Explore topic-wise InterviewSolutions in Current Affairs.

This section includes 7 InterviewSolutions, each offering curated multiple-choice questions to sharpen your Current Affairs knowledge and support exam preparation. Choose a topic below to get started.

1.

Write the shape and bond angle of sp hybrid orbitals?

Answer»

Linear, 1800

2.

Write the shape and bond angle of sp2 hybridized orbitals.

Answer»

Trigonal planar and bond angle 1200

3.

Draw and explain diagram showing hybridization in ethane.

Answer»

In the formation of ethene molecule, one of the sp2 hybrid orbitals of carbon atom overlaps axially with sp2 hybridised orbital of another carbon atom to form C–C sigma bond. While the other two sp2 hybrid orbitals of each carbon atom are used for making sp2 –s sigma bond with two hydrogen atoms. The unhybridised orbital (2px or 2py) of one carbon atom overlaps sidewise with the similar orbital of the other carbon atom to form weak π bond.

4.

What is the shape and hybridization of NH4 + ion ?

Answer»

sp3 tetrahedral

5.

According to a survey of 50,000 students under mid day meal in five districts of Rajasthan, 60% students like Pulses—Chapati, 20% like vegetable and chapatti and rest like khichdi. Determine percentage of students who like khichdi?

Answer»

Total no. of students = 50,000

Number of students like Pulses and Chapatti = 60% of 50,000

= 50,000 x 60/100 = 30,000

Number of students like Vegetable and Chapatti = 25% of 50,000

= 50,000 x 25/100

= 12,500

Remaining students

= 50,000 – (30,000 + 12,500)

= 50,000 – 42,500

= 7,500

∴ No. of students like Khichdi = 7,500

∴ Percentage of students like Khichdi

= 7500/50000 x 100

= 15%

Hence, 15% students like Khichdi.

6.

If interest is calculated quarterly then time period is(a) double(b) four times(c) triple(d) No change

Answer»

(b) four times

7.

80% students are good at maths out of 25 students. How many students are not good at maths?(a) 10(b) 5(c) 15(d) 20

Answer»

The correct option is (b) 5.

8.

Fill in the blanks:1. The concession on printed rate is called___2. Sale tax is added to cost price of item is called___3. The interest compounded on total amount of previous year (A = P + I) is called___4. The direct ratios is x and y then x1/y1 = ___5. If the speed of a vehicle is increased then the time taken for travelling same distance is___

Answer»

1. discount

2. VAT

3. Compound Interest

4. x2/y2

5. decreased.

9.

Complete the following statements:1. Commodity which satisfies human wants is called __________2. Actual transfer of goods from one person to another is termed as __________3.Income which is paid as per terms and condition is called as __________4. Anything which has market value and can be exchanged for money is termed as __________5. Adam Smith defines Economics as __________6. Scarcity oriented definition of economics was given by __________7. Wealth oriented definition of economics was given by __________8. Kautilya authored the Indian Political treatise called __________9. Paul Samuelson referred economics as __________10. Psychology, Sociology, Political Science are the examples of __________11. Kautilya is also known by the name __________12. The term microeconomics and macroeconomics was coined by __________

Answer»

1. Utility

2. Physical transfer

3. Contractual Income

4. Wealth

5. Science of wealth

6. Lionel Robbins

7. Alfred Marshall

8. the ‘Arthashastra’

9. “Queen of Social Science”

10. Social Science/Abstract Science

11. Chanakya or Vishnugupta

12. Sir Ragnar Frisch

10.

Choose the wrong pair:Group ‘A’Group ‘B’1. Alfred Marshall(a) Welfare Oriented Definition2. Lionel Robbins(b) Scarcity Oriented Definition3. Adam Smith(c) Scarcity Oriented Definition4. J. M. Keynes(d) Modern School of Thought

Answer»

Wrong Pair:

Adam Smith – Scarcity Oriented Definition

11.

