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? |
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Answer» Linear, 1800 |
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| 2. |
Write the shape and bond angle of sp2 hybridized orbitals. |
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Answer» Trigonal planar and bond angle 1200 |
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| 3. |
Draw and explain diagram showing hybridization in ethane. |
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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. |
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| 4. |
What is the shape and hybridization of NH4 + ion ? |
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Answer» sp3 tetrahedral |
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| 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? |
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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. |
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| 6. |
If interest is calculated quarterly then time period is(a) double(b) four times(c) triple(d) No change |
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Answer» (b) four times |
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| 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 |
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Answer» The correct option is (b) 5. |
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| 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___ |
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Answer» 1. discount 2. VAT 3. Compound Interest 4. x2/y2 5. decreased. |
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| 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 __________ |
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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 |
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| 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 |
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Answer» Wrong Pair: Adam Smith – Scarcity Oriented Definition |
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| 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 |
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Answer» Correct pair (c) 1 – c; 2 – b; 3 – d; 4 – a |
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| 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 |
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Answer» Wrong Pair: Prof. Alfred Marshall – Father of Economics |
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| 13. |
How Chemical Equation will balance. |
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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. |
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| 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 |
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Answer» Correct Answer is: (a) Advertising |
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| 15. |
….. was the chief port of the Kingdom of Cholas. (a) Marakannam (b) Arikamedu (c) Kayal (d) Kaveripumpattinam |
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Answer» (d) Kaveripumpattinam |
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| 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 |
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Answer» Kautilya Who advocated maintenance of perfect balance between State management and people’s welfare through trading activities. |
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| 17. |
Who wrote “Arthasasthra”? (a) Kautilya (b) Chanakiya (c) Thiruvalluvar (d) Elangovadigal |
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Answer» Correct Answer is: (a) Kautilya |
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| 18. |
What type of boats were used to cross rivers for domestic trade? |
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Answer» Boats like ‘Padgu’ , ‘Thimil’ , ‘Thom’ , ‘Ambu’ ‘Odampunai’ , etc., were used to cross rivers for domestic trade. |
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| 19. |
What type of boats were used to cross oceans for foreign trade? |
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Answer» Boats like ‘Kalam’ , ‘Marakalam’ , ‘Vangam’ , ‘Navai’ , etc., were used for crossing oceans for foreign trade. |
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| 20. |
…….. in Sangam period was both internal and external. (a) Sales (b) Education (c) Industry (d) Trade |
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Answer» Trade in Sangam period was both internal and external. |
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| 21. |
Day market was called as ……………(a) Nalangadi (b) Angadi (c) Business (d) Trade |
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Answer» Day market was called as Nalangadi. |
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| 22. |
How Can Students Join The d-and f-Block Elements Class 12 Mock Test? |
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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 |
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| 23. |
Class 12 Chemistry MCQ Questions of The d-and f-Block Elements with Answers? |
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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 2. Electronic configuration of alkaline earth element is (a) ns2 3. Lanthanoid contraction is due to increase in (a) atomic number 4. Which one of the following is diamagnetic ion? (a) Co2+ 5. Colour of transition metal ions are due to absorption of some wavelength. This results in (a) d-s transition 6. The melting points of Cu, Ag and Au follow the order (a) Cu > Ag > Au 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+ 8. Zr and Hf have almost equal atomic and ionic radii because of (a) diagonal relationship 9. Malachite is an ore of (a) Silver 10. The chief ore of Hg is (a) Pyrolusite 11. The transition element which shows the highest oxidation state is: (a) Iron 12. Gun metal is (a) Cu + Zn 13. German silver is an alloy of: (a) Fe, Cr, Ni 14. The composition of duralumin is (a) Al 94%, Mg 6% 15. Which of the following is not an ore of iron? (a) limonite 16. Lanthanide for which +II and +III oxidation states are common is (a) La 17. F2 is formed by reacting K2MnF6 with (a) SbF5 18. Which of the following radioisotopes is used as anticancerous? (a) Na -24 19. The colour of copper sulphide is (a) Blue 20. What is wrong about transition metals? (a) Diamagnetic 21. The last electron in d-block elements goes to (a) (n-1) d 22. The elements which exhibit both vertical and horizontal similarities are (a) inert gas elements 23. An atom has electronic configuration 1s2 2s2 2p6 3s2 3p6 3d3 4s2 in which group would it be placed? (a) Fifth 24. Number of unpaired electrons in Ni2+ (Z = 28) is (a) 4 25. Transition metals usually exhibit highest oxidation states in their (a) chlorides 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 e−s. 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: K2MnF6 + 2SbF5 → 2KSbF6 + MnF3 + \(\frac{1}{2}\)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 |
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| 24. |
Differentiate homospory and heterospory with example. |
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Answer»
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| 25. |
Important Class 10 Science MCQ Questions of Control and Coordination and Answers with detailed explanation? |
