List of practice Questions

A firm anticipates an expenditure of ₹10,000 for a new equipment at the end of 5 years from now. How much should the firm deposit at the end of each quarter into a sinking fund earning interest 10% per year compounded quarterly to provide for the purchase?
[Use (1.025)20=1.7]
An equilateral triangle of side \( 4\sqrt{3} \) cm formed out of a sheet is converted into a rectangle such that there is no loss of the area of the triangle. Then the least perimeter of the rectangle (in cm) will be:
\(\text{The area of the region enclosed between the curves } 4x^2 = y \text{ and } y = 4 \text{ is:}\)
The direction cosines of the line which is perpendicular to the lines with direction ratios 1,-2,-2 and 0, 2, 1 are:
Which of the following cannot be the direction ratios of the straight line \(\frac{x - 3}{2} = \frac{2 - y}{3} = \frac{z + 4}{-1}\)?
The value of \(\int_{\frac{\pi}{6}}^{\frac{\pi}{3}} \frac{dx}{1 + \tan^{18}x}\) is:
Subject to constraints: 2x + 4y ≤ 8, 3x + y ≤ 6, x + y ≤ 4, x, y ≥ 0; The maximum value of Z = 3x + 15y is:

If \( y = e^{{2}\log_e t} \) and \( x = \log_3(e^{t^2}) \), then \( \frac{dy}{dx} \) is equal to:

The area of the region bounded by the lines \( \frac{x}{7\sqrt{3a}} + \frac{y}{b} = 4 \), \( x = 0 \), and \( y = 0 \) is:
A flower vase costs 36,000. With an annual depreciation of 2,000, its cost will be 6,000 in how many years?
Let \( f : \mathbb{R} \to \mathbb{R} \) be defined as \( f(x) = 10 - x^2 \), then:
If \[ y = \frac{1}{\sqrt{1 - 4 \sin^2 x \cos^2 x}}, \] then $\frac{dy}{dx}$ is:
The cost of a machinery is ₹8,00,000. Its scrap value will be one-tenth of its original cost in 15 years. Using the linear method of depreciation, the book value of the machine at the end of the 10th year will be:
Match List-I with List-II:
List-IList-II
The derivative of \( \log_e x \) with respect to \( \frac{1}{x} \) at \( x = 5 \) is(I) -5
If \( x^3 + x^2y + xy^2 - 21x = 0 \), then \( \frac{dy}{dx} \) at \( (1, 1) \) is(II) -6
If \( f(x) = x^3 \log_e \frac{1}{x} \), then \( f'(1) + f''(1) \) is(III) 5
If \( y = f(x^2) \) and \( f'(x) = e^{\sqrt{x}} \), then \( \frac{dy}{dx} \) at \( x = 0 \) is(IV) 0
Choose the correct answer from the options given below :
An objective function $Z = ax + by$ is maximum at points $(8, 2)$ and $(4, 6)$. If $a \geq 0$ and $b \geq 0$ and $ab = 25$, then the maximum value of the function is:
Evaluate the integral $\int_0^1 \frac{a - bx^2}{(a + bx^2)^2} , dx$:
If the lengths of the three sides of a trapezium other than the base are 10 cm each, then the maximum area of the trapezium is:
A random variable X has the following probability distribution:
X-2-1012
P(X)0.20.10.30.20.2

The variance of X will be:
If \( x = at^4 \) and \( y = 2at^2 \), then \( \frac{d^2y}{dx^2} \) is equal to:
The area (in square units) of the region bounded by curves \( y = x \) and \( y = x^3 \) is:
The particular solution of the differential equation \((y - x^2) dy = (1 - x^3) dx\) with \(y(0) = 1\), is:
The value of \[ \int_{-1}^{1} \tan^{-1} x \, dx \] is:
\(\text{Evaluate } \int e^x \left( \frac{2x + 1}{2 \sqrt{x}} \right) dx:\)
\(\text{ If } f(x), \text{ defined by } f(x) = \begin{cases}  kx + 1 & \text{if } x \leq \pi \\  \cos x & \text{if } x > \pi  \end{cases} \text{ is continuous at } x = \pi, \text{ then the value of } k \text{ is:}\)
Relation \( R \) on the set \( A = \{1, 2, 3, \ldots, 13, 14\} \) defined as \( R = \{(x, y) : 3x - y = 0\} \) is: