List of practice Questions

Let y = y(x) be the solution of the differential equation dy/dx = (y + 1) ((y + 1)e^{x²/2 - x} - 1), 0<x<2.1, with y(2) = 0. Then the value of dy/dx at x=1 is equal to :
The area bounded by the curve 4y² = x²(4 - x)(x - 2) is equal to :
Let g(x) = ∫₀ˣ f(t) dt, where f is continuous function in [0, 3] such that 1/3 ≤ f(t) ≤ 1 for all t ∈ [0, 1] and 0 ≤ f(t) ≤ 1/2 for all t ∈ (1, 3]. The largest possible interval in which g(3) lies is :
Let S1 be the sum of first 2n terms of an arithmetic progression. Let S2 be the sum of first 4n terms of the same arithmetic progression. If (S2 - S1) is 1000, then the sum of the first 6n terms of the arithmetic progression is equal to :
Let S1 : x² + y² = 9 and S2 : (x - 2)² + y² = 1. Then the locus of center of a variable circle S which touches S1 internally and S2 externally always passes through the points :
Let the system of linear equations 4x + λy + 2z = 0 ; 2x - y + z = 0 ; μx + 2y + 3z = 0, λ, μ ∈ R has a non-trivial solution. Then which of the following is true ?
Let f : R - {3} → R - {1} be defined by f(x) = (x - 2)/(x - 3). Let g : R → R be given as g(x) = 2x - 3. Then, the sum of all the values of x for which f⁻¹(x) + g⁻¹(x) = 13/2 is equal to.
Let a complex number be w = 1 - √3 i. Let another complex number z be such that |z w| = 1 and arg(z) - arg(w) = π/2. Then the area of the triangle with vertices origin, z and w is equal to :
Define a relation R over a class of n × n real matrices A and B as "ARB iff there exists a non-singular matrix P such that P A P⁻¹ = B". Then which of the following is true ?
Consider a hyperbola H: x² - 2y² = 4. Let the tangent at a point P(4, √6) meet the x-axis at Q and latus rectum at R(x₁, y₁), x₁>0. If F is a focus of H which is nearer to the point P, then the area of ΔQFR is equal to :
Let a and b be two non-zero vectors perpendicular to each other and |a| = |b|. If |a × b| = |a|, then the angle between the vectors (a + b + (a × b)) and a is equal to :
If f(x) and g(x) are two polynomials such that the polynomial P(x) = f(x³) + x · g(x³) is divisible by x² + x + 1, then P(1) is equal to ________.
Let I be an identity matrix of order 2 × 2 and P = [2 -1; 5 -3]. Then the value of n ∈ N for which Pⁿ = 5I - 8P is equal to ________.
The term independent of x in the expansion of [ (x + 1)/(x^{2/3} - x^{1/3} + 1) - (x - 1)/(x - x^{1/2}) ]¹⁰, x ≠ 1, is equal to ________.
If ∑_{r=1}^{10} r! (r³ + 6r² + 2r + 5) = α (11!), then the value of α is equal to ________.
Let P(x) be a real polynomial of degree 3 which vanishes at x = -3. Let P(x) have local minima at x = 1, local maxima at x = -1 and ∫_{-1}^{1} P(x) dx = 18, then the sum of all the coefficients of the polynomial P(x) is equal to ________.
Let y=y(x) be the solution of the differential equation x dy - y dx = √(x² - y²) dx, x ≥ 1, with y(1) = 0. If the area bounded by the line x=1, x=e^{π}, y=0 and y=y(x) is α e^{2π} + β, then the value of 10(α + β) is equal to ________.
Let P be a plane containing the line (x-1)/3 = (y+6)/4 = (z+5)/2 and parallel to the line (x-3)/4 = (y-2)/(-3) = (z+5)/7. If the point (1, -1, α) lies on the plane P, then the value of |α| is equal to ________.
Let the mirror image of the point (1, 3, a) with respect to the plane r · (2 i - j + k) - b = 0 be (-3, 5, 2). Then, the value of |a + b| is equal to ________.
Let n C_r denote the binomial coefficient of x^r in the expansion of (1 + x)^n. If ∑_{k=0}^{10} (2² + 3k) n C_k = α 3^{10} + β 2^{10}, α, β ∈ R, then α + β is equal to ________.
Let f: R → R satisfy the equation f(x + y) = f(x) · f(y) for all x, y ∈ R and f(x) ≠ 0 for any x ∈ R. If the function f is differentiable at x=0 and f'(0)=3, then lim_{h → 0} (1/h) (f(h) - 1) is equal to ________.

A simple pendulum of mass 'm', length 'l' and charge '+q' suspended in the electric field produced by two conducting parallel plates as shown. The value of deflection of pendulum in equilibrium position will be: 

 

Find the truth table for the function Y of A and B represented in the following figure. 

 

A particle of mass M originally at rest is subjected to a force whose direction is constant but magnitude varies with time according to the relation
$F = F_0 \left[1 - \left(\frac{t}{T}\right)^2\right]$
Where $F_0$ and T are constants. The force acts only for the time interval 2T. The velocity v of the particle after time 2T is:

Match List I with List II. 

Choose the correct answer from the options given below: