AP Inter 1st Year Chemistry Question Paper March 2017

Access to a variety of AP Inter 1st Year Chemistry Model Papers and AP Inter 1st Year Chemistry Question Paper March 2017 helps students overcome exam anxiety by fostering familiarity. question patterns.

AP Inter 1st Year Chemistry Question Paper March 2017

Note : Read the following instructions carefully.

  1. Answer all questions of Section ‘A’. Answer any six questions in Section ‘B’ and any two questions in Section ‘C’.
  2. In Section ‘A’, questions from Sr. Nos. 1 to 10 are of “Very Short Answer Type’1. Each question carries two marks. Every answer may be limited to 2 or 3 sentences. Answer all these questions at one place in the same order.
  3. In Section ‘B1, questions from Sr. Nos. 11 to 18 are of “Short Answer Type”. Each question carries four marks. Every answer may be limited to 75 words.
  4. In Section ‘C’, questions from Sr. Nos. 19 to 21 are of “Long Answer Type”. Each question carries eight marks. Every answer may be limited to 300 words.
  5. Draw labelled diagrams wherever necessary for questions in Sections ‘3’ and ‘C’.

Section – A

Note : Answer all questions.

Question 1.
Write any two adverse effects of global warming.
Answer:
Adverse effects of global warming :

  1. Unseasonal raining.
  2. Agricultural sector will be badly affected due to global warming.
  3. Increase the infections diseases.
  4. Melting of polar ice caps and causing of sealevels to rise.

Question 2.
Define the sink and receptor.
Answer:

  1. Sink : The medium which retains and interacts with long lived pollutant is called sink.
  2. Receptor: The medium which is effected by a pollutant is called receptor.

Question 3.
Write the effect of temperature on surface tension and viscosity. Give reason to that.
Answer:
With increase of temperature surface tension decreases.
With increase of temperature viscousity also decreases.

Reasons :
Due to increase in K.E. of molecules.
Due to decrease in intermolecular forces.

AP Inter 1st Year Chemistry Question Paper March 2017

Question 4.
Calculate the oxidation number of ‘Cr’ in K2Cr2O7.
Answer:
K2Cr2O7
2(+1) + 2x + 7(-2) = 0 ‘
2 + 2x – 14 = 0
2x = 12
x = +6

Question 5.
Define the ionic product of water.
Answer:
Ionic product of water : The product of concentrations of H+ ion and OH ion in water is called ionic product of water (Kw). Kw value at 25°C is 1.008 × 10-14 mole2/lit2

Question 6.
What is plaster of Paris?
Answer:
The hemihydrate of calcium sulphate is called plastor of paris.
The formula of plaster of paris is CaSO2\(\frac{1}{2}\) H20.

Question 7.
Write any four uses of CO2 gas.
Answer:
Uses of CO2 gas :

  1. Solid CO2 (dryice) is used in ice creams and frozen food.
  2. CO2 gas is used in the waste water treatment as a cooling medium.
  3. CO2 gas is used in exting shing fire.
  4. CO2 gas is used in the food industry, oil industry etc.
  5. CO2 gas is useful for plants and in the photo synthesis.

Question 8.
Why are alkali metals not found in the free state in nature?
Answer:
Alkali metals are not found in the free state in nature because they readily lose their valency electron to form M+ ion (a nonvalent ion).

AP Inter 1st Year Chemistry Question Paper March 2017

Question 9.
Why the graphite is good conducter of electrocity?
Answer:
In graphite carbon undergoes sp2 hybridisation. Each carbon forms three σ – bonds with three neighbouring carbon atoms. Fourth electron forms π – bond and it is delocalised. Due to the presence of these moving (or) free electrons graphite acts as good conductor.

Question 10.
What is the type of hybridization of each carbon in the following compound?
HC = C – CH = CH2
Answer:
AP Inter 1st Year Chemistry Question Paper March 2017 - 1
1st carbon – sp – hybridisation
2nd carbon – sp – hybridisation
3rdcarbon – sp2 – hybridisation
4th carbon – sp2 – hybridisation

Section – B

Question 11.
State and explain Graham’s law of diffusion.
Answer:
Graham’s law of diffusion : At a given temperature and pressure, the rate of diffusion of a gas is inversely proportional to the square root of density, vapour density or molecular weight.
\(r \propto \frac{1}{\sqrt{d}} ; r \propto \frac{1}{\sqrt{V D}} ; r \propto \frac{1}{\sqrt{M}}\)

If r1 and r2 are the rates of diffusion of two gases d1 and d2 are their densities respectively, then
\(\frac{r_1}{r_2}=\sqrt{\frac{d_2}{d_1}}\)
This equation can be written as :
\(\frac{r_1}{r_2}=\sqrt{\frac{d_2}{d_1}}=\sqrt{\frac{V D_2}{V D_1}}=\sqrt{\frac{M_2}{M_1}}\)

Comparison of the volumes of the gases that diffuse in the same time. Let V1 and V2 are the volumes of two gases that diffuse in the same time ‘t’.
\(\frac{r_1}{r_2}=\frac{\frac{v_1}{t_1}}{\frac{v_2}{t_2}}\)
When time of flow is same then : \(\frac{r_1}{r_2}=\frac{v_1}{v_2}\)
When volume is the same then : \(\frac{r_1}{r_2}=\frac{t_2}{t_1}\)

Applications :

  1. This principle is used in the separation of isotopes like U235 and U238.
  2. Molar mass of unknown gas can be determined by comparing the rate of diffusion of a known gas molecular mass.
  3. Ansil’s alarms which are used in coal mines to detect the explosive marsh gas works on the principle of diffusion.

Question 12.
A carbon compound contains 12.8% carbon, 2.1 % hydrogen, 85.1% bromine. The molecular weight of the compound is 187.9. Calculate the molecular formula (At. wt C = 12, H = 1,Br = 80).
Answer:
AP Inter 1st Year Chemistry Question Paper March 2017 - 2
∴ Emperical formula of the compound = C1 H2 Br
Molecular formula = n (Emperical formula)
n = \(\frac{\text { Molecular wt }}{\text { Emperical wt }}=\frac{187.9}{94}\)
Given molecular wt = 187.9
∴ Molecular firmula = 2 (CH2Br)
Emperical Wt = 94 (CH2 Br)
= C2H4Br2

AP Inter 1st Year Chemistry Question Paper March 2017

Question 13.
What is hydrogen bond ? Explain the different types of hydrogen bonds with examples.
Answer:
Hydrogen bond is a weak electrostatic bond formed between partially positive charged hydrogen atom and an highly electronegative atom of the same molecule or another molecule.

Hydrogen- bond is formed when the Hydrogen is bonded to small, highly electronegative atoms like F, O and N. A partial positive charge will be on hydrogen atom and partial negative charge on the electronegative atom.

The bond dissociation energy of hydrogen bond is 40 KJ/mole. Hydrogen bond is represented with dotted lines (……). Hydrogen bond is stronger than Vander Waals’ forces and weaker than covalent bond.

Hydrogen bonding is of two types. (1) Intermolecular hydrogen bond and (2) Intramolecular hydrogen bond.

1) Intermolecular hydrogen bond:
If the hydrogen bond is formed between two polar molecules it is called intermolecular hydrogen bond, i.e., the hydrogen bond is formed between hydrogen atom of one molecule and highly electronegative atom of another molecule is known as intermolecular hydrogen bond.
Ex.: Water (H2O); HF: NH3; p – nitrophenol, CH3COOH, ethyl alcohol
AP Inter 1st Year Chemistry Question Paper March 2017 - 3
Water molecule forms an associated molecule through inter- molecular hydrogen bond. Due to molecular association water possess high boiling point.
AP Inter 1st Year Chemistry Question Paper March 2017 - 15

2) Intramolecular hydrogen bond:
If the hydrogen bond is formed within the molecule it is known as intramolecular hydrogen bond.
Ex. : o – nitrophenol; o – hydroxy benzaldehyde.
AP Inter 1st Year Chemistry Question Paper March 2017 - 4

Abnormal behaviour due to hydrogen bond :

  1. The physical state of substance may alter. They have high melting and boiling points.
  2. Ammonia has higher boiling point than HCl eventhough nitrogen and chlorine have same electronegativity values (3.0). Ammonia forms an associated molecule through inter- molecular hydrogen bond.
  3. p – hydroxy benzaldehyde have higher boiling point than o- hydroxy benzaldehyde. This is due to intermolecular hydrogen bonding in para isomer.
  4. Ethyl alcohol is highly soluble in water due to association and co-association through intermolecular hydrogen bonding

Question 14.
State and explain Hess’s law of constant heat summation. Give an example.
Answer:
Hess’s law states that the total amount of heat evolved or absorbed in a chemical reaction is always same whether the reaction is carried out in one step (or) in several steps.

Illustration:
This means that the heat of reaction depends only on the initial and final stages and not on the intermediate stages through which the reaction is carried out. Let us consider a reaction in which ∆ gives D. The reaction is brought out in one step and let the heat of reaction be ∆H
A → D; ∆H
Suppose the same reaction is brought out in three stages as follows
AP Inter 1st Year Chemistry Question Paper March 2017 - 5
A → B : ∆H1
B → C : ∆H2
C → D : ∆H3

The net heat of reaction is ∆H1 + ∆H2 + ∆H3.
According to Hess law AH = ∆H1 + ∆H2+ ∆H3.
Ex: Consider the formation of CO2. It can be prepared in two ways.

1) Direct method : By heating carbon in excess of O2.
C(s) + O2(g) → CO2(g); ∆H1 = -393.5 kj

2) Indirect method : Carbon can be converted into CO2 in the following two steps.
C(s) + \(\frac{1}{2}\)O2 → CO2(g); ∆H1 = -110.5 Kj
CO(g) + \(\frac{1}{2}\)O2 → CO2(g); ∆H2 = -283.02 kj
Total ∆H = -393.52 Kj (∆H1 + ∆H2)
The two ∆H values are same.

AP Inter 1st Year Chemistry Question Paper March 2017

Question 15.
What are homogenous and heterogenous equilibria? Give two examples of each.
Answer:
Homogenous:
It the physical states of the participating substances are same they the equilibrium is homogeneous equilibrium.
e.g : H2(g) + I2(g) ⇌ 2HI(g)
N2(g) + 3H2(g) ⇌ 2NH3(g)

Heterogenous:
If the physical states of all (or) some of the participating substances are same then the equilibrium is Het-erogeneous equilibrium.
AP Inter 1st Year Chemistry Question Paper March 2017 - 6

Question 16.
Write any four uses of dihydrogen (H2).
Answer:
Hydrogen as a fuel :

  1. The heat of combustion of hydrogen is high i.e about 242kj/mole. Hence hydrogen is used as industrial fuel,
  2. The energy released by the combustion of dihydrogen is more than the petrol (3 times).
  3. Hydrogen is major constituent in fuel gases like coal gas and water gas.
  4. Hydrogen is also used in fuel cells for the generation of electric power. 5% dihydrogen is used in CNG for running four-wheeler vehicles.

By hydrogen economy principle the storage and transportation of energy in the form of liquid (or) gaseous state. Here energy is transmitted in the form of dihydrogen and not as electric power.

Question 17.
Explain the structure of diborane.
Answer:
Diborane is an electron deficient compound. It has ’12’ valency electrons for bonding purpose instead of ’14’ electrons. In diborane each boron atom undergoes sp3 hybridization out of the four hybrid orbitals one is vacant.

Each boron forms two, a – bonds (2 centred – 2 electron bonds) bonds with two hydrogen atoms by overlapping with their’1s’orbital. The remaining hybrid orbitals of boran used for the forma¬tion of B-H-B bridge bonds

In the formation of B-H-B bridge, half filled sp3 hybrid orbitaj of one boron atom and vacant sp3 hybrid orbital of second boron atom overlap with 1s orbital of H-atom. These three centred two electron bonds are also called as banana bonds. These bonds are present above and below the plane of BH2 units.

Diborane contains two coplanar BH2 groups. The four hydrogen atoms are called terminal hydrogen atoms and the remaining two hydrogens are called bridge hydrogen atoms.
AP Inter 1st Year Chemistry Question Paper March 2017 - 7
Bonding in diborane, Each B atom uses sp3 hybrids for bonding. Out of the four sp3 hybrids on each B atom, one is without an electron shown in broken lines. The terminal B-H bonds are ormal 2-centre-2-electron bonds but the two bridge bonds are 3-centre-2- electron bonds. The 3-centre-2-electron bridge bonds.
AP Inter 1st Year Chemistry Question Paper March 2017 - 8

Question 18.
Define the dipole moment. Why the BF3 molecule dipole moment is zero?
Answer:
Dipolemoment : The product of magnetude of charge and the distance between the two poles (bond length) is called dipolemoment (p).

Units – Debyl
μ = q × d
q = charge;
d = bond length

BF3 molecule is a symmetrical molecule. In symmetrical molecules bonds moments vectorial sum is zero. SO BF3 has zero dipole moment.

AP Inter 1st Year Chemistry Question Paper March 2017

Question 19.
What are the postulates of Bohr’s model of hydrogen atom? Discuss the importance of this model to explain various series of line spectra in hydrogen atom.
Answer:
Niels Bohr quantitatively gave the general features of hydrogen atom structure and it’s spectrum. His theory is used to evaluate several points in the atomic structure and spectras. The postulates of Bohr atomic model for hydrogen as follows Postulates:
1. The electron in the hydrogen atom can revolve around the nucleus in a circular path of fixed radius and energy. These paths are called orbits (or) stationary states. These circular orbits are concentric (having same center) around the nucleus.

2. The energy of an electron in the orbit does not change with time. When an electron moves from lower stationary state to higher stationary state absorption of energy takes place.

3. When an electron moves from higher stationary state to lower stationary state emission of energy takes place.

4. When an electronic transition takes place between two sta¬tionary states that differ in energy by
∆E = E2 – E1 = hυ
∴ The frequency of radiation absorbed (or) emitted υ = \(\frac{E_2-E_1}{h}\)
E1 and E2 are energies of lower, higher energy states respectively.

4. The angular momentum of an electron is given by mvr = \(\frac{\mathrm{nh}}{2 \pi}\)
An electron revolve only in the orbits for which it’s angular momentum is integral multiple of \(\frac{h}{2 \pi}\)

Line spectra of Hydrogen – Bohr’s Theory :
In case of hydrogen atom line spectrum is observed and this can be explained by using Bohr’s Theory.
According to Bohr’s postulate when an electronic transition takes place between two stationary states that differ in energy is given by
ΔE = Ef – Ei
Ef = final orbit energy
Ei = initial orbit energy
AP Inter 1st Year Chemistry Question Paper March 2017 - 9

  1. In case of absorption spectrum nf ni → energy is absorbed (+Ve)
  2. In case of emission spectrum ni > nf → energy is emitted (-ve)
  3. Each spectral line in absorption (or) emission spectrum associated to the particular transition in hydrogen atom
  4. In case of large no.of hydrogen atoms large no.of transitions possible they results in large no.of spectral lines.

Img-1
The series of lines observed in hydrogen spectra are

‘n’ value Series Region
1 Lyman series UV region
2 Balmer series visible
3 Pachen series Near I.R.
4 Brackette series I.R.
5 pfund series Far I.R.

Question 20.
Write the classification of elements into s, p, d and f blocks in long form of periodic table.
Answer:
According to the electronic configuration of elements, the elements have been classified into four blocks. The basis for this classification is the entry of the differentiating electron into the subshell. They are classified into s, p, d and f blocks.

‘s’ block elements : If the differentiating electron enters into ‘s’ orbital, the elements belongs to ‘s’ block. In every group there are two ‘s’ block elements. As an ‘s’ orbital can have a maximum of two electrons, ‘s’ block has two groups IA and IIA.
AP Inter 1st Year Chemistry Question Paper March 2017 - 10

‘p‘ block elements : If the differentiating electron enters into ‘p’ orbital, the elements belongs to ‘p’ block. ‘p’ block contains six elements in each period. They are MIA to VII A and zero group elements. The electronic configuration of “p” block elements varies from ns2np1 to ns2np6.

‘d’ block elements :
It the differentiating electron enters into (n – 1) d – orbitals the elements belongs to’d’ block. These elements are in between ‘s’ and ‘p’ blocks. These elements are also known as transition elements. In these elements n and (n – 1) shells are incompletely filled. The general electronic configuration of’d’ block elements is (n – 1) d1-10 ns1-2. This block consists of NIB to VIIB, VIII, IB and MB groups.

f block elements :
If the differentiating electron enters into f orbitals of antipenultimate shell (n – 2) of atoms of the elements belongs to T block. They are in sixth and seventh periods in the form of two series with 14 elements each. They are known as lanthanides and actinides and are arranged at the bottom of the periodic table. The general electronic configuration is (n – 2) f1-14 (n -1) d0-1 ns2.

In these shells the last three shells (ultimate, penultimate and anti penultimate) are incompletely filled. Lanthanides belongs to 4f series. It contains Ce to Lu. Actinides belong to 5f series. It contains Th to Lr.

Advantages of this kind of classification :
As a result of this classification of elements were placed in correct positions in the periodic table. It shows a gradual gradation in physical and chemical properties of elements. The metallic nature gradually decreases and non – metallic nature gradually increases from ‘s’ block to ‘p’ block. This classification gave a special place for radioactive elements.

AP Inter 1st Year Chemistry Question Paper March 2017

Question 21.
a) Wurtz’s reaction
Answer:
Wurtuz reactions : Alkylhalides reacts with sodium metal in presence of dry ether to form alkanes.
AP Inter 1st Year Chemistry Question Paper March 2017 - 11

b) Polymerization of ethylene
Answer:
Polymerization : A large molecular weight complex compound which is formed by the repeated combination of smaller units is called polymer. The process of formation of polymer is called polymerisation.
Ex : Ethylene undergo polymerisation at 200°C and 1500 – 200 atm pressure gives polythene.
AP Inter 1st Year Chemistry Question Paper March 2017 - 12

c) Addition of water to acetylene
Answer:
Addition of water: Acetylene undergoes addition reaction with water in the presence of mercuric sulphate and sulphuric acid. Vinyl alcohol formed in the reaction undergoes rearrangement gives acetaldehyde.
AP Inter 1st Year Chemistry Question Paper March 2017 - 13

d) Nitration of benzene
Answer:
Nitration : Benzene when heated with a mixture of cone. H2SO4 and cone. HNO3 below 60°C to give Nitrobenzene.
AP Inter 1st Year Chemistry Question Paper March 2017 - 14

AP Inter 1st Year Chemistry Model Paper Set 3 with Solutions

Utilizing AP Inter 1st Year Chemistry Model Papers Set 3 helps students overcome exam anxiety by fostering familiarity.

AP Inter 1st Year Chemistry Model Paper Set 3 with Solutions

Time: 3 Hours
Maximum Marks: 60

Note : Read the following instructions carefully.

  1. Answer all questions of Section – A. Answer ANY SIX questions in Section – B and ANY TWO questions in Section – C.
  2. In Section – A, questions from Sr. Nos. 1 to 10 are of Very short answer type. Each question carries TWO marks. Every answer may be limited to 2 or 3 sentences. Answer all these questions at one place in the same order.
  3. In Section – B, questions from Sr. Nos. 11 to 18 are of Short answer type. Each question carries FOUR marks. Every answer may be limited to 75 words.
  4. In Section – C, questions from Sr. Nos. 19 to 21 are of Long answer type. Each question carries EIGHT marks. Every answer may be limited to 300 words.
  5. Draw labelled diagrams, wherever necessary for questions in Section – B and Section – C.

Section – A

Note : Answer ALL questions.

Question 1.
Define COD and BOD.

Question 2.
State the Hess Law of Constant Heat Summation.

Question 3.
Name any two man-made silicates.

Question 4.
Write Van der Waals1 equation.

AP Inter 1st Year Chemistry Model Paper Set 3 with Solutions

Question 5.
What are intensive and extensive properties ?

Question 6.
How is Nitrobenzene prepared from Benzene ?

Question 7.
Name two adverse effects caused by acid rain.

Question 8.
A solution is prepared by adding 2 gm. of a substance A to 18 gm. of water. Calculate the mass percent of the solute.

Question 9.
Graphite is a good conductor. Explain.

Question 10.
What is Buffer Solution?

Section – B

Note : Answer ANY SIX questions.

Question 11.
What is Hydrogen bond? Explain the different types of Hydrogen bonds with examples.

Question 12.
Deduce (a) Boyle’s law and (b) Dalton’s law from kinetic gas equation.

AP Inter 1st Year Chemistry Model Paper Set 3 with Solutions

Question 13.
Discuss the application of Le-Chatelier’s principle for the industrial synthesis of Ammonia.

Question 14.
Explain the structure of Diborane.

Question 15.
Write any two oxidising and two reducing properties of hydrogen peroxide.

Question 16.
What is Plaster of Paris ? Write a short note on it.

Question 17.
Balance the following Redox Reaction by ion-electron method.
AP Inter 1st Year Chemistry Model Paper Set 3 with Solutions 1

Question 18.
Predict the shapes of the following molecules, making use of Valence Shell Electron Pair Repulsion (VSEPR) Theory.
(a) XeF4
(b) BrF5
(c) ClF3
(d) H2O

Section – C

Note : Answer ANY TWO questions.

Question 19.
Write about four quantum numbers and explain the significance of these quantum numbers.

AP Inter 1st Year Chemistry Model Paper Set 3 with Solutions

Question 20.
Write an essay on s, p, d and f block elements.

Question 21.
Give two methods of preparation of acetylene. How does it react with water and halogens?

AP Inter 1st Year Chemistry Model Paper Set 1 with Solutions

Utilizing AP Inter 1st Year Chemistry Model Papers Set 1 helps students overcome exam anxiety by fostering familiarity.

AP Inter 1st Year Chemistry Model Paper Set 1 with Solutions

Time: 3 Hours
Maximum Marks: 60

Note : Read the following instructions carefully.

  1. Answer all questions of Section – A. Answer ANY SIX questions in Section – B and ANY TWO questions in Section – C.
  2. In Section – A, questions from Sr. Nos. 1 to 10 are of Very short answer type. Each question carries TWO marks. Every answer may be limited to 2 or 3 sentences. Answer all these questions at one place in the same order.
  3. In Section – B, questions from Sr. Nos. 11 to 18 are of Short answer type. Each question carries FOUR marks. Every answer may be limited to 75 words.
  4. In Section – C, questions from Sr. Nos. 19 to 21 are of Long answer type. Each question carries EIGHT marks. Every answer may be limited to 300 words.
  5. Draw labelled diagrams, wherever necessary for questions in Section – B and Section – C.

Section – A

Note : Answer ALL questions.

Question 1.
Define sink and receptor.

Question 2.
Which is called “Milk of Magnesia” ? Give its use.

Question 3.
What is Lewis acid ? Give one example.

AP Inter 1st Year Chemistry Model Paper Set 1 with Solutions

Question 4.
Write any two important uses of caustic soda.

Question 5.
Calculate kinetic energy in calories of 5 moles of nitrogen at 27°C.

Question 6.
What are silicones ? Give one example.

Question 7.
What are the oxidation number of the underlined elements in each of the following species? a) KMnO4 b) MnO4-2

Question 8.
What is meant by dry ice ? Give its application.

Question 9.
An Alkyne ‘A’ undergo cyclic polymerisation by passing through red hot iron tube to give ‘B’. What are ‘A’ and ‘B’?

Question 10.
Mention the harmful effects caused due to depletion of ozone layer.

Section – B

Note : Answer ANY SIX questions.

Question 11.
Deduce : a) Graham’s law and b) Dalton’s law from kinetic gas equation.

Question 12.
Balance the following redox equation in acidic medium by ion- electron method :
Fe2+(aq) + Cr2O72- → Fe(aq)3+ + Cr(aq)3+

Question 13.
State ane explain the Hess’s law of constant heat summation with an example.

Question 14.
Derive the relation between Kp and for the equilibrium reaction :
N2(g) + 3H2(g) ⇌ 2NH3(g)

AP Inter 1st Year Chemistry Model Paper Set 1 with Solutions

Question 15.
What is the cause for permanent hardness of water ? Explain the removal of hardness of water by Calgon method.

Question 16.
Explain the structure of Ethylene.

Question 17.
State Fajan’s rules with suitable examples.

Question 18.
Explain the structure of diborane.

Section – C

Note : Answer ANY TWO of the following questions.

Question 19.
What are the postulates of Bohr’s model of hydrogen atom? Write its limitations. Give any two differences between emission and absorption spectra.

AP Inter 1st Year Chemistry Model Paper Set 1 with Solutions

Question 20.
What is a periodic proeprty ? How the following properties vary in a group and in a period ? Explain.
a) Ionisation enthalpy
b) Electronegativity
c) Electron gain enthalpy

Question 21.
Describe any two methods of preparation of benzene with corresponding equations. Explain the following benzene reactions:
a) Halogenation
b) Alkylation
c) Acylation
d) Nitration

AP Inter 1st Year Maths 1A Question Paper March 2018

Thoroughly analyzing AP Inter 1st Year Maths 1A Model Papers and AP Inter 1st Year Maths 1A Question Paper March 2018 helps students identify their strengths and weaknesses.

AP Inter 1st Year Maths 1A Question Paper March 2018

Time: 3 Hours
Maximum Marks: 75

Note: This question paper consists of three sections A, B, and C.

Section – A
(10 × 2 = 20 Marks)

I. Very Short Answer Type Questions.

  • Answer all the questions.
  • Each question carries two marks.

Question 1.
Find the domain of the real-valued function f(x) = \(\sqrt{x^2-25}\).
Solution:
Given f(x) = \(\sqrt{x^2-25}\)
f(x) ∈ R ⇔ x2 – 25 ≥ 0
⇔ (x + 5) (x – 5) ≥ 0
⇔ x ∈ (-∞, -5) ∪ (5, ∞)
∴ Domain of f = (-∞, -5) ∪ (5, ∞).

Question 2.
If f : R → R, g : R → R are defined by f(x) = 3x – 1, g(x) = x2 + 1 then find (fog) (2).
Solution:
Given f(x) = 3x – 1 and g(x) = x2 + 1
(fog) (2) = f[g(2)]
= f[22 + 1]
= f[4 + 1]
= f(5)
= 3(5) – 1
= 15 – 1
= 14
∴ (fog) (2) = 14

Question 3.
Define a symmetric matrix. Give one example of order 3 × 3.
Solution:
Symmetric Matrix: A square matrix ‘A’ is said to be a symmetric matrix if AT = A.
Example: A = \(\left[\begin{array}{ccc}
1 & 2 & 0 \\
2 & -3 & -1 \\
0 & -1 & 4
\end{array}\right]\)

AP Inter 1st Year Maths 1A Question Paper March 2018

Question 4.
Find the inverse of the matrix \(\left[\begin{array}{cc}
1 & 2 \\
3 & -5
\end{array}\right]\).
Solution:
AP Inter 1st Year Maths 1A Question Paper March 2018 Q4

Question 5.
If the vectors \(-3 \bar{i}+4 \bar{j}+\lambda \bar{k}\) and \(\mu \bar{i}+8 \overline{\mathrm{j}}+6 \overline{\mathrm{k}}\) are collinear then find λ and μ.
Solution:
AP Inter 1st Year Maths 1A Question Paper March 2018 Q5

Question 6.
Find the vector equation of the plane passing through the points (0, 0, 0), (0, 5, 0) and (2, 0, 1).
Solution:
Let A = (0, 0, 0)
B = (0, 5, 0)
C = (2, 0, 1)
The vector equation of the plane passing through the points A(\(\bar{a}\)), B(\(\bar{b}\)), C(\(\bar{c}\)) is
AP Inter 1st Year Maths 1A Question Paper March 2018 Q6

Question 7.
Find the angle between the vectors \(\bar{i}+2 \bar{j}+3 \bar{k}\) and \(3 \bar{i}-\bar{j}+2 \bar{k}\).
Solution:
AP Inter 1st Year Maths 1A Question Paper March 2018 Q7

Question 8.
Find the value of sin 330° cos 120° + cos 210° sin 300°.
Solution:
sin 330° . cos 120° + cos 210° . sin 300°
= sin (360° – 30°) . cos (180° – 60°) + cos(180° + 30°) . sin(360° – 60°)
= (-sin 30°) . (-cos 60°) + (-cos 30°) . (-sin 60°)
= \(\left(\frac{-1}{2}\right)\left(\frac{-1}{2}\right)+\left(\frac{-\sqrt{3}}{2}\right)\left(\frac{-\sqrt{3}}{2}\right)\)
= \(\frac{1}{4}+\frac{3}{4}\)
= 1
∴ sin 330° . cos 120° + cos 210° . sin 300° = 1

Question 9.
Find the extreme values of cos 2x + cos2x.
Solution:
Let f(x) = cos 2x + cos2x
= cos 2x + \(\frac{1+\cos 2 x}{2}\)
= cos 2x + \(\frac{1}{2}\) + \(\frac{1}{2}\) cos 2x
= \(\frac{1}{2}\) + (1 + \(\frac{1}{2}\)) cos 2x
= \(\frac{1}{2}\) + \(\frac{3}{2}\) cos 2x
We know -1 ≤ cos 2x ≤ 1
⇒ \(\frac{-3}{2} \leq \frac{3}{2} \cos 2 x \leq \frac{3}{2}\)
⇒ \(\frac{1}{2}-\frac{3}{2} \leq \frac{1}{2}+\frac{3}{2} \cos 2 x \leq \frac{1}{2}+\frac{3}{2}\)
⇒ -1 ≤ f(x) ≤ 2
∴ Minimum value = -1, Maximum value = 2

AP Inter 1st Year Maths 1A Question Paper March 2018

Question 10.
For any x ∈ R show that cosh 2x = 2 cosh2x – 1.
Solution:
L.H.S = cosh 2x
= cosh2x + sinh2x
= cosh2x + cosh2x – 1 {∵ cosh2x – sinh2x = 1}
= 2 cosh2x – 1
= R.H.S
∴ L.H.S = R.H.S
Hence cosh 2x = 2 cosh2x – 1

Section – B
(5 × 4 = 20 Marks)

II. Short Answer Type Questions.

  • Answer any five questions.
  • Each question carries four marks.

Question 11.
If A = \(\left[\begin{array}{rr}
7 & -2 \\
-1 & 2 \\
5 & 3
\end{array}\right]\) and B = \(\left[\begin{array}{rr}
-2 & -1 \\
4 & 2 \\
-1 & 0
\end{array}\right]\) then find AB’ and BA’.
Solution:
AP Inter 1st Year Maths 1A Question Paper March 2018 Q11
AP Inter 1st Year Maths 1A Question Paper March 2018 Q11.1

Question 12.
If \(\bar{a}, \bar{b}, \bar{c}\) are non-coplanar vectors, prove that the following four points are co-planar.
\(-\overline{\mathrm{a}}+4 \overline{\mathrm{b}}-3 \overline{\mathrm{c}}, 3 \overline{\mathrm{a}},+2 \overline{\mathrm{b}}-5 \overline{\mathrm{c}},-3 \overline{\mathrm{a}}+8 \overline{\mathrm{b}}-5 \overline{\mathrm{c}} \text { and }-3 \overline{\mathrm{a}}+2 \overline{\mathrm{b}}+\overline{\mathrm{c}}\)
Solution:
AP Inter 1st Year Maths 1A Question Paper March 2018 Q12
AP Inter 1st Year Maths 1A Question Paper March 2018 Q12.1
AP Inter 1st Year Maths 1A Question Paper March 2018 Q12.2

Question 13.
Let \(\bar{a}\) and \(\bar{b}\) be vectors, satisfying |\(\bar{a}\)| = |\(\bar{b}\)| = 5 and (\(\bar{a}\), \(\bar{b}\)) = 45°. Find the area of the triangle having \(\bar{a}-2 \bar{b}\) and \(3 \bar{a}+2 \bar{b}\) as two of its sides.
Solution:
Given |\(\bar{a}\)| = |\(\bar{b}\)| = 5 and (\(\bar{a}\), \(\bar{b}\)) = 45°
Let \(\overline{\mathrm{AB}}=\overline{\mathrm{a}}-2 \overline{\mathrm{b}}\)
AP Inter 1st Year Maths 1A Question Paper March 2018 Q13

Question 14.
If A is not an integral multiple of \(\frac{\pi}{2}\) then prove that
(i) tan A + cot A = 2 cosec 2A
(ii) cot A – tan A = 2 cot 2A
Solution:
AP Inter 1st Year Maths 1A Question Paper March 2018 Q14
AP Inter 1st Year Maths 1A Question Paper March 2018 Q14.1

AP Inter 1st Year Maths 1A Question Paper March 2018

Question 15.
Solve the equation √3 sin θ – cos θ = √2.
Solution:
AP Inter 1st Year Maths 1A Question Paper March 2018 Q15

Question 16.
Prove that \(\sin ^{-1}\left(\frac{4}{5}\right)+\sin ^{-1}\left(\frac{5}{13}\right)+\sin ^{-1}\left(\frac{16}{65}\right)=\frac{\pi}{2}\).
Solution:
Let \(\sin ^{-1}\left(\frac{4}{5}\right)\) = A and \(\sin ^{-1}\left(\frac{5}{13}\right)\) = B
Then sin A = \(\frac{4}{5}\) and sin B = \(\frac{5}{13}\)
AP Inter 1st Year Maths 1A Question Paper March 2018 Q16
AP Inter 1st Year Maths 1A Question Paper March 2018 Q16.1
AP Inter 1st Year Maths 1A Question Paper March 2018 Q16.2

Question 17.
If a = (b – c) sec θ, prove that tan θ = \(\frac{2 \sqrt{b c}}{b-c} \sin \frac{A}{2}\).
Solution:
Given a = (b – c) sec θ
⇒ sec θ = \(\frac{a}{b-c}\)
We know tan2θ = sec2θ – 1
AP Inter 1st Year Maths 1A Question Paper March 2018 Q17

Section – C
(5 × 7 = 35 Marks)

III. Long Answer Type Questions.

  • Attempt any five questions.
  • Each question carries seven marks.

Question 18.
Let f: A → B, g: B → C be bijections then prove that gof: A → C is a bijection.
Solution:
Given f: A → B, g: B → C be bijections.
(i) To prove gof: A → C, is one-one
Let a1, a2 ∈ a Then f(a1), f(a2) ∈ B
(gof)(a1) = (gof)(a2)
⇒ g[f(a1)] = g[f(a2)]
⇒ f(a1) = f(a2) {∵ g is one-one}
⇒ a1 = a2 {∵ f is one-one}
∴ gof: A → C is one-one

(ii) To prove gof: A → C is onto
Let c ∈ c
Since g: B → C is onto
∴ There exists b ∈ B such that g(b) = c
Since f: A → B is onto
∴ There exists a ∈ A such that f(a) = b
c = g(b) = g[f(a)] = (gof) (a)
∴ There exists a ∈ A such that (gof) (a) = c
∴ gof: A → C is onto
Hence gof: A → C is a bijective function.

AP Inter 1st Year Maths 1A Question Paper March 2018

Question 19.
Using mathematical induction, prove that \(\frac{1}{1.3}+\frac{1}{3.5}+\frac{1}{5.7}+\ldots .+\frac{1}{(2 n-1)(2 n+1)}=\frac{n}{2 n+1}\) for all n ∈ N.
Solution:
Let p(n) be the statement that
AP Inter 1st Year Maths 1A Question Paper March 2018 Q19
AP Inter 1st Year Maths 1A Question Paper March 2018 Q19.1
∴ p(k + 1) is true.
∴ By the principle of finite mathematical induction p(n) is true for all n ∈ N.
∴ \(\frac{1}{1.3}+\frac{1}{3.5}+\frac{1}{5.7}+\ldots .+\frac{1}{(2 n-1)(2 n+1)}=\frac{n}{2 n+1}\) for all n ∈ N.

Question 20.
Show that \(\left|\begin{array}{ccc}
a+b+2 c & a & b \\
c & b+c+2 a & b \\
c & a & c+a+2 b
\end{array}\right|\) = 2(a + b + c)3.
Solution:
AP Inter 1st Year Maths 1A Question Paper March 2018 Q20
= 2(a + b + c) (1) (b + c + a) (c + a + b)
= 2(a + b + c)3
= R.H.S.
∴ L.H.S = R.H.S
Hence \(\left|\begin{array}{ccc}
a+b+2 c & a & b \\
c & b+c+2 a & b \\
c & a & c+a+2 b
\end{array}\right|\) = 2(a + b + c)3

Question 21.
Solve the following system of equations by Gauss-Jordan Method:
2x – y + 3z = 9, x + y + z = 6 and x – y + z = 2
Solution:
Given that the system of equations are
2x – y + 3z = 9
x + y + z = 6
x – y + z = 2
The given system of equations can be expressed as AX = B
AP Inter 1st Year Maths 1A Question Paper March 2018 Q21
AP Inter 1st Year Maths 1A Question Paper March 2018 Q21.1
∴ x = 1, y = 2, z = 3

Question 22.
For any four vectors \(\bar{a}, \bar{b}, \bar{c}\) and \(\bar{d}\), prove that
(i) \((\overline{\mathrm{a}} \times \overline{\mathrm{b}}) \times(\overline{\mathrm{c}} \times \overline{\mathrm{d}})=[\overline{\mathrm{a}} \overline{\mathrm{c}} \overline{\mathrm{d}}] \overline{\mathrm{b}}-[\overline{\mathrm{b}} \overline{\mathrm{c}} \overline{\mathrm{d}}] \overline{\mathrm{a}}\)
(ii) \((\overline{\mathrm{a}} \times \overline{\mathrm{b}}) \times(\overline{\mathrm{c}} \times \overline{\mathrm{d}})=[\overline{\mathrm{a}} \overline{\mathrm{b}} \overline{\mathrm{d}}] \overline{\mathrm{c}}-[\overline{\mathrm{a}} \overline{\mathrm{b}} \overline{\mathrm{c}}] \overline{\mathrm{d}}\)
Solution:
AP Inter 1st Year Maths 1A Question Paper March 2018 Q22
AP Inter 1st Year Maths 1A Question Paper March 2018 Q22.1

Question 23.
If A, B, C are the angles in a triangle, then prove that cos A + cos B + cos C = 1 + 4\(\sin \left(\frac{A}{2}\right) \sin \left(\frac{B}{2}\right) \sin \left(\frac{C}{2}\right)\).
Solution:
Since A, B, C are the angles of a triangle.
∴ A + B + C = 180°
⇒ \(\frac{A}{2}+\frac{B}{2}+\frac{C}{2}\) = 90°
L.H.S. = cos A + cos B + cos C
AP Inter 1st Year Maths 1A Question Paper March 2018 Q23
∴ L.H.S = R.H.S
Hence cos A + cos B + cos C = 1 + 4\(\sin \left(\frac{A}{2}\right) \sin \left(\frac{B}{2}\right) \sin \left(\frac{C}{2}\right)\)

AP Inter 1st Year Maths 1A Question Paper March 2018

Question 24.
Show that in any triangle ABC, r + r3 + r1 – r2 = 4R cos B.
Solution:
AP Inter 1st Year Maths 1A Question Paper March 2018 Q24
AP Inter 1st Year Maths 1A Question Paper March 2018 Q24.1
AP Inter 1st Year Maths 1A Question Paper March 2018 Q24.2

AP TS Inter 1st Year Chemistry Model Papers 2024-2025 – TS AP Inter 1st Year Chemistry Previous Question Papers

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AP TS Inter 1st Year Chemistry Model Papers 2024-2025 with Answers

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AP Inter 1st Year Maths 1A Model Paper Set 3 with Solutions

Thoroughly analyzing AP Inter 1st Year Maths 1A Model Papers Set 3 helps students identify their strengths and weaknesses.

AP Inter 1st Year Maths 1A Model Paper Set 3 with Solutions

Time: 3 Hours
Maximum Marks: 75

Note: The Question Paper consists of three sections A, B, and C.

Section – A
(10 × 2 = 20 Marks)

I. Very Short Answer Questions.

  • Answer All questions.
  • Each Question carries Two marks.

Question 1.
Find the domain and range of the real-valued function f(x) = \(\frac{x^2-4}{x-2}\).

Question 2.
If f : R → R, g : R → R are defined by f(x) = 3x – 2 and g(x) = x2 + 1, then find (g o f)(x – 1).

Question 3.
If A = \(\left[\begin{array}{lll}
1 & 4 & 7 \\
2 & 5 & 8
\end{array}\right]\), B = \(\left[\begin{array}{ccc}
-3 & 4 & 0 \\
4 & -2 & -1
\end{array}\right]\), then show that (A + B)T = AT + BT.

AP Inter 1st Year Maths 1A Model Paper Set 3 with Solutions

Question 4.
If A = \(\left[\begin{array}{cc}
2 & 4 \\
-1 & k
\end{array}\right]\) and A2 = 0, then find the value of k.

Question 5.
Show that the vectors \(\bar{i}+\bar{j}, \bar{j}+\bar{k},-\bar{k}+\bar{i}\) are linearly dependent.

Question 6.
If the vectors \(\overline{\mathrm{a}}=2 \overline{\mathrm{i}}+5 \overline{\mathrm{j}}+\overline{\mathrm{k}}\) and \(\overline{\mathrm{b}}=4 \overline{\mathrm{i}}+m \overline{\mathrm{j}}+n \overline{\mathrm{k}}\) are collinear, then find the values of m and n.

Question 7.
Find the radius of the sphere whose equation is r2 = \(2 \bar{r} \cdot(4 \bar{i}-2 \bar{j}+2 \bar{k})\).

Question 8.
Find the period of the function f(x) = \(2 \sin \left(\frac{\pi x}{4}\right)+3 \cos \left(\frac{\pi x}{3}\right)\).

Question 9.
If tan θ = \(\frac{b}{a}\), then prove that a cos 2θ + b sin 2θ = a.

Question 10.
If tanh x = \(\frac{1}{4}\), then prove that x = \(\frac{1}{2} \cdot \log _e\left(\frac{5}{3}\right)\).

Section – B
(5 × 4 = 20 Marks)

II. Short Answer Questions.

  • Answer any Five questions.
  • Each Question carries Four marks.

Question 11.
If θ – φ = \(\frac{\pi}{2}\), show that \(\left[\begin{array}{cc}
\cos ^2 \theta & \cos \theta \sin \theta \\
\cos \theta \sin \theta & \sin ^2 \theta
\end{array}\right]\) \(\left[\begin{array}{cc}
\cos ^2 \phi & \cos \phi \sin \phi \\
\cos \phi \sin \phi & \sin ^2 \phi
\end{array}\right]\) = 0

AP Inter 1st Year Maths 1A Model Paper Set 3 with Solutions

Question 12.
Prove that the four points \(4 \bar{i}+5 \bar{j}+\bar{k},-(\bar{j}+\bar{k}), 3 \bar{i}+9 \bar{j}+4 \bar{k}\) and \(-4 \bar{i}+4 \bar{j}+4 \bar{k}\) are coplanar.

Question 13.
Prove that \(\sin ^4 \frac{\pi}{8}+\sin ^4 \frac{3 \pi}{8}+\sin ^4 \frac{5 \pi}{8}+\sin ^4 \frac{7 \pi}{8}=\frac{3}{2}\).

Question 14.
Solve cot2x – (√3 + 1) cot x + √3 = 0 in 0 < x < \(\frac{\pi}{2}\).

Question 15.
If \(\cos ^{-1}\left(\frac{p}{a}\right)+\cos ^{-1}\left(\frac{q}{b}\right)=\alpha\), then prove that \(\frac{p^2}{a^2}-2 \frac{p q}{a b} \cos \alpha\) + \(\frac{q^2}{b^2}=\sin ^2 \alpha\).

Question 16.
In ΔABC, If a = (b + c) cos θ, then prove that sin θ = \(\frac{2 \sqrt{b c}}{b+c} \cos \frac{A}{2}\).

Question 17.
Expand sin 5θ, cos 5θ in the powers of sin θ and cos θ.

Section – C
(5 × 7 = 35 Marks)

III. Long Answer Questions.

  • Answer any Five questions.
  • Each Question carries Seven marks.

Question 18.
If f : A → B and g : B → C is bijective, then prove that (gof)-1 = (f-1og-1).

Question 19.
Show that 3 . 52n+1 + 23n+1 is divisible by 17 for all n ∈ N, by using mathematical induction.

Question 20.
Show that \(\left|\begin{array}{lll}
b+c & c+a & a+b \\
c+a & a+b & b+c \\
a+b & b+c & c+a
\end{array}\right|=2\left|\begin{array}{lll}
a & b & c \\
b & c & a \\
c & a & b
\end{array}\right|\).

AP Inter 1st Year Maths 1A Model Paper Set 3 with Solutions

Question 21.
Solve the following equation by using the matrix inversion method.
3x + 4y + 52 = 18
2x – y – 82 = 13
5x – 2y + 72 = 20

Question 22.
\(\bar{a}\), \(\bar{b}\) and \(\bar{c}\) are non-zero and non-collinear vectors and θ ≠ 0, π is the angle between \(\bar{b}\) and \(\bar{c}\). If \((\bar{a} \times \bar{b}) \times \bar{c}=\frac{1}{3}|\bar{b}||\bar{c}| \bar{a}|\), then find sin θ.

Question 23.
Suppose (α – β) is not an odd multiple of \(\frac{\pi}{2}\), m is a non-zero real number such that m ≠ -1 and \(\frac{\sin (\alpha+\beta)}{\cos (\alpha-\beta)}=\frac{1-m}{1+m}\), then prove that \(\tan \left(\frac{\pi}{4}-\alpha\right)=m \tan \left(\frac{\pi}{4}+\beta\right)\).

Question 24.
Show that in a ΔABC, \(\frac{r_1}{b c}+\frac{r_2}{c a}+\frac{r_3}{a b}=\frac{1}{r}-\frac{1}{2 R}\).

AP Inter 1st Year Maths 1A Model Paper Set 2 with Solutions

Thoroughly analyzing AP Inter 1st Year Maths 1A Model Papers Set 2 helps students identify their strengths and weaknesses.

AP Inter 1st Year Maths 1A Model Paper Set 2 with Solutions

Time: 3 Hours
Maximum Marks: 75

Note: The Question Paper consists of three sections A, B, and C.

Section – A
(10 × 2 = 20 Marks)

I. Very Short Answer Questions.

  • Answer All questions.
  • Each Question carries Two marks.

Question 1.
Find the domain of f(x) = \(\sin \left(\log \left(\frac{\sqrt{4-x^2}}{1-x}\right)\right)\).

Question 2.
Find the inverse functions. If f: (0, ∞) → R defined by f(x) = log2(x).

Question 3.
If \(\left[\begin{array}{ccc}
0 & 1 & 4 \\
-1 & 0 & 7 \\
-x & -7 & 0
\end{array}\right]\) is a skew symmetric matrix, than find x.

Question 4.
Find the rank of the matrix \(\left[\begin{array}{lll}
1 & 1 & 1 \\
1 & 1 & 1 \\
1 & 1 & 1
\end{array}\right]\).

AP Inter 1st Year Maths 1A Model Paper Set 2 with Solutions

Question 5.
ABCDEF is a regular hexagon with centre ‘O’, show that \(\overline{\mathrm{AB}}+\overline{\mathrm{AC}}+\overline{\mathrm{AD}}+\overline{\mathrm{AE}}+\overline{\mathrm{AF}}=3(\overline{\mathrm{AD}})=6(\overline{\mathrm{AO}})\).

Question 6.
Find the vector equation of the plane passing through the points (1, -2, 5), (6, -5, -1) and (-3, 5, 0).

Question 7.
If \(\stackrel{\rightharpoonup}{a}+\stackrel{\rightharpoonup}{b}+\bar{c}=0,|\bar{a}|=3,|\bar{b}|=5\), and |\(\bar{c}\)| = 7, then find the angle between \(\bar{a}\) and \(\bar{b}\).

Question 8.
Find the extreme values of 5 cos x + 3 cos (x + \(\frac{\pi}{3}\)) + 8.

Question 9.
Sketch the region enclosed by y = sin x, y = cos x, and x-axis in the interval [0, π].

Question 10.
Prove that cosh (3x) = 4 cosh3x – 3 cosh x.

Section – B
(5 × 4 = 20 Marks)

II. Short Answer Questions.

  • Answer any Five questions.
  • Each question carries Four marks.

Question 11.
If A = \(\left[\begin{array}{cc}
\cos \theta & -\sin \theta \\
-\sin \theta & \cos \theta
\end{array}\right]\), then show that from all the positive Integers n, An = \(\left[\begin{array}{cc}
\cos n \theta & -\sin n \theta \\
-\sin n \theta & \cos n \theta
\end{array}\right]\).

Question 12.
Find the equation of the line parallel to the vector \(2 \overline{\mathrm{i}}-\overline{\mathrm{j}}+2 \overline{\mathrm{k}}\) and which passes through the point A whose position vector is \(3 \bar{i}+\bar{j}-\bar{k}\). If P is a point on this line such that AP = 15, find the position vector of P.

Question 13.
Let \(\bar{a}\) and \(\bar{b}\) be vector, satisfying \(|\bar{a}|=|\bar{b}|\) = 5 and \((\bar{a}, \bar{b})\) = 45°. Find the area of the triangle having \(\bar{a}-2 \bar{b}\) and \(3 \bar{a}+2 \bar{b}\) as two its sides.

AP Inter 1st Year Maths 1A Model Paper Set 2 with Solutions

Question 14.
Prove that \(\cos \frac{\pi}{11} \cdot \cos \frac{2 \pi}{11} \cdot \cos \frac{3 \pi}{11} \cdot \cos \frac{4 \pi}{11} \cdot \cos \frac{5 \pi}{11}=\frac{1}{32}\).

Question 15.
If α, β are the solutions of the equation a cos θ + b sin θ = c, where a, b, c ∈ R and if a2 + b2 > 0, cos α ≠ cos β and sin α ≠ sin β, then show that cos α + cos β = \(\frac{2 a c}{a^2+b^2}\), cos α . cos β = \(\frac{c^2-b^2}{a^2+b^2}\).

Question 16.
Solve the equation \({Tan}^{-1}\left(\frac{x-1}{x-2}\right)+{Tan}^{-1}\left(\frac{x+1}{x+2}\right)=\frac{\pi}{4}\).

Question 17.
In ΔABC, If a = 5, b = 4 and cos (A – B) = \(\frac{31}{32}\), then show that c = 6.

Section – C
(5 × 7 = 35 Marks)

III. Long Answer Questions.

  • Answer any Five questions.
  • Each Question carries Seven marks.

Question 18.
Determine whether the function f : R → R defined by \(\left\{\begin{array}{cc}
x, & \text { if } x>2 \\
5 x-2, & \text { if } x \leq 2
\end{array}\right.\) is an infection a surjection or a bijection.

Question 19.
Using mathematical induction, prove that 12 + (12 + 22) + (12 + 22 + 32) + ….. upto n terms = \(\frac{n(n+1)^2(n+2)}{12}\), ∀ n ∈ N.

Question 20.
Show that \(\left[\begin{array}{ccc}
1 & a^2 & a^3 \\
1 & b^2 & b^3 \\
1 & c^2 & c^3
\end{array}\right]\) = (a – b) (b – c) (c – a) (ab + bc + ca).

Question 21.
Examine whether the following system of equations is consistent or inconsistent and if consistent find the complete solutions.
x + y + 2 = 6, x – y + 2 = 2, 2x – y + 32 = 9

Question 22.
If \(\overline{\mathrm{a}}=\overline{\mathrm{i}}-2 \overline{\mathrm{j}}-3 \overline{\mathrm{k}}, \quad \overline{\mathrm{b}}=2 \overline{\mathrm{i}}+\overline{\mathrm{j}}+\overline{\mathrm{k}}\) and \(\bar{c}=\bar{i}+\bar{j}+2 \bar{k}\), then find \(|(\bar{a} \times \bar{b}) \times \bar{c}|\) and \(|\overline{\mathrm{a}} \times(\overline{\mathrm{b}} \times \overline{\mathrm{c}})|\).

AP Inter 1st Year Maths 1A Model Paper Set 2 with Solutions

Question 23.
If A + B + C + D = 360°, prove that cos 2A + cos 2B + cos 2C + cos 2D = 4 cos (A + B) . cos (A + C) . cos (A + D).

Question 24.
In ΔABC, Prove that \(r_1^2+r_2^2+r_3^2+r^2\) = 16R2 – (a2 + b2 + c2).

AP Inter 1st Year Maths 1A Model Paper Set 1 with Solutions

Thoroughly analyzing AP Inter 1st Year Maths 1A Model Papers Set 1 helps students identify their strengths and weaknesses.

AP Inter 1st Year Maths 1A Model Paper Set 1 with Solutions

Time: 3 Hours
Maximum Marks: 75

Note: The Question Paper consists of three sections A, B, and C.

Section – A
(10 × 2 = 20 Marks)

I. Very Short Answer Questions.

  • Answer All Questions.
  • Each Question Carries Two marks.

Question 1.
Find the range of the real-valued function \(\sqrt{[x]-x}\).

Question 2.
Let f = {(1, a), (2, c), (4, d), (3, b)} and g-1{(2, a), (4, b), (1, c), (3, d)} then show that (gof)-1 = f-1og-1.

Question 3.
Define the trace of a matrix.

AP Inter 1st Year Maths 1A Model Paper Set 1 with Solutions

Question 4.
If A = \(\left[\begin{array}{cc}
-2 & 1 \\
5 & 0 \\
-1 & 4
\end{array}\right]\) and B = \(\left[\begin{array}{ccc}
-2 & 3 & 1 \\
4 & 0 & 2
\end{array}\right]\) then find 2A + BT and 3BT – A.

Question 5.
ABCDE is a pentagon. If the sum of the vectors \(\overline{\mathrm{AB}}, \overline{\mathrm{AE}}, \overline{\mathrm{BC}}, \overline{\mathrm{DC}}, \overline{\mathrm{ED}}\) and \(\bar{AC}\) is λ(\(\bar{AC}\)), then find the value of λ.

Question 6.
\(\bar{a}, \bar{b}, \bar{c}\) are pair wise non-zero and non collinear vectors. If \(\bar{a}+\bar{b}\) is collinear with \(\bar{c}\) and \(\bar{b}+\bar{c}\) is collinear with \(\bar{a}\), then find the vector \(\bar{a}+\bar{b}+\bar{c}\).

Question 7.
Find the angle between the planes \(\overline{\mathrm{r}} \cdot(2 \overline{\mathrm{i}}-\overline{\mathrm{j}}+2 \overline{\mathrm{k}})=3\) and \(\bar{r} \cdot(3 \bar{i}+6 \bar{j}+\bar{k})=4\).

Question 8.
Prove that tan 50° – tan 40° = 2 tan 10°.

Question 9.
Find the value of cot \(67 \frac{1}{2}^{\circ}\).

Question 10.
For any n ∈ R, prove that (cosh x – sinh x)n = cosh (nx) – sinh (nx).

Section – B
(5 × 4 = 20 Marks)

II. Short Answer Questions.

  • Answer any Five questions.
  • Each Question carries Four marks.

Question 11.
If I = \(\left[\begin{array}{ll}
1 & 0 \\
0 & 1
\end{array}\right]\) and E = \(\left[\begin{array}{ll}
0 & 1 \\
0 & 0
\end{array}\right]\) then show that (aI + bE)3 = a3I + 3a2bE.

AP Inter 1st Year Maths 1A Model Paper Set 1 with Solutions

Question 12.
The median AD of ∆ABC is bisected at E and BE is produced to meet the side AC in F. Show that \(\overline{\mathrm{AF}}=\frac{1}{3}(\overline{\mathrm{AC}})\) and \(\overline{\mathrm{EF}}=\frac{1}{4} \overline{\mathrm{BF}}\).

Question 13.
Let \(\overline{\mathrm{a}}=\overline{\mathrm{i}}+\overline{\mathrm{j}}+\overline{\mathrm{k}}\) and \(\overline{\mathrm{b}}=2 \overline{\mathrm{i}}+3 \overline{\mathrm{j}}+\overline{\mathrm{k}}\). Find
(i) The projection vector of \(\bar{b}\) on \(\bar{a}\) and its magnitude.
(ii) The component vector of \(\bar{b}\) in the direction of a and perpendicular to \(\bar{a}\).

Question 14.
If 8α is not an integral multiple of π, then prove that tan α + 2 tan 2α + 4 tan 4α + 8 cot 8α = cot α.

Question 15.
If \(\tan \left(\frac{\pi}{2} \sin \theta\right)=\cot \left(\frac{\pi}{2} \cos \theta\right)\), then prove that \(\sin \left(\theta+\frac{\pi}{4}\right)= \pm \frac{1}{\sqrt{2}}\).

Question 16.
Prove that \(\tan \left(\frac{\pi}{4}+\frac{1}{2} \cos ^{-1} \frac{a}{b}\right)+\tan \left(\frac{\pi}{4}-\frac{1}{2} \cos ^{-1} \frac{a}{b}\right)=\frac{2 b}{a}\).

Question 17.
In ∆ABC, prove that \(\left[\frac{b-c}{b+c}\right] \cot \left(\frac{A}{2}\right)+\frac{b+c}{b-c} \tan \left(\frac{A}{2}\right)\) = 2 cosec (B – C).

Section – C
(5 × 7 = 35 Marks)

III. Long Answer Questions.

  • Answer any Five questions.
  • Each question carries Seven marks.

Question 18.
If f = {(4, 5), (5, 6), (6, -4)} and g = {(4, -4), (6, 5), (8, 5)} then find
(i) f – g
(ii) fg
(iii) \(\frac{f}{g}\)
(iv) |f|
(v) 2f + 4g

Question 19.
Using mathematical induction, for all n ∈ N, prove that 2.3 + 3.4 + 4.5 + …… up to n terms = \(\frac{n\left(n^2+6 n+11\right)}{3}\).

Question 20.
Show that \(\left|\begin{array}{lll}
a & b & c \\
b & c & a \\
c & a & b
\end{array}\right|^2=\left|\begin{array}{ccc}
2 b c-a^2 & c^2 & b^2 \\
c^2 & 2 a c-b^2 & a^2 \\
b^2 & a^2 & 2 a b-c^2
\end{array}\right|\)

Question 21.
Solve the following system of equations by using the Gauss-Jordan method.
2x + 4y – z = 0, x + 2y + 2z = 5, 3x + 6y – 7z = 2.

Question 22.
Let \(\overline{\mathrm{OA}}=\overline{\mathrm{a}}, \overline{\mathrm{OB}}=10, \overline{\mathrm{a}}+2 \overline{\mathrm{b}}\) and \(\overline{\mathrm{OC}}=\overline{\mathrm{b}}\) where O, A, B and C are non-collinear points. Let λ denote the area of the quadrilateral OABC and Let µ denote the area of the parallelogram with \(\bar{OA}\) and \(\bar{OC}\) as adjacent sides. Prove that λ = 6µ.

AP Inter 1st Year Maths 1A Model Paper Set 1 with Solutions

Question 23.
In ΔABC, prove that \(\cos \frac{A}{2}+\cos \frac{B}{2}+\cos \frac{C}{2}=4 \cos \left(\frac{\pi-A}{4}\right) \cos \left(\frac{\pi-B}{4}\right)\) \(\cos \left(\frac{\pi-C}{4}\right)\).

Question 24.
If r1 = 36, r2 = 18 and r3 = 12, then prove that a = 30, b = 24, c = 18 and R = 15.

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TS Inter 2nd Year Maths 2B Question Paper March 2018

Access to a variety of AP Inter 2nd Year Maths 2B Model Papers and TS Inter 2nd Year Maths 2B Question Paper March 2018 allows students to familiarize themselves with different question patterns.

TS Inter 2nd Year Maths 2B Question Paper March 2018

Time : 3 Hours
Max. Marks : 75

Section – A
(10 × 2 = 20)

I. Very Short Answer type questions.

  1. Attempt all questions.
  2. Each question carries two marks.

Question 1.
Find the equation of the circle whose centre is (-1, 2) and which passes through (5, 6).
Solution:
Let C = (-1, 2) and P = (5, 6)
r = CP = \(\sqrt{(5+1)^2+(6-2)^2}\)
= \(\sqrt{36+16}\)
= \(\sqrt{52}\)
∴ The equation of the circle whose centre is (-1, 2) and which passes through (5, 6) is
(x – h)2 + (y – k)2 = r2
⇒ (x + 1)2 + (y – 2)2 = \((\sqrt{52})^2\)
⇒ x2 + 2x + 1 + y2 – 4y + 4 = 52
⇒ x2 + y2 + 2x – 4y – 47 = 0

Question 2.
If the length of the tangent from (2, 5) to the circle x2 + y2 – 5x + 4y + k = 0 is \(\sqrt{37}\), then find k.
Solution:
Let
S ≡ x2 + y2 – 5x + 4y + k.
Length or the tangent = \(\sqrt{511}\)
⇒ \(\sqrt{37}\) = \(\sqrt{(2)^2+(5)^2-5(2)+4(5)+k}\)
⇒ \(\sqrt{37}\) = \(\sqrt{4+25-10+20+k}\)
⇒ 37 = k + 39
⇒ k = + 37 – 39
⇒ k = -2

TS Inter 2nd Year Maths 2B Question Paper March 2018

Question 3.
If the angle between the circles x2 + y2 – 12x – 6y + 41 = 0 and x2 + y2 + kx + 6y – 59 = 0 is 45°, find k.
Solution:
Given circle equations are
x2 + y2 – 12x – 6y + 41 = 0 ………… (1)
x2 + y2 + kx + 6y – 59 = 0 ………. (2)
Here 2g = -12 ⇒ g= -6
2f = -6 ⇒ f = -3 and c = 41
2g1 = k ⇒ g = -6
2f = -6 ⇒ f = -3 and c1 = -59.
TS Inter 2nd Year Maths 2B Question Paper March 2018 1

Question 4.
Find the equation of the parabola whose vertex is (3, -2) and focus is (3, 1).
Solution:
Given vertex = (3, -2)
Focus = (3, 1)
Here the abcissae of the vertex and focus are equal to 3.
∴ The axis of the parabola is x = 3, a line parallel to y – axis and focus is above the vertex.
a = distance between vertex and focus
= \(\sqrt{(3-3)^2+(1+2)^2}\)
= 3
Hence equation of the parabola is
(x – h)2 = 4a (y k)
⇒ (x – 3)2 = 4.3 (y + 2)
⇒ (x – 3)2 = 12(y + 2).

Question 5.
If 3x – 4y + k = 0 is a tangent to x2 – 4y2 = 5, find the value of k.
Solution:
Given line equation is 3x – 4y + k = 0
⇒ 4y = 3x + k
y = \(\frac{3}{4}\)x + \(\frac{k}{4}\) ……… (1)
Hence m = \(\frac{3}{4}\), c = \(\frac{k}{4}\)
Given Hyperbola equation is x2 – 4y2 = 5
⇒ \(\frac{x^2}{5}\) – \(\frac{y^2}{5 / 4}\) = 1
Here a2 = 5, b2 = \(\frac{5}{4}\)
If (1) is a tangent to the Hyperbola x2 – 4y2 = 5 then
c2 = a2m2 – b2
⇒ \(\frac{k^2}{16}\) = 5.\(\frac{9}{16}\) – \(\frac{5}{4}\)
⇒ \(\frac{k^2}{16}\) = \(\frac{45-20}{16}\)
⇒ k2 = 25
⇒ k = ±5

Question 6.
Evaluate : \(\int \frac{e x}{e^x+1}\) dx
Solution:
I = \(\int \frac{e x}{e^x+1}\)dx
Let ex + 1 = t
ex dx = dt
= \(\int \frac{1}{t}\)dt
= log |t| + c
= log |ex + 1| + c
∴ \(\int \frac{e x}{e^x+1}\)dx = log|ex + 1| + c

Question 7.
Evaluate : ∫cos3x sin x dx
Solution:
I = ∫cos3 x sinx dx
Let cosx = t
-sinx dx = dt
sinx dx = – dt
= ∫t3(-dt)
TS Inter 2nd Year Maths 2B Question Paper March 2018 2

Question 8.
Evaluate : \(\int_0^2\)|1 – x| dx
Solution:
TS Inter 2nd Year Maths 2B Question Paper March 2018 3

Question 9.
Evaluate : \(\int_0^{\pi / 2}\)xsin x dx
Solution:
TS Inter 2nd Year Maths 2B Question Paper March 2018 4

Question 10.
Find the general solution of \(\frac{d y}{d x}\) = ex+y
Solution:
Given differential equation is
\(\frac{d y}{d x}\) = ex+y …….. (1)
⇒ \(\frac{\mathrm{dy}}{\mathrm{dx}}\) = ex . ey
⇒ \(\frac{d y}{e^y}\) = ex dx
⇒ exdx – e-ydy = 0
Integrating
∫exdx – ∫e-ydy = c
⇒ ex – \(\frac{e^{-y}}{-1}\) = c
⇒ ex + e-y = c
∴ The general solution of (1) is
ex + e-y = c

Section – B

II. Short Answer Type questions.

  1. Attempt any five questions.
  2. Each question carries four marks.

Question 11.
Find the area of the triangle formed by the normal at (3, -4) to the circle
x2 + y2 – 22x – 4y + 25 = 0 with the co-ordinate axes.
Solution:
Given circle equation is
s ≡ x2 + y2 – 22x – 4y + 25 = 0
Here 2g = – 22 ⇒ g = – 11
2f = – 4 ⇒ f = – 2
(x1, y1 = (3, -2)
The equation of the normal at (3, -2) or
the circle s = 0 is (x – x1) (y1 + f) – (y – y1) (x1 + g) = 0
⇒ (x – 3) (-4 – 2) – (y + 4) (3 – 11) = 0
⇒ (x – 3) (- 6) – (y + 4) (- 8) = 0
– 6x + 18 + 8y + 32 = 0
⇒ 6x – 8y – 50 = 0
⇒ 3x – 4y – 25 = 0
The area of the triangle formed by the normal at (3, -2) to the circle s = 0 with co-ordinate axis is \(\frac{1}{2} \cdot \frac{c^2}{|a b|}\) sq. units
= \(\frac{1}{2} \frac{(-25)^2}{|3.4|}\)
= \(\frac{625}{24}\) sq.units.

TS Inter 2nd Year Maths 2B Question Paper March 2018

Question 12.
Find the equation and length of the common chord of the two circles:
x2 + y2 + 3x + 5y + 4 = 0 and
x2 + y2 + 5x + 3y + 4 = 0.
Solution:
Let s ≡ x2 + y2 + 3x + 5y + 4 = 0
s1 ≡ x2 + y2 + 5x + 3y + 4 = 0
The radical axis of the circles s = 0, s1 = 0 is s – s1 = 0
⇒ – 2x + 2y = 0
⇒ x – y = 0 ………….. (1)
TS Inter 2nd Year Maths 2B Question Paper March 2018 5
d = The perpendicular distance from c to the line (1)
= \(\frac{\left|\frac{-3}{2}+\frac{5}{2}\right|}{\sqrt{1+1}}\)
= \(\frac{1}{\sqrt{2}}\)
The length of the common chord of the two circles s = 0, s1 = 0 is 2\(\sqrt{r^2-d^2}\)
= 2\(\sqrt{\frac{18}{4}-\frac{1}{2}}\)
= 2\(\sqrt{\frac{18-2}{4}}\)
= 2.2
= 4 units.

Question 13.
Find the equation of the ellipse referred to its major and minor axes as the co-ordinate axes X, Y – respectively with latus rectum of length 4, and distance between foci 4\(\sqrt{2}\)
Solution:
Let \(\frac{x^2}{a^2}\) + \(\frac{y^2}{b^2}\) = 1 be the required ellipse equation.
Given length of the latus rectum = 4
⇒ \(\frac{2 b^2}{a}\) = 4
⇒ 2b2 = 4a
⇒ b2 = 2a
Given distance between foci is 4\(\sqrt{2}\)
⇒ 2ae = 4\(\sqrt{2}\)
⇒ ae = 2\(\sqrt{2}\)
we know b2 = a2(1 – e2)
⇒ 2a = a2 – a2e2
⇒ 2a = a2 – (2\(\sqrt{2}\))2
⇒ 2a = a2 – 8
⇒ a2 – 2a – 8 = 0
⇒ a2 – 4a + 2a – 8 = 0
⇒ a(a – 4) + 2(a – 4) = 0
⇒ (a – 4) (a + 2) = 0
sine a > 0
∴ a = 4
b2 = 2a = 2.4 = 8
Required ellipse equation is
\(\frac{x^2}{a^2}\) + \(\frac{y^2}{b^2}\) = 1
⇒ \(\frac{x^2}{16}\) + \(\frac{y^2}{8}\) = 1
⇒ x2 + 2y2 = 16

Question 14.
Find the eccentricity, length of latus rectum, foci and the equations of directrices of the ellipse :
9x2 + 16y2 – 36x + 32y – 92 = 0.
Solution:
Given ellipse equation is
9x2 + 16y2 – 36x + 32y – 92 = 0
⇒ 9(x2 – 4x + 4) + 16 (y2 + 2y + 1) = 92 + 36 + 16
⇒ 9 (x – 2)2 + 16 (y + 1 )2 = 144
⇒ \(\frac{(x-2)^2}{16}\) + \(\frac{(y+1)^2}{9}\) = 1
Here α = 2, β = -1
Also a2 = 16, b2 = 9
TS Inter 2nd Year Maths 2B Question Paper March 2018 6
TS Inter 2nd Year Maths 2B Question Paper March 2018 7

Question 15.
Show that angle between the two asymptotes of a hyperbola
\(\frac{x^2}{a^2}-\frac{y^2}{b^2}\) = 1 is 2 tan-1 (b/a) (or) 2 sec-1(e).
Solution:
Given Hyperbola equation is \(\frac{x^2}{a^2}\) – \(\frac{y^2}{b^2}\) = 1
Equations of the asymptotes are
\(\frac{x}{a}\) + \(\frac{y}{b}\) = 0 and \(\frac{x}{a}\) – \(\frac{y}{b}\) = 0
If 2θ is the angle between the asymptotes
then tanθ = \(\frac{b}{a}\) = slope of the asymptotes
∴ θ = tan-1\(\left(\frac{b}{a}\right)\)
∴ Angle between the asymptotes = 2θ
= 2tan-1\(\left(\frac{b}{a}\right)\)
we know that sec2θ = 1 + tan2θ
= 1 + \(\frac{b^2}{a^2}\)
= \(\frac{a^2+b^2}{a^2}\)
= e2
⇒ secθ = e
⇒ θ = sec-1(e).
∴ Angle between the asymptotes = 2 tan-1\(\left(\frac{b}{a}\right)\) or sec-1(e).

Question 16.
Find the area bounded between the curves y = x2, y = \(\sqrt{x}\).
Solution:
Given curve equations are y = x2 ………. (1)
y = \(\sqrt{x}\) ……. (2)
From (1) & (2)
TS Inter 2nd Year Maths 2B Question Paper March 2018 8

Question 17.
Solve : \(\frac{d y}{d x}\) + 1 = ex+y.
Solution:
Given differential equation is
\(\frac{d y}{d x}\) + 1 = ex+y …… (1)
Let x + y = t
then 1 + \(\frac{d y}{d x}\) = \(\frac{d t}{d x}\)
From (1)
\(\frac{\mathrm{dt}}{\mathrm{dx}}\) = et
⇒ \(\frac{d t}{e^t}\) = dx
⇒ dx – e-tdt = 0
Integrating
∫dx – ∫e-t dt = c
⇒ x – \(\frac{\mathrm{e}^{-\mathrm{t}}}{-1}\) = c
⇒ x + e-t = c
⇒ x + e-(x+y) = c
∴ The general solution of (1) is
x + e-(x+y) = c.

Section – C

II. Long Answer Type questions.

  1. Attempt any five questions.
  2. Each question carries seven marks.

Question 18.
Find the equation of a circle which passes through (4, 1), (6, 5) and having the centre on :
4x + 3y – 24 = 0.
Solution:
Let the equation of the required circle
x2 + y2 + 2gx + 2fy + c = 0 …… (1)
If (1) Passes through (4, 1) then
16 + 1 + 8g + 2f + c = 0
8g + 2f + c = -17 …… (2)
If (1) passes through (6, 5) then
36 + 25 + 12g + 10f + c = 0
12g + 10f + c = -61 ……… (3)
Since centre (-g, -f) lies on 4x + 3y – 24 = 0
⇒ 4 (-g) + 3 (-f) – 24 = 0
⇒ – 4g – 3f – 24 = 0
⇒ 4g + 3f + 24 = 0 …….. (4)
(3) – (2) ⇒ 4g + 8f = – 44
⇒ -4g + 8f + 44 = 0 …….. (5)
(5) – (4) ⇒ 5f + 20 = 0
⇒ f + 4 = 0
⇒ f = -4
from (5)
4g + 8(-4) + 44 = 0
⇒ 4g + 12 = 0
⇒ g + 3 = 0
⇒ g = -3
from (2)
8(-3) + 2(-4) + c = – 17
-24 – 8 + c = -17
⇒ c = 15
∴ Required circle equation is
x2 + y2 + 2(-3) x + 2(-4)y + 15 = 0
⇒ x2 + y2 – 6x – 8y + 15 = 0

Question 19.
Show that the circles:
x2 + y2 – 6x – 9y + 13 = 0.
x2 + y2 – 2x – 16y = 0
touch each other. Find the point of contact and the equation of common tangent at their point of contact.
Solution:
Given circle equations are
s ≡ x2 + y2 – 6x – 9y + 13 = 0
s1 ≡ x2 + y2 – 2x – 16y = 0
centres A = (3, \(\frac{9}{2}\)) and B = (1, 8)
TS Inter 2nd Year Maths 2B Question Paper March 2018 9
AB = |r1 – r2|
∴ The circles touch each other internally
The point of contact p divides AB externally in the ratio
TS Inter 2nd Year Maths 2B Question Paper March 2018 24
Equation of the common tangent is s – s1 = 0
⇒ -4x + 7y + 13 = 0
⇒ 4x – 7y – 13 = 0

Question 20.
Derive the equation of parabola in the standard form, that is y2 = 4ax.
Solution:
To study the nature of the curve, we prefer its equation in the simplest possible form we proceed us follows to derive such an equation.
Let S be the focus, I be the directrix as shown in fig. Let Z be the projection of ‘S’ on I and ‘A’ be the midpoint of SZ. A lies on the parabola because SA = AZ. A is called the vertex of the parabola. Let YAY be the straight line through A and parallel to the directrix. Now take ZX as the -axis and YY as the Y-axis.
Then A is the origin (0, 0). Let S = (a, 0), (a > 0). Then Z – (-a, 0) and the equation of the directrix is x + a = 0.
TS Inter 2nd Year Maths 2B Question Paper March 2018 11
If P(x, y) is a point on the parabola and PM is the perpendicular distance from P to the directrix l, then \(\frac{S P}{P M}\) = e = 1.
∴ (SP)2 = (PM)2
⇒ (x – a)2 + y2 = (x + a)2
∴ y2 = 4ax.
Conversely if P (x, y) is any point such that y2 = 4ax then
SP = \(\sqrt{(x-a)^2+y^2}\) = \(\sqrt{x^2+a^2-2 a x+4 a x}\)
= \(\sqrt{(x+a)^2}\) = |x + a | = PM.
Hence P (x, y) is on the locus. In other words, a necessary and sufficient condition for the point P(x, y) to be on the parabola is that y2 = 4ax.
Thus the equation of the parabola is y2 = 4ax.

Question 21.
Evaluate: \(\int_a^b \sqrt{(x-a)(b-x)}\)dx
Solution:
TS Inter 2nd Year Maths 2B Question Paper March 2018 13
TS Inter 2nd Year Maths 2B Question Paper March 2018 14

TS Inter 2nd Year Maths 2B Question Paper March 2018

Question 22.
Evaluate : \(\int \frac{d x}{(x+1) \sqrt{2 x^2+3 x+1}}\)
Solution:
TS Inter 2nd Year Maths 2B Question Paper March 2018 15
TS Inter 2nd Year Maths 2B Question Paper March 2018 16
TS Inter 2nd Year Maths 2B Question Paper March 2018 17

Question 23.
Evaluate : \(\int \frac{d x}{4 \cos x+3 \sin x}\)
Solution:
TS Inter 2nd Year Maths 2B Question Paper March 2018 18
TS Inter 2nd Year Maths 2B Question Paper March 2018 19
TS Inter 2nd Year Maths 2B Question Paper March 2018 20
TS Inter 2nd Year Maths 2B Question Paper March 2018 21

Question 24.
Solve : (1 + y2)dx = (tan-1y – x) dy.
Solution:
Given differential equation is
TS Inter 2nd Year Maths 2B Question Paper March 2018 22
The general solution of (1) ¡s
TS Inter 2nd Year Maths 2B Question Paper March 2018 23

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AP Inter 1st Year Zoology Notes Chapter 8 Ecology and Environment

AP Inter 1st Year Zoology Notes Chapter 8 Ecology and Environment

Students can go through AP Inter 1st Year Zoology Notes 8th Lesson Ecology and Environment will help students in revising the entire concepts quickly.

AP Inter 1st Year Zoology Notes 8th Lesson Ecology and Environment

→ Ecology is a subject which deals with the study of the interactions among organisms and between the organisms and their physical environment.

→ The term ‘ecology’ was introduced by ‘Ernst Haeckel’

→ Ecology has two main branches Autecology, Synecology

→ A biome is a large community of plants and animals that occupies a vast region.

→ All the habitate zones on the earth constitue that ecosphere or bisophere

→ Environment is the sum total of biotic and abiotic factors present around the organisms influencing them in various ways.

→ With in a community, each organisms occupies a particular biological role or niche.

→ The duration of the light hours/exposure to the light in a day is known as photoperiod.

→ When water reaches 4°C, it becomes more dense and heavy.

AP Inter 1st Year Zoology Notes Chapter 8 Ecology and Environment

→ Biological control methods adapted in agricultural pest control are based on the ability of the predators to regulate prey population.

→ The plant Calotropis produces highly poisonous Cardia glycosides.

→ An ecosystem is a functional unit of nature.

→ When the path of food energy is linear, the components resemble the links of a chain, and it is called food chain’. The transfer of energy through a food chain is known as energy flow.

→ The differences in the temperature from thermal layers in water are called ‘Thermal stratification’.

→ The zone of water where there is rapid decrease in temperature is called Thermocline.

→ Increase in the concentration of the pollutant or toxicant of successive tropic levels is an aquatic food chain is called Bio-magnification.

AP Inter 1st Year Zoology Notes Chapter 8 Ecology and Environment

→ In an ecosystem the main trophic levels are

  1. producers,
  2. consumers,
  3. decomposers.

→ Each has 3 main divisions :

  1. Atmosphere,
  2. Hydrosphere,
  3. Lithiosphere.

→ Atmosphere has 4 main layers.

  1. Trophosphere,
  2. Stratosphere,
  3. Ionosphere,
  4. Exosphere.

→ Three zones of a fresh water lakes are

  1. Litoral zone,
  2. Limnotic zone,
  3. Profundal zone.

→ Autecology : Ecology of individual species.

→ Basking : Exposing the body to sun light, to gain temperature.

→ Benthos : It refers to all the attached, creeping or burrowing organisms that inhabit the bottom of rivers, lakes and sea.

→ Biomass : The total mass of living material within a specified area at a given time.

→ Blubber : It is a specialized subcutaneous layer of fat found only in marine mammals. It is almost continuous across the body of marine mammals but absent on appendages.

→ Brakish water : An intermediate zone between freshwater and marine water.

→ Camouflage : Concealing coloration (e.g., melanism) and morphology (e.g. stick insects) as defence against predation.

AP Inter 1st Year Zoology Notes Chapter 8 Ecology and Environment

→ Chemoautotrophs : These comprise bacteria that obtain energy from the oxidation of simple inorganic compounds and can use the energy released to assimilate CO2 and transfer the energy into organic compounds. E.g. : Thiobacilli species.

→ Climate : The climate of an area can be described by its mean values of temperature, rainfall, wind speed.

→ Community : The total living biotic component of an ecosystem, including plants, animals and microbes.

→ Competitive exclusion : It is often known as Gause principle.

→ Cyclomorphosis : Cyclic change in phenotype, such as seasonal changes in morphology, particularly conspicuous among cladoceran, crustacean and rotifers.

→ Detritus : Non living organic matter. Usually refers to particulate matter to that of plant rather than animal origin. E.g. : Leaf litter.

→ Diapause : It is a condition of arrested growth or reproductive development common in many organisms, particularly insects, during unfavourable conditions.

→ Dimictic Lake : The fake that undergoes two periods of complete vertical mixing, usually in the spring and the fall. During.the summer dimictic lakes are thermally stratified.

→ Edaphic factors : The physical, chemical and biotic characteristics of the soil that influence plant growth and distribution.

→ Estuary : It is a place where river joins the sea. The water in an estuary is subjected to seasonal variations in salinity. The water is called brackish water. The animals living there are euryhaline.

→ Gemmules : These are the internal buds that appear in the asexual reproduction of sponges. Gemmules are made up of amoebocytes and covered by a layer of spicules and can survive in unfavourable conditions.

AP Inter 1st Year Zoology Notes Chapter 8 Ecology and Environment

→ Leaching : Removal of soluble components by flowing water from soil.

→ Mycorrhizae : The symbiotic association of mycelium of a fungus with the roots of a seed plant.

→ Osmotrophic nutrition : Intake of pre-digested food material through the body surface.

→ Pedonic forms : The organisms which depend on substratum (or support) in an aquatic ecosystem.

→ Periphyton : The communities of tiny organisms like protozoa, insect larvae, snails that live on the surfaces of aquatic plants.

→ Savannas : Grassland region with scattered trees in subtropical and tropical regions.

→ Standing crop : The mass of vegetation in a given area at one particular time. Although most of often applied to plant material, the term includes animal biomass.

→ Denitrificaion : Denitrification is a microbially facilitated process of nitrate reduction that may ultimately produce molecular nitrogen (N2) through a series of intermediate gaseous nitrogen oxide products.

→ Decomposer : Decomposers (or saprotrophs) are organisms that break down dead or decaying organisms, and in doing so carry out the natural process of decomposition.

→ Sedimentary : Formed by the accumulation and consolidation of mineral and organic fragments that have been deposited by water, ice, or wind.

→ Asymptote : A line which approaches nearer to some curve than assignable distance, but, though infinitely extended, would never meet it.

→ Chemo-autotroph : An organism (typically a bacterium or a protozoan) that obtains energy through chemical process, which is by the oxidation of electron donating molecules from the environment, rather than by photosynthesis.

AP Inter 1st Year Zoology Notes Chapter 8 Ecology and Environment

→ Mortality : Death rate or the number of individuals that died in a population in a unit time.

→ Natality : Birth rate or the number of individuals produced in a population in unit time.

→ Acid rains : Acid rain is a rain or any other form of precipitation that is unusually acidic.

→ Algal bloom : An algal bloom is a rapid increase or accumulation in the population of algae (typically microscopic) in an aquatic system.

→ Biodegradable : Capable of being broken down especially into harmless products by the action of living things (as microorganisms).

→ Biological Oxygen Demand (BOD) : The amount of dissolved oxygen needed by aerobic biological organisms in a body of water to break down organic material present in a given water sample at certain temperature over a specific time period.

→ Chemical Oxygen Demand (COD): A test procedure, based on the chemical decomposition of organic and inorganic contaminants dissolved or suspended in water.

→ Chloro Fluoro Carbons (CFC) : Any of various halocarbon compounds consisting of carbon, hydrogen, chlorine, and fluorine, once used widely as aerosol propellants and refrigerators. Chlorofluorocarbons are believed to cause depletion of the atmospheric ozone layer.

→ Deforesation : The removal of a forest or stand of trees where the land is thereafter converted to a non-forest use.

→ Eutrophication : A process whereby water bodies, such as lakes, estuaries or slow-moving streams receive excess nutrients that stimulate excessive plant growth (algae, periphyton attached algae, and nuisance plants and weeds).

→ Fungicides : Substance or preparation, as a spray or dust, used for destroying fungi.

→ Herbicides : Chemicals used to kill unwanted weeds.

AP Inter 1st Year Zoology Notes Chapter 8 Ecology and Environment

→ Incinerator : a furnace or apparatus for burning trash, garbage, etc., to ashes.

→ Land fills : Landfill is a carefully engineered depression in the ground (or built on top of the ground, resembling a football stadium) into which wastes are dumped.

→ Pesticides : Chemicals used to kill pests, especially insects.

→ Photochemical smog: A type of air pollution produced when sunlight acts upon motor vehicle exhaust gases to form harmful substances such as ozone (O3), aldehydes and peroxy acetyl nitrate (PAN).

→ Polyblend : Physical mixture of two or more polymers. Such blends usually yield products with favorable properties of both components.

→ Scrubber : Scrubbers are commonly used to eliminate potentially harmful dust and pollutants from exhausts. In scrubbers, a liquid, in general water added with active chemicals is sprayed in to the air flow. Aerosol and gaseous pollutants in the air stream are removed by either absorption or chemical reactions with the water solution.

→ Sewage : Domestic waste water containing various solid and liquid waste materials including human excreta,

→ Soil erosion : The washing away of soil by the flow of water or wind

→ Thermal Pollution : Water Pollution caused by hot water coming out from industries, thermal power plants etc., which is harmful to aquatic organisms.

AP Inter 1st Year Zoology Notes Chapter 8 Ecology and Environment

→ Ultraviolet-B (UV-B) : One of the three types of invisible light rays (together with ultraviolet-A and ultraviolet-C) given off by the sun. Although ultraviolet-C is the most dangerous type of ultraviolet light In terms of its potential to harm life on earth, it cannot penetrate earth’s protective ozone layer. Therefore, it posses no threat to human, animal or plant life on earth as long as ozone layer is intact. Ultraviolet-A and ultraviolet- B, on the other.hand, do penetrate the ozone layer in attenuated form and reach the surface of the planet. UV-A rays cause cells to age and can cause some damage to cells DNA. They are linked to long-term skin damage such as wrinkles, but are also thought to play a role in some skin cancers. UVB rays can cause direct damage to the DNA and are the main rays that cause sunburns. They are also believed to cause most skin cancers.

→ Environment is the major factor for the evolution and continuation of life.

→ Ecology is the study of environment and its habitants.

→ Living being cannot survive without energy.

→ Solar energy is the primary source of energy for all the living forms either directly or indirectly.

→ Capture and storing of energy by chlorophyll is the major step in evolution.

→ Ozone is beneficial when it is in the outer layer of atmosphere. It protects the life from UV rays of Sun. It is dangerous when it is close to the earth.

→ Pollutants are there since the beginning of earth, but they are balanced by non-pollutants.

→ Man is solely responsible for fitting the balance towards pollutants and may be responsible for his own extinction unless remedial steps are taken.

→ UV rays kill micro organisms. UV rays convert the sterols in the skin to vitamin D. [IPE]

→ Mutualism is a type of interaction between different species in which both are benefited.
Ex: Bees and flowering plant. [IPE]

AP Inter 1st Year Zoology Notes Chapter 8 Ecology and Environment

→ Summer Stratification: During summer in temporate lakes, the formation of three layers of water is called Summer stratification.

→ DFC is detritus food chain. It is an important food chain in terrestrial ecosystem.

→ Detritus is formed from leaf litter, dead bodies, and faeces of animals.

→ Green house effect: ‘Green house effect’ is a naturally occuring phenomenon, that is responsible for heating of the Earth’s surface and atmosphere. [IPE]

→ Global Warming: Rise of temperature above normal level in the atmosphere is called global warming. This happens due to the increase in the emission of green house gases. [IPE]

→ Lake Ecosystem zones: (I) Littoral zone (II) Limnetic zone (III) Profundal zone. [IPE]

→ Food chains of Ecosystem: [IPE]

  1. Grazing food chain
  2. Parasite food chain
  3. Detritus food chain

→ Major air pollutants:

  1. Carbonmonoxide
  2. Carbondioxide
  3. Sulphurdioxide [IPE]
  4. Nitrogen oxides
  5. Aerosols
  6. Noise pollution.

→ National Aquatic Animal of India is River dolphin. [NEET-2016]

→ The principle of competitive exclusion was stated by GF.Gause. [NEET-2016]

→ The upright pyramid of number is absent in forest. [2012 PMT]

→ Stratification is not a functional unit of an ecosystem. [2012 PMT]

→ About 70% of total global carbon is found in oceans.

→ The highest DDT concentration in aquatic food chain shall occur in seagull. [NEET-2016]

→ A lake which is rich in organic waste may result in mortality of fish due to lack of oxygen. [NEET-2016]

→ Biochemical Oxygen Demand (BOD) may not be a good index for pollution for water bodies receiving effluents from petroleum industry. [NEET-2016]

→ Depletion of Ozone gas in the atmosphere can lead to an increased incidence of skin Cancers [NEET-2016]

→ Increase in concentration of the toxicant at successive trophic levels is known as biomagnification. [NEET-2015]

AP Inter 1st Year Zoology Notes Chapter 8 Ecology and Environment

→ Lichens are most suitable indicators of SO2 pollution in the environment. [NEET-2015]

→ Eutrophication of water bodies leading to killing of fishes is mainly due to non-availability of oxygen. [NEET-2015]

→ Acid rain is caused by increase in the atmospheric concentration of SO2NO2. [NEET-2015]

→ Ozone layer is present in stratosphere. [NEET-2014]

→ A location with luxuriant growth of lichens on the trees indicates that the location is not polluted. [NEET-2014]

→ Global warming can be controlled by reducing deforestation, cutting down use of fossil fuel. [NEET-2013]

→ Number of hotspots of biodiversity in the world have been identified till date by Norman Myers is 34. [NEET-2016]

→ Kyoto protocol was endorsed at CoP-3. [NEET-2013]

→ Theregion of biosphere reserve which is legally protected and where no human activity is allowed is known as core zone. [NEET-2017]

→ Alexander von Humboldt described for the first time population growth equation. [NEET-2017]

→ Thiobacillus is a group of bacteria helpful in carrying out Denitrification. [NEET-2019]

→ Gases mainly responsible for green house effect are Carbondioxide & Methane [NEET-2019]

→ The most suitable method for disposal of nuclear waste is ’burying the waste within rocks and deep below the earth’s surface’. [NEET-2019]

AP Inter 1st Year Zoology Notes Chapter 8 Ecology and Environment

→ Polyblend, a fine powder of recycled modified plastic, has proved to be a good material for construction of roads. [NEET-2019]

→ The protocol that aimed for reducing emission of chlorofluorocarbons in the atmosphere is Montreal protocol

→ Group of biocontrol agents:Trichoderma, Baculovirus, Bacillus Thruingiensis [NEET-2019]

→ Equipment that is essentially required for growing microbes on a large scale, for industrial production of enzymes is Bioreactor. [NEET-2019]

→ The amount of nutrients, such as carbon, nitrogen, phosphorus and calcium present in the soil at any given time, is referred as standing state. [NEET-2021]