Download Revision - Gr 11

Buds Public School , Dubai
Mathematics - Linear In equalities ,Permutation
and combinations
Grade : 11 A B
1. Solve the following inequalities for real x :
2𝑥−1
3𝑥−2
2−𝑥
𝑥
5𝑥−2
2−𝑥
a) 3 ≥ 4 − 5
b) 4 < 3 − 5
8𝑥
c)
3𝑥−1
5
𝑥−1
≤ 2−
𝑥+1
3
𝑥+3
d) 4𝑥−7 < 2, 𝑥 − 7 < 0
e) |𝑥 + 2| > |3𝑥 − 5|
f) 𝑥+2 > 1 , 𝑥−1 > 2
2. Solve the following system of inequalities graphically :
a) 𝑥 + 2𝑦 ≤ 8 , 2𝑥 + 𝑦 ≤ 8 , 𝑥 ≥ 0 , 𝑦 ≥ 0
b) 4𝑥 + 3𝑦 ≤ 60 , 𝑦 ≥ 2𝑥, 𝑥 ≥ 3, 𝑥 ≥ 0 , 𝑦 ≥ 0
c) 𝑥 + 2𝑦 + 3 ≤ 0 𝑎𝑛𝑑 𝑥 + 2𝑦 − 4 ≥ 0
d) 𝑥 + 2𝑦 ≤ 10 , 𝑥 + 𝑦 > 1, 𝑥 − 𝑦 ≤ 0, 𝑥 ≥ 0 , 𝑦 ≥ 0
𝑥
5𝑥−2
7𝑥−3
3. Solve the inequality : 4 < 3 − 5 and show the graph of the solution.
4. Solve the following system of inequalities graphically ; a) 3𝑥 − 7 < 5 + 𝑥 b) 11 − 5𝑥 ≤ 1
And represent the solutions on the number line .
18
7
5. Draw the graph of the in equation 7𝑥 + 18 ≥ −𝑥 + 5 and 3 (𝑥 − 3) ≤ 8𝑥 + 3
6. Find all pairs of consecutive odd natural numbers , both of which are larger than 10 such that
their sum is less than 40 .
7. In how many ways 3 prizes be distributed among 4 boys ,when
i) no boy gets more than one prize ?
ii) a boy may get any number of prizes ? iii) no boy gets all the prizes ?
8. In how many ways six persons be seated in a row ?
9. How many 3 digit numbers can be formed from the digits 1,2,3,4 and 5 assuming that
i) repetition of digits is allowed ? ii) repetition of digits is not allowed ?
10. A coin is tossed three times the outcomes are recorded . How many possible outcomes are
there ?
11.How many 5 digit telephone numbers can be constructed using the digits 0 to 9 if each
number starts from 67 and no digits appears more than once ?
12. Evaluate :
1
1
𝑛!
𝑟!(𝑛−𝑟)!
𝑥
13. If 6! + 7! =
8!
, 𝑤ℎ𝑒𝑛 𝑛 = 8 , 𝑟 = 2 .
, find x
14. Find 6P3 and 6 C 3 . Are they equal ?
15. From a committee of 8 persons , in how many ways can we choose a chairman and a vice
chairman assuming one person can not hold more than one position ?
16. A bag contains 5 black and 6 red balls . Determine the number of ways in which 2 black and
3 red balls can be selected . If n C8 = n C2 . find n .
17 If 22P r+1 : 20P r+2 = 11: 52 , find r .
18. Write the number of ways in which 7 men and 7 women can sit together on a round table
such that no two women sit together .
19. In how many ways can five children stand in a queue?
20. How many words , with or without meaning can be formed by using the letters of the word
‘TRIANGLE ‘ ?
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Buds Public School , Dubai
Mathematics - Binomial Theorem
Grade : 11 A B
1. Using Binomial theorem expand the following : a) (101)5 b) (1 − 2𝑥)5
d) 514
e) (0.998)8
f)
(999)4 g) (𝑥 + 3𝑦)3
h) (3𝑥 2 − 2𝑦)4
𝑥
2
c) (2 − 𝑥)5 .
1
i) (𝑥 + )11
𝑦
2. Using Binomial Theorem , prove that 6𝑛 − 5𝑛 always leaves the remainder 1 when divided by 25 .
3. Find the value of the following :
1
a) (2𝑥 − 3𝑦)4 b) (0.99)8
c) (x - 𝑥)4 i) (√2 + 1)6 − (√2 − 1)6
3
3
4. Expand : a) ( √𝑥 − √𝑎)6
𝑥
2
1
b) (𝑥 + 1 − 𝑥)3
c) (√3 + √2)6 − (√3 − √2)6
5. Expand a) (1 + 2 − 𝑥)4 , x≠ 0
b) (1 − 𝑥 + 𝑥 2 )4
6. Find the general term of the following :
1
1
a) (2𝑥 − 𝑥 2 )54 b)
(x + 𝑥)8
1
7. Find the 7 th term in the expansion of (𝑥 − 𝑥 2 )40
8 Find the coefficient of 𝑥10 in the expansion of
1
1
(2𝑥 − 𝑥 2 )20
9. Write the general term of the expansion (𝑥 + 𝑦)11
1
10. Find the 10 th term in the binomial expansion of (2𝑥 2 + 𝑥)12
1
11. Find the coefficient of 𝑥 32 𝑎𝑛𝑑 𝑥 −17 in the expansion (𝑥 4 − 𝑥 3 )15
12. Find the coefficient of 𝑥 6 𝑦 3 in the binomial expansion of (𝑥 + 2𝑦)9
x
3a
13. Find the 4 th term in the expansion of (a − x2 )12.
x
3a
14. Find the 9 th tem in the expansion of (a − x2 )12.
15. Find the general term of the following :
1
1
a) (2𝑥 − 2 )54 b)
(x + )8
𝑥
16. Find the 7
th
𝑥
1
term in the expansion of (𝑥 − 𝑥 2 )40
1
17 Find the coefficient of 𝑥10 in the expansion of (2𝑥 − 𝑥 2 )20
18. If 3rd , 4th ,5th and 6th terms in the expansion of (𝑥 + 𝑎)𝑛 be respectively a, b,c and d , prove
𝑏 2 −𝑎𝑐
5𝑎
That 2
=
𝑐 −𝑏𝑑
3𝑐
19. If the coefficient of three consecutive terms in the expansion of
(1 + 𝑥)𝑛 𝑏𝑒 76,95𝑎𝑛𝑑 76.Find n
20. The ratio of the sum of n terms of two A.P ‘s is (7𝑛 + 1): (4𝑛 + 27). Find the ratio of their
mth terms .
2
4
Find the sum up to n terms and 5 terms of the geometric series 1 + 3 +9 , ….
Find the 12 th term of a G.p whose 8th term 192 and the common ratio is 2 .
Insert 4 AM ‘s between 4 and 19.
Find three numbers in G.P whose sum is 13 and the sum of whose squares is 91.
Find the sum of the following series :
a) 5 + 55 + 555 +…..to n terms . b) 1.22 + 2. 32 + 3. 42 + … . 𝑛 𝑡𝑒𝑟𝑚𝑠 .
c) 1.2.4 + 2.3.7+ 3.4.10 + ….. n terms d) 2𝑛3 + 3𝑛2 − 1
26. If 𝑎2 + 𝑏 2 , 𝑎𝑏 + 𝑏𝑐 𝑎𝑛𝑑 𝑏 2 + 𝑐 2 are in G.P , prove that a ,b , c are in G.P.
1,1 1
27. If a,b,c are in A.P , b , c, d are in G.P and 𝑐 𝑑 , 𝑒 𝑎𝑟𝑒 𝑖𝑛 𝐴. 𝑃 , prove that a, c, e are G.P
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21.
22.
23.
24.
25
Sequence and Series
1. Find the sum of even numbers from 1000 to 2000
−11
2. How many terms of the A.P -6 , 2 ,−5 , … are needed to give tha sum –25?
3. If the sum of a certain number of terms of the A.P 25,22,19,… is 116 . Find the last term .
4. Find the sum up to n terms of the A.P whose k th term is 5𝐾 + 1
5. Insert 7 terms between 8 and 26 such that the resulting sequence is an A. P
6. Which term is the Sequences 2,2√2: , 4 is 125?
7. How many terms of the A.P -6 ,
−11
2
,−5 , … are needed to give tha sum –25?
8. If the sum of a certain number of terms of the A.P 25,22,19,… is 116 . Find the last term .
9. The fourth term of a G,P is square of its second term and the first term is -3 .
−2
−7
10. For what values of x , the numbers 7 , 𝑥 , 2 are G.P ?
11. Find the sum upto n terms of the sequence 6,66,666,…..
2 4
12. Find the sum up to n terms and 5 terms of the geometric series 1 + 3 +9 , ….
13. Find the 12 th term of a G.p whose 8th term 192 and the common ratio is 2 .
14. The fourth term of a G,P is square of its second term and the first term is -3 .
Straight Lines
1.Find the slope of the line whose inclination with x – axis is a) 30° 𝑏) 0° 𝑐) 90° d) 135°
2. Find the slope of the line passing through the points (0, 2)𝑎𝑛𝑑 ( √3 , 3)
3. Calculate the slope of the line passing through the points (3,2) and (4, 1) . Also find the
inclination of the line with x-axis .
4.. Find the slope of a line which makes the following angle with x-axis .
a) 150° 𝑏) 240° 𝑐) 315° 𝑑) − 120°
5. Find the slope of the line passing through the points . a) A (-1,7) , B(0,3) b) R(𝑎2 , 𝑏), (𝑏 2 , 𝑎)
6. Slope of a line joining the points (7,3) and ( k, 2) is -4 . Find the value of k .
7. Find the angle between the line joining the points (3,-1) and (2,3) and the points (5,2) and (9,3)
8. In fig . time and distance graph of a linear motion is given . Two positions of time and distance are
recorded as , when T = 0 , D = 2 and when T = 3 , D = 8 . Using the concept of slope find the law of
motion . (ie) how distance depends upon time .
𝑎 𝑏
9. If three points (h,0) , (a , b) and (0,k) lie on a line , show that ℎ +𝑘 = 1
10. Find the angle between the x-axis and the line joining the points (3,-1)and (4,-2)
𝜋
1
11. The acute angle between two lines is 4 and slope of one of the lines is 2. Find the slope of the
other line .
12. Write the equation for the X – axis and Y – Axis .
1
13. If the line passing through the point (-4,3) with slope 2 . Form the equation of the line .
14. Find the equation of a line parallel to x-axis at a distance of 3 units above x-axis .
15. A line passes through point (1,5) and cuts off intercept 7 units on x-axis . Find the slope of the
line .
16. Reduce the general equation of a line 𝑎𝑥 + 𝑏𝑦 + 𝑐 = 0 in tangent form and hence find the
slope of the line and the y –intercept .
17. Find the value c and m so that the line 𝑦 = 𝑚𝑥 + 𝑐 may pass through the points (-2,3) and
(4,-3)
18. Find the slope of the line whose inclination with x – axis is a) 30° 𝑏) 0° 𝑐) 90° d) 135°
19. Find the slope of the line passing through the points (0, 2)𝑎𝑛𝑑 ( √3 , 3)
20. Calculate the slope of the line passing through the points (3,2) and (4, 1) . Also find the inclination of
the line with x-axis .
