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GATE ME 2023 Set 1

All 65 solved GATE ME 2023 Set 1 questions in exam order. Open a question, commit to an answer, and learn from the step-by-step solution. One question at a time.

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Questions

65

Paper marks

100

Question formats

3

MCQ · NAT · MSQ

Revision mode

Self-paced

No timer. Focus on understanding.

Difficulty mixEasy 14Medium 39Hard 12

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65 of 65 questions

General Aptitude (GA)

10
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    Question

    He did not manage to fix the car himself, so he _______ in the garage.
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    Question

    Planting : Seed : : Raising : _____
    (By word meaning)
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    Question

    A certain country has 504 universities and 25951 colleges. These are categorised into Grades I, II, and III as shown in the given pie charts.
    What is the percentage, correct to one decimal place, of higher education institutions (colleges and universities) that fall into Grade III?
    Two pie charts showing the distribution of Universities and Colleges into Grades I, II, and III
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    Question

    The minute-hand and second-hand of a clock cross each other ______ times between 09:15:00 AM and 09:45:00 AM on a day.
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    Question

    The symbols O, *, Δ, and □ are to be filled, one in each box, as shown below.
    A 4x4 grid partially filled with symbols O, *, Δ, and a square. The top-left cell is marked with a '?'. The grid is divided into four 2x2 subgrids by bold lines.
    The rules for filling in the four symbols are as follows.
    1) Every row and every column must contain each of the four symbols.
    2) Every 2×2 square delineated by bold lines must contain each of the four symbols.
    Which symbol will occupy the box marked with ‘?’ in the partially filled figure?
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    Question

    In a recently held parent-teacher meeting, the teachers had very few complaints about Ravi. After all, Ravi was a hardworking and kind student. Incidentally, almost all of Ravi’s friends at school were hardworking and kind too. But the teachers drew attention to Ravi’s complete lack of interest in sports. The teachers believed that, along with some of his friends who showed similar disinterest in sports, Ravi needed to engage in some sports for his overall development.
    Based only on the information provided above, which one of the following statements can be logically inferred with certainty?
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    Question

    Consider the following inequalities p2−4q<4p^2 - 4q < 43p+2q<63p + 2q < 6where pp and qq are positive integers.
    The value of (p+q)(p + q) is _______.
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    Question

    Which one of the sentence sequences in the given options creates a coherent narrative?
    (i) I could not bring myself to knock.
    (ii) There was a murmur of unfamiliar voices coming from the big drawing room and the door was firmly shut.
    (iii) The passage was dark for a bit, but then it suddenly opened into a bright kitchen.
    (iv) I decided I would rather wander down the passage.
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    Question

    How many pairs of sets (S,T)(S, T) are possible among the subsets of {1,2,3,4,5,6}\{1, 2, 3, 4, 5, 6\} that satisfy the condition that SS is a subset of TT?
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    Question

    An opaque pyramid (shown below), with a square base and isosceles faces, is suspended in the path of a parallel beam of light, such that its shadow is cast on a screen oriented perpendicular to the direction of the light beam. The pyramid can be reoriented in any direction within the light beam. Under these conditions, which one of the shadows P, Q, R, and S is NOT possible?
    An opaque pyramid and four potential shadow shapes labeled P, Q, R, and S
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Mechanical Engineering (ME)

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    Question

    A machine produces a defective component with a probability of 0.0150.015. The number of defective components in a packed box containing 200200 components produced by the machine follows a Poisson distribution. The mean and the variance of the distribution are
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    Question

    The figure shows the plot of a function over the interval [−4,4][-4, 4]. Which one of the options given CORRECTLY identifies the function?
    A graph of a function over the interval [-4, 4
    ]
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    Question

    With reference to the Economic Order Quantity (EOQ) model, which one of the options given is correct?
    EOQ model graph showing Cost per unit vs Order quantity with four curves P1 (red dashed U-shape), P2 (blue dash-dot linear), P3 (blue dashed hyperbolic), and P4 (green solid horizontal)
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    Question

    Which one of the options given represents the feasible region of the linear programming model:Maximize 45X1+60X2\text{Maximize } 45X_1 + 60X_2X1≤45X_1 \leq 45X2≤50X_2 \leq 5010X1+10X2≥60010X_1 + 10X_2 \geq 60025X1+5X2≤75025X_1 + 5X_2 \leq 750
    Graph showing regions P, Q, R, and S defined by linear constraints
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    Question

