AP PGECET 2026 Electrical Engineering Question Paper: Download Answer Key with Solutions

Minimum number of 2-input NAND gates required for implementing the logic \( F = AB + A'C \) is:

• (A) 3
• (B) 4
• (C) 5
• (D) 6

The Universal gate is:

• (A) AND
• (B) OR
• (C) NAND
• (D) XOR

The MSB frequency of a 4-bit ripple counter with 16 MHz clock is

• (A) 1 MHz
• (B) 2 MHz
• (C) 4 MHz
• (D) 8 MHz

A JK flip-flop with J=K=1 behaves as:

• (A) Reset
• (B) Set
• (C) Toggle
• (D) Hold

The Race-around occurs when:

• (A) Clock width small
• (B) Clock width \(>\) propagation delay
• (C) Input zero
• (D) Output constant

The number of Select lines required for a 16:1 MUX is

• (A) 2
• (B) 3
• (C) 4
• (D) 5

The Karnaugh map is used for:

• (A) Timing
• (B) Simplification
• (C) Storage
• (D) Speed

A full adder can be built using:

• (A) One Half-Adder (HA)
• (B) Two HAs + OR
• (C) XOR only
• (D) Three HAs

A Decoder with n inputs has the following number of outputs:

• (A) n
• (B) \( 2^n \)
• (C) \( n^2 \)
• (D) \( \log n \)

Setup time is the time:

• (A) After clock
• (B) Before the clock edge
• (C) During clock
• (D) Between outputs

Mealy output depends on:

• (A) State
• (B) State and Input
• (C) Input
• (D) Previous state

In a moving coil instruments, the deflecting torque is proportional to:

• (A) \( I^2 \)
• (B) \( I \)
• (C) \( V \)
• (D) Power

A Moving iron instrument measures:

• (A) DC only
• (B) AC only
• (C) AC and DC
• (D) Pulsating DC

In an analogue instrument, the Controlling torque is provided by:

• (A) Springs only
• (B) Damping
• (C) Spring or gravity
• (D) Gravity only

In measuring instruments, The Damping torque is used to:

• (A) Increase Sensitivity
• (B) Reduce Oscillations
• (C) Increase Deflection
• (D) Measure Current

In a Voltmeter the sensitivity is given by:

• (A) \( \Omega/V \)
• (B) \( V/\Omega \)
• (C) \( A/V \)
• (D) \( W/V \)

To minimize loading, the voltmeter should have:

• (A) Low Resistance
• (B) High Resistance
• (C) High Current
• (D) Low Voltage

A Permanent Magnet Moving Coil (PMMC) instrument can be used to measure:

• (A) DC quantities only
• (B) AC quantities only
• (C) Both AC and DC quantities
• (D) Only high-frequency AC

Which bridge is best suited for the measurement of a high-quality factor (Q) inductor?

• (A) Maxwell's Bridge
• (B) Hay's Bridge
• (C) Anderson's Bridge
• (D) Schering Bridge

A 100V voltmeter has an accuracy of \( \pm \)2% of full-scale deflection (FSD). If it reads 50V, what is the limiting error?

• (A) \( \pm \)2%
• (B) \( \pm \)1%
• (C) \( \pm \)4%
• (D) \( \pm \)5%

The "Creeping" phenomenon occurs in which type of instrument?

• (A) PMMC Ammeter
• (B) Induction type Energy Meter
• (C) Digital Multimeter
• (D) Dynamometer Wattmeter

A digital voltmeter (DVM) with a \( 3 \frac{1}{2} \) digit display can show a maximum count of:

• (A) 999
• (B) 1999
• (C) 3999
• (D) 9999

A Dynamometer-type instrument is primarily used as a "Transfer Instrument" because:

• (A) It has high sensitivity
• (B) It has a linear scale
• (C) It gives the same calibration for both AC and DC
• (D) It is very inexpensive

A Q-meter works on the principle of:

• (A) Mutual Induction
• (B) Series Resonance
• (C) Parallel Resonance
• (D) Piezoelectric effect

Which of the following is an "Active" transducer?

• (A) Strain Gauge
• (B) Thermistor
• (C) Thermocouple
• (D) LVDT

The Lissajous patterns are used to measure:

• (A) Voltage and Current
• (B) Power and Energy
• (C) Frequency and Phase shift
• (D) Resistance and Inductance

The bridge used for the measurement of an unknown capacitance is:

• (A) Wheatstone Bridge
• (B) Wien Bridge
• (C) Schering Bridge
• (D) Kelvin Double Bridge

For a Type-1 system, the steady-state error to a unit ramp input is:

• (A) Zero
• (B) 1/Kv
• (C) Infinite
• (D) 1/(1+Kp)

In the Routh-Hurwitz criterion, a system is stable if:

• (A) All elements in the first row are positive
• (B) All elements in the first column have the same sign
• (C) All roots are on the right-half plane
• (D) The determinant is zero

The Root Locus always starts at the _______ and ends at the _______.

• (A) Zeros; Poles
• (B) Poles; Zeros
• (C) Origin; Infinity
• (D) Real axis; Imaginary axis

Boost converter output \( V_o \) is:

• (A) Less than \( V_{in} \)
• (B) Equal to \( V_{in} \)
• (C) Greater than \( V_{in} \)
• (D) Zero

A Gain Margin of 0 dB or a Phase Margin of 0 degrees indicates the system is:

• (A) Stable
• (B) Unstable
• (C) Marginally stable
• (D) Highly damped

Lead compensation is used primarily to:

• (A) Increase the steady-state accuracy
• (B) Reduce the bandwidth
• (C) Improve the transient response and increase speed
• (D) Eliminate noise

In State-Space representation, the "State Vector" consists of:

• (A) Only the input variables
• (B) The minimum set of variables that describe the system's future behavior
• (C) The output variables only
• (D) The coefficients of the transfer function

A system is "Controllable" if:

• (A) All states can be reached from an arbitrary initial state using an input
• (B) The output can be determined by observing the input
• (C) The eigenvalues are all negative
• (D) The transfer function has no zeros

The State Transition Matrix \( \phi(t) \) is given by:

• (A) \( \mathcal{L}^{-1}[sI - A] \)
• (B) \( \mathcal{L}^{-1}[(sI - A)^{-1}] \)
• (C) \( [sI - A]^{-1} \)
• (D) \( e^{-At} \)

If the eigenvalues of the system matrix A are -2 and -5, the system is:

• (A) Stable
• (B) Unstable
• (C) Oscillatory
• (D) Critically stable

Which plot is used to determine stability by looking at the encirclements of the point (-1, j0)?

• (A) Bode Plot
• (B) Root Locus
• (C) Nyquist Plot
• (D) Nichols Chart

Adding a Zero to the transfer function generally:

• (A) Increases the overshoot and improves the speed of response
• (B) Decreases the overshoot
• (C) Makes the system unstable
• (D) Has no effect on the transient response

A diode current increases from 1 mA to 10 mA. Dynamic resistance changes by:

• (A) 10\( \times \) increase
• (B) 10\( \times \) decrease
• (C) Same
• (D) 100\( \times \) decrease

A DC generator has flux doubled and speed halved. New EMF is

• (A) Doubled
• (B) Halved
• (C) Same
• (D) Zero

An autotransformer converts 200V to 100V supplying same load. Copper saving is

• (A) 25%
• (B) 50%
• (C) 75%
• (D) 100%

A synchronous generator is over-excited and running in parallel with an identical generator at the same terminal voltage. The over-excited generator will:

• (A) Supply real power only
• (B) Supply reactive power only
• (C) Supply both real and reactive power
• (D) Absorb reactive power

A 6-pulse HVDC rectifier produces which harmonics in AC line current?

• (A) 5th, 7th, 11th, 13th ...
• (B) 6th, 12th, 18th ...
• (C) 3rd, 9th, 15th ...
• (D) Only 1st

A 12-pulse HVDC link: line voltage = 220 kV (LL), \(\alpha = 30^\circ\). Approximate DC voltage:

• (A) 220 kV
• (B) 238 kV
• (C) 325 kV
• (D) 200 kV

An overcurrent relay operates at 1.2pu for 2 s. Fault current = 3pu. What is the expected operation time if relay is inverse-time type?

• (A) \(<\)2s
• (B) 2s
• (C) \(>\)2s
• (D) Depends on system voltage

If \(A^2 = 0\), then:

• (A) \(e^{At} = I\)
• (B) \(e^{At} = I + At\)
• (C) \(e^{At} = At\)
• (D) Infinite series required

An AC voltage controller reduces the RMS voltage across a resistive load to 0.5 of its original value. What is the reduction in power?

• (A) 50%
• (B) 75%
• (C) 25%
• (D) No change

The property of state transition matrix, \(\Phi(t_1 + t_2) =\)

• (A) \(\Phi(t_1) + \Phi(t_2)\)
• (B) \(\Phi(t_1) \cdot \Phi(t_2)\)
• (C) \(\Phi(t_1) - \Phi(t_2)\)
• (D) \(\Phi(t_1) / \Phi(t_2)\)

In a connected electrical network graph, which statement about Kirchhoff's Current Law (KCL) is most accurate?

• (A) KCL is derived from Faraday's law of electromagnetic induction and applies only to mesh analysis.
• (B) KCL is applicable only to resistive networks under steady state DC conditions and is not valid for AC circuits.
• (C) KCL is valid only for networks consisting of linear and time-invariant elements.
• (D) KCL is based on the principle of conservation of charge and states that the algebraic sum of currents at a node is zero

A series RL circuit consists of a resistor \(R=10 \ \Omega\) and an inductor \(L=1 H\), connected to a DC voltage source of \(V=20 V\). The switch is closed at \(t=0\), and the initial current is zero. Determine the expression for the current \(i(t)\) through the inductor for \(t>0\), assuming ideal components.

• (A) \(i(t)=2e^{-10t} A\)
• (B) \(i(t)=2(1-e^{-0.1t}) A\)
• (C) \(i(t)=2(1-e^{-10t}) A\)
• (D) \(i(t)=2e^{-0.1t} A\)

A series RLC circuit has \(R = 10 \ \Omega\), \(L = 1 H\), \(C = 100 \ \mu F\). At resonance, the impedance is:

• (A) \(0 \ \Omega\)
• (B) \(10 \ \Omega\)
• (C) \(100 \ \Omega\)
• (D) Depends on frequency

In an induction motor, maximum torque occurs when:

• (A) \(R_2 = X_2\)
• (B) \(R_2 = s X_2\)
• (C) \(\frac{R_2}{s} = X_2\)
• (D) \(\frac{R_2}{X_2} = s_m\)

A balanced three-phase, star-connected source with a line-to-line RMS voltage of \(400 V\) supplies a balanced star-connected load. Each phase of the load has a resistance of \(20 \ \Omega\). Neglect the line impedance. Calculate the total active power consumed by the load.

• (A) \(6 kW\)
• (B) \(4 kW\)
• (C) \(12 kW\)
• (D) \(8 kW\)

In a Capacitor-start motor, main winding current is \(2.0 pu\), auxiliary winding current is \(2.0 pu\) and the phase difference is \(90^\circ\). Then the starting torque is:

• (A) \(2.0 pu\)
• (B) \(4.0 pu\)
• (C) \(6.0 pu\)
• (D) \(8.0 pu\)

A stepper motor with a step angle \(1.8^\circ\) requires ______ steps per one revolution.

• (A) 180
• (B) 200
• (C) 360
• (D) 100

Which of the following is true about hunting in synchronous machines?

• (A) Occurs at low speed only
• (B) Occurs when mechanical load fluctuates
• (C) Is due to sudden change in field current
• (D) Is due to interaction between rotor inertia and synchronizing torque

A series RL circuit has \(V = 10 V\), \(R = 5 \ \Omega\), \(L = 2 H\). With \(i(t) = (V/R)(1 - e^{-t/\tau})\), \(\tau = L/R\). Find \(i\) at \(t = 0.5 s\).

• (A) \(1.26 A\)
• (B) \(1.428 A\)
• (C) \(2.00 A\)
• (D) \(0.95 A\)

A series RLC circuit has \(R=20 \ \Omega\), \(L=1 H\), \(C=0.01 F\). Using \(\zeta = R / (2\sqrt{L/C})\), the response is:

• (A) Underdamped
• (B) Critically damped
• (C) Overdamped
• (D) Sustained oscillations

A \(4 A\) current source feeds \(8 \ \Omega\) and \(4 \ \Omega\) in parallel. Power in \(8 \ \Omega\) is:

• (A) \(32 W\)
• (B) \(16 W\)
• (C) \(14.22 W\)
• (D) \(10.67 W\)

A 24 V source with internal resistance 6 \(\Omega\) feeds a load. Maximum power to load is:

• (A) 24 W
• (B) 48 W
• (C) 36 W
• (D) 12 W

\(v(t) = 50\sqrt{2} \sin(\omega t)\) is applied to \(Z = 3 + j4 \ \Omega\). Average power is:

• (A) 200 W
• (B) 250 W
• (C) 300 W
• (D) 400 W

With dependent sources, Thevenin's resistance is found by:

• (A) Open-circuit method
• (B) Short-circuit method
• (C) Applying a test source (V/I)
• (D) Ignoring sources

The Maximum power transfer in an AC network occurs when:

• (A) \(RL = RS\)
• (B) \(ZL = ZS^*\)
• (C) \(ZL = ZS\)
• (D) \(RL = 0\)

The Newton–Raphson method is preferred because it,

• (A) is simplest
• (B) has Quadratic Convergence
• (C) uses fewer equations
• (D) avoids iteration

In a per-unit system, the base impedance (\(Z_{base}\)) is defined as:

• (A) \(V_{base} \times I_{base}\)
• (B) \(V_{base} / I_{base}\)
• (C) \(I_{base} / V_{base}\)
• (D) \(V_{base}^2 \times P_{base}\)

The Swing Equation describes the relationship between:

• (A) Voltage and Current
• (B) Power Angle and Rotor Dynamics
• (C) Fault Current and Time
• (D) Real Power and Reactive Power

In a Load Flow study, which parameters are known at a PQ (Load) bus?

• (A) P and V
• (B) V and \(\delta\)
• (C) P and Q
• (D) Q and \(\delta\)

The "Equal Area Criterion" is used to determine:

• (A) Economic Dispatch
• (B) Fault Current Magnitude
• (C) Transient Stability Limits
• (D) Transmission Line Losses

If the supply frequency increases, the Skin Effect in a conductor:

• (A) Increases
• (B) Decreases
• (C) Remains the same
• (D) Becomes zero

A 100 MVA, 11 kV generator has a reactance of 0.2 pu. What is the reactance in ohms?

• (A) \(0.242 \ \Omega\)
• (B) \(2.42 \ \Omega\)
• (C) \(24.2 \ \Omega\)
• (D) \(1.21 \ \Omega\)

The Differential protection is primarily used for:

• (A) Long Transmission Lines
• (B) Transformers and Generators
• (C) Distribution Feeders
• (D) Lightning Protection

In Economic Load Dispatch, the most efficient operation occurs when:

• (A) All units operate at max capacity.
• (B) Incremental fuel costs of all units are equal.
• (C) Total fuel consumption is maximized.
• (D) All units carry equal loads.

What is the primary purpose of a FACTS device like a STATCOM?

• (A) To break circuit arcs.
• (B) To provide fast-acting reactive power compensation.
• (C) To measure harmonic distortion.
• (D) To convert AC to DC

HVDC transmission is preferred over HVAC for very long distances because:

• (A) DC equipment is cheaper.
• (B) There are no charging current losses or skin effect.
• (C) It doesn't require transformers.
• (D) It is easier to tap off power.

Which relay is most affected by power swings?

• (A) Mho Relay
• (B) Differential Relay
• (C) Over-current Relay
• (D) Buchholz Relay

The "Corona Effect" in a transmission line is most likely to occur during

• (A) High-voltage transmission in fair weather
• (B) Low-voltage transmission in humid weather
• (C) High-voltage transmission in rainy or foggy weather
• (D) Low-voltage transmission in dry weather

A symmetrical 3-phase fault occurs at a bus. If the pre-fault voltage is 1.0pu and the Thevenin impedance is j0.2pu, the fault current in per-unit is:

• (A) 0.2pu
• (B) 2.0pu
• (C) 5.0pu
• (D) 10.0pu

If a transmission line carries 100 MW at unity power factor, the reactive power is:

• (A) 0 MVAR
• (B) 50 MVAR
• (C) 1000 MVAR
• (D) 10 MVAR

In a single-phase transformer, the core is laminated primarily to reduce:

• (A) Copper loss
• (B) Hysteresis loss
• (C) Eddy current loss
• (D) Friction loss

A 200/100 V transformer has a primary resistance of 0.1 \(\Omega\). What is this resistance when referred to the secondary side?

• (A) 0.4 \(\Omega\)
• (B) 0.2 \(\Omega\)
• (C) 0.05 \(\Omega\)
• (D) 0.025 \(\Omega\)

For two three-phase transformers to operate in parallel, which of these is an absolute requirement?

• (A) Same KVA rating
• (B) Same cooling method
• (C) Same phase sequence
• (D) Same percentage impedance

An auto-transformer is most efficient when the transformation ratio (\(K = V_2/V_1\)) is:

• (A) Near zero
• (B) Near unity (1.0)
• (C) Exactly 0.5
• (D) Greater than 10

In a DC Machine, the function of the commutator is to:

• (A) Convert AC to DC in a generator
• (B) Convert DC to AC in a motor
• (C) Convert AC to DC in a generator and DC to AC in a motor
• (D) Increase the speed of the machine

The effect of "Armature Reaction" in a DC generator mainly results in:

• (A) Cross-magnetization and demagnetization
• (B) Increased terminal voltage
• (C) Improved commutation
• (D) Reduction in mechanical friction

A 4-pole, 50 Hz induction motor has a synchronous speed of:

• (A) 1000 rpm
• (B) 1500 rpm
• (C) 3000 rpm
• (D) 750 rpm

If the air gap of an induction motor is increased, then,

• (A) Efficiency will increase
• (B) Magnetizing current will increase
• (C) Power factor will improve
• (D) Speed will increase

The starting torque of a capacitor-start single-phase induction motor is:

• (A) Zero
• (B) Low
• (C) High
• (D) Same as a shaded-pole motor

A synchronous motor can be used for power factor correction when it is:

• (A) Under-excited only
• (B) Over-excited only
• (C) Operating at no load only
• (D) Both Over-excited and Operating at no load

In a synchronous generator, "Voltage Regulation" is negative when the load is:

• (A) Purely resistive
• (B) Inductive (Lagging)
• (C) Capacitive (Leading)
• (D) Unity power factor

The "Hunting" in synchronous machines can be prevented by using:

• (A) Slip rings
• (B) Damper windings
• (C) Carbon brushes
• (D) Compensating windings

A Servo Motor differs from a standard motor because it:

• (A) Cannot be used for AC
• (B) Operates only at very high speeds
• (C) Features a feedback mechanism for precise control
• (D) Does not have a rotor

What happens if the field winding of a running DC Shunt motor suddenly opens?

• (A) The motor stops immediately
• (B) The motor speed remains constant
• (C) The motor attains dangerously high speed
• (D) The motor starts running in reverse

A signal x(t) = cos(2\(\pi\)t) + sin(3\(\pi\)t) is:

• (A) Periodic only
• (B) Aperiodic only
• (C) Both periodic and even
• (D) Neither periodic nor odd

If x[n] = u[n] and h[n] = u[n - 2], find the value of y[5].

• (A) 3
• (B) 4
• (C) 5
• (D) 6

If energy of x(t) = e\(^{-2t}\)u(t) is 0.25, find a:

• (A) 1
• (B) 3
• (C) 2
• (D) 1/4

If x(t) \(\leftrightarrow\) X(\(\omega\)), then FT of tx(t) is:

• (A) j \(\frac{dX(\omega)}{d\omega}\)
• (B) -j \(\frac{dX(\omega)}{d\omega}\)
• (C) jX(\(\omega\))
• (D) \(\frac{X(\omega)}{\omega}\)

A signal contains frequencies up to 4\(\pi\) rad/sec. Minimum sampling frequency is:

• (A) 4\(\pi\)
• (B) 8\(\pi\)
• (C) 2\(\pi\)
• (D) 16\(\pi\)

Given X(s) = \(\frac{1}{s+2}\), ROC: Re(s) \(>\) -2. Signal is:

• (A) e\(^{-2t}\)u(t)
• (B) -e\(^{-2t}\)u(-t)
• (C) e\(^{2t}\)u(t)
• (D) e\(^{-2t}\)u(-t)

Find the z-transform of x[n] = \(\delta\)[n] + \(\delta\)[n + 1]:

• (A) 1 + z\(^{-1}\)
• (B) z + 1
• (C) 1 + z
• (D) z\(^{-1}\)

If the Impulse response is \(h[n] = (0.5)^n u(n)\) then, the system is

• (A) unstable
• (B) stable
• (C) marginally stable
• (D) time-varying

The z-transform of \(a^n u(n)\) is:

• (A) \(\frac{z}{z-a}\)
• (B) \(\frac{1}{1-az}\)
• (C) \(\frac{1}{z-a}\)
• (D) \(\frac{1}{1+az}\)

Which of the following system is linear?

• (A) \(y(t) = x^2(t)\)
• (B) \(y(t) = 3x(t) + 2\)
• (C) \(y(t) = x(t) + x(t - 1)\)
• (D) \(y(t) = \sin(x(t))\)

A band limited signal with \(f_{max} = 5Hz\) is sampled at 8kHz. The result is:

• (A) Perfect reconstruction
• (B) Aliasing
• (C) No distortion
• (D) Infinite bandwidth

Aliasing occurs when,

• (A) Sampling rate is higher than Nyquist rate
• (B) Sampling rate is lower than Nyquist rate
• (C) Signal is periodic
• (D) Signal is even

The SCR triggers when:

• (A) Gate pulse with forward bias
• (B) Reverse voltage
• (C) Temperature rises
• (D) Current zero

3 - \(\phi\) fully controlled rectifier \(V_{avg} \propto\):

• (A) \((3V_m/\pi)\cos\alpha\)
• (B) \((3\sqrt{3}V_m/\pi)\cos\alpha\)
• (C) \((6V_m/\pi)\cos\alpha\)
• (D) \((V_m/\pi)\cos\alpha\)

The FIFO structure represents a:

• (A) Stack
• (B) Queue
• (C) Tree
• (D) Graph

The Output of the statement \(int\ x = 5;\ printf("%d, x++");\) is:

• (A) 5
• (B) 6
• (C) 0
• (D) Error

The Advantages of Renewable energy sources are:

• (A) High cost
• (B) Pollution
• (C) Sustainability
• (D) Low efficiency

Consider the following system of equations:
\(x - 2y + 3z = -1\)
\(x - 3y + 4z = 1\)
\(-2x + 4y - 6z = k\)

The value of \(k\) for which the system has infinitely many solutions is _______

• (A) 0
• (B) 1
• (C) 2
• (D) -1

If a \(2 \times 2\) matrix \(A\) has eigenvalues 1 and 4 with the corresponding eigenvectors \(\begin{pmatrix} 1
-1 \end{pmatrix}\) and \(\begin{pmatrix} 2
1 \end{pmatrix}\), respectively, then \(A\) is _______

• (A) \(\begin{pmatrix} -4 & -8
  5 & 9 \end{pmatrix}\)
• (B) \(\begin{pmatrix} 9 & -8
  5 & -4 \end{pmatrix}\)
• (C) \(\begin{pmatrix} 2 & 2
  1 & 3 \end{pmatrix}\)
• (D) \(\begin{pmatrix} 3 & 2
  1 & 2 \end{pmatrix}\)

Let \(f(x)\) be a differentiable function at \(x = a\) with \(f'(a) = 4\) and \(f(a) = 8\). Then, \(\lim_{x\to a} \frac{xf(a) - af(x)}{x - a} = \)_______

• (A) \(4(1 - 2a)\)
• (B) \(4(2a - 1)\)
• (C) \(4(2 - a)\)
• (D) \(4(a - 2)\)

The directional derivative of \(f(x,y) = x^2 + 3xy\) at the point (1,2) in the direction of the vector \(3\hat{i} + 4\hat{j}\) is

• (A) \(\frac{22}{5}\)
• (B) \(\frac{24}{5}\)
• (C) \(\frac{12}{5}\)
• (D) \(\frac{36}{5}\)

If \(y(x)\) satisfies the differential equation \((\sin x)\frac{dy}{dx} + y \cos x - 1 = 0\) subject to the condition \(y\left(\frac{\pi}{2}\right) = 0\), then \(y\left(\frac{\pi}{6}\right) = \)_______

• (A) \(\frac{\pi}{3}\)
• (B) \(\frac{-2\pi}{3}\)
• (C) \(\frac{-\pi}{6}\)
• (D) \(\frac{-\pi}{3}\)

Which one of the following functions is analytic over the entire complex plane?

• (A) \(e^{\frac{1}{z}}\)
• (B) \(\cos z\)
• (C) \(\frac{2}{1-z}\)
• (D) \(\ln z\)

Two dice are thrown. Let \(A\) be the event that the sum of the points on the faces is 9. Let \(B\) be the event that at least one of them is 6. Then, \((\bar{A} \cup \bar{B}) = \)_______

• (A) \(\frac{17}{18}\)
• (B) \(\frac{1}{18}\)
• (C) \(\frac{1}{9}\)
• (D) \(\frac{8}{9}\)

Let \(C\) be the circle \(|z| = \frac{3}{2}\) in the complex plane that is oriented in the counter-clockwise direction. The value of \(a\) for which \(\int_C \left(\frac{z+1}{z^2 - 3z + 2} + \frac{a}{z-1}\right) dz = 0\) is _______

• (A) 1
• (B) -1
• (C) 2
• (D) -2

The regression lines of a sample are \(x + 6y = 6\) and \(3x + 2y = 10\). Then the sample means and the correlation coefficient are

• (A) \(\bar{x} = 3, \bar{y} = \frac{1}{2}, r = -\frac{1}{3}\)
• (B) \(\bar{x} = 3, \bar{y} = \frac{1}{2}, r = \frac{1}{3}\)
• (C) \(\bar{x} = \frac{1}{2}, \bar{y} = 3, r = -\frac{1}{3}\)
• (D) \(\bar{x} = \frac{1}{2}, \bar{y} = 3, r = \frac{1}{3}\)

Consider the fixed-point iteration \(x_{k+1} = g(x_k)\) with \(g(x) = \frac{x}{3} + \frac{4}{3x}\). Which root-finding problem is this equivalent to?

• (A) \(x - \frac{1}{3} + \frac{4}{3x^2} = 0\)
• (B) \(\frac{x}{3} + \frac{4}{3x} = 0\)
• (C) \(\frac{1}{3} - \frac{4}{3x^2} = 0\)
• (D) \(x^2 - 2 = 0\)