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, ...

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

### 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...

### Mark the tick against the correct answer in the following: The principal value of is

A.

B.

C.

D.

Solution: Option(C) is correct. To Find: The Principle value of $\cos ^{-1}\left(\frac{-1}{\sqrt{2}}\right)$ Let the principle value be given by $\mathrm{x}$ Now, let $x=\cos...

### Find the (iii) coordinates of the foci, (iv) eccentricity

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...

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

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...

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

Given \[{{\mathbf{x}}^{\mathbf{2}}}+\text{ }\mathbf{4}{{\mathbf{y}}^{\mathbf{2}}}=\text{ }\mathbf{100}\] Divide by \[100\] to both the sides, we get...

### Find the (iii) coordinates of the foci, (iv) eccentricity

Given \[{{\mathbf{x}}^{\mathbf{2}}}+\text{ }\mathbf{4}{{\mathbf{y}}^{\mathbf{2}}}=\text{ }\mathbf{100}\] Divide by \[100\] to both the sides, we get...

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

Given \[{{\mathbf{x}}^{\mathbf{2}}}+\text{ }\mathbf{4}{{\mathbf{y}}^{\mathbf{2}}}=\text{ }\mathbf{100}\] Divide by \[100\] to both the sides, we get...

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

Given: \[\mathbf{16}{{\mathbf{x}}^{\mathbf{2}}}+\text{ }\mathbf{25}{{\mathbf{y}}^{\mathbf{2}}}=\text{ }\mathbf{400}\] Divide by \[400\] to both the sides, we get...

### Find the (iii) coordinates of the foci, (iv) eccentricity

Given: \[\mathbf{16}{{\mathbf{x}}^{\mathbf{2}}}+\text{ }\mathbf{25}{{\mathbf{y}}^{\mathbf{2}}}=\text{ }\mathbf{400}\] Divide by \[400\] to both the sides, we get...

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

Given: \[\mathbf{16}{{\mathbf{x}}^{\mathbf{2}}}+\text{ }\mathbf{25}{{\mathbf{y}}^{\mathbf{2}}}=\text{ }\mathbf{400}\] Divide by \[400\] to both the sides, we get...

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

Given: \[\frac{{{x}^{2}}}{49}+\frac{{{y}^{2}}}{36}=1\]…(i) Since, \[49\text{ }>\text{ }36\] So, above equation is of the form, \[\frac{{{x}^{2}}}{{{a}^{2}}}+\frac{{{y}^{2}}}{{{b}^{2}}}=1\] …(ii)...

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

Given: \[\frac{{{x}^{2}}}{49}+\frac{{{y}^{2}}}{36}=1\]…(i) Since, \[49\text{ }>\text{ }36\] So, above equation is of the form, \[\frac{{{x}^{2}}}{{{a}^{2}}}+\frac{{{y}^{2}}}{{{b}^{2}}}=1\] …(ii)...

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

Given: \[\frac{{{x}^{2}}}{{{a}^{2}}}+\frac{{{y}^{2}}}{{{b}^{2}}}=1\]…(i) Since, \[25>9\] So, above equation is of the form, \[\frac{{{x}^{2}}}{{{a}^{2}}}+\frac{{{y}^{2}}}{{{b}^{2}}}=1\]…(ii)...

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

Given: \[\frac{{{x}^{2}}}{{{a}^{2}}}+\frac{{{y}^{2}}}{{{b}^{2}}}=1\]…(i) Since, \[25>9\] So, above equation is of the form, \[\frac{{{x}^{2}}}{{{a}^{2}}}+\frac{{{y}^{2}}}{{{b}^{2}}}=1\]…(ii)...

### Prove that

Answer:

### Prove that:

Answer: Taking LHS

### Prove that:

Answer:

### Prove that:

Answer: Taking LHS

### If , prove that

Answer: cosx + cosy = 1/3--------- i sinx + siny = 1/3----------ii dividing ii by i we get Using the formula, sinA + sinB = 2sin(A+B)/2 . cos(A-B)/2 cosA + cosB = 2cos(A+B)/2 ....

### Prove that:

Answer; Taking LHS = 3/16

### Prove that:

= 3/16

### Prove that:

Answer: = 1/16

### Prove that:

Answer: = cot 5x Using the formulas, 2cosAsinB = sin (A + B) – sin (A - B) 2sinAsinB = cos (A - B) – cos (A + B)

### Prove that:

Answer: = tan 2x Using the formulas, 2cosAsinB = sin (A + B) – sin (A - B) 2cosAcosB = cos (A + B) + cos (A - B) 2sinAsinB = cos (A - B) – cos (A + B)

### Prove that:

Answer: =Sin3x+sin2x-sinx = (sin3x- sinx)+sin2x = ( 2cos(3x+x)/2 . sin(3x-x)/2 ) + sin2x = 2cos2xsinx +sin2x = 2cos2xsinx + 2sinxcosx = 2sinx (cos2x + cosx ) = 2sinx (2cos(2x+x)/2 . cos(2x-x)/2 ) =...

### Prove that:

Answer: Using the formula, sinA + sinB = 2sin(A+B)/2 . cos(A-B)/2 sinA - sinB = 2cos(A+B)/2 . sin(A-B)/2

### Prove that:

Answer: Using the formula, cosA - cosB = -2sin(A+B)/2 . sin(A-B)/2 cosA + cosB = 2cos(A+B)/2 . cos(A-B)/2

### Prove that:

Answer: Using the formula, cosA - cosB = -2sin(A+B)/2 . sin(A-B)/2 sinA - sinB = 2cos(A+B)/2 . sin(A-B)/2

### Prove that:

Using the formula, cosA - cosB = -2sin(A+B)/2 . sin(A-B)/2 sinA - sinB = 2cos(A+B)/2 . sin(A-B)/2

### Prove that (sin 3x + sin x) sin x + (cos 3x – cos x) cos x = 0

Answer: = (sin 3x + sin x) sin x + (cos 3x – cos x) cos x = (2sin(3x+x)/2 . cos(3x-x)/2) sin x + (-2sin(3x+x)/2 . sin(3x-x)/2) cosx = (2sin2x cosx) sinx-(2sin2x sinx) cosx = 0. Using the formula,...

### Prove that cot 4x (sin 5x + sin 3x) = cot x (sin 5x – sin 3x)

Answer: L.H.S cot 4x (sin 5x + sin3x) = cot 4x (2sin(5x+3x)/2 . cos (5x-3x)/2 ) = cot 4x (2 sin4x cosx) = (cos4x / sin 4x).(2 sin4x cosx) = 2cos4xcosx R.H.S cot x (sin 5x - sin3x) = cot x...

### Prove that:

Answer: = sin 6x / cos 6x = tan 6x Using the formula, sinA + sinB = 2sin(A+B)/2 . cos(A-B)/2 cosA + cosB = 2cos(A+B)/2 . cos(A-B)/2

### Prove that:

Answer: Using the formula, cosA - cosB = -2sin(A+B)/2 . sin(A-B)/2 sinA - sinB = 2cos(A+B)/2 . sin(A-B)/2

### Prove that:

Answer: = tan4x Using the formula, sinA + sinB = 2sin(A+B)/2 . cos(A-B)/2 cosA + cosB = 2cos(A+B)/2 . cos(A-B)/2

### Prove that:

Answer: = sin x / cos x = tanx Using the formula, sinA - sinB = 2cos(A+B)/2 . sin(A-B)/2 cosA + cosB = 2cos(A+B)/2 . cos(A-B)/2

### Prove that

Answer: = cos x / sin x = cot x Using the formula, sinA + sinB = 2sin (A+B)/2 . cos (A-B)/2 cosA - cosB = -2sin (A+B)/2 . sin (A-B)/2

### Express each of the following as an algebraic sum of sines or cosines : (iii) 2cos 7x cos 3x (iv) 2sin 8x sin 2x

Answer: iii) 2cos7xcos3x = cos (7x+3x) + cos (7x – 3x) =cos10x + cos 4x Using, 2cosAcosB = cos (A+ B) + cos (A - B) iv)2sin8xsin2x = cos (8x - 2x) – cos (8x + 2x) = cos6x – cos10x Using, 2sinAsinB =...

### Express each of the following as an algebraic sum of sines or cosines : (i) 2sin 6x cos 4x (ii) 2cos 5x din 3x

Answer: i) 2sin 6x cos 4x = sin (6x+4x) + sin (6x-4x) = sin 10x + sin 2x Using, 2sinAcosB = sin (A+ B) + sin (A - B) ii) 2cos 5x sin3x = sin (5x + 3x) – sin (5x – 3x) = sin8x – sin2x Using,...

### Express each of the following as a product. 3. cos 7x + cos 5x 4. cos2x – cos 4x

Answer:

### Express each of the following as a product. 1. sin 10x + sin 6x 2. sin 7x – sin 3x

Answer:

### Prove that:

### Prove that

Answer: In this question the following formulas will be used:

### Prove that

Answer: In this question the following formulas will be used: sin (A - B) = sinA cos B - cosA sinB cos (A - B) = cosAcosB+ sinAsinB

### Prove that cos x + cos (120 – x) + cos (120 + x) = 0

Answer: In this question the following formulas will be used: cos (A + B) = cosAcosB – sinAsinB cos (A - B) = cosAcosB+ sinAsinB = cos x + cos 120 cosx – sin120sinx + cos 120 cosx+sin120sinx = cosx...

### Prove that sin(150 + x) + sin (150 – x) = cos x

Answer: In this question the following formula will be used: Sin( A +B)= sinA cos B + cosA sinB Sin( A - B)= sinA cos B - cosA sinB = sin150 cosx + cos 150 sinx + sin150 cosx – cos150 sinx =2sin150...

### 7. Construct a ABC in which BC = 8 cm, B 45 and C 60 . Construct another

### 6. Construct a ABC in which AB = 6 cm, A 30and AB 60 . Construct

Sol: Steps of Construction with base AB’ = 8 cm. Step 1: Draw a line segment AB = 6cm. Step 2: At A, draw ÐXAB = 30°. Step 3: At B, draw ÐYBA = 60°. Suppose AX and BY intersect at C. Thus, DABC is...

### 5. Construct a ABC with BC = 7 cm, B 60 and AB = 6 cm. Construct another triangle

### Construct a triangle with sides 5 cm, 6 cm, and 7 cm and then another triangle whose sides

### 3. Construct a PQR , in which PQ = 6 cm, QR = 7 cm and PR =- 8 cm. Then, construct

### Draw a line segment of length 7.6 cm and divide it in the ratio 5:8. Measure the two parts.

### Draw a line segment AB of length 7 cm. Using ruler and compasses, find a point P on AB

### In how many ways can the letters of the word ‘PENCIL’ be arranged so that N is always next to E? be arranged so that N is always next to E?

### A child has plastic toys bearing the digits 4, 4 and 5. How many 3-digit numbers can he make using them?

### In how many ways can 5 boys and 3 girls be seated in a row so that each girl is between 2 boys?

### How many 5-digit numbers can be formed by using the digits 0, 1 and 2?

### In how many ways can 4 letters be posted in 5 letter boxes?

Answer : Given: We have 4 letters and 5 letter boxes To Find: Number of ways of posting letters. One letter can be posted in any of 5 letter boxes. We have to assume that all the letters are...

### How many words can be formed by the letters of the word ‘SUNDAY’?

### How many permutations of the letters of the word ‘APPLE’ are there?

### How many different words can be formed by using all the letters of the word ‘ALLAHABAD’?

### In how many ways can the letters of the word ‘PERMUTATIONS’ be arranged if each word starts with P and ends with S?

### In how many ways can the letters of the word ‘CHEESE’ be arranged?

### How many numbers divisible by 5 and lying between 4000 and 5000 can be formed from the digits 4, 5, 6, 7, 8 if repetition of digits is allowed?

### How many 3-digit numbers above 600 can be formed by using the digits 2, 3, 4, 5, 6, if repetition of digits is allowed?

### . How many 3-digit numbers are there with no digit repeated?

### find the value of x.

### Suppose of men and of women have grey hair. A grey-haired person is selected at random. What is the probability of this person being male? Assume that there is an equal number of males and females.

Let MG : Men having grey hair WG: Women having grey hair G: Having grey hair Given an equal number of males and females. So let's assume both the probability be $\frac{1}{2}$ We want to find $P(M G...

### 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...

### In a certain college, of boys and of girls are taller than meters. Furthermore, of the students are girls. If a student is selected at random and is taller than meters, what is the probability that the selected student is a girl?

Let, T :students taller than $1.75$ B: Boys in class G: Girls in class We want to find $P(G \mid T)$, i.e. probability that selected taller is a girl $\begin{array}{l} \mathrm{P}(\mathrm{G} \mid...

### A company manufactures scooters at two plants, and B. plant A produces and plant B produces of the total product. of the scooters produced at pant and of the scooters produced at plant are of standard quality. A scooter produced by the company is selected at random, and it is found to be of standard quality. What is the probability that it was manufactured at plant A?

Let S : Standard quality We want to find $\mathrm{P}(\mathrm{A} \mid \mathrm{S})$, i.e. probability that selected standard scooter is from plant A $\mathrm{P}(\mathrm{A} \mid...

### In a bulb factory, three machines, , manufacture and of the total production respectively. Of their respective outputs, and are defective. A bulb is drawn at random from the total product, and it is found to be defective. Find the probability that it was manufactured by machine .

Let D : Bulb is defective We want to find $P(C \mid D)$, i.e. probability that the selected defective bulb is manufactured by $C$ $\mathrm{P}(\mathrm{C} \mid \mathrm{D})=\frac{\mathrm{P}(\mathrm{C})...

### The vertices of a triangle ABC are A(3, 2, 0), B(5, 3, 2) and C(-9, 6, -3). The bisector AD of ∠ A meets BC at D, find the fourth vertex D.

Answer: Given, A(3, 2, 0) B(5, 3, 2) C(-9, 6, -3)

### If the three consecutive vertices of a parallelogram be A(3, 4, -3), B(7, 10, -3) and C(5, -2, 7), find the fourth vertex D.

Answer: The vertices of the parallelogram be A(3, 4, -3), B(7, 10, -3) and C(5, -2, 7), and the fourth coordinate be D(a,b,c). The property of parallelogram is the diagonal bisect each other. The...

### Two vertices of a triangle ABC are A(2, -4, 3) and B(3, -1, -2), and its centroid is (1, 0, 3). Find its third vertex C.

Answer: The centroid of a triangle $ = \left( {\frac{{{x_2} + {x_1} + {x_1}}}{3},\frac{{{y_2} + {y_1} + {y_2}}}{3},\frac{{{z_2} + {z_1} + {z_3}}}{3}} \right)$ The points are A(2, -4, 3) and B(3, -1,...

### If the origin is the centroid of triangle ABC with vertices A(a, 1, 3), B(-2, b, – 5) and C(4, 7, c), find the values of a, b, c.

Answer: The centroid of a triangle $ = \left( {\frac{{{x_2} + {x_1} + {x_1}}}{3},\frac{{{y_2} + {y_1} + {y_2}}}{3},\frac{{{z_2} + {z_1} + {z_3}}}{3}} \right)$ The points are A(a,1,3) and B(-2, b,...

### The midpoints of the sides of a triangle are (1, 5, -1), (0, 4, -2) and (2, 3, 4). Find its vertices.

Answer: The midpoints of the sides of a triangle are (1, 5, -1), (0, 4, -2) and (2, 3, 4). Let its vertices be A(x1,y1,z1), B(x2,y2,z2), C(x3,y3,z3) . The mid point of AB is (1,5,-1). $\frac{{{x_2}...

### 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...

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

Solution: Given: $S_{1}$ and $S_{2}$ are two swiches whose probabilities of working be given by $\mathrm{P}\left(\mathrm{S}_{1}\right)=\frac{4}{5} \text { and }...

### An anti-aircraft gun can take a maximum of 4 shots at an enemy plane moving away from it. The probabilities of hitting the plane at the first, second, third and fourth shots are and respectively. What is the probability that at least one shot hits the plane?

Given:Let $A, B, C$ and $D b e$ first second third and fourth shots whose probability of hitting the plane is given i.e, $\mathrm{P}(\mathrm{A})=0.4, \mathrm{P}(\mathrm{B})=0.3,...

### 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 line segment having the end points A(-1, -3, 4) and B(4, 2, -1) is divided by the xz-plane. Also, find the coordinates of the point of division.

Answer: Let the plane XZ divides the points A(-1, -3, 4) and B(4, 2, -1) in ratio k:1. On XZ plane, Y co- ordinate of every point be zero. Formula - Using the above formula, $\begin{array}{l} =...

### Find the coordinates of the point where the line joining A(3, 4, 1) and B(5, 1, 6) crosses the xy-plane.

Answer: Let the plane XY divides the points A(3,4,1) and B(5, 1, 6) in ratio k:1. On XY plane, Z co- ordinate of every point be zero. Formula - Using the above formula, $\begin{array}{l} = \left(...

### 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...

### Neelam has offered physics, chemistry and mathematics in Class XII. She estimates that her probabilities of receiving a grade in these courses are and respectively. Find the probabilities that Neelam receives exactly 2 A grades.

Given : let $A, B$ and $C$ represent the subjects physics,chemistry and mathematics respectively ,the probability of neelam getting $A$ grade in these three subjects is given i.e, $P(A)=0.2,...

### Neelam has offered physics, chemistry and mathematics in Class XII. She estimates that her probabilities of receiving a grade in these courses are and respectively. Find the probabilities that Neelam receives

(i) all A grades

(ii) no A grade

Given : let $A, B$ and $C$ represent the subjects physics,chemistry and mathematics respectively ,the probability of neelam getting $A$ grade in these three subjects is given i.e, $P(A)=0.2,...

### A can hit a target 4 times in 5 shots, B can hit 3 times in 4 shots, and can hit 2 times in 3 shots. Calculate the probability that

(i) and all hit the target

(ii) and hit and does not hit the target.

Given : let $A, B$ and $C$ chances of hitting a target is given i.e, $P(A)=\frac{4}{5}, P(B)=\frac{3}{4}$ and $P(C)=\frac{2}{3}$ $\Rightarrow \mathrm{P}(\bar{A})=\frac{1}{5},...

### The probabilities of solving a problem are and , respectively. If all the three try to solve the problem simultaneously, find the probability that exactly one of them will solve it.

Given : let $A, B$ and $C$ be three students whose chances of solving a problem is given i.e, $P(A)=\frac{1}{3}, P(B)=\frac{1}{4}$ and $P(C)=\frac{1}{6}$ $\Rightarrow...

### 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...

### A problem is given to three students whose chances of solving it are and , respectively. Find the probability that the problem is solved.

Given : let $A, B$ and $C$ be three students whose chances of solving a problem is given i.e, $P(A)=\frac{1}{4}, P(B)=\frac{1}{5}$ and $P(C)=\frac{1}{6}$ $\Rightarrow...

### Given the probability that A can solve a problem is , and the probability that B can solve the same problem is \%, find the probability that

(i)at least one of and will solve the problem

(ii)none of the two will solve the problem

Given : Here probability of $A$ and $B$ that can solve the same problem is given, i.e., $P(A)=\frac{2}{3}$ and $P(B)=\frac{3}{5} \Rightarrow P($ $\bar{A})=\frac{1}{3}$ and...

### In how many different ways can 20 different pearls be arranged to form a necklace?

Answer : We know that necklace in the form of a circle, So we need to arrange 20 pearls in Circle 20 pearls can be arranged by 19! Now each pearl have the same neighbour in the clockwise and...

### In how many ways can 8 persons be seated at a round table so that all shall not have the same neighbour in any two arrangement?

Answer : By using the formula (n-1)! (Mention in Solution-1) So 8 persons can be arranged by 7! Now each person have the same neighbour in the clockwise and anticlockwise arrangement Total number of...

### A and B appear for an interview for two vacancies in the same post. The probability of A’s selection is and that of B’s selection is Find the probability that

(i) none is selected

(ii) at least one of them is selected.

Given : $A$ and $B$ appear for an interview, then $P(A)=\frac{1}{6}$ and $P(B)=\frac{1}{4} \Rightarrow P(\bar{A})=\frac{5}{6}$ and $P(\bar{B})=\frac{3}{4}$ Also, $A$ and $B$ are independent. A and...

### In how many ways can 11 members of a committee sit at a round table so that the secretary and the joint secretary are always the neighbour of the president?

### A and B appear for an interview for two vacancies in the same post. The probability of A’s selection is and that of B’s selection is Find the probability that

(i) both of them are selected

(ii) only one of them is selected

Given : $A$ and $B$ appear for an interview, then $P(A)=\frac{1}{6}$ and $P(B)=\frac{1}{4} \Rightarrow P(\bar{A})=\frac{5}{6}$ and $P(\bar{B})=\frac{3}{4}$ Also, $A$ and $B$ are independent. A and...

### There are 5 men and 5 ladies to dine at a round table. In how many ways can they sit so that no ladies are together?

### In how many ways can 6 persons be arranged in

(i)a line,

(ii) a circle?

### Arun and Ved appeared for an interview for two vacancies. The probability of Arun’s selection is , and that of Ved’s rejection is Find the probability that at least one of them will be selected.

Given : let A denote the event 'Arun is selected' and let B denote the event 'ved is selected'. Therefore, $\mathrm{P}(\mathrm{A})=\frac{1}{4}$ and $\mathrm{P}(\bar{B})=\frac{2}{3} \Rightarrow...

### 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,...

### Show that the points A(1, 2, 3), B(-1, -2, -1), C(2, 3, 2) and D(4, 7, 6) are the vertices of a parallelogram. Show that ABCD is not a rectangle.

Answer: (x1,y1,z1) = (1, 2, 3) (x2,y2,z2) = (-1, -2, -1) (x3,y3,z3) = (2, 3, 2) (x4,y4,z4) = (4, 7, 6) $\begin{array}{l} Length AB = \sqrt {{{({x_2} - {x_1})}^2} + {{({y_2} - {y_1})}^2} + {{({z_2}...

### Show that the points P(2, 3, 5), Q(-4, 7, -7), R(-2, 1, -10) and S(4, -3, 2) are the vertices of a rectangle.

Answer: (x1,y1,z1) = (2, 3, 5) (x2,y2,z2) = (-4, 7, -7) (x3,y3,z3) = (-2, 1, -10) (x4,y4,z4) = (4, -3, 2) $\begin{array}{l} Length PQ = \sqrt {{{({x_2} - {x_1})}^2} + {{({y_2} - {y_1})}^2} +...

### How many 4-digit numbers can be formed with the digits 0, 2, 3, 4, 5 when a digit may be repeated any numbers of time in any arrangement?

### 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...

### Show that the points P(1, 3, 4), Q(-1, 6, 10), R(-7, 4, 7) and S(-5, 1, 1) are the vertices of a rhombus.

Answer: (x1,y1,z1) = (1, 3, 4) (x2,y2,z2) = (-1, 6, 10) (x3,y3,z3) = (-7, 4, 7) (x4,y4,z4) = (-5, 1, 1) $\begin{array}{l} Length PQ = \sqrt {{{({x_2} - {x_1})}^2} + {{({y_2} - {y_1})}^2} + {{({z_2}...

### Show that D(-1, 4, -3) is the circumcentre of triangle ABC with vertices A(3, 2, -5), B(-3. 8, -5) and C(-3, 2, 1).

Answer: (x1,y1,z1) = (3, 2, -5) (x2,y2,z2) = (-3, 8, -5) (x3,y3,z3) = (-3, 2, 1) (x4,y4,z4) = (-1, 4, -3) $\begin{array}{l} Length AD = \sqrt {{{({x_4} - {x_1})}^2} + {{({y_4} - {y_1})}^2} +...

### Show that the following points are collinear : A(-2, 3, 5), B(1, 2, 3) and C(7, 0, -1)

Answer: (x1,y1,z1) = (-2, 3, 5) (x2,y2,z2) = (1, 2, 3) (x3,y3,z3) = (7, 0, -1) $\begin{array}{l} Length AB = \sqrt {{{({x_2} - {x_1})}^2} + {{({y_2} - {y_1})}^2} + {{({z_2} - {z_1})}^2}} \\...

### Show that the following points are collinear : A(3, -5, 1), B(-1, 0, 8) and C(7, -10, -6)

Answer: (x1,y1,z1) = (3, -5, 1) (x2,y2,z2) = (-1, 0, 8) (x3,y3,z3) = (7, -10, -6) $\begin{array}{l} Length AB = \sqrt {{{({x_2} - {x_1})}^2} + {{({y_2} - {y_1})}^2} + {{({z_2} - {z_1})}^2}}...

### How many 3-digit numbers are there when a digit may be repeated any numbers of time?

### Show that the following points are collinear : P(3, -2, 4), Q(1, 1, 1) and R(-1, 4, 2)

Answer: (x1,y1,z1) = (3, -2, 4) (x2,y2,z2) = (1, 1, 1) (x3,y3,z3) = (-1, 4, -2) $\begin{array}{l} Length PQ = \sqrt {{{({x_2} - {x_1})}^2} + {{({y_2} - {y_1})}^2} + {{({z_2} - {z_1})}^2}} \\...

### Find the equation of the curve formed by the set of all points which are equidistant from the points A(-1, 2, 3) and B(3, 2, 1).

Answer: Let us take, C(x,y,z) point equidistant from points A(-1, 2, 3) and B(3, 2, 1). ∴ AC = BC $\sqrt {{{(x + 1)}^2} + {{(y - 2)}^2} + {{(z - 3)}^2}} = \sqrt {{{(x - 3)}^2} + {{(y - 2)}^2} +...

### Find the point on the y-axis which is equidistant from the points A(3, 1, 2) and B(5, 5, 2).

Answer: Let us take, C(0,y,0) point which lies on y axis and is equidistant from points A(3, 1, 2) and B(5, 5, 2). ∴ AC = BC $\sqrt {{{(0 - 3)}^2} + {{(y - 1)}^2} + {{(0 - 2)}^2}} = \sqrt {{{(0 -...

### Find the point on the z-axis which is equidistant from the points A(1, 5, 7) and B(5, 1, -4).

Answer: Let us take, C(0,0,z) point which lies on z axis and is equidistant from points A(1, 5, 7) and B(5, 1, -4). ∴ AC = BC $\sqrt {{{(0 - 1)}^2} + {{(0 - 5)}^2} + {{(z - 7)}^2}} = \sqrt {{{(0 -...

### 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 coordinates of the point which is equidistant from the points A(a, 0, 0), B(0, b, 0), C(0, 0, c) and O(0, 0, 0).

Answer: Let us take, D(x,y,z) point equidistant from points A(a, 0, 0), B(0, b, 0), C(0, 0, c) and O(0, 0, 0). ∴ AD = OD $\sqrt {{{(x - a)}^2} + {{(y - 0)}^2} + {{(z - 0)}^2}} = \sqrt {{{(x -...

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

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

### 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...

### The letters of the word ‘INDIA’ are arranged as in a dictionary. What are the 1st, 13th, 49th and 60th words?

### How many 6 – digit numbers can be formed by using the digits 4, 5, 0, 3, 4, 5?

### How many 7 – digit numbers can be formed by using the digits 1, 2, 0, 2, 4, 2, 4?

### How many numbers can be formed with the digits 2, 3, 4, 5, 4, 3, 2 so that the odd digits occupy the odd places?

### 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

### (i)Find the number of different words by using all the letters of the word, ‘INSTITUTION’. In how many of them

(ii) are the three T’ s together

(iii) are the first two letters the two N’ s?

### In how many ways can the letters of the word ‘INTERMEDIATE’ be arranged so that: (i) The vowels always occupy even places? (ii) The relative orders of vowels and consonants do not change?

### (i)How many arrangements can be made by using all the letters of the word ‘MATHEMATICS’?

(ii) How many of them begin with C?

(iii) How many of them begin with T?

### 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 words can be formed from the letters of the word ‘SERIES’, which start with S and end with S?

### Find the number of arrangements of the letters of the word ‘ALGEBRA’ without altering the relative positions of the vowels and the consonants.

### How many words can be formed by arranging the letters of the word ‘INDIA’, so that the vowels are never together?

### How many words can be formed by arranging the letters of the word ‘ARRANGEMENT’, so that the vowels remain together?

### 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...

### There are 3 blue balls, 4 red balls and 5 green balls. In how many ways can they are arranged in a row?

### In how many ways can the letters of the expression x2y2z4 be arranged when written without using exponents?

### Find the total number of permutations of the letters of each of the words given below:

(i) APPLE

(ii) ARRANGE

(iii) COMMERCE

(iv) INSTITUTE

(v) ENGINEERING

(vi) INTERMEDIATE

### Find the number of ways in which m boys and n girls may be arranged in a row so that no two of the girls are together; it is given that m > n.

### when a group photograph is taken, all the seven teachers should be in the first row, and all the twenty students should be in the second row. If the tow corners of the second row are reserved for the two tallest students, interchangeable only between them, and if the middle seat of the front row is reserved for the principal, how many arrangements are possible?

### 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...

### In an examination, there are 8 candidates out of which 3 candidates have to appear in mathematics and the rest in different subjects. In how many ways can they are seated in a row if candidates appearing in mathematics are not to sit together?

### How many numbers divisible by 5 and lying between 3000 and 4000 can be formed by using the digits 3, 4, 5, 6, 7, 8 when no digit is repeated in any such number?

### 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 ways in which the letters of the word ‘MACHINE’ can be arranged such that the vowels may occupy only odd positions.

### In how many arrangements of the word ‘GOLDEN’ will the vowels never occur together?

### In how many ways can the letters of the word ‘FAILURE’ be arranged so that the consonants may occupy only odd positions?

### How many words can be formed out of the letters of the word ‘ORIENTAL’ so that the vowels always occupy the odd places?

### In how many ways can the letters of the word ‘HEXAGON’ be permuted? In how many words will the vowels be together?

### Find the number of permutations of the letters of the word ‘ENGLISH’. How many of these begin with E and end with I?

### How many words beginning with C and ending with Y can be formed by using the letters of the word ‘COURTESY’?

### How many words can be formed from the letters of the word ‘SUNDAY’? How many of these begin with D?

### Find the number of different 4-letter words (may be meaningless) that can be formed from the letters of the word ‘NUMBERS’,

### Find the number of words formed (may be meaningless) by using all the letters of the word ‘EQUATION’, using each letter exactly once.

### In how many ways can 6 pictures be hung from 4 picture nails on a wall?

### If there are 6 periods on each working day of a school, in how many ways can one arrange 5 subjects such that each subject is allowed at least one period?

### Ten students are participating in a race. In how many ways can the first three prizes be won?

### Five letters F, K, R, R and V one in each were purchased from a plastic warehouse. How many ordered pairs of letters, to be used as initials, can be formed from them?

### There are 6 items in column A and 6 items in column B. A student is asked to match each item in column A with an item in column B. How many possible, correct or incorrect answers are there to this question?

### It is required to seat 5 men and 3 women in a row so that the women occupy the even places. How many such arrangements are possible?

### Six students are contesting the election for the president ship of the students, union. In how many ways can their names be listed on the ballot papers?

### In how many ways can 4 different books, one each in chemistry, physics, biology and mathematics, be arranged on a shelf?

### In how many ways can 6 women draw water from 6 wells if no well remains unused?

### In how many ways can 7 people line up at a ticket window of a cinema hall?

### In how many ways can 5 persons occupy 3 vacant seats?

### 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...

### Prove that 1 + 1. 1P1 + 2. 2P2 + 3. 3P3 + …. n. nPn = n+1Pn+1.

### Prove that 9P3 + 3 × 9P2 = 10P3.

### Evaluate:

### Evaluate:

### Evaluate:

### A. Evaluate:

### 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...

### A customer forgets a four-digit code for an automated teller machine (ATM) in a bank. However, he remembers that this code consists of digits 3, 5, 6, 9. Find the largest possible number of trials necessary to obtain the correct code.

Answer : Given: code consists of digits 3, 5, 6, 9. To find: the largest possible number of trials necessary to obtain the correct code. The customer remembers that this 4 digit code consists of...