Entrance Questions

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11. What is the limit of a function at a point where the function is defined but not continuous?
A) The limit does not exist
B) The limit is equal to the function's value at that point
C) The limit is always zero
D) The limit is infinite

12. What is the limit of a function as x approaches a point where the function has a jump discontinuity?
A) The limit is equal to the jump size
B) The limit does not exist
C) The limit is always zero
D) The limit is infinite

13. Which of the following statements is true about the Intermediate Value Theorem?
A) It guarantees the existence of a limit for all functions
B) It states that a continuous function takes on every value between any two points
C) It is used to find vertical asymptotes of functions
D) It only applies to differentiable functions

14. What type of discontinuity occurs when a function approaches different values from the left and right at a point?
A) Removable discontinuity
B) Infinite discontinuity
C) Jump discontinuity
D) Oscillating discontinuity

15. Which of the following functions is not continuous at x = 0?
A) f(x) = x^2
B) g(x) = 1/x
C) h(x) = |x|
D) k(x) = sin(x)

16. What is the limit of the function f(x) = 1/x as x approaches infinity?
A) The limit does not exist
B) The limit is 0
C) The limit is 1
D) The limit is infinity

17. What is the limit of the function f(x) = sin(x)/x as x approaches 0?
A) The limit does not exist
B) The limit is 0
C) The limit is 1
D) The limit is infinity

18. Which of the following is a condition for a function to be continuous on a closed interval [a, b]?
A) The function is differentiable at every point in the interval
B) The function has no vertical asymptotes in the interval
C) The function is defined at every point in the interval
D) The limit of the function exists at every point in the interval

19. What is the limit of the function f(x) = x^2 - 4x + 4 as x approaches 2?
A) 0
B) 2
C) 4
D) 8

20. Which of the following functions is continuous for all real numbers?
A) f(x) = 1/x
B) g(x) = x^2
C) h(x) = |x|
D) k(x) = sin(x)

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Someone requested for questions from Grade 12 Unit 2. So here are questions from Limits and Continuity

1. What does the limit of a function represent?
A) The maximum value of the function
B) The average value of the function
C) The value the function approaches as the input approaches a certain value
D) The derivative of the function

2. Which symbol is used to denote a limit?
A) ∫
B) ∑
C) lim
D) ∆

3. What does it mean for a function to be continuous at a point?
A) The function is defined at that point
B) The function has a removable discontinuity at that point
C) The limit of the function exists at that point
D) The limit of the function equals the function's value at that point

4. What is the limit of a constant function?
A) The constant value itself
B) Zero
C) Infinity
D) It does not exist

5. Which of the following is a necessary condition for a function to be continuous at a point?
A) The function is differentiable at that point
B) The function has no vertical asymptotes at that point
C) The limit of the function exists at that point
D) The function's value at that point equals the limit of the function at that point

6. What is the limit of a function as x approaches infinity called?
A) Infinite limit
B) Limit at infinity
C) Horizontal asymptote
D) Vertical asymptote

7. Which of the following is a way to determine if a function is continuous at a point?
A) Calculate the derivative at that point
B) Check if the function is defined at that point
C) Evaluate the limit of the function at that point
D) Look for vertical asymptotes at that point

8. What is the limit of a rational function as x approaches a vertical asymptote?
A) The function approaches infinity
B) The limit does not exist
C) The function approaches zero
D) The function approaches a constant value

9. Which of the following is true about the continuity of a function at a point?
A) The function must have a hole at that point
B) The function must be defined at that point
C) The limit of the function must be infinite at that point
D) The function must be decreasing at that point

10. What is the limit of a piecewise function at a point where the pieces meet?
A) The average of the limits of the individual pieces
B) The limit of the function does not exist
C) The limit is equal to the value of the function at that point
D) The limit is always zero


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🧩Algebra- Revision Notes on Probability🧩
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The sum of all the probabilities in the sample space is 1.

The probability of an event which cannot occur is 0.

The probability of any event which is not in the sample space is zero.

The probability of an event which must occur is 1.

The probability of the sample space is 1.

The probability of an event not occurring is one minus the probability of it occurring.

The complement of an event E is denoted as E' and is written as P (E') = 1 - P (E)

P (A∪B) is written as P (A + B) and P (A ∩ B) is written as P (AB).

If A and B are mutually exclusive events, P(A or B) = P (A) + P (B)

When two events A and B are independent i.e. when event A has no effect on the probability of event B, the conditional probability of event B given event A is simply the probability of event B, that is P(B).

