Evaluate the limit: \lim_{x \to 2} \frac{x^3 - 8}{x^2 - 4}.
Mathematics
Question · Easy
Find the derivative dy/dx of the parametric curve defined by x = a(t - \sin t) and y = a(1 - \cos t) at t = \pi/2.
Question · Medium
An open box is to be made from a square sheet of cardboard of side 12 cm by cutting equal squares of side x from the corners and folding up the sides. Find the value of x (in cm) that maximizes the volume of the box.
Question · Hard
Evaluate the definite integral: \int_{0}^{\pi/2} \frac{\sin^{3/2} x}{\sin^{3/2} x + \cos^{3/2} x} dx.
Find the area of the region bounded by the curves y^2 = 4x and x^2 = 4y.
Solve the differential equation: dy/dx + y \tan x = \sec x.
If A is a 3x3 non-singular matrix such that |A| = 5, find the determinant of the adjoint of A, i.e., |adj(A)|.
For what value of k does the system of equations: x + y + z = 1, x + 2y + 4z = k, x + 4y + 10z = k^2 have a solution?
If \omega is a complex cube root of unity, evaluate the value of (1 - \omega + \omega^2)^5 + (1 + \omega - \omega^2)^5.
If the roots of the quadratic equation x^2 - px + q = 0 differ by unity, which of the following relations is correct?
If the 5th and 12th terms of an Arithmetic Progression (AP) are 30 and 65 respectively, find the sum of the first 20 terms of the AP.
How many 4-digit numbers can be formed using the digits 1, 2, 3, 5, 7, 8 without repetition such that the number is divisible by 5?
Find the coefficient of x^4 in the expansion of (x/2 - 3/x^2)^10.
Find the general solution of the trigonometric equation: \sin x + \cos x = 1.
Evaluate the value of \tan(2 \tan^{-1}(1/5) - \pi/4).
Find the distance between the parallel lines 3x - 4y + 5 = 0 and 3x - 4y - 10 = 0.
Find the equation of the tangent to the circle x^2 + y^2 = 25 at the point (3, 4).
Find the equation of the tangent to the parabola y^2 = 8x which is parallel to the line 2x - y + 3 = 0.
Find the eccentricity of the ellipse 9x^2 + 25y^2 = 225.
Find the equation of the hyperbola whose asymptotes are 3x \pm 4y = 0 and which passes through the point (5, 9/4).
Find the angle between the vectors \vec{a} = 2\hat{i} + \hat{j} - \hat{k} and \vec{b} = \hat{i} - \hat{j} + 2\hat{k}.
Find the volume of the parallelepiped whose coterminous edges are represented by the vectors \vec{a} = \hat{i} + 2\hat{j} - \hat{k}, \vec{b} = 2\hat{i} + 3\hat{k}, and \vec{c} = 3\hat{i} - \hat{j} + 2\hat{k}.
Find the shortest distance between the lines \frac{x-1}{2} = \frac{y-2}{3} = \frac{z-3}{4} and \frac{x-2}{3} = \frac{y-4}{4} = \frac{z-5}{5}.
Find the equation of the plane passing through the point (1, 2, 3) and perpendicular to the planes x + 2y + 3z = 4 and 2x - 3y + z = 5.
A bag contains 5 red and 3 blue balls. If two balls are drawn at random one by one without replacement, what is the probability that both are red?
In a factory, Machine A produces 60% of the total output and Machine B produces 40%. Of their output, 2% from Machine A and 1% from Machine B are defective. A product is chosen at random and found to be defective. Find the probability that it was produced by Machine A.
An unbiased coin is tossed 6 times. Find the probability of getting at least 4 heads.
If the variance of the numbers 2, 4, 5, 6, 8, 17 is V, find the variance of 12, 14, 15, 16, 18, 27.
If A is a symmetric matrix and B is a skew-symmetric matrix of the same order, then the matrix AB - BA is always a:
Find the values of a and b such that the function f(x) defined as f(x) = x^2 + 3x + a for x <= 1 and f(x) = bx + 2 for x > 1 is differentiable at x = 1.
Evaluate the value of the determinant: |[1, a, b+c; 1, b, c+a; 1, c, a+b]|.
Find the angle between the pair of straight lines represented by the equation x^2 - 3xy + 2y^2 = 0.
In a triangle ABC, if the sides are a = 4, b = 5, and c = 6, find the value of \cos A.
Which of the following points lies in the feasible region defined by the inequalities: x + y <= 4, x >= 0, y >= 0?
A projectile is thrown with an initial velocity of 20 m/s at an angle of 30 degrees to the horizontal. Calculate its maximum height. (Take g = 10 m/s^2)
Physics
Two blocks of masses 2 kg and 3 kg are connected by a light string passing over a frictionless pulley. Find the acceleration of the system when released from rest. (Take g = 10 m/s^2)
A bullet of mass 10 g moving with a speed of 400 m/s penetrates a sand bag and is brought to rest in 0.1 s. Calculate the average retarding force acting on the bullet.
A shell of mass 3 kg, initially at rest, explodes into three fragments of masses 1 kg, 1 kg, and 1 kg. If the first two fragments fly off perpendicular to each other with speeds of 3 m/s and 4 m/s respectively, find the speed of the third fragment.
A solid cylinder of mass M and radius R rolls down an inclined plane of inclination \theta without slipping. Find the acceleration of the cylinder down the incline.
Calculate the depth below the surface of the Earth where the acceleration due to gravity becomes half of its value at the surface. (Take radius of Earth = R)
A circular track of radius 100 m is banked for a speed of 20 m/s. What is the angle of banking of the track? (Take g = 10 m/s^2)
A steel wire of length 2 m and cross-sectional area 1 mm^2 is stretched by a force of 100 N. Calculate the change in length of the wire. (Young's modulus of steel = 2 * 10^11 N/m^2)
Find the capillary rise of water in a glass tube of diameter 1 mm. (Surface tension of water = 0.075 N/m, density of water = 1000 kg/m^3, contact angle = 0, g = 10 m/s^2)
A cylindrical tank of height H is completely filled with water. A small hole is made at a depth h below the free surface of water. Find the value of h (in terms of H) for which the horizontal range of the water jet emerging from the hole is maximum.
A mass of 2 kg is suspended from a vertical spring of force constant k = 200 N/m. Calculate the time period of vertical oscillations of the mass.
A train whistle emits a sound of frequency 400 Hz. If the train is approaching a stationary observer with a speed of 30 m/s, calculate the frequency of the sound heard by the observer. (Speed of sound in air = 330 m/s)
According to the kinetic theory of gases, at what temperature (in Kelvin) will the root mean square (RMS) velocity of oxygen molecules be double of its value at 27 degrees Celsius?
One mole of an ideal gas expands isothermally at a temperature of 300 K from an initial volume of 1 L to a final volume of 10 L. Calculate the work done by the gas. (Take R = 8.31 J/mol K, \ln 10 = 2.30)
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