21. Find the slope of a line which makes the following angle with x-axis .
a) 150° 𝑏) 240° 𝑐) 315° 𝑑) − 120°
22. Find the slope of the line passing through the points . a) A (-1,7) , B(0,3) b) R(𝑎2 , 𝑏), (𝑏 2 , 𝑎)
23. Slope of a line joining the points (7,3) and ( k, 2) is -4 . Find the value of k .
3
24. Write the equation for the line whose slope is ½ and its y-intercept is 2.
25. Find the equation of the line ,which makes intercepts -3 and 2 on the x-axes and y-axes respectively .
26.Find the equation of a line whose perpendicular distance from the origin is 4units and the angle which
the normal makes with positive direction of x-axis is 15°
27. A line passes through point (1,5) and cuts off intercept 7 units on x-axis . Find the slope of the line .
28. Reduce the general equation of a line 𝑎𝑥 + 𝑏𝑦 + 𝑐 = 0 in tangent form and hence find the
slope of the line and the y –intercept .
29. Find the value c and m so that the line 𝑦 = 𝑚𝑥 + 𝑐 may pass through the points (-2,3) and
(4,-3)
30. The equation of a line is 3x-4y+10 = 0 Find its slope , x-and y – intercepts.
31. Reduce the equation √3 𝑥 + 𝑦 − 8 = in to the normal form . Find the values of 𝑎𝑛𝑑 𝜔 . 32.Find the
equation of a line perpendicular to the line x-2y + 3= 0 and passing through the point (1,-2) .
33. Find the distance of the point (-1,1) from the line 12(x+6) = 5(y-2) .
𝑥
𝑦
34. Find the points on the x-axis whose distances from the line 3 + 4 = 1 are 4 units .
35. Find the distance between parallel lines 15x+8y = 34 , and 15x+8y+31 = 0
36. Find the equation of the line parallel to the line 3x-4y+2 = 0 and passing through the point
(-2,3).
37.. Find the angle between the lines √3 𝑥 + 𝑦 = 1 𝑎𝑛𝑑 𝑥 + √3 𝑦 = 1
Conics Sections
38. Find the centre , radius of the following circles
a) (𝑥 + 5)2 +(𝑦 − 3)2 = 36.
b) 2𝑥 2 +2𝑦 2 − 𝑥 = 0
39. Find the equation of the circle passing through the points (2,3) and (-1,1) and whose centre and
whose centre on the line 𝑥 − 3𝑦 − 11 = 0
40 Find the equation of the circle passing through (0,0) and making intercept a and b on the
co ordinate axis .
41. Find the coordinate of the focus , axis , equation of the directrix and length of the latus rectum .
the following parabolas a) 𝑦 2 =12x b) 𝑥 2 − 9𝑦
42. Find the equation of the parabola whose focus is at (6,0) and directrix is x = -6
43. Find the equation of the parabola whose focus is at (2,0) and directrix is x = - 2
44. Find the coordinate of the foci, vertices the length of major axis , the minor axis , the
eccentricity and length of the latus rectum of the ellipse .
𝑥2
of
𝑦2
a) 36 + 16 = 1 b) 4𝑥 2 + 9𝑦 2 = 36
45. Find the equation of an ellipse whose vertices (±5,0), 𝑓𝑜𝑐𝑖 (±4,0)
46. Find the equation of an ellipse whose ends of major axis (±3,0) , ends of minor axis (0,±2)
47.Find the equation of the that satisfies the following conditions :
a) b = 3 , a = 5 , centre at the origin : foci on the x-axis .
b) major axis on the x-axis and passes through the points (4,3) and (6,2) .
48. Find the equation of the hyperbola where foci are (0,±12) 𝑎𝑛𝑑 the length of the latus rectum is 36.
49. Find the coordinate of the foci, vertices the length of major axis , the minor axis , the eccentricity and
length of the latus rectum of the hyperbola .
𝑥2
𝑦2
a)
− = 1 b) 4𝑥 2 − 9𝑦 2 = 36 c) 49𝑥 2 − 16𝑦 2 = 784
36
16
50. Find the equation of a hyperbola whose vertices (±3,0), 𝑓𝑜𝑐𝑖 (±4,0)
51. Find the equation of the hyperbola whose foci (±4,0) , the latus rectum is of length 12.
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