short Question (integration)

1)  ∫ eˣ (1- cotx + cot²x) dx

A) eˣ cot x + c B) eˣ cosec x + c C) - eˣ cot x+ c D) - eˣ cosec x+ c

2) ∫ dx/√(e²ˣ -1)

A) sin⁻¹ (eˣ)  B) cos⁻¹ (eˣ)  C) tan⁻¹ (eˣ)  D) sec⁻¹ (eˣ)

3) ∫ sin x/sin(x - a)  dx.

A) (x -a) cos a + sin a log|sin(x - a)|

B)  (x -a) cos x + log|sin(x - a)|

C) sin(x - a) + sin x

D) cos (x -a) cos x 

4) ∫ dx/√(2x - x²)

A) sin⁻¹(x +1) B) √(2x - x²) C) - √(x - x²) D) si(x -1)

5)  ∫ xeˣ/(x+1)² dx

A) eˣ/(x+1)²  B) eˣ/(x+1) C) - eˣ/(x+1) D) none

6) ∫ cos⁻¹(1/x) dx

A) x sec⁻¹x + log|x + √(x²- 1)|

B) x sec⁻¹x - sin⁻¹x 

C) x sec⁻¹x - log|x + √(x²- 1)| 

D) x sec⁻¹x - 2 log|x + √(x²- 1)|

7)  ∫x/(x² + 4x +5)

A) 1/2 log|x² + 4x + 5| + 2 tan⁻¹x 

B) 1/2 log|x² + 4x + 5| - tan⁻¹(x +2)

C) 1/2 log|x² + 4x + 5| + tan⁻¹(x+2)

D) 1/2 log|x² + 4x + 5| - 2 tan⁻¹(x +2)

8)  ∫ (1+ x + √(x+x²))/(√x+ √(1+ x))

A) √(x+1)/2 B) 2√(x+1)³/2

C)  √(x+1)   D) 2√(x+1)³

9)  ∫ₑ√x dx

A) ₑ√x B) 2(√x -1)ₑ√x C) 1/2 ₑ√x  D) 2(√x +1)ₑ√x

10)  ∫ x sin x dx= - cos x+ m, then the value of m is

A) sin x+ m B) cos x+ c D) cos x- sin x+ c D) x cos x+ c

11) ∫ ₐ√x dx

A) 2 log a. ₐ√x+ c

B) log a. ₐ√x+ c  

C) ₐ√x/log a + c

D) 2ₐ√x/log a + c

12)  ∫ ₑtan⁻¹x

A) tan⁻¹x+ c B) 1/(1+ x²) + c C) ₑtan⁻¹x + c D) 2x ₑtan⁻¹x/(1+ x²)² + c

13)  ∫ dx/(sin x - cos x + √2)

A) 1/√2 cot(x/2 + π/8)+ c

B) - 1/√2 cot(x/2 + π/8)+ c 

C) 1/√2 tan(x/2 + π/8)+ c

D) - 1/√2 tan(x/2 + π/8)+ c

14) ∫ ₑˡᵒᵍ ᵗᵃⁿ ˣ dx

A) log(tan x)+ c B) ₑ ᵗᵃⁿ ˣ+ c C) log(cos x)+ c D) log(sec x)+ c

15)  ∫(1+ x - x ⁻¹)ₑ(x + x⁻¹) dx

A) (x+1) ₑ(x + x⁻¹) +c

B)  x. ₑ(x + x⁻¹) +c 

C) (x- 1) ₑ(x + x⁻¹) +c

D) - x ₑ(x + x⁻¹) +c

16) If ∫f(x) dx =f(x), then ∫{f(x)}² dx is

A) 1/2 {f(x)}²+ c B) {f(x)}³+ c C) 1/3 {f(x)}³ + c    D) {f(x)}²+ c

17)  ∫ √tan x/(sin x cos x) 

A) 2√secx B) 2/√tan x  C) 2√cosx D) 2√tan x 

18) ∫ dx/(eˣ + e⁻ˣ).

A) log(eˣ + e⁻ˣ)+ c B) tan⁻¹(eˣ) + c

C)  tan⁻¹(e²ˣ) + c D) eˣ - e⁻ˣ + c

19) If ¢(x)= f(x)+ x f'(x), then the value of ∫ ¢(x) is

A) 1/2 f(x)+c B) xf(x)+ c C) 2/2 f(x)+ c D) 2x f(x)+ c

20) ∫ cosec⁴x dx

A) - cot x - 1/3 cot³x + c

B) - cot x + 1/3 cot³x + c

C) tan x + 1/3 tan³x + c

D) - cot x + 1/3 cot³x + c 

21) ∫ dx/{2√x(x+1)}

A) 1/2  tan⁻¹√x+ c

B) 2  tan⁻¹√x+ c

C) tan⁻¹(2√x) + c

D) 1/2  tan⁻¹√x+ c

22) ∫ sin³x cos x dx.

A) 1/4  cos⁴x+ c B) -1/4  sin⁴x+ c

C)  -1/4  cos⁴x+ c  D) 1/4  sin⁴x + c

23) ∫ √(1+ sin(x/4)) dx

A) 8(sin(x/8) + cos(x/8))+ c

B) 8(cos(x/8) - sin(x/8))+ c   

C) 8(sin(x/8) - cos(x/8))+ c 

D) 4(sin(x/8) - cos(x/8))+ c

24) ∫ dx/(x²+ 4x+ 13).

A) 1/3  tan⁻¹{(x+2)/3}+ c

B) log(x²+ 4x+13)+ c

C) 1/6. Log|(x+3)/(x-1)| +c

D) (x+2)/(x²+ 4x+13)²+ c

25) ∫ eˣ{(1+ sin x)/(1+ cos x)} dx

A) eˣ sec²(x/2)+ c

B)  eˣ sec(x/2)+ c

C) eˣ tan(x/2)+ c

D) eˣ tan x+ c

26)  ∫ ₐx/2 /√(ₐ-x - ₐx)

A) 1/(log a) sin⁻¹(aˣ)+ c

B) 1/(log a) tan⁻¹(aˣ)+ c

C) log (aˣ - 1)+ c 

D)  sin⁻¹(aˣ)+ c

27)  ∫ sin x/sin(x - a) = Ax + B log|sin (x - a)| + c, then the value of (A, B) is

A) (cos a, sin a)  B) (- sin a, cos a)

C) (sin a , cos a)  D) (- cos a, sin a)

28) ∫ x³ log x dx

A) 1/8 (x⁴ log x - 4x⁴)+ c

B) 1/16 (4x⁴ log x - x⁴)+ c 

C) 1/16 (4x⁴ log x + x⁴)+ c

D) 1/4 x⁴ log x - x⁴+ c

29)  ∫eˣ/{(eˣ+2)(eˣ+1)}.

A) log {(eˣ+1)/(eˣ+2)}+ c

B) log {(eˣ+2)/(eˣ+1)}+ c

C) {(eˣ+1)/(eˣ+2)}+ c

D) log {{(eˣ+3)/(eˣ+1)}+ c

30) ∫ dx/(cos x- sin x)

A) 1/√2  log|tan(x/2  - 3π/8)|+ c

B) 1/√2  log|cot(x/2)|+ c

C) 1/√2  log|tan(x/2  + 3π/8)|+ c

D) 1/√2  log|tan(x/2  - π/8)|+ c

31) ∫ ³√(x - x³)/x⁴ 

A) -3/8 ³√(1/x² - 1)⁴+ c

B) 3/8 ³√(1/x² - 1)⁴+ c 

C)  1/8 ³√(1 - 1/x²)⁴+ c

D) 1/x² ³√(x - x³)⁴+ c 

32)  ∫ ₑx log a. eˣ dx 

A) (ae)ˣ+ c

B) eˣ/(1+ log a) + c

C) eˣ/log a + c

D) (ae)ˣ/lig(ae) + c

33) ∫ [(log x -1)/{1+ (log x)²}]² dx

A) xeˣ/(1+ x²) + c

B) x/(1+(log x)²)²+ c

C) log x/(1+ (log x)²)+ c

D) x/(x²+1)+ c

34)  ∫ f'(x)/( f(x) log{f(x)}) dx is

A) log[log {f(x)}]+ c

B) f(x)/log{f(x)}+ c

C)  f(x)log{f(x)}+ c

D) 1/log[log{f(x)}]  + c

35) ∫ ₑ - log x dx

A) 1/x + c B)  - 1/x + c C) log|x |+ c D) log x + c

36) ∫ √x ₑ√x dx

A) 2√x -  ₑ√x - 4 √x ₑ√x + c

B) (1- 4√x)  ₑ√x + c

C) (2x + 4√x) ₑ√x + c

D) (2x - 4√x +4) ₑ√x + c

37)  ∫ dx/(sin x cos x).

A) log|sin x| + c

B) log|tan x| + c

C)  log|cos x| + c

D) log|cot x| + c

38) ∫ {eˣ(1+ sin x)}/(1+ cos x).

A) sin(x/2)      B) cos(x/2)

C) tan(x/2)     D) cot(x/2)

39) ∫(eˣ - e⁻ˣ)/{(eˣ + e⁻ˣ) log(eˣ + e⁻ˣ)}

A) 2 log(eˣ + e⁻ˣ)+ c

B) 2 log(eˣ - e⁻ˣ)+ c

C)  2 log[log(eˣ + e⁻ˣ)]+ c

D) log[log(eˣ + e⁻ˣ)]+ c 

40) ∫ dx/{x(x⁷+1)}.

A) log|x⁷/(x⁷+1)|+ c

B) 1/7 log|x⁷/(x⁷+1)|+ c

C) log|(x⁷+1)/x⁷|+ c

D) 1/7 log|(x⁷+1)/x⁷|+ c

41) ∫ cos[2 cot⁻¹√{(1- x)/(1+ x)}] dx

A) x/2 + c 

B) 1/2 sin[2cot⁻¹√{(1- x)/(1+ x)}] + c

C) x²/2+ c D) - x²/2 + c

42)  ∫ (cos x+ 1)/(cot x - tan x).

A) -1/2 B) -1/4 C) -1/8 D) 1/8

43) ∫ dx/{sin(x- a) sin(x - b)}.

A) 1/sin(a-b) log|sin(x- a)/sin(x - b)|+ c

B)  1/sin(b-a) log|sin(x- a)/sin(x - b)|+ c

C) 1/sin(a-b) log|sin(x- b)/sin(x - a)|+ c

D)  1/sin(b-a) log|sin(x- b)/sin(x - a)|+ c

44) ∫ √(eˣ -1) dx

A) 2[ √(eˣ -1) + tan⁻¹√(eˣ -1)]+ c

B) √(eˣ -1) - tan⁻¹√(eˣ -1)+ c

C) √(eˣ -1) + tan⁻¹√(eˣ -1)+ c

D) 2[ √(eˣ -1) - tan⁻¹√(eˣ -1)+ c

45) If m= ∫ sin⁻¹x dx and n=sin⁻¹√( 1-x²) dx then

A) m= n B) n= πm/2 C) m+ n= πx/2 D) m+ n = π/2

46) ∫ cos⁻³⁾⁷x cos⁻¹¹⁾⁷x dx

A) -7/4 tan⁻⁴⁾⁷x + c

B) 4/7 tan⁴⁾⁷x+ c

C) log|sin⁴⁾⁷x| + c

D) log|cos³⁾⁷x|+ c

47) ∫ dx/{x √(x⁴ -1)} = 1/2 f(x) + c, then the value of f(x) is

A) tan⁻¹(x²) B) sec⁻¹(x²) C) tan⁻¹(x²/2) D) sec⁻¹(x²/2)

48) ∫ dx/{xⁿ(1+ xⁿ)¹⁾ⁿ 

A) 1/(n -1) . 1/xⁿ⁻¹ (1+ xⁿ)¹⁻¹⁾ⁿ+ c

B) -1/(n -1) . 1/xⁿ⁻¹ (1+ xⁿ)¹⁻¹⁾ⁿ+ c

C) 2/(n -1) . 1/xⁿ (1+ xⁿ)¹⁻¹⁾ⁿ+ c

D) 2/(n -1) . 1/xⁿ⁻¹ (1+ xⁿ)¹⁻¹⁾ⁿ+ c

49)  ∫ sinx/(sinx+ cosx)= x/2 + k log|sinx + cosx| + c, then the value of k is

A) 2 B) -2 C) 1/2 D) -1/2

50) ∫ dx/(cosx + cos k) = f(k) log|{cos(x-k)/2}/cos{(x+ k)/2}|+ c then the value of f(k) is

A) sin k B) cos k C) cosec k D) sec k

51)  ∫√{(sinx - sin³x)/(1- sin³x)} dx = 2/3 f(x) + c, then the value of f(x) is

A) √sin³x B) √cos³x C) sin⁻¹(√cos³x) D) sin⁻¹(√sin³x)

52)  ∫{x⁴⁹ tan⁻¹(x⁵⁰)}/(1+ x¹⁰⁰).

A) 1/50  B) -1/50 C) 1/100 D) -1/100

53) ∫ dx/(xⁿ⁺¹ + x).

A) 1/n log|xⁿ +1|+ c

B) 1/n log|xⁿ/(xⁿ +1)|+ c

C) 1/n log|1/(xⁿ +1)|+ c

D) 1/n log|(xⁿ +1)/xⁿ|+ c

54) ∫ dx/(x+ x⁵)= f(x)+ c, then the value of x⁴dx/(x+ x⁵) is

A) log|x| - f(x)+ k

B) f(x) + log|x| + k

C)  f(x) - log|x|+ k D) none

55) ∫ (sin x+ cos x)/√sin 2x

A) log|cosx - sinx + √sin 2x|+ c

B) log|sin x - cosx + √sin 2x|+ c

C) sin⁻¹(sinx - cosx) + c 

D) sin⁻¹√sun 2x+ c

56)  ∫(1+ tan x)/(e⁻ˣ cos x)

A) (e⁻ˣ tan x) + c

B) (e⁻ˣ sec x) + c

C) (eˣ sec x) + c

D) (eˣ tan x) + c

              ₓ

57) ∫ ₂2²  ₂2ˣ. 2ˣ ex

58) ∫ tan⁻¹x dx= x tan⁻¹x+ f(x) + c then the value of f(x) is

A) log(1+ x²) B) 1/2 log(1+ x²) C) - log(1+ x²)  D) -1/2 log(1+ x²)  

59) ∫ √x/√(a³ - x³) = 2/3 f(x)+ c then the value of  f(x) is

A) sin⁻¹{√x³} B) cos⁻¹{√(x/a)³  C) sin⁻¹{³√(x/a)²  D) cos⁻¹{³√x/a)²

60)  ∫ eˣ{(1- x)/(1+ x²)}² dx= k. eˣ/(1+ x²) + c, then the value of k is

A) 1/2  B) -1 C) 1 D) 2

61) ∫(x²-1)/(x⁴+1) = k log|(x²+1- √2 x)/(x²+ 1+ √2 x)|+ c, then the value of k is

A) 1/√2 B) 1/2 C) √2 D) 1/2√2

62)  ∫ {log(1- x)}/x²= (1- 1/x) log(1- x)+ f(x)+ c, then the value of f(x) is

A) 2/x  B) -2/x  C) log x  D) - log x

Quick Revision - X(22/23)

SECTION FORMULA

1) Determine the coordinates of the middle points of the sides of the triangle whose vertices have coordinates are (3,2), (-1, -2) and (-5,-4).         (1,0),(-3,-3),(-1,-1)

2) find the coordinates of the centroid of the triangles whose vertices are 

A) (-4,5),(8,2),(2,-1).                 (2,2)

B) (3,4),(- 1,7),(10,10).       (4,7)

3) find the co-ordinates of the point which divides the line segment PQ joining the points P(6,4) and Q(7,-5) in the ratio 3:2.         33/5, -7/5

4) Find the co-ordinate of the point which divide the line segment joining the points (9,5) and (-7,-3) in the ratio 5:3.            (-1,0)

5) Write down the coordinates of the point which divides the augment joining the points (1,-2)  and (6,8) in the ratio 2:3.      (3,2)

6) Find the coordinates of the point of trisection of the line segment joining the points P(-2,3) and Q(3,-1) that is nearer to P.   (-/3, 5/3)

7) A line segment directed from (-3,2) to (1,-4) is trebled. Find the coordinates of the terminal point.    (9,-16)

8) The line segment joining the point (2,-2) and (4,6) is extended each way a distance equal to half its own length; find the coordinates of its terminal points.  (5,10),(1,-6)

9) Show that the points (0,2),(4,1) and (16,-2) lie in a straight line.

10) Show that the points (-4,0),(6,3) and (36,12) lie in a straight line.

11) if (19,8),(15,-6),(-11,- 12) are the three vertices of a parallelogram and The Fourth vertex lies in the second quadrant, find the coordinates of the fourth vertex.   (-7,2)

12) Show that the points A(8,12), B(-2,7) and C(2,9) lie on a straight line. Also, find the ratio in which the line segment AB is divided at C.    (3:2)

13) If the point (9,2) divides the line segment joining the points P(6,8) and (x,y) in the ratio 3:7, find the coordinates of Q.             (16,-12)

14) If the point (6,3) divides the segment of the line from (4,5) to (x, y) in the ratio 2:5, find the co-ordinates (x, y) of Q. What are the coordinates of the mid point of PQ.     (11,-2),(15/2,3/2)

15) Find the ratio in which the points (-2,2) divides the line segment joining the points (-4,6) and (1/2, -3).                       4:5 

16) Find the ratio in which point (-5, -20) divides the line joining the points (4,7) and (1,-2).        3:2

17) find the ratio in which the point (-1,0) divides the line segment joining the points (- 7,-3) and (9,5).     3:5

18) Prove that the points (2,1),(0,0),(-1,2) and (1,3) form a square.

Ratio And Proportion

1) Find the duplicate ratio of 3:4 and the sub duplicate ratio of 169: 256.                      9:16, 13:16

2) Find the ratio compounded of the ratios 6a: 5b, 2ab: 3c² and c: a.    4a: 5c

3) Find the ratio compound of the Triplicate ratio of 2a: 3b and sub duplicate ratio of 9:64.          a³: 9b³

4) If the ratio (x+7):2(x+14) be equal to the duplicate ratio of 5:8, find the value of x.                      18

5) Show that the ratio x : y is the duplicate of the ratio (x+z):(y+z) if z² - xy= 0.

6) If x: y = 3: 4, find the value of (7x - 4y):(3x+y).                                    5:13

7) If (x - y):(x+ y)= 7:11, find the value of x: y.                                  9:2

8) If (a+ b):(a - b)= 5:2, find the value of b: a .                                3:7

9) If (2x+ 5y):(3x +5y)= 9: 10. Find x : y.                                                   5:7

10) For what value of x will the ratio (23 +x): (19+ x) be equal to 2.    -15

11) If the ratio of (5+x) and (37+x) be equal to the ratio 1, and 3. Find x.                                     11

11) What number must be added to each term of the ratio 5: 37 to make it equal to the ratio 1:3.                 11

12) Two numbers are in the ratio 3: 4 and if 7 is added to each term of the ratio, then the new ratio is 4: 5. Find the numbers.                    21, 28

13) Two numbers are in the ratio 7: 9 and if 10 be subtracted from each term of the ratio, then the new ratio is 8:11. Find the numbers.      42, 54

14) What number must be added to each term of the ratio a : b to make it equal to c: d.        (ad - bc)/(c - d)

15) If  x/a = y/b = z/c, 

Prove:

A) (x² - yz)/(a² - bc) = (y² - zx)/(b² - ac) = (z² - xy)/(c² - ab).

B) (x² + a²)/(x+ a) + (y² +b²)/(y+ b) + (z² + c²)/(z+ c( = {(x+ y+ z)² +(a+ b + c)²}/{(x+ y+ z) + (a+ b+ c)}.

C) (3x² + 5y² + 4z²)/(3a²+ 5b²+ 4c²) = (x²+y² +z²)/(a²+ b²+ c²).

16) If the work done by (x -1) men in (x - 1) days is to the work done by (x+2) men in (x -1) days be in the ratio 9:10, find x.                           8

17)  The monthly income of two person are the ratio 4:5 and their monthly expenditures are in the ratio 7:9. If each saves ₹50 per month, find their monthly incomes.     ₹400, ₹500

18) The monthly salary of two persons are in the ratio 3:5. If each receives an increase of ₹20 in the monthly salary, the ratio is altered to 13 :21, find their salaries.   240, 400

19) find in what ratio will the total wages of the workers of a factory be increased or decreased if there be a reduction in the number of workers in the ratio 15 :11 and an increament in the wages in the ratio 22: 25.                                        6:5

20)  An employer raises hourly rate of wages in the ratio 5:6, but reduces the working hours per week in the ratio 7:8. What will be the increase or decrease in the weekly wages bill ? if it was Rs 2400 previously, what it will be now ?     21:20 increase, ₹2520

21) The railway fare in a certain year increases in the ratio 22: 25 but the number of passengers decreases in the ratio 13 :11. Find in what ratio will the total income from Passengers' fare increases or decrease.      26:25, decrease

22) The Railway fare in a certain year increases in the ratio 14:25 but the number of passengers decreases in the ratio 15:7. Find in what ratio will the total income increase or decrease.     6:5, decrease

23) The ratio of prices of two articles was 16:23. Two years later when the price of the first had risen by 10% and that of second by ₹477, the ratio of their prices becomes 11:20. Find the original prices of the two articles.                     848, 1219

24) The profit of a trader increases every year in the ratio 4:5 for the first 5 years but from the sixth year in the ratio 4:3 in the 3 subsequent years. If his profit in the first year was ₹2400, find out the change in his profit in the eighth year.    71.92

25) The prime cost of an article was three times the value of the raw materials used. The cost of  raw materials increases in the ratio 3:7 and productive wages increases in the ratio 4:9. Find the present Prime cost of an article which could formerly be made for ₹18.     41

26) Factory on cost of the articles produced by a factory is taken as the sum of the cost of material, cost of labour and factory overhead charges. These three costs are in the ratio 15: 6 :5. Later the cost of material is decreased in the ratio 10:9 and the cost of labour is increased in the ratio 2:3, find the subsequent altered ratio in the factory overhead charges so as to keep the factory on-Cost unaltered.      10:7 , decreased.

27) A mixture of 30 litres contains milk and water in the ratio 7 to 3; how much water must be added to the mixture so that the ratio of milk to water may be 3: 7.       40 litres

28) A mixture of 30 litres contains milk and water in the ratio of 7:3. How much mixture must be replaced by water to make the ratio 3:7  ?                                     17:14

29) water and milk are mixed in the ratio 2:7 in one vessel and in the ratio 2:9 in the another vessel. Determine the volume of the mixture of the second vessel that should be mixed with 225c.c. of the mixture in the first vessel to form a new mixture containing milk and water in the ratio 4:1.         275cc

30) Two mixture contains milk and honey with ratio of 7:2 and 5 : 1. In what ratio these two mixture should we mixed so that the resulting mixture may contain milk honey in  the ratio 9:2 ?                             3:8

31) Find the third proportional

A) 16 and 20.                              25

B) a²b and ab.                                 b

32) Find the fourth proportional to

A) 2, ,3, 4.                                          6

B) 2, 5 , 22.                                      55

C)  4a², 7ab, 8ab².                       14b³

33) Find the mean proportional between 

A) 3 and 12.                                   6

B) 30a²b a8120a²b³.             60a²b²

34) If x be the mean proportional between find (x-3) and (x -6), find x.   2

35) If m/n = p/q = √(7/15) find the value of

A) (n+ m)/(n - m).         (11+√105)/4

B) (q²+ p²)/(q² - p²).                  11/4

C) (m²+ n²)/(n²+ q²).                  7/15

36) what number must be added to each of the four numbers 9,11, 15, 18 so that the sums will be proportional.                                  3

37)  What number must be subtracted from each of the four numbers 15,18, 21, 27 so that the difference will be proportional.   9

38) if 3, x, 1083 are in continued proportion, find x.                      ±57

39) If 2,x , 18 are in continued proportion, find x.        ±6

40) Divide 345 into 4 parts proportional to 3,5, 6, 9.    45, 75, 90, 135

41) if x :y :z = 2:7 :11 and 4x- 5y + 3z= 60,  find x, y, z.       20, 70, 110

42) If a: b= c: d prove:

A) a²+ c² : b² + d²= ac: bd.

B) ac(a+ c): bd(b +d)= (a+c)³: (b+d)³

C) (a+ c)³ : (b + d)³= a(a - c)²: b(b - d)².

D) pa²+ qb² : pa²² - qb²= pc²+ qd²: pc² - qd².

43) If a, b, c are in continued proportion, prove that:

A) (a+ b)²: (b+ c)² =(a²+ b²):(b²+ c²)

B) (a²+ ab+b²) : (b² + bc+ c²)= a: c.

44) If a, b, c, d are in continued proportion, prove that:

A) a : (b+ d)= c³: (c²a +d³).

B) (a+ b) : (c+ d)= (a²+ b²+ c²): (b² + c² +d²).

C) (2a+ 3d) : (3a - 4d)= (2a³ + 3b³): (3a³ - 4b³).

D) (a²+ b²+ c²)(b²+ c²+ d²) = (ab+ bc+ cd)².

45) If x: y: z = a: b : c, prove that

A) (x²+ y²+ z²)(a²+ b²+ c²) = ax+ by + cz)².

B) (x²+ 5y²+ 4z²) : (3a² + 5b²+ 4c²) = (x²+ y² +z²): (a²+b² + c²).

46) If x/{(b+c)(b+ c - 2a)} = y/{(c - a)(c+ a - 2b)} = z/{(a- b)(a+ b - 2c)}, Prove that x+ y+ z = 0.

47) If x/(q+ r - p) = y/(r+ p - q) = z/(p+ q - r), Prove that (q-r)x+ (r - p)y+ (p -q)z = 0.

48) If (a+b+c+ d)(a- b - c + d) = (a- b+ c - d)(a+ b - c - d), = Prove that a, b, c ,d are proportional.

49) If y be the mean proportional between x and z, prove that x²y²z²{1/x³ +1/y³+ 1/z³) x³+ y³+ z³.

TRIGONOMETRIC RATIOS 

1) If x = 30°, verify the following:

A) cos 2x = 2 cos²x - 1 = 1 - 2 sin²x = cos²x - sin²x.

B) sin 2x = 2 sinx cosx.

C) sin3x = 3 sinx - 4 sin³x.

2) Find the value of the following:

A) sec30 tan 60+ sin45 cosec45+ cos30 cot60.                            7/2

B) cot²60+ 3 cos² 60- 3/2 sec²45 - 8 sun²60.                                  - 311/12

C) 4/3 cot²30+ 3 sin²60 - 2 cosec²60 - 3/4 tan²30              10/3

3) If x= 60 and y= 30 verify

A) sin(x+y)= sinx cosy + cosx siny

B) tan(x+y)= (tanx + tany)/(1- tanx tany)

C) sin(x+y). Sin(x - y)= sin²x - sin²y = cos²y - cos²x.

4) Show that

A) √{(1+ cos30)/(1- cos30){= sec 60 + tan 60.

B) √{(1+ cos30)/(1- cos30)} - cosec 30 = cosec 30 - √{(1- cos30)/(1+ cos30).

C) (√3+1)(3- cot 30)= tan³60 - 2 sin 60.

D) sec²45. Tan²30 - tan²45. sec²30 = tan²30 - tan²45.

E) (tan 60+ tan 30)(1- cot60. Cot30)+ (cot60+ cot30)(1- tan60 tan30)= 0.

5) If x and y are two positive acute angles such that cos(x - y)= 1 and cot(x + y)= 0, find x and y.           45

6) The angle C of a triangle ABC is obtuse and sin(B+ 2A)= tan(B - A) = 1. Find the angles A, B, C.   15,60,105

7) Solve:

A) sinx = cosx.                                45

B) 2 cosx+ 5 tanx =4 secx.            30

C) tanx +√3 cotx =√3+ 1.        45, 60

D) sinx+ cosx = √2.                    45  

E) 7 sin²x + 3cos²x = 4.                 30

F) 3 sec⁴x - 10 sec²x + 8 = 0.         30, 45

MEASUREMENT OF ANGLES

1) Express the following angles in both centesimal and circular measure:

A) 30°

B) 15°15'15"

C) 40°29"

2) The angles A of a parallelogram ABCD is 45°. Find the other angles in sexagesimial

3) The angles of a triangle in the ratio 1:4:5. Find them in centesimal.  20, 80, 100

4) Find in sexagesimial measure of each interior angle of a regular pentagon.                            108°

5) The sum of two angles is equal to 112°. If the number of grades in one is twice the number of degrees in the other, find the angles in grades.                80, 44.44

Quadratic Equation

1) x² - x - 12= 0. 4, -3

2) 4x² - 16x +15= 0. 3/2, 5/2

3) 4x² - 4x - 3= 0. -1/2,3/2

4) x² - 0.5x + 0.06= 0. 0.2, 0.3

5) 2x² - 3x +1= 0. 1, 1/2

6) 2x² - 3x - 1= 0. 1/4 (3±√17)

7) x² - 2√3x - 3x- 13= 0. √3±4

8) x² +3x - (a-1)(a+2)= 0. 1-a, a+2

9) x² - (p+ 1/p) x + 1= 0. a, 1/a      

10) x² - {a/(a+ b) + (a+b)/a} x + 1= 0. -a/(a+ b), -(a+b)/a

11) (x+2)/(x-2) + (x -2)/(x+2)= 11/2. ±2 √(15/7)

12) 1/(x+1) + 1/(x +3) = 1/(x +4) + 1/(x+6). (-7± √5)/2

13) 2x² - 3x -5= 0. 5/2, -1

14) √(x+2) + √(x -3) = 5. 7

15) 2√(x+5) - √(2x +8) = 2 ±4

16) √(2x+1) + √(3x +4) = 7 4, 480

17) 4x²+6x + √(2x²+ 3x +4) = 13. 1, -5/2

18) √(2x²- x+10) - √(2x²- x +3) =1. 2, -3/2

19) √{(x+1)/(x -1)} + √{(x -1)/(x+1)} = 5/√6. ±5

20) √{(1-x)/x} + √{x/(1- x)} = 13/6. 4/13, 9/13

21) √(x+5) + √(x +12)= √(2x +41) -21, 4

22) √(x² 8x+15) + √(x²+ 2x -15)= √(4x² - 18x +18) 3, 17/3

23) {√(x²+4) + √(x +1)}/{√(x² +4)- √(x +1)}= 3. 0, 4

24) 1/(x+ a+ b) = 1/x + 1/a + 1/b. - a, - b

                 

Height and Distance

Type -1

1) What is the ratio between the height of a vertical pole and  length of its shadow, when the elevation of the sun is 

A) 30°.    1:√3

B) 45°.      1:1

C) 60°.       √3:1

2) The length of the shadow of x metres high vertical tower is x√3. What is the elevation of the sun ?   30°

3)  What is the angle of the elevation of the sun, when the length of shadow of a vertical Pole is equal to its height ?   45°

4) The height of a tree is √3 times, the length of its Shadow. Find the angle of the elevation the sun.   60

5) The angle of the elevation of the top of a tower from a point on the ground and at a distance of 160 m from its foot, is found to be 60°. Find the height of the tower.   277.12m

6) A kite is attached to a string. Find the length of the string, when the height of the kite 60°m and the string makes an angle of 30° with the ground.         120m

7) A boy, 1.6 m tall, is 20m away from a tower and observes the angle of the elevation of the top of the tower to be 

A) 45°

B) 60°

 Find the height of the tower in each case.           21.6m, 36.24m

8) A vertical flagstaff stands on a horizontal plane. From a point 80 m from its foot, the angle of elevation of its top  is found to be 30°. Find the height of the flagstaff.        46.19m

9) The upper part of a tree, broken over by the wind, makes on angle of 45° with ground; and the distance from the root to the point where the top touches the ground, is 15m. What was the height of the tree before it was broken.             36.21m

10) The angle of Bangalore elevation of the top of an unfinished tower at a point distance 80m from its base is 30°. How much higher must the Tower be raised so that its angle of elevation at the same point may be 60°.       92.37m

11) The angle of elevation of the top of a tower, which is incomplete 45° from a point on the level ground and at a distance of 150 m from the base of the tower. How much higher should it be raised so that the elevation of the top of the tower may become 60° at the same point?  109.8m

12) At a  particular time, when the sun's altitude is 30°, the length of the shadow of a vertical Tower is 45°m. Calculate

A)  the height of the tower.

B) the length of the shadow of the same Tower, when the sun's altitude is 

A) 45°.        25.98m

A)  60°.        25.98m 

Type -2

1) The length of the shadow of a vertical tower on level ground increases by 10m, when the altitude of the sun changes from 45° to 30°. Calculate the height of the tower.         13.66m

2) An observer on the top of a cliff,  200 m above the sea level, observes the angle of a depression of the two ships to be 45° and 30° respectively. Find the distance between the ships, if the ships are

A)  on the same side of the cliff.

B) on the opposite sides of the cliff.      146.4m, 546.4m

3) A man on the top of a vertical observation Tower of a car moving at uniform speed coming directly towards it, if it takes 12 minutes for the angle of depression to change from 30° to 45°. How soon after this will be the car reach the observation Tower ?    16.39 minutes

4) Find the height of a tree when it is found that walking away from it 20 m, in a horizontal line through its base, the elevation of its top changes from 60° to 30°.     17.32m

5) A person standing on the bank of a river observes the angle of elevation of the top of a tree, on a opposite bank, to be 60°. When he retires 30m from the bank, he finds the angle of the elevation to be 30°. Find the height of the tree and the breadth of the river.     25.98m, 15m

6) Find the height of a building, when it is found that on walking towards it 40 m in a horizontal line through its base the angular elevation of its top changes from 30° to 45°.            54.64m

7) The shadow of a tower standing on a level ground is found to be 40m longer, when Sun's altitude decreases from 45°  to 30°. Find the height of the tower.       54.64m

8) Two pillars of equal heights stands on either side of a roadway, which is 150m wide. At a point in the roadway between the pillars the elevations of the tops of the pillars are 60° and 30°, find the height of the pillars and the position of the point.    64.95m, 37.5m

9) The angle of the elevation of the tower of the top of a tower is observed to be 60°. At a point 30m vertically above the first point of the observation, the elevation is found to be 45°. Find

A) the height of the tower.   70.98m

B) its horizontal distance from the points from the points of observation.         40.98m

10) From the top of a cliff, 60m high the angles of depression of the top and bottom of a tower are observed to be 30° and 60°. Find the height of the tower.      40m

11) The angle of elevation of the top P of a vertical Tower PQ from a point X is 60°, at a point Y, 40 m vertically above X, the angle of elevation is 45°.

A) Find the height of the tower PQ.

B) Find the distance XQ.  95, 55

(Give your answers to the nearest metre)

12) A man on a cliff observes a boat at an angle of depression 30°  which is sailing towards the shore to the pointing immediately beneath him.  3 minutes later the angle of depression of the boat is found to be 60°  assuming that the boat sail at a uniform speed, determine:

A) how much more time it will take to reach the shore.   1.5min

B)  the speed of the boat in metres per second. If the height of the cliff is 500m.          3.21 m/s

13) A man in a Boat rowing away from a Lighthouse 150m high, takes 2 minutes to change the angle of elevation of the top of the Lighthouse from 60°  to 45°.  Find the speed of the boat.   0.53 m/sec

14) A person standing on the bank of a river observes that the angle of elevation of the top of a tree standing on the opposite bank is 60°. When he moves 40 m away from the bank, he finds the angle evelvation to be 30°. Find

A) the height of the tree, correct to 2 decimal places.     34.64m

B) the width of the River.     20m

15) The horizontal distance between the two Towers is 75m and the angular depression of the top of the first tower as seen from the top of the second, which is 160m high, is 45°.  Find the height of the first tower.         85m

16) The length of the shadow of a tower standing on a level plane is found to be 2y metre longer when the sun's altitude is 30°  than when it was 45°. Prove that the height of the tower is y(√3+1) m.

17) An aeroplane flying horizontally 1 km above the ground is observed at an elevation at 60°. After 10 seconds its elevation is observed to the 30°, find the uniform speed of the Aeroplane in km per hour.     415.67 km/hr

Trigonometric Identity

1) cos x/(secx - tanx)= 1 + sinx

2) (1+ cos x - sin²x)/{sinx(1+ cos x)})= cotx

3) (cosecx - sinx)(secx - cosx)(tanx + cotx)= 1 

4) sinx/(1+ cos x) + (1+ cosx)/sin x = 2 cosec x

5) (1+cos x)/(1 - cosx)= (cosec x + cot x)²

6) (1+sin x)/(1 - sinx)= (sec x + tan x)²

7) √{(1- sin x)/(1 + sinx)= sec x - tan x 

8) (1+cos x + sinx)/(1 - cosx + sinx)= sin x/(1- cos x) = (1 + cos x)/sinx

9) (secx+tan x)/(cosec c + cotx) - (sec x - tanx )/(cosecx - cotx) = 2(secx - cosecx)

10) (secx- cos x)(cosecx - sinx)= tanx/ (1+ tan²x)

11) (1+secx + tanx)(1 - secx + tanx)= 2 tan x

12) (1+ cosecx + cotx)(1 - cosecx + cotx)= 2 cot x

13) sec⁴x + tan⁴x = 1+ 2 sec²x tan²x.

14) sinx/(1- cot x) + cos x/(1 - tanx) = sinx + cos x

15) (sin x - secx)² + (cosx - cosec x)² = (1- secx cosecx)²

16) (1+ tanx)/(1- tan x) - (1 - tanx)/(1 + tanx)= 4 sinx cosx/(1- 2 sin²x)

17) tanx/(1+tan²x)² + cotx/(1 + cot²x)²= sin x cosx

18) 1/(cosecx + cotx) - 1/sinx = 1/sinx - 1/(cosecx - cotx).

19) √{(1+ cos)/(1+ cosx)} = cosecx - cot x)

20) (1 + secx + tanx)= 2/(1- cosecx + cot x)

21) (secx - tanx)/(secx + tanx)= 1 - 2 secx tan x + 2 tan²x.

22) (1 + tan²x)/(1 + cot²x)= {(1- tan x)/(1- cotx)}²

23) {(1+sin x)/(1- sinx)} - sec x = secx- √{(1 - sinx)/(1 + sinx).

24) sin⁶x + cos⁶x = 1 - 3sin²x cos²x.

25) cosec⁶x - cot⁶x = 1 + 3 cosec²x cot²x.

26) (cosx + cosy)/(sinx - sin y) = (sinx + siny)/(cosy - cos x).

27) sec²x tan²y - tan²x sec²y = 3tan²y - tan²x.

28) If sinx = (sinx+ sin y)/(1+ sin x sin y), Prove cos x = = cos x cos y/( 1 + sinx sin y).

Short Question for X

MIXED QUESTIONS 

solve 

1)A) 3x² - x -7=0 and give your answer Correct to 2 decimal places.

B) solve: (4x²-1) - 3(2x +1) + x(2x+1)= 0

C) x² - 1/x² = 29/10(x - 1/x)

D) √(x+15) = x +3, x belongs N

E) √{x(x-3)}=√10

F) √(6x-5) - √(3x -2) = 2

2) If -3 is the root of x² - kx -27 =0, find k.

3) If 4, x,36,y are in Continued Proportion, find x and y.

4) The Mean of 12,18,x,13,19,22 is 16, find x.

5) Solve 3x²+8x+1=0. Give your answer correct to two decimal places.

6) From the top of a building AB, 60 metres high, the angles of depression of the top and bottom of a vertical lamp post CD are observed to be 30° and 60°. Find

a) the horizontal distance between AB and CD.

b) the height of the lamp post CD.

7) SOLVE: 24x² - 334x + 135=0

8) An open cylindrical vessel is made of steel. The internal diameter is 14cm, the internal depth is 20.6cm and the metal is everywhere 4mm thick. Calculate

a) the internal volume

b) the volume of the metal correct to the nearest cm³.

9) John sends his servant to the market to buy oranges worth Rs15. The servant having eaten three oranges on the way. John pays 25 paise per orange more than the market price. Taking x to be the number of oranges which John receives, form a quadratic equation in x. Hence, find the value of x.

10) 

X: 0-5  5-10  10-15  15-20  20-25  25-30

 F:  3     7          15       24      16        8

Find Mean correct to 2 decimal.

11) A man is standing on a level ground, observes that a pole 30m away subtends an angle of 50° at his eye, which is 2.0m above the ground level. Calculate the height of the pole. (Give your answer correct to a length of a metre).

12) solve: 

a) (4x²-1) -3(2x +1) +x(2x+1)= 0

b) x² - 1/x² = 29/10(x - 1/x)

c) √(x+15) = x +3, x belongs N

d) √{x(x-3)}=√10

e) √(6x-5) - √(3x -2) = 2

        

           MENSURATION

1) If the area of the base of a right circular cone= 792/7cm² and height is 8cm,then the volume of the cone ?

2) A hemisphere and a cone have equal base. If their heights are also equal then the ratio of their curved surface area will be.

3) Total surface area of a cube is 5 square units; if diagonal be d units, then find the relation of S and d

4) A cone of height 15cm and base diameter 30cm is curved out of a wooden sphere of radius 15cm. The% of wasted wood is.

5) The length, breadth and height of a cuboidal hole are 40m, 12m, and 16m. The number of planks having a height of 5m, breadth of 4m and thickness of 2m, that can be kept in that hole, is-

6) The surface area of a cube is 256m², the volume of the cube is-

7) If the length of radii of two solid right circular cylinder are in the ratio 2:3 and their heights are in the ratio 5:3, then the ratio of their lateral surface area is 

8) In a right circular cylinder, if the length of radius is halved and height is doubled, volume of the cylinder is

9) If the length of radii of a right circular cylinder is doubled and height is halved, then the lateral surface area will be

10) The ratio of the volume of two cubes is 1:27, the ratio of total surface areas of two cubes is-

11) The volume of a solid sphere having the radius of 2r units length is

12) If the numerical value of curved surface area of a solid sphere is three times of its volume, then the length of its radius will be.

13) Keeping the radius of a right circular cone is same, if the height of it is increased twice, the volume of it will be increased by

14) A solid sphere of r units is melted and from it a solid right circular cone is made. The base radius of cone is

15) By melting a solid right circular cone, a solid right circular cylinder of same radius is made whose height is 5cm. The height of the cone will be

16) If the diameter of the base of a cylinder is 5.6 and height is 1.5, then find the volume of the cylinder.

17) Find the maximum length of a pencil that can be kept in a rectangular box of dimensions 8cmx6cmx2cm.

18) If the numerical value of the surface area of a cube is equal to the numerical value of the volume of that cube then find the total surface area of the cube.

19) Two right circular cylinder of equal volume have their heights in the ratio 1:2. Find the ratio of their radii

20) The total surface area of a cube and a sphere are equal. Find the ratio between their volumes.

21) The length of a rectangular paper is l units and breadth is b units. The rectangular paper is rolled and a cylinder is formed whose perimeter is equal to the length of the paper. Find the lateral surface area of the cylinder.

22) If the numerical value of volume and lateral surface area of a right circular cylinder are equal then find the length of diameter.

23) If the length of each edge of a cube is thrice of that 1st cube then the volume of this cube is 9 times more than that of the 1st cube.  T/F

24) In rainy season, the height of rainfall in 5cm land is 5 hectre, the volume of rain water is 2000m³. T/F

25) If length of radius of a right circular cone is decreased by half and it's height is increased by thrice of it.Then volume be same.       T/F

26) If the height, slant height and diameter of a cone are h,i, d respectively, then the value of (l² - h²)/d² is ¼          T/F

27) If the lateral surface area of right circular cylindrical pillar is 264m² and volume is 924m³ find the radius.

28) If the lateral surface area of right circular cylinder is c square units, radius is r unit and volume is v cubic units. Find the value of cr/v.

29) If curved surface area of a sphere is S and volume is V, find the value of S³/V³

30) The curved surface area of a circular cone is√5 times of its base area. Find the ratio of the height and the length of radii of the cone.

31) If the volume of a right circular cone is V cubic unit, base area is A sq.unit and height is H unit, find the value of AH/V.

32) The numerical values of the volume and the lateral surface area of a right circular cone are equal. If the height and the radius of the cone are H unit and r unit respectively. Then find 1/h² + 1/r².

                   STATISTICS                

1) The median of 

a)11, 29, 17, 21, 13, 31, 39,19 

b) 1,5,9,3,8,7

2) The mode of 1,2,3,4,5,6,7 is

3) Median of a frequency distribution can be obtained from ----

4) If the median after arranging in ascending order the data 8, 9, 12, 17, x+2, x+4, 30,31,34,39, is 15, then value of x is.

5) If a:2= b: 5= c: 8 , then 50% of a = 20% of  = ---- % of c.

6) If mean proportional of (x -2) and (x-3) is x, then the value of x is…..

      RATIO AND PROPORTION

1)If a:b = m; b:c = m and d:c = m then a:b:c:d = ?

2) If ax² + bx + c then find the sum and product of the roots.

3) The fourth proportional of 3,4,6

4) a is a positive number and if a: 27/64:¾:a, then the value of a is-

5) a:b=m: n and b:c = p:q,then a:c is

 

2) If the quotient when 3x³ - 2x²+7x-5 is divided by x+3 is given as 3x² -11x +a. Find a. 40           

3) If the quotient on dividing x³ - 3x² + 4x +5 by x-3 is x² - a, find a                                                     

4) If a² - 3a +4 is a factor of a³+a²- la +m, find l and m. -16, 16

5) Given that x- 1 is a factor of x²+ax +1 and show that x- a is a factor of x³ + 3x² +3x +2

6) If 2x-1, 2x-3 are the factor of 8x³+ ax² +46x+b, find a, b. Then factorise Completely. -36, -15

7) What should be substracted from x³ - 3x² - 10x+25 so that x-2 may be a factor.

2) Find the value of k given that 3x³ + 4x² -6x +k  is divisible by x+1

3)Use graph for this question.  Take 1cm= 1 unit on both axes.

i) Plot point P(2,3) and Q(3, 1)

ii) Reflect P in x-axis to P'. Reflect P' in y-axis axis to P". Write coordinates of P'' and P".

iii) Reflect Q in y-axis to Q' and reflect Q' in the origin to Q". write coordinates of Q' and Q". 

iv) write the geometrical name of PQQ"P'.

4) Given A= 2    0  and B= p    q

                      0    5               o     r

i) Compute A+B

ii) AB

iii) Given A+B= AB,  find the value of p, q, r.

5) Solve the inequation

- 1/3 < x/2 -4/3≤ 1/6 , x belongs to R . Graph the solution set on a number line.

8) a cylindrical water tank,  base radius 1.4 metre and height 2.1 metre is filled with with the help of a pipe of radius 7cm. calculate the time(in minutes) required to fill the tank, given that water flows at the rate of 2m/s in the pipe.

9) use graph paper for this question.

Monthly wages of some factory workers are given in the following table .

with 2cm= Rs 400 starting the origin at Rs4000 and to 2cm=10 workers on the y-axis, draw the Ogive. estimate the median from the graph.

Wages in Rs.     No. Of workers.

4000-4400                8

4400-4800               12 

4800-5200               20

5200-5600               25 

5600-6000               17 

6000-6400               10

1) A point P(3,-4) is reflected in X-axis..

I) write the coordinates of P'', the image of P..

ii) PP' is joined. To which coordinates axis PP' parallel to?

2) When expression ax²+bx-6 is divided by x-1, x+1, the remainder are -10, 4. Find a,b.

3) Given a/b= c/d, prove (2a-c)/(2a+c) = (2b-d)/(2b+d)

4) Calculate i) the Arithmetic mean ii) median iii) mode for

11,10,,11,13,13,12,15,17,14,12,13,14

5) 2/5≤ x - (1+ 2x/5)< 4/5, x belongs to R and show the number line.

6) A bus moving at its usual speed covers distance between town X and Y, which are 550km apart, in 1 hour less than it takes to cover the same distance, when it is raining and the bus has to reduce the speed by 5km/hr. Calculate the time taken to cover the distance between X and Y, when it is raining.

2) Find the value of k given that 

3x³ + 4x² -6x +k is divisible by x+1

3)Use graph for this question. Take 1cm= 1 unit on both axes.

i) Plot point P(2,3) and Q(3, 1)

ii) Reflect P in x-axis to P'. Reflect P' in y-axis axis to P". Write coordinates of P'' and P".

iii) Reflect Q in y-axis to Q' and reflect Q' in the origin to Q". write coordinates of Q' and Q". 

iv) write the geometrical name of PQQ"P'.

5) Solve the inequation

 - 1/3 < x/2 -4/3≤ 1/6 , x belongs to R . Graph the solution set on a number line.

8) a cylindrical water tank, base radius 1.4 metre and height 2.1 metre is filled with with the help of a pipe of radius 7cm. calculate the time(in minutes) required to fill the tank, given that water flows at the rate of 2m/s in the pipe.

1) Using the Remainder Theorem, find the remainder when 7x²-3x+8 is divided by (x- 4).                   

2) If x²,4 and 9 are in Continued Proportion, find x.                      

4) (i) If 7 is the mean of 5,3,0.5,4.5,b,8.5,9.5 find b.      

(ii) if each observation is decreased in value by 1 unit,what would the new mean be ?      

6) solve by formula

  (x+3)/(2x+3) =(x+1)/(3x+2).   

7) From the following table, find:

(i) The average wage of a worker, give your answer, correct to the nearest paise.

(ii) The modal class.

Wages in Rs. No of workers

Below 10.               15

Below 20.               35

Below 30.               60

Below 40.               80

Below 50.               96

Below 60.              127

Below 70.              190

Below 80.              200.              

8) prove.√{(1+cos x)/(1-cos x)}=Cosec x + Cot x.                   

9) A(4,5), B(2,6) and C(2,-3) are the vertices of a triangle in the co-ordinate plane.

a) Write down the coordinates 

of A', the reflection of A in the y-axis. A" the reflection of A in the origin.

b) Write down the coordinates of B', the images of B by reflection in the y-axis.

c) Name the image of line AB under reflection in the x-axis. What type of figure is formed by the line AB and its image A' , B' ?            

12) In a cricket match Ram took 3 wickets less than twice the number of wickets taken by Anshu. If the product of the number of wickets taken by them is 20, find the number of wickets taken by each.                                                     

15) When 7x² -3x+8 is divided by x -4, find the remainder.

