SHORT QUESTIONS -XI
COMPLEX NUMBERS(1)
1) Simplify:
a) i²⁸. 1
b) i²⁵³. i
c) i⁻¹³. - i
d) i⁹ + 1/i⁵. 0
e) (1+ i)²+ (1- i)². 0
f) ω³⁷. ω
g) ω⁵ + ω¹⁰. -1
2) Express in the form A+ iB, where A and B are real: (i+ i²+ i³ + i⁴)/(1+ i). 0+ i0
3) Express (1+ 3i)/(2- 5i) in the form of x + it, where x, y are real. -13/29+ 11i/29
4) If (2+ i)/(2- 3i)= A + iB, find the value of (A²+ B)². 5/13
5) If x= 1 + i and y= 1- i, find the value of x²+ xy + y². 2
6) If a, b, c, d are real and ad - bc=0, show that (a+ ib)/(c + id) is a real quantity.
7) If a, b, c, d are real and a+ ib = c + id, show that a - ib = c - i d.
8) If ³√(x + iy) = 2+ 3i, find the value of x + 5y. -1
9) If x + 2y + 3i = 3+ (2y - x)i, find the values of x and y. 0, 3/2
10) Find the conjugate of:
a) - 5i. 5i
b) 1/(3+ i). (3+ i)/10
11) If x, y are real and the complex numbers (x + 3i) and (-2+ iy) are conjugate to each other, find x, y. -2,-3
12) Are the numbers (2+ 3i) and (-2+ 3i) conjugate to each other? No
13) Are the numbers 1/(3+ i) and 1/(3- i) conjugate to each other? Yes
14) Find the modulus of:
a) (a - ib)². a²+ b²
b) (1+ 2i)/(2- i). 1
15) Find amplitude of:
a) sin120°- I cos 120°. 30°
16) Find amplitude and modulus of:
a) -5i. 5, - π/2
b) 1+ i. √2, π/4
c) (1+ i)/(1- i). 1, π/2
17) Express in modulus -amplitude form:
a) 1+ i. √(cosπ/4+ i sinπ/4)
b) -1. 1(cosπ + i sinπ)
c) -2i. 2[cos(-π/2) + i sin(-π/2)]
18) The modulus and amplitude of a complex number are √2 and 45° respectively; find the complex number. 1+ i
19) Find the square roots of:
a) 5 - 12 i. ±(3- 2i)
b) (1+ i)/(1- i). ±(1/√2) (1+ i)
20) Find the fourth roots of 1. ±1,±i
21) Find the cube roots of
a) -1. -1,-ω, ω²
b) 27. 3, 3ω, 3ω²
22) Find the sum and product of two complex roots of unity. -1,1
23) If one imaginary cube root of 1 and ω, show that
a) (aω+ b + ω²)/(bω² + a + ω) = ω.
b) (1- ω+ ω²)=4.
c) (1- 2ω + ω²)(1- 2ω²+ ω)=9.
d) (1- ω- ω²)³ - (1+ ω- ω²)³ = 16.
e) (a +ω+ ω²)(a + ω² + ω⁴)(a+ω⁴ + ω⁸)....to n factors= (a - 1)ⁿ
24) Find the least positive integeral value of n for which ω²ⁿ⁺⁵ =1. 2
25) If one imaginary cube root of 1 be ω, find |ω| and amp ω. 1, 2π/3 or -2π/3
26) Show that {(1+ √-3)/2}¹⁹ + {(-1-√-3)/2}¹⁹= -1.
27) If m= (-1- √-3)/2 and n= (-1+√3)/2, show that m²+ mn + n²=0.
28) If |z|= 1, find the value of |z⁻¹|. 1
29) If |z - 3| ≤ 1, find the greatest value of |z|. 4
30) Is the statement i > 0 correct? No
THEORY OF QUADRATIC EQUATIONS
1) If m,n are the roots of x(x -3)=4, what is the value of m²+ n²? 17
2) if m and n are the roots of x(2x -1)= 1, what is the value of m²- n²? ±3/4
3) If the roots of x²- px + q=0 are m and n, Find the value of 1/m + 1/n. p/q
4) What is the sum of the reciprocal of the roots of the equation 3x²- 6x + 4 =0 ? 3/2
5) If m and n are the roots of the equation ax² + bx + c =0, what is the value of am²/(bm + c) - an²/(bn + c) ? 0
6) If one root of the equation 2x²- 5x + k =0 be twice other, find the value of k. 25/9
7) If m, n are the roots of x²- px + q=0, find the value of 1/(p - m) + 1/(p - n). p/q
8) Show that the AM of the roots of the equation x²- 2ax + b² =0 is equals to the GM of the roots of the equation x²- 2bx + a² =0.
