CLASS XI - (x series)
THEORY OF QUADRATIC EQUATIONS
1) Determine the nature of roots of the following equations:
a) 9x²- 12√2 x +8=0. irrational and equal
b) x²- 5 x +6=0. rational and unequal
c) 2x² + 4x -3 =0. irrational and unequal
d) 3x²- 2 x + 5=0. imaginary and unequal
2) For what value(or values) of m will the equation x²- 2(5+ 2m)x + 3(7+ 10m) =0 have
a) equal roots. 2 or 1/2
b) reciprocal roots. -2.3
3) Find the value of m, if the roots of x²- (5+2m)x + (10+ 2m) =0 equal in magnitude but opposite in sign . -5/2
4) Both the roots of the quadratic equation x²- (a+1)x + a+ 4 =0 are negative. Calculate the value of a. -4< a <-1
5) Form the quadric equation whose roots are 3 and -5. x² +2 x -15=0
6) Form the quadratic equations whose one root
a) 2+ √3. x²- 4x + 1 =0
b) 3- 4i. 3px²- 6x + 25=0
7) If m,n be the roots of the equation ax² + bx + c =0, Find the value of the following:
a) m²+ n². (b²- 2ac)/a²
b) m²- n². ±b√(b²- 4ac)/a²
c) m³- n³. (3abc - b³)/a³
d) m⁴- n⁴. (b⁴+ 2a²c² - 4ab²c)/a⁴
e) mn⁻¹ + nm⁻¹. (b²- 2ca)/ca
f) m/n²+ n/m². (3abc - b³)/ab²
8) If m, n be the roots of x²- px + q=0, form the equation whose roots are mn + m+ n and mn - m - n. x²- 2qx + (q²- p²) =0
9) if the roots on the equation 3x²- 6x + 4 =0 are m, n, find the value of (m/n + n/m)+ 2(1/m + 1/n) + 3mn. 8
10) if mn are the roots of the equation x²- 2 x + 3 =0, find the equation whose roots are m/n² and n/m². 9x²+10x + 3=0
11) The roots of the equation px²- 2 (p+ 2)x + 3p =0 are m,n If m - n =2, calculate the value of m, n and p. -1,-3,-2/3 or 3,1,2
12) The ratio of the roots of the equation x²- mx + m+ 2=0 is 2. Find the values of the parameter m. 6 or -3/2
13) If one root of the equation x² + bx + 8 =0be 4 and the roots of the equation x² + bx + c =0 are equal, find the value of c. What will be the equation whose roots are inverse of roots of the first equation? 8x²- 6x + 1=0.
14) Find the condition that the roots of the equation ax² + bx + c =0 may differ by 5. b²- 4ac = 25a²
15) The ratio of the roots of ax² + bx + c =0 is 3:4. Prove that 12b²= 49ac.
16) If the roots of the equation px² + qx + q =0 be the ratio m: n, show that √(m/n) + √(n/m) + √(q/p) = 0.
17) if one roots of ax² + bx + c =0 be the square of the others, prove that b³+ a²c + ac² = 3abc.
18) If the difference between the roots ax² + bx + c =0 be equal to the difference between the roots of px² + qx + r=0, show that p²(b²- 4ac)= a²(q²- 4pr).
19) prove that if x²- p x + q=0 and x²+ qx + p =0 have common root, then either p= q or, p+ q+1=0.
20) If the equations x²- 5x + y =0 and x² + mx + 3 =0 have a common root , find the value of m. -4 or -7/2
21) If the equation ax² + bx + c. =0 and bx²+ cx + a=0 have a common root, prove that, a³+ b³+ c³= 3abc.
22) Prove that if the equation x² + bx + ca =0 and x² + cx + ab =0 (b ≠ c) have only one root common , then their roots will be satisfy t²+ at + bc=0.
QUADRATIC EXPRESSION OR FUNCTION
1) Prove that expression 14x²- 9x + 1 is positive for all real values of x, except when x lies between 1/7 and 1/2.