Choose the correct pair:Group ‘A’Group ‘B’1. Economic Wants(a) Travelling by Bus2. Non-Economic Wants(b) Air, Sunshine, etc.3. Individual Wants(c) Food, Medicine, etc.4. Collective Wants(d) Doctor using StethoscopeOptions: (a) 1 – d; 2 – c; 3 – a; 4 – b (b) 1 – b; 2 – c; 3 – d; 4 – a (c) 1 – c; 2 – b; 3 – d; 4 – a (d) 1 – a; 2 – d; 3 – b; 4 – c

Answer»

Correct pair

(c) 1 – c; 2 – b; 3 – d; 4 – a

12.

Choose the wrong pair:Group ‘A’Group ‘B’1. Physics, Chemistry, etc(a) Natural Science2. Kautilya(b) Arthshastra3. Psychology, Sociology, etc(c) Social Science4. Prof. Alfred Marshall(d) Father of Economics

Answer»

Wrong Pair: 

Prof. Alfred Marshall – Father of Economics

13.

How Chemical Equation will balance.

Answer»

Steps involved in balancing a Chemical Equation:

Step 1: Write the correct skeleton equation.

Step 2: Start with the compound that has the maximum atoms or maximum kinds of atoms and the atoms present in it are balanced first.

Step 3: Balance elements that appear only once on each side of the arrow first. Then balance elements that appear more than once on a side.

Step 4: Elementary substances are balanced last of all.

Step 5: If required the whole equation is multiplied by some suitable number in order to make all the coefficients whole numbers.

14.

Which one helps in announcing the arrival of new products and their uses to the people? (a) Advertising (b) Banking (c) Trade (d) Transport

Answer»

Correct Answer is: (a) Advertising

15.

….. was the chief port of the Kingdom of Cholas. (a) Marakannam (b) Arikamedu (c) Kayal (d) Kaveripumpattinam

Answer»

(d) Kaveripumpattinam

16.

….. Who advocated maintenance of perfect balance between State management and people’s welfare through trading activities. (a) Kautilya (b) Pandiya (c) Hebrew (d) Chola

Answer»

Kautilya Who advocated maintenance of perfect balance between State management and people’s welfare through trading activities.

17.

Who wrote “Arthasasthra”? (a) Kautilya (b) Chanakiya (c) Thiruvalluvar (d) Elangovadigal

Answer»

Correct Answer is: (a) Kautilya

18.

What type of boats were used to cross rivers for domestic trade?

Answer»

Boats like ‘Padgu’ , ‘Thimil’ , ‘Thom’ , ‘Ambu’ ‘Odampunai’ , etc., were used to cross rivers for domestic trade.

19.

What type of boats were used to cross oceans for foreign trade?

Answer»

Boats like ‘Kalam’ , ‘Marakalam’ , ‘Vangam’ , ‘Navai’ , etc., were used for crossing oceans for foreign trade.

20.

…….. in Sangam period was both internal and external. (a) Sales (b) Education (c) Industry (d) Trade

Answer»

Trade in Sangam period was both internal and external.

21.

Day market was called as ……………(a) Nalangadi (b) Angadi (c) Business (d) Trade

Answer»

Day market was called as Nalangadi.

22.

How Can Students Join The d-and f-Block Elements Class 12 Mock Test?

Answer»

Try our new mock tests series with multiple attempts and calculate your performance unless you get fully satisfied with your preparation level. The d-and f-block elements are an important section of chemistry in class 12. Start practicing The d-and f-Block Elements Class 12 Mock Test and score maximum in the actual board exam of class 12. Give mock tests daily and evaluate your performance and understand that where you want to give more efforts.

Mock test for class 12 Chemistry are prepared as per the newest syllabus and Exam Pattern. Students can solve these Class 12 Mock test with Answers and assess their preparation level.

Students are advised to practice the Class 12 online Mock test of The d-and f-Block Elements to reconsider the essential ideas and find out about the kind of questions that may appear on the board exam. Students can practice these The d-and f-Block Elements Mock test with Answers and evaluate their level of preparation.

Click here for regular practice: - Class 12 The d-and f-Block Elements Mock Test

23.

Class 12 Chemistry  MCQ Questions of The d-and f-Block Elements with Answers?

Answer»

Students can solve these The d-and f-Block Elements MCQ Questions with Answers and evaluate their level of preparation. Students are advised to practice the Class 12 Chemistry MCQ Questions of The d-and f-Block Elements with Answers to reconsider the essential ideas and find out about the kind of questions that may appear on the board exam.

MCQ Questions for class 12 Chemistry with Answers are prepared as per the newest syllabus and Exam Pattern. Students can solve these Class 12 MCQ Questions with Answers and assess their preparation level.

Practice MCQ Question for Class 12 Chemistry chapter-wise

1. Shape of d-orbital is

(a) spherical
(b) dumb bell
(c) double dumb bell
(d) none of these

2. Electronic configuration of alkaline earth element is

(a) ns2
(b) ns1
(c) np6
(d) ns0

3. Lanthanoid contraction is due to increase in

(a) atomic number
(b) effective nuclear charge
(c) atomic radius
(d) valence electrons

4. Which one of the following is diamagnetic ion?

(a) Co2+
(b) Ni2+
(c) Cu2+
(d) Zn2+

5. Colour of transition metal ions are due to absorption of some wavelength. This results in

(a) d-s transition
(b) s-s transition
(c) s-t/transition
(d) d-d transition

6. The melting points of Cu, Ag and Au follow the order

(a) Cu > Ag > Au
(b) Cu >Au > Ag
(c) Au > Ag > Cu
(d) Ag > Au > Cu

7. In which of the following pairs of ions, the lower oxidation state in aqueous solution is more stable than the other?

(a) Ti+, Ti3+
(b) Cu+, Cu3+
(c) Cr2+, Cr3+
(d) V2+, VO2+ (V4)

8. Zr and Hf have almost equal atomic and ionic radii because of

(a) diagonal relationship
(b) lanthanoid contraction
(c) actinoid contraction
(d) belonging to the same group

9. Malachite is an ore of 

(a) Silver
(b) Mercury
(c) Magnesium
(d) Copper

10. The chief ore of Hg is

(a) Pyrolusite
(b) Bauxite
(c) Galena
(d) Cinnabar

11. The transition element which shows the highest oxidation state is: 

(a) Iron
(b) Vanadium
(c) Manganese
(d) Chromium

12. Gun metal is

(a) Cu + Zn
(b) Cu + Sn + Zn
(c) Cu + Sn
(d) Zn + Sn

13. German silver is an alloy of:

(a) Fe, Cr, Ni
(b) Ag, Cu Au
(c) Cu, Zn, Ni
(d) Cu, Zn, Sn

14. The composition of duralumin is

(a) Al 94%, Mg 6%
(b) Cu 56%, Zn 24%, Ni 20%
(c) Cu 95%, Al 5%
(d) Al 95%, Cu 4%, Mn 0.5%, Mg 0.5%

15. Which of the following is not an ore of iron?

(a) limonite
(b) Cassiterite
(c) magnetite
(d) none of these

16. Lanthanide for which +II and +III oxidation states are common is

(a) La
(b) Pr
(c) Ce
(d) Eu

17. F2 is formed by reacting K2MnF6 with

(a) SbF5
(b) MnF3
(c) KSbF6
(d) MnF4

18. Which of the following radioisotopes is used as anticancerous?

(a) Na -24
(b) C -14
(c) U -235
(d) Co -60

19. The colour of copper sulphide is

(a) Blue
(b) Black
(c) Red
(d) Green

20. What is wrong about transition metals?

(a) Diamagnetic
(b) Paramagnetic
(c) Form complexes
(d) Shows variable oxidation state

21. The last electron in d-block elements goes to

(a) (n-1) d
(b) nd
(c) np
(d) (n-1) s

22. The elements which exhibit both vertical and horizontal similarities are

(a) inert gas elements
(b) representative elements
(c) rare elements
(d) transition elements