9) What is the condition that the equation ax² + bx + c =0 will have
a) one root zero. c= 0
b) both roots zero? b= c= 0
10) For what value of m, the roots of the equation x²- (3+ 2m)x + (10+ 2m) =0 are equal in magnitude but opposite in signs ? -3/2
11) For what value of p, the roots of the equation 3x²- 2(7+ 9p)x + (8- 5p)=0 are reciprocal to one another ? 1
12) For what value of k, the sum of the roots of the equation x² + (2k +5)x - k - 3 =0 will be equal to their product. -2
13) if the sum and the product of the roots of the equation ax²- 5x + c =0 are 10 and 10 respectively, What are the value of a and c ? 1/2,5
14) The difference of the roots of the equation x²+ px + 12 =0 is 1; What is the value of p ? ±7
15) If m and n are the roots of the equation x² + mx + n =0, find the numerical value of m and n. (m≠ n, m≠0, n≠0). 1, -2
16) Form a quadratic equation whose roots are 1/2 and -3/2. 4x²+ 4x -3=0
17) Find a quadratic equation whose roots are equal to the roots of the equation x² +4x + 5 =0 but in opposite signs. x²- 4x + 5=0
18) Find the equation whose roots are the reciprocals of the roots of the equation ax²+ bx + c =0. cx²+ bx + a=0
19) Form a quadratic equation with rational coefficient whose one root is 2√5. x²- 20=0
20) Form a quadratic equations with real coefficients whose one root is 2/i. x²+ 4=0
21) If 2 - i√3 be a root of of a quadratic equation with real coefficients, what is the difference of its roots. i 2√3
22) If one root of the equation x²+ px + q =0, (p,q are real) is 2 + 3i, find the value of p and q. -4,13
23) If one root of the equation x²+ px + q =0 (p,q are rational ) is 3 + √2, find the value of p and q. -6,7
24) Is it possible to have one real root and the other imaginary root of a quadratic equation with real coefficients? Give reasons for your answer. No
24) If one root of the equation x² - (2p +1)x +3 p=0 be 2, find the other root . 3
25) find one root of the equation x + 1/x = k (where k is a constant) be i, find the value of k. What is the other root of the equation? - i
26) If the roots of x² -2x +4 =0 are m and n, find the value of m⁵+ n⁵ ? 32
27) If the roots of x² -5x +8 =0 are m, n and the roots of x²+ px + q =0 are m²+ n² and mn/2, find p and q. -13,36
PERMUTATION(1)
1) In how many ways can the letters of the word BANANA be arranged ? 60
2) How many different permutations can be made out of the letters of the expression x³y²z⁴ when written at a full length ? 1260
3) If ⁴⁻ˣP₂ = 6, find the value of x. 1
4) In how many ways can 10 coins of 10 paise and 5 coins of 5 paise can be arranged in a line so that two coins of 5 paise do not come together ? 462
5) The number of different messages by 5 signals with 3 dots and 2 dashes is
a) 20 b) 1000 c)10 d)
6) At the end of a conference each of the representatives exchange their signatures with the others. If the total number of signature be 420, how many representatives were there in the conference? 21
COMBINATION(1)
1) Show that, 3. ⁿC₃ = n ⁿ⁻¹C₂.
2) If ⁿC₈ = ⁿC₆, find the value of ⁿC₂. 91
3) If ¹⁰Cᵣ = ¹⁰Cᵣ₊₂, find the value of r. 4
4) Which one is greater between ⁿPᵣ and ⁿCᵣ ? ⁿPᵣ> ⁿCᵣ
5) If ⁿCᵣ = x. (ⁿPᵣ), find x. 1/r!
6) If ⁿPᵣ= 120. ⁿCᵣ, find the value of r. 5
7) If ⁿPᵣ = 5040(ⁿ⁻¹Cᵣ)+ ⁿ⁻¹Cᵣ₋₁, find the value of r. 7
8) Find the value of
a) ⁶C₂ + ⁶C₃ + ⁷C₄ + ⁸C₅. 126
b) ⁹C₁ + ⁹C₃ + ⁹C₅ + ⁹C₇. 255
9) Show that, ⁴C₂+ 2. ⁴C₃ + ⁴C₄ = ⁶C₄.
10) In how many ways can 9 alike things be divided
a) into two groups. 4
b) between two persons . 8
11) A polygon has 44 diagonals. The number of its sides is
a) 11 b) 7 c) 8 d) 9
12) In a football championship, there were played 36 matches. Every two teams played one match with each other. The number of teams, participating in the championship, is
a) 8 b) 9 c) 18 d) none
13) There are 6 points on a circle. How many triangles can be formed by joining those points ? Find the number of chords passing through the points. 20, 15
14) In a plane 13 straight lines meet at a point. Find the number of angles in between the straight lines. 156
PROBABILITY(1)
1) A dice is rolled once. Find the probability of each of the following cases:
a) getting an odd number. 1/2
b) getting a number more than 4. 1/3
c) getting a number less than 4. 1/2
d) getting even number greater than 3. 1/3
2) 2 coins are thrown together. Find the probability of each of the following cases:
a) getting two heads. 1/4
b) getting only one head. 1/2
c) getting at least one head. 3/4
3) A card is drawn from a pack of 52 playing cards. What is the probability that the card is
a) a spade . 1/4
b) an ace. 1/13
c) a black colour. 1/2
d) not hearts . 3/4
e) not ace. 12/13
f) A king of red colour. 1/26
4) A dice and a coin are thrown simultaneously. What is the probability of getting head and 4. 1/12
5) Two dice are thrown simultately. Find the probability that the sum of the numbers on the faces is
a) even. 1/2
b) odd. 1/2
6) Two dice are thrown simultaneously. The probability that the members on both the dice are same, is
a) 1/6 b) 1/18 c) 1/36 d) none
7) a dice is thrown. What is the odds in favour of getting 3 ? What is odds against getting 3 ? 1:5, 5:1
8) The odds in favour of an event is 5:13. What is the probability of occurance of the event ? 5/18
9) The probability of occurrence of an event is 3/11, find the odds in favour of the event. 3:8
BINOMIAL THEOREM
1) Write down the general in the expansion of (a+ x)ⁿ when n is a major positive integer. ⁿCᵣaⁿ⁻ʳxʳ
2) Write down the general term in the expansion of (x²+ 1/x)²ⁿ when n is positive integer. ²ⁿCᵣx⁴ⁿ⁻³ʳ
3) What is the sum of the indices of a and x in the 5th term in the expansion of (a+ x)¹²? 12
4) What is the number of terms in expansion of (a + x)²⁰? 21
5) What is the number of terms in the expension of (x + y)⁵(x - y)⁵? 6
6) How many different terms can be obtained by simplifying the expansion of (a + 2b)⁵⁰ + (a - 2b)⁵⁰? 26
7) The n-th term in the expansion of (x + 1/x)¹⁰ is independent of x; what is the value of r ? 6
8) If the coefficients of the 16th and 26th terms in the expansion of (1+ x)ⁿ are equal, what is the value of n ? 40
9) If the coefficients of the 3r-th and (r+2)-th terms in the expansion of (1+ x)ⁿ are equal, then
a) n= 2r b) n = 3r c) n= 2r +1 d) none
10) If the coefficient of x⁶ in the expansion of (1+ x)ⁿ is 177100, what is the coefficient of xⁿ⁻⁶ in that expansion ? 177100
11) Find the number of terms contaning positive powers of x in the expansion of (2x - 1/3x²)¹². 4
ARITHMETIC PROGRESSION (1)
2) The sum of the first 5 terms of an AP is 20, find its third term. 4
3) The 10th term an AP is 44 and the common difference is 4, find the first term of the series. 8
4) The 15th term of an AP and the 32nd term is 67, find the first term and the common difference. 5,2
5) The first two terms of an AP are 5 and 8 and the last term is 80, find the number of terms. 26
6) 3K + 1, 7k and 10K + 8 are in AP, what will be the common difference. 35
7) The third term of an AP is twice that of its first term and the fourth term is 15, find the common difference of the AP. 3
8) Find one arithmetic mean between -5 and 15. 5
9) Find the arithmetic mean between (x -y)² and (x + y)². x²+ y²
10) 2, x, 8, y are in AP, find the values of x and y. 5, 11
11) In an AP the first term is (-5), the last term is 25 and the number of terms is 10. What is the sum of the series ? 100
12) A polygon has 25 sides , the lengths of which are in AP. If the perimeter of the polygon be 1100cm and the length of the largest side is 10 times that of the smallest, find the length of the smallest side. 8cm
13) The sum of how many terms of the AP 21 + 18 + 15 + 12 ....will be zero? 15
14) find the sum of the first 100 natural numbers. 5050
15) If each term of a series in AP be multiply by 3. Would the series so obtained be again in AP? Yes
16) Let Sₙ denote the sum of first n terms of an AP. If S₂ₙ = 3Sₙ then find the values of S₃ₙ/Sₙ. 6
GEOMETRIC PROGRESSION (1)
1) The product of the first three terms of a GP is 27/8; find the middle term. 3/2
2) The first term of a GP is 5 and 4th term is 320, find the comma ratio. 4
3) The second term of a GP is - 24 and fifth term is 81, find its first term. 16
4) The first two terms of a GP are 9 and 3. Which term of the series is 1/243 ? 8th
5) A GP has first term is 6 and last term is 96. If each term is twice its proceeding term, find the number of terms. 5
6)2x, 2x+1 and 2x+3 are in GP, find the common ratio. 2
7) If (3a+1), (6a-4),(3a -2) are in GP. find the value of a. 1
8) the third term of a GP is equals to the square of the first term and 4th term is 32, find the common ratio. 2
9) The sum of the first two terms of a GP is 16 and sum of next two terms is 144. Find the common ratio of the series. ±3
10) If the r-th term of a series is 4.3ʳ⁻², show that the series is a GP.
11) The sum of first n terms of a series is 4(3ⁿ -1). Show that the series is a GP.
12) Find the Geometric mean between 3/4 and 16/27. ±2/3
13) If three positive numbers a, b, c are in GP., then show that, a+ c > 2b
14) if x, y, z are such that (x²+ y²)(y²+ z²)= (xy + yz)², then prove that x,y,z are in GP.
15) If p, q, r are in GP, find the value of (q -p)/(q- r)+ (q+ p)/(q +r). 0
16) Can any term of a GP be zero? No
17) is there any two different real numbers such that their AM and GM are equal ? No
18) What type of series will be formed by the reciprocal of the terms of a GP? Geometric series
19) Can three different quantities be in AP and GP at the same time? No
20) The third term of a GP is 4, the product of the first 5 terms is:
a) 4² b) 4³ c) 4⁴ d) 4⁵
21) The sum of the first n terms of a GP is denoted by Sₙ. If S₂ₙ = 4Sₙ, then what is the value of S₃ₙ/Sₙ ? 13
MISCELLANEOUS -(1)
1) If a, b, c are in AP show that, (a- c)²= 4(b²- ac).
2) There are (p - 1) arithmetic between a and b. The common difference of the AP is
a) (a+ b)/(p -1) b) (b - a)/p c) (b - a)/(p -1) d) (b - a)/(p -2)
3) The fourth term of a GP is the square of the first term. If the 7th term of the progression is 216, find the first term. 6
4) the seventh term of an AP is 40. Find the sum of the first 13 terms ? 520
5) Show that the conjugate complex number of (1+ i)⁵ is -4(1- i).
6) if m, n are the roots of x²+ x +1= 0, find the value of m/(m +1) + n/(n +1). 1
7) In the equation 4x²+ 2bx +c = 0 if b= 0, find the relation between the roots of the equation. Roots are equal in magnitude but opposite in signs
8) If one root of the equation x - 1/x = k is 1+ √2, find the value of k. 2
9) Form the quadratic equation whose roots are p+ q and p - q. x² - 2px +p²- q²= 0
10) Form the quadratic equation with rational coefficient whose one root of 5 + 2√6. x² -10x +1= 0
11) if the roots of m, n of ax²+ 2x +1= 0 satisfy the relation 1/m + 1/n = 1/(m + n), find the value of a. 4
12) Form the quadratic equation in x such that arithmetic mean of its roots is A and geometric mean is G. x²- 2Ax +G² = 0
13) Find the values of n and r, when ⁿPᵣ = ⁿPᵣ₊₁ and ⁿCᵣ = ⁿCᵣ₋₁. 3,2
14) Which polygon has the same number of diagonals as sides ? Pentagon
15) If the coefficient of xʳ and xʳ⁺¹ in the expansion of (1+ x)²ⁿ⁻¹ are equal, Show that, r= n -1.
16) Find the series of which the sum of the first n terms is 2ⁿ⁺¹ - (n +2). 1+3+7+15+...
17) Find the sum of all odd numbers between 50 and 150. 5000
18) The sum of the areas of n square is n². If the areas of the squares can be put in AP., what is the length of the side of the largest square ? √(2n -1)
19) Find the value of: 5²5⁴5⁶....5²ˣ = (0.04)⁻³⁶. 8
20) Find the value of x: (x +1)+ (x +4)+ (x +7)+....+(x+28)= 195. 5
21) If x, 12, y, 27 are in GP, find the value of x and y. ±8,±18
22) Find the 7 term of the sequence 1,4,13,40,.... 1093
23) Find the 10th term of the sequence 9, 16, 27, 42,.... 216
24) Find the coefficient of x⁹⁹ in the expression (x -1)(x -2)(x -3)....(x -100). -5050
25) Show that the three cube roots of I are, - i, (i+√3)/2, (i - √3)/2.