    A cuboidal part has to be accurately positioned first, arresting six degrees of freedom and then clamped in a fixture, to be used for machining. Locating pins in the form of cylinders with hemi-spherical tips are to be placed on the fixture for positioning. Four different configurations of locating pins are proposed as shown. Which one of the options given is correct?
    Four 3D diagrams (P1, P2, P3, P4) showing a cuboidal block positioned by different arrangements of hemi-spherical tipped locating pins
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    Question

    The effective stiffness of a cantilever beam of length LL and flexural rigidity EIEI subjected to a transverse tip load WW is
    A cantilever beam of length L fixed at one end and subjected to a transverse load W at the free end.
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    Question

    The options show frames consisting of rigid bars connected by pin joints. Which one of the frames is non-rigid?
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    Question

    The S-N curve from a fatigue test for steel is shown. Which one of the options gives the endurance limit?
    S-N curve showing fatigue strength S vs number of cycles N. The curve starts at Sut and decreases through points 1, 2, 3 until it becomes horizontal at point 4 corresponding to stress S4.
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    Question

    Air (density = 1.2 kg/m31.2\text{ kg/m}^3, kinematic viscosity = 1.5×10−5 m2/s1.5 \times 10^{-5}\text{ m}^2\text{/s}) flows over a flat plate with a free-stream velocity of 2 m/s2\text{ m/s}. The wall shear stress at a location 15 mm15\text{ mm} from the leading edge is τw\tau_w. What is the wall shear stress at a location 30 mm30\text{ mm} from the leading edge?
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    Question

    Consider an isentropic flow of air (ratio of specific heats = 1.41.4) through a duct as shown in the figure.
    The variations in the flow across the cross-section are negligible. The flow conditions at Location 1 are given as follows: P1=100 kPaP_1 = 100\text{ kPa}, ρ1=1.2 kg/m3\rho_1 = 1.2\text{ kg/m}^3, u1=400 m/su_1 = 400\text{ m/s}The duct cross-sectional area at Location 2 is given by A2=2A1A_2 = 2A_1, where A1A_1 denotes the duct cross-sectional area at Location 1. Which one of the given statements about the velocity u2u_2 and pressure P2P_2 at Location 2 is TRUE?
    A diverging duct showing flow from Location 1 to Location 2 with increasing area.
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    Question

    Consider incompressible laminar flow of a constant property Newtonian fluid in an isothermal circular tube. The flow is steady with fully-developed temperature and velocity profiles. The Nusselt number for this flow depends on
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    Question

    A heat engine extracts heat (QHQ_H) from a thermal reservoir at a temperature of 1000 K1000\text{ K} and rejects heat (QLQ_L) to a thermal reservoir at a temperature of 100 K100\text{ K}, while producing work (WW). Which one of the combinations of [QH,QL and WQ_H, Q_L \text{ and } W] given is allowed?
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    Question

    Two surfaces P and Q are to be joined together. In which of the given joining operation(s), there is no melting of the two surfaces P and Q for creating the joint?
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    Question

    A beam is undergoing pure bending as shown in the figure. The stress (σ\sigma)-strain (ϵ\epsilon) curve for the material is also given. The yield stress of the material is σY\sigma_Y. Which of the option(s) given represent(s) the bending stress distribution at cross-section AA after plastic yielding?
    Diagram showing a beam in pure bending and its elastic-perfectly plastic stress-strain curve
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    Question

    In a metal casting process to manufacture parts, both patterns and moulds provide shape by dictating where the material should or should not go. Which of the option(s) given correctly describe(s) the mould and the pattern?
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    Question

    The principal stresses at a point P in a solid are 7070 MPa, −70-70 MPa and 00. The yield stress of the material is 100100 MPa. Which prediction(s) about material failure at P is/are CORRECT?
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    Question

    Which of the plot(s) shown is/are valid Mohr’s circle representations of a plane stress state in a material? (The center of each circle is indicated by OO.)
    Four Mohr's circle plots labeled M1, M2, M3, and M4
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    Question

    Consider a laterally insulated rod of length LL and constant thermal conductivity. Assuming one-dimensional heat conduction in the rod, which of the following steady-state temperature profile(s) can occur without internal heat generation?
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    Question