2) Show that for all real values of x, the expression 3x²- 12 x + 17 is positive.
3) Find for the real values of x, the maximum value of 3 - 20x - 25x². -2/5
4) Find for all the real values of x, the minimum value of 3x²- 5x + 8. 5/6
QUADRATIC INEQUALITIES AND THEIR SOLUTIONS
1) Solve: x² + 4x - 5 > 0. x< -5 or x > 1
2) Solve: (x +1)/(4x -1) < 0. -1< x < -1/4
3) Determine the range of the value of x for which (x² +x + 1)/(x²+2) < 1/3, x being real. -1< x <1/2
4) Find the range of real values of x for which (x -2)/(3x+4) < (x -4)/(3x -2). -4/3< x < 2/3
5) Find all real values of x satisfy x²- 3x + 2 ≥ 0 and x²- 3x -4 ≤ 0. -1≤ x ≤ 1 or 2 ≤ x ≤ 4
6) Find the range of x in the equation (x²- 5x + 7)/(x²- 7x + 12) < 1/2. 1<x <2 and 3< x < 4
7) If x is real, prove that the value of the Expression (x²- x + 1)/(x²+x + 1) lies between 1/3 and 3.
8) If x is real, find the maximum in the minimum value of (x +2)/(2x²+3x + 6). 1/3 and -1/13
9) If 3p²= 5p +2 and 3q²= 5q +2 where p≠ q, obtain the equation whose roots are 3p - 2q and 3q - 2p. 3x²- 5x -200 =0
10) Find the quadratic equation one of whose root is 2ab/{(a+ b)+ √(a²+ b²)}. x²- 2(a+ b)x + 2ab =0.
11) If 3b³+ 9a²c + ac²= 9abc, prove that the square of one roots of the quadratic equation ax² + bx + c =0 is three times other.
ARITHMETIC PROGRESSION
1) Determine the AP whose first time is 5 and common difference is - 2.
2) Find 10th term of the AP where first term is 5 in the common difference is 2. 23
3) The 20th term of an AP is 79. If the first term is 3, find 10th term. 39
4) Which term of the AP 2 + 5 + 8 + .....92 ? 31st term
5) Is 90 a term of the series 4 + 7 + 10 + 13 + .....? No
6) Determine 2nd term and r-th term of the AP whose 6th term is 12 and 8th term is 22. -8, 5r - 18
7) Insert 5 arithmetic means between 17 and 29. 19,21,23,25,27
8) Insert 4 arithmetic means (a- b)² and a²+ b²+ 8ab. a²+ b², (a+ b)², a²+ b²+ 4ab, a²+ b²+ 6ab
9) Find the middle term or terms of the following APs
a) 5,8,11,.....95. 50
b) 88,80,72,....,-64? 16,8
10) If a, b, c are in AP show that 1/bc, 1/ca, 1/ab are also in AP.
11) If 1/(b + c), 1/(c + a), 1/(a+ b) are in AP prove that a², b², c² are also in AP.
12) Find the value of k for which k²- 7k, k²+ 9 and 6 are in AP. -3 or -4
13) If a, b, c are in AP, show that (b + c)² - a², (c + a)² - b², (a+ b)² - c² are also in AP.
14) Find three numbers in AP, whose sum is 15 and sum of their products in pair is 71. 3,5,7 or 7,5,3
15) Find the four terms in AP whose sum is 20 and the sum of whose squares is 120. 2,4,6,8 or 8,6,4,2
16) Find the sum of the following series:
a) 2 + 7 + 12.... up to 20 terms. 990
b) (n -1)/2 + n/n + (n +1)/n........up to n terms. (3/2) (n -1)
17) Find the sum of the series 7 + 10 + 13 + ....+64. 710
18) The first and the last term of an AP having finite number of terms are respectively -2 and 124 and sum of the AP is 6161. Find the number of terms in the AP. 101
19) Find the sum of all integers between 50 and 500 are divisible by 7. 17696
20) Find the sum of all perfect square between 200 to 800. 6699
21) Sum to n terms the series 3.7+ 5.10 + 7.13 +.......(n/2) (4n²+ 17n +21)
22) Sum to n terms the series 1/2.5 + 1/5.8 + 1/8.11+.....? n/2(3n +2)
23) Find the increasing AP, the sum of whose first three terms is 27 and the sum of their squares is 275. 5,9,13,17,...
