Practice questions
88551 published questions with worked solutions.
- Let f : [1, ) R be a differentiable function. If 6 _1^x f(t) ,dt = 3xf(x) + x^3 - 4 for all x 1, then the value of f(2) - f(3) is
- If the line x + 2y = 1, where R , does not meet the hyperbola x^2 - 9y^2 = 9, then a possible value of is:
- If the image of the point P(1, 2, a) in the line x-6 3 = y-7 2 = 7-z 2 is Q(5, b, c), then a^2 + b^2 + c^2 is equal to
- Let the set of all values of r, for which the circles (x+1)^2+(y+4)^2=r^2 and x^2+y^2-4x-2y-4=0 intersect at two distinct points...
- If A = bmatrix 2 & 3 3 & 5 bmatrix , then the determinant of the matrix (A^ 2025 - 3A^ 2024 + A^ 2023 ) is
- If the domain of the function f(x) = ^ -1 ( 5-x 3+2x ) + 1 _e(10-x) is (- , ] [ , ) - , then 6( + + + ) is equal to
- The value of _ - /2 ^ /2 ( 1 [x]+4 )dx, where [ ] denotes the greatest integer function, is
- The coefficient of x^ 48 in (1+x)+2(1+x)^2+3(1+x)^3+ +100(1+x)^ 100 is equal to:
- If the chord joining the points P_1(x_1, y_1) and P_2(x_2, y_2) on the parabola y^2 = 12x subtends a right angle at the vertex of...
- The number of solutions of ^ -1 4x + ^ -1 6x = 6 , where - 1 2 6 < x < 1 2 6 is equal to
- Let the solution curve of the differential equation xdy - ydx = x^2+y^2 ,dx, x > 0, y(1) = 0, be y = y(x). Then y(3) is equal to
- If the sum of the first four terms of an A.P. is 6 and the sum of its first six terms is 4, then the sum of its first twelve...
- Let = -1+i 3 2 and = -1-i 3 2 , i = -1 . If (7-7 +9 )^ 20 +(9+7 -7 )^ 20 +(-7+9 +7 )^ 20 +(14+7 +7 )^ 20 = m^ 10 , then m is...
- Let A be a 3 3 matrix such that A+A^T = O. If A bmatrix 1 -1 0 bmatrix = bmatrix 3 3 2 bmatrix , A^2 bmatrix 1 -1 0 bmatrix =...
- If ( x)^ -11/2 ( x)^ -5/2 dx = - p_1 q_1 ( x)^ 9/2 - p_2 q_2 ( x)^ 5/2 - p_3 q_3 ( x)^ 1/2 + p_4 q_4 ( x)^ -3/2 + C, where p_i...
- If ^2 48^ - ^2 12^ ^2 24^ - ^2 6^ = + 5 2 , where , N , then + is equal to _____.
- Let ABC be a triangle. Consider four points p_1,p_2,p_3,p_4 on the side AB, five points p_5,p_6,p_7,p_8,p_9 on the side BC and...
- A solid sphere of mass 5 kg and radius 10 cm is kept in contact with another solid sphere of mass 10 kg and radius 20 cm. The...
- 7.9 MeV -particle scatters from a target material of atomic number 79. From the given data the estimated diameter of nuclei of...
- The escape velocity from a spherical planet A is 10 km/s. The escape velocity from another planet B whose density and radius are...