    Two meshing spur gears 1 and 2 with diametral pitch of 8 teeth per mm and an angular velocity ratio ∣ω2∣/∣ω1∣=1/4|\omega_2|/|\omega_1| = 1/4, have their centers 30 mm apart. The number of teeth on the driver (gear 1) is _______ .
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    Question

    The figure shows a block of mass m=20 kgm = 20\text{ kg} attached to a pair of identical linear springs, each having a spring constant k=1000 N/mk = 1000\text{ N/m}. The block oscillates on a frictionless horizontal surface. Assuming free vibrations, the time taken by the block to complete ten oscillations is _________ seconds. (Rounded off to two decimal places)
    Take π=3.14\pi = 3.14.
    A block of mass m attached to two parallel springs with spring constant k on a horizontal surface
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    Question

    A vector fieldB(x,y,z)=xi^+yj^−2zk^\mathbf{B}(x, y, z) = x \hat{i} + y \hat{j} - 2z \hat{k}is defined over a conical region having height h=2h = 2, base radius r=3r = 3 and axis along zz, as shown in the figure. The base of the cone lies in the xx-yy plane and is centered at the origin.
    If n\mathbf{n} denotes the unit outward normal to the curved surface SS of the cone, the value of the integral∫SB⋅n dS\int_S \mathbf{B} \cdot \mathbf{n} \, dSequals _________ . (Answer in integer)
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    Question

    A linear transformation maps a point (x,y)(x, y) in the plane to the point (x^,y^)(\hat{x}, \hat{y}) according to the rulex^=3y,y^=2x.\hat{x} = 3y, \quad \hat{y} = 2x.Then, the disc x2+y2≤1x^2 + y^2 \leq 1 gets transformed to a region with an area equal to
    _________ . (Rounded off to two decimals)
    Use π=3.14\pi = 3.14.
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    Question

    The value of kk that makes the complex-valued functionf(z)=e−kx(cos⁡2y−isin⁡2y)f(z) = e^{-kx} (\cos 2y - i \sin 2y)analytic, where z=x+iyz = x + iy, is _________.
    (Answer in integer)
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    Question

    The braking system shown in the figure uses a belt to slow down a pulley rotating in the clockwise direction by the application of a force P. The belt wraps around the pulley over an angle α=270\alpha = 270 degrees. The coefficient of friction between the belt and the pulley is 0.3. The influence of centrifugal forces on the belt is negligible.
    A braking system diagram showing a pulley with a belt wrapped around it. The belt has tensions $T_1$ and $T_2$ on either side. A lever is used to apply force P to tighten the belt.
    During braking, the ratio of the tensions T1T_1 to T2T_2 in the belt is equal to __________. (Rounded off to two decimal places)
    Take π=3.14\pi = 3.14.
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    Question

    Consider a counter-flow heat exchanger with the inlet temperatures of two fluids (1 and 2) being T1,in=300T_{1, in} = 300 K and T2,in=350T_{2, in} = 350 K. The heat capacity rates of the two fluids are C1=1000C_1 = 1000 W/K and C2=400C_2 = 400 W/K, and the effectiveness of the heat exchanger is 0.5. The actual heat transfer rate is _____ kW.
    (Answer in integer)
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    Question

    Which one of the options given is the inverse Laplace transform of 1s3−s\frac{1}{s^3 - s}?u(t)u(t) denotes the unit-step function.
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    Question

    A spherical ball weighing 2 kg is dropped from a height of 4.9 m onto an immovable rigid block as shown in the figure. If the collision is perfectly elastic, what is the momentum vector of the ball (in kg m/s) just after impact?
    Take the acceleration due to gravity to be g=9.8 m/s2g = 9.8 \text{ m/s}^2. Options have been rounded off to one decimal place.
    A spherical ball of mass 2 kg falling vertically from a height of 4.9 m onto a rigid block with a surface inclined at 30 degrees to the horizontal. A coordinate system with unit vectors $\hat{i}$ and $\hat{j}$ is shown.
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    Question

    The figure shows a wheel rolling without slipping on a horizontal plane with angular velocity ω1\omega_1. A rigid bar PQ is pinned to the wheel at P while the end Q slides on the floor.
    What is the angular velocity ω2\omega_2 of the bar PQ?
    A wheel of radius 3 m rolling on a floor. Point P is at a distance of 2 m from the center O. A bar PQ of length such that Q is on the floor and the horizontal distance from the vertical line through P to Q is 8 m. The wheel has angular velocity $\omega_1$.
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    Question

    A beam of length LL is loaded in the xyxy-plane by a uniformly distributed load, and by a concentrated tip load parallel to the zz-axis, as shown in the figure. The resulting bending moment distributions about the yy and the zz axes are denoted by MyM_y and MzM_z, respectively.
    Which one of the options given depicts qualitatively CORRECT variations of MyM_y and MzM_z along the length of the beam?
    A cantilever beam of length L with a uniformly distributed load q in the xy-plane and a concentrated tip load P parallel to the z-axis.
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    Question

    The figure shows a thin-walled open-top cylindrical vessel of radius rr and wall thickness tt. The vessel is held along the brim and contains a constant-density liquid to height hh from the base. Neglect atmospheric pressure, the weight of the vessel and bending stresses in the vessel walls.
    Which one of the plots depicts qualitatively CORRECT dependence of the magnitudes of axial wall stress (σ1\sigma_1) and circumferential wall stress (σ2\sigma_2) on yy?
    A thin-walled cylindrical vessel of radius r and thickness t, held at the brim, filled with liquid to height h. A coordinate y is defined as depth from the brim.
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    Question

    Which one of the following statements is FALSE?
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    Question

    Consider a fully adiabatic piston-cylinder arrangement as shown in the figure. The piston is massless and cross-sectional area of the cylinder is AA. The fluid inside the cylinder is air (considered as a perfect gas), with γ\gamma being the ratio of the specific heat at constant pressure to the specific heat at constant volume for air. The piston is initially located at a position L1L_1. The initial pressure of the air inside the cylinder is P1≫P0P_1 \gg P_0, where P0P_0 is the atmospheric pressure. The stop S1S_1 is instantaneously removed and the piston moves to the position L2L_2, where the equilibrium pressure of air inside the cylinder is P2≫P0P_2 \gg P_0.
    What is the work done by the piston on the atmosphere during this process?
    An adiabatic piston-cylinder containing air at pressure P1. The piston is at L1, held by stop S1. A second stop S2 is at L2. The external environment is at atmospheric pressure P0.
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    Question

    A cylindrical rod of length hh and diameter dd is placed inside a cubic enclosure of side length LL. SS denotes the inner surface of the cube. The view-factor FS−SF_{S-S} is
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    Question

    In an ideal orthogonal cutting experiment (see figure), the cutting speed VV is 1 m/s1\text{ m/s}, the rake angle of the tool α=5∘\alpha = 5^\circ, and the shear angle, ϕ\phi, is known to be 45∘45^\circ.
    Applying the ideal orthogonal cutting model, consider two shear planes PQPQ and RSRS close to each other. As they approach the thin shear zone (shown as a thick line in the figure), plane RSRS gets sheared with respect to PQPQ (point R1R1 shears to R2R2, and S1S1 shears to S2S2).
    Assuming that the perpendicular distance between PQPQ and RSRS is δ=25 μm\delta = 25\text{ }\mu\text{m}, what is the value of shear strain rate (in s−1\text{s}^{-1}) that the material undergoes at the shear zone?
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    Question

    A CNC machine has one of its linear positioning axes as shown in the figure, consisting of a motor rotating a lead screw, which in turn moves a nut horizontally on which a table is mounted. The motor moves in discrete rotational steps of 5050 steps per revolution. The pitch of the screw is 5 mm5\text{ mm} and the total horizontal traverse length of the table is 100 mm100\text{ mm}. What is the total number of controllable locations at which the table can be positioned on this axis?
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    Question

    Cylindrical bars P and Q have identical lengths and radii, but are composed of different linear elastic materials. The Young’s modulus and coefficient of thermal expansion of Q are twice the corresponding values of P. Assume the bars to be perfectly bonded at the interface, and their weights to be negligible.
    The bars are held between rigid supports as shown in the figure and the temperature is raised by ΔT\Delta T. Assume that the stress in each bar is homogeneous and uniaxial. Denote the magnitudes of stress in P and Q by σ1\sigma_1 and σ2\sigma_2, respectively.
    Which of the statement(s) given is/are CORRECT?
    Cylindrical bars P and Q between rigid supports
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    Question

    A very large metal plate of thickness dd and thermal conductivity kk is cooled by a stream of air at temperature T∞=300 KT_\infty = 300\text{ K} with a heat transfer coefficient hh, as shown in the figure. The centerline temperature of the plate is TPT_P. In which of the following case(s) can the lumped parameter model be used to study the heat transfer in the metal plate?
    Metal plate cooling diagram
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    Question