23. An atom has electronic configuration 1s2 2s2 2p3s3p3d4s2 in which group would it be placed?

(a) Fifth
(b) Fifteenth
(c) Second
(d) Third

24. Number of unpaired electrons in Ni2+ (Z = 28) is

(a) 4
(b) 2
(c) 6
(d) 8

25. Transition metals usually exhibit highest oxidation states in their

(a) chlorides
(b) fluorides
(c) bromides
(d) iodides

Answer :

1. Answer : (c) double dumb bell

Explanation: The ml values for five d orbitals are -2, -1, 0, +1, and +2 i.e., we can say d-subshell has five orientations. All these d-orbitals have the same energy and are called degenerate orbitals. Hence, we can say d-orbitals have double dumbbell-shaped.

2. Answer : (a) ns2

Explanation: The general electronic configuration of alkaline earth elements is ns2. These elements have two valence electrons in the outermost s shell.

3. Answer : (b) effective nuclear charge

Explanation: Lanthanoid contraction effect results from poor shielding of nuclear charge (nuclear attractive force on electrons) by 4f electrons so effective nuclear charge increases; the 6s electrons are drawn towards the nucleus, thus resulting in a smaller atomic radius.

4. Answer : (d) Zn2+

Explanation: Zn2+ = [Ne 3s2 3p6 4s0 3d10]  It is diamagnetic, so no unpaired es.

5. Answer : (d) d-d transition

Explanation: Colour of transition metal ions are due to absorption of the same wavelength. This results in d−d transition.

6. Answer : (b) Cu >Au > Ag

Explanation: Hence the melting point decreases from copper to silver, but the melting point of gold becomes even higher than silver due to its higher density that happened due to contraction.

7. Answer : (a) Ti+, Ti3+

Explanation: Ti+ ions are more stable than Ti3+ and thus, Ti3+ ions change to Ti+ ions thereby acting as oxidising agents.

8. Answer : (b) lanthanoid contraction

Explanation: Zr and Hf have almost similar atomic and ionic radii it can be explained on the basis of shielding effect . Due to lanthanide contraction. Moreover, the electrons which are present in inner shells, shields outer electrons from nuclear charge, after that they experience a low effective nuclear charge.

9. Answer : (d) Copper

Explanation: Malachite, a minor ore but a widespread mineral of copper, basic copper carbonate, Cu2CO3(OH)2. Because of its distinctive bright green colour and its presence in the weathered zone of nearly all copper deposits, malachite serves as a prospecting guide for that metal.

10. Answer : (d) Cinnabar

Explanation: Cinnabar is an ore of mercury.

11. Answer : (c) Manganese

Explanation: Manganese has the highest oxidation state because the number of unpaired electrons in the outermost shell is more, i.e. 3d5 4s2.

12. Answer : (b) Cu + Sn + Zn

Explanation: Gunmetal, also called G Metal, variety of bronze, formerly used for ordnance. Modern admiralty gunmetal is composed of 88 percent copper, 10 percent tin, and 2 percent zinc and is used for gears and bearings that are to be subjected to heavy loads and low speeds.

13. Answer : (c) Cu, Zn, Ni

Explanation: German Silver is an alloy of copper, zinc and nickel, sometimes also containing lead and tin.

14. Answer : (d) Al 95%, Cu 4%, Mn 0.5%, Mg 0.5%

Explanation: Duralumin is an alloy 95% aluminium, 4% copper, 0.5% manganese and 0.5% magnesium.

15. Answer : (b) Cassiterite

Explanation: Cassiterite was the main tin ore throughout ancient history and remains the most important source of tin today. Thus we can say that cassiterite isn't an ore of iron.

16. Answer : (d) Eu

Explanation: Lanthanides show variable oxidation states. The most stable oxidation state of Lanthanides is +3. They also show +2 and +4 oxidation states due to the presence of either half filled or completely filled or empty 4f sub shell. Due to stable half filled configuration Eu shows +2 configuration.

17. Answer : (a) SbF5

Explanation: F2 is the formed by reacting K2MnF6 with Sb5. Reaction is as follows:

K2MnF+ 2SbF→ 2KSbF+ MnF\(\frac{1}{2}\)F2

18. Answer : (d) Co -60

Explanation: Co-60 is used as anti-cancerous among the given radioactive isotopes. It emits \(\beta\)− particles and energetic gamma rays, for which reason it is used in radiation therapy. 

