic Equations

  151. The area (in sq units) of the region bounded by the curves y = 2^x and y = |x+1| in the first quadrant is

    Definite Integrals and Areas

  152. Let y = y(x) be the solution of the differential equation, dy/dx + ytan x = 2x + x^2 tan x, x ∈ (-(π)/2, (π)/2 ), such that y(0) = 1. Then

    Differential Equations

  153. Let f:[0,1] → R be such that f(xy) = f(x) · f(y), for all x, y ∈ [0,1] and f(0) ≠ 0. If y = y(x) satisfies the differential equation, dy/dx = f(x) with y(0) = 1, then y (1/4 ) + y (3/4 ) is equal to

    Differential Equations · with solution

  154. Lines are drawn parallel to the line 4x - 3y + 2 = 0, at a distance 3/5 from the origin. Then which one of the following points lies on any of these lines?

    Straight Lines and Pair of Lines

  155. The product of three consecutive terms of a GP is 512. If 4 is added to each of the first and the second of these terms, the three terms now form an AP. Then, the sum of the original three terms of the given GP is

    Sequences and Series · with solution

  156. The sum of an infinite geometric series with positive terms is 3 and the sum of the cubes of its terms is 27/19. Then, the common ratio of this series is

    Sequences and Series

  157. If a circle of radius R passes through the origin O and intersects the coordinate axes at A and B, then the locus of the foot of perpendicular from O on AB is

    Circles

  158. The tangent and the normal lines at the point ( √(3), 1) to the circle x^2 + y^2 = 4 and the X-axis form a triangle. The area of this triangle (in square units) is

    Circles · with solution

  159. Two circles with equal radii are intersecting at the points (0, 1) and (0, -1). The tangent at the point (0, 1) to one of the circles passes through the centre of the other circle. Then, the distance between the centres…

    Circles · with solution

  160. The area (in sq units) of the smaller of the two circles that touch the parabola, y^2 = 4x at the point (1, 2) and the X-axis is

    Conic Sections · with solution

  161. If the line x - 2y = 12 is tangent to the ellipse (x^2)/(a^2) + (y^2)/(b^2) = 1 at the point (3, (-9)/2 ), then the length of the latus rectum of the ellipse is

    Conic Sections · with solution

  162. Let S and S' be the foci of an ellipse and B be any one of the extremities of its minor axis. If triangle S'BS is a right angled triangle with right angle at B and area (triangle S'BS) = 8 sq units, then the length of a…

    Conic Sections

  163. If the eccentricity of the standard hyperbola passing through the point (4, 6) is 2, then the equation of the tangent to the hyperbola at (4, 6) is

    Conic Sections · with solution

  164. If a directrix of a hyperbola centred at the origin and passing through the point (4, -2√(3)) is 5x = 4√(5) and its eccentricity is e, then

    Conic Sections · with solution

  165. ABC is a triangular park with AB = AC = 100 m. A vertical tower is situated at the mid-point of BC. If the angles of elevation of the top of the tower at A and B are cot^-1(3√(2)) and cosec^-1(2√(2)) respectively, then…

    Trigonometry · with solution

  166. If the lengths of the sides of a triangle are in AP and the greatest angle is double the smallest, then a ratio of lengths of the sides of this triangle is

    Trigonometry · with solution

  167. Let S = \θ ∈ [-2π, 2π] : 2cos^2θ + 3sinθ = 0\, then the sum of the elements of S is

    Trigonometry

  168. Consider three boxes, each containing 10 balls labelled 1, 2, ..., 10. Suppose one ball is randomly drawn from each of the boxes. Denote by n_i, the label of the ball drawn from the ith box, (i = 1, 2, 3). Then, the…

    Permutations and Combinations · with solution

  169. If α = cos^-1 (3/5 ), β = tan^-1 (1/3 ), where 0 < α, β < (π)/2, then α - β is equal to

    Inverse Trigonometric Functions · with solution

  170. Some identical balls are arranged in rows to form an equilateral triangle. The first row consists of one ball, the second row consists of two balls and so on. If 99 more identical balls are added to the total number of…

    Permutations and Combinations

  171. If the length of the perpendicular from the point (β, 0, β) (β ≠ 0) to the line, x/1 = (y-1)/0 = (z+1)/(-1) is √(3/2), then β is equal to

    Three Dimensional Geometry

  172. A plane which bisects the angle between the two given planes 2x - y + 2z - 4 = 0 and x + 2y + 2z - 2 = 0, passes through the point

    Three Dimensional Geometry

  173. The term independent of x in the expansion of (1/60 - (x^8)/81 ) · (2x^2 - 3/x )^6 is equal to

    Binomial Theorem

  174. Minimum number of times a fair coin must be tossed so that the probability of getting atleast one head is more than 99% is

    Probability · with solution

  175. If there of the six vertices of a regular hexagon are chosen at random, then the probability that the triangle formed with these chosen vertices is equilateral is

    Probability · with solution

  176. Let A and B be two non-null events such that A ⊂ B. Then, which of the following statements is always correct.

    Probability · with solution

  177. Let A = beginbmatrix cosα & -sinα \ sinα & cosα endbmatrix (α ∈ R) such that A^32 = beginbmatrix 0 & -1 \ 1 & 0 endbmatrix. Then, a value of α is

    Matrices and Determinants

  178. If Δ_1 = beginvmatrix x & sinθ & cosθ \ -sinθ & -x & 1 \ cosθ & 1 & x endvmatrix and Δ_2 = beginvmatrix x & sin 2θ & cos 2θ \ -sin 2θ & -x & 1 \ cos 2θ & 1 & x endvmatrix, x ≠ 0, then for all θ ∈ (0, (π)/2 )

    Matrices and Determinants · with solution

  179. If the system of linear equations x + y + z = 5 x + 2y + 2z = 6 x + 3y + λ z = μ, (λ, μ ∈ R), has infinitely many solutions, then the value of λ + μ is

    Matrices and Determinants · with solution

  180. Let A and B be two invertible matrices of order 3 × 3. If det(ABA^T) = 8 and det(AB^-1) = 8, then det(BA^-1B^T) is equal to

    Matrices and Determinants · with solution

  181. Let sum_k=1^10 f(a+k) = 16(2^10-1), where the function f satisfies f(x+y) = f(x)f(y) for all natural numbers x, y and f(1) = 2. Then, the natural number 'a' is

    Sets Relations and Functions · with solution

  182. The number of functions f from \1, 2, 3, …, 20\ onto \1, 2, 3, …, 20\ such that f(k) is a multiple of 3, whenever k is a multiple of 4, is

    Permutations and Combinations

  183. Let f(x) = 15 - |x - 10|; x ∈ R. Then, the set of all values of x, at which the function, g(x) = f(f(x)) is not differentiable, is

    Limits Continuity and Differentiability · with solution

  184. If lim_x → 1 (x^4 - 1)/(x - 1) = lim_x → k (x^3 - k^3)/(x^2 - k^2), then k is

    Limits Continuity and Differentiability · with solution

  185. Let [x] denote the greatest integer less than or equal to x. Then, lim_x → 0 (tan(π sin^2 x) + (|x| - sin(x[x]))^2)/(x^2)

    Limits Continuity and Differentiability

  186. If e^y + xy = e, the ordered pair (dy/dx, (d^2y)/(dx^2) ) at x = 0 is equal to

    Limits Continuity and Differentiability

  187. If the function f(x) = begincases a|π - x| + 1, x ≤ 5 \ b|x - π| + 3, x > 5 endcases is continuous at x = 5, then the value of a - b is

    Limits Continuity and Differentiability · with solution

  188. If both the roots of the quadratic equation x^2 - mx + 4 = 0 are real and distinct and they lie in the interval [1, 5] then m lies in the interval

    Complex Numbers and Quadratic Equations · with solution

  189. If a variable line, 3x + 4y - λ = 0 is such that the two circles x^2 + y^2 - 2x - 2y + 1 = 0 and x^2 + y^2 - 18x - 2y + 78 = 0 are on its opposite sides, then the set of all values of λ is the interval

    Circles · with solution

  190. In a triangle, the sum of lengths of two sides is x and the product of the lengths of the same two sides is y. If x^2 - c^2 = y, where c is the length of the third side of the triangle, then the circumradius of the…

    Trigonometry

  191. If f(x) is a non-zero polynomial of degree four, having local extreme points at x = -1, 0, 1, then the set S = \x ∈ R : f(x) = f(0)\ contains exactly

    Application of Derivatives · with solution

  192. The tangent to the curve y = x^2 - 5x + 5, parallel to the line 2y = 4x + 1, also passes through the point

    Application of Derivatives · with solution

  193. If the sum and product of the first three terms in an AP are 33 and 1155, respectively, then a value of its 11th term is

    Sequences and Series · with solution

  194. Let a_1, a_2, …, a_10 be a GP. If a_3/a_1 = 25, then a_9/a_5 equals

    Sequences and Series · with solution

  195. If ∫ x^5 e^-4x^3 dx = 1/48 e^-4x^3 f(x) + C, where C is a constant of integration, then f(x) is equal to

    Indefinite Integration · with solution

  196. The value of ∫limits_0^2π [sin 2x(1 + cos 3x)] dx, where [t] denotes the greatest integer function, is

    Definite Integrals and Areas

  197. If (z-α)/(z+α) (α ∈ R) is a purely imaginary number and |z| = 2, then a value of α is

    Complex Numbers and Quadratic Equations · with solution

  198. The area (in sq units) of the region A = \(x, y): (y^2)/2 ≤ x ≤ y + 4 \ is

    Definite Integrals and Areas · with solution

  199. If y = y(x) is the solution of the differential equation dy/dx = (tan x - y)sec^2 x, x ∈ (-(π)/2, (π)/2 ), such that y(0) = 0, then y (-(π)/4 ) is equal to

    Differential Equations · with solution

  200. The region represented by |x - y| ≤ 2 and |x + y| ≤ 2 is bounded by a

    Straight Lines and Pair of Lines

  201. Three circles of radii a, b, c (a < b < c) touch each other externally. If they have X-axis as a common tangent, then

    Circles

  202. The straight line x + 2y = 1 meets the coordinate axes at A and B. A circle is drawn through A, B and the origin. Then, the sum of perpendicular distances from A and B on the tangent to the circle at the origin is

    Circles

  203. Let C_1 and C_2 be the centres of the circles x^2 + y^2 - 2x - 2y - 2 = 0 and x^2 + y^2 - 6x - 6y + 14 = 0 respectively. If P and Q are the points of intersection of these circles, then the area (in sq units) of the…

    Circles · with solution

  204. The equation of a tangent to the parabola, x^2 = 8y, which makes an angle θ with the positive direction of X-axis, is

    Conic Sections · with solution

  205. Let the length of the latus rectum of an ellipse with its major axis along X-axis and centre at the origin, be 8. If the distance between the foci of this ellipse is equal to the length of its minor axis, then which one…

    Conic Sections · with solution

  206. If the tangent to the parabola y^2 = x at a point (α, β), (β > 0) is also a tangent to the ellipse x^2 + 2y^2 = 1, then α is equal to

    Conic Sections

  207. If the vertices of a hyperbola be at (-2, 0) and (2, 0) and one of its foci be at (-3, 0), then which one of the following points does not lie on this hyperbola?

    Conic Sections

  208. The equation of a tangent to the hyperbola 4x^2 - 5y^2 = 20 parallel to the line x - y = 2 is

    Conic Sections · with solution

  209. The number of natural numbers less than 7,000 which can be formed by using the digits 0, 1, 3, 7, 9 (repetition of digits allowed) is equal to

    Permutations and Combinations

  210. If cos(α + β) = 3/5, sin(α - β) = 5/13 and 0 < α, β < (π)/4, then tan(2α) is equal to

    Trigonometry · with solution

  211. Two poles standing on a horizontal ground are of heights 5 m and 10 m, respectively. The line joining their tops makes an angle of 15° with the ground. Then, the distance (in m) between the poles, is

    Trigonometry

  212. If sin^4 α + 4cos^4 β + 2 = 4√(2)sinαcosβ; α, β ∈ [0, π], then cos(α + β) - cos(α - β) is equal to

    Trigonometry

  213. Considering only the principal values of inverse functions, the set A = \x ≥ 0: tan^-1(2x) + tan^-1(3x) = (π)/4 \

    Inverse Trigonometric Functions

  214. Let α = 3i + j and β = 2i - j + 3k. If β = β_1 - β_2, where β_1 is parallel to α and β_2 is perpendicular to α, then β_1 × β_2 is equal to

    Vector Algebra

  215. Let a = i + j + √(2) k, b = b_1i + b_2j + √(2) k and c = 5i + j + √(2) k be three vectors such that the projection vector of b on a is a. If a + b is perpendicular to c, then |b| is equal to

    Vector Algebra

  216. If the line (x-2)/3 = (y+1)/2 = (z-1)/(-1) intersects the plane 2x + 3y - z + 13 = 0 at a point P and the plane 3x + y + 4z = 16 at a point Q, then PQ is equal to

    Three Dimensional Geometry · with solution

  217. There are m men and two women participating in a chess tournament. Each participant plays two games with every other participant. If the number of games played by the men between themselves exceeds the number of games…

    Permutations and Combinations

  218. The smallest natural number n, such that the coefficient of x in the expansion of (x^2 + 1/(x^3) )^n is ^nC_23, is

    Binomial Theorem · with solution

  219. Two integers are selected at random from the set \1, 2, ……, 11\. Given that the sum of selected numbers is even, the conditional probability that both the numbers are even is

    Probability · with solution

  220. The minimum number of times one has to toss a fair coin so that the probability of observing atleast one head is atleast 90% is

    Probability · with solution

  221. Let P = beginbmatrix 1 & 0 & 0 \ 3 & 1 & 0 \ 9 & 3 & 1 endbmatrix and Q = [q_ij] be two 3 × 3 matrices such that Q - P^3 = I_3. Then, dfracq_21 + q_31q_32 is equal to

    Matrices and Determinants

  222. If the system of equations 2x + 3y - z = 0, x + ky - 2z = 0 and 2x - y + z = 0 has a non-trivial solution (x, y, z), then x/y + y/z + z/x + k is equal to

    Matrices and Determinants

  223. If beginbmatrix 1 & 1 \ 0 & 1 endbmatrix beginbmatrix 1 & 2 \ 0 & 1 endbmatrix beginbmatrix 1 & 3 \ 0 & 1 endbmatrix ·s beginbmatrix 1 & n-1 \ 0 & 1 endbmatrix = beginbmatrix 1 & 78 \ 0 & 1 endbmatrix, then the inverse…

    Matrices and Determinants · with solution

  224. If A = beginbmatrix cosθ & -sinθ \ sinθ & cosθ endbmatrix, then the matrix A^-50 when θ = (π)/12, is equal to

    Matrices and Determinants

  225. Let f: R → R be defined by f(x) = x/(1+x^2), x ∈ R. Then, the range of f is

    Sets Relations and Functions · with solution

  226. Let (-2 - 1/3i )^3 = (x+iy)/27 (i = √(-1)), where x and y are real numbers, then y - x equals

    Complex Numbers and Quadratic Equations

  227. Let S be the set of all points in (-π, π) at which the function, f(x) = min\sin x, cos x\ is not differentiable. Then, S is a subset of which of the following?

