Class 11

### Find the (i) lengths of major axes, (ii) coordinates of the vertices

Given: $\mathbf{4}{{\mathbf{x}}^{\mathbf{2}}}+\text{ }\mathbf{9}{{\mathbf{y}}^{\mathbf{2}}}=\text{ }\mathbf{1}$ $\frac{{{x}^{2}}}{\frac{1}{4}}+\frac{{{y}^{2}}}{\frac{1}{9}}=1$….(i) Since, ...

### If , find the values of tan 2x

Answer: Replacing x by 2x, we get tan 2x = sin 2x / cos 2x Putting the values of sin 2x and cos 2x, we get

### If , find the values of cos 2x

Answer: We know that, cos 2x = 2cos2 x – 1 Putting the value, we get

### If , find the values of sin 2x

Answer: We know that, sin2x = 2 sinx cosx …(i) Here, we don’t have the value of sin x. So, firstly we have to find the value of sinx We know that, cos2 x + sin2 x = 1 Putting the values, we get...

### If , find the values of tan 2x

Answer: We know that: Replacing x by 2x, we get tan 2x = sin 2x / cos 2x Putting the values of sin 2x and cos 2x, we get

### If , find the values of cos 2x

Answer: We know that, cos 2x = 1 – 2sin2 x Putting the value, we get

### If , find the values of sin 2x

Answer: Given: $\sin x=\frac{\sqrt{5}}{3}$ To find: sin2x We know that, sin2x = 2 sinx cosx …(i) Here, we don’t have the value of cos x. So, firstly we have to find the value of cosx We know that,...

### Find the (v) length of the latus rectum of each of the following ellipses.

Given: $\mathbf{9}{{\mathbf{x}}^{\mathbf{2}}}+\text{ }\mathbf{16}{{\mathbf{y}}^{\mathbf{2}}}=\text{ }\mathbf{144}$ Divide by $144$ to both the sides, we get...

### In a class, of the boys and of the girls have an IQ of more than In this class, of the students are boys. If a student is selected at random and found to have and IQ of more than 150, find the probability that the student is a boy.

Let, I : students having IQ more than 150 B : Boys in the class G: Girls in the class We want to find $P(B \mid I)$ i.e. probability that selected student having IQ greater than 150 is a boy...

### If (n + 1)! = 12 × [(n – 1)!], find the value of n.

Answer : To Find: Value of n Given: (n+1)! = 12× [(n-1)!] Formula Used: n! = (n) × (n-1) × (n-2) × (n-3)............. 3 × 2 × 1 Now, (n+1)! = 12× [(n-1)!] ⇒ (n+1) × (n) × [(n-1)!] = 12 × [(n-1)!] ⇒...

### A coin is tossed. If a head comes up, a die is thrown, but if a tail comes up, the coin is tossed again. Find the probability of obtaining a head and an even number.

Given : let $\mathrm{H}$ be head, and $\mathrm{T}$ be tails where as $1,2,3,4,5,6$ be the numbers on the dice which are thrown when a head comes up or else coin is tossed again if its tail....

### A coin is tossed. If a head comes up, a die is thrown, but if a tail comes up, the coin is tossed again. Find the probability of obtaining(i) two tails(ii) a head and the number 6

Given : let $\mathrm{H}$ be head, and $\mathrm{T}$ be tails where as $1,2,3,4,5,6$ be the numbers on the dice which are thrown when a head comes up or else coin is tossed again if its tail....

### Let and be two the switches and let their probabilities of working be given by and Find the probability that the current flows from terminal A to terminal , when and are installed in parallel, as shown below:

Solution: Given: $\mathrm{S}_{1}$ and $\mathrm{S}_{2}$ are two swiches whose probabilities of working be given by $\mathrm{P}\left(\mathrm{S}_{1}\right)=\frac{2}{3}$ and...

### Find the coordinates of the point which divides the join of A(3, 2, 5) and B(-4, 2, -2) in the ratio 4 : 3.

Answer: The coordinates of point R that divides the line segment joining points P (x1, y1, z1) and Q (x2, y2, z2) in the ratio m: n are point A( 3, 2, 5 ) and B( -4, 2, -2 ), m and n are 4 and 3....

### Let A(2, 1, -3) and B(5, -8, 3) be two given points. Find the coordinates of the point of trisection of the segment AB.

Answer: The coordinates of point R that divides the line segment joining points P (x1, y1, z1) and Q (x2, y2, z2) in the ratio m: n are point A( 2, 1, -3 ) and B( 5, -8, 3 ), m and n are 2 and 1....

### Find the coordinates of the point that divides the join of A(-2, 4, 7) and B(3, – 5, 8) extremally in the ratio 2 : 1.

Answer: The coordinates of point R that divides the line segment joining points P (x1, y1, z1) and Q (x2, y2, z2) externally in the ratio m: n are point A( -2, 4, 7 ) and B( 3, -5, 8 ), m and n are...

### Find the ratio in which the point R(5, 4, -6) divides the join of P(3, 2, -4) and Q(9, 8, -10).

Answer: Let the ratio be k:1 in which point R divides point P and point Q, where m and n are k and 1. The point which this formula gives is already given. R(5,4,-6) and the joining points are P(3,...

### A machine operates only when all of its three components function. The probabilities of the failures of the first, second and third components are and , respectively. What is the probability that the machine will fail?

Given: let $A, B$ and $C$ be the three components of a machine which works only if all its three compenents function.the probabilities of the failures of $A, B$ and $C$ respectively is given i.e,...

### Find the ratio in which the point C(5, 9, -14) divides the join of A(2, -3, 4) and B(3, 1, -2).

Answer: Let the ratio be k:1 in which point R divides point P and point Q, where  m and n are k and 1. The point which this formula gives is already given. R(5,9,-14) and the joining points are P(2,...

### Find the ratio in which the plane x – 2y + 3z = 5 divides the join of A(3, -5, 4) and B(2, 3, -7). Find the coordinates of the point of intersection of the line and the plane.

Answer: The plane x – 2y + 3z = 5 divides the join of A(3, -5, 4) and B(2, 3, -7) in ratio k:1. The point which will come by section formula will be in the plane. Putting that in the plane equation...

### A town has two fire-extinguishing engines, functioning independently. The probability of availability of each engine when needed is What is the probability that(i) neither of them is available when needed?(ii) an engine is available when needed?

Given: Let $A$ and $B$ be two fire extinguishing engines. The probability of availability of each of the two fire extinguishing engines is given i.e., $\mathrm{P}(\mathrm{A})=0.95$ and...

### An article manufactured by a company consists of two parts and . In the process of manufacture of part X. 8 out of 100 parts may be defective. Similarly, 5 out of 100 parts of may be defective. Calculate the probability that the assembled product will not be defective.

Given: $X$ and $Y$ are the two parts of a company that manufactures an article. Here the probability of the parts being defective is given i.e, $\mathrm{P}(\mathrm{X})=\frac{8}{100}$ and...

### In how many different ways can a garland of 16 different flowers be made?

Answer : It is also in the form of a circle, So we need to arrange 16flowers in Circle 16 flowers can be arranged by 15! Now each flower have the same neighbour in the clockwise and anticlockwise...

### There are 4 candidates for the post of a chairman, and one is to be elected by votes of 5 men. In how many ways can the vote be given?

Answer : Let suppose 4 candidates be C1, C2, C3, C4 and 5 men be M1, M2, M3, M4, M5 Now M1 choose any one candidates from four (C1, C2, C3, C4) and give the vote to him by any 4 ways Similarly, M2...

### Kamal and Vimal appeared for an interview for two vacancies. The probability of Kamal’s selection is , and that of Vimal’s selection is Find the probability that only one of them will be selected.

event 'vimal is selected'. Therefore, $\mathrm{P}(\mathrm{A})=\frac{1}{3}$ and $\mathrm{P}(\mathrm{B})=\frac{1}{5}$ Also, $A$ and $B$ are independent .A and not $B$ are independent, not $A$ and $B$...

### In how many ways can 4 prizes be given to 3 boys when a boy is eligible for all prizes?

Answer : Let suppose 4 prizes be P1, P2, P3, P4 and 3 boys be B1, B2, B3 Now P1 can be distributed to 3 boys(B1, B2, B3) by 3 ways, Similarly, P2 can be distributed to 3 boys(B1, B2, B3) by 3 ways,...

### Let and be the events such that and or .State whether A and B are(i) mutually exclusive(ii) independent

Given: $A$ and $B$ are the events such that $P(A)=\frac{1}{2}$ and $P(B)=\frac{7}{12}$ and $P(\operatorname{not} A$ or $\operatorname{not} B)=\frac{1}{4}$ To Find: i)if A and B are mutually...

### In how many ways can 3 letters can be posted in 2 letterboxes?

Answer : Let Suppose Letterbox be B1, B2 and letters are L1, L2, L3 So L1 can be posted in any 2 letterboxes (B1, B2) by 2 ways Similarly, L2 can be posted in any 2 letterbox (B1, B2) by 2 ways...

### Find the point in xy-plane which is equidistant from the points A(2, 0, 3), B(0, 3, 2) and C(0, 0, 1).

Answer: The general point on xy plane is D(x, y, 0). Consider this point is equidistant to the points A(2, 0, 3), B(0, 3, 2) and C(0, 0, 1). ∴ AD = BD \$\sqrt {{{(x - 2)}^2} + {{(y - 0)}^2} + {{(0 -...

### In how many ways can 5 bananas be distributed among 3 boys, there being no restriction to the number of bananas each boy may get?

Answer : As there is 5 banana, So suppose it as B1, B2, B3, B4, B5 And Let the Boy be A1, A2, A3 So B1 can Be distributed to 3 Boys (A1, A2, A3) by 3 ways, Similarly, B2, B3, B4, B5 Can be...

### A child has 6 pockets. In how many ways, he can put 5 marbles in his pocket?

Answer : The first marble can be put into the pockets in 6 ways, i.e. Choose 1 Pocket From 6 by 6C1=6 Similarly second, third, Fourth, fifth & Sixth marble. Thus, the number of ways in which the...

### How many five – digit numbers can be formed with the digits 5, 4, 3, 5, 3?

Answer : To find: Number of 5 - digit numbers that can be formed 2 numbers are of 1 kind, and 2 are of another kind Total number of permutations = 30 number can be formed

### In how many ways can the letters of the word ‘ASSASSINATION’ be arranged so that all S’s are together?

Answer : To find: number of ways letters can be arranged such that all S’s are together Let all S’s be represented by a single letter Z New word is AAINATIONZ Number of arrangements = Letters can be...

### How many different words can be formed with the letters of the word ‘CAPTAIN’? In how many of these C and T are never together?

Answer : To find: number of words such that C and T are never together Number of words where C and T are never the together = Total numbers of words - Number of words where C and T are together...

### In how many ways can the letters of the word ‘PARALLEL’ be arranged so that all L’s do not come together?

Answer : To find: number of words where L do not come together Let the three L’s be treated as a single letter say Z Number of words with L not the together = Total number of words - Words with L’s...

### How many different signals can be transmitted by arranging 2 red, 3 yellow and 2 green flags on a pole, if all the seven flags are used to transmit a signal?

Answer : To find: Number of distinct signals possible Total number of fags = 7 2 are of 1 kind, 3 are of another kind, and 2 are of the 3rd kind ⇒ Number of distinct signals Hence 210 different...

### A child has three plastic toys bearing the digits 3, 3, 5 respectively. How many 3 – digit numbers can he make using them?

Answer : To find: number of 3 digit numbers he can make If all were distinct, he could have made 3! = 6 numbers But 2 number are the same So the number of possibilities He can make 3 three - digit...

### In how many ways can 5 children be arranged in a line such that (i) two of them, Rajan and Tanvy, are always together? (ii) two of them, Rajan and Tanvy, are never together,

Answer : (i) two of them, Rajan and Tanvy, are always together Consider Rajan and Tanvy as a group which can be arranged in 2! = 2 ways. The 3 children with this 1 group can be arranged in 4! = 24...

### How many permutations can be formed by the letters of the word ‘VOWELS’, when (i) there is no restriction on letters; (ii) each word begins with E; (iii) each word begins with O and ends with L; (iv) all vowels come together; (v) all consonants come together?

Answer : (i) There is no restriction on letters The word VOWELS contain 6 letters. The permutation of letters of the word will be 6! = 720 words. Each word begins with Here the position of letter E...

### Find the number of permutations of 10 objects, taken 4 at a time.

Answer : To find: the number of permutations of 10 objects, taken 4 at a time. Formula Used: Total number of ways in which n objects can be arranged in r places (Such that no object is replaced) is...

### In how many ways can 3 prizes be distributed among 4 girls, when (i)no girl gets more than one prize? (ii)a girl may get any number of prizes? (iii)no girl gets all the prizes?

Answer : (i)To distribute 3 prizes among 4 girls where no girl gets more than one prize the possible number of permutation possible are:4P3=24 To distribute 3 prizes among 4 girls where a girl may...