19. Answer : (b) Black

Explanation: When hydrogen sulphide gas is passed through a blue solution of copper sulphate, a black precipitate of copper sulphide is obtained and the sulphuric acid so formed remains in the solution.

20. Answer : (a) Diamagnetic

Explanation: Transition elements have one or more unpaired electrons. Due to the presence of these unpaired electrons, they show paramagnetism. Thus, transition metals are not diamagnetic. They are good conductors of heat and electricity. They form complexes and show variable oxidation state as d orbitals are gradually filled up in the transition series.

21. Answer : (a) (n-1) d

Explanation: In these elements, the last electron enters the d orbital of the penultimate shell i.e. the last electron goes to (n-1) d orbital. Hence these elements are named as d-block elements.

22. Answer : (d) transition elements

Explanation: Elements which exhibit both vertical and horizontal similarities are transition elements. They show vertical similarity because of same electronic configuration. They show horizontal similarity because of similar size.

23. Answer : (a) Fifth

Explanation: The outermost shell and one shell prior to it in d subshell, has total 5 electron so it will be placed in fifth group in transition element or 'd' block of periodic table.

24. Answer : (b) 2

Explanation: Nickel has two unpaired electrons in the 3d sublevel. When nickel ionizes to the +2 state it loses the two electrons in the 4s, so there are still two unpaired electrons in Ni2+.

25. Answer : (b) fluorides

Explanation: The highest oxidation states of transition metals are shown by fluorides and oxides as fluorine and oxygen are the most electronegative elements. The highest oxidation state shown by any transition metal is +8.

Click here to practice  MCQ Question for The d-and f-Block Elements Class 12 

24.

Differentiate homospory and heterospory with example.

Answer»
HomosporyHeterospory
Production of one type of spores. e.g..SelaginellaProduction of two types of spores. e.g. Lycopodium 

25.

Important Class 10 Science MCQ Questions of Control and Coordination and Answers with detailed explanation?

Answer»

These MCQ Questions Online Tests cover the whole of Control and Coordination. Our experts are prepared MCQ Questions for Class 10 based on the latest exam pattern. Students can refer to these Class 10 Science MCQ Questions of Control and Coordination with Answer for their exam preparation. 

MCQ Questions of Control and Coordination are taken from all important topics of the chapter and answer just as an explanation is given underneath each question.

Practice Class 10 Science MCQ Questions of Control and Coordination

1. Which of the following acts as both endocrine and exocrine glands?

(a) Adrenal
(b) Pituitary
(c) Ovaries
(d) Pancreas

2. Which hormone regulates the ionic balance in the body?

(a) Glucagon
(b) Thyroxine
(c) Testosterone
(d) Vasopressin

3. Impulse flows from ___.

(a) dendrite to axon in same neuron
(b) axon to dendrite
(c) nodes of ranvier to axon
(d) none

4. ____ are multipolar and conduct impulse out from spinal cord.

(a) neuroglia
(b) neurolemma
(c) schwann cells
(d) motor neuron

5. Which of the following statements is correct about receptors?

(a) Gustatory receptors detect taste while olfactory receptors detect smell
(b) Both gustatory and olfactory receptors detect smell
(c) Auditory receptors detect smell and olfactory receptors detect taste
(d) Olfactory receptors detect taste and gustatory receptors smell

6. In a neuron, conversion of electrical signal to a chemical signal occurs at/in

(a) cell body
(b) axonal end
(c) dendritic end
(d) axon

7. Posture and balance of the body is controlled by

(a) cerebrum
(b) cerebellum
(c) medulla
(d) pons

8. Spinal cord originates from

(a) cerebrum
(b) medulla
(c) pons
(d) cerebellum

9. The gap between two neurons is called a:

(a) dendrite
(b) synapse
(c) axon
(d) impulse

10. The brain is responsible for:

(a) thinking
(b) regulating the heart beat
(c) balancing the body
(d) all of the above

11. Cartisons are secreted by:

(a) pancreatic gland
(b) adrenal gland
(c) thyroid gland
(d) pituitary gland

12. Female sex hormone is:

(a) estrogen
(b) androgen
(c) insulin
(d) adrenalin

13. The main function of the plant hormone called abscisic acid is to:

(a) Increase the length of cells
(b) Promote cell division
(c) Inhibit growth
(d) Promote growth of stem and roots

14. The growth of tendrils in pea plants is due to the:

(a) Effect of sunlight on the tendril cells facing the sun
(b) Effect of gravity on the part of tendril hanging down towards the earth
(c) Rapid cell division and elongation in tendril cells that are away from the support
(d) Rapid cell division and elongation in tendril cells in contact with the support

15. The plant hormone which triggers the fall of mature leaves and fruits from the plant body is:

(a) Auxin
(b) Gibberellin
(c) Abscisic acid
(d) Cytokinin

16. The stimulus in the process of thigmotropism is:

(a) Touch
(b) Gravity
(c) Light
(d) Chemical

17. The number of pairs of nerves which arises from the spinal cord is:

(a) 21
(b) 31
(c) 41
(d) 51

18. Reflex actions are mediated through

(a) brain
(b) effectors
(c) spinal cord
(d) receptors

19. The highest coordinating centre in the human body is

(a) spinal cord
(b) heart
(c) brain
(d) kidney

20. Iodine is necessary for the synthesis of which of the following hormone?

(a) Adrenaline
(b) Auxin
(c) Thyroxine
(d) Insulin

Answer key

1. (d) Pancreas
2. (d) Vasopressin
3. (a) dendrite to axon in same neuron
4. (d) motor neuron
5. (a) Gustatory receptors detect taste while olfactory receptors detect smell
6. (b) axonal end
7. (b) cerebellum
8. (b) medulla
9. (b) synapse
10. (d) all of the above
11. (b) adrenal gland
12. (a) estrogen
13. (c) Inhibit growth
14. (c) Rapid cell division and elongation in tendril cells that are away from the support
15. (c) Abscisic acid
16. (a) Touch
17. (b) 31
18. (c) spinal cord
19. (c) brain
20. (c) Thyroxine 

Click here to practice more MCQ Questions from Chapter Control and Coordination Class 10 Science

26.

Compare Eusporangiate and Leptosporangiate. 

Answer»
Eusporangiate Leptosporangiate 
Development of sporangium from group of initials Development of sporangium from single initial 

27.

Where are Eustachian valve and Thebisan valve located?

Answer»

Eustachian valve is present at the opening of inferior vena cava while Thebeisan valve is present near the opening of coronary sinus.

28.

Give definition of the following :Erythropoiesis

Answer»

The process of formation of RBCs is called erythropoiesis.

29.

What is leucopenia and erythrocytopenia ?

Answer»

The decrease in the number of white blood cells is called leucopenia whereas decrease in the number of red blood cells is called erythrocytopenia.

30.

Give definition of the following :Hematocrit

Answer»

The hematocrit is ratio of the volume of RBCs to total blood volume of blood.

31.

In the given figure, if lines PQ and RS intersect at point T, such that ∠PRT = 40º, ∠RPT = 95º and ∠TSQ = 75º, find ∠SQT.

Answer»

Using angle sum property for ΔPRT, we obtain

PRT + RPT + PTR = 180º

40º + 95º + PTR = 180º

PTR = 180º − 135º

PTR = 45º

STQ = PTR = 45º (Vertically opposite angles)

STQ = 45º

By using angle sum property for ΔSTQ, we obtain 

STQ+ SQT + QST = 180º 

45º + SQT + 75º = 180º 

SQT = 180º − 120º

SQT = 60º

32.

Write the order of the product matrix.\(M = \begin{bmatrix}1\\[0.3em]2 \\[0.3em]3\end{bmatrix}\)|2 3 4|

Answer»

Order is 3 × 3 because it is product of two matrices having order 3 × 1 and 1 × 3.

33.

If A is a 3 × 3 matrix, whose elements are given by  \(a_{ij}= \frac{1}{3}|-3i+j|\) , then write the value of a23.

Answer»

\(a_{23} = \frac{1}{3}|3\times2+3|\)

\(= \frac{1}{3}|-6+3|\)

\(=\frac{1}{3}\times3\) = 1

34.

If  \(\begin{bmatrix}cos\,\alpha & -sin\,\alpha \\[0.3em]sin\,\alpha & cos\,\alpha \\[0.3em]\end{bmatrix}\), then for what value of α, A is an identity matrix?

Answer»

If A is identity matrix, 

Then,  A = I2

\(\begin{bmatrix} cos\,\alpha & -sin\,\alpha \\[0.3em] sin\,\alpha & cos\,\alpha \\[0.3em] \end{bmatrix} \)\(\begin{bmatrix} 1 & 0 \\[0.3em] 0 & 1 \\[0.3em] \end{bmatrix} \)

On equating, we get

⇒ cos α = 1,sin α = 0

⇒ α = 0

35.

If \(​​\begin{bmatrix}7y & 5 \\[0.3em]2x\,-\,3y& -3 \\[0.3em]\end{bmatrix}\)=\(​​\begin{bmatrix}-21 & 5 \\[0.3em]11& -3 \\[0.3em]\end{bmatrix}\),then find the value of x.

Answer»

Using equality of two matrices,

7y = – 21

⇒ y = – 3

2x – 3y = 11

⇒ 2x + 9 = 11

⇒ x = 1

∴ x = 1, y = – 3

36.

For a 2 × 2 matrix, A = [aij], whose elements are given by aij = \(\frac{i}{j}\), write the value of a12.

Answer»

aij = \(\frac{i}{j}\) 

⇒ a12\(\frac{1}{2}\)  [ Here i = 1 and j = 2]

37.

Show that f(x) = 1/(1 + x2) is increasing for all x ≤ 0.

Answer»

It is given that

f(x) = 1/(1 + x2)

By differentiating w.r.t. x

f’(x) = –2x/(1 + x2)2 ≤ 0

Therefore, f(x) is increasing function for all x ≤ 0.

38.

Prove that f(x) = 3x is strictly increasing on R.

Answer»

Consider f(x) = 3x

The domain of f(x) is R.

f’(x) = 3x ln(3)

3x is always greater than 0 and ln(3) is also + ve.

Overall f’(x) is >0 means strictly increasing in its domain i.e. R.

39.

Show that f(x) = (x - 1/x) is increasing all x ∈ R, where x ≠ 0.

Answer»

f(x) = (x - 1/x)

domain of function is R - {0}

f'(x) = (1+1/x2)

f’(x) ∀ x ∈ R is greater than 0.

40.

Show that f(x) = x3 - 15x2+75x - 50 is increasing on R.

Answer»

Consider f(x) = x- 15x2+75x - 50

Domain of the function is R.

f’(x) = 3x- 30x+75

= 3(x- 10+25)

= 3(x - 5)(x - 5)

= 3(x - 5)2

f’(x) = 0 for x = 5

for x<5

f’(x)>5

and

for x>5

f’(x)>5

we can see throughout R the derivative is +ve but at x = 5 it is 0 so it is increasing.

41.

Show that f(x) = [x3 + 1/x3] is decreasing on ]-1, 1[.

Answer»

It is given that

f(x) = [x3 + 1/x3]

By differentiating w.r.t. x

f’(x) = 3x2 – 3x-4

By taking 3 as common

f’(x) = 3[x2 – 1/x4]

Taking LCM

f’(x) = 3[(x6 – 1)/x4] = 3[(x2)3 – 1]/x4

On further calculation

f’(x) = [3(x2 – 1) (x4 + x2 + 1)]/x4

So we get

f’(x) = [3(x – 1) (x + 1) (x4 + x2 + 1)]/x4 < 0 for x ∈ (- 1, 1).

Therefore, f(x) is decreasing function on ]-1, 1[.

42.

Show that f(x) = [3/x + 5] is decreasing for all x ∈ R, where x ≠ 0.

Answer»

It is given that

f(x) = [3/x + 5]

By differentiating w.r.t. x

f’(x) = 3(-1/x2) = -3/x2 < 0 for all value of R

Therefore, f(x) is decreasing for all x ∈ R, where x ≠ 0.

43.

The function f(x) = 4 - 3x+3x2 - x3 isA. decreasing on RB. increasing on RC. strictly decreasing on RD. strictly increasing on R

Answer»

f(x) = -x3+3x- 3x+4.

f’(x) = -3x2+6x - 3

f’(x) = -3(x- 2x+1)

f’(x) = -3(x -1)2

As f’(x) has –ve sign before 3

⇒ f’(x) is decreasing over R.

44.

Show that f(x) = x/sin x is increasing on ]0, π/2[.

Answer»

It is given that

f(x) = x/sin x

By differentiating w.r.t. x

f’(x) = (sin x. 1 – x. cos x)/(sin x)2 = (sin x – x cos x)/(sin x)2

We know that

f’(x) > 0

By substituting the values

(sin x – x cos x)/(sin x)2 > 0

So we get

(sin x – x cos x) > 0

Here tan x > x which is true on ] 0, π/2[.

45.

Show that the height of a closed cylinder of given volume and the least surface area is equal to its diameter.

Answer»

Let r be the radius of the base and h the height of a cylinder.

The surface area is given by,

S = 2 π r2 + 2 π rh

h = S - 2πr2/2πr ………(1)

Let V be the volume of the cylinder.

Therefore, V = πr2h

V = πr2 \((\frac{S-2\pi r^3}{2\pi r})\) …….Using equation 1

V = \(\frac{Sr-2\pi r^3}{2}\)

Differentiating both sides w.r.t r, we get,

\(\frac{dV}{dr}=\frac{S}{2}\) - 3πr2 ……….(2)

For maximum or minimum, we have,

 \(\frac{dV}{dr}\) = 0

⇒ \(\frac{S}{2}\) - 3πr2 = 0

⇒ S = 6πr2

2πr2 + 2πrh = 6πr2

h = 2r

Differentiating equation 2, with respect to r to check for maxima and minima, we get,

 \(\frac{d^2V}{dr^2}\) = - 6πr < 0

Hence, V is maximum when h = 2r or h = diameter

46.

Show that f(x) = 1/(1+x2) is decreasing for all x ≥ 0

Answer»

Consider f(x) = \(\frac{1}{(1+x^2)}\),

f'(x) = \(-\frac{2x}{(1+x^2)^2}\)

for x ≥ 0,

f’(x) is -ve.

hence function is decreasing for x ≤ 0

47.

Statements which are assumed to be true without proof are A) axioms B) counter examples C) conjectures D) open sentences

Answer»

Correct option is: A) axioms

48.

A sentence which is clearly true or false but not both is called a A) Axiom B) Theorem m C) Conjecture D) Statement

Answer»

Correct option is: D) Statement

49.

Show that the function f(x) = x2 is(a) strictly increasing on [0, ∞[(b) strictly decreasing on ]- ∞, 0[(c) neither strictly increasing nor strictly decreasing on R

Answer»

(a) It is given that f(x) = x2

By differentiating w.r.t x

f’(x) = 2x > 0 for x ∈ (0, ∞)

Here f’(x) > 0 which means 2x > 0

So we get x > 0

(b) We know that

If f’(x) < 0 we get x < 0

(c) It is given that f(x) = x2 is strictly increasing on [0, ∞[ and strictly decreasing on ]- ∞, 0[

So it is neither strictly increasing nor strictly decreasing on the whole real line.

50.

Show that the function f(x) = 5x – 2 is a strictly increasing function on R.

Answer»

It is given that

f(x) = 5x – 2

By differentiating w.r.t. x

f’(x) = 5 > 0 for all x > 0

Therefore, f(x) is strictly increasing function for all x > 0.