List of practice Questions

A simply supported horizontal beam is subjected to a distributed transverse load varying linearly from \( q_0 \) at A to zero at B, as shown in the figure. Which one of the following options is correct?
\includegraphics[width=0.5\linewidth]{27.png}

A uniform symmetric cross-section cantilever beam of length \( L \) is subjected to a transverse force \( P \) at the free end, as shown in the figure. The Young’s modulus of the material is \( E \) and the moment of inertia is \( I \). Ignoring the contributions due to transverse shear, the strain energy stored in the beam is \_\_\_\_\_.
\includegraphics[width=0.5\linewidth]{29.png}

In the given figure, plate ABCD in its undeformed configuration (solid line) is a rhombus with all the internal angles being 90°. The lengths of the undeformed diagonals are 20 cm. ABCD deforms as shown by the dotted lines. Upon deformation, diagonal AC reduces to 19.96 cm and BD increases to 20.04 cm. In the given x-y coordinate system, the engineering shear strain \( \gamma_{xy} \) is equal to \_\_\_\_\_.
\includegraphics[width=0.5\linewidth]{30.png}

Consider a pair of point vortices with clockwise circulation \( \Gamma \) each. The distance between their centers is \( a \), as shown in the figure. Assume two-dimensional, incompressible, inviscid flow. Which one of the following options is correct?
\includegraphics[width=0.5\linewidth]{20.png}

The eigenvalues of the matrix \[ \begin{bmatrix} 1 & 2
0 & 3 \end{bmatrix} \] are \_\_\_\_\_ .

The partial differential equation \[ \frac{\partial^2 u}{\partial x^2} + 4 \frac{\partial^2 u}{\partial x \partial y} + 2 \frac{\partial^2 u}{\partial y^2} = 0 \] is \_\_\_\_\_\_.

Let the function \( f(x) \) be defined as: \[ f(x) = \begin{cases} A + x, & {if } x<2
1 + x^2, & {if } x \geq 2 \end{cases} \] If the function \( f(x) \) is continuous at \( x = 2 \), the value of \( A \) is \_\_\_\_\_.

If \( i = \sqrt{-1} \), \[ \frac{(i+1)^3}{i-1} = \_\_\_\_\_\_\_\_. \]

For a two-dimensional incompressible flow over a flat plate, the laminar boundary layer thickness at a distance \( x \) from the leading edge is \( \delta \). If \( Re_x \) is the Reynolds number defined based on length scale \( x \), \[ \frac{\delta}{x} \propto \_\_\_\_\_\_. \]

Weight of a person can be expressed as a function of their age. The function usually varies from person to person. Suppose this function is identical for two brothers, and it monotonically increases till the age of 50 years and then it monotonically decreases. Let \( a_1 \) and \( a_2 \) (in years) denote the ages of the brothers and \( a_1<a_2 \). Which one of the following statements is correct about their age on the day when they attain the same weight?

A regular dodecagon (12-sided regular polygon) is inscribed in a circle of radius \( r \) cm as shown in the figure. The side of the dodecagon is \( d \) cm. All the triangles (numbered 1 to 12 in the figure) are used to form squares of side \( r \) cm, and each numbered triangle is used only once to form a square. The number of squares that can be formed and the number of triangles required to form each square, respectively, are:
\includegraphics[width=0.5\linewidth]{8.png}

If a real variable \(x\) satisfies \(3^{x^2} = 27 \times 9^x\), then the value of \(\frac{2^{x^2}}{(2^x)^2\) is:}

The number of patients per shift (X) consulting Dr. Gita in her past 100 shifts is shown in the figure. If the amount she earns is \({\rupee} 1000(X - 0.2)\), what is the average amount (in \rupee) she has earned per shift in the past 100 shifts?
\includegraphics[width=0.5\linewidth]{10.png}

Solve the system of equations: \[ 2x + 3y + z = 0 \] \[ x + y = 0 \] \[ y + z = 0 \] The given system of equations has:

We \_\_\_\_\_\_\_ tennis in the lawn when it suddenly started to rain.
Select the most appropriate option to complete the above sentence.