24) Find four terms in arithmetic progression if their sum and product are respectively 16 and 105.
25) Find the r-th term of an AP, the sum of whose first n terms in 2n + 3n². 6r -1
26) The sum of n terms of an AP is 3n²+ 5n. Find which term of the AP is 152. 25
27) The first and the last term of an AP are respectively -4 and 146, and the sum of AP = 7171. Find the number of terms of the AP and also its common difference. 101, 1.5
28) if p-th, qth and rth terms of an AP are a, b and c respectively, prove that a(x- r)+ b(r - p)+ c(p - q)=0.
29) if a,b,c be respectively the sums of p, q and r terms of an AP prove that :
a(q -p)/p + b(r - p)/q + c(p - q)/r =0.
30) The sum of p terms of a series in 2p²+ p. Prove that the series is in AP.
31) If a, b, c, are in AP, show that
a) a²(b + c), b²(c + a), c²(a+ b) are in AP.
b) a(b + c)/bc , b(c +a)/ca, c(a+ b)/ab are in AP.
32) If S₁, S₂, S₃ be the sums of n, 2n and 3n terms respectively of an AP, show that S₃ = 3(S₂ - S₁).
33) The sum of m terms of an AP is n and the sum of n terms is m. Find the sum of (m + n) terms. -(m + n)
34) Sum to n terms of the series 1 + 3 + 6 + 10+..... n(n+1)(n+2)/6
35) The sum of the digits of a three digited number is 12. The digits are in AP, if the digits are reversed, then the number is diminished by 396. Find the number. 642
36) A man is employed in a company on Rs800 per month, and is promised an increment of Rs25 per year. Find the total amount which he receives in 13 years and his pay in the last year. Rs1100
37) A man borrows Rs1200 at the total interest of Rs68, he repays the entire amount in 12 installments, each installment being less than the preceding one by Rs20. Find the first installment. Rs224
38) a9 sum of Rs6240 is paid off in 30 installments , such that each installments is Rs10 more that the preceding installment. Calculate the value of the first installment. Rs63
39) A man arranges to pay off a debt of Rs7200 by 20 installments which form an AP. When 15 of the installment are paid, he finds that one-third of his debts still remained unpaid. Find the amount of his 16th installment. Rs448
40) A circle is completilly divided into n sectors in such a way that the angles of the sectors are in arithmetic progression. If the smallest of these angles is 8° and the largest 72°, calculate n and the angle in the fourth sector. 32°
41) The interior angles of a polygon are in AP. The smallest angle is 52° and the common difference is 8°. Find the number of sides of the polygon. 3
42) the last term of an arithmetic progression 2, 5, 8, 11.... is x. The sum of the terms of the AP is 155. find x. 29
43) a₁ , a₂ , a₃ , a₄ , a₅ are first five terms of an AP such that a₁ + a₃ + a₅ = -12 and a₁. a₂. a₃ = 8. Find the first term and the common difference. 2, -3
44) sum of the series 1/n + (n +1)/n + (2n +1)/n + ......+ (n²- n +1)/n.
GEOMETRIC PROGRESSION
1) Determine the GP whose first term is 3 and common ratio is -2. 3,-6,18,-36,...
2) Find the 10th term of the GP whose first term is 3 and common ratio is -2. -1536
3) The 5th term of a GP is 162 and first term is 2. Find the common ratio. 3 or -3
4) The 10th term of a GP is -2560 and the first term is 5. Find the 5th and n-th term of the GP. 80, 5(-2)ⁿ⁻¹
5) Which term of the series 4 + 12 + 36 + 108+.... is 2916? 7th
6) Us 3000 a term of the series GP. 3, 15, 45, 135...? No
7) Determine the first 3 terms and 9th term of a GP whose 4th term is 24 and 7th term is 192. 3,6,12, 768
8) Insert 6 geometric means between 8 and 1/16. 4,2,1,1/2,1/4,1/8
9) Insert 5 geometric means between 3 and 81. 3√3,9,9√3,27,27√3 or -3√3,9, -9√3, 27, - 27√3
10) Find the sum of the following series:
a) 2+6+18+54+.... to 10 terms . 59048
b) 1- 1/2 + 1/4 - 1/8 +,,,, to 12 terms. 4095/2048
11) Find the sum of the series 3 + 6 + 12 +....+ 3072. 6141
12) Find the sum of the first n terms of the following :
a) 4 + 44+ 444+....... 40(10ⁿ -1)/81 -4n/9
b) .3 + .33 + .333+.... n/3 -1/27(1- 1/10ⁿ).
13) Find the sum of the infinite of GP 1, 1/2, 1/4, 1/8,... 2
14) A man saves Rs100 in the first day and in each of the succeeding days he saved 9/10th of what he saved in the previous day. Show that his total savings will not exceed Rs1000 however long he may live.
15) If S₁, S₂, S₃,....Sₙ are the sum of infinite geometric series whose first terms are 1,2,3.....n and whose common ratios are 1/2, 1/3, 1/4,.....1/(n +1) respectively, then find the value of S₁²+ S₂²+ .....S²₂ₙ₋₁. {n(2n+1)(4n +1)-3}/3
16) The side of a given square is equal to a. The mid-points of its sides are join to form a new square. Again, the mid-points of the sides of this new square are joined to form another new square. This process is continued indefinitely. Find the sum of the areas of the squares and the sum of the perimeter of the squares. 2a², 4√2(√2+1)a
17) The sum of an Infinity GP is 16 and the sum of the squares of its terms is 153-3/5. Find the common ratio and the fourth terms of the progression. 1/4, 3/16
ARITHMETICO- GEOMETRIC SERIES
1) Find the sum of 1.3 + 2.3²+ 3.3²+..... n. 3ⁿ.
2) Find the sum of n terms of the series 1+ 11/5 + 76/25 + 501/125+....
3) if 3x+1, 7x, and 10x + 8 be in GP, Find the value of x. 2 or -4/19
4) If a, b, c, d are in GP show that a²- b², b²- c², c²- d² will also be in GP.
5) if a, b, c, d are in GP show that (b - c)²+ (c - a)²+ (d - b)²= (a - d)².
6) The first three terms of a GP are x, x+3, and x+9. Find the value of x and the sum of the first 8 terms. 3, 765
7) Calculate the least number of terms of the GP 5+10+29+..... whose sum would exceed 1000000. 18
8) Show that in a GP, the product of any two terms equidistance from the beginning and the end is equals to the product of the first and the last terms.
9) If a,b,c are in GP and x,y be the arithmetic means between a,b and b,c respectively. show that a/x + c/y = 2.
10) if 1/(x+ y), 1/2y, 1/(y+ z) are the three consecutive terms of an AP. prove that x, y, z are the 3 consecutive terms of a GP.
11) If the arithmetic mean between a and b is twice as last as their Geometric mean, prove that the ratio between the numbers can be written as 2+ √3: 2 -√3.
12) if one arithmetic mean A. and two geometric mean G₁ and G₂ be inserted between any two numbers, then show that G³₁ + G³₂ = 2AG₁G₂.
13) If S be the sum, P the product and R the sum of reciprocal of first n terms in a GP. Show that P²Rⁿ = Sⁿ.
14) Find the sum of the infinite series 1+ (1+ b)r+ (1+ b + b²)r²+ ...
15) Find the four positive integers x, y, z, w such that y, z,w are in AP; x,y,z are in GP and z+ w=10, x+ y= 3. 1,2,4,6
16) If Sₙ be the sum of an infinite GP series whose first term is n and the common ratio is 1/(n+1), find the sum S₁ + S₂ +....+ Sₙ. n(n+3)/2
17) The sum of an infinit GP is 15 and the sum of their squares is 45. Find the series.
18) The product of three number in GP is 729 and the sum of their squares is 819. Find the numbers. 27,9,3 or -27,9,-3
19) The product of four positive numbers in GP is 729 and the sum of two intermediate terms is 12. Find the numbers. 1,3,9,27 or 27,9,3,1
20) The sum of the four terms in GP is 16. and the arithmetic mean of the first and the last numbers is 18. Find the numbers . 32,16,8,4 or 4,8,12,32
21) Three numbers , whose sum is 70 are in GP. If each of the extremes is multiplied by 4 and the mean by 5, the numbers will be in AP. Find the numbers. 10,20,40 or 40,20,10
22) Three numbers are in AP and their sum is 21. If 1,5,15 be added to them respectively, they form a GP. Find the numbers. 5,7,9
23) Find three number a,b,c between 2 and 18 such that
a) their sum is 25.
b) the number 2, a, b are consecutive terms of an AP
c) the numbers b,c 18 are consecutive terms of a GP. 5,8,12
24) A bouncing tennis ball rebounds each time to a height equal to one half of the height of the previous balance. If is dropped from a height of 16 m, find the total distance it has travelled when it hits the ground for the 10th time. 47-15/16m
25) A man borrows Rs16380 without interest and repays the loan in 12 monthly installments , each installment (beginning with the second) being twice the preceding one. Find of the last installment. Rs8192
26) If a, b, c are in AP a, x, b and b, y, c are in GP. Show that x², b², y² are in AP.
27) a, b, c are in AP and b, c, a are in GP show that 1/c, 1/a, 1/b are in AP.
28) If p,q,r are in AP show that p-th, qth and rth terms of a GP are in GP.
29) Find the first term of a GP whose sum to infinite term is 8 and second term is 2. 4
30) If one geometric mean G and two arithmetic means a and b be inserted between two quantities. Show that (2a - b)(2b - a)= G².
31) If a, b, c are three distinct real numbers and they are in GP if a + b + c = xb then show x< -1 or x > 3.
FUNCTION
2) 2x -1 when x≤ 2
f(x)= x²-1 when 2 < x ≤ 3
2x²+ 1 when x > 3
Find f-1), f(2), f(2.5), f(3), f(3.5). -3, 1. 5.25, 8, 8
3) If f(x)= 3 cos²x - 2sinx. Find f(π/3, f(4π/3), f(-π/6). 3/4 - √3, 3/4 + √3, 13/4
4) If f: x--> log₃(x²+ 2x +3), find f(4). 3
5) If y= f(x)= (2- x)/(5+ 3x) and z= f(y), express z in terms of x. (7x+8)/12x+31)
6) If f(2x- 1)= (3x-1)/(x+1), find f{f(4)} and f{f(1- 3x)}. 23/17, (8-15x)/(8-9x)
7) If f(x)= x(x - a)/(b - a) + x(x - b)/(a- b), show that f(a)+ f(b)= f(a+ b).
8) If f(x)= (ax+ b).(bx - a), find f{f(1/x)}. 1/x
9) If f(x)= log{(1- x)/(1+ x)}, show that f(p)+ f(q)= f{(p+ q)/(1+ PQ)}.
10) If y= f(x)=(2x +1)/(3x - m) and f(y)=x, find m. 2
11) Show that x(2ˣ +1)/(2ˣ -1) is an even function.
12) Show that log{√(1+ x²) - x} is an odd function of x.
13) If f(x)= ax²+ bx + c, find a, b so that f(x +1)= f(x)+ x +1 may hold identically. 1/2, 1/2
14) If 3 f(x) - 2f(x) = 10x - 3, find f(5 - 2x). 7 - 4x
15) Find the domain of definition of the followings:
a) (2x+1)/(x²- 6x +8). R --- {2,4}
b) √(x²- 8x +15). - ∞ < x ≤ 3 and 5≤ x < ∞
c) (x -2)/√(9- x²). -3< x < 3
d) log{(4+ x)/(4- x)}. -4< x < 4
e) 1/√{(logx)² - 5 logx +6)}. 0< x< 100 and x > 1000
16) Find the range of the following functions:
a) x/(1+ x²). 0< y ≤1/2
b) 1/(3- cos2x). 1/4≤ y≤1/2
c) √(x - x²). 0≤ y ≤ 1/2
CIRCLE
Find the equation of the circle
a) whose centre is at the origin and radius is √3 units. x²+ y²= 3
b) Whose centre is at the point (3,7) and radius is 5 units. x²+ y² -6x -14y+ 33=0
2) The equation of two diameters of a circle passing through the point (-2,2) are 3x + y= 5 and x+ y +1=0 respectively. Find the equation of the circle. x²+ y² -6x +8y- 36=0
3) Find the equation of the circle which passes through the point (4,2) and touches both the coordinates axes. How many such circles are possible ? 2
4) Find the equation of the circle which touches the x-axis at a distance of 12 units from the origin and cuts off a chord of length 24 units from the y-axis. x²+ y² ±24x ±24√2y+ 144=0
5) Find the centres and the radii of the following circles:
a) x²+ y² -5x +2y+ 5=0. (5/2,-1), 3/2
b) 3x²+ 3y² -5x -6y+ 4=0. 5/6,1
6) Prove that the centeres of the three circles x²+ y² =1, x²+ y² -6x -2y-6 =0 and x²+ y² -12x +4y=9 are collinear and their radii are in AP.
7) Find the equation of circle which passes through the points (1,1),( 3,2) and (5,4). x²+ y² +3x -17y+ 12=0
8) A circle has its centre on the line 5x- 2y +1=0 and its cut the x-axis at the two points whose abscissa are - 5 and 3; find the equation of the circle. x²+ y² +2x +4y-15=0
9) The ends of the diameter of a circle are the points (4,-2) and (-1,3); find the equation the circle. Find the equation of the diameter of this circle through origin. x= 3y
10) Find the equation of the circle circumscribing the triangle formed by the straight line 2x + 3y = 6 with the coordinates axes . What is the length of the diameter of the circle ? x²+ y² -3x -2y=0. √13 units
11) Find the equation of the circle which is concentric with the circle x²+ y² -2x +4y-11 =0 and passing through the point of intersection of the lines 2x - 3y= 2 and x+ 2y = 8. x²+ y² -2x +4y-20=0.
12) Find the equation of the circle which is concentric with the circle x²+ y² -2x +4y+ 17 =0 and touches the line 3x + 4y +20= 0. x²+ y² -2x +4y- 4=0
13) Find the length of the chord of the circle x²+ y² =20 intercepted on the straight line 3x + 4y +10= 0. 8 units
14) Find the equations of the chords of the circle x²+ y² -2x=0 which pass through the origin and are of lengths √2 units. y= ± x
15) A circle touches both the axes in the positive direction and also touches the straight line 3x+ 4y -6=0. Find the equation of the circle. x²+ y² -6x -6y+ 9 =0 and 4(x²+ y²) -4(x +y) + 1=0
16) Show that the circles x²+ y² -8x -4y -16 =0 andnx²+ y² -2x +4y+ 4=0 touch each other internally and find the co-ordinates of their point of contact.
17) Show that the circle x²+ y² -2x - 15=0 lies entirely inside the circle x²+ y² -x - 30=0.
18) Show that the two circles x²+ y² +2ax + c²=0 and x²+ y² + 2by+ c² =0 will touch each other if 1/a² + 1/b² = 1/c².
19) Show that the point (3,-2) lies inside the circle x²+ y² -4x +2y-11=0. Find the equation of the chord of the circle which is bisected at the point (3,-2). x- y=5
20) Show that the straight line 4x + 3y - 31=0 touches the circle x²+ y² -6x +4y= 12. Find the coordinates the point of contact.
21) A circle touches the line 3x - 2y = 6 at the point (4,3) and passes through the point (-2,1). Find the equation of the circle. 7(x²+ y²) +4x - 82y+ 55 =0
22) Show so that the point (2,-5) lies on the circle x²+ y² -2x +4y =5. Find the equation of the tangent to the circle at the point (2,-5).
23) Find the co-ordinates of the point on the circle x²+ y² -2x + 6y - 58 =0 the tangents at which are perpendicular to 4x - y= 2. (-1,-11) and (3,5)
24) Find the centre and radius of the circle x²+ y² -4x +2y-20 =0. Find also the point on the circle which is nearest to the point (2,3/2). (2,-1), 5 units, (2,4)
25) Show that for all the values of p, the circle x²+ y² -x(3p+4) -y(p -2)+ 10p =0 passes through the point (3,1). If p varies, find the locus of the centre of the above circle.
26) show that the equation the locus of the midpoints of the chords of the circle x²+ y² = r² passing through (h,k) is x²+ y²= hx + ky.
27) A is a point on the circle x²+ y²= 36. if the ordinate AN is divided at P internally in the ratio 2:1, find the equation of the locus of the point P. x²+ 9y² =36
28) The abscissa of the two points A and B are the roots of the equation x²- b² +2ax =0 and their ordinates are the roots of the equation x² + 2px - q²= 0. Find equation and the radius of the circle with AB as the diameter. x²+ y² +2ax + 2py - b² - q²=0
29) Let x²+ y² -4x -2y -11=0 be a given circle. Find the area of the quadrilateral formed by the pair of tangents drawn from the point (4,5) with the pair of corresponding radii. 8 sq. Units
30) Find the co-ordinates of the point on the circle x²+ y² -4x +2y-8 =0 which is nearest to the straight line 3x - 2y +18=0. (-1,1)
31) Find the equation of the circle which touches both the axes at a distance of +5 units from the origin. x²+ y² -10x -10y+ 25 =0
32) The circle x²+ y² -8x -6y+ 16=0
a) touches x-axis b) touches the y-axis c) does not touch any axis
33) Identify the equations of the circle from the following equations (give reason for your answer)
a) x²+ y² +3xy -6x+ 5y -8=0.
b) 2x²+ 3y² +2x -3y+ 5=0.
c) x² - y² + 5x -3y+ 6=0.
34) If the following equations represents circle, find their nature:
a) x²+ y² -4x + 6y+ 13 =0. Point circle
b) 3x²+ 3y² +8x + 6y+ 3=0. Real circle
c) 2x²+ 2y² -7x -6y+ 12=0. No
35) Can any circle be drawn through the points (1,1),(6,-2) and (-4,4)? No
36) Find the equation of a diameter of the circle x²+ y² -6x +2y=15 which passes through the point (8,-2). x+ 5y+2=0
37) show that the point (-1,-2) lies on the circle x²+ y² -x -y 8=0. Find the coordinates of the other end of the diameter through the point (-1,-2).
38) Find the radius of the circle which passes through the origin and the points (a,0) and (0,b). (1/2) √(a²+ b²)
39) Find the radius of the circle passing through (12,5) and concentric with the circle x²+ y²= 1. 13 units
40) Find whether the point (2,1) lies inside, on or outside the circle x²+ y² -4x +6y+ 8 =0. Inside
41) Can any tangent be drawn to the circle x²+ y² = 36 at or from the point (4,-4)? No
42) Show that the straight line 4x - 7y +28=0 does not interesect the circle x²+ y² -6x -8y+ 23 =0.
43) Find the co-ordinates of the points which lie on the circle x²+ y² -6x -2y+ 6 =0 and are equidistance from the axes. (1,1) and (3,3)
44) If the parametric equation of the circle be x= 2+3 cos k, y= -5+ 3 sin k; then find the equation of the circle. x²+ y²- 4x + 10y +20= 0
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