Solutions: The following is a graphical way for finding zeroes:- When solving any polynomial equation, the total number of zeroes equals the total number of times the curve meets the x-axis. i) The...
and 
will be independent, if 
and 
are mutually exclusive ![P\left(A^{\prime} B^{\prime}\right)=[1-P(A)][1-P(B)]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-ed48ea5f986832c34caa5efbac8759a1_l3.png "Rendered by QuickLaTeX.com")
Solution: Answer: (B) $P\left(A^{\prime} B^{\prime}\right)=[1-P(A)][1-P(B)]$Concept: Two events are independent, statistically independent, or stochastically independent if the occurrence of one...
Solution: Answer: D. $1 / 36$ Explanation: Given A pair of dice is rolled. Hence the number of outcomes $=36$ The chance of getting an even prime number on each die is represented by the...
of the students read Hindi newspaper, 
read English newspaper and 
read both Hindi and English newspapers. A student is selected at random. Given: The letters $H$ and $E$ stand for the number of students who read the Hindi newspaper and the English daily, respectively. Hence, $\mathrm{P}(\mathrm{H})=$ Probability of students who read...
(i) \[\mathbf{43}.\mathbf{123456789}\] (ii) \[\mathbf{0}.\mathbf{120120012000120000}\ldots \ldots \] $\left( III \right)$ $43.\overline{123456789}$ (i) \[\mathbf{43}.\mathbf{123456789}\] Since ,it...
and 
independent? : ‘the card drawn is a king’ Solution: Given: A deck of 52 cards. Concept: Two events are independent, statistically independent, or stochastically independent if the occurrence of one does not affect the probability of...
and are 
and 
respectively. If both try to solve the problem independently, find the probability that (i) The problem is solved Given, $P(A)=$ Probability of solving the problem by $A=1 / 2$ Concept: $\mathrm{P}(\mathrm{B})=$ Probability of solving the problem by $\mathrm{B}=1 / 3$ Because $A$ and $B$ both are...
Solution: Given A box with ten black and eight red balls. Total number of balls in box = 18 (i) Both balls are red. Probability of getting a red ball in first draw $=8 / 18=4 / 9$ As the ball is...
Given: A die is tossed three times. Then the sample space $S={1,2,3,4,5,6}$ Let $\mathrm{P}(\mathrm{A})=$ probability of getting an odd number in first throw. $\Rightarrow P(A)=3 / 6=1 / 2 .$ Let...
and are such that 
and 
not or 
. State whether and are independent?Solution: Given: $P(A)=1 / 2, P(B)=7 / 12$ and $P($ not $A$ or $\operatorname{not} B)=1 / 4$Concept: Two events are independent, statistically independent, or stochastically independent if the...
and are two events such that 
and 
, find not and not B).Solution: Given $P(A)=1 / 4, P(B)=1 / 2$ and $P(A \cap B)=1 / 8$ Concept: $P($ not $A$ and $\operatorname{not} B)=P\left(A^{\prime} \cap B^{\prime}\right)$ As, $\left{A^{\prime} \cap B^{\prime}=(A...
and be independent events with 
and 
. Find (i) 
Solution: Given: $\mathrm{P}(\mathrm{A})=0.3$ and $\mathrm{P}(\mathrm{B})=0.4$ (i) $P(A \cap B)$Given $P ( A )=0.3$ and $P ( B )=0.4$ When A and B are independent.Two events are independent,...
and be independent events with and . Find (i) Given $P ( A )=0.3$ and $P ( B )=0.4$ (i) $P(A \cap B)$ When A and B are independent. Two events are independent, statistically independent, or stochastically independent if the occurrence of one...
and are such that 
and 
. Find 
if they are (i) mutually exclusive (ii) independent.Solution: Given: $\mathrm{P}(\mathrm{A})=1 / 2, \mathrm{P}(\mathrm{A} \cup \mathrm{B})=1 / 5$ and let the probability of event be $\mathrm{P}(\mathrm{B})=\mathrm{p}$ (i) Mutually exclusive When A...
and be events with 
and 
. Are and independent?Solution: Given: $P(E)=3 / 5, P(F)=3 / 10$ and $P(E \cap F)=1 / 5$Evaluating the value of parameter & comparing with given, $P(E) . P(F)=3 / 5 \times 3 / 10=9 / 50 \neq 1 / 5$ $\Rightarrow P(E...
(i) ${}^{13}/{}_{3125}$ (ii) ${}^{17}/{}_{8}$ (iii) ${}^{64}/{}_{455}$ (iv) ${}^{15}/{}_{1600}$ (v) ${}^{29}/{}_{343}$ (vi)$23/{{2}^{3}}{{5}^{2}}$ (vii)...
in red and 
in green is tossed. Let be the event, ‘the number is even,’ and be the event, ‘the number is red’. Are A and B independent?Solution: The dice sample space will be be $S={1,2,3,4,5,6}$ Let $A$ be the event, the number is even, sample space of the event: $\Rightarrow \mathrm{A}={2,4,6}$ $\Rightarrow P(A)=3 / 6=1 / 2$ Now,...
The general trends in chemical properties of the group – 15 are as follows:(i) Reactivity towards hydrogen:The elements present in group 15 undergo a reaction with hydrogen to form hydrides of type...
be the event 
on the die’. Check whether and are independent events or not.Given: A fair coin and an unbiased die are tossed. Let A be the event head appears on the coin. So the sample space of the event will be: $\Rightarrow A={(\mathrm{H}, 1),(\mathrm{H}, 2),(\mathrm{H},...
Solution: Given: A box of oranges. Let A, B, and C represent the events that occur when the first, second, and third drawn oranges are all excellent. Now, $P(A)=P$ (good orange in first draw) $=12 /...
Given: A pack of 52 cards. As we all know, there are a total of 26 black cards. Let $A$ and $B$ represent the first and second drawn cards being black, respectively. Now if the probability of event...
and 
, find 
if and are independent events.Solution: Given: $P(A)=3 / 5$ and $P(B)=1 / 5$ As A and B are independent events.So we know that for independent events,$\Rightarrow P(A \cap B)=P(A) . P(B)$Substituiting the values we get, $=3 / 5...
The process of adsorption is always exothermic. There are two ways to understand this which are as follows :- (I) The residual forces on the adsorbent's surface are reduced as a result of...
Activation of adsorbent is the process of increasing the adsorption power of the adsorbent. The following methods are used to activate adsorbents: (I) Increasing the adsorbent surface area. This can...
$\left( 1 \right)$ ${}^{1}/{}_{\sqrt{2}}$ $(2)7\sqrt{5}$ $(3)6+\sqrt{2}$ $\left( 1 \right)$ Let us assume, ${}^{1}/{}_{\sqrt{2}}$ is rational. Then, we can find co-prime x and y$(y\ne 0)$ such that...
is irrational ?Let us assume $3+2\sqrt{5}$ is rational. Then we can find co-prime x and y($y\ne 0$ ) such that $3+2\sqrt{5}=\frac{x}{y}$ Rearranging , we get $2\sqrt{5}=\frac{x}{y}-3$...
is irrational number?Let, $\sqrt{5}$ is rational number. Therefore ,$\sqrt{5}$ can be written as ${}^{x}/{}_{y}$ (where,x & y are coprime) $y\sqrt{5}=x$ Squaring both sides ${{(y\sqrt{5})}^{2}}={{x}^{2}}$ $$...
Nitrogen is a chemically less reactive element because of the high stability of its molecular form ${{N}_{2}}$. In nitrogen molecule, the two nitrogen atoms form a triple bond among themselves....
Soln: Pollination Fertilization Pollen grains are transferred from the anther to the stigma of a flower. It is the result of the union of male and female gametes. Achieved by natural forces such as...
Since, Both Sonia and Ravi move in the same direction and at the same time, The method to find the time when they will be meeting again at the starting point is LCM(Lowest Common Multiple) of 18...
and 
are composite numbers?Numbers are of two types –prime numbers(two factors) & composite numbers(more then 2 factors). $(7\times 11\times 13)+13$ Taking 13 as common factor =$13[(7\times 11)+1]$ =$13(77+1)$...
can end with the digit 0 for any natural number n?If last digit of any number is 0,it should be divisible by 10. It also divisible by 2 and 5 as $10=2\times 5$ Prime factorisation of ${{6}^{n}}={{(2\times 3)}^{n}}$ By observation we can say...
As we know that, HCF×LCM=Product of the two given numbers(Multiplication of two given number) Since, HCF(Highest Common Factor) is given 9 of two numbers 306 & 657 Therefore using formula...
(i) damage to the immune system (ii) damage to lungs (iii) skin cancer (iv) peptic ulcers (a) (i) and (ii) (b) (ii) and (iv) (c) (i) and (iii) (d) (iii) and (iv) Correct option – ( c ) Explanation...
(a) unidirectional (b) bidirectional (c) multidirectional (d) no specific direction Correct option – ( a ) Explanation for correct option – the flow of energy in an ecosystem is always in one...
(a) food web (b) ecological pyramid (c) ecosystem (d) food chain Correct option – ( a ) Explanation for correct option – food web is composed of organisms of higher trophic level which eat several...
(a) heat energy (b) light energy (c) chemical energy (d) mechanical energy Correct option – ( c ) Explanation for correct option – energy that is transmitted from one trophic level to another one is...
(a) decomposers (b) producers (c) herbivores (d) carnivores Correct option – ( b ) Explanation for correct option – producers are those which produce their own food with the help of solar energy ,...
(a) chlorofluorocarbon compounds (b) carbon monoxide (c) methane (d) pesticides Correct option – ( a ) Explanation for correct option - in the stratosphere chlorofluorocarbon compounds are...
(a) eutrophication (b) pollution (c) biomagnification (d) accumulation Correct option – ( c ) Explanation for correct option – biomagnification refers to the rise in concentration of adverese...
Grass → Grasshopper → Frog → Snake → Hawk (a) 5 k J (b) 50 k J (c) 500 k J (d) 5000 k J Correct option – ( d ) Explanation for correct option – at a particular trophic level the available energy...
(a) all living organisms (b) non-living objects (c) both living organisms and non-living objects (d) sometimes living organisms and sometimes non-living objects Correct option – ( c ) Explanation...
(a) carnivores (b) herbivores (c) decomposers (d) producers Correct option – ( a ) Explanation for correct option – carnivores are placed at the third place in the trophic level . Explanation for...
(a) Pond (b) Crop field (c) Lake (d) Forest Correct option – ( b ) Explanation for correct option – crop field is artificial ecosystem . It is an area in which plants are grown selectively ....
(i) 140 (ii) 156 (iii) 3825 (iv) 5005 (v) 7429 i) 140 By Taking the LCM(Lowest Common Multiple) of 140, we will get the product of its prime factor. Therefore, $140=2\times 2\times 5\times...
and are events such that 
, then 
but 
Solution:Answer: D. $P(A)=P(B)$ Explanation: Given: $\mathrm{P}(\mathrm{A} \mid \mathrm{B})=\mathrm{P}(\mathrm{B} \mid \mathrm{A})$We are aware that, according to the concept of conditional...
, then is
Answer: C. Not defined Explanation: We are aware that, according to the concept of conditional probability, $\mathrm{P}(\mathrm{A} \mid \mathrm{B})=\frac{\mathrm{P}(\mathrm{A} \cap...
Let $E$ be the event that 'the coin shows a tail' and $F$ be the event that 'at least one die shows a 3 '. $\Rightarrow \mathrm{E}={1 \mathrm{~T}, 2 \mathrm{~T}, 4 \mathrm{~T}, 5 \mathrm{~T}}$ and...
Let $E$ be the event that 'the coin shows a tail' and $F$ be the event that 'at least one die shows a 3 '. $Rightarrow mathrm{E}={1 mathrm{~T}, 2 mathrm{~T}, 4 mathrm{~T}, 5 mathrm{~T}}$ and...
The sample space for the experiment in question is shown in the table below. $\mathrm{S}=\left{\begin{array}{c}(3,1),(3,2),(3,3),(3,4),(3,5),(3,6), \ (6,1),(6,2),(6,3),(6,4),(6,5),(6,6) \ 1...
Solution: Let $\mathrm{E}$ be the event that 'the sum of numbers on the dice is $4^{\prime}$ and $\mathrm{F}$ be the event that 'the two numbers appearing on throwing the two dice are different. So...
Solution: There are two sorts of questions in this section: True/False and Multiple Choice Questions (T/F or MCQ), and each of these types is further subdivided into Easy and Difficult categories,...
Solution Assume $B$ denotes boy and $G$ denote girl. Then, the sample space of the given experiment is $S={G G, G B, B G, B B}$ Let $\mathrm{E}$ be the event that 'both are girls'. $\Rightarrow...
and 
. Find (i) 
and 
and 
and 
Solution: The sample space for the given event is $S={1,2,3,4,5,6}$ Sample space of events,$E={1,3,5}, F={2,3}$ and $G={2,3,4,5} \ldots \ldots \ldots$ (i)The events probability of events are:...
