List of practice Questions

A hard rubber roller is cut with a ‘V-shaped’ profile at three locations as shown in the Top View in the image on the left. The roller is then dipped in blue colour and rolled over a white plane. Which option shows the correct impression on the plane?

Which option(s) is/are NOT a part of the image on the left?

Four objects that are symmetrical along vertical planes are placed in four containers in the orientation shown below. Molten wax is poured in each of these containers up to the level indicated on the objects. Assuming that wax does not stick to the objects, which of the objects will leave its exact negative impression on the set wax when they are vertically lifted?

Coloured strips are printed on a transparent sheet as shown below. The bottom half of this sheet is folded backwards along the PQ axis to form a semi-circle. Which of the option(s) will be a part of the resultant semi-circle?

A table made out of cardboard is shown on left. If dotted lines indicate folding, which of the option(s) is/are the unfolded view(s) of the table?

A plant pruner (cutter) with two studs P and Q are shown on the left in open and closed positions. Which options(s) will allow the pruner to return to its open position?

A square paper of side 10 cm is folded sequentially and a circular cut of radius 1 cm is made at the corners as shown below. After unfolding the sheet completely, what will be the total area of all the pieces which have been cut-out from the original square sheet? Assume the value of $\pi$ = 3.14.

How many unique patterns are there in the image?

How many spheres are rotating in the space shown below?

Nine tiles containing parts of English letters in different Text Fonts, are shown below. What is the maximum number of COMPLETE letters that can be formed, by rearranging the tiles in a 3x3 grid?

A plane passes through three vertices of a cube and divides the cube into two parts, a green part and a blue part, and they remain together, as shown below. Eight such cubes are assembled to create a larger cube, where blue portion is on the inside as shown on the right. Calculate the volume of blue part in the larger cube, if the edge of the original cube is 1 cm.

A board game, and two views of a unique dice are shown below. This unique dice has only three faces with numbers one, two and three. Consider all the ladders are advantages that allow the player to jump from initial to final point. What is the minimum number of dice throws that is required to move from Point A to Point B using all given advantages on the board?

A coloured sheet is folded into an octahedron as shown on the left. Which of the options show(s) the CORRECT rotated views of the octahedron?

Five geometric blocks with their dimensions are given below. Calculate the length of the path as shown in the image below, between points A and B.

Shown below is an ice-cream cone completely filled with ice-cream. Assume the top portion (visible portion of ice-cream) is a perfect hemisphere. What is the total mass of the ice cream in grams? The density of ice-cream = 0.9 grams/cubic centimeter. Consider the value of $\pi$ = 3.14.

$\int \frac{1}{(1 - \cos x)}(1 + \sec x) dx = $

$\int \frac{10e^x}{(2e^x + 5)^3} dx$ is equal to

Let $f(x) = \cos(5x)\cos(3x) - \sin(5x)\sin(3x), 0 \leq x \leq \frac{\pi}{4}$. Then $f$ attains its minimum at $x =$

The point $P$ lies on the curve with equation \[ y = x\sqrt{2\log_e(x)}, \qquad x>0. \] If \[ \frac{dy}{dx}=2 \] at $x=k$, then the value of $k$ is equal to:

Let $f(x) = \frac{\sin x}{x}$ for $x \neq 0$. Then the value of $f'(\frac{\pi}{2})$ is equal to

Let $f(x)=2\sqrt{\,2\sin x+2\sqrt{2}\cos x\,},\; x\in\mathbb{R}$. Then the value of $f\!\left(\frac{\pi}{12}\right)$ is equal to

If \[ f(x)= \begin{cases} x^2, & \text{if } x\leq 2,\\[4pt] 4x-\alpha, & \text{if } x>2, \end{cases} \] is continuous at $x=2$, then the value of $\alpha$ is equal to:

Let the functions $f$ and $g$ be defined by $f(x)=3\sin x,\; -\frac{\pi}{2}\le x\le \frac{\pi}{2}$ and $g(x)=6-3x^2,\; x\in\mathbb{R}$. Then $f^{-1}(g(x))=$

If $\lim_{x \to 3} \frac{(2x - k)\tan(x - 3)}{x^2 - 6x + 9} = 2$, then the value of 'k' is equal to

The value of $\displaystyle \int_{0}^{1} \frac{t}{(t+1)^3}\,dt$ is equal to: