Prasanna

Kerala Plus Two Physics Model Question Paper 3

Time: 2 Hours
Cool off time : 15 Minutes

General Instructions to candidates:

• There is a ‘cool off time’ of 15 minutes in addition to the writing time of 2 hrs.
• Your are not allowed to write your answers nor to discuss anything with others during the ‘cool off time’.
• Use the ‘cool off time’ to get familiar with the questions and to plan your answers.
• Read questions carefully before you answering.
• All questions are compulsory and only internal choice is allowed.
• When you select a question, all the sub-questions must be answered from the same question itself.
• Calculations, figures and graphs should be shown in the answer sheet itself.
• Malayalam version of the questions is also provided.
• Give equations wherever necessary.
• Electronic devices except non programmable calculators are not allowed in the Examination Hall.

Questions 1 – 7 carry 1 score each. Answer any six questions.

Question 1.
Potentiometer measures potential more accurately because

a. It measures potential in the open circuit
b. It uses sensitive galvanometer for null deflection
c. It uses high resistance potentiometer wire
d. It measures potential in the closed circuit

Question 2.
Tesla is the unit of

Question 3.
Write the time constant of C-R circuit ?

Question 4.
Green-house effect is the heating up of earth atmosphere due to____
a. green plants
b.infra-red rays
c. x-rays
d. ultraviolet rays

Question 5.
Rainbow is formed due to a combination of

a. dispersion and total internal reflection
b. refraction and absorption
c. dispersion and focusing
d. refraction and scattering

Question 6.
Ratio of intensities of two waves are given by 4:1. Then find the ratio of the amplitudes of the two waves.

Question 7.
The spectrum obtained from a sodium vapor lamp is an example of

a. band spectrum
b. continuous spectrum
c. emission spectrum
d. absorption spectrum

Questions 8 to 15 carry 2 score each. Answer any 7 questions.

Question 8.
Copper is one of the suitable materials to make connecting wires due to its low resistivity.
a. What do you mean by resistivity?
b. A copper wire is in the form of a cylinder and has a resistance R. It is stretched till its thickness reduces by half of its initial size. Find its new resistance in terms of R.

Question 9.
It was james Clark Maxwell who modified Ampere’s circuital theorem by introducing the concept of displacement current.

a. What do you mean by displacement current?

Question 10.
Match the following

Question 11.
A body of mass m is charged negatively. State whether the following statements are true or false.
a. During charging, there is change in mass of the body.
b. The body can be charged to 2.5 e where e is the charge of an electron.
c. While charging the body by induction new charges are created in it.
d. The force between two charged object is less when there is a medium between them (than in vacuum).

Question 12.
a. A receiver in a communication system must have
i. pick-up antenna
ii. demodulator
iii. amplifier
iv. all of these
b. Which of the following statements is wrong?
i. The attenuation of surface waves in creases with increase in frequency.
ii. The phenomenon involved in sky wave propagation is similar to total internal reflection.
iii. Space wave mode of propagation is used in satellite communication.
iv. Sky wave propagation is useful only in the range of frequencies 30 MHz to 40 MHz.

Question 13.
a. The core of a transformer has the following properties :
i. core is laminated.
ii. hysterisis loop is narrow.
Explain the significance of each property.
b. What is meant by resonance in an LCR circuit ?

Question 14.
a. Permanent magnets should have
i. high retentivity and low coercivity.
ii. low retentivity and high coercivity.
iii. high retentivity and high coercivity.
iv. low retentivity and low coercivity.
b. Distinguish between Para, Ferro an Dia magnetism.

Question 15.

a. From the above figures which one corresponds to a typical semiconductor.
b. Dimension of temperature coefficient α is……..

Questions 16 to 22 carry 3 score each. Answer any 6 questions.

Question 16.
a. Work function of a metal is the
i. energy required by an electron to get absorbed in the metal surface.
ii. minimum energy required by an electron to escape from the metal surface.
iii. energy required by an electron to be retained in the metal surface.
iv. maximum energy required by an electron to escape from the metal surface.
b. Write Einstein’s Photo-Electric Equation and explain the terms in it.
c. All Photo electrons are not emitted with the same energy as the incident photons. Why?

Question 17.
Kirchhoff’s rules are very useful for analysis of electrical circuits.
a. State Kirchhoff s junction rule.
b. Find the effective resistance of the circuit given below :

Question 18.
A fascinating behaviour of the series RLC circuit is the phenomenon of resonance.
a. Explain Resonance in an LCR circuit.
b. Draw a graphical representation of variation of current amplitude in i_m with frequency to ω
c. What do you mean by sharpness of resonance? Explain it.

Question 19.
a. When a low flying aircraft passes over head, we sometimes notice a slight shaking of the picture on our TV screen. Identify the optical phenomenon behind it.
b. In electromagnetic spectra, the wave length and frequencies are inversely related. A radio can tune into any station in the 7.5 MHz to 12 MHz band. Determine the corresponding wavelength band

Question 20.
Various propagation modes are used in communication.
a. Mention two communication system that uses space wave propagation.
b. Why modulation is necessary in communication ?
c. If a signal of frequency cos is used to ω_s modulate a carrier wave of frequency ω_c
which are the frequencies contained in the modulated signal other than ω_c?

Question 21.
Match the following quantities using the analogy between mechanical and electrical quantities.

Question 22.
In a nuclear reactor the chain reaction is carried out under controlled conditions.
a. Name the material that is used as control rods in a nuclear reactor.
b. Average energy of a neutron produced in fission of ₉₂²³⁵U nucleus is…….
c. Write down the reactions involved in the conversion of to ₉₂²³⁸U to ₉₄²³⁹Pu in a nuclear reactor

Questions from 23 – 26 carry 4 scores. Answer any 3 questions.

Question 23.
To emit a free electron from a metal surface a minimum amount of energy must be supplied.
a. This energy is called….
b. State three methods to supply this energy to the free electron.
c. When light of frequency 7.21×10¹⁴ Hz is incident on a metal surface, the maximum speed of the ejected electrons is 6×105 m/s. Calculate the threshold frequency for the metal.

Question 24.
Seema constructed a series LCR circuit in the laboratory as shown in the diagram. She found that the voltages across the inductor and capacitor are equal when the circuit is connected to an ac source.


a. State the condition at which the voltages across L and C become equal.

b. Obtain an expression for the frequency at which this situation occurs in a series LCR circuit.

c. Find the voltmeter and ammeter readings in the circuit.

Question 25.
Brewster’s law gives a relationship between angle of polarisation and refractive index of the material.

a. State Brsewster’s law

b. A ray of light is allowed to incident on the surface of a glass plate and it is found that the reflected and refracted rays are mutually perpendicular as shown in the figure. The refracted ray is deviated from its initial path by an angle of 30°
Determine the refractive index of the glass plate.

Question 26.
a. A ray of light travels from a denser to a rarer medium then, the ray
i. doesn’t bend at all.
ii. bends towards the normal.
iii. bends away from the normal.
iv. goes along the normal.
b. Draw a diagram showing the path of a monochromatic light through a triangular prism.
c. Using this diagram obtain the relation

Questions from 27 – 29 carry 5 scores. Answer any 2 questions.

Question 27.
a. What do you mean by barrier potential of a diode ?
b. With the help of a diagram explain the working of a full wave rectifier.

Question 28.
Various phenomena exhibited by light can be explained using the wave theory of light.
a. Name the phenomenon which proves the transverse nature of light.
b. What are the differences between interference and diffraction?
c. A plane wave front is incident on a single slit. Discuss the diffraction pattern formed by the slit. Represent the variation of intensity graphically.

Question 29.
A moving charge can produce a magnetic field.
a. How does a current loop behaves like a magnetic dipole ?
b. Draw the magnetic field lines for a current loop to support your answer.
c. i. What is a cyclotron ?
ii. Write down the expression for cyclotron frequency.

Answers

Answer 1.
a

Answer 2.
magnetic field

Answer 3.
CR

Answer 4.
b

Answer 5.
a

Answer 6.

Answer 7.
c

Answer 8.
a. Resistivity of the material of a conductor is defined as the resistance of the conductor of unit length and of unit area of cross-section.

Answer 9.
a. Displacement current is that current which comes into existence, in addition to the conduction current whenever the electric flux changes with time.
b. Ampere

Answer 10.

Answer 11.
a. true
b. false
c. false
d. true

Answer 12.
a. (iv) all of these
b. (iv) Sky wave propagation is useful only in the range of frequencies 30 to 40MHz

Answer 13.
a. (i) For reducing eddy current. Eddy current heats the core and energy will be lost.
(ii) For reducing hysterisis loss.
b. In LCR, at a particular frequency X_L = X_C. This condition is called resonance. This frequency is called resonance frequency.

Answer 14.
a. iii. high retentivity and high coercivity.
b. Paramagnetism: Feebly magnetise along applied field.
Diamagnetism: Feebly magnetise opposite to magnetic field.
Ferromag netism: Strongly magnetise along magnetic field.

Answer 15.
a. Fig.(2)
b. α = coefficient of temp

Answer 16.
a. ii. minimum energy required by an electron to escape from the metal surface.

c. All the photoelectrons emitted from the metal do not have the same energy. Photoelectric current is zero when the stopping potential is sufficient to repel | even the most energetic photoelectrons, with the maximum kinetic energy (K_max),So that K_{max = eV0}

Answer 17.
a. In an electric circuit, the sum of currents | entering the junction is equal to the sum of currents leaving the junction.
b. Here 3Ω, 6Ω are parallel and 6Ω and 12Ω are parallel.

Answer 18.
a. In LCR, at a particular frequency X_L = X_C. This frequency is called resonance frequency.

c. Sharpness of resonance is depends of quality factor or Q factor.
Q factor is equal to the ratio inductive reactance or capacitative reactance to the impedence at resonance.

Answer 19.
a. Interference

Answer 20.
a. TV Broadcast, Satellite, FM, RADAR,
b. 1. Size of the antenna: We need an antenna having a size comparable to the wave length of the signal.
2. Effective power radiated by antenna,

Answer 21.

Answer 22.

Answer 23.
a. Work function.
b. Thermionic emission, field emission and photo electric emission.

Answer 24.
a. p.d across L and C are equal when inductive reactance = Capacitive reactance (X_L = X_C), i.e., at resonance.
b. At resonance, X_L = X_C

Answer 25.
a. Tangent of angle of polarization is equal to the reflective index.
b. i + r = 90°
θ + r = 90°
θ = 90° – r = 90 – 30 = 60°
n = tan 60° = √3 = 1.732

Answer 26.
a. iii bends away from the normal
b.

Answer 27.
a. During the formation of a p-n junction, holes diffuse from p to n side and electrons diffuse from n to p side. Thus there is excess of positively charged ions in n side and negatively charged ions in p side. This sets up a potential difference across the junction which is called barrier potential. It stops further diffusion.

• During +ve half cycle of a.c, the end A is positive and end B is negative with respect to the centre tap. Diode D1 gets forward biased and conducts current while D2 is reverse biased and does not conduct.
• During -ve half cycle, the end Abe comes negative and end B becomes positive with respect to the centre tap. The diode D gets reverse biased and does not conduct. The diode D2 gets forward biased and conducts current.
• The output current is unidirectional and pulsating which may be filtered by connecting a capacitor parallel to load R_L.

Answer 28.
a. polarization
b.

Answer 29.
a. One face north and other face south.
i.e.,


Cyclotron uses the fact that the frequency of revolution of the charged particle in a magnetic field is independent of its energy. The particles move most of the time inside two semicircular disc-like metal containers, D1 and D2, which are called dees.
The whole assembly is evacuated to minimise collisions between the ions and the air molecules. A high frequency alternating voltage is applied to the dees. Positive ions or positively charged particles (e.g., proton) are released at the centre P. They move a semi-circular path in one of the dees and arrive in the gap between the dees in a time interval T/2; where T, the period of revolution, is given T = 1/ u_c = 2πm/qB
u_c = qB / 2πm. This frequency is called the cyclotron frequency for obvious reasons and is denoted by u_c.