    Limits Continuity and Differentiability

  228. For each t ∈ R, let [t] be the greatest integer less than or equal to t. Then, lim_x → 1^+ ((1-|x|+sin|1-x|)sin ((π)/2[1-x] ))/(|1-x|[1-x])

    Limits Continuity and Differentiability

  229. lim_x → 0 dfracsin^2 x√(2) - √(1 + cos x) equals

    Limits Continuity and Differentiability · with solution

  230. If f(x) = [x] - [x/4 ], x ∈ R where [x] denotes the greatest integer function, then (a) lim_x → 4+ f(x) exists but lim_x → 4- f(x) does not exist (b) f is continuous at x = 4 (c) Both lim_x → 4- f(x) and lim_x → 4+ f(x)…

    Limits Continuity and Differentiability

  231. If f(1) = 1, f'(1) = 3, then the derivative of f(f(f(x))) + (f(x))^2 at x = 1 is

    Limits Continuity and Differentiability

  232. Let f:[0,2] → R be a twice differentiable function such that f''(x) > 0, for all x ∈ (0,2). If φ(x) = f(x) + f(2-x), then φ is

    Application of Derivatives · with solution

  233. If ∫ dfracdxx^3(1+x^6)^2/3 = xf(x)(1+x^6)^1/3 + C where, C is a constant of integration, then the function f(x) is equal to

    Indefinite Integration · with solution

  234. A square is inscribed in the circle x^2 + y^2 - 6x + 8y - 103 = 0 with its sides parallel to the coordinate axes. Then, the distance of the vertex of this square which is nearest to the origin is

    Circles

  235. The number of integral values of m for which the equation (1 + m^2)x^2 - 2(1 + 3m)x + (1 + 8m) = 0, has no real root is

    Complex Numbers and Quadratic Equations

  236. The value of the integral ∫limits_0^1 xcot^-1(1-x^2+x^4)dx is

    Definite Integrals and Areas

  237. Let z be a complex number such that |z| + z = 3 + i (where i = √(-1)). Then, |z| is equal to

    Complex Numbers and Quadratic Equations · with solution

  238. The area (in sq units) of the region A = \(x, y) : x^2 ≤ y ≤ x + 2\ is

    Definite Integrals and Areas · with solution

  239. If cos x dy/dx - y sin x = 6x, (0 < x < (π)/2 ) and y ((π)/3 ) = 0, then y ((π)/6 ) is equal to

    Differential Equations · with solution

  240. Let S_k = (1+2+3+…+k)/k. If S_1^2 + S_2^2 + … + S_10^2 = 5/12A, then A is equal to

    Sequences and Series

  241. Let the sum of the first n terms of a non-constant AP a_1, a_2, a_3, … be 50n + (n(n-7))/2A, where A is a constant. If d is the common difference of this AP, then the ordered pair (d, a_50) is equal to

    Sequences and Series · with solution

  242. If the two lines x + (a-1)y = 1 and 2x + a^2y = 1, (a ∈ R - \0, 1\) are perpendicular, then the distance of their point of intersection from the origin is

    Straight Lines and Pair of Lines · with solution

  243. If a circle C passing through the point (4, 0) touches the circle x^2 + y^2 + 4x - 6y = 12 externally at the point (1, -1), then the radius of C is

    Circles

  244. If the circles x^2 + y^2 - 16x - 20y + 164 = r^2 and (x-4)^2 + (y-7)^2 = 36 intersect at two distinct points, then

    Circles · with solution

  245. If the area of an equilateral triangle inscribed in the circle, x^2 + y^2 + 10x + 12y + c = 0 is 27√(3) sq units, then c is equal to

    Circles

  246. Equation of a common tangent to the circle, x^2 + y^2 - 6x = 0 and the parabola, y^2 = 4x, is

    Conic Sections · with solution

  247. If a hyperbola has length of its conjugate axis equal to 5 and the distance between its foci is 13, then the eccentricity of the hyperbola is

    Conic Sections · with solution

  248. Consider a class of 5 girls and 7 boys. The number of different teams consisting of 2 girls and 3 boys that can be formed from this class, if there are two specific boys A and B, who refuse to be the members of the same…

    Permutations and Combinations · with solution

  249. If ^nC_4, ^nC_5 and ^nC_6 are in AP, then n can be

    Permutations and Combinations · with solution

  250. Two vertical poles of heights, 20 m and 80 m stand apart on a horizontal plane. The height (in m) of the point of intersection of the lines joining the top of each pole to the foot of the other, from this horizontal…

    Trigonometry · with solution

  251. Let f_k(x) = 1/k(sin^k x + cos^k x) for k = 1, 2, 3 … Then, for all x ∈ R, the value of f_4(x) - f_6(x) is equal to

    Trigonometry · with solution

  252. Let α and β be the roots of the quadratic equation x^2 sinθ - x(sinθcosθ + 1) + cosθ = 0 (0 < θ < 45°) and α < β. Then sum_n=0^∞ (α^n + ((-1)^n)/(β^n) ) is equal to

    Complex Numbers and Quadratic Equations

  253. If x = sin^-1(sin 10) and y = cos^-1(cos 10), then y - x is equal to

    Inverse Trigonometric Functions · with solution

  254. If lines x = ay + b, z = cy + d and x = a'z + b', y = c'z + d' are perpendicular, then

    Straight Lines and Pair of Lines · with solution

  255. Let a = 3i + 2j + xk and b = i - j + k, for some real x. Then |a × b| = r is possible if

    Vector Algebra · with solution

  256. A perpendicular is drawn from a point on the line (x-1)/2 = (y+1)/(-1) = z/1 to the plane x + y + z = 3 such that the foot of the perpendicular Q also lies on the plane x - y + z = 3. Then, the coordinates of Q are

    Three Dimensional Geometry · with solution

  257. If some three consecutive coefficients in the binomial expansion of (x+1)^n in powers of x are in the ratio 2:15:70, then the average of these three coefficients is

    Binomial Theorem

  258. An unbiased coin is tossed. If the outcome is a head, then a pair of unbiased dice is rolled and the sum of the numbers obtained on them is noted. If the toss of the coin results in tail, then a card from a…

    Probability

  259. In a game, a man wins ₹100 if he gets 5 or 6 on a throw of a fair die and loses ₹50 for getting any other number on the die. If he decides to throw the die either till he gets a five or a six or to a maximum of three…

    Probability

  260. If the system of linear equations x - 2y + kz = 1, 2x + y + z = 2, 3x - y - kz = 3 has a solution (x, y, z), z ≠ 0, then (x, y) lies on the straight line whose equation is

    Matrices and Determinants

  261. Let A = beginbmatrix 0 & 2q & r \ p & q & -r \ p & -q & r endbmatrix. If AA^T = I_3, then |p| is

    Matrices and Determinants · with solution

  262. Let α and β be the roots of the equation x^2 + x + 1 = 0. Then, for y ≠ 0 in R, beginvmatrix y+1 & α & β \ α & y+β & 1 \ β & 1 & y+α endvmatrix is equal to

    Matrices and Determinants

  263. For x ∈ R - \0, 1\, let f_1(x) = 1/x, f_2(x) = 1 - x and f_3(x) = 1/(1-x) be three given functions. If a function, J(x) satisfies (f_2 circ J circ f_1)(x) = f_3(x), then J(x) is equal to

    Sets Relations and Functions · with solution

  264. Let N be the set of natural numbers and two functions f and g be defined as f, g: N → N such that f(n) = begincases (n+1)/2 & if n is odd \ n/2 & if n is even endcases and g(n) = n - (-1)^n. Then, fog is

    Sets Relations and Functions

  265. Let K be the set of all values of x, where the function f(x) = sin|x| - |x| + 2(x - π)cos|x| is not differentiable. Then, the set K is equal to

    Limits Continuity and Differentiability

  266. lim_x → (π)/4 (cot^3 x - tan x)/(cos (x + (π)/4 )) is

    Limits Continuity and Differentiability · with solution

  267. Let f: [-1, 3] arrow R be defined as f(x) = begincases |x| + [x], & -1 ≤ x < 1 \ x + |x|, & 1 ≤ x < 2 \ x + [x], & 2 ≤ x ≤ 3 endcases where, [t] denotes the greatest integer less than or equal to t. Then, f is…

    Limits Continuity and Differentiability

  268. For each x ∈ R, let [x] be the greatest integer less than or equal to x. Then, lim_x → 0^- (x([x] + |x|) sin[x])/(|x|) is equal to

    Limits Continuity and Differentiability

  269. A helicopter is flying along the curve given by y - x^3/2 = 7, (x ≥ 0). A soldier positioned at the point (1/2, 7 ) wants to shoot down the helicopter when it is nearest to him. Then, this nearest distance is

    Application of Derivatives · with solution

  270. Let A = \θ ∈ (-(π)/2, π ) : (3 + 2isinθ)/(1 - 2isinθ) is purely imaginary \ Then, the sum of the elements in A is

    Complex Numbers and Quadratic Equations

  271. If the function f given by f(x) = x^3 - 3(a-2)x^2 + 3ax + 7, for some a ∈ R is increasing in (0, 1] and decreasing in [1, 5), then a root of the equation, frac(x-1)^-14(x-1)^2 = 0 (x ≠ 1) is

    Application of Derivatives · with solution

  272. The height of a right circular cylinder of maximum volume inscribed in a sphere of radius 3 is

    Application of Derivatives · with solution

  273. ∫ (sin 5x/2)/(sin x/2) dx is equal to (where, C is a constant of integration)

    Indefinite Integration · with solution

  274. If the plane 2x - y + 2z + 3 = 0 has the distances 1/3 and 2/3 units from the planes 4x - 2y + 4z + λ = 0 and 2x - y + 2z + μ = 0, respectively, then the maximum value of λ + μ is equal to

    Three Dimensional Geometry

  275. Let f(x) = dfracx√(a^2 + x^2) - dfracd - x√(b^2 + (d-x)^2), x ∈ R, where a, b and d are non-zero real constants. Then,

    Application of Derivatives

  276. The integral ∫ dfrac3x^13 + 2x^11(2x^4 + 3x^2 + 1)^4 dx is equal to (where C is a constant of integration)

    Indefinite Integration

  277. The value of int_0^π/2 (sin^3 x)/(sin x + cos x) dx is

    Definite Integrals and Areas

  278. The number of integral values of m for which the quadratic expression, (1 + 2m)x^2 - 2(1 + 3m)x + 4(1 + m), x ∈ R, is always positive, is

    Complex Numbers and Quadratic Equations · with solution

  279. The solution of the differential equation xdy/dx + 2y = x^2(x ≠ 0) with y(1) = 1, is

    Differential Equations · with solution

  280. If a, b and c be three distinct real numbers in GP and a + b + c = xb, then x cannot be

    Sequences and Series · with solution

  281. Slope of a line passing through P (2, 3) and intersecting the line, x + y = 7 at a distance of 4 units from P, is

    Straight Lines and Pair of Lines · with solution

  282. The sum of all two digit positive numbers which when divided by 7 yield 2 or 5 as remainder is

    Sequences and Series · with solution

  283. Let x, y be positive real numbers and m, n positive integers. The maximum value of the expression fracx^m y^n(1 + x^2m)(1 + y^2n) is

    Sequences and Series

  284. If the angle of elevation of a cloud from a point P which is 25 m above a lake be 30° and the angle of depression of reflection of the cloud in the lake from P be 60°, then the height of the cloud (in meters) from the…

    Trigonometry · with solution

  285. The value of cos(π)/(2^2)·cos(π)/(2^3)·s·scosdfracπ2^10·sindfracπ2^10 is

    Trigonometry · with solution

  286. The sum of all values of θ ∈ (0, (π)/2 ) satisfying sin^2 2θ + cos^4 2θ = 3/4 is

    Trigonometry · with solution

  287. If sum_r=1^25 \^50C_r · ^50-rC_25-r\ = K(^50C_25), then, K is equal to

    Permutations and Combinations

  288. A tetrahedron has vertices P(1,2,1), Q(2,1,3), R(-1,1,2), O(0,0,0). The angle between the faces OPQ and PQR is

    Three Dimensional Geometry · with solution

  289. If the fourth term in the binomial expansion of (2/x + x^log_8 x )^6 (x > 0) is 20 × 8^7, then the value of x is

    Binomial Theorem · with solution

  290. Let the numbers 2, b, c be in an AP and A = beginbmatrix 1 & 1 & 1 \ 2 & b & c \ 4 & b^2 & c^2 endbmatrix If det(A) ∈ [2, 16], then c lies in the interval

    Matrices and Determinants · with solution

  291. Let A = \x ∈ R : x is not a positive integer\. Define a function f : A → R as f(x) = 2x/(x-1), then f is

    Sets Relations and Functions · with solution

  292. Let f(x) = begincases -1, & -2 ≤ x < 0 \ x^2 - 1, & 0 ≤ x ≤ 2 endcases and g(x) = |f(x)| + f(|x|). Then, in the interval (-2, 2), g is

    Limits Continuity and Differentiability

  293. For x > 1, if (2x)^2y = 4e^2x-2y, then (1 + log_e 2x)^2 dy/dx is equal to

    Limits Continuity and Differentiability · with solution

  294. lim_x → 0 (xcot(4x))/(sin^2 x cot^2(2x)) is equal to

    Limits Continuity and Differentiability · with solution

  295. Let f: R → R be a function defined as f(x) = begincases 5, & if x ≤ 1 \ a + bx, & if 1 < x < 3 \ b + 5x, & if 3 ≤ x < 5 \ 30, & if x ≥ 5 endcases Then, f is

    Limits Continuity and Differentiability

  296. If S_1 and S_2 are respectively the sets of local minimum and local maximum points of the function, f(x) = 9x^4 + 12x^3 - 36x^2 + 25, x ∈ R, then

    Application of Derivatives

  297. If θ denotes the acute angle between the curves, y = 10 - x^2 and y = 2 + x^2 at a point of their intersection, then |tanθ| is equal to

    Application of Derivatives

  298. Let S(α) = \(x, y) : y^2 ≤ x, 0 ≤ x ≤ α\ and A(α) is area of the region S(α). If for λ, 0 < λ < 4, A(λ) : A(4) = 2 : 5, then λ equals

    Definite Integrals and Areas · with solution

  299. If Q(0, -1, -3) is the image of the point P in the plane 3x - y + 4z = 2 and R is the point (3, -1, -2), then the area (in sq units) of triangle PQR is

    Three Dimensional Geometry

  300. The shortest distance between the line y = x and the curve y^2 = x - 2 is

    Application of Derivatives · with solution

  301. If ∫ dfracx+1√(2x-1) dx = f(x)√(2x-1) + C, where C is a constant of integration, then f(x) is equal to

    Indefinite Integration · with solution

  302. Let f(x) = ∫limits_0^x g(t)dt, where g is a non-zero even function. If f(x+5) = g(x), then ∫limits_0^x f(t)dt equals

    Definite Integrals and Areas

  303. If λ be the ratio of the roots of the quadratic equation in x, 3m^2x^2 + m(m-4)x + 2 = 0, then the least value of m for which λ + 1/(λ) = 1, is

    Complex Numbers and Quadratic Equations

  304. Let y = y(x) be the solution of the differential equation, (x^2 + 1)^2 dy/dx + 2x(x^2 + 1)y = 1 such that y(0) = 0. If √(a) y(1) = (π)/32, then the value of 'a' is

    Differential Equations · with solution

  305. The sum of the following series 1 + 6 + (9(1^2 + 2^2 + 3^2))/7 + (12(1^2 + 2^2 + 3^2 + 4^2))/9 + (15(1^2 + 2^2 + ·s + 5^2))/11 + … up to 15 terms is

    Sequences and Series

  306. If a point R(4, y, z) lies on the line segment joining the points P(2, -3, 4) and Q(8, 0, 10), then the distance of R from the origin is

    Three Dimensional Geometry · with solution

  307. Let a_1, a_2, …, a_30 be an AP, S = sum_i=1^30 a_i and T = sum_i=1^15 a_(2i-1). If a_5 = 27 and S - 2T = 75, then a_10 is equal to

    Sequences and Series · with solution

  308. Equation of a common tangent to the parabola y^2 = 4x and the hyperbola xy = 2 is

    Conic Sections

  309. Consider a triangular plot ABC with sides AB = 7 m, BC = 5 m and CA = 6 m. A vertical lamp-post at the mid-point D of AC subtends an angle 30° at B. The height (in m) of the lamp-post is

    Trigonometry · with solution

  310. If 0 ≤ x ≤ (π)/2, then the number of values of x for which sin x - sin 2x + sin 3x = 0, is

    Trigonometry

  311. If sum_i=1^20 ( frac^20C_i-1^20C_i + ^20C_i-1 )^3 = k/21, then k equals

    Permutations and Combinations · with solution

  312. If the fourth term in the binomial expansion of (sqrtfrac1x^1+log_10x+x^1/12 )^6 is equal to 200, and x>1, then the value of x is

    Binomial Theorem

  313. Two cards are drawn successively with replacement from a well shuffled deck of 52 cards. Let X denote the random variable of number of aces obtained in the two drawn cards. Then, P(X=1) + P(X=2) equals

    Probability · with solution

  314. The set of all values of λ for which the system of linear equations x - 2y - 2z = λ x x + 2y + z = λ y and -x - y = λ z has a non-trivial solution

    Matrices and Determinants

  315. Let f:(-1, 1) → R be a function defined by f(x) = max\-|x|, -√(1-x^2)\. If K be the set of all points at which f is not differentiable, then K has exactly

    Limits Continuity and Differentiability · with solution

  316. If xlog_e(log_e x) - x^2 + y^2 = 4 (y > 0), then dy/dx at x = e is equal to

    Limits Continuity and Differentiability

  317. lim_y → 0 dfrac√(1 + √(1 + y^4)) - √(2)y^4 \hspace{2cm}

    Limits Continuity and Differentiability

  318. The maximum area (in sq. units) of a rectangle having its base on the X-axis and its other two vertices on the parabola, y = 12 - x^2 such that the rectangle lies inside the parabola, is

    Application of Derivatives · with solution

  319. If ∫ dfrac√(1-x^2)x^4 dx = A(x)(√(1-x^2))^m + C, for a suitable chosen integer m and a function A(x), where C is a constant of integration, then (A(x))^m equals

    Indefinite Integration

  320. If f(x) = (2 - xcos x)/(2 + xcos x) and g(x) = log_e x, (x > 0) then the value of the integral int_-π/4^π/4 g(f(x)) dx is

    Definite Integrals and Areas · with solution

  321. If a curve passes through the point (1, -2) and has slope of the tangent at any point (x, y) on it as (x^2 - 2y)/x, then the curve also passes through the point

    Differential Equations · with solution

  322. Suppose that the points (h, k), (1, 2) and (-3, 4) lie on the line L_1. If a line L_2 passing through the points (h, k) and (4, 3) is perpendicular to L_1, then k/h equals

    Straight Lines and Pair of Lines

  323. Let 0 < θ < (π)/2. If the eccentricity of the hyperbola (x^2)/(cos^2θ) - (y^2)/(sin^2θ) = 1 is greater than 2, then the length of its latus rectum lies in the interval

    Conic Sections · with solution

  324. Let P be the plane, which contains the line of intersection of the planes, x + y + z - 6 = 0 and 2x + 3y + z + 5 = 0 and it is perpendicular to the XY-plane. Then, the distance of the point (0, 0, 256) from P is equal to

    Three Dimensional Geometry

  325. The sum of the coefficients of all even degree terms is x in the expansion of (x + √(x^3 - 1))^6 + (x - √(x^3 - 1))^6, (x > 1) is equal to

    Binomial Theorem · with solution

  326. If beginvmatrix a-b-c & 2a & 2a \ 2b & b-c-a & 2b \ 2c & 2c & c-a-b endvmatrix = (a+b+c)(x+a+b+c)^2, x ≠ 0 and a+b+c ≠ 0, then x is equal to

    Matrices and Determinants

  327. An ordered pair (α, β) for which the system of linear equations (1 + α)x + β y + z = 2 α x + (1 + β)y + z = 3 α x + β y + 2z = 2 has a unique solution, is

    Matrices and Determinants

  328. Let f(x) = begincases max\|x|, x^2\, & |x| ≤ 2 \ 8 - 2|x|, & 2 < |x| ≤ 4 endcases Let S be the set of points in the interval (-4, 4) at which f is not differentiable. Then, S

    Limits Continuity and Differentiability

  329. Let f: R → R be a function such that f(x) = x^3 + x^2 f'(1) + xf''(2) + f'''(3), x ∈ R. Then, f(2) equals

    Limits Continuity and Differentiability

  330. The area (in sq units) of the region A = \(x, y) ∈ R × R mid 0 ≤ x ≤ 3, 0 ≤ y ≤ 4, y ≤ x^2 + 3x\ is

    Definite Integrals and Areas · with solution

  331. If one real root of the quadratic equation 81x^2 + kx + 256 = 0 is cube of the other root, then a value of k is

    Complex Numbers and Quadratic Equations · with solution

  332. A plane passing through the points (0, -1, 0) and (0, 0, 1) and making an angle (π)/4 with the plane y - z + 5 = 0, also passes through the point

    Three Dimensional Geometry

  333. Let f and g be continuous functions on [0, a] such that f(x) = f(a - x) and g(x) + g(a - x) = 4, then int_0^a f(x) g(x) dx is equal to

    Definite Integrals and Areas

  334. The area (in sq units) of the region bounded by the parabola, y = x^2 + 2 and the lines, y = x + 1, x = 0 and x = 3, is

