List of practice Questions

If \(\tan^{-1}\left(\frac{2}{3 - x + 1}\right) = \cot^{-1}\left(\frac{3}{3x + 1}\right)\), then which one of the following is true?

A particle moves along the curve \(6x = y^3 + 2\). The points on the curve at which the \(x\) coordinate is changing 8 times as fast as \(y\) coordinate are:

The corner points of the feasible region for an L.P.P. are (0, 10), (5, 5), (5, 15), and (0, 30). If the objective function is Z = αx + βy, α, β > 0, the condition on α and β so that maximum of Z occurs at corner points (5, 5) and (0, 20) is:

The value of \( \lambda \) for which the lines \(\frac{2 - x}{3} = \frac{3 - 4y}{5} = \frac{z - 2}{3}\) and  \(\frac{x - 2}{-3} = \frac{2y - 4}{3} = \frac{2 - z}{\lambda}\) are perpendicular is:

Two pipes A and B can fill a tank in 32 minutes and 48 minutes respectively. If both the pipes are opened simultaneously, after how much time B should be turned off so that the tank is full in 20 minutes?

If $t = e^{2x}$ and $y = \ln(t^2)$, then $\frac{d^2 y}{dx^2}$ is:

The area of the region bounded by the lines $x + 2y = 12$, $x = 2$, $x = 6$, and the $x$-axis is:

If $f(x) = x^2 + bx + 1$ is increasing in the interval $[1, 2]$, then the least value of $b$ is:

Let [x] denote the greatest integer function. Then match List-I with List-II:

For an investment, if the nominal rate of interest is\(10\%\)compounded half-yearly, then the effective rate of interest is:

Consider the following LPP: Maximise \( Z = 9x + 3y \) Subject to the constraints: \[ x + 3y \leq 60, \quad x - y \leq 0, \quad x \geq 0, \quad y \geq 0 \] If \( x = A, y = B \) is the optimum solution of the given LPP, then the value of \( A + B \) is:

The degree and order of the differential equation \[ \left( \frac{d^2 y}{dx^2} \right)^{\frac{4}{5}} = 10 \frac{dy}{dx} + 2 \] are:

In a 600 m race, the ratio of the speeds of two participants A and B is 4:5. If A has a head start of 200 m, then the distance by which A wins is:

\(\text{ If } f(x) = \sin x + \frac{1}{2} \cos 2x \text{ in } \left[ 0, \frac{\pi}{2} \right], \text{ then:}\)
(A) \(f'(x) = \cos x - \sin 2x\)
(B)The critical points of the function are \(x = \frac{\pi}{6}\) and \(x = \frac{\pi}{2}\)
(C) The minimum value of the function is 2
(D) The maximum value of the function is \(\frac{3}{4}\)

A mixture contains apple juice and water in the ratio 10 : x. When 36 litres of the mixture and 9 litres of water are mixed, the ratio of apple juice and water becomes 5 : 4. The value of x is:

The area (in square units) bounded by the curve y = |x−2| between x = 0, y = 0, and x = 5 is:

If \( \vec{a}, \vec{b} \) and \( \vec{c} \) are three vectors such that \( \vec{a} + \vec{b} + \vec{c} = 0 \), where \( \vec{a} \) and \( \vec{b} \) are unit vectors and \( |\vec{c}| = 2 \), then the angle between the vectors \( \vec{b} \) and \( \vec{c} \) is:

Ms. Sheela creates a fund of 100,000 to provide scholarships to needy children. The scholarship is provided at the beginning of each year, and the fund earns an interest of r% annually. If the scholarship amount is 8,000, find r.

In a 700 m race, Amit reaches the finish point in 20 seconds and Rahul reaches in 25 seconds. Amit beats Rahul by a distance of:

The equation of the tangent to the curve x^5/2 + y^5/2 = 33 at the point(1, 4) is:

Two pipes A and B together can fill a tank in 40 minutes. Pipe A is twice as fast as pipe B. Pipe A alone can fill the tank in:

The interval, in which the function \( f(x) = \frac{3}{x} + \frac{x}{3} \) is strictly decreasing, is:

The value of \( I = \int_{0}^{1.5} \left\lfloor x^2 \right\rfloor dx \), where [ ] denotes the greatest integer function, is:

The ratio in which a grocer must mix two varieties of tea worth ₹60 per kg and ₹65 per kg so that by selling the mixture at ₹68.20 per kg, he may gain 10% is:

Vibhuti bought a car worth ₹10,25,000 and made a down payment of ₹4,00,000. The balance is to be paid in 3 years by equal monthly installments at an interest rate of 12% p.a. The EMI that Vibhuti has to pay for the car is:
(Use \( (1.01)^{-36} = 0.7 \))