Mathematics questions
18011 published questions with worked solutions.
- Let y = y(x) be the solution of the differential equation dy dx = 2x(x + y)^3 - x(x + y) - 1, y(0) = 1. Then ( 1 2 + y ( 1 2 )...
- Let C : x² + y² = 4 and C′ : x² + y² − 4λx + 9 = 0 be two circles. If the set of all values of λ so that the circles C and C′...
- If x = x(t) is the solution of the differential equation (t + 1)dx = (2x + (t + 1)^4)dt, x(0) = 2, then, x(1) equals _____.
- Let P = z ∈ C: |z + 2 − 3i| ≤ 1 and Q = z ∈ C: z(1 + i) + z(1 − i) ≤ −8 . Let in P ∩ Q, |z − 3 + 2i| be maximum and minimum at...
- The number of elements in the set S = (x, y, z) : x, y, z ∈ ℤ, x + 2y + 3z = 42, x, y, z ≥ 0 equal ______.
- Let 3, 7, 11, 15, …, 403 and 2, 5, 8, 11, …, 404 be two arithmetic progressions. Then the sum of the common terms in them, is...
- Let A = 1, 2, 3, …, 20 . Let R_1 and R_2 two relation on A such that R_1 = (a, b) : b is divisible by a , R_2 = (a, b) : a is an...
- If _ - /2 ^ /2 8 2 x ,dx (1 + e^ x )(1 + ^4x) = + _e(3 + 2 2) where α, β are integers, then α² + β² equals_____.
- If the coefficient of x^ 30 in the expansion of (1 + 1x )^6(1 + x^2)^7(1 - x^3)^8; x ≠ 0 is α, then |α| equals_______.
- Let the line L : 2x + y = pass through the point of the intersection P (in the first quadrant) of the circle x^2 + y^2 = 3 and...
- Let the line of the shortest distance between the lines L_1 : r = ( i + 2 j + 3 k) + ( i - j + k) and L_2 : r = (4 i + 5 j + 6 k)...
- If the function f(x) = cases 72^x - 9^x - 8^x + 1 2 - 1 + x , & x 0 a _e2 _e3, & x = 0 cases is continuous at x = 0, then the...
- If λ > 0, let θ be the angle between the vectors a = i + j - 3 k and b = 3 i - j + 2 k. If the vectors a + b and a - b are...
- Let C be a circle with radius 10 units and centre at the origin. Let the line x + y = 2 intersects the circle C at the points P...
- Let a relation R on N × N be defined as: (x_1, y_1) R (x_2, y_2) if and only if x_1 ≤ x_2 or y_1 ≤ y_2. Consider the two...
- Let three real numbers a, b, c be in arithmetic progression and a + 1, b, c + 3 be in geometric progression. If a > 10 and the...
- Let A = bmatrix 1 & 2 0 & 1 bmatrix and B = I + adj(A) + (adj A)² + … + (adj A)^ 10 . Then, the sum of all the elements of the...
- The value of 1 2^2 + 2 3^2 + + 100 (101)^2 1^2 2 + 2^2 3 + + 100^2 101 is
- Let f(x) = _0^x(t + (1 - e^t))dt, x ∈ R. Then _ x 0 f(x) x^3 is equal to
- The area (in sq. units) of the region described by (x, y) : y² ≤ 2x, and y ≥ 4x − 1 is