List of practice Questions

A tank with internal dimension 10 cm x 10 cm x 10 cm (length x width x height) is fully filled with spherical marbles, each marble having a diameter of 1 cm. Each layer of marbles should have equal number of marbles. Water is then poured into the tank. What is the volume of water in cubic centimeters required to fill the voids? (Use \( \pi = \frac227 \))
Image P shows an object as seen from the top. Which of the objects given below will appear as shown in image P when viewed from the top? Consider all the objects are made with opaque solids.
Q, R, S and T are cut-out shapes used to create a composition shown in image P. These cut-out shapes could be rotated and combined without overlapping to create the composition. These cut-out shapes can also be repeated. What is/are the correct combination(s) that can be used to create the composition shown in image P?
Given below is a pattern P, composed of three different shapes 1, 2, and 3. Which options show the pattern(s) that is/are possible if the shape 2 is cut in half along the vertical axis, the shape 3 is rotated clockwise at the center by 45 degrees, and the shape 1 remains unchanged? Overlapping of shapes is not allowed.
Given below are five images P, Q, R, S and T. Each of these is rotated about its center. Which of the following statement(s) is/are TRUE? (Ignore minor variations due to pixelated appearance)
A composition made of letters is shown below. What is the total number of letters used to create this image?
A sharpened pencil with eraser is shown in image P. How many sharpened pencils with erasers are in image Q?
Several kettles are shown below. What is the total number of usable kettles that are designed to store and pour hot water?
Given below are 8 triangular pieces of a puzzle made from the shape P. What is the three-digit number formed when these pieces are arranged correctly?
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Given below is a drawing on a sheet of paper. Make a cut along the solid arcs and fold the resulting semicircular rings at 90 degrees along the dotted lines. What is the number of raised semicircular rings after following these steps?
A mobile screen has a dimension of 10 cm x 4 cm (height x width). A mobile user is viewing a picture in portrait mode and it occupies the entire screen. The user rotates the screen to landscape mode. After rotation, the height of the picture is now the width of the screen and its width gets reduced proportionally, leaving equal blank spaces on both sides of the picture. What is the area (in square centimeters) of the blank space on the left-hand side of the picture?
What number will replace the question mark?
In the image shown below, PQ = QR = QS = 2 units. Also, QT = TS = QU = UR. The points T and U are the centre points of the semicircles. What is the total area of shaded portions in square units?
What is the number that you get when you divide the difference between the number of triangles and number of circles by the number of rectangles for the image shown below?
Shown below is the front and top views of two cubes. The red and blue cubes are made up of metal and wax respectively. Red cube is heated and is hot enough to melt wax in the blue cube. As shown in the front and top views, the red cube is moved along the path PQRS creating sharp edges and flat surfaces on the blue cube. What is the total number of surfaces the blue cube will have after the red cube reaches the point S?
One of the pots does not belong to the group of pots shown below. What is the number written on that pot?
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What is the number of different patterns on the capsules shown?
How many times does the most used black coloured shape appear in the composition shown below?
If the sum of first 4 terms of an A.P. is \(6\) and the sum of first 6 terms is \(4\), then the sum of first 12 terms of the A.P. is
If \[ A=\begin{bmatrix}2 & 3\\3 & 5\end{bmatrix}, \] then find the value of \[ \left|A^{2025}-3A^{2024}+A^{2023}\right|. \]
If the domain of the function \[ f(x)=\frac{1}{\ln(10-x)}+\sin^{-1}\!\left(\frac{x+2}{2x+3}\right) \] is \[ (-\infty,-a]\ \cup\ (-1,b)\ \cup\ (b,c), \] then find the value of \( (b+c+3a) \).
Let \( M = \{1,2,3,\ldots,16\} \) and \( R \) be a relation on \( M \) defined by \( xRy \) if and only if \( 4y = 5x - 3 \). Then, the number of ordered pairs required to be added to \( R \) to make it symmetric is
The number of values of \(x\) satisfying \[ \tan^{-1}(4x)+\tan^{-1}(6x)=\frac{\pi}{6}, \quad x\in\left[-\frac{1}{2\sqrt{6}},\,\frac{1}{2\sqrt{6}}\right] \] is
The value of \[ \int_{\frac{\pi}{2}}^{\pi} \frac{dx}{[x]+4} \] where \([\,\cdot\,]\) denotes the greatest integer function, is
Let \( A = \begin{bmatrix} 3 & -4 \\ 1 & -1 \end{bmatrix} \) and \( B = \begin{bmatrix} 29 & 49 \\1 & 2 \end{bmatrix} \). If \( (A^5 + B)\begin{bmatrix} x\\ y \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \end{bmatrix} \), then find \( (x, y) \).