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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 2. Which hormone regulates the ionic balance in the body? (a) Glucagon 3. Impulse flows from ___. (a) dendrite to axon in same neuron 4. ____ are multipolar and conduct impulse out from spinal cord. (a) neuroglia 5. Which of the following statements is correct about receptors? (a) Gustatory receptors detect taste while olfactory receptors detect smell 6. In a neuron, conversion of electrical signal to a chemical signal occurs at/in (a) cell body 7. Posture and balance of the body is controlled by (a) cerebrum 8. Spinal cord originates from (a) cerebrum 9. The gap between two neurons is called a: (a) dendrite 10. The brain is responsible for: (a) thinking 11. Cartisons are secreted by: (a) pancreatic gland 12. Female sex hormone is: (a) estrogen 13. The main function of the plant hormone called abscisic acid is to: (a) Increase the length of cells 14. The growth of tendrils in pea plants is due to the: (a) Effect of sunlight on the tendril cells facing the sun 15. The plant hormone which triggers the fall of mature leaves and fruits from the plant body is: (a) Auxin 16. The stimulus in the process of thigmotropism is: (a) Touch 17. The number of pairs of nerves which arises from the spinal cord is: (a) 21 18. Reflex actions are mediated through (a) brain 19. The highest coordinating centre in the human body is (a) spinal cord 20. Iodine is necessary for the synthesis of which of the following hormone? (a) Adrenaline Answer key 1. (d) Pancreas Click here to practice more MCQ Questions from Chapter Control and Coordination Class 10 Science |
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| 26. |
Compare Eusporangiate and Leptosporangiate. |
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Answer»
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| 27. |
Where are Eustachian valve and Thebisan valve located? |
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Answer» Eustachian valve is present at the opening of inferior vena cava while Thebeisan valve is present near the opening of coronary sinus. |
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| 28. |
Give definition of the following :Erythropoiesis |
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Answer» The process of formation of RBCs is called erythropoiesis. |
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| 29. |
What is leucopenia and erythrocytopenia ? |
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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. |
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| 30. |
Give definition of the following :Hematocrit |
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Answer» The hematocrit is ratio of the volume of RBCs to total blood volume of blood. |
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| 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. |
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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 |
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| 32. |
Write the order of the product matrix.\(M = \begin{bmatrix}1\\[0.3em]2 \\[0.3em]3\end{bmatrix}\)|2 3 4| |
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Answer» Order is 3 × 3 because it is product of two matrices having order 3 × 1 and 1 × 3. |
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| 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. |
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Answer» \(a_{23} = \frac{1}{3}|3\times2+3|\) \(= \frac{1}{3}|-6+3|\) \(=\frac{1}{3}\times3\) = 1 |
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| 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? |
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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 |
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| 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. |
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Answer» Using equality of two matrices, 7y = – 21 ⇒ y = – 3 2x – 3y = 11 ⇒ 2x + 9 = 11 ⇒ x = 1 ∴ x = 1, y = – 3 |
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| 36. |
For a 2 × 2 matrix, A = [aij], whose elements are given by aij = \(\frac{i}{j}\), write the value of a12. |
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Answer» ∵ aij = \(\frac{i}{j}\) ⇒ a12 = \(\frac{1}{2}\) [ Here i = 1 and j = 2] |
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| 37. |
Show that f(x) = 1/(1 + x2) is increasing for all x ≤ 0. |
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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. |
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| 38. |
Prove that f(x) = 3x is strictly increasing on R. |
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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. |
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| 39. |
Show that f(x) = (x - 1/x) is increasing all x ∈ R, where x ≠ 0. |
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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. |
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| 40. |
Show that f(x) = x3 - 15x2+75x - 50 is increasing on R. |
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Answer» Consider f(x) = x3 - 15x2+75x - 50 Domain of the function is R. f’(x) = 3x2 - 30x+75 = 3(x2 - 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. |
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| 41. |
Show that f(x) = [x3 + 1/x3] is decreasing on ]-1, 1[. |
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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[. |
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| 42. |
Show that f(x) = [3/x + 5] is decreasing for all x ∈ R, where x ≠ 0. |
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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. |
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| 43. |
The function f(x) = 4 - 3x+3x2 - x3 isA. decreasing on RB. increasing on RC. strictly decreasing on RD. strictly increasing on R |
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Answer» f(x) = -x3+3x2 - 3x+4. f’(x) = -3x2+6x - 3 f’(x) = -3(x2 - 2x+1) f’(x) = -3(x -1)2 As f’(x) has –ve sign before 3 ⇒ f’(x) is decreasing over R. |
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| 44. |
Show that f(x) = x/sin x is increasing on ]0, π/2[. |
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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[. |
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| 45. |
Show that the height of a closed cylinder of given volume and the least surface area is equal to its diameter. |
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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 |
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| 46. |
Show that f(x) = 1/(1+x2) is decreasing for all x ≥ 0 |
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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 |
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| 47. |
Statements which are assumed to be true without proof are A) axioms B) counter examples C) conjectures D) open sentences |
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Answer» Correct option is: A) axioms |
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| 48. |
A sentence which is clearly true or false but not both is called a A) Axiom B) Theorem m C) Conjecture D) Statement |
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Answer» Correct option is: D) Statement |
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| 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 |
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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. |
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| 50. |
Show that the function f(x) = 5x – 2 is a strictly increasing function on R. |
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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. |
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