26) If z= 1+ sin k + I cos k, find amp z. π/4 - k/2
27) If z= 1+ cos 2k + i sin 2k, π/2< k < 3π/2, find amp z. k - π
28) If az + ib(conjugate of z)= 0, where z= x + it and a+ b ≠ 0, find the value of x + y. 0
29) If z₁ , z₂ are conjugate of each other and z₃ , z₄ are conjugate of each other, show that, amp (z₁/z₄) = amp (z₃/z₂).
30) Show that, amp (z) - amp (-z)= π, when amp(z)>0
= -π when amp(z)<0.
31) Explain with the reasons which one of the following is correct ?
a) 2+ 3i > 1+ 4i b) 6+ 2i > 3+ 3i
c) 5+ 8i > 5 + 7i d) none
32) If |z₁ | = |z₂ |= 1 and amp z₁ + amp z₂ = 0, show that z₁z₂ = 1.
33) If the complex number z= x + iy satisfy the equation |(z - 5i)/(z + 5i)| = 1, then z lie on
a) the x-axis b) the line y = 5 c) a circle passing through the origin
34) Solve : |z|+ z = 1+ 3i, where z is a complex number. -4+ 3i
35) For any complex number z the minimum value |z| + |z -1| is
a) 1 0 c) 1/2 d) 3/2
36) For what value of n the equation (4- n)x²+ (4+ 2n)x + 1 + 8n =0 has equal integral roots ? 3
37) if f(x)= 3x³- 2x²+ 6x +11, find a quadratic equation with real coefficients whose one root is f(i). x²- 26x + 178= 0
38) The roots of the equation x² -(a -2)x - a +1=0 are m and n. If a is real, find the least value of m²+ n². 1
39) If the product of the roots of the equation x² - 3kx + 2e²ˡᵒᵍᵏ - 1 =0 be 7, find the value of k for which the roots are real. 2
40) What is the condition that the roots of the equation 2x² +3x +p(p -1) =0 are of opposite signs ? 0< p< 1
41) If a, b are the roots of x² - 2x +2 =0, then the least positive integeral value of n for which aⁿ/bⁿ = 1 is
a) 2 b) 3 c) 4 d) none
42) The number of solutions of the equation |x|² - 3|x| + 2 =0 is
a) 4 b) 1 c) 3 d) 2
43) In how many ways can 3A's, 2B's and 1C's be arranged in one line so that 2A's never occur together? 12
44) The number of 5 digit telephone numbers, none of their digits being repeated, is
a) 50 b) ¹⁰P₅ c) 5¹⁰ d) 10⁵
45) The number of 10 digit numbers formed with the digit 1 and 2, is
a) ¹⁰C₁ + ⁹C₂ b) 2¹⁰ c) ¹⁰C₂ d) 10!
46) A, B, C, D, E have been asked to deliver a lecture in a meeting. In how many ways can their lectures be arranged so that C delivers lecture just before A? 24
47) How many different signals can be given by using any number of flags from 6 flags of different colours ? 1956
48) The number of ways in which 5 unlike rings a man can wear on the four fingers of one hand is
a) 120 b) 625 c) 1024 d) none
49) Show that the product of r successive natural numbers is divisible by r!.
50) If n> 7, prove that ⁿ⁻¹C₃ + ⁿ⁻¹C₄ > ⁿC₃.
51) The value of ⁴⁷C₄ +⁵ ∑ᵢ₌₁ ⁵²⁻ⁱC₃ is
a)⁴⁷C₄ b) ⁵²C₃ c) ⁵²C₄ d) none
52) ¹⁵C₁ + ¹⁵C₃ + ¹⁵C₅+.....¹⁵C₁₅ = ?
a) 15! b) 15. 2⁸ c) 2¹⁴ d) 2¹⁵
53) In a football championships, there were played 153 matches. Every 2 teams played one match with each other. The number of teams, participating in the championship is
a) 17 b) 18 c) 9 d) none
54) Everybody in a room shakes hands with everybody else. The total number of hands shakes is 66. The total number of persons in the room is
a) 11 b) 12 c) 13 d) 14
55) The number of ways to form a team of 11 players out of 22 players where two particular players are included and 4 particular players are never included in the team.
a) ¹⁶C₁₁ b) ¹⁶C₅ c) ¹⁶C₉ d) ²⁰C₉
56) The total number of factors of 1998(including 1 and 1998) is
a) 18 b) 16 c) 12 d) 10
57) In how many ways can 9 different things be divided into three groups of 2, 3 and 4 things respectively ? 1260
58) In how many ways can 12 different things be divided equally into four groups ? 15400
59) Of the four numbers 25, 150, 170, 210 -- which one is the number of diagonals of a polygons of 20 sides ? 170
60) if a polygon has 54 diagonals, find the number of sides of the polygon. 12
61) In how many ways can the result( win, loss or draw) of 3 successive football matches be decided ? 27
62) There are 10 electric bulbs in a hall. Each of them can be lightened separately. The number of ways for lighting the hall is
a) 10² b) 1023 c) 2¹⁰ d) 10!
63) How many quadrilaterals can be formed with 7 side of the lengths 1 cm, 2cms, 3cms, 4cms, 5 cms, 6 cms and 7 cms? 32
64) How many different algebraic expressions can be formed by combining the letters a, b, c, d, e, f with signs +, -, all the letters taken together ? 64
65) Find the total number of ways in which six + and four - signs can be arranged in a line such that no two - signs occur together. 35
66) Find the coefficient of a⁶b³ in expansion of (2a - b/3)⁹. -1792/9
TRIGONOMETRICAL RATIOS OF ANGLES
1) Find the value of
a) sin 1755°. -1/√2
b) cot(-870). √3
c) cot 660+ tan(-1050). 0
2) Show that, sec(-1680) sin330= 1
3) Show: tan130 tan 140=1.
4) Find the value of m² sin(π/2)- n² sin(3π/2)+ 2mn secπ. (m - n)²
5) If 6k=11π, the value of 2 cosk + 3 tank is
a) 1 b) 0 c) √3 d) 2√3
6) If cotx= cos60+ sin30, the value of cosx + cos(x - 90) is
a) 1 b) √2 c) 1/√2 d) 0
7) If 3sin²x+ 5 cos²x = 4 and π/2< x <π, find the value of sin2x. -1
8) If tanx= -4/3, find the value of sinx. ±4/5
9) If sinx= -2/3 and 270< x <360, then find the value of sin(x -270) tan(360- x). 2/3
10) If x lies in the second quadrant, express sinx in terms of tanx. tanx/√(1+ tan²x)
11) If A,B,C,D are four angles of a quadrilateral, show that, sin(A+ B)+ sin(C+ D)= 0.
12) If A,B,C are the angles of a triangle, show that tan (C- A)/2= cot(A+ B/2).
13) If A+ B= 60°, show that, sin(120°- A)= cos(30° - B).
14) For what values of x the equation 2 sink = x + 1/x is possible. ±1
15) Find the value of tan(nπ+ π/4). 1
16) If x= 100, determine the sign of the expression sinx + cosx. Positive
17) If sin(A - B)=√3/2 and sin A= 1/√2, find the least value of B. 75°
18) If cosx= cosy (x≠ y), find a possible value of cos(x+ y). 1
19) Show that: tan1+ tan2 + tan3.........tan87 tan88 tan89= 1.
20) Find the value of cos1+ cos2+ cos3+.........+ cos180. -1
COMPOUND ANGLES
1) From the formula of sin(A+ B) find the formula of cos(A+ B).
2) With the help of formula of sin(A+ B) show that, sin²A+ cos²A= 1.
3) By the formula of cos(A+ B), prove that, cos2B= cos²B - sin²B.
4) Using the formula of cos(A+ B), show that, sin²A + cos²A= 1.
5) If x+ y= π/4, Express tan x in terms of tan y. (1- tany)/(1+ tany)
6) If tanx= 1/2 and x + y=π/4, find the value of tan y. 1/3
7) If tan(A- B)= 1/3 and tanB= 1/2, what is the value of A ? (A is an acute angle). π/4
8) if tanx= 5/6 and tany= 1/11, find the value of x+ y. π/4
9) The value of tangent of two acute angles are 3 and 1/2; what is the difference of those two angles ? π/4
10) The values of contangent of two positive acute angles are 1/2 and 1/3; find the sum of those two angles. 3π/4
11) If tan 18°26'= 1/3, find the angle whose tangent is 1/2. 26°34'
12) if sinx cosy = 1/4 and 3 tanx = tany, show that, sin(x+ y)= 1.
13) if sinx - cosy = sin(π/7) and cosx + siny = cos(π/7), show that, sin(x - y)= 1/2
14) if sinx= 3/5 and cosy = 15/17 and π/2< x <π, 0 < y< π/2, in which quadrant (x + y) lies ? 2nd quadrant
15) If A+ B+ C=π, then the value of cosecA(sinB cosC + cosB sinC) is
a) -1 b) 0 c) 1 d) none
16) The value of sin50 sin70 - cos50 cos70 is
a) cos20 b) - cos20 c) -1/2 d) 1/2
17) The value of cos²(45+ x( - sin²(45- x) is
a) 0 b) 1 c) cos2x d) sin2x
18) The value of tan27+ tan 18+ tan27 tan18 is
a) greater than 1 b) less than 1 c) equals to 1 d) none
19) If (1+ tanx)(1+ tany)= 2, the value of (x + y) is
a) π/4 b) π/2 c) 3π/4 d) none
20) If x+ y + z =π and tanx = 1, tany= 2, the value of tanz is
a) -1 b) -2 c) -3 d) 3
MULTIPLE ANGLES
1) If cosx = 4/5, find the values of cos2x, sin2x, tan2x(0°< x<90°). 7/25,24/25,24/7
2) If sinA= 3.5, find the value of cos2A. 7/25
3) If sinx= 3/5, and x lies in the second quadrant , find the value of sin2x. -24/25
4) if cos⁴x - sin⁴x = (1/2) cos²x, then what is the value of cos2x. 1/3
5) If 2 cosk= a+ 1/a, show that, cos2x= (1/2)(a²+ 1/a²).
6) If cos2x= 3/5, find the values of cosx and sinx. ±2/√5, ±1/√5
7) Find the values of cos15° and sin15° using the value of cos30°. (√3+1)/2, (√3-1)/2
8) Express tanx in terms of cos2x, when π/2< x <3π/4. -√{(1- cos2x)/(1+ cos2c)}
9) if cos2y = 12/13 and 270< 2y <360°, find the value of tany. -1/5
10) if tan2x= 3/4, what is the value of tanx ? (π/2< x<3π/4). -3
11) If tanA= 2/3, find the value of
a) sin2A. 12/13
b) cos2A. 5/13
c) tab2A. 12/5
12) If cosx= 2/√5, find the value of cos3x. 2/5√5
13) if cos A = 3/5, find the value of sin3A. (0°< A< 90°). 44/125
14) If cosx= -4/5 and sin2x= 24/25, in which quadrant x lies ? 3rd
15) if sinx + cosx =1, find the value of sin2x. 0
16) If siny - cosy = √2 siny, show that cot2y =1.
17) if cotx - tanx = 2, show that cot2x =1.
18) If 0< x < π/2 and sinx + cosx =√2, what is the value of sin3x. 1/√2
19) Find the simplest value of 6 sin20- 8 sin³20. √3
20) find the simplest value of prove that if then what is the value of find the greatest value find the greatest value of
SUBMULTIPLE ANGLES
1) Find the value of
a) tan(45/2)+ cot(45/2). 2√2
b) cot(π/8) - tan(π/8). 2
c) cos(π/5) cos(2π/5). 1/4
d) sin(π/10) + sin(13π/10). -1/2
2) Show that 4(cos6+ sin24)= √3+ √15.
3) If cos(m/2) = 12/13, find the value of cosm and tan m. (0< m<90). 119/169, 120/169, 120/119
4) Using the value of sin18, find the value of cos36. (√5+1)/4
5) Using the value of cos36, find the value of sin18. (√5-1)/4
6) If t lies in the fourth quadrant and cos t= 7/25, then find the values of cos(t/2), sin(t/2), tan(t/2). -4/5,3/5,-3/4
7) If sink = -3/5 and 180< k <270, find the value of sin(k/2) and cos(k/2). 3/√10, -1/√10
8) If tan(x/2) = 3/4, find the value of sin x, cosx and ranx. 24/25,7/25, 24/7
9) If cos k= tan²(k/2), find the value of cos k. √2 -1
10) Show that 3 sinx + 4 cosx + 5 is a perfect square quantity.
GENERAL SOLUTIONS OF TRIGONOMETRICAL EQUATIONS
CIRCLE
1) Find the equation of the circle whose centre is at (3,-4) and radius is equals to the radius of the circle x² + y² = 36. x² + y² - 6x + 8y = 11
2) The centre recycle through the origin at the point (-2,3); find the equation of the circle. x² + y² +4x -6y = 0
3) The radius of the circle √(1+ a²)(x² + y²) - 2bx -2aby = 0
a) b b) a c) ab d) √(1 + a²)
4) Find the length of the greatest chord of the circle x² + y² +3x -2y = 3. 5 units
5) A circle touches the axes at (1, 0) and (0, 1). Find the equation of the circle. x² + y² - 2x -2y +1= 0
6) Point out the equation of circle from the following equations
a) x² + 2y² +3x - y+6 = 0.
b) x² + y² - 3x + 2y = 5.
c) x² - y² +4x - 7y+3 = 0
d) x² + y² - 2xy + +4x - 4y = 5
7) If the following equations represent a circle, find their nature:
a) x² + y² = 0. Point circle
b) x² + y² + 25 = 0. Imaginary circle
c) 2x² + 2y² - 4x -6y = 5. Centre of the circle (1,3/2), r= √23/2
d) 3x² + 3y² +8x + 6y +9= 0. Imaginary circle
e) x² + y² +2x - 8y +17= 0. Point circle
8) What is the nature of the locus represent by the equation y(x - y+8)= x(x + y- 6)? Circle
9) Find the radius of the circle 5(x² + y²)= 8x -6y +15.! 2 units
10) If the centre of a circle is at the point (a cosk, a sink) and radius is a units then the circle passes through the origin. T/F
11) When does the equation x² + y² +2gx + 2fy + c= 0 represent a point Circle. T/F
12) Can any circle be drawn through the points ?
a) (0,0), (5,0),(0,-3). Yes
b) (-2,1),( 3,-4) and (-5,4). No
13) Find the equation of the diameter of the circle 3x² + 3y² - 12x + 4y +7 = 0 which passes through (2,-3).
14) one end of a diameter of the circle x² + y² - 8x -4y +15= 0 is ( 2,1) then the other end is
a) (0,0) b) (6,3) c) ( 4,2) d) (3,6)
15) Which one of the following is the equation of a diameter of the circle x² + y² - 6x - 2y = 0 ?
a) 2x + y=0 b) x+ 2y =0 c) x+3y= 0 d) 3x + y=0
16) Under what condition the straight line x+ y+1=0 will be the diameter of the circle x² + y² +2gx + 2fy + c= 0 ? 2x + y=3
17) Examine whether a segment of the straight line 2x + 3y+3= 0 is a diameter of the circle x² + y² + 6x - 2y = 0.
18) find the equation of the normal passing through the origin to the circlenx² + y² - 4x + 6y = 12. 3x+ 2y=0
19) Find find the equation the line passes through the centre of the circle x² + y² +2x + 2y = 23 and is perpendicular to the line x - y +8=0. x+ y +2=0
20) The straight line 3x - 4y +7=0 is a tangent to the circle x² + y² +4x + 2y +4= 0 at the point P. Find the equation of the normal to the circle at the point P. Rx+ 3y +11=0
21) The circle x² + y² - 4x + 2y = 11 intercepts a chord on the line x - 2y =3. Find the equation of the diameter which bisects this chord. Wx + y=3
22) Show that the circle x² + y² +4(x + y) = 0 touches the both the axes.
23) The hypotenuse of the right angled triangle ABC is BC. if the coordinates of the B and C are (3,7) and (9,1) then find centre and radius of the circum-circle of the triangle. (6,4), 3√2
24) Find the equation of the image of the circle x² + y² - x + 8y+10 = 0 with respect to the x-axis. x² + y² - x - 8y +10= 0
25) Find the equation of circle which passes through (-1,1) and is concentric with the circle x² + y² +4x = 0. x² + y² +4x + 2 = 0
26) The radius of the circle which is concentric with 3x² + 3y² - 2x -6y = 4 is 5 units . Find the equation of the circle. 9x² + 9y² - 6x - 18y = 215
27) Find the length of intercept on the x-axis by the circle x² + y² - 7x + 6 = 0. 5 units
28) Find the position (inside/on/outside) of the point (5,7) with respect to the circle x² + y² + 6x -2y = 15. Outside
29) The point (0,0) lies
a) on b) in c) outside the circumference of the circle x² + y² +2x -2y = 2.
30) Find the position of the point (-3,-2) with respect to the circle whose equation is x² + y² - 3x + 2y = 19. Inside
31) The straight line x cosk + y sink = 4 does not interesect the circle x² + y² = 25. T/F
32) Express the equation of the circle x² + y² +6x -10y = 15 in the parametric form. x= -3+ 7 cosk, y= 5+ 7 sink
33) If the length of the diameter of the circle 2x² + 2y² +4x -6y - k= 0 is 4, find k. 3/2
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