    The smallest perimeter that a rectangle with area of 4 square units can have is ______ units.
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    Question

    Consider the second-order linear ordinary differential equationx2d2ydx2+xdydx−y=0,x≥1x^2 \frac{d^2y}{dx^2} + x \frac{dy}{dx} - y = 0, x \geq 1with the initial conditionsy(x=1)=6,dydx∣x=1=2.y(x = 1) = 6, \quad \left. \frac{dy}{dx} \right|_{x=1} = 2.The value of yy at x=2x = 2 equals _________.
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    Question

    The initial value problemdydt+2y=0,y(0)=1\frac{dy}{dt} + 2y = 0, \quad y(0) = 1is solved numerically using the forward Euler’s method with a constant and positive time step of Δt\Delta t.
    Let yny_n represent the numerical solution obtained after nn steps. The condition ∣yn+1∣≤∣yn∣|y_{n+1}| \leq |y_n| is satisfied if and only if Δt\Delta t does not exceed _____________.
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    The atomic radius of a hypothetical face-centered cubic (FCC) metal is (2/10)(\sqrt{2}/10) nm. The atomic weight of the metal is 24.092 g/mol. Taking Avogadro’s number to be 6.023×10236.023 \times 10^{23} atoms/mol, the density of the metal is ________ kg/m3^3.
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    A steel sample with 1.5 wt.% carbon (no other alloying elements present) is slowly cooled from 1100 °C to just below the eutectoid temperature (723 °C). A part of the iron-cementite phase diagram is shown in the figure. The ratio of the pro-eutectoid cementite content to the total cementite content in the microstructure that develops just below the eutectoid temperature is ______.
    (Rounded off to two decimal places)
    Iron-cementite phase diagram for a steel sample.
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    A part, produced in high volumes, is dimensioned as shown. The machining process making this part is known to be statistically in control based on sampling data. The sampling data shows that D1 follows a normal distribution with a mean of 20 mm and a standard deviation of 0.3 mm, while D2 follows a normal distribution with a mean of 35 mm and a standard deviation of 0.4 mm. An inspection of dimension C is carried out in a sufficiently large number of parts.
    To be considered under six-sigma process control, the upper limit of dimension C should be __________ mm.
    (Rounded off to one decimal place)
    Diagram of a part with dimensions D1, D2, and C
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    A coordinate measuring machine (CMM) is used to determine the distance between Surface SPSP and Surface SQSQ of an approximately cuboidal shaped part. Surface SPSP is declared as the datum as per the engineering drawing used for manufacturing this part. The CMM is used to measure four points P1,P2,P3,P4P1, P2, P3, P4 on Surface SPSP, and four points Q1,Q2,Q3,Q4Q1, Q2, Q3, Q4 on Surface SQSQ as shown. A regression procedure is used to fit the necessary planes.
    The distance between the two fitted planes is ___________ mm.
    (Answer in integer)
    3D coordinate plot showing points P1-P4 on surface SP and Q1-Q4 on surface SQ
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    A solid part (see figure) of polymer material is to be fabricated by additive manufacturing (AM) in square-shaped layers starting from the bottom of the part working upwards. The nozzle diameter of the AM machine is a/10a/10 mm and the nozzle follows a linear serpentine path parallel to the sides of the square layers with a feed rate of a/5a/5 mm/min.
    Ignore any tool path motions other than those involved in adding material, and any other delays between layers or the serpentine scan lines.
    The time taken to fabricate this part is ___________ minutes.
    A 3D stepped square pyramid structure with three blocks of dimensions 3a x 3a x 1.5a, 2a x 2a x 0.5a, and a x a x a.