(b) Find the conditional probability of obtaining the sum 8 , given that the red die resulted in a number less than 
Solution:Assuming: B denote black coloured die and R denote red coloured die. Therefore, the sample space for the specified experiment will be as follows:...
Solution: We assume that $\mathrm{M}$ denote mother, $\mathrm{F}$ denote father and $\mathrm{S}$ denote son. Then, the sample space for the given experiment will be: S= { MFS, SFM, FSM, MSF, SMF,...
Solution: We assume that $mathrm{M}$ denote mother, $mathrm{F}$ denote father and $mathrm{S}$ denote son. Then, the sample space for the given experiment will be: S= $\{$ MFS, SFM, FSM, MSF, SMF,...
Solution: We assume that $\mathrm{M}$ denote mother, $\mathrm{F}$ denote father and $\mathrm{S}$ denote son. Then, the sample space for the given experiment will be: S= ${$MFS, SFM, FSM, MSF, SMF,...
and 5 appears respectively on first two tosses.In the sample space, there are 216 outcomes, with each element of the sample space having three entries and taking the form $(x, y, z)$ where $1 \leq x, y, z \leq 6$. Considering the event, E: 4...
and 5 appears respectively on first two tosses.Solution: In the sample space, there are 216 outcomes, with each element of the sample space having three entries and taking the form $(x, y, z)$ where $\1 leq x, y, z leq 6$. Considering the event,...
and 5 appears respectively on first two tosses.Solution: In the sample space, there are 216 outcomes, with each element of the sample space having three entries and taking the form $(x, y, z)$ where $1 \leq x, y, z \leq 6$. Considering the...
: one coin shows head Solution: Determining the sample space of the given experiment is $\mathrm{S}=\{\mathrm{HH}, \mathrm{HT}, \mathrm{TH}, \Pi\}\}$ (i) Given that, E: tail appears on one coin And F: one coin shows head...
: head on third toss, 
: heads on first two tosses (ii) : at least two heads, : at most two heads (iii) : at most two tails, 
at least one tailSolution: The sample space of the given experiment will be: S={HHH, HHT, HTH, THH, HTT, THT, TTH, TTT} (i) Given that, E: head-on third toss And F: heads on first two tossesSo writing the sample...
and 
, findSolution: Given: $\mathrm{P}(\mathrm{A})=\frac{6}{11}, \mathrm{P}(\mathrm{B})=\frac{5}{11}, \mathrm{P}(\mathrm{A} \cup \mathrm{B})=\frac{7}{11}$ (i) We know that $\mathrm{P}\left(\mathrm{A}^{*}...
, if 
and 
.Solution: i) Given relation $2 \mathrm{P}(\mathrm{A})=\mathrm{P}(\mathrm{B})=\frac{5}{13}$ and $\mathrm{P}(\mathrm{A} \mid \mathrm{B})=\frac{2}{5}$ $\Rightarrow \mathrm{P}(\mathrm{B})=\frac{5}{13},...
and 
, findSolution:Given $\mathrm{P}(\mathrm{A})=0.8, \mathrm{P}(\mathrm{B})=0.5$ and $\mathrm{P}(\mathrm{B} \mid \mathrm{A})=0.4$ (i) By definition of conditional probability, we know that,...
, if 
and 
Solution: Given: $P(B)=0.5$ and $P(A \cap B)=0.32$ We know this because conditional probability is defined as follows: $\mathrm{P}(\mathrm{A} \mid \mathrm{B})=\frac{\mathrm{P}(\mathrm{A} \cap...
and are events such that 
and 
, find and Solution: Given: $P(E)=0.6, P(F)=0.3$ and $P(E \cap F)=0.2$ We know this because conditional probability is defined as follows: $P(A \mid B)=\frac{P(A \cap B)}{P(B)}$ By substituting the values we...
Given, Number of army contingent members=616 Number of army band members = 32 When we find the HCF(616,32) then we get maximum number of columns in which they can march. We use Euclid’s...
We know that in an isosceles triangle, length of two sides is equal. We know that formula to find the distance (d) between two points $\left( {{x}_{1}},{{y}_{1}} \right)$and $\left(...
All three points are collinear if the total of the lengths of any two line segments equals the length of the third line segment. Consider the following values, A = (1, 5) B = (2, 3) and C = (-2,...
Let a be any positive integer and b=6.Then,by Euclid’s algorithm, $a=6q+r$ for some integer $q\ge 0$ and r=0,1,2,3,4,5 because $0\le r<6$ . Therefore, $a=6q$ or $6q+1$ or $6q+2$ or $6q+3$...
(i) (2, 3), (4, 1) (ii) (-5, 7), (-1, 3)(iii) (a, b), (- a, – b) We know that formula to find the distance (d) between two points $\left( {{x}_{1}},{{y}_{1}} \right)$and $\left( {{x}_{2}},{{y}_{2}}...
(1)We have 225>135 So, we apply the division lemma to 225 & 135 to obtain 225=135×1+90 Here remainder 90≠0, we apply the division lemma again to 135 and 90 to obtain 135=90×1+45 We consider...
Solution - travelling of the electrical impulses takes place in the neurons . When it is transmitted to another neuron it takes the form of the neurotransmitter . Neurotransmitter are...
Solution : the function of control and coordination is regulated by the hormonal and nervous system . All the organs and organ system in the body are under the control of the brain . Brain controls...
Solution – reflex action is a rapid and involuntary response to a stimulus . Examples of reflex action are – in bright light the pupil of the changes in size . ...
Solution – some of the plant hormones are gibberellin – it is responsible for the growth of the stemauxin – it causes the cell to elongateabscisic acid – the growth of the plant is halted and...
(a) Thyroxin (b) Insulin (c) Adrenaline (d) Growth hormone (e) Testosterone. Solution – (a) Thyroxin : it is important in the role of digestion , heart and muscle function , development of brain and...
Solution – Brain and spinal cord makes up the central nervous system . peripheral nervous system comprises of nerves which are present outside the spinal cord . Central nervous system is very well...
Solution – major parts of the brain along with their functions are – forebrain - it has an important role in information processing , sensory and associative functions and voluntary motor...
Solution - neurons transmit information among the different areas of the brain and rest of the nervous system . Structure of the neuron – Cell body have dendrites which are responsible to receives...
(b) Dwarfism results due to deficiency of which hormone? (a) Which hormone is responsible for the changes noticed in females at puberty? (c) Blood sugar level rises due to deficiency of which...
Solution – electrical impulse releases chemicals at the end of the axon . These chemicals cross the synapse and in the dendrite similar electrical impulses are created .
Solution – our metabolism is controlled by thyroid hormone . Low levels of iodine results in low thyroid function and also causes abnormal enlargement of the thyroid gland ( hypothyroidism ) known...
Solution – when a plant shows some growth movement in response to a stimulus then it is known at tropic movement . Movement can be either direction that is it can be towards the stimulus or away...
Column AColumn B(a) Olfactory receptors(i) Tongue(b) Thermo receptors (temperature receptors)(ii) Eye(c) Gustatoreceptors(iii) Nose(d) Photoreceptors(iv) Skin solution Column AColumn B(a) Olfactory...
Figure Solution – DendriteCell bodyAxonAxon terminal
Figure Solution – figure ( a ) is more accurate . It is so because the plant is showing the ideal condition that is the roots are showing positive geotrophism and the shoot system is showing...
figure Solution – Pineal glandPituitary glandThyroid glandThymus
(a) elongation of cells (b) growth of stem (c) promotion of cell division (d) falling of senescent leaves. Solution – (a) elongation of cells – Auxin (b) growth of stem – gibberellin (c) promotion...
Solution – Sensory neuron Spinal cord Motor neuron Muscle Flow of electrical signals: Receptors in hand ==> Sensory neurons ==. Spinal cord (CNS) ==>Motor neurons ==> Effector - muscles in...
(a) reproductive and endocrine systems (b) respiratory and nervous systems (c) endocrine and digestive systems (d) nervous and endocrine systems Correct option – ( d ) Explanation for correct option...
(a) oestrogen (b) testosterone (c) insulin (d) growth hormone Correct option – ( b ) Explanation for correct option – Testosterone is the male sex hormone produces in testes of males which promotes...
(a) his blood pressure was low (b) his heart was beating slowly (c) he was suffering from goitre (d) his sugar level in blood was high Correct option – ( d ) Explanation for correct option –...
Can you identify these? The given cells are, 1.Mitochondria which is the power house of the cell. 2.Centrosome which is present in the animal cell alone and contains one or two centrioles. 3.Golgi...