    Definite Integrals and Areas

  335. Let y = y(x) be the solution of the differential equation, xdy/dx + y = xlog_e x, (x > 1). If 2y(2) = log_e 4 - 1, then y(e) is equal to

    Differential Equations · with solution

  336. The total number of irrational terms in the binomial expansion of (7^1/5 - 3^1/10)^60 is

    Binomial Theorem

  337. If the system of linear equations 2x + 2y + 3z = a 3x - y + 5z = b x - 3y + 2z = c where a, b, c are non-zero real numbers, has more than one solution, then

    Matrices and Determinants · with solution

  338. Let f be a differentiable function from R to R such that |f(x) - f(y)| ≤ 2|x - y|^3/2, for all x, y ∈ R. If f(0) = 1, then int_0^1 f^2(x) dx is equal to

    Limits Continuity and Differentiability · with solution

  339. The maximum value of the function f(x) = 3x^3 - 18x^2 + 27x - 40 on the set S = \x ∈ R : x^2 + 30 ≤ 11x\ is

    Application of Derivatives · with solution

  340. If 5, 5r, 5r² are the lengths of the sides of a triangle, then r cannot be equal to

    Sequences and Series · with solution

  341. If the line (x-1)/2 = (y+1)/3 = (z-2)/4 meets the plane, x + 2y + 3z = 15 at a point P, then the distance of P from the origin is

    Three Dimensional Geometry

  342. Let A(4, -4) and B(9, 6) be points on the parabola, y^2 = 4x. Let C be chosen on an arc AOB of the parabola, where O is the origin, such that the area of triangle ACB is maximum. Then, the area (in sq. units) of…

    Application of Derivatives · with solution

  343. If f(x) = ∫ (5x^8 + 7x^6)/((x^2 + 1 + 2x^7)^2) dx, \ (x ≥ 0), and f(0) = 0, then the value of f(1) is

    Definite Integrals and Areas · with solution

  344. The integral int_1^e [ (x/e )^2x - (e/x )^x ]log_e x dx is equal to

    Definite Integrals and Areas

  345. If y(x) is the solution of the differential equation dy/dx + ((2x+1)/x )y = e^-2x, x > 0, where y(1) = 1/2e^-2, then

    Differential Equations

  346. If a straight line passing through the point P(-3, 4) is such that its intercepted portion between the coordinate axes is bisected at P, then its equation is

    Straight Lines and Pair of Lines · with solution

  347. The ratio of the 5th term from the beginning to the 5th term from the end in the binomial expansion of (2^1/3 + dfrac12(3)^1/3 )^10 is

    Binomial Theorem

  348. Let A = beginbmatrix 2 & b & 1 \ b & b^2+1 & b \ 1 & b & 2 endbmatrix, where b > 0. Then, the minimum value of (det(A))/b is

    Matrices and Determinants · with solution

  349. The number of values of θ ∈ (0, π) for which the system of linear equations x + 3y + 7z = 0, -x + 4y + 7z = 0, (sin 3θ)x + (cos 2θ)y + 2z = 0 has a non-trivial solution, is

    Matrices and Determinants

  350. The area (in sq units) in the first quadrant bounded by the parabola, y = x^2 + 1, the tangent to it at the point (2, 5) and the coordinate axes is

    Definite Integrals and Areas · with solution

  351. The value of λ such that sum of the squares of the roots of the quadratic equation, x^2 + (3 - λ)x + 2 = λ has the least value is

    Complex Numbers and Quadratic Equations

  352. For x^2 ≠ nπ + 1, n ∈ N (the set of natural numbers), the integral ∫ x√((2sin(x^2-1)-sin 2(x^2-1))/(2sin(x^2-1)+sin 2(x^2-1))) dx is equal to (where C is a constant of integration)

    Indefinite Integration

  353. The integral int_π/6^π/4 dx/(sin 2x(tan^5 x + cot^5 x)) equals

    Definite Integrals and Areas

  354. Let f be a differentiable function such that f'(x) = 7 - 3/4 (f(x))/x, (x > 0) and f(1) ≠ 4. Then, lim_x → 0^+ x f (1/x )

    Limits Continuity and Differentiability

  355. If the straight line, 2x - 3y + 17 = 0 is perpendicular to the line passing through the points (7, 17) and (15, β), then β equals

    Straight Lines and Pair of Lines · with solution

  356. The vector equation of the plane through the line of intersection of the planes x + y + z = 1 and 2x + 3y + 4z = 5, which is perpendicular to the plane x - y + z = 0 is

    Three Dimensional Geometry · with solution

  357. The sum of the real values of x for which the middle term in the binomial expansion of ((x^3)/3 + 3/x )^8 equals 5670 is

    Binomial Theorem · with solution

  358. Let d ∈ R, and A = beginbmatrix -2 & 4+d & (sinθ)-2 \ 1 & (sinθ)+2 & d \ 5 & (2sinθ)-d & (-sinθ)+2+2d endbmatrix, θ ∈ [0, 2π]. If the minimum value of det(A) is 8, then a value of d is

    Matrices and Determinants

  359. If the system of equations x + y + z = 5, x + 2y + 3z = 9, x + 3y + α z = β has infinitely many solutions, then β - α equals

    Matrices and Determinants · with solution

  360. The maximum volume (in cu.m) of the right circular cone having slant height 3m is

    Application of Derivatives · with solution

  361. The area (in sq units) of the region bounded by the curve x^2 = 4y and the straight line x = 4y - 2 is

    Definite Integrals and Areas · with solution

  362. The number of all possible positive integral values of α for which the roots of the quadratic equation 6x^2 - 11x + α = 0 are rational numbers is

    Complex Numbers and Quadratic Equations

  363. Let α and β be two roots of the equation x^2 + 2x + 2 = 0, then α^15 + β^15 is equal to

    Complex Numbers and Quadratic Equations · with solution

  364. The value of the integral int_-2^2 (sin^2 x)/( [x/(π) ] + 1/2) dx (where, [x] denotes the greatest integer less than or equal to x) is

    Definite Integrals and Areas · with solution

  365. The area of the region A = \(x, y) 0 ≤ y ≤ |x| + 1 and -1 ≤ x ≤ 1\ in sq. units, is

    Definite Integrals and Areas

  366. If dy/dx + 3/(cos^2 x) y = 1/(cos^2 x), x ∈ ((-π)/3, (π)/3 ) and y ((π)/4 ) = 4/3, then y (-(π)/4 ) equals

    Differential Equations · with solution

  367. The length of the perpendicular from the point (2, -1, 4) on the straight line, (x+3)/10 = (y-2)/(-7) = z/1 is

    Three Dimensional Geometry

  368. The positive value of λ for which the coefficient of x^2 in the expression x^2 (√(x) + (λ)/(x^2) )^10 is 720, is

    Binomial Theorem · with solution

  369. If the system of linear equations x - 4y + 7z = g, 3y - 5z = h, -2x + 5y - 9z = k is consistent, then

    Matrices and Determinants · with solution

  370. If in a parallelogram ABDC, the coordinates of A, B and C are respectively (1, 2), (3, 4) and (2, 5), then the equation of the diagonal AD is

    Straight Lines and Pair of Lines

  371. The area (in sq units) bounded by the parabola y = x^2 - 1, the tangent at the point (2, 3) to it and the Y-axis is

    Definite Integrals and Areas

  372. The magnitude of the projection of the vector 2i + 3j + k on the vector perpendicular to the plane containing the vectors i + j + k and i + 2j + 3k, is