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    An optical flat is used to measure the height difference between a reference slip gauge A and a slip gauge B. Upon viewing via the optical flat using a monochromatic light of wavelength 0.5 μm0.5\ \mu\text{m}, 12 fringes were observed over a length of 15 mm15\ \text{mm} of gauge B. If the gauges are placed 45 mm45\ \text{mm} apart, the height difference of the gauges is ______________ μm\mu\text{m}.
    Diagram showing an optical flat placed on two slip gauges A and B on a flat surface, with dimensions 45 mm between gauges and 15 mm width for gauge B.
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    Ignoring the small elastic region, the true stress (σ\sigma) – true strain (ϵ\epsilon) variation of a material beyond yielding follows the equation σ=400ϵ0.3\sigma = 400\epsilon^{0.3} MPa. The engineering ultimate tensile strength value of this material is ________ MPa. (Rounded off to one decimal place)
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    The area moment of inertia about the yy-axis of a linearly tapered section shown in the figure is _____________ m4m^4.
    (Answer in integer)
    A linearly tapered section symmetric about the x-axis, with a total height of 3 m at the left end (x=0) and 6 m at the right end (x=12 m). The y-axis is located at the left end.
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    A cylindrical bar has a length L=5 mL = 5\text{ m} and cross section area S=10 m2S = 10\text{ m}^2. The bar is made of a linear elastic material with a density ρ=2700 kg/m3\rho = 2700\text{ kg/m}^3 and Young’s modulus E=70 GPaE = 70\text{ GPa}. The bar is suspended as shown in the figure and is in a state of uniaxial tension due to its self-weight.
    The elastic strain energy stored in the bar equals _________ J. (Rounded off to two decimal places)
    Take the acceleration due to gravity as g=9.8 m/s2g = 9.8\text{ m/s}^2.
    A cylindrical bar of length L suspended vertically from a rigid support, with gravity g acting downwards.
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    A cylindrical transmission shaft of length 1.5 m1.5\text{ m} and diameter 100 mm100\text{ mm} is made of a linear elastic material with a shear modulus of 80 GPa80\text{ GPa}. While operating at 500 rpm500\text{ rpm}, the angle of twist across its length is found to be 0.5 degrees0.5\text{ degrees}.
    The power transmitted by the shaft at this speed is _______ kW. (Rounded off to two decimal places)
    Take π=3.14\pi = 3.14.
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    Consider a mixture of two ideal gases, X and Y, with molar masses M‾X=10 kg/kmol\overline{M}_X = 10 \text{ kg/kmol} and M‾Y=20 kg/kmol\overline{M}_Y = 20 \text{ kg/kmol}, respectively, in a container. The total pressure in the container is 100 kPa100 \text{ kPa}, the total volume of the container is 10 m310 \text{ m}^3 and the temperature of the contents of the container is 300 K300 \text{ K}. If the mass of gas-X in the container is 2 kg2 \text{ kg}, then the mass of gas-Y in the container is ____ kg. (Rounded off to one decimal place)
    Assume that the universal gas constant is 8314 J kmol−1K−18314 \text{ J kmol}^{-1}\text{K}^{-1}.
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    The velocity field of a certain two-dimensional flow is given byV(x,y)=k(xi^−yj^)\mathbf{V}(x, y) = k(x\hat{i} - y\hat{j})where k=2 s−1k = 2 \text{ s}^{-1}. The coordinates xx and yy are in meters. Assume gravitational effects to be negligible.
    If the density of the fluid is 1000 kg/m31000 \text{ kg/m}^3 and the pressure at the origin is 100 kPa100 \text{ kPa}, the pressure at the location (2 m,2 m)(2 \text{ m}, 2 \text{ m}) is _____________ kPa.
    (Answer in integer)
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    Consider a unidirectional fluid flow with the velocity field given byV(x,y,z,t)=u(x,t)i^\mathbf{V}(x, y, z, t) = u(x, t) \hat{i}where u(0,t)=1u(0, t) = 1. If the spatially homogeneous density field varies with time tt asρ(t)=1+0.2e−t\rho(t) = 1 + 0.2e^{-t}the value of u(2,1)u(2, 1) is ______________. (Rounded off to two decimal places)
    Assume all quantities to be dimensionless.
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    The figure shows two fluids held by a hinged gate. The atmospheric pressure is Pa=100 kPaP_a = 100\text{ kPa}. The moment per unit width about the base of the hinge is ________ kNm/m\text{kNm/m}. (Rounded off to one decimal place)
    Take the acceleration due to gravity to be g=9.8 m/s2g = 9.8\text{ m/s}^2.
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    An explosion at time t=0t = 0 releases energy EE at the origin in a space filled with a gas of density ρ\rho. Subsequently, a hemispherical blast wave propagates radially outwards as shown in the figure.
    Let RR denote the radius of the front of the hemispherical blast wave. The radius RR follows the relationship R=ktaEbρcR = k t^a E^b \rho^c, where kk is a dimensionless constant. The value of exponent aa is __________.
    (Rounded off to one decimal place)
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