CTET January 2012 Question Paper-2

Solution: 4 – (2 – 9)⁰ + 3² ÷ 1 +3 = 4 – 1 + 9 + 3 = 15

Explain: If the difference between the numerator and denominator of every fraction is same then the fraction with the least value numerator will be least fraction and the greatest value numerator will be greatest fraction.

Here difference between the numerator and denominator of every fraction is 1

And  the fraction with the least value numerator is 10/11

So least fraction is 10/11

Explain: Let the remainder is R

$
\,\,R\,\,of\,\,\frac{3^{98}}{5}=R\,\,of\,\,\left[ \frac{\left( 3^4 \right) ^{24}\times 3^2}{5} \right]
\\
=\,\,R\,\,of\,\,\left[ \frac{\left\{ R\,\,of\,\,\frac{\left( 3^4 \right) ^{24}}{5}\times R\,\,of\,\,\frac{3^2}{5} \right\}}{5} \right]
\\
Now\,\,R\,\,of\,\,\frac{\left( 3^4 \right) ^{24}}{5}\,\,=\,\,R\,\,of\,\,\frac{81^{24}}{5}\,\,
\\
=\,\,R\,\,of\,\, \frac{\left( 80+1 \right) ^{24}}{5}
\\
=\text{1 }\left( u\sin g\,\,R\,\,of\,\,\frac{\left( ax+1 \right) ^n}{a}=1 \right)
\\
and\,\,R\,\,of\,\,\frac{3^2}{5}\,\,=\,\,4
\\
\therefore \,\,R\,\,of\,\,\frac{3^{98}}{5}\,\,
\\
=\,\,R\,\,of\,\,\frac{1\times 4}{5}\,\,=\,\,4
$

Solution: y decreases by = $\frac{10}{110}\times \text{100 }=\,\,\frac{100}{11}%$

Trick:  If two positive numbers a and b are inversely proportion and a increase by x%, then y decreases by $\frac{x}{100+x}\times \text{100%}$

Explain: Any digit from 2 to 9 is used 20 times to write from 1 to 100. Only 1 is used 21 times.  ∴ 2 is used to write from 11 to 100 = 20 – 1 = 19

Any digit from 1 to 9 is used 20 times to write from 101 to 200. Only 2 is used 21 times.

∴ 2 is used to write from 101 to 199 = 21 – 1 = 20

∴2 is used to write from 11 to 199 = 19 + 20 = 39

Solution: x² – y² + x + y – z² +2yz – z

= x² – (y² – 2yz + z²) +x + y – z

= (x – y + z) ( x + y – z) + ( x + y ­– z)

= ( x + y ­– z) ( x – y + z +1 )

∴ ( x – y + z +1 ) is a factor of given expression.

Solution: value of one number is 1 – $\frac{1}{n}$ and (n – 1) numbers =1

Sum = $\left( n-1 \right) \times 1+\,\,\text{1 }-\,\,\frac{1}{n}=\,\,n-\frac{1}{n}$

Means = ( sum/ total numbers)

= $\frac{n-\frac{1}{n}}{n}=\,\,1-\frac{1}{n^2}$

Solution: 4 × 1 + 8 ÷ 2 = 4 + 4 = 8

Solution: A = $\frac{3}{4}\div \frac{5}{6}=\frac{3\times 6}{4\times 5}=0.9$

B = 3 ÷ [(4 ÷ 5) ÷ 6]

= $
\frac{3}{\left( \frac{4}{5}\div 6 \right)}=\frac{3}{\frac{4}{5\times 6}}
\\
=\frac{3\times 5\times 6}{4}=22.5
$

C = [3 ÷ (4 ÷ 5)] ÷ 6

= $\frac{\left( 3\div \frac{4}{5} \right)}{6}=\frac{\frac{3\times 5}{4}}{6}=\frac{3\times 5}{4\times 6}=0.625$

D = 3 ÷ 4(5 ÷ 6)

= $\frac{3}{4\left( \frac{5}{6} \right)}=\frac{3\times 6}{4\times 5}=0.9$

Solution: 3.75 × 10^–7 = 0.000000375,

3¾ × 10^–7= 3.75 × 10^–7 = 0.000000375,

375 × 10^–9 = 0.000000375,

all are same but 3/8 × 10^–7 = 0.0000000375

Solution: If diameter of a cylindrical increased then radius also same increased.