    Three Dimensional Geometry

  373. If the third term in the binomial expansion of (1 + x^log_2 x)^5 equals 2560, then a possible value of x is

    Binomial Theorem

  374. The system of linear equations x + y + z = 2, 2x + 3y + 2z = 5 2x + 3y + (a^2 - 1)z = a + 1

    Matrices and Determinants · with solution

  375. The tangent to the curve, y = xe^x^2 passing through the point (1, e) also passes through the point

    Application of Derivatives · with solution

  376. If int_0^π/3 fractanθ√(2ksecθ) dθ = 1 - frac1√(2), (k > 0), then the value of k is

    Definite Integrals and Areas · with solution

  377. Two sides of a parallelogram are along the lines, x + y = 3 and x - y + 3 = 0. If its diagonals intersect at (2, 4), then one of its vertex is

    Straight Lines and Pair of Lines · with solution

  378. The equation of a plane containing the line of intersection of the planes 2x - y - 4 = 0 and y + 2z - 4 = 0 and passing through the point (1, 1, 0) is

    Three Dimensional Geometry

  379. The coefficient of t^4 in the expansion of ((1-t^6)/(1-t) )^3 is

    Binomial Theorem

  380. If an angle between the line, (x+1)/2 = (y-2)/1 = (z-3)/(-2) and the plane, x - 2y - kz = 3 is cos^-1 (frac2√(2)3 ), then value of k is

    Three Dimensional Geometry

  381. The shortest distance between the point (3/2, 0 ) and the curve y = √(x), (x > 0), is

    Application of Derivatives · with solution

  382. Let S be the set of all real values of λ such that a plane passing through the points (-λ^2, 1, 1), (1, -λ^2, 1), (1, 1, -λ^2) also passes through the point (-1, -1, 1). Then S is equal to

    Three Dimensional Geometry

  383. The value of int_-π^π |cos x|^3 dx is

    Definite Integrals and Areas · with solution

  384. If the line 3x + 4y - 24 = 0 intersects the X-axis at the point A and the Y-axis at the point B, then the incentre of the triangle OAB, where O is the origin, is

    Straight Lines and Pair of Lines · with solution

  385. A point P moves on the line 2x - 3y + 4 = 0. If Q(1, 4) and R(3, -2) are fixed points, then the locus of the centroid of triangle PQR is a line

    Straight Lines and Pair of Lines · with solution

  386. The perpendicular distance from the origin to the plane containing the two lines, (x+2)/3 = (y-2)/5 = (z+5)/7 and (x-1)/1 = (y-4)/4 = (z+7)/7, is

    Three Dimensional Geometry

  387. Two lines (x-3)/1 = (y+1)/3 = (z-6)/(-1) and (x+5)/7 = (y-2)/(-6) = (z-3)/4 intersect at the point R. The reflection of R in the xy-plane has coordinates

    Three Dimensional Geometry · with solution

  388. If the point (2, α, β) lies on the plane which passes through the points (3, 4, 2) and (7, 0, 6) and is perpendicular to the plane 2x - 5y = 15, then 2α - 3β is equal to

    Three Dimensional Geometry

  389. The plane containing the line (x-3)/2 = (y+2)/(-1) = (z-1)/3 and also containing its projection on the plane 2x + 3y - z = 5, contains which one of the following points?

    Three Dimensional Geometry · with solution

  390. The direction ratios of normal to the plane through the points (0, -1, 0) and (0, 0, 1) and making an angle π/4 with the plane y - z + 5 = 0 are

    Three Dimensional Geometry

  391. The plane which bisects the line segment joining the points (-3, -3, 4) and (3, 7, 6) at right angles, passes through which one of the following points?

    Three Dimensional Geometry

  392. The plane passing through the point (4, -1, 2) and parallel to the lines (x+2)/3 = (y-2)/(-1) = (z+1)/2 and (x-2)/1 = (y-3)/2 = (z-4)/3 also passes through the point

    Three Dimensional Geometry · with solution

  393. Let A be a point on the line r = (1 - 3μ)i + (μ - 1)j + (2 + 5μ)k and B(3, 2, 6) be a point in the space. Then, the value of μ for which the vector overrightarrowAB is parallel to the plane x - 4y + 3z = 1 is

    Three Dimensional Geometry

  394. The equation of the plane containing the straight line x/2 = y/3 = z/4 and perpendicular to the plane containing the straight lines x/3 = y/4 = z/2 and x/4 = y/2 = z/3 is

    Three Dimensional Geometry

  395. The plane through the intersection of the planes x + y + z = 1 and 2x + 3y - z + 4 = 0 and parallel to Y-axis also passes through the point

    Three Dimensional Geometry · with solution

Chemistry

  1. An example of a disproportionation reaction is

    Redox and Electrochemistry

  2. 5 moles of AB_2 weight 125 × 10^-3 kg and 10 moles of A_2B_2 weight 300 × 10^-3 kg. The molar mass of A (M_A) and molar mass of B (M_B) in kg mol^-1 are

    Some Basic Concepts of Chemistry

  3. Among the following, the energy of 2s-orbital is lowest in

    Atomic Structure · with solution

  4. Which one of the following about an electron occupying the 1s-orbital in a hydrogen atom is incorrect? (The Bohr radius is represented by a_0)

    Atomic Structure

  5. The minimum amount of O_2(g) consumed per gram of reactant is for the reaction (Given atomic mass : Fe = 56, O = 16, Mg = 24, P = 31, C = 12, H = 1)

    Some Basic Concepts of Chemistry

  6. In an acid-base titration, 0.1 M HCl solution was added to the NaOH solution of unknown strength. Which of the following correctly shows the change of pH of the titration mixture in this experiment? (2019 Main, 9 April…

    Equilibrium

  7. If p is the momentum of the fastest electron ejected from a metal surface after the irradiation of light having wavelength λ, then for 1.5 p momentum of the photoelectron, the wavelength of the light should be (Assume…

    Atomic Structure

  8. The electrons are more likely to be found [figure]

    Atomic Structure

  9. At 300 K and 1 atmospheric pressure, 10 mL of a hydrocarbon required 55 mL of O_2 for complete combustion and 40 mL of CO_2 is formed. The formula of the hydrocarbon is

    Some Basic Concepts of Chemistry

  10. What is the work function of the metal, if the light of wavelength 4000 Å generates photoelectron of velocity 6 × 10^5 ms^-1 from it? (Mass of electron = 9 × 10^-31 kg Velocity of light = 3 × 10^8 ms^-1 Planck's…

    Atomic Structure

  11. The ratio of the shortest wavelength of two spectral series of hydrogen spectrum is found to be about 9. The spectral series are

    Atomic Structure · with solution

  12. The ground state energy of hydrogen atom is -13.6 eV. The energy of second excited state of He^+ ion in eV is

    Atomic Structure · with solution

  13. The graph between |ψ|^2 and r (radial distance) is shown below. This represents [figure]

    Atomic Structure · with solution

  14. 10 mL of 1 mM surfactant solution forms a monolayer covering 0.24 cm² on a polar substrate. If the polar head is approximated as a cube, what is its edge length?