If events A and B are not independent, then the probability of the intersection of A and B (the probability that both events occur) is defined by P (A and B) = P (A) P (B|A).

A and B are independent if P (B/A) = P(B) and P(A/B) = P(A).

If E1, E2, ......... En are n independent events then P (E1 ∩ E2 ∩ ... ∩ En) = P (E1) P (E2) P (E3)...P (En).

Events E1, E2, E3, ......... En will be pairwise independent if P(Ai ∩ Aj) = P(Ai) P(Aj) i ≠ j.

P(Hi | A) = P(A | Hi) P(Hi) / ∑i P(A | Hi) P(Hi).

If A1, A2, ……An are exhaustive events and S is the sample space, then A1 U A2 U A3 U ............... U An = S

If E1, E2,….., En are mutually exclusive events, then P(E1 U E2 U ...... U En) = ∑P(Ei)

If the events are not mutually exclusive then P (A or B) = P (A) +P (B) – P (A and B)

Three events A, B and C are said to be mutually independent if P(A∩B) = P(A).P(B), P(B∩C) = P(B).P(C), P(A∩C) = P(A).P(C), P(A∩B∩C) = P(A).P(B).P(C)

The concept of mutually exclusive events is set theoretic in nature while the concept of independent events is probabilistic in nature.

If two events A and B are mutually exclusive,

P (A ∩ B) = 0 but P(A) P(B) ≠ 0 (In general)

⇒ P(A ∩ B) ≠ P(A) P(B)

⇒ Mutually exclusive events will not be independent.

The probability distribution of a count variable X is said to be the binomial distribution with parameters n and abbreviated B (n,p) if it satisfies the following conditions:

The total number of observations is fixed

The observations are independent.

Each outcome represents either a success or a failure.

The probability of success i.e. p is same for every outcome.

Some important facts related to binomial distribution:

(p + q)n = C0Pn + C1Pn-1q +...... Crpn-rqr +...+ Cnqn

The probability of getting at least k successes out of n trials is

P(x > k) = Σnx = k nCxpxqn-x

Σnx = k nCxqn-xpx = (q + p)n = 1

Mean of binomial distribution is np

Variance is npq

Standard deviation is given by (npq)1/2, where n

Sum of binomials is also binomial i.e. if X ~ B(n, p) and Y ~ B(m, p) are independent binomial variables with the same probability p, then X + Y is again a binomial variable with distribution X + Y ~ B(n + m, p).

If X ~ B(n, p) and, conditional on X, Y ~ B(X, q), then Y is a simple binomial variable with distributionY ~ B( n, pq).

The Bernoulli distribution is a special case of the binomial distribution, where n = 1. Symbolically, X ~ B (1, p) has the same meaning as X ~ Bern (p).

If an experiment has only two possible outcomes, then it is said to be a Bernoulli trial. The two outcomes are success and failure.

Any binomial distribution, B (n, p), is the distribution of the sum of n independent Bernoulli trials Bern (p), each with the same probability p.

The binomial distribution is a special case of the Poisson Binomial Distribution which is a sum of n independent non-identical Bernoulli trials Bern(pi). If X has the Poisson binomial distribution with p1 = … = pn = p then X ~ B(n, p).

A cumulative binomial probability refers to the probability that the binomial random variable falls within a specified range (e.g., is greater than or equal to a stated lower limit and less than or equal to a stated upper limit).

✅ Electrolytic Cells and Electrolysis

✍️ Electrolysis is a process of passing a direct current through the electrodes to achieve a chemical reaction. It is not possible to achieve a chemical reaction when the chosen electrolyte is in a solid-state.

✍️ Aqua regia also known as royal water is a yellow-orange mixture of concentrated nitric acid and hydrochloric acid in the ratio 1:3. It is used by an alchemist to dissolve noble metals like gold and silver.

✍️ Electrodes which do not take part in the chemical reaction during electrolysis are known as inert electrodes. Gold, silver and graphite do not take part in the process, but graphite is preferred because gold and silver electrodes are expensive.

✍️ In the electrolysis of NaCl, if the electrolyte is molten NaCl, then the only ions formed after dissociation are Na+ and Cl– ions. The cathode being a negatively charged electrode attracts the positive Na+ ions and neutralizes it to form Sodium metal.