Let initial radius (r₁)= 100r and height (h₁) = 100h

Volume(v₁) = π (100r)².100h = 1000000πr²h

Now Volume (V₂) = π (125r)².h₂ = 1000000πr²h

⇒ h₂ = 1000000h/15625 = 64h = 100h – 36h

Height decreased 36%

Solution: ∠D + ∠E = 90° ⇒ ∠D + 2∠D = 90° ⇒∠D = 30°

∴∠E = 60°  ∴ ∠A = 60° and ∠B = 60° ( CA = CB )

∴∠ACD = 60° + 60° = 120°

Explain: probability of getting an orange P(A) = 3/10

probability of getting a banana P(B) = 5/10

probability of getting an orange or a banana P( A or B) = P(A) + P(B) = 3/10 + 5/10= 4/5

Explain:

= (3 + 4)² + (2 + 4) – 4² + 4

= 49 + 6 – 20 = 35

Explain: 6x + 6y + x + y = 63 (put xy = 6)

⇒ 7(x + y) = 63 ⇒ x + y =9

Now x² + y² = (x + y)² – 2xy = 81 –12 = 69

Explain: Arrange the given value in ascending order 6, 6, 8, 8, 9 , 9, 9, 14

Hence given sequence have even number then middle value are 8 and 9

∴ Median = $\frac{8+9}{2}=8.5$

Mode = value that appears maximum number of times = 9

Range = maximum value – minimum value = 14 – 6 = 8

Required mean = $\frac{8.5+9+8}{3}=8.5$

Explain: Sales in January = 3.75 million

Sales in February = 3.5 million

Sales in March = 4.25 million

Sales in April = 4.75 million

Sales in May = 3.75 million

Sales in June = 4.25 million

Sales Change January – February = 3.75 – 3.5 = 0.25 million

February – March = 4.25 – 3.5 = 0.75 million

April – May = 4.75 – 3.75 = 1 million

May – June = 4.25 – 3.75 = 0.50 million

From the given data, it can be observed that change in sales is highest from April to May.

Explain: PQ = TR, PR = TQ and QR is the common side of two triangles.

As all three sides of both triangles are equal, the given triangles are congruent

The angle corresponding to ∠P is ∠Q and that is corresponding to ∠R.

Therefore, ∆PQR and ∆TRQ are congruent.

Explain: Any number is divisible by a given set of numbers if that number is divisible by the LCM of the numbers.

LCM of 10, 12, 15 and 18 = 180; 900 is the least number that is divisible by 180. It is also a perfect square.

Hence, 900 is the required number.

Explain: A parallelogram does not have any line of symmetry.

Explain: Ankur has a problem in comprehending the language of the question, though he can solve equations. Providing assistance to such students (through pictures or suitable manipulators) in understanding problems and making them solve a number of similar problems will be helpful. Slowly, they will learn to think abstract terms like other students in the class.

Explain: Technology integration using technology to enhance the understanding of students in a particular subject. For subjects like coordinate geometry, Geo-algebra very effective. The teacher can demonstrate it in the class, or the students can experience the joy learning themselves by using this software on à computer.

Explain:  Summative assessment is the ‘assessment of learning. i.e., to assess how much the student has learnt. This is done by a paper-pencil test that consists of a fixed number of questions to be completed in a given time period. The students are graded on the basis of the scores they obtain in the test.

Explain: Most of the time, he saw the right-angled triangle as he had drawn, but he could not analyze that his teacher also drew a right-angled triangle: only the position of the angle had changed.

Explain: Repeated use of terms through activities like crossword or jigsaw puzzles, quizzes, etc., helps students remember difficult terms and their meanings.

Explain: The students will get an idea of ratio and proportion when they see their teacher demonstrating ‘copy and paste’ and ‘copy and enlarge’ or copy and reduce” and explaining how many times the image has been enlarged or reduced. She may draw the attention of the students towards the proportionate increase or decrease in the size of the image.

Explain: It is an inductive method wherein a number of examples are given and students find the common feature in all of them and generalize the result. So, it is a method of constructing a formula with the help of a sufficient number of concrete examples. An inductive approach is based on psychological principles: particular to general, concrete to abstract, known to unknown, simple to complex, etc.

Explain: Manish knows that there are students with different learning styles in his class. He uses a combination of different methods to teach the concept of symmetry so that all the students are benefitted from one or the other.

Explain: The problem-solving method is a gradual and systematic guiding through the hierarchy of mathematical notions, ideas and techniques. This method makes students retrieve their prior information and apply it to the problems based on new or varying situations.

Explain: He has confused it with multiplication where (–) (–) become (+).

Explain: Litmus paper is used for testing the acidic or basic nature of a substance. There are two color litmus papers, i.e., red and blue. If red litmus turns blue on dipping in a solution, it shows the given solution is basic. f blue litmus turns red on dipping in a solution, it shows the given solution is acidic, Baking soda is basic in nature so it would convert the red litmus to blue.

Explain: There are nine sub-divisions (0.1-0.9) between two markings. If there are five mall divisions between two markings, then it shows that each sub-division covers a reading of 0.2. So, if there are five small divisions between 55°C and 56°C then each small division can read up to 55.2°C and so on.