16) Calculate the length of the tangent drawn to a circle of diameter 8cm from a point 5cm away from the centre of the circle.

17) If x²,4 and 9 are in Continued Proportion, find the value of x.

18) If x belongs to Z, find the solution set for the inequation 5<2x-3≤14 and graph the solution on a number line.

19) Find p and q if g(x)=x+2 is a factor of f(x)=x³-px+x+q and 

21) The volume of a cylinder 14cm long is equal to that of a cube having an edge 11cm. Calculate the radius of the cylinder.

23) The point P(a,b) is reflected in the x-axis to obtain point Q(3,-4). Find a and b 

23) The mean of Numbers 6,y,7,x and 14 is 8 Express y in terms of x.

24) Solve: x²-5x-2=0. Give your answer correct to 3 significant figures.

25)(8a+5b)/(8c+5d)=(8a-5b)(8c-5d) prove a/b= c/d.

26) Solve: 1<3x-3≤12, show the number line also.

27) find mean median and mode of 12,11,10,12,13,14,13,15,13.

28) The work done by (2x-3) men in (3x+1) days and the work done by (3x+1) men in (x+8) days are in the ratio of 11:15. Find x.

30) prove: Sinx + cosx =

Sinx/(1-Cotx) + Cosx/(1-tanx).

31) In an auditorium, seats were arranged in rows and columns. The number of rows was equal to the number of seats in each row. When the number of rows was doubled and the number of seats in each row was reduced by 15, the total number of seats increased by 400

Find a) The Number of rows in the original arrangement.

b) The Number of seats in the auditorium after rearrangement.

33) A man standing on the bank of a river observes that the angle of elevation of a tree on the opposite bank is 60°. When he moves 40 m away from the bank, he finds the angle of elevation to be 30°. Find

a) the width of the river.

b) the height of the tree.

34) Using remainder Theorem, find the remainder when 7x²-3x+8 Divided

by x- 4.

35) Find the length of the tangent drawn to a circle of radius 4cm from a point 5cm away from the centre of the circle.

36) Find the values of p and q if x+2 is the factor of x³- px+x+q and f(2)=4.

37) If 7 is the mean of 5, 3, 0.5, 4.5,

       b,8.5, 9.5 find b.

39) Solve: (x+3)/(2x+3) =(x+1)/(3x+2)

40) A boy standing on a vertical cliff in a jungle observes two rest-houses in line with him on opposite sides deep in the jungle below. If their angles of depression are30° and 60° and the distance between them is 222m, find the height of the cliff.

41) Find mean and mode

X: Bel10 -20 -30 -40 -50 -60 -70 

Age: 15 35 60 80 96 127 190

42) Find the value of k, if x - k is a factor of x³-kx²++x+4.

43) If x:y=4:3 find (5x+8y):(6x- 7y)

44) Solve: 2x-5≤5x+4<11

   where x belong to R

45) 5,6,8,9,10,11,11,12,13, 13, 14,

  14,15,15,15, 16,1618,19,20. Find

   Mean, Median, Mode.

46) Prove 

1/(SinA +CosA) +1/(SinA-CosA)=

2SinA)(2Sin²A -1)

47) An aeroplane travelled a distance of 800 km at an average speed of x km/hr. On return journey, the speed was increased by 40 km/hr. Write down an expression for the time taken for:

a) the onward journey

b) the return journey

If the return journey took 40 minutes less than the onward journey, write down an equation in x and find its value.

48) For what value of k, the polynomial x²+ (4- k)x + 2 is Divisible by x- 2 ?

49) Find the remainder when 2x³-3x²+7x-8 divided by x - 2.

50) If a/b= c/d prove

(3a- 5b)/(3a+5b)= (3c-5d)/(3c+5d)

51) Two numbers are in the ratio of 7:11. If 15 is added to each Number, the ratio becomes 5:7. Find the numbers.

52) Preeti deposited Rs1500 per month in a bank for 8 months under the Recurring Deposit Scheme. What will be the maturity value of her deposits, if the rate of interest is 12% p.a and interest is calculated at the end of every month.

53) Solve: 3x²-5x=1. Up to 2 decimal.

54) If 2 tan²A -1=0 prove

Cos3A= 4cos³A - 3cosA

55) SinA(1+tanA) +cosA(1+ cotA)=

  Cosec A + SecA

57) A vertical tower 40m high. A man standing at some distance from the tower knows that the cosine of the angle of elevation is 60° . How far is he standing from the foot of the tower ?

58) Factorise with the help of Factor Theorem f(x)= 6x³ - 7x²- 7x+6. Hence , find the values of x when f(x)=0.

59) A bag contains 4 white and 3 green balls. One ball is drawn at random. Find the probability that it is white.

60) What is the probability that the sum of the faces is not less than 10 when two unbiased dice are thrown

63) The midpoint of AB is P(-2,4). The coordinates of the point A and B are (a,0),(0,b) . Find A, B

65) Solve: 30-4(2x-1)> - 8.

66) Solve y - √(3y -6) =2.

67) Point P(a,b) is reflected in x-axis to P(5,-2).

a) write down the value of a, b

b) P" is the image of P when reflected in the y-axis. Write down the coordinates of P"

c) Name the single transformation that maps P to P".

68) If a, b, c are continued Proportion prove 

(a²+ b²)/b(a+c)= b(a+c)/(b²+c²).

70) If -5 is a root of the x²+ kx - 130=0

Find k, Hence find the other root.

71) An open cylindrical vessel of internal diameter 49cm and height 64cm stands on a horizontal platform. Inside this is placed a solid metallic right circular cone whose base has a diameter of 10.5cm and whose height is 12cm. Calculate the volume of water required to fill the tank. (π=22/7)

72) The perimeter of a rectangular plot is 180m and its area is 1800m². If the length is x m, Express the breadth in terms of x. Hence, form an Equation in x. Solve the Equation and find the length and the breadth of the rectangle.

73) (1+tan²x)/(1+cot²x)=sin²x/cos²x.

75) The angle of elevation of a cloud from a point 50m above a lake is 30° and the measure of the angle of depression of its reflection in the lake is 69°, find the height of the cloud.

76) A solid cylinder of radius 14cm and height 21cm is melted down and recast into spheres of radius 3.5cm each. Calculate the number of spheres that can be made.

77) If -3 is the root of x² - kx -27 =0, find k.

78) If 4, x,36,y are in Continued Proportion, find x and y.

79) The Mean of 12,18,x,13,19,22 is 16, find x.

80) Solve 3x²+8x+1=0. Give your answer correct to two decimal places.

81) Mrs.X deposited Rs.1500 per month in bank for 1 year 6 months under the Recurring Deposit Scheme. If the maturity value is Rs.30420, find the rate of interest p.a

82) Solve (11-2x)/5 ≥(9 -3x)/8 +3/4 and x belongs to N.

83) A(2,3), B(0,4) and C(0,-5) are the vertices of a triangle in the Cartesian plane.

a) write the co-ordinates of A' , the reflection of A in the x-axis and A" , the reflection of A in the origin.

b) write down the coordinates of B' , the image of By reflection in y-axis.

86) From the top of a building AB, 60 metres high, the angles of depression of the top and bottom of a vertical lamp post CD are observed to be 30° and 60°. Find

a) the horizontal distance between AB and CD.

b) the height of the lamp post CD.

87) SOLVE: 24x² - 334x + 135=0

88) An open cylindrical vessel is made of steel. The internal diameter is 14cm, the internal depth is 20.6cm and the metal is everywhere 4mm thick. Calculate

a) the internal volume

b) the volume of the metal correct to the nearest cm³.

89) The line joining A(-3,4) and

B(a,9) is divided in the ratio 2:3 at P, the point where the line segment AB intersects y-axis. Find

a) the value of a

b) the coordinates of P.

90) John sends his servant to the market to buy oranges worth Rs15. The servant having eaten three oranges on the way. John pays 25 paise per orange more than the market price. Taking x to be the number of oranges which John receives, form a quadratic equation in x. Hence, find the value of x.

92) A man is standing on a level ground, observes that a pole 30m away subtends an angle of 50° at his eye, which is 2.0m above the ground level. Calculate the height of the pole. (Give your answer correct to a length of a metre).

