Answer : Given: three jobs, I, II and III to be assigned to three persons A, B and C. To find: In how many ways this can be done. Condition: one person is assigned only one job and all are capable...

### How many 6-digit telephone numbers can be constructed using the digits 0 to 9, if each number starts with 67 and no digit appears more than once?

### How many natural numbers less than 1000 can be formed from the digits 0, 1, 2, 3, 4, 5 when a digit may be repeated any number of times?

Answer : To find: number of natural numbers less than 1000 that can be formed from the digits 0, 1, 2, 3, 4, 5 when a digit may be repeated any number of times For forming a 3 digit number less than...

### How many 6-digit numbers can be formed from the digits 0, 1, 3, 5, 7, 9 when no digit is repeated? How many of them are divisible by 10?

### How many 3-digit numbers can be formed by using the digits 0, 1, 3, 5, 7 while each digit may be repeated any number of times?

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

### How many numbers can be formed from the digits 1, 3, 5, 9 if repetition of digits is not allowed?

Answer : To find: number of numbers that can be formed from the digits 1, 3, 5, 9 if repetition of digits is not allowed Forming a 4 digit number:4! Forming a 3 digit number:4C3 × 3! Forming a 2...

### How many 4-digit numbers are there, when a digit may be repeated any number of times?

Answer : To find: Number of 4 digit numbers when a digit may be repeated any number of times The first place has possibilities of any of 9 digits. (0 not included because 0 in starting would make...

### How many 3-letters words can be formed using a, b, c, d, e if (i) Repetition of letters is not allowed? (ii) Repetition of letters is allowed

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

Answer : Each letter has 4 possible letter boxes option. So the number of ways in which 5 letters can be posted in 4 letter boxes =4 × 4 × 4 × 4 × 4=45 (Each 4 for each letter.)

### In how many ways 6 rings of different types can be worn in 4 fingers?

Answer : Given:6 rings and 4 fingers. Each ring has 4 different fingers that they can be worn. So total number of ways in which 6 rings of different types can be worn in 4 fingers =4 × 4 × 4 × 4 × 4...

### A gentleman has 6 friends to invite. In how many ways can be send invitation cards to them, if he has 3 servants to carry the cards?

Answer : Given: A gentleman has 6 friends to invite. He has 3 servants to carry the cards. Each friend can be invited by 3 possible number of servants. So the number of ways of inviting 6 friends...

### Find the total number of ways of answering 5 objective-type question, each question having 4 choices.

Answer : Given: 5 objective-type question, each question having 4 choices. To find: the number of ways of answering them. Each objective-type question has 4 choices. So the total number of ways of...

### In how many ways can the following prizes be given away to a class of 20 students : first and second in mathematics; first and second in chemistry; first in physics and first in English?

Answer : Given: 20 students. The number of ways of giving first and second prizes in mathematics to a class of 20 students=20 × 19. (First prize can be given to any one of the 20 students but the...

### For a set of five true or false questions, no student has written the all correct answer and no two students have given the same sequence of answers. What is the maximum number of students in the class for this to be possible?

Answer : Given: a set of five true – false questions. To find: the maximum number of students in the class. Condition: no student has written the all correct answer and no two students have given...

### A sample of 3 bulbs is tested. A bulb is labeled as G if it is good and D if it is defective. Find the number of all possible outcomes.

Answer : A bulb can be good or defective, so there are 2 different possibilities of a bulb. So number of all possible outcomes (of all bulbs)=2 × 2 × 2=8

### From among the 36 teachers in a school, one principal and one vice- principal are to be appointed. In how many ways can this be done?

Answer : Given: 36 teachers are there in a school. To find: Number of ways in which one principal and one vice-principal can be appointed. There are 36 options of appointing principal and 35 option...

### A mint prepares metallic calendars specifying months, dates and days in the form of monthly sheets (one plate for each month). How many types of February calendars should it prepare to serve for all the possibilities in the future years?

Answer : To find: types of February calendars that can be prepared. There are two factors to develop FEBRUARY metallic calendars The day on the start of the year of which possibility=7 Whether the...

### 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?

### How many arithmetic progressions with 10 terms are there whose first term in the set {1, 2, 3} and whose common difference is in the set {2, 3, 4}?

Answer : Given: Two sets: {1, 2, 3} & {2, 3, 4} To find: number of A.P. with 10n terms whose first term is in the set {1, 2, 3} and whose common difference is in the set {2, 3, 4} Number of...

### Given, A = {2, 3, 5} and B = {0, 1}. Find the number of different ordered pairs in which the first entry is an element of A and the second is an element of B.

Answer : This is the example of Cartesian product of two sets. The pairs in which the first entry is an element of A and the second is an element of B are : (2,0),(2,1),(3,0),(3,1),(5,0),(5,1) ⇒ 3 ×...

### How many 4-letter codes can be formed using the first 10 letters of the English alphabet, if no letter can be repeated?

Answer : Given: first 10 letters of the English alphabet. In 4 letter code for first position there are 10 possibilities for second position there are 9 possibilities, for third position there are 8...

### Find the number of different signals that can be generated by arranging at least 2 flags in order (one below the other) on a vertical staff, if five different flags are available.

### (ac and the outcomes are recorded. How many possible outcomes are there?

(b) How many possible outcomes if the coin is tossed.

(i)four times?

(ii) five times?

(iii) n times?

Answer : (a) A coin is tossed three times So possible number of outcomes=23=8 (HHH,HHT,HTH,HTT,THH,THT,TTH,TTT) (b) i) A coin is tossed four times So possible number of outcomes=24=16...

### How many 8-digit telephone numbers can be constructed using the digits 0 to 9 if each number starts with 270 and n digit appears more than once?

### In how many ways can a vowel, a consonant and a digit be chosen out of the 26 letters of the English alphabet and the 10 digits?

### In a school, there are four sections of 40 students each in XI standard. In how many ways can a set of 4 student representatives be chosen, one from each section?

### In a textbook on mathematics there are three exercises A, B and C consisting of 12, 18 and 10 questions respectively. In how many ways can three questions be selected choosing one from each exercise?

### In How many ways can 5 ladies draw water from 5 taps, assuming the no tap remains unused?

Answer : To find: number of ways in which 5 ladies draw water from 5 taps. Condition: no tap remains unused The condition given is that no well should remain unused. So possible number of ways are:...

### In How many ways can 4 people be seated in a row containing 5 seats?

### There are 12 steamers plying between A and B. In how many ways could the round trip from A be made if the return was made on (i) the same steamer? (ii) a different steamer?

Answer : Given: 12 steamers plying between A and B. To find: number of ways the round trip from A can be made. The steamer which will go from A to B will be returning back, since the given condition...

### A, B and C are three cities. There are 5 routes from A to B and 3 routes from B to C. How many different routes are there from A to C via B?

Answer : Given: 5 routes from A to B and 3 routes from B to C. To find: number of different routes from A to C via B. Let E1 be the event : 5 routes from A to B Let E2 be the event : 3 routes from B...

### Solve for :

Solution: To find: value of $x$ Given: $\sin ^{-1} \frac{\mathrm{g}}{\mathrm{x}}+\sin ^{-1} \frac{15}{\mathrm{x}}=\frac{\pi}{2}$ We know $\sin ^{-1} \mathrm{x}+\cos ^{-1} \mathrm{x}=\frac{\pi}{2}$...

### There are 10 buses running between Delhi and Agra. In how many ways can a man go from Delhi to Agra and return by a different bus?

Answer : Given: 10 buses running between Delhi and Agra. To Find: Number of ways a man can go from Delhi to Agra and return by a different bus. There are 10 buses running between Delhi and Agra so...

### Prove that

### Prove that (n + 2) × (n!) + (n + 1) ! = (n!). (2n + 3)

### Evaluate when n = 15 and r = 12.

### Prove that:

Solution: To Prove: $\sin ^{-1} \frac{1}{\sqrt{17}}+\cos ^{-1} \frac{9}{\sqrt{85}}=\tan ^{-1} \frac{1}{2}$ Formula Used: $\tan ^{-1} x+\tan ^{-1} y=\tan ^{-1}\left(\frac{x+y}{1-x y}\right)$ where $x...

### find the value of n.

### find the value of n.

### If (n + 3) ! = 56 × (n + 1) !, find the value of n.

### If (n + 2) ! = 2550 × n!, find the value of n.

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

### Which of the following are true of false?

(i) (2 + 3) ! = 2 ! + 3!

(ii) (2 × 3)! = (2!) × (3!)

### Write the following products in factorial notation:

(i) 6 × 7 × 8 × 9 × 10 × 11 × 12

(ii) 3 × 6 × 9 × 12 × 15

### find the value of x.

### Prove that

### Prove that LCM {6!, 7!, 8!} = 8!

### Compute:

### Prove that:

Solution: To Prove: $\sin ^{-1} \frac{1}{\sqrt{5}}+\sin ^{-1} \frac{2}{\sqrt{5}}=\frac{\pi}{2}$ Formula Used: $\sin ^{-1} \mathrm{x}+\sin ^{-1} \mathrm{y}=\sin ^{-1}\left(\mathrm{x} \times...

### Prove that:

Solution: To Prove: $\tan ^{-1} 2-\tan ^{-1} 1=\tan ^{-1} \frac{1}{3}$ Formula Used: $\tan ^{-1} \mathrm{x}-\tan ^{-1} \mathrm{y}=\tan ^{-1}\left(\frac{\mathrm{x}-\mathrm{y}}{1+\mathrm{xy}}\right)$...

### Prove that:

Solution: To Prove: $\tan ^{-1}\left(\frac{\sin x}{1+\cos x}\right)=\frac{x}{2}$ Formula Used: 1) $\sin A=2 \times \sin \frac{A}{2} \times \cos \frac{A}{2}$ 2) $1+\cos A=2 \cos ^{2} \frac{A}{2}$...

### Prove that:

Solution: To Prove: $\cos ^{-1}\left(1-2 x^{2}\right)=2 \sin ^{-1} x$ Formula Used: $\cos 2 A=1-2 \sin ^{2} A$ Proof: $\operatorname{LHS}=\cos ^{-1}\left(1-2 x^{2}\right) \ldots(1)$ Let $x=\sin A...

### Prove that:

Solution: To Prove: $\tan ^{-1}\left(\frac{3 x-x^{3}}{1-3 x^{2}}\right)=3 \tan ^{-1} x$ Formula Used: $\tan 3 A=\frac{3 \tan A-\tan ^{3} A}{1-3 \tan ^{2} A}$ Proof: $\text { LHS }=\tan...

### Prove that:

Solution: To Prove: $\tan ^{-1} x+\cot ^{-1}(x+1)=\tan ^{-1}\left(x^{2}+x+1\right)$ Formula Used: 1) $\cot ^{-1} \mathrm{x}=\tan ^{-1} \frac{1}{\mathrm{x}}$ 2) $\tan ^{-1} \mathrm{x}+\tan ^{-1}...

### Prove that:

Solution: To Prove: $\tan ^{-1}\left(\frac{1+x}{1-x}\right)=\frac{\pi}{4}+\tan ^{-1} x$ Formula Used: $\tan \left(\frac{\pi}{4}+A\right)=\frac{1+\tan A}{1-\tan A}$ Proof: $\mathrm{LHS}=\tan...

### Find the principal value of each of the following :

Solution: $\cot ^{-1}(-1)=\pi-\cot ^{-1}(1)$ [Formula: $\left.\cot ^{-1}(-x)=\pi-\cot ^{-1}(x)\right]$ $\begin{array}{l} =\pi-\frac{\pi}{4} \\ =\frac{3 \pi}{4} \end{array}$

### If |z| = 2 and arg find z.

### Write z = (–1 + i ) in polar form.

### Write z = (1 – i) in polar form.

### Write –3i in polar form.

### Write 2i in polar form.

### Write –9 in polar form.

### Write the principal argument of (1 + i)2.

### Find the principal argument of (–2i).

### If |z| = 6 find z.

### Prove that arg (z) + arg = 0

### , find z-1.

### If z = (1 – i), find z-1.

### Find the conjugate of

### Solve for x: x2 – 5ix – 6 = 0.

### Solve for x: (1 – i) x + (1 + i) y = 1 – 3i.

### Express in the form (a + ib).

### Express in the form (a + ib).

### Express (2 – 3i)3 in the form (a + ib).

### Find the least positive integer n for which

### Evaluate: .

### Evaluate (1 + i10 + i20 + i30).

### Find the sum (i + i2 + i3 + i4 +…. up to 400 terms)., where n N

### Find the sum (in+ in+1 + in+2 + in+3), where n N.

### Evaluate .

### Evaluate (i4n+1 – i4n–1)

### Evaluate

### Evaluate (i57 + i70 + i91 + i101 + i104).

### Evaluate

### solve the equation

### solve the equation

### solve the equation

### solve the equation

### solve the equation

### solve the equation

### solve the equation

### solve the equation

### solve the equation

### solve the equation

### solve the equation

### solve the equation

### solve the equation

### 2×2 + 3ix + 2 = 0

### x2 + 3ix + 10 = 0

### x2 + 13 = 4x

### 21×2 – 28x + 10 = 0

### 3×2 + 7ix +6 = 0

### solve the quadratic equation

### 3×2 + 5 = 7x

### 17×2 – 8x + 1 = 0

### solve the quadratic equation

### 27×2 + 10x + 1 = 0

### 8×2 + 2x + 1 = 0

### 25×2 – 30x + 11 = 0

### solve the equation

### x2 + 3x + 5 = 0

### 2×2 – 4x + 3 = 0

### x2 + 2x + 2 = 0

### x2 – x + 2 = 0

### x2 + x + 1 = 0

### 2×2 + 1 = 0

### x2 + 5 = 0

### x2 + 2 = 0

Answer : This equation is a quadratic equation. Solution of a general quadratic equation ax2 + bx + c = 0 is given by: Given: ⇒x2 + 2 = 0 ⇒x2 = -2 ⇒x = ± √ (-2) But we know that √ (-1) = i ⇒ x = ±√2...

### Find the modulus of each of the following complex numbers and hence express each of them in polar form: (sin 120° – i cos 120°)

### Find the modulus of each of the following complex numbers and hence express each of them in polar form:

### Find the modulus of each of the following complex numbers and hence express each of them in polar form: (i25)3

### Find the modulus of each of the following complex numbers and hence express each of them in polar form:

### Find the modulus of each of the following complex numbers and hence express each of them in polar form:

### Prove that:

Answer:

### Prove that:

Answer:

### Prove that:

Answer:

### Prove that:

Answer:

### If , where , find the values of (iii) tan (x + y)

Answer: (iii)Here first we will calculate value of tanx and tany,

### If , where , find the values of (i) sin (x + y) (ii) cos (x – y)

Answer: Given: $\cos x=\frac{3}{5}\,and\,\cos y=-\frac{24}{25}$ We will first find out value of sinx and siny, (i)sin(x + y) = sinx.cosy + cosx.siny (ii)cos(x - y) = cosx.cosy +...

### If , where , find the values of (iii) tan (x – y)

Answer: Given: $\sin x=\frac{12}{13}\,and\,\sin y=\frac{4}{5}$ (iii)Here first we will calculate value of tanx and tany

### If , where , find the values of (i) sin (x + y) (ii) cos (x + y)

Answer: Given: $\sin x=\frac{12}{13}\,and\,\sin y=\frac{4}{5}$ Here we will find values of cosx and cosy (i) sin(x + y) = sinx.cosy + cosx.siny (ii)cos(x + y) = cosx.cosy +...

### If x and y are acute angles such that , prove that

Answer: Given: $\cos x=\frac{13}{14}\,and\,\cos y=\frac{1}{7}$ Now we will calculate value of sinx and siny Hence, Cos(x - y) = cosx.cosy + sinx.siny

### Find the modulus of each of the following complex numbers and hence express each of them in polar form: 2 – 2i

### Find the modulus of each of the following complex numbers and hence express each of them in polar form: –1 + i

### Find the modulus of each of the following complex numbers and hence express each of them in polar form: 1 – i

### Find the modulus of each of the following complex numbers and hence express each of them in polar form: 2i

### Find the modulus of each of the following complex numbers and hence express each of them in polar form: –i

### Find the modulus of each of the following complex numbers and hence express each of them in polar form: –2

### Find the modulus of each of the following complex numbers and hence express each of them in polar form: 4

### If a and b are real numbers such that a2 + b2 = 1 then show that a real value of x satisfies the equation,

### If z1 = (1 + i) and z2 = (–2 + 4i), prove that

### For all z C, prove that

### If z1 is a complex number other than –1 such that |z1| = 1 and z2 = then show that z2 is purely imaginary.

### is purely imaginary and z = –1, show that |z| = 1.

### If z2 + |z|2 = 0, show that z is purely imaginary.

### Find real values of x and y for which (x4 + 2xi) – (3×2 + iy) = (3 – 5i) + (1 + 2iy).

### Express (1 – 2i)–3 in the form (a + ib).

### If (x + iy)1/3 = (a + ib) then prove that 4 (a2 – b2).

### If (x + iy)3 = (u + iv) then prove that 4 (x2 – y2).

### Express each of the following in the form (a + ib) and find its multiplicative inverse:

### Express each of the following in the form (a + ib) and find its conjugate.

### Find the complex number z for which |z| = z + 1 + 2i.

### Solve the system of equations, Re(z2) = 0, |z| = 2.

### purely an imaginary number and z ≠ -1 then find the value of |z|.

### If (1 + i)z = (1 – i) then prove that

### If z = (2 – 3i), prove that z2 – 4z + 13 = 0 and hence deduce that 4z3 – 3z2 + 169 = 0.

### Find the real values of x and y for which the complex number (-3 + iyx2) and (x2 + y + 4i) are conjugates of each other.

### Find the real values of x and y for which (x – iy) (3 + 5i) is the conjugate of (-6 – 24i).

### Find the real values of x and y for which:

### Find the real values of x and y for which:

Answer : Given: ⇒ x + 3i = (1 – i)(2 + iy) ⇒ x + 3i = 1(2 + iy) – i(2 + iy) ⇒ x + 3i = 2 + iy – 2i – i2y ⇒ x + 3i = 2 + i(y – 2) – (-1)y [i2 = -1] ⇒ x + 3i = 2 + i(y – 2) + y ⇒ x + 3i = (2 + y) +...