Explain: Rhizobium is a bacterium found in the root nodules of the leguminous plants. It is capable of fixing the nitrogen present in the soil into the forms that can be easily absorbed by the plants.

Explain: Insoluble calcium carbonate is formed when carbon dioxide is passed through lime water. This gives a milky appearance to the lime water. the reaction that occurs is as follows:

CO₂ (aq) +Ca(OH)₂ (ag) → CaCO₃ (s) + H₂O

Explain: The amount of heat produced by a metallic conductor on passing the electric current depends upon three following:

Material: as less heat is produced by good conductors

Thickness: as production of heat decreases with increase in the thickness

Length: as production of heat increase with increase in the length

Explain: PET (polyethylene terephthalate) is commonly used polyester for making clothing fibers, food and liquid containers. It is commonly referred to as polyester (common name of PET). Acrylic and rayon are artificial fibers whereas polyamide is a heteropolymer that naturally occurs in wool and silk.

Explain: The minerals required for proper functioning of muscles are sodium, magnesium and potassium. Iodine plays a role in functioning of thyroid gland. Iron helps in formation of blood cells while fluorine helps in preventing the dental caries. Calcium is important for making the teeth and bones strong. Deficiency of sodium and potassium causes muscle weakness and muscle Cramps, respectively.

Explain: Antibiotics are given for fighting bacterial diseases such as cholera and typhoid. Dengue, hepatitis A and influenza are viral diseases.

Explain: Combustion is the process in which substances are burnt in air or Oxygen with the evolution of light and heat. Complete combustion Occurs when a fuel is burnt with enough oxygen to Support the process of burning and converting all of the carbon into carbon dioxide. Water and energy are also released during this process.

Explain: Bacteria includes the smallest species o organisms on Earth with a wide variation in their sizes. They possess a rigid cell wall to protect themselves. Cyan bacteria are not the smallest bacteria.

Explain: Fermentation is a process in which sugar (glucose) is converted into alcohol (ethanol) with a release of carbon dioxide. It is a form of anaerobic respiration that occurs naturally in yeast. The reaction that occurs is as follows.

C₆ H₁₂ O₆ → 2 C₂ H₅OH +CO₂

Explain: The calorific value of a fuel is the amount of heat generated when 1 kg of the fuel burns completely. It is commonly expressed in terms of J/kg, however, its SI unit is joule per Kilogram 0/kg).

Explain: Slow combustion refers to the process of combustion occurring at low temperatures that produces less amount of heat. Respiration and burning of a candle are examples of slow combustion as they occur at low temperatures. Rusting of iron is an oxidation reaction but it is not combustion as energy is not released during this process. Similarly, in photosynthesis, there is no release of energy thus, it is also not a combustion reaction.

Explain: Rayon is a manmade fiber which is also referred to as artificial silk due to its appearance that resembles silk. It is made from the cellulose obtained from wood pulp.

Explain: The frequency of sound determines the pitch of the sound, whereas the amplitude of sound determines the loudness of the sound. If the frequency of sound is more, the pitch will be higher and vice versa. Similarly, a high amplitude sound is louder than the sound of low amplitude.

Explain: Roughage (dietary fibers) is the part of the plant food products that humans are unable to digest. The dietary fibers contain cellulose that cannot be digested by humans.

Explain: Seismograph is a device that records the seismic waves (waves of energy travelling in the layers of Earth) that originate from the focus (epicenter) of the earthquake.

Explain: Carbon monoxide is a poisonous gas that interferes with the binding of oxygen to hemoglobin, which is required for the transport of oxygen in the body. It has a higher affinity to bind with hemoglobin than oxygen and may cause suffocation if inhaled.

Explain: Greenhouse gases are gases that cause the greenhouse effect. The main greenhouse gas in the atmosphere is carbon dioxide. Other greenhouse gases are methane, nitrous oxide, fluorinated Bases, and water vapors.

Explain: Litmus paper is used for testing the acidic or basic nature of a substance. There are two color litmus papers, i.e. red and blue. If red litmus turns blue on dipping in a solution, it shows the given solution Is basic. If blue litmus turns red on dipping in a solution, it shows the given solution is acidic. Sodium chloride is neutral; so, there will be no change in color of the litmus paper. Lime water and washing soda are basic; thus, they will change the red litmus blue. Vinegar is acidic; so, it will change the blue litmus red.

Explain: A combination of project cum laboratory method should be used by the teacher. In both of these methods, students have to use process skills like identifying a problem, defining the problem, making a hypothesis, testing hypothesis by different ways experimentation, selection of apparatus, setting apparatus, taking readings, measurements, recording data, data analysis, making conclusions, comparing results etc. These skills are not used in lecture, questioning, demonstration or assignments.

Explain: Out of the four options given, top promote rational thinking is the major objective as t prepares the person to solve problems and take logical decisions without getting impulsive or illogical.