    Surface Chemistry · with solution

  15. In order to oxidise a mixture of one mole of each of FeC_2O_4, Fe_2(C_2O_4)_3, FeSO_4 and Fe_2(SO_4)_3 in acidic medium, the number of moles of KMnO_4 required is

    Redox and Electrochemistry

  16. Which of the graphs shown below does not represent the relationship between incident light and the electron ejected from metal surface? [figure]

    Atomic Structure

  17. For any given series of spectral lines of atomic hydrogen, let ΔV = V_max - V_min be the difference in maximum and minimum frequencies in cm^-1. The ratio ΔV_Lyman / ΔV_Balmer is

    Atomic Structure · with solution

  18. For a reaction, N_2(g) + 3H_2(g) → 2NH_3(g), identify dihydrogen (H_2) as a limiting reagent in the following reaction mixtures.

    Some Basic Concepts of Chemistry

  19. 100 mL of a water sample contains 0.81 g of calcium bicarbonate and 0.73 g of magnesium bicarbonate. The hardness of this water sample expressed in terms of equivalents of CaCO_3 is (molar mass of calcium bicarbonate is…

    Some Basic Concepts of Chemistry

  20. The percentage composition of carbon by mole in methane is

    Some Basic Concepts of Chemistry · with solution

  21. 8 g of NaOH is dissolved in 18 g of H_2O. Mole fraction of NaOH in solution and molality (in mol kg^-1) of the solution respectively are

    Solutions · with solution

  22. 25 mL of the given HCl solution requires 30 mL of 0.1 M sodium carbonate solution. What is the volume of this HCl solution required to titrate 30 mL of 0.2 M aqueous NaOH solution?

    Some Basic Concepts of Chemistry

  23. If the de-Broglie wavelength of the electron in n^th Bohr orbit in a hydrogenic atom is equal to 1.5 π a_0 (a_0 is Bohr radius), then the value of n/Z is

    Atomic Structure

  24. The de-Broglie wavelength (λ) associated with a photoelectron varies with the frequency (v) of the incident radiation as, [v_0 is threshold frequency]

    Atomic Structure

  25. The volume strength of 1 M H_2O_2 is (Molar mass of H_2O_2 = 34 g mol^-1)

    Some Basic Concepts of Chemistry

  26. In the reaction of oxalate with permanganate in acidic medium, the number of electrons involved in producing one molecule of CO_2 is

    Redox and Electrochemistry · with solution

  27. Which of the following combination of statements is true regarding the interpretation of the atomic orbitals? I. An electron in an orbital of high angular momentum stays away from the nucleus than an electron in the…

    Atomic Structure

  28. The amount of sugar (C_12H_22O_11) required to prepare 2 L of its 0.1 M aqueous solution is

    Solutions · with solution

  29. For the following reaction, the mass of water produced from 445 g of C_7H_10O_6 is : 2C_7H_10O_6(s) + 16O_2(g) arrow 14CO_2(g) + 110 H_2O(l)

    Some Basic Concepts of Chemistry

  30. For emission line of atomic hydrogen from n_i = 8 to n_f = n, the plot of wave number (ν) against (1/(n^2) ) will be (The Rydberg constant, R_H is in wave number unit)

    Atomic Structure

  31. A solution of sodium sulphate contains 92 g of Na^+ ions per kilogram of water. The molality of Na^+ ions in that solution in mol kg^-1 is

    Solutions

Physics

  1. The area of a square is 5.29 cm^2. The area of 7 such squares taking into account the significant figures is

    Units and Measurements · with solution

  2. The stream of a river is flowing with a speed of 2 km/h. A swimmer can swim at a speed of 4 km/h. What should be the direction of the swimmer with respect to the flow of the river to cross the river straight ?

    Kinematics · with solution

  3. Two particles are projected from the same point with the same speed u such that they have the same range R, but different maximum heights h_1 and h_2. Which of the following is correct?

    Kinematics

  4. A particle starts from origin O from rest and moves with a uniform acceleration along the positive X-axis. Identify all figures that correctly represent the motion qualitatively. (a = acceleration, v = velocity, x =…

    Kinematics

  5. A particle is moving with speed v = b√(x) along positive X-axis. Calculate the speed of the particle at time t = τ (assume that the particle is at origin at t = 0).

    Kinematics

  6. A ball is thrown upward with an initial velocity v_0 from the surface of the earth. The motion of the ball is affected by a drag force equal to mγ v^2 (where, m is mass of the ball, v is its instantaneous velocity and γ…

    Kinematics

  7. A shell is fired from a fixed artillery gun with an initial speed u such that it hits the target on the ground at a distance R from it. If t_1 and t_2 are the values of the time taken by it to hit the target in two…

    Kinematics

  8. Ship A is sailing towards north-east with velocity v = 30i + 50j km/h, where i points east and j north. Ship B is at a distance of 80 km east and 150 km north of Ship A and is sailing towards west at 10 km/h. A will be…

    Kinematics

  9. The trajectory of a projectile near the surface of the earth is given as y = 2x - 9x^2. If it were launched at an angle θ_0 with speed v_0, then (Take, g = 10 ms^-2)

    Kinematics

  10. In a simple pendulum, experiment for determination of acceleration due to gravity (g), time taken for 20 oscillations is measured by using a watch of 1 second least count. The mean value of time taken comes out to be 30…

    Experimental Physics

  11. A person standing on an open ground hears the sound of a jet aeroplane, coming from north at an angle 60° with ground level. But he finds the aeroplane right vertically above his position. If v is the speed of sound,…

    Kinematics · with solution

  12. The bob of a simple pendulum has mass 2g and a charge of 5.0μC. It is at rest in a uniform horizontal electric field of intensity 2000 V/m. At equilibrium, the angle that the pendulum makes with the vertical is (take g…

    Electrostatics

  13. In SI units, the dimensions of (ε_0)/(mu_0) is

    Units and Measurements

  14. A plane is inclined at an angle α = 30° with respect to the horizontal. A particle is projected with a speed u = 2 ms^-1, from the base of the plane, making an angle θ = 15° with respect to the plane as shown in the…

    Kinematics · with solution

  15. A particle moves from the point (2.0i + 4.0j) m at t = 0 with an initial velocity (5.0i + 4.0j) ms^-1. It is acted upon by a constant force which produces a constant acceleration (4.0i + 4.0j) ms^-2. What is the…

    Kinematics

  16. A passenger train of length 60 m travels at a speed of 80 km/hr. Another freight train of length 120 m travels at a speed of 30 km/hr. The ratio of times taken by the passenger train to completely cross the freight…

    Kinematics

  17. Let l, r, c, and v represent inductance, resistance, capacitance and voltage, respectively. The dimension of l/rcv in SI units will be

    Units and Measurements

  18. In a car race on a straight path, car A takes a time t less than car B at the finish and passes finishing point with a speed 'v' more than that of car B. Both the cars start from rest and travel with constant…

    Kinematics

  19. The position vector of particle changes with time according to the relation r(t) = 15t^2i + (4 - 20t^2)j. What is the magnitude of the acceleration (in ms^-2) at t = 1?

    Kinematics · with solution

  20. If speed (V), acceleration (A) and force (F) are considered as fundamental units, the dimension of Young's modulus will be

    Units and Measurements

  21. A body is projected at t = 0 with a velocity 10 ms^-1 at an angle of 60° with the horizontal. The radius of curvature of its trajectory at t = 1 s is R. Neglecting air resistance and taking acceleration due to gravity g…

    Kinematics

  22. The density of a material in SI units is 128 kg m^-3. In certain units in which the unit of length is 25 cm and the unit of mass is 50 g, the numerical value of density of the material is

    Units and Measurements

  23. In form of G (universal gravitational constant), h (Planck constant) and c (speed of light), the time period will be proportional to

    Units and Measurements

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