Plus Two Physics Previous Year Question Papers and Answers

Kerala Plus Two Physics Model Question Paper 2

Time: 2 Hours
Cool off time : 15 Minutes

General Instructions to candidates:

• There is a ‘cool off time’ of 15 minutes in addition to the writing time of 2 hrs.
• Your are not allowed to write your answers nor to discuss anything with others during the ‘cool off time’.
• Use the ‘cool off time’ to get familiar with the questions and to plan your answers.
• Read questions carefully before you answering.
• All questions are compulsory and only internal choice is allowed.
• When you select a question, all the sub-questions must be answered from the same question itself.
• Calculations, figures and graphs should be shown in the answer sheet itself.
• Malayalam version of the questions is also provided.
• Give equations wherever necessary.
• Electronic devices except non programmable calculators are not allowed in the Examination Hall.

Questions 1 and 2 carry 1 score each. Answer both.

Question 1.
Electromagnets are made of soft iron because soft iron has

a. low retentivity and high coercive force
b. high retentivity and high coercive force
c. low retentivity and low coercive force
d. high retentivity and low coercive force

Question 2.
Draw the symbol of NOR gate

Answer any six questions from question number 3 to 10. Each question carries 2 scores.

Question 3.
a. The electric field of a plane electromagnetic wave traveling in the +ve Z-direction is described by
a. E_x = E₀ sin (kz + ωt)
b. E_x = E₀ sin (kz – ωt)
c. E_x = E₀ sin (2kz)
d. E_x = E₀ sin (kz)
b. We feel excessive sweating on a cloudy day. Why ?

Question 4.
For many purposes, it is necessary to change an alternating voltage from one value to another. This is done with a transformer.
a. The basic principle behind a transformer is…..
b. Give an expression for the voltage and current in a transformer.

Question 5.
a. Name the physical quantity which has its unit joule coulomb^-1. Is it a vector or a scalar?
b. Two plane sheets of charge densities +σ and -σ are kept in air as shown in figure. What are the electric field intensities at point A and B?

Question 6.
Name the electromagnetic waves used for the following applications.
a. Imaging of bones in human body.
b. Mobile phone communication.
c. Remote control of T.V.sets.
d. For sterilizing surgical instruments.

Question 7.
a. Range of an electronic communication system is the
i. distance to the nearest TV station.
ii. distance to the nearest radio station.
iii. largest distance the signal can travel.
iv. largest distance between a source and destination up to which the signal is received with sufficient strength.
b. If the height of TV transmitting antenna is increased its coverage increases. Why?

Question 8.
Match the following in 3 columns.

Question 9.
a. The resistance ‘R’ of a conductor depends on its length ‘I’, area of cross- section ‘A’ and the resistivity of the material ‘ρ’. The correct expression connecting R, I, A and ‘ρ’ is

b. The voltage – current graphs for two resistors of the same material and same radii with lengths L1 and L2 are shown in the figure. If L1 > L2 state with reason, which of these graphs represents voltage current charge for L1?

Question 10.
De Broglie proposed the wave nature of electrons suggesting matter waves.
Find the momentum, speed and de Broglie wavelength of an electron with kinetic energy of 120 eV.

Answer any five questions from question number 11 to 16. Each question carries 3 scores.

Question 11.
a. The work function of a metal is 6 eV. If two photons each having energy 4 eV strike with the metal surface
i. will the emission be possible ?
ii. why ?
b. The waves associated with matter is called matter waves. Let λ_e and λ_p be the de- Broglie wave lengths associated with electron and proton respectively. If they are accelerated by same potential, then

Question 12.
The region around a charge where its effect can be felt is called the electric field.
a. The electric field lines corresponding to an electric field is shown below.

b. Some equipotential surfaces are shown in the figure. What can you say about the magnitude and direction of the electric field ?

Question 13.
Given below is the graph between frequency (υ) of the incident light and maximum kinetic energy (E_k) of emitted photoelectrons.

a. Define the terms – work functions and threshold frequency.
b. Find the values of
i. threshold frequency and
ii.work function from the graph

Question 14.
Consider the logic circuit using NOR gates given below:

a. Write the truth table for the circuit.
b. Identify the logic operation which this circuit is performing.

Question 15.
The phenomenon in which electric current is generated by varying magnetic field is called electromagnetic induction.
a. Which law helps to detect the direction of induced current? State the law.
b. Explain what are eddy currents.
c. Write any two uses of eddy currents.

Question 16.
Biasing is provided for maintaining proper current flow across a p-n junction.

a. In a biased p-n junction the net flow of holes is from ‘n’ region to ‘p’ regional


b. For the device shown below draw the V-I characteristics when the potential is applied between the terminals A and B.

Answer any four questions from question number 17 to 21. Each question carries 4 scores.

Question 17.
a. Find the equivalent resistance of the circuit.

b. Determine the current through the 75Ω resistance.

Question 18.
a. A convex lens
i. is thicker at the edges than at the middle.
ii. is thicker at the middle than at the edges.
iii. diverges rays of light.
iv. is of uniform thickness everywhere.
b. With the help of a ray diagram sketch the image formation of a convex lens when the object is between C and F.
c. Derive the lens maker’s formula.

Question 19.
Remya makes the following circuit to measure the emf of a cell.

She says that the voltmeter reading will give the emf of the cell.

a. The Physics teacher says that it is not :possible to measure the emf of the cell in this way. Justify this statement.
b. Explain with a circuit diagram the method to measure the emf of a cell with a potentiometer if you are given another standard cell, (whose e.m.f. is known)

Question 20.
a. What do you mean by Q value of a nuclear reaction?
b. Write down the expression for Q value in the case of α decay.
c. Two nuclei have mass numbers in the ratio 1 : 64. What is the ratio of their nuclear radii ?

Question 21.
A capacitor is a device used for storing charge as well as energy. It is based on a property called capacitance.
a. Write the SI unit of capacitance and define it.
b. You are given two capacitors of 2µF and 3µF. What are the maximum and minimum values of capacitance that can be obtained by combining them?
c. Obtain an expression for the electrostatic energy stored in a capacitor.

Answer any three questions from question number 22 to 25. Each question carries 5 scores.

Question 22.
To emit a free electron from a metal surface a minimum amount of energy must be supplied.
a. This energy is called …….
b. State three methods to supply this energy to the free electron.
c. When light of frequency 7.21×10¹⁴ Hz is incident on a metal surface, the maximum speed of the ejected electrons is 6×10⁵ m/ s. Calculate the threshold frequency for the metal.

Question 23.
Figure shows a graph between angle of incidence and the respective deviation of light when it is passed through an optical element.

a. Name this optical element.
b. What will happen to the angle of minimum deviation D if it is completely immersed in water? Justify your answer.
c. Obtain an expression for the refractive index of the material of this optical element.

Question 24.
The idea of ‘Electric field lines ’ is useful in pictorially mapping the electric field around charges.
a. Give any two properties of electric lines of force.
b. State Gauss’s theorem in electrostatics.
c. Using the theorem, derive an expression for electric field due to a uniformly charged spherical shell.
i. at a point outside the shell
ii. at a point inside the shell
d.

A point charge of +10 μc is at a distance of 5cm directly above the centre of a square of side 10 cm as shown in the figure. What is the electric flux through the square?

Question 25.
Gauss’s law can be used to determine the electric field due to a charge distribution, a. Below are some statements about Gauss’s law. Say whether they are true or false.
i. Gauss’s law is valid only for symmetrical charge distributions.
ii. The electric field calculated by Gauss’s law is the field due to charges inside the Gaussian surface.
b. Apply Gauss’s law to find the electric field due to an infinitely long plane sheet of charge.
c. “There can be no net charge in a region in which the electric field is uniform at all points”. Do you agree with this statement? Justify your answer.

Answers

Answer 1.
d

Answer 2.

Answer 3.
a. b. E_x = E₀ sin (kz – ωt)
b. The clouds prevents IR radiation from passing through and thus keeps the earth’s surface warm.

Answer 4.
a. Mutual induction

Answer 5.
a. Electric potential. It is a scalar quantity.

Answer 6.
a. x-rays
b. Radio waves
c. Infrared rays
d. Ultraviolet rays

Answer 7.
a. iv. largest distance between a source and destination up to which the signal is received with sufficient strength.
b. We know range, d = √2hR ; Where h is the height of the antenna. Therefore increasing height increases the range ie., coverage.

Answer 8.

Answer 9.
a. \(R=\frac { pl }{ A }\)
b. B represents voltage change for L. Resistance of B is greater than that for A, since resistance increases with length.

Answer 10.
De Broglie wavelength of electrons

from V², we can calculate V so we get the value of P also.

Answer 11.
a. (i) No
(ii) Free electrons are bound to the metal with some force. The minimum energy j required to liberate an electron from the metal is called work function, here the energy of one photon is less than work function. So emission will not be possible.
b. (i) λ_e > λ_p

Answer 12.
a. E_A = E_C > E_B
b. The direction of electric field at any point on the equatorial line of the dipole will be opposite to the dipole moment. The magnitudes of electric fields are equal

Answer 13.
a. Work functions: It is the minimum amount of energy required by an electron to just escape from the metal surface of the metal.
Threshold frequency: The minimum value of the frequency of incident radiation which can emit photo electrons from a material.
b. i.Threshold frequency,v₀ = 10 x 10¹⁴ Hz
ii. Work function = hv₀
= 6.63 x 10^-34 x 10 x 10¹⁴
= 6.63 x 10^-19 J

Answer 14.
a. Truth Table

Answer 15.
a. Lenz’s law. This law states that the direction of the induced e.m.f or current is such as to oppose the cause that produced it
b. The currents induced in the body of a conductor, when the magnetic flux linked with the conductor changes are called eddy currents or Focault’s currents
c. i. Electric Brakes
ii. Speedometer

Answer 16.
a. Reverse

Answer 17.

Answer 18.
a. ii. is thicker at the middle than at the edges.

Answer 19.
a. The voltmeter reads terminal potential difference. Emf cannot be measured in this way since the cell is connected to a resistance. Emf is the potential different across the terminals of a cell in a open circuit.

Answer 20.
a. The disintegration energy or the Q- value of a nuclear reaction is the difference between the initial mass energy and the total mass energy of the decay products.
b. For α -decay

Answer 21.
a. SI unit of capacitance is farad. Capacity of a conductor is said to be one farad if one coulomb of charge on it raises its potential by one volt.
b. If the capacitors are in series,

Answer 22.
a. Work function.
b. Thermionic emission, field emission and photo electric emission.
c. u = 7.21 x 10¹⁴ Hz, V = 6 x 10⁵ m/s

Answer 23.
a. Prism
b. The angle of minimum deviation decreases.
Refractive index of glass with respect to water is 1.5. But refractive index of glass with respect to water is 1.33.

c.

Answer 24.
a. i. Electric lines of force will never intersect.
ii.The tangent to the line of force at any point gives the direction of electric field at that point.
b. The electric flux through any closed surface in free is equal to 1/ε₀ times the total charge enclosed by the surface.

Answer 25.
a. i. False
ii. False

c. Yes, there can be no net charge in a region in which the electric field is uniform at all points. Consider a region with uniform electric field and a cubical gaussian surface in it whose face is perpendicular to electric field. Now from one face electric field enters which is considered to be +ve from the possible face it gets out which is considered to be -ve. So net flux through the gaussian surface is zero.

Plus Two Physics Previous Year Question Papers and Answers

Kerala State Board New Syllabus Plus Two Physics Previous Year Question Papers and Answers.

Kerala Plus Two Physics Model Question Paper 1 with Answers

Time: 2 Hours
Cool off time: 15 Minutes
Maximum: 60 Score

General Instructions to candidates:

• There is a ‘cool off-time’ of 15 minutes in addition to the writing time of 2 hrs.
• You are not allowed to write your answers nor to discuss anything with others during the ‘cool off time’.
• Use the ‘cool off time’ to get familiar with the questions and to plan your answers.
• Read questions carefully before you answering.
• All questions are compulsory and the only internal choice is allowed.
• When you select a question, all the sub-questions must be answered from the same question itself.
• Calculations, figures, and graphs should be shown in the answer sheet itself.
• Malayalam version of the questions is also provided.
• Give equations wherever necessary.
• Electronic devices except non-programmable calculators are not allowed in the Examination Hall.

Questions 1 – 7 Carry 1 Score each. Answer any six questions. (6 × 1 = 6)

Question 1.
Write the expression for intensity of electric field near to the surface of a charged conductor with uniform charge density σ.
Answer:
E = \(\frac{\sigma}{\varepsilon_{0}}\)

Question 2.
The unit of electrical resistivity is ……….
Answer:
Ωm (ohm meter)

Question 3.
E.m.f can be induced in a coil placed in an external magnetic field by:
a) Changing the intensity of the magnetic field
b) Changing the area of the coil
c) Changing the orientation of the coil
d) All of the above
Answer:
d) All of the above

Question 4.
Write the energy and momentum associated with a moving photon.
Answer:
E = hν, P = mc

Question 5.
The value of relative permeability of a diamagnetic material is;
a) μ_r > 1
b) μ_r = 1
c) μ_r < 1
d) μ_r = 1
Answer:
c) μ_r < 1

Question 6.
The electric field amplitude of an electromagnetic wave is 15V/m. Find the magnetic field amplitude of the wave.
Answer:
B = \(\frac{E}{C}=\frac{15}{3 \times 10^{8}}\) = 5 × 10^-8T

Question 7.
A convex lens is placed in a medium having a refractive index greater than that of the lens. The lens now behaves as;
a) Converging lens
b) Diverging lens
c) Plane glass plate
d) None of the above
Answer:
b) Diverging lens

Questions 8 to 15 carry 2 scores each. Answer any 7 questions. (7 × 2 = 14)

Question 8.
a) ‘Electrostatic field is always normal to the surface of a charged conductor’. Justify the statement.
b) The value of elelctric potential at the surface of a charged conductor is 10V. Find the value of the intensity of the electric field and potential at a point interior to it.
Answer:
a) The surface of the charged conductor is an equipotential surface. In equipotential surface, the potential is the same and hence the electric field is always normal to the surface of a charged conductor.
b) E = 0

Question 9.
A galvanometer with a coil of resistance 12Ω shows a full-scale deflection for a current of 2.5 mA. How can it be converted into an ammeter of range 7.5 A?
Answer:
G= 12Ω, I_g= 2.5 mA = 2.5 × 10^-3A, I = 7.5 A
S = \(\frac{I_{g} \times G}{I-I_{g}}=\frac{2.5 \times 10^{-3} \times 12}{7.5-2.5 \times 10^{-3}}=4 \times 10^{-3} \Omega\)

Question 10.
The curves shown in figure are drawn for different magnetic materials. Among the three curves, name the curve that,

a) Represent the material usually used for making permanent magnets.
b) Represent the material usually used for making electromagnets.
Answer:
a) A
b) C

Question 11.
A capacitor C, a variable resistance R, and a bulb B are connected in series to a.c. mains in the circuit as shown. The bulb glows with some brightness. How the glow of the bulb change if.

a) A dielectric slab is introduced between the plates of the capacitor.
b) The resistance R is increased keeping the same capacitance.
Answer:
a) The brightness of bulb increases.
(When dielectric slab is introduced, capacitance increases hence capacitive reactance decreases, then bulb glows with more brightness.)
b) The brightness of bulb decreases.

Question 12.
James Clerk Maxwell modified Ampere’s circuital theorem by introducing the concept of displacement current.
a) What do you mean by displacement current?
b) Write down the equation for displacement current.
Answer:
a) The current due to time varying electric field is called displacement current.
b) i_d = \(\varepsilon_{0} \frac{\mathrm{d} \varphi}{\mathrm{dt}}\)

Question 13.
An object AB is kept in front of a concave mirror as shown in figure.

a) Complete the ray diagram showing the image formation of object.
b) How will the position and intensity of image be affected if the lower half of the mirror’s reflecting surface is painted black?
Answer:
a)

b) The position of image does not charge. But the intensity decreases.

Question 14.
Assuming that the two diodes D₁ and D₂ used in the electric circuit as shown in figure are ideal. Find out the value of current flowing through 2.5Ω resistor.

Answer:
Since D₂ is reverse biased, no current flows through 3Ω. We need to consider only the other two resistance 3Ω and 2.5Ω, which are connected in series. Total resistance, R = 3Ω + 2.5Ω = 5.5Ω
v = 10V
I = \(\frac{V}{R}=\frac{10}{5.5}\) = 1.81 A

Question 15.
a) Mention the function of the following used in the communication system
(i) Transducer
(ii) Transmitter
b) Figure shows the block diagram of an AM transmitter. Identify boxes X and Y.

Answer:
a) i) Transducer: The device that converts one form of energy into another is called transducer.
ii) Transmitter – Atransmittertransmits the information

b) x → Modulator
y → Power Amplifier

Questions 16 to 22 carry 3 scores each. Answer any 6 questions. (6 × 3 = 18)

Question 16.
Three capacitors of capacitances 2µF, 3µF and 4µF are connected in series.
a) Find the equivalent capacitance of the combination
b) The plates of a parallel plate capacitor have an area 20 cm² each are separated by a distance of 2.5 mm. The capacitor is charged by connecting it to a 400V supply. How much electrostatic energy is stored in the capacitor?
Answer:
a) C₁ = 2μF = 2 × 10^-6F
C₂ = 3μF = 3 × 10^-6F

b) A = 20cm² = 20 × 10^-4m²
d = 2.5 mm = 2.5 × 10^-3m
ε₀ = 8.85 × 10^-12C²/Nm²
v = 400v

Question 17.
A circuit using a potentiometer and a battery of negligible internal resistance is set up as shown to develop a constant potential gradient along the wire PQ. Two cells of e.m.fs E₁ and E₂ are connected in series as in figure in combination 1 and 2. The balance points are obtained respectively at 400 cm and 240 cm from point P. Find

a) The ratio between E₁ and E₂
b) Balancing length for the cell with emf E₁ only.
Answer:
a) From the connection (1), we get
E₁ + E₂ = k × 400 ………(1)
k = constant
From the connection (2), we get
E₂ – E₁ = 240 K ……….(2)
Adding (1) and (2), we get
2E₂ = 640 K
E₂ = 320 K
Subtituting for E₂ in equation (1)
E₁ + 320 K = 400 K
E₁ = 80 K
The ratio \(\frac{E_{1}}{E_{2}}=\frac{1}{4}\)
b) l₁ be the balancing length of E₁
E₁ = kl₁
80k ∝ kl₁
l₁ = 80 cm

Question 18.
A conducting rod PQ of length ‘l’ connected to a resistance ‘R’ is moved at a uniform speed ‘V’, normal to a uniform magnetic field ‘B’.

a) Deduce the expression for e.m.f induced in the conductor.
b) Find the magnitude and direction of current through the conductor.
Answer:
a) flux, Φ = BA
Here, Area A = ldx
dx is small displacement rod in time Δ T
Φ = B × l × dx

Question 19.
In tuner circuits, we use the phenomenon of resonance.
a) Write the condition of resonance in a series LCR circuit.
b) A series LCR circuit uses L = 0.1 H, C = 10μF and R = 100Ω. Find the value of frequency at which the amplitude of current is maximum.
Answer:
a) Lω = \(\frac{1}{\mathrm{C} \omega}\)
b) L = 0.1 H, C = 10μF = 10 × 10^-6F

Question 20.
The focal length of a lens has dependent on its radii of curvatures and refractive index. Derive lens maker’s formula.

Question 21.
a) Write the expression for the de Broglie wavelength associated with a charged particle having charge ‘q’ and mass ‘m’, when it is accelerated by a potential of ‘V’ volts.
b) A proton and an electron have the same kinetic energy. Which one has a greater value of de Broglie wavelength and why?
Answer:

For electron, λ_e = \(\frac{h}{\sqrt{2 m_{e} K E_{e}}}\)
But KE_p = KE_e and m_p > M_e, hence λ_e > λ_p
Electron has greater de Broglie wavelength.

Question 22.
Find the binding energy per nucleon of \({ }_{20}^{40} \mathrm{Ca}\) nucleus.
Given m\({ }_{20}^{40} \mathrm{Ca}\) = 39.962589u.
mp = 1.00783u, mn = 1.00867u.
Take 1 amu = 931 MeV/c²
Answer:
Mass detect, Δm = Zm_p + (A – Z) m_n – M
Δm = 20 m_p + 20 m_n – 39.962589
= 20 × 1.00783 + 20 × 1.00867 – 39.962589
= 0.367411 u
Binding energy = 0.367411 × 931 = 342.06 MeV

Questions from 23 – 26 carry 4 scores. Answer any 3 questions. (3 × 4 = 12)

Question 23.
Gauss’s theorem is useful for finding the intensity of elelctric field.
a) Write Gauss’s law in its mathematical form
b) Using the law, prove that the intensity of the electric field at a point due to a uniformly charged infinite plane sheet is independent of the distance from it.
Answer:
\(\phi=\oint \overrightarrow{\mathrm{E}} \cdot \mathrm{d} \overrightarrow{\mathrm{s}}=\frac{\mathrm{q}}{\varepsilon_{0}}\)

Question 24.
a) Express Ohm’s law in terms of current density, electrical resistivity, and intensity of the electric field.
b) Explain the variation of resistance of a semiconductor with temperature. Also, draw the graph showing the variation of resistivity of silicon with temperature.
Answer:
a) J = σE
b) As temperature increases, the number density of electrons (η) of semiconductor increases.
ρ = \(\frac{m}{h e^{2} \tau}\)
Hence resistivity decreases with increase in temperature.

Question 25.
The relation between the magnetic field and current is given by Biot-Savart’s law.
a) Write the expression for the magnetic field at point along the axis of a circuit loop of radius ‘R’ carrying a current ‘I’.
b) From the above expression. Find the value of magnetic field at the centre of the loop.
c) Sketch the magnetic field lines for current carrying circular loop
Answer:
a) B = \(\frac{\mu_{0} \mathrm{NIR}^{2}}{2\left(\mathrm{x}^{2}+\mathrm{R}^{2}\right)^{3 / 2}}\)
I → Current
N → Number of turns
X → distance tot he point from the center of loop
b) B = \(\frac{\mu_{0} N I}{2 a}\)
c)

Question 26.

a) Identify the diode in the circuit and write the use of the resistance Rs.
b) Explain how the diode helps to stabilize the output voltage of the circuit.
c) Name the type of biasing used in this diode
Answer:
a) Zener diode, Voltage regulator
b) Solar Cell: Solar cell is a junction diode used to convert solar energy into electrical energy.

Circuit details:

Its p-region is thin and transparent and is called emitter. The n-region is thick and is called base. The output is taken across R_L.

Working: When light falls on this layer, electrons from the n-region cross to the p-region and holes in the p-region cross into the n-region. Thus a voltage is developed across R_L.
Solar cells are used to charge storage batteries during the daytime.
c) Reverse bias

Questions 27 to 29 carry 5 scores. Answer any 2 questions. (2 x 5 = 10)

Question 27.
In the figure, PQ is the incident ray on the equilateral glass prism ABC.

a) Complete the ray diagram showing the passage of light and mark the angle of deviation.
b) Derive an expression for the refractive index of the material of the prism.
Answer:

b)

ABC is a section of a prism. AB and AC are the refracting faces, BC is the base of the prism. ∠A is the angle of the prism.

A ray PQ incidents on the face AB at an angle i₁. QR is the refracted ray inside the prism, which makes two angles r₁ and r₂ (inside the prism). RS is the emergent ray at angle ‘i₂’

The angle between the emergent ray and incident ray is the deviation ‘d’.
In the quadrilateral AQMR,
∠Q + ∠R = 180°
[since N₁M and NM are normall
i.e. ∠A + ∠M = 18O° ………(1)
In the ΔQMR,
∴ r₁ + r₂ + ∠M = 180°……..(2)
Comparing eq (1) and eq (2)
r₁ + r₂ = ∠A ……….(3)
From the ΔQRT,
(i₁ – r₁) + (i₂ – r₂) = d
[since exterior angle equal sum of the opposite interior angles]
(i₁ + i₂) – (r₁ + r₂) = d
but, r₁ + r₂ = A
∴ (i₁ + i₂) – A = d
(i₁ + i₂) = d + A ……..(4)
It is found that for a particular angle of incidence, the deviation is found to be minimum value ‘D.
At the minimum deviation position,
i₁ = i₂ =i, r₁ = r₂ = r and d=D
Hence eq (3) can be written as,


It is found that when the angle of incidence increases deviation (d) decreases and reaches a minimum value and then increases. This minimum value of the angle of deviation is called the angle of minimum deviation.

Question 28.
According to Christian Huygens’s wave theory, light emanating from a source as wavefronts.
a) What is the shape of wavefront emerging from a linear source?
b) Derive the mathematical expression for the bandwidth of interference bands obtained in young’s double-slit experiment with the help of a suitable diagram.
Answer:
a) Cylinder
b)

S₁ and S₂ are two coherent sources having wavelength A. Let ‘d’ be the distance between two coherent sources. A screen is placed at a distance D from sources. ‘O’ is a point on the screen equidistant from S₁ and S₂.
Hence the path difference, S₁O – S₂O = 0
So at ‘O’ maximum brightness is obtained.
Let ‘P’ be the position of nm bright band at a distance x_n from O. Draw S₁A and S₂B as shown in the figure.
From the right angle ΔS₁AP



This is the width of the bright band. It is the same for the dark band also.

Question 29.
a) The radius of nth stationary orbit of hydrogen atom is:
r_n = \(\frac{\mathrm{n}^{2} \mathrm{~h}^{2} \varepsilon_{0}}{\pi \mathrm{me}^{2}}\)
Using Bohr postulates, obtain an expression for the energy of electron in the stationary states of H-atom.
b) Draw the energy level diagram showing how the spectral lines corresponding to Balmerseries occur due to the transition between the energy levels.
Answer:
a) Consider a hydrogen atom which has one electron revolving around the nucleus having one proton. The centripetal force for an electron is provided by the electrostatic force between electron and proton

b) Energy level diagram of the hydrogen atom

Note: An electron can have any total energy above E = 0 EV. In such situations electron is free. Thus there is a continuum of energy states above E = 0 EV.

Kerala State Board New Syllabus Plus Two Physics Previous Year Question Papers and Answers.

Kerala Plus Two Physics Previous Year Question Paper March 2017 with Answers

Time: 2½ Hours
Cool off time : 15 Minutes

General Instructions to Candidates

• There is a ‘cool off time’ of 15 minutes in addition to the writing time of 2½ hrs.
• You are not allowed to write your answers nor to discuss anything with others during the ‘cool off time’.
• Use the ‘cool off time’ to get familiar with the questions and to plan your answers.
• Read questions carefully before you answering.
• All questions are compulsory and the only internal choice is allowed.
• When you select a question, all the sub-questions must be answered from the same question itself.
• Calculations, figures, and graphs should be shown in the answer sheet itself.
• Malayalam version of the questions is also provided.
• Give equations wherever necessary.
• Electronic devices except non-programmable calculators are not allowed in the Examination Hall.

Question 1.
A concave lens always produces ………….. images
i. real
ii. virtual
iii. magnified
iv. None of these

Question 2.
A Zener diode is always operated in …….. bias.

Question 3.
Momentum of a photon with wavelength λ is……

Question 4.
Write down the truth table of a NOR gate.

Question 5.
a. How many electrons constitute an electric charge of -16 µC?
(i) 10¹³
(ii) 10¹⁴
(iii) 10¹⁵
(iv) 10¹²
b. An electric dipole is a pair of equal and opposite point charges +q and -q separated by a distance r. Write an expression for its dipole moment.
c. When an electric dipole is subjected to a uniform electric field, what will happen?

Question 6.
A message signal of frequency 10kHz and peak voltage 10V is used to modulate a carrier of frequency 1 MHz and peak voltage 20V. Find the modulation index.

Question 7.
a. Resistivity of a conductor depends upon
(i) its material
(ii) its cross-sectional area
(iii) its length
(iv) All of the above
b. Calculate the current flowing through the following circuit :

c. A potentiometer is a device to measure emf of a cell. Explain how the emfs of two cells can be compared using a potentiometer.

Question 8.
a. Choose the wrong option.
(i) Volt = Weber / second
(ii) Weber = Henry x Ampere
(iii) Joule = Henry x Ampere²
(iv) Volt = Weber x Second
b. The current in a coil of self-inductance 0.1 H varies from 2A to 5A in a time of 1ms. Find the induced emf across the coil.

Question 9.
a. Sound waves do not exhibit
i. interference
ii. diffraction
iii. polarization
iv. reflection
b. Describe Young’s double-slit experiment to determine the bandwidth of the interference pattern.
OR
a. The intensity of scattered light I in Rayleigh scattering is proportional to……
b. Explain the diffraction pattern obtained due to a single slit and represent graphically the variation of intensity with angle of diffraction.

Question 10.
a. Define half-life period of a radioactive nucleus. Write down the relation connecting half-life period and mean life.
b. Define 1 amu. Calculate its energy equivalent in MeV.

Question 11.
Photoelectric current does not depend on energy of the radiation, but on its intensity. Explain?

Question 12.
a. Speed of light in glass is 2 x 10⁸ m/s. The Refractive index of glass is……
b. For an equilateral prism made of a material of refractive index √2 . find the angle of minimum deviation for a ray of monochromatic light.
c. Draw the ray diagram of a simple microscope that uses a single convex lens. Derive an expression for its linear magnification.

Question 13.
a. A dielectric slab is placed between the plates of a parallel plate capacitor. Its capacitance
i. becomes zero
ii. remains the same
iii. decreases
iv. increases
b. Derive an expression for energy stored in a capacitor.

Question 14.
a. At resonance, in an LCR circuit, the emf and current are
(i) in phase
(ii) out of phase
(iii) having a phase difference of \(\frac { \pi }{ 2 } \)
(iv) having a phase difference of \(\frac { \pi }{ 6 } \)
b. In the following circuit, find the value of V

OR
a. In a circuit carrying an ideal coil with negligible resistance, the power dissipated is …….
b. In the following circuit, find the impedance.

Question 15.
Explain hysteresis and draw hysteresis curve for a ferromagnetic substance.

Question 16.
Choose the appropriate values for X-rays from the given table.

Question 17.
a. Unit of wavenumber is
i. Hz
ii. eV
iii. m
iv. m^-1
b.Energy of ground state of a hydrogen atom is -13.6 eV. What is its ionization potential?

Question 18.
The current amplification factor for CB configuration of a transistor is 0.9. Find out the current amplification ion factor for CE configuration.

Question 19.
a. An electric charge q is moving with a velocity v in the direction of a magnetic field B. The magnetic force acting on the charge is

b. starting from Biot – Savart law, obtain an expression for the magnetic field at an axial point of a circular coil carrying current.
OR
a. An ammeter is a current measuring device which is always connected in an electric circuit.
b. Describe a cyclotron and obtain an expression for cyclotron frequency.

Kerala Plus Two Maths Model Question Paper 5

Time : 2 1/2 Hours
Cool off time : 15 Minutes
Maximum : 80 Score

General Instructions to Candidates :

• There is a ‘Cool off time’ of 15 minutes in addition to the writing time.
• Use the ‘Cool off time’ to get familiar with questions and to plan your answers.
• Read questions carefully before you answering.
• Read the instructions carefully.
• When you select a question, all the sub-questions must be answered from the same question itself.
• Calculations, figures and graphs should be shown in the answer sheet itself.
• Malayalam version of the questions is also provided.
• Give equations wherever necessary.
• Electronic devices except non programmable calculators are not allowed in the Examination Hall.

QUESTIONS

Question 1 to 7 carry 3 scores each. Answer any six questions only

Question 1.
a. Let R be a relation on the set {1,2,3} given by R= {(1,1), (2,2), (1,2), (2,1), (2,3)}. Which among the following element to be inclu¬ded to R so that R becomes Symmetric?
i. (3,3)
ii. (3,2)
iii. (1,3)
iv. (3,1)

b. If * is defined by a * b=a-b² and ⊕ is de-fined by a ⊕ is b=a²+b, where a and b are integers. Then find the value of (3⊕4)*5.

Question 2.

Question 3.
Find rate of change of area of a circle.
a. With respect to the radius, when r = 10cm
b. With respect to the time when the radius is increasing at the rate of 0.7cm/s. Given that r=5cm.

Question 4.

Question 5.
Find area of a circle with centre (0, 0) and radius “a” using integration.

Question 6.
Consider the differential equation \(\frac { dy }{ dx } \) = \(\frac { x+y }{ x } \)
a. Write the order of the differential equation.
b. Solve above given differential equation.

Question 7.
Following Table shows a brief description about manufacturing process of a company. Time required in hours per unit of the product and maximum availability of machines is also given in the table

a. Write the objective function.
b. Whether it is a maximisation case or a minimisation case Justify,
c. Write the contraints.

Question 8 to 17 carry 4 scores each. Answer any eight questions only

Question 8.
a. Afunctionf: A → B whereA= {1,2,3} and B= {4,5,6} defined by f (1) = 5, f (2) = 6, f (3) = 4, Check whether f is a bijection. If it is bijection, write f¹ as set of ordered pairs.2
b. The operation table for an operation * is given below. Given that I is the identify element, then which among the following is true regarding the elements in first column?

Question 9.

Question 10.
a. Find th relation between ‘a’ and ‘b’ so that the function defined by

b. “All continuous function are not differ-entiable”. Justify this statement with an example.

Question 11.
a. Find the equation of the tangent to the curve y = x² – 2x + 7 at (2,7)
b. Find the maximum value of the function.

Question 12.

Question 13.
Consider the differential equation \(\frac { xdy }{ dx } \) + y = \(\frac { 1 }{ { x }^{ 2 } } \)
a. Find the integrating factor
b. Solve the above differential equation.

Question 14.

Question 15.

Question 16.
a. Find the cartesian equation of a line passing through the origin and (5,-2,3)
b. The point P(x,y,z) lies in the First Octant and its distance from origin is 12 units. If the position vector of P makes angles 45°, 60° with x and y axes respectively, find co-ordinates of P.

Question 17.
Solve graphically :

Question 18 to 24 carry 6 scores each. Answer any 5 questions only

Question 18.

Question 19.
a. Without expanding prove that

b. Consider the following system of equations 2x – 3y + 5z = 11, 3x + 2y – 4z = -5, x + y-2z = -3
i. Express the system in the form Ax = B.
ii. Solve the system by matrix method a.

Question 20.

Question 21.
Evaluate the following.

Question 22.
Consider the parabolas y² = 4x and x² = 4y
a. Draw rough figure for the above parabolas.
b. Find the point of intersection of the two parabolas.
c. Find the area bounded by these two pa-rabolas.

Question 23.
a. Find the shortest distance between the lines whos vector equations are

b. If a plane meets positive x axis at a distance of 2 units from origin, positive y axis at a distance of 3 units from orign and positive z axiz at a distance of 4 units from origin. Find the equation of the plane.
c. Find the prependicular distance of (0,0,0) from the plane obtained in part (b).

Question 24.
a. A die is thrown twice let the event A be ‘odd number on First throw’ and B be ‘odd number on the second throw’ check whether A and B are independent.
b. Coloured balls are distributed in three boxes as shown in the following table.

A box is selected at random and a ball is taken out. If the ball taken is of red colour, What is the probabability that the other ball in the box is also of red colour?

ANSWERS

Answer 1.

Answer 2.

Answer 3.

Answer 4.

Answer 5.

Answer 6.

Answer 7.
Lex x = Machine G and y = Machine H
a. Objective function: Z= 20x + 30y
b. It is a maximization problem.
c. Constraints are:
3x + 4y < 10 ; 5x + 6y < 1 5 ; x > 0 ; y > 0

Answer 8.
a. A = {1,2,3}
B = {4,5,6}
f(1) = 5 ; f (2) = 6 ; f (3) = 4
Since it is one-one as well as onto, f is bijective.

∴ f = {(1,5), (2,6), (3,4)}
∴ f^-1 ={(5,1), (6,2), (4,3)}

b. (ii) 1,2,3
1 * 1 = 2,
2 * 1=2
Further 3 * 2 = 3
2 * 3 = 3
∴ * is commutative.

Answer 9.

Answer 10.

Answer 11.

Answer 12.

Answer 13.

Answer 14.

Answer 15.

Answer 16.

Answer 17.

Answer 18.

Answer 19.

Answer 20.

Answer 21.

Answer 22.

Answer 23.

Answer 24.

Plus Two Maths Previous Year Question Papers and Answers

Kerala Plus Two Maths Model Question Paper 4

Time : 2 1/2 Hours
Cool off time : 15 Minutes
Maximum : 80 Score

General Instructions to Candidates :

• There is a ‘Cool off time’ of 15 minutes in addition to the writing time.
• Use the ‘Cool off time’ to get familiar with questions and to plan your answers.
• Read questions carefully before you answering.
• Read the instructions careully.
• When you select a question, all the sub-questions must be answered from the same question itself.
• Calculations, figures and graphs should be shown in the answer sheet itself.
• Malayalam version of the questions is also provided.
• Give equations wherever necessary.
• Electronic devices except non programmable calculators are not allowed in the Examination Hall.

QUESTIONS

Question 1 to 7 carry 3 scores each. Answer any six questions only

Question 1.
a. The slope of the tangent to the curve given by
x = 1- cos θ , y = θ – sin θ at θ = \(\frac { \pi }{ 2 }\) is
(i) 0
(ii) -1
(iii) 1
(iv) Not defined

b. Find the intervals in which the function f (x) = x² – 4x + 6 is strictly decreasing.

Question 2.

Question 3.

Question 4.
a. Write the Cartesian equation of the straight line through the point (1, 2, 3) and along the vector 3\(\widehat { i }\) + \(\widehat { j }\) + 2\(\widehat { k }\).
b. Write a general point on this straight line.
c. Find the distance from (1, 2, 3) to the plane 2x + 3y – z + 2 = 0.

Question 5.

Question 6.
Solve the system of linear equations :
x + 2y + z = 8
2x + y – z = 1
x – y + z = 2
x + 2y + z = 8
2x + y – z = 1

Question 7.

Question 8 to 17 carry 4 scores each. Answer any eight questions only

Question 8.

Question 9.

Question 10.
i. Find the equation of the plane through the points (3,-1,2), (5,2,4) and (-1,-1,6).
ii. Find the perpendicular distance from the point (6,5,9) to this plane.

Question 11.

a. Express the euqations of the lines into vector form.
b. Find the shortest distance between the lines.

Question 12.

Question 13.
a. Find the equation of a plane with intercepts 2, 3 and 4 on X, Y and Z axes respectively.
b. Find the distance of the point (-1,-2, 3) from the plane r.(2\(\widehat { i }\) – \(\widehat { j }\)) + 4 \(\widehat { k }\)  = 4.

Question 14.

Question 15.

Question 16.

Question 17.
a. For two independent events A and B, which of the following pair of events need not be independent?
i. A’, B’
ii. A, B’
iii. A’, B
iv. A-B, B-A

Question 18 to 24 carry 6 scores each. Answer any 5 questions only

Question 18.
Consider the following L.P.P.
Maximize Z = 3x+2y
Subject to the constraints
x+2y < 10
3x+y < 15
x, y > 0
a. Draw its feasible region.
b. Find the comer points of the feasible region.
c. Find the maximum value of Z.

Question 19.
a. y=a cos x+b sin x is the solution of the differential equation

Question 20.

Question 21.

Question 22.

Question 23.

Question 24.
a. If \(\overline { a } \),\(\overline { b } \),\(\overline { c } \),\(\overline { d } \) respectively are the position vectors representing the vertices A, B, C, D of a parallelogram, then write \(\overline { d } \) in terms of \(\overline { a } \), \(\overline { b } \) and \(\overline { c } \) .

b. Find the projection vector of \(\overline { b } \) = \(\widehat { i }\) + 2 \(\widehat { j }\) + \(\widehat { k }\) along the vector \(\overline { a } \) = 2i +j + 2k. Also write \(\overline { b } \) as the sum of a vector along \(\overline { a } \) and a vector perpendicular to \(\overline { a } \) .

c. Find the area of a parallelogram for which the vectors 2 \(\widehat { i }\) + \(\widehat { j }\) and 3 \(\widehat { i }\) + \(\widehat { j }\) + 4 \(\widehat { k }\) are adjacent sides.

ANSWERS

Answer 1.

The point x = 2 divides the real line into two disjoint intervals namly (-∞, 2) and (2,∞). In the interval(-∞,2), f'(x)=2x-4 < 0
∴ f is strictly decreasing in this interval.

Answer 2.

Answer 3.

Answer 4.

Answer 5.

Answer 6.

Answer 7.

Answer 8.

Answer 9.

Answer 10.

Answer 11.

Answer 12.

Answer 13.

Answer 14.

Answer 15.

Answer 16.

Answer 17.

Answer 18.

Answer 19.

Answer 20.

Answer 21.

Answer 22.

Answer 23.

Answer 24.

Plus Two Maths Previous Year Question Papers and Answers

Kerala Plus Two Maths Model Question Paper 3

Time : 2 1/2 Hours
Cool off time : 15 Minutes
Maximum : 80 Score

General Instructions to Candidates :

• There is a ‘Cool off time’ of 15 minutes in addition to the writing time.
• Use the ‘Cool off time’ to get familiar with questions and to plan your answers.
• Read questions carefully before you answering.
• Read the instructions careully.
• When you select a question, all the sub-questions must be answered from the same question itself.
• Calculations, figures and graphs should be shown in the answer sheet itself.
• Malayalam version of the questions is also provided.
• Give equations wherever necessary.
• Electronic devices except non programmable calculators are not allowed in the Examination Hall.

QUESTIONS

Question 1 to 7 carry 3 scores each. Answer any six questions only

Question 1.
a. Let f : R → R be a function defined by f (x) = x³ + 5. Then f¹ (x) is
i. (x+5)^1/3
ii. (x-5)^1/3
iii. (5-x)^1/3
iv. 5-x

b. Let * be a binary operation defined on Q
a*b = a-b + ab. Check whether
i. It is commutative?
ii. Is * associative ?

Question 2.

Question 3.

Question 4.

Question 5.
Prove that the function f given by f (x)= log sin x is strictly increasing on ( 0,\(\frac { \pi }{ 2 } \))

Question 6.

Question 7.

Question 8 to 17 carry 4 scores each. Answer any eight questions only

Question 8.
a. Show that the relation R in set of real numbers defined as R = {(a,b): a < b²} is neither reflexive nor symmetric not transitive.
b. Show that the operation * on Q, defined by a*b = a+b-ab is commutative, and ex-its and identity elements find it.

Question 9.
a. The principal value of the expression cos-1 cos (680) is …………..

Question 10.

Question 11.

Question 12.

Question 13.

Question 14.

Question 15.
a. Find the distance between the planes x-y + z-5 = 0 and 2x-2y + 2z = 0.
b. Write the vector equation corresponding to Cartesian equation of a line

Question 16.
Find the shortest distance between the lines

Question 17.

Question 18 to 25 carry 6 scores each. Answer any 5 questions only

Question 18.

Question 19.
a. Use differential to approximate (0.999)^1/10
b. A window is in the form of rectangle sur-mounted by a semicircular opening. The total perimeter of the windows is 1 Om. Find the dimensions of the window to admit maximum light through the whole opening.

Question 20.
Find the area lying above x axis and included between the circle x² + y² = 8x and inside of the parabola y² = 4x. Also draw a neat diagram.

Question 21.
Evaluate:

Question 22.
Minimize and maximize Z = 5x + 10 y subject to x + 2y < 120, x + y > 60, x – 2y > 0, x, y > 0
a. Draw the feasible region
b. Find the comer points
c. Find the maximum and minimum profit.

Question 23.
a. Find the distance of the point (-1, -5, -10) from the point of intersection of the line

Question 24.
a. Two numbers are selected at random (with-out replacement) from the Pt six positive integers.

Let X denote the larger of the two numbers obtained. Find E(X) and Var(X)

b. A card from a pack of 52 cards is lost from the remaining X cards of the pack, two cards are drawn and are found to be both spades. Find the probability of the lost card being a spade.

ANSWERS

Answer 1.
a. ii
b. a * b = a – b + ab
b*a = b- a + ba = b – a + ab
∴ * is not commutative.
(a,b) * c = d * c = d- c + dc
= a- b + ab – c + ac – bc + abc
= a- b + ab – c + ac – bc + abc
= a – b – c + ab – bc + ca + abc
a * (b * c) = a * d = a- d + ad
= a – (b – c + be) + a (b – c + be)
= a- b + c – bc + ab – ac + abc
= a – b + c ab – be + ca + abc
∴ * is not associative.

Answer 2.

Answer 3.
We are giving that

Answer 4.

Answer 5.

Answer 6.

Answer 7.

Answer 8.
a. R = {(a,b): a < b²}
Relation R is defined in the set of real numbers.
i. Reflexive
Consider a ∈ R
If a ∈ R ⇒ a < a² which is false (a, a) ∈ R
R is not reflexive.

ii. Symmetric
Let a,b ∈ R and
(a,b) ∈ R ⇒ a < b² and b < a²,
which is false ⇒ (a,b) ∈ R, but (b,a) ∈ R
∴ R is not symmetric.

iii. Transitive
Let a, b, c ∈ R

Answer 9.

Answer 10.

Answer 11.

Answer 12.

Answer 13.

Answer 14.

Answer 15.

Answer 16.

Answer 17.

Answer 18.

Answer 19.

Answer 20.
The given equation of the circle x² + y² = 8x can be expressed as (x – 4)² + y² = 16. Thus, the centre of the circle is (4,0) and radius is 4. Its intersection with the parabola y² = 4x gives
x² + 4x = 8x
or
x² – 4x = 0
or
x(x-4) = 0
or
x = 0, x = 4
Thus, the point of intersection of these two curves are 0 (0,0) and P (4,4) above the x-axis.

From the above the required area of the region OPQCO included between these two curves above x-axis is
= (area of the region OCPO)
+ (area of the region PCQP)

Answer 21.

Answer 22.
a. The feasible region determine! by the constraints,
x + 2y < 120, x + y > 60, x – 2y > 0,
x > 0 and y > 0
is as follows.

b. The comer points of the feasible are region are A(60,0), C(60,30) and D (40,20).
The values of Z at these comer points are as follows.

a. The minimum value of Z is 300 at (60,0) and the maximum value of Z is 600 at all the points on the line segment joining (120,0) and (60,30)

Answer 23.

Answer 24.
a. The two positive integers can be sele-cted from the fist six positive integers without replacement in 6 x 5 = 30 ways.

X represents the larger of the two numbers obtained. Therefore, X can take the value of 2,3,4,5 or 6.

For X=2, the possible observations are (1,1)and(2,1)
∴ P (x = 2) = \(\frac { 2 }{ 30 } \) = \(\frac { 1 }{ 15 } \)

For X = 3 the possible observations are (1,3), (2,3), (3,1) and (3,2).
∴ p (x = 3) = \(\frac { 4 }{ 30 } \) = \(\frac { 2 }{ 15 } \)

For x = 4 the possible observations are

(1,4), (2,4), (3,4), (4,3), (4,2) and (4,1).
∴ p (x = 4) = \(\frac { 6 }{ 30 } \) = \(\frac { 1 }{ 5 } \)

For X = 5, the possible observations are (1.5) , (2,5), (3,5), (4,5), (5,4), (5,3), (5,2) and (5,1).
∴ p (x = 5) = \(\frac { 8 }{ 30 } \) = \(\frac { 4 }{ 15 } \)

For X = 6, the possible observations are (1.6), (2,6), (3,6), (4,6), (5,6), (6,4), (6,3), (6,2) and (6,1)
∴ p (x = 6) = \(\frac { 10 }{ 30 } \) = \(\frac { 1 }{ 3 } \)

Therefore, the required probability distribution is as follows.

b. Let E and E, be the respective events of choosing a spade card and a card which is not spade. Out of 52 cards, 13 cards are spade and 39 cards are not spade.

Plus Two Maths Previous Year Question Papers and Answers

Kerala State Board New Syllabus Plus Two Maths Previous Year Question Papers and Answers.

Plus Two Maths Model Question Papers Paper 1 with Answers

Time : 2 1/2 Hours
Cool off time : 15 Minutes
Maximum : 80 Score

General Instructions to Candidates:

• There is a ‘Cool off time’ of 15 minutes in addition to the writing time.
• Use the ‘Cool off time’ to get familiar with questions and to plan your answers.
• Read questions carefully before you answering.
• Read the instructions carefully.
• When you select a question, all the sub-questions must be answered from the same question itself.
• Calculations, figures and graphs should be shown in the answer sheet itself.
• Malayalam version of the questions is also provided.
• Give equations wherever necessary.
• Electronic devices except non-programmable calculators are not allowed in the Examination Hall.

Questions 1 to 7 carry 3 Scores each. Answer any six questions only. (6 × 3 = 18)

Question 1.
i) Let R be a relation on the set {1, 2, 3} given by R = {(1, 1), (2, 2), (1, 2), (2, 1), (2, 3)}. Which among the following element to be included to R so that R becomes Symmetric?
a) (3, 3)
b) (3, 2)
c) (1, 3)
d) (3, 1)
ii) If * is defined by a*b = a – b² and \(\oplus\) is defined by a \(\oplus\) b = a² + b , where a and b are integers. Then find the value of (3 \(\oplus\) 4) * 5
Answer:
i) b) (3, 2)
ii) (3 \(\oplus\) 4) * 5 = (32 + 4) * 5
= 13 * 5
= 13 – 5² = -12

Question 2.
If X + Y = \(\left[\begin{array}{ll}
5 & 2 \\
0 & 9
\end{array}\right]\) and X – Y = \(\left[\begin{array}{cc}
3 & 6 \\
0 & -1
\end{array}\right]\)find 2X – 3Y.
Answer:

Question 3.
Find rate of change of area of a circle.
a) with respect to the radius, when r = 10cm
b) with respect to the time when the radius is increasing at the rate of 0.7 cm/s. Given that r = 5cm
Answer:
i) Area of a circle A = πr²

Question 4.
i) Integrate \(\int \frac{(1+\log x)^{2}}{x} d x\)

Answer:

Question 5.
Find the area of a circle with centre (0, 0) and radius ‘a’ using integration.
Answer:
Equation of the circle is x² + y² = a²
y = \(\sqrt{a^{2}-x^{2}}\)
Area of the circle will 4 time the area under the curve from 0 to a.
Area = \(4 \int_{0}^{a} y d x=4 \int_{0}^{a} \sqrt{a^{2}-x^{2}} d x\)

Question 6.
Consider the differential equation \(\frac{d y}{d x}=\frac{x+y}{x}\)
i) Write the order of the differential equation.
ii) Solve the above given differential equation.
Answer:
i) Order = 1.

Question 7.
The following table shows a brief description of the manufacturing process of a company. The time required in hours per unit of the product and maximum availability of machine is also given in the table:

i) Write the objective function.
ii) Whether it is a maximization case or a minimization case. Justify.
iii) Write the constraints.
Answer:
Let x units = Machine G and y units = Machine H
i) Objective function: Z = 20x + 30y
ii) It is a maximization problem.
iii) Constraints are:
3x + 4y ≤ 10; 5x + 6y ≤ 15; x, y ≥ 0

Questions 8 to 17 carry 4 scores each. Answer any 8 questions. (8 × 4 = 32)

Question 8.
i) A function f : A → B, where A = {1, 2, 3} and B = {4, 5, 6} defined by f(1) = 5, f(2) = 6, f(3) = 4, Check whether f is a bijection. If it is a bijection, write f^-1 as set of ordered pairs.
ii) The operation table for an operation * is given below. Given that 1 is the identity element. Then which among the following is true regarding the element in the first column?

a) 3, 2, 2
b) 1, 2, 3
c) 1, 1, 2
d) 2, 2, 2
Check whether * is commutative.
Answer:
i) A = {1, 2, 3}; B = {4, 5, 6}
f(1) = 5, f(2) = 6, f(3) = 4
Hence f= {(1, 5),(2, 6),(3, 4)}
f^-1 = {(5, 1), (6, 2), (4, 3)}
ii) b) 1, 2, 3
Since; 1*2 = 2 and 2*1 =2
3 * 2 = 3 and 2*3 = 3
Therefore * is commutative

Question 9.
i) If Sin^-1 x = y, then
(a) 0 ≤ y ≤ π
b) \(-\frac{\pi}{2}\) ≤ y ≤ \(\frac{\pi}{2}\)
c) 0 < y < π
d) \(-\frac{\pi}{2}\) ≤ y ≤ \(\frac{\pi}{2}\)
ii) Find the principal value of sin^-1 (\(\frac{1}{2}\))
iii) sin^-1x = \(\frac{3}{4}\) find the value of cos^-1 x
Answer:
i) \(-\frac{\pi}{2}\) ≤ y ≤ \(\frac{\pi}{2}\)
ii) sin^-1 \(\frac{1}{2}\) = \(\frac{\pi}{6}\)
iii) sin^-1x + cos^-1x = \(\frac{\pi}{2}\)
\(\frac{3}{4}\) + cos^-1x = \(\frac{\pi}{2}\) ⇒ cos^-1x = \(\frac{\pi}{2}\) – \(\frac{3}{4}\)

Question 10.
i) Find the relation between ‘a’ and ‘b’ so that the function defined by
f(x) = \(\left\{\begin{array}{ll}
a x+1, & x \leq 3 \\
b x+3, & x>3
\end{array}\right.\) is continuous at x = 3
ii) “All continuous function are not differentiable.” Justify your answer with an example.
Answer:
i) Since f(x) is continuous at x = 3
\(\lim _{x \rightarrow 3^{-}}\) f(x) = \(\lim _{x \rightarrow 3^{+}}\) f(x) = f(3)
\(\lim _{x \rightarrow 3}\) (ax + 1) = \(\lim _{x \rightarrow 3}\) (bx + 3) = 3a +1
⇒ 3a + 1 = 3b + 3 = 3a +1
⇒ 3a + l = 3b + 3 ⇒ 3a – 3b – 2 = 0

ii) Consider the function f(x) = |x|.
Let us check the continuity and differentiability at x = 0.

Left derivative * Right derivative
Therefore not differentiable at x = 0.

Question 11.
i) Find the equation to the tangent to the curve y = x² – 2x + 7 at (2, 7)
ii) Find the maximum value of the function?
Answer:
f(x) = sin x + cos x, 0 < x < \(\frac{\pi}{2}\)
i) y = x² – 2x + 7 ⇒ \(\frac{d y}{d x}\) = 2x – 2
Slope at x = 2 is 2 × 2 – 2 – 2
Equation of the tangent is
(y – y₁) = m(x – x₁)
⇒ (y – 7) = 2(x- 2)
⇒ y – 7 = 2x – 4
⇒ 2x – y + 3 = 0

ii) c) f(x) = sin x + cos x
f'(x) = cos x – sin x
f”(x) = -sinx-cosx
For turning points;
f'(x) = cos x – sin x = 0 ⇒ tan x = 1

Question 12.
Integrate \(\int \frac{x+2}{2 x^{2}+6 x+5} d x\)
Answer:
Put x + 2 = A(4x + 6) + B
4A = 1 ⇒ A = \(\frac{1}{4}\)

Question 13.
Consider the differential equation
x \(\frac{d y}{d x}\) + y = \(\frac{1}{x^{2}}\)
i) Find the integrating factor.
ii) Solve the above differential equation.
Answer:

Question 14.
lf the vectors \(\overline{P Q}\) = -3i + 4j + 4k and \(\overline{P R}\) = -5i + 2j + 4k are the sides of a ΔPQR
i) Find the angle between \(\overline{P Q}\) and \(\overline{P R}\)
ii) Find the length of the median through the vertex P.
Answer:

ii)

With respect to the initial P the position vector of Q and R will be \(\overline{P Q}\) = -3i + 4j + 4k
and \(\overline{P R}\) = -5i + 2j + 4k respectively.
Since M is the midpoint of QR. The position vector of M will be

Question 15.
i) If \(\bar{a}\) = 5i – j – 3k and \(\bar{b}\) = i + 3j + 5k, then show that the vectors \(\bar{a}+\bar{b}, \bar{a}-\bar{b}\) are perpendicular.
ii) If \(\bar{a}\) = i – 2j + 3k, \(\bar{b}\) = 2i + 3j – 4k and
\(\bar{c}\) = i – 3j + 5k, then check whether \(\bar{a}\), \(\bar{b}\), \(\bar{c}\) are coplanar.
Answer:

Question 16.
i) Find the Cartesian equation of the line passing through origin and (5, -2, 3) ii) The point P(x, y, z) lies in the first octant and its distance from the origin is 12 units. If the position vector of P makes angles 45°, 60°with x and y axes respectively, find coordinates of P.
Answer:

Since P lies in the first octant, we take n = \(\frac{1}{2}\)
Therefore the coordinate of P is

Question 17.
Solve graphically Maximise Z = 5x + 3y Subject to the constraints 3x + 5y ≤ 15; 5x + 2y ≤ 10; x ≥ 0, y ≥ 0.
Answer:
In the figure the shaded region OABC is the fesible region. Here the region is bounded.
The corner points are
O(0, 0), A(2, 0), B(\(\frac{20}{19}, \frac{45}{19})\), C(0, 3)

Given Z = 5x + 3y

Since maximum value of Z occurs at B, the soluion is z = \(\frac{235}{19}\), (\(\frac{20}{19}\), \(\frac{45}{19}\))

Questions 18 to 24 carry 6 scores each. Answer any 5 questions. (5 × 6 = 30)

Question 18.
A = \(\left[\begin{array}{ccc}
3 & 3 & -1 \\
-2 & -2 & 1 \\
-4 & -5 & 2
\end{array}\right]\)
a) Find A^T
b) Express A as the sum of a symmetric and skew symmetric matrix.
ii) If A^T = \(\left[\begin{array}{cc}
\cos x & \sin x \\
-\sin x & \cos x
\end{array}\right]\), verift that A^T A = I
Answer:

Question 19.
i) Without expanding prove that
\(\left|\begin{array}{ccc}
x+y & y+z & z+x \\
z & x & y \\
1 & 1 & 1
\end{array}\right|\) = 0
ii) Consider the following system of equations
2x – 3y + 5z = 11; 3x + 2y – 4z = -5; x + y – 2z = -3
a) Express the system in Ax = B form.
b) Solve the system by matrix method.
Answer:

Question 20.
Find \(\frac{d y}{d x}\) of the following
i) x² + 2xy + 2y² = 1
ii) y^x = 2^x
iii) x = cos θ; y = sin θ at θ = \(\frac{\pi}{4}\)
Answer:
i) x² + 2xy + 2y² = 1
Differentiating w.r.to x;

ii) y^x = 2^x
Take log on both sides;
x log y = x log 2
Differentiating w.r.to x;

Question 21.
Evaluate the following

Answer:

Question 22.
Consider the parabolas y² = 4x, x² = 4y
i) Draw a rough figure for the above parabolas.
ii) Find the point of intersection of the two parabolas.
iii) Find the area bounded by these two parabolas.
Answer:

ii) solving the two conics we get the point of intersection as (0, 0) and (4, 4).
iii) Area of the enclosed region

Question 23.
i) Find the shortest distance between the lines whose vector equations are
\(\bar{r}\) = (i + 2j + 3k) + λ(i – 3j + 2k) and \(\bar{r}\) =(4i + 5j + 6k) + μ(i – 3j + 2k)
ii) If a plane meets positive x axis at a distance of 2 units from the origin, positive y axis at a distance of 3 units from the origin and positive z axis at a distance of 4 units from the origin. Find the equation of the plane.
iii) Find the perpendicular distance of (0, 0, 0) from the plane obtained in part (ii)
Answer:
i) \(\overline{a_{1}}\) = i + 2j + 3k; \(\bar{b}\) = i – 3j + 2k
\(\overline{a_{2}}\) = 4i + 5j + 6k Both lines are parallel.

Question 24.
i) A die is thrown twice let the event A be ‘odd number on first throw’ and B be ‘odd number on the second throw’ check whether A and B are independent.
ii) Coloured balls are distributed in three boxes as shown in the following table:

A box is selected at random and a ball is taken out. If the ball taken is of red colour, What is the probability that the other ball in the box is also of red colour?
Answer:
i) n(S) = 36
A = {(1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6), (3, 1), (3, 2), (3, 3), (3, 4), (3, 5), (3, 6), (5, 1), (5, 2), (5, 3), (5, 4), (5, 5), (5, 6)}
B = {(1, 1), (2, 1), (3, 1), (4, 1), (5, 1), (6, 1), (1, 3), (2, 3), (3, 3), (4, 3), (5, 3), (6, 3), (1, 5), (2, 5), (3,5), (4, 5), (5, 5), (6, 5)}
(A ∩ B) = {(1, 1), (1, 3), (1, 5), (3, 1), (3, 3), (3, 5), (5, 1), (5, 3), (5, 5)}

Therefore A and B are independent.

ii) Let B₁, B₂, B₃ are the event of getting the boxes.
P(B₁) = P(B₂) = P(B₃) = \(\frac{1}{3}\)
Let E be the event of getting a red ball.

Kerala State Board New Syllabus Plus Two Maths Previous Year Question Papers and Answers.

Kerala Plus Two Maths Previous Year Question Paper March 2017 with Answers

Time: 2½ Hours
Cool off time : 15 Minutes

General Instructions to Candidates

• There is a ‘cool off time’ of 15 minutes in addition to the writing time of 2½ hrs.
• You are not allowed to write your answers nor to discuss anything with others during the ‘cool off time’.
• Use the ‘cool off time’ to get familiar with the questions and to plan your answers.
• Read questions carefully before you answering.
• All questions are compulsory and the only internal choice is allowed.
• When you select a question, all the sub-questions must be answered from the same question itself.
• Calculations, figures, and graphs should be shown in the answer sheet itself.
• Malayalam version of the questions is also provided.
• Give equations wherever necessary.
• Electronic devices except non-programmable calculators are not allowed in the Examination Hall.

Question 1.
a. Let R be a relation defined on A = {1,2, 3} by R = {(1,3), (3, 1), (2, 2)}.R is
a. Reflexive
b. Symmetric
c. Transitive
d. Reflexive but not transitive

b. Find fog and gof if f(x) = |x + 1| and g(x) = 2x – 1.

c. Let * be a binary operation defined on N x N by (a, b) * (c, d) = (a + c, b + d). Find the identity element for * if it exists.

Question 2.

Question 3.

Question 4.

Question 5.

Question 6.
Slope of the normal to the curve y² = 4x at (1, 2) is
a. 1
b. \(\frac { 1 }{ 2 } \)
c. 2
d.-1

b. Find the interval in which 2x³ + 9x² + 12x – 1 is strictly increasing.
OR
a. The rate of change of volume of a sphere with respect to its radius when the radius is 1 unit
a. 4π
b. 2π
c. π
d. \(\frac { \pi }{ 2 } \)

b. Find two positive numbers whose sum is 16 and the sum of whose cubes are minimum.

Question 7.

Question 8.

Question 9.

Question 10.

Question 11.

Question 12.
a. The angle between the vectors \(\widehat { i } \) + \(\widehat { j } \) and \(\widehat { j } \) + \(\widehat { k } \) is
(i) 60°
(ii) 30°
(iii) 45°
(iv) 90°

Question 13.
a. The line x – 1 = y = z is perpendicular to the line

b. Find the shortest distance between the lines

Question 14.

Question 15.
Consider the linear programming problem :
Maximize Z = 50x + 40y
Subject to the constraints
x + 2y > 10
3x + 4y < 24
x > 0, y > 0
a. Find the feasible region.
b. Find the comer points of the feasible region.
c. Find the maximum value of Z.
Maximize Z = 50x + 40y
Subject to the constraints
x + 2y > 10
3x + 4y < 24
x > 0, y > 0

Question 16.
If A and B are two events such that A ⊂ B and P(A) ≠ then P (A/B) is
a. \(\frac { P(A) }{ P(B) } \)
b. \(\frac { P(B) }{ P(A) } \)
c. \(\frac { 1 }{ P(A) } \)
d. \(\frac { 1 }{ P(B) } \)

b. There are two identical bags. Bag I contains 3 red and 4 black balls while Bag II contains 5 red and 4 black balls. One ball is drawn at random from one of the bags.
i. Find the probability that the ball is drawn is red.
ii. If the ball drawn is red what is the probability that it was drawn from the bag I?
OR
Consider the following probability distribution of a random variable X.

Kerala State Board New Syllabus Plus Two Maths Previous Year Question Papers and Answers.

Kerala Plus Two Maths Previous Year Question Paper March 2018 with Answers

Time : 2 1/2 Hours
Cool off time : 15 Minutes
Maximum : 80 Score

General Instructions to Candidates :

• There is a ‘Cool off time’ of 15 minutes in addition to the writing time.
• Use the ‘Cool off time’ to get familiar with questions and to plan your answers.
• Read questions carefully before you answering.
• Read the instructions carefully.
• When you select a question, all the sub-questions must be answered from the same question itself.
• Calculations, figures and graphs should be shown in the answer sheet itself.
• Malayalam version of the questions is also provided.
• Give equations wherever necessary.
• Electronic devices except non-programmable calculators are not allowed in the Examination Hall.

Question 1 to 7 carry 3 scores each. Answer any six questions. (6 × 3 = 18)

Question 1.
If f(x) = \(\frac{x}{x-1}\), x ≠ 1
i) Find fof (x)
ii) Find the inverse of f.
Answer:

Question 2.
Using elementary row operations, find the inverse of the matrix \(\left[\begin{array}{cc}
1 & 2 \\
2 & -1
\end{array}\right]\)
Answer:

Question 3.
i) f(x) is a strictly increasing function, if f'(x) is ………..
a) positive
b) negative
c) 0
d) None of these,
ii) Show that the function f given by f(x) = x³ – 3x² + 4x, x ∈ R is strictly increasing.
Answer:
i) a) Positive.
ii) f(x) = x³ – 3x² + 4x
= 3 (x² – 2x + 1 – 1 + \(\frac{4}{3}\)) = 3((x + 1)2 + \(\frac{1}{3}\)) > 0

Question 4.
i) \(\int_{0}^{a}\)f(a – x)dx = …………

Answer:

Question 5.
Find the area of the region bounded by the curve y² = x’, x-axis and the lines x = 1 and x = 4.
Answer:

Question 6.
Find the general solution of the differential equation x \(\frac{d y}{d x}\) + 2y = x² log x

Solution is

Question 7.
A manufacturer produces nuts and bolts. It takes 1 hour of work on Machine A and 3 hours on Machine B to produce a package of nuts. It takes 3 hours on Machine A and 1 hour on Machine B to produce a package of bolts. He earns profit of Rs. 17.50 per package on nuts and Rs. 7.00 per package on bolts. Formulate the above LPp if the Machine operates for at most 12 hours a day.
Answer:
Let x packet of nuts and y packets of bolts.
Maximise: Z=17. 5x + 7y
Subject to
x + 3y ≤ 12; 3x + y ≤ 12; x, y ≥ O

Questions 8 to 17 carry 4 scores each. Answer any 8. (8 × 4 = 32)

Question 8.
Let A = N × N and ‘*’ be a binary operation on A defined by
(a, b)*(c, d)=(a + c, b + d)
i) Find (1, 2) * (2, 3)
ii) Prove that * is commutative.
iii) Prove that * is associative.
Answer:
i) (1, 2)*(2, 3)=(1 + 2, 2 + 3) = (3, 5)
ii) (c, d)*(a, b) = (c + a, d + b)
= (a + c, b + d) = (a, b)* (c, d)
iii) (a, b)*[(c, d)*(e, f)] = (a, b)*(c + e, d + f)
= (a + c + e, b + d + f)
[(a, b) * (c, d)] * (e, f) = (a + c, b + d) * (e, f)
= (a + c + e, b + d + f)

Question 9.
i) Identify the function from the above graph

a) tan^-1x
b) sin^-1x
c) cos^-1x
d) cos ec^-1x
ii) Find the domain and range of the function represented in above graph.
iii) Prove that \(\tan ^{-1} \frac{1}{2}+\tan ^{-1} \frac{2}{11}=\tan ^{-1} \frac{3}{4}\)
Answer:
i) sin^-1x

Question 10.
i) \(\frac{d\left(a^{x}\right)}{d x}\) = ………..
a) a^x
b) log(a^x)
c) a^x log a
d) xa^x-1
ii) Find \(\frac{d y}{d x}\) if x^y = y^x
Answer:
i) a^x loga
ii) Given; y^x = x^y, taking log on both sides;
x log y = y log x,
Differentiating with respect to x;

Question 11.
i) Find the slope of the tangent to the curve y = (x – 2)² at x = 1.
ii) Find a point at which the tangent to the curve y = (x – 2)² is parallel to the chord joining the point A (2, 0) and B (4, 4)
iii) Find the equation of the tangent to the above curve and parallel to the line AB.
y² = 4ax, a > 0 and x² = 4ay, a > 0
Answer:
i) \(\frac{d y}{d x}\) = 2(x – 2) ⇒ Slope = 2(1 – 2) = -2
ii) Slope of AB = \(\frac{y_{2}-y_{1}}{x_{2}-x_{1}}=\frac{4-0}{4-2}=2\)
Here 2(x – 2) = 2 ⇒ x = 3, y = 1
iii) Equation of the tangent line is
y – 1 = 2 (x – 3) ⇒ 2x – y = 5

Question 12.
\(\int_{0}^{2}\)(x² +1)dx as the limit of a sum.
Answer:
Here; f(x) = x² + 1; a = 0, b = 2 ⇒ h = \(\frac{2}{n}\)
\(\int_{0}^{2}\)(x² +1)dx
\(\lim _{h \rightarrow 0}\) h[f(0) + f(0 + h) + f(0 + 2h)+…+ f(0 + (n – 1)h)]
\(\lim _{h \rightarrow 0}\) h (1 + (h² + 1)+((2h) ² + 1)+…+[(n – 1)h²] + 1]

Question 13.
Consider the following figure:

i) Find the point of intersection ‘P’ of the circle x² + y² = 50 and the line y = x
ii) Find the area of the shaded region.
Answer:
i) point of intersection is x² + x² = 50 ⇒ x² = 25 ⇒ ±5; y = ±5
point P is (5, 5)

Question 14.
i) The degree of the differential equation \(x y\left(\frac{d^{2} y}{d x^{2}}\right)^{2}+x^{4}\left(\frac{d y}{d x}\right)^{3}-y \frac{d y}{d x}=0\) is ………
a) 4
b)3
c)2
d)1
ii) Find the general solution of the differential equation
sec² x tan ydx + sec² y tan xdy = 0
Answer:
i) c
ii) Given; sec²x tan ydx + sec²y tan xdy = 0

log tanx = -log tany + c
log tan x + log tan y = c

Question 15.
i) Prove that for any vector \(\bar{a}, \bar{b}, \bar{c}\)
\([\bar{a}+\bar{b}, \bar{b}+\bar{c}, \bar{c}+\bar{a}]=2\left[\begin{array}{lll}
\bar{a} & \bar{b} & \bar{c}
\end{array}\right]\)
ii) Show that if \(\bar{a}+\bar{b}, \bar{b}+\bar{c}, \bar{c}+\bar{a}\) are coplanar then \(\bar{a}, \bar{b}, \bar{c}\) are also coplanar.
Answer:

Question 16.
i) Find the equation of a plane which makes x,y,z intercepts respectively as 1, 2, 3.
ii) Find the equation of a plane passing through the point (1, 2, 3) which is parallel to above plane.
Answer:
\(\frac{x}{1}+\frac{y}{2}+\frac{z}{3}\) ⇒ 6x + 3y + 2z = 6
ii) Equation of the parallel plane passing
6(x – 1) + 3( y – 2) + 2(z – 3) = 0
⇒ 6x + 3y + 2z – 18 = 0

Question 17.
Solve the LPP graphically:
Minimise z = -3x + 4y
Subject to constraints:
x + 2y ≤ 8; 3x + 2y ≥ l2; x, y ≥ 0
Answer:
x + 2y = 8

3x + 12y = 12

The corner points are O(0, 0), A(4, 0), B(2, 3), C(0, 4)

Z attains minimum at (4,0).

Questions from 18 to 24 carry 6 scores each. Answer any 5. (5 × 6 = 30)

Question 18.
i) Find x and y if \(x\left[\begin{array}{l}
2 \\
3
\end{array}\right]+y\left[\begin{array}{c}
-1 \\
1
\end{array}\right]=\left[\begin{array}{c}
10 \\
5
\end{array}\right]\)
ii) Express the matrix \(\left[\begin{array}{ccc}
2 & -2 & -4 \\
-1 & 3 & 4 \\
1 & -2 & -3
\end{array}\right]\) as the sum of a symmetric and a skew symmetric matrices.
Answer:
Given; 2x – y = 10; 3x + y = 5
5x = 15 ⇒ x = 3; y = -4

Question 19.
i) Prove that \(\left|\begin{array}{ccc}
a & b & c \\
a+2 x & b+2 y & c+2 z \\
x & y & z
\end{array}\right|=0\)
ii) If A = \(\left[\begin{array}{ccc}
1 & -1 & 2 \\
0 & 2 & -3 \\
3 & -2 & 4
\end{array}\right]\), B = \(\left[\begin{array}{ccc}
-2 & 0 & 1 \\
9 & 2 & -3 \\
6 & 1 & -2
\end{array}\right]\)
a) Prove that B = A^-1
b) Using Asolve the system of linear equations given below:
x – y + 2z = 1; 2y – 3z = 1; 3x – 2y + 4z = 2
Answer:

iii) Matrix form of the system of linear equation is

Question 20.
i) Prove that the function defined by f(x) = cos x² is a continuous function.
ii) a) If y = \(e^{a \cos ^{-1} x}\), -1 ≤ x ≤ 1, show that
\(\frac{d y}{d x}=\frac{-a e^{a \cos ^{-1} x}}{\sqrt{1-x^{2}}}\)
b) Hence prove that
\(\left(1-x^{2}\right) \frac{d^{2} y}{d x^{2}}-x \frac{d y}{d x}-a^{2} y=0\)
Answer:
i) f(x) = cosx; g(x) = x² both are continuous.
Composition two continuous function is again continuous.
fog(x) = f(g(x)) = cos(x²)

Question 21.
Evaluate the following:

Answer:

Question 22.
i) lf \(\bar{a}\) = 3i + 2j + 2k, \(\bar{b}\) = i + 2j – 2k
a) Find \(\bar{a}\) + \(\bar{b}\), \(\bar{a}\) – \(\bar{b}\)
b) Find a unit vector perpendicular to both \(\bar{a}\) + \(\bar{b}\), \(\bar{a}\) – \(\bar{b}\)
ii) Consider the points A(1, 2, 7); B(2, 6, 3); C (3, 10, -1)
a) Find \(\overline{A B}, \overline{B C}\)
b) Prove that A, B, C are collinear points.
Answer:

Question 23.
i) Find the angle between the lines \(\frac{x-2}{2}=\frac{y-1}{5}=\frac{z+3}{-3}\) and \(\frac{x+2}{-1}=\frac{y-4}{8}=\frac{z-5}{4}\)
ii) Find the shortest distance between the pair of lines
\(\bar{r}\) = (i + 2j + 3k) + λ(i – 3j + 2k)
\(\bar{r}\) = (4i + 5j +6k) + μ(2i + 3j + k)
Answer:

Question 24.
i) The probability distribution of a random variable is given by P(x). What is Σp(x)?
ii) The following is a probability distribution function of a random variable.

a) Find k
b) P(x > 3)
c) P(-3 < x < 4)
d) P(x < -3)
Answer:
Σp(x) = 1

ii) a) k + 2k + 3k + 4k + 5k + 7k
+ 8k + 9k + 10k + 11k + 12k = 1
72k = 1 ⇒ k = \(\frac{1}{72}\)
b) P(x > 3) = P(4) + P(5) = 23k = \(\frac{23}{72}\)
c) P(-3 < x < 4) = P(-2) + P(-1) + P(0) + P(1) + P > (2)+ P > (3) = 43k = \(\frac{43}{72}\)
d) P(x < -3) = P(-5) + P(-4) = 3k = \(\frac{3}{72}\)

Read More:

ICICIGI Pivot Point Calculator

Kerala Plus Two Maths Model Question Paper 2

Time : 2 1/2 Hours
Cool off time : 15 Minutes
Maximum : 80 Score

General Instructions to Candidates :

• There is a ‘Cool off time’ of 15 minutes in addition to the writing time.
• Use the ‘Cool off time’ to get familiar with questions and to plan your answers.
• Read questions carefully before you answering.
• Read the instructions careully.
• When you select a question, all the sub-questions must be answered from the same question itself.
• Calculations, figures and graphs should be shown in the answer sheet itself.
• Malayalam version of the questions is also provided.
• Give equations wherever necessary.
• Electronic devices except non programmable calculators are not allowed in the Examination Hall.

QUESTIONS

Question 1 to 7 carry 3 scores each. Answer any six questions only

Question 1.

Question 2.

Question 3.
Consider the differential equation \(\frac { { d }^{ 2 }{ y } }{ { dx }^{ 2 } } +y=0\)
a. write its order and degree
b. Verify that y = a cos x + b sin x, where a, b ∈ R is a solution of the given differential equation

Question 4.
Consider the differential equation x \(\frac { dy }{ dx } \) + 2y = x2 ; x ≠ 0
a. What is its integrating factor?
b. Obtain its general solution.

Question 5.
The foot of the perpendicular drawn from origin to a plane is (4,-2, 5).
a. How far is the plane from the origin?
b. Find a unit vector perpendicular to that plane.
c. Obtain the equation of the plane in general form.

Question 6.

Question 7.

Question 8 to 17 carry 4 scores each. Answer any eight questions only

Question 8.

Question 9.

Question 10.

Question 11.
Consider the points A (2, 2, -1), B (3, 4, 2) and C (7, 0, 6). Find the vector and Cartesian equation of the plane passing through these points.

Question 12.
a. Find the cartesian equation of the plane through the point (1,2,-3) and perpen-dicular to the vector 2\(\widehat { i } \) – \(\widehat { j } \) + 2\(\widehat { k } \).
b. Find the angle between the above plane and the line \(\frac { x-1 }{ 2 } \) = \(\frac { y-3 }{ 3 } \) = \(\frac { z }{ 6 } \)

Question 13.
a. Let R be the relation on the set N of . natural numbers given by R = {(a,b): a – b > 2, b > 3}.
Choose the correct answer.
A. (4,1) ∈ R
B. (5,8) ∈ R
C. (8,7) ∈ R
D.(10,6) ∈ R

b. If f (x) = 8x³ and g (x) = x^1/3, find g (f(x)) and f (g (x)).

c. Let * be a binary operation on the set Q of rational numbers defined by a*b= \(\frac { ab }{ 3 } \).

Check whether * is commutative and associative?

Question 14.

The probability distribution of a random variable X taking values 1, 2, 3, 4, 5 is given.
a. Find the value of P.
b. Find the mean of X.
c. Find the variance of X.

Question 15.
a. Consider the family of all circles having their centre at the point (1,2). Write the equation of the family.
Write the corresponding differential equation.
b. Write the integrating factor of the differential equation,

Question 16.
Consider the functions: f (x) = |x|-1 and g(x) = 1- | x|
a. Sketch their graphs and shade the closed region between them.
b. Find the area of their shaded region.

Question 17.
a. What is the value of sin^-1 (sin 160°)?

Question 18 to 24 carry 6 scores each. Answer any 5 questions only

Question 18.

Question 19.
integrate the following :

Question 20.
a. Choose the correct statement related to the matrices A = \(\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}\) and B = \(\begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}\)

Question 21.
a. Which of the following functions is always increasing?
i. x + sin 2x
ii. x – sin 2x
iii. 2x + sin 3x
iv. 2x – sin x
b. The radius of a cylinder increases at a rate of 1 cm/s and its height decreases at a rate of 1 cm/s. Find the rate of change of its volume when the radius is 5 cm and the height is 15 cm.
If the volume should not change even when the radius and height are changed, what is the relation between the radius and height?
c. Write the equation of tangent at (1,1) on the curve 2x² + 3y² = 5.

Question 22.
Consider the linear programming problem :
Minimize Z = 3x + 9y
Subject to the constraints :
x+3y < 60 ; x + y > 10 ; x < y ; x > 0 , y > 0
a. Draw its feasible region.
b. Find the vertices of the feasible region.
c. Find the minimum value of Z subject to the given constraints.
Minimize Z = 3x + 9y
Subject to the constraints :
x + 3y < 60 ; x + y > 10 ; x < y ; x > 0, y > 0

Question 23.
Consider the following figure :

a. Find the point of intersection P, of the circle, x²+ y² = 32 and the line y = x.
b. Express the area of the shaded portion as a Sum of two definite integrals.
c. Find the area of the shaded portion.

Question 24.

ANSWERS

Answer 1.

Answer 2.

Answer 3.

Answer 4.

Answer 5.

Answer 6.

Answer 7.

Answer 8.

Answer 9.

Answer 10.

Answer 11.

Answer 12.

Answer 13.

Answer 14.

Answer 15.

Answer 16.

Answer 17.

Answer 18.

Answer 19.

Answer 20.

Answer 21.

Answer 22.

Answer 23.

Answer 24.

c. 1. f (x) is continuous at (-2, 2)
2. f (x) is differentiable at (-2, 2)
3. If f (b) = f (a) then there exist a point at (-2, 2) such that f¹ (c) = 0
f (b) = f (-2) = (-2)² + 2 = 6
f (a) = f (2) = 2² + 2 = 6
∴ Then there exist a point C such that
f¹ (c) = 0
f¹ (x) = 2x
f¹ (x) = 2c = 0
∴ c = 0 at (-2,2)
∴ Rolle’s theorem verified.

Plus Two Maths Previous Year Question Papers and Answers

Read More:

GMRINFRA Pivot Point Calculator