TRIGONOMETRIC FUNCTIONS 

   ******************************

PROVE

----------

1) Cos⁴A - Sin⁴A= Cos² - Sin²A

2) (1+TanA)²+(1 - Tan)²= 2Sec²A

3)Cot⁴A +Cot²A=cosec⁴A - Cosec²A

4) (secθ+tanθ)/(cscθ+cotθ) 

        = (cscθ-cotθ)/(secθ-tanθ)

5) (1+sinθ)/(1-sinθ)=(secθ+tanθ)²

6) (secθcotθ)= cscθ

7) tanθ+cotθ = secθ cscθ

8) cosθ/(secθ - tanθ) = 1+sinθ

9) (1+cosθ-sin²)/{sinθ(1+cosθ)} =cotθ

10) (tanθ+cotθ)sinθ.cosθ = 1

11) cosθ=cotθ/cscθ=cotθ/√(1+cot²θ)

12) sin⁴θ - cos⁴θ= sin²θ - cos²θ

13) sec²β - sec⁴β = -(tan²β + tan⁴β)

14) (cscθ-sinθ)(secθ-cosθ) 

       (tanθ+cotθ) = 1

15) (cot θ + tan β) /(cot β+tan θ)

       = cotθtanβ

16) sinα/(1+cosα)  + (1+cosα)/sinα 

      = 2cscα

17) 1+ 1/cos(α)  = tan²α/(secα-1)

18) (1+cosα)/(1-cosα)=(csc + cotα)²

19) (3 - 4sin²A)/Cos²A = 3 -Tan²A

20) (TanA+SecA -1)/(TanA-SecA+1)

       = (1+SinA)/Cos A.

21) Sec²ATan²B - Tan²ASec²B =

        Tan²B-Tan²A

22) SinA(1+TanA)+CosA(1+CotA)=

       SecA+CosecA

23) (1-SinA+CosA)² 

                    =2(1-SinA)(1+CosA)

24) CosA/(SecA-TanA)= 1+SinA

25)(1+CosA+sinA)/(1-CosA+SinA)=

        SinA/(1-cosA)=(1+cosA)/SinA

26) (secA-cosA)(cosecA-sinA) =

                 TanA/(1+tan²A)

27) If tanA + sinA=α and tanA - sinA=β prove α² - β²=4√(αβ)

MENSURATION

1) The base radius of a cylinder is 7 cm. Its height is 14 cm. Find 

A) its curved surface area.               616

B) its whole sarface area.                9243

C) the volume. (π=22/7).            2156

2) A hollow metallic cylinder with internal and external diameter 7cm and 10.5cm and length 20cm. Find

A) volume.                              962.24

B) total surface area (π=3.142). (Up to 2 d.p.                     1296.25

3) The radius of the base of a right circular cone 7cm. Its vertical height is 24cm. Calculate

A) slant height.                      5cm

B) curved surface area.              550

C) volume.                  1232

4) The diameter of a sphere is 7cm. Find

A) Surface of the spear.            154

B) volume up to 2d.p.           179.67

5) The surface area of a sphere is 1256 cm². It is cut into two hemisphere. Find

A) radius if there sphere.                   10

B) Total surface area of hemisphere.   942

C) volume of the hemisphere, correct to 2d.p. (π=3.14)      2093.33

6) A Quadrant form a circular metallic sheet of radius 10 cm is folded to a form a cone. Calculate

A) radius of the base of the cone.      2.5

B) area of the base of the cone.    19.63

C) volume of the cone to 2 decimal places. (π= 3.14).             63.32

7) A solid metalic cylinder, base radius 2 cm and height 9 cm, is melted to form a sphere. Calculate:

A) the radius of the sphere.       3

B) surface of sphere. (π=3.14).    113.04

8) Water flows through a pipe, whose cross section is a circular of radius 3.5 cm at 12 km/hr. Calculate the quantity of water discharged in 15 minutes. Give your answer in m³. (π=3.14).    11.5395

9) An iron sphere of a diameter 7 cm is dropped into a cylindrical can of diameter 14 cm. and is completely submerged in the water contained in the can. Calculate the rise in the level of water in the cylindrical can.    10/9 cm

10) What length of a solid cylinder 3 cm in diameter must be taken to recast it into a hollow cylinder of external diameter 14cm, 5 cm thick and 10 cm long?        

11) 80 equal metallic pellets, spherical in shape, each of radius 1cm are melted and cast into a single sphere. Calculate the radius of the new(single) sphere, correct to two decimal places.    4.31

12) A right circular cylinder is surmounted by a hemisphere at one circular end. 

The radius of the common end is 7cm. The maximum height of the solid is 15cm. Calculate the volume of the solid, correct to 2 dp.      1950.67

13) The diameter of the base of a cylinder and the height of a cylinder are equal. The volume of the cylinder is equal to the volume of a sphere of radius 6cm. Calculate the radius of the base of the cylinder correct to 1 d.p.   5.2

14) Two right circular cones x and y are such that 

A) radius of cone X is twice the radius of cone y

B) volume of cone x is half the volume of cone Y. 

Calculate: vertical height of cone y/ vertical height of cone x.            8

15) A solid cylinder, base radius 7cm and height 24cm, is  surmounded by a cone of same base radius and same vertical height, at one end. A hemisphere surmounts the cylinder at the other end. Calculate

A) volume of the solid, correct to 2 decimal places.              5646.67

B) surface area of the solid.        1914

16) A hollow cylindrical pipe 50cm long, whose external diameter is 7cm and the internal diameter is 5cm, is melted and recast into a right circular cone, whose base radius is 10cm. Calculate the height of the cone.b.  B     9cm

17) An exhibition tent is in the form of a cylindrical surmounted by a cone. The total height of the tent above the ground is 85m and the height of the cylindrical part is 50m. If the diameter of the box is 168m, find the quantity of Canvas required to make the tent. Allow 20% extra for fold and stitching. Give your answer to nearest m².           60502m²

18) From a solid cylinder 8 cm high and base radius 6cm, a conical cavity of 4 cm deep and base radius 3 cm is allowed out. Calculate

A) the volume the remaining solid correct to 2d.p.          867.43

B) the total surface area of the remaining solid to 2 dp.         546.86

19) Earth dug out to make a cylindrical well 2m in diameter and 7m deep is spread all round the tank uniformly to a width of 1m, to form an embankment. Calculate the height of the embankment.     7/3 cm

20) An inverted conical vessel of radius 6 cm and height 8cm is filled with water. A sphere is lowered into the vessel. Find

A) the radius of the sphere given that when it touches the sides, the highest point of the sphere is in the level with the base of the cone.                     3cm

B) volume of the water that flows out of conical vessel consequent to lowering the sphere in it.                   113.04cm³

21) From a sphere of radius 10 cm, a right circular cylinder diameter of whose base is 12cm, is carved out. Calculate the volume of the right circular cylinder correct two decimal places.     1810.29

QUADRATIC EQUATIONS

1) Solve: √{x/(1- x)} + √{(1-x)/x}= 13)6.

2) Solve: (x+1)(x+2)(x+3)(x+4)= 120.

3) Nature of the roots of the equation x² - x -3= 0 is

A) Real, imaginary, rational, irrational

4) Value of m of x² - 2mx + 7m -12= 0 if the roots are equal:

A) 3 B) 4 C) 3 or 4 D) -3

5) Value of m of x² - 2mx + 7m -12= 0 if the roots are reciprocal.

A) 12 B) 7 C) 12/7 D) none 

6) If one root of 2x² - 5x + k= 0 be the double the other roots , find k

A) 2 B) 7 C) 9 D) 25/9

7) If one root of 2x² - 5x + k= 0 be the double the other roots , find k

A) 2 B) 7 C) 9 D) 25/9

8) If the roots of the equation x² - bx + c= 0 differ by 2, then which of the following is true?

A)  c²= 4(c+1)  B) b²= 4(c+1)  C) c²= b+4 D) b²= 4(c+2) 

9) The sum of a number and its reciprocal is 17/4. Number is

A) 7 B) 8 C) 6 D) none

SOLVE:

1) 1/(x+1)  + 2/(x+2) = 4/(x+4), x≠ -1,-2,-4

2) (x-1)(x+2) + (x-3)(x-4)= 10/3, x≠ -2,4

3) abc² + (b² - ac)x - bc = 0.

4) Find the value of k which the equation (k+ 4)x² + (k+1)x + 1= 0 has equal roots.

5) (x-a)/(x-b) + (x-b)/(x-a) = a/b + b/a

6) a/(x-a) + b/(x-b) = 2c/(x - c).

7) x² - x - a(a+1)=0

8) a(x²+1) - x(a²+1)= 0

9) x² - 4√2 x +6= 0

10) ax²+ (4a² - 3b)x - 12ab = 0.

11) x² - (√3+ 1)x + √3= 0.

12) 2/x² - 5/x +2= 0

13) a²x² - 3abx + 2b²= 0.

14) 4x²+ 4bx - (a² - b²)= 0.

15) If the roots of the equation (a²+