✍️ Na2SO4 dissociates into Na+ and SO42- ions in the electrolysis of aqueous Na2SO4. Na+ has much lower reduction potential than water and hence Na+ ions are not reduced at the cathode. Instead, reduction of water occurs giving out hydrogen gas at the cathode.

✍️ In the electrolysis of aqueous CuSO4, Cu2+, SO42+, H+ and OH– are the ions formed after dissociation. Copper ions have much higher reduction potential than water. Hence, these ions are easily reduced and deposited as Cu at the cathode.

✍️ Electroplating is a process that uses direct electric current to carry metal ions from anode and carry them through the electrolyte containing the metal ion to the cathode to get a coherent metal coating.

✍️ The electrolyte in electrolysis should contain the metal to be coated, gold in this case. AuCN is used because it is exceptionally stable and doesn’t resist the flow of Au+ ions from anode to cathode.

✍️ The two electrodes that are used in a Daniell cell are zinc (as anode) and copper (as cathode) electrodes which are dipped in a solution containing its own ions, generally zinc sulphate and copper sulphate.

✍️ Yes, the distance between the electrodes is directly proportional to the resistance between them. As the distance between the two electrodes increases, the resistance offered by the electrolyte increases and therefore reduces the voltage between them.


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✅ Important Points to Remember: Electrochemistry – Galvanic Cells


✍️ A galvanic cell is a type of electrochemical cell that converts chemical energy into electrical energy. The electrochemical cell which converts electrical energy into chemical energy is called electrolytic cell.

✍️ Electrochemical cells are also called galvanic or voltaic cells, after the names of Luigi Galvani and Alessandro Volta who were the first to perform experiments on the conversion of chemical energy into electrical energy.

✍️ In a salt bridge, the electrolytes like KCl, KNO3 or NH4NO3 are preferred because their ions have almost equal transport number, viz., 0.5, i.e., they move with almost the same speed when an electric current flows through the

✍️ Galvanic cells are used to convert chemical energy into electrical energy. Two electrodes are usually set up in two separate beakers. The electrolytes taken in the two beakers are different. Galvanic cells are based upon spontaneous redox reactions. A salt bridge is used to set up this cell.

✍️ An anode is an electrode where oxidation takes place. An anode is a negative pole in a galvanic cell. In an electrolytic cell, the anode acts as the positive pole. Cathodes are electrodes where reduction takes place.

✍️ Greater the oxidation potential of metal, the more easily it can lose electrons and hence greater is its reactivity. As a result, a metal with greater oxidation potential can displace metals with lower oxidation potentials from their salt solutions.



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🌀Chemical Properties of Alkali Earth Metals:



1. Reaction with water :

Mg + H2O → MgO + H2

or, Mg + 2H2O → Mg (OH)2 + H2

Ca + 2H2O → Ca(OH)2 + H2

2. Formation of oxides and nitrides

Be + O2 (air) +Δ→ 2BeO

3Be + N2 (air) +Δ → Be3N2

Mg + air + Δ → MgO + Ng3N2

3. Formation of Nitrides

3M + N2 + Δ → M3N2

Be3N2 + Δ → 3Be + N2

Ba3N2 + 6H2O + Δ → 3Ba (OH)2 + 2NH3

Ca3N2 + 6H2O + Δ → 3Ca (OH)2 + 2NH3

4. Reaction with hydrogen:

M + H2 + Δ → MH2

Both BeH2 and MgH2 are covalent compounds having polymeric structures in which H – atoms between beryllium atoms are held together by three
centre – two electron (3C - 2e) bonds as shown below:

5. Reaction with carbon – (Formation of carbides)

When BeO is heated with carbon at 2175 – 2275 K a brick red coloured carbide of the formula Be2C is formed

2BeO +2C \xrightarrow[]{2175 - 2275 K}Be_2C +2CO

It is a covalent compound and react water forming methane.

Be2C + 4H2O → 2Be (OH)2 + CH4

6. Reaction with Ammonia:

Like alkali metal, the alkaline earth metals dissolve in liquid ammonia to give deep blue black solution from which ammoniates [ M (NH3)6 ]2+ can be recovered.

Anamolous Behaviour of Beryllium:
Be is harder than other members of its group.

Be is lighter than Mg.

Its melting and boiling points are higher than those of Mg & other members.

Be does not react with water while Mg reacts with boiling water.

BeO is amphoteric while MgO is weakly basic.

Be forms covalent compounds whereas other members form ionic compounds.

Beryllium carbide reacts with water to give methane whereas carbides of other alkaline earth metals gives acetylene gas.

Be2C + 4H2O → 2Be (OH)2 + CH4

Mg2C2 + 2H2O → Mg (OH)2 + C2H2

CaC2 + 2H2O → Ca (OH)2 + C2H2

Beryllium does not exhibit coordination number more than four as it has four orbitals in the valence shell. The other members of this group has coordination number

Diagonal relationship of Be with Al:
Unlike groups – 2 elements but like aluminium, beryllium forms covalent compounds.

The hydroxides of Be, [Be(OH)2] and aluminium [Al(OH)3] are amphoteric in nature, whereas those of other elements of group – 2 are basic in nature.

The oxides of both Be and Al i.e. BeO and Al2O3 are high melting insoluble solids.

BeCl2 and AlCl3 have bridged chloride polymeric structure.

The salts of beryllium as well as aluminium are extensively hydrolysed.

Carbides of both the metal reacts with water liberating methane gas.

Be2C + 4H2O → 2Be (OH)2 + CH4

AI4C3 + 12H2O → 4Al (OH)3 + 3CH4

The oxides and hydroxides of both Be and Al are amphoteric and dissolve in sodium hydroxide as well as in hydrochloric acid.
BeO + 2HCI → BeCI2 + H2O

BeO + 2NaOH → Na2BeO2 + H2O

Al2O3 + 6HCI → 2AICI3 + H2O

AI2O3 + 2NaOH → 2NaAIO2 + H2O

Like Al, Be is not readily attacked by acids because of the presence of an oxide film.
Calcium Carbonate (CaCO3):
It occurs in nature as marble, limestone, chalk, coral, calcite, etc. It is prepared as a white powder, known as precipitated chalk, by dissolving marble or limestone in hydrochloric acid and removing iron and aluminium present by precipitating with NH3, and then adding ammonium carbonate to the solution; the precipitate is filtered, washed and dried.

CaCl2 + (NH4)2CO3 →CaCO3 + 2NH4Cl
It dissolves in water containing CO2, forming Ca(HCO3)2 but is precipitated from solution by boiling.

CaCO3 + H2O + CO2 ↔️ Ca(HCO3)2


@Euee_Tips

1. Answer: C. going
Explanation: The correct form here is a gerund "going" after "remember" since it indicates an action that took place in the past.

2. Answer: B. to read
Explanation: The verb "advised" is followed by the infinitive "to read" to show the purpose of the advice.

3. Answer: A. interested
Explanation: "Interested" is an adjective that describes someone's feelings, and in this case, it is the correct form to use after "seems."

4. Answer: C. walking
Explanation: "Continued" is followed by the gerund form "walking" to show that the action was ongoing.

5. Answer: C. waiting
Explanation: "Can't stand" is followed by the gerund form "waiting" to express a general dislike for waiting in lines.

6. Answer: B. to submit
Explanation: "Expects" is followed by the infinitive "to submit" to indicate what the teacher expects the students to do.

7. Answer: C. reading
Explanation: "Enjoys" is followed by the gerund form "reading" to show the activity that she finds enjoyable.

8. Answer: B. to learn
Explanation: "Decided" is followed by the infinitive "to learn" to express the decision to engage in learning a new language.

9. Answer: B. to discuss
Explanation: "Agreed" is followed by the infinitive "to discuss" to indicate what the manager agreed to do.

10. Answer: C. going
Explanation: "Suggested" is followed by the gerund form "going" to show the activity recommended by the friends.



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Multiplechoice Questions On Gerunds And Infinitives:

1. I remember __ to that concert last year.

A. go
B. went
C. going
D. to go
2. She advised __ the document carefully before signing it.

A. read
B. to read
C. reading
D. reads
3. He seems __ in the project you proposed.

A. interested
B. to interest
C. interesting
D. interest
4. They continued __ despite the heavy rain.

A. walk
B. to walk
C. walking
D. walks
5. I can't stand __ in long lines at the store.

A. wait
B. to wait
C. waiting
D. waits
6. The teacher expects __ their assignments on time.

A. submit
B. to submit
C. submitting
D. submits
7. She enjoys __ novels in her free time.

A. read
B. to read
C. reading
D. reads
8. They decided __ a new language this summer.

A. learn
B. to learn
C. learning
D. learns
9. The manager agreed __ the proposal at the next meeting.

A. discuss
B. to discuss
C. discussing
D. discusses
10. My friends suggested __ to the new restaurant downtown.

A. go
B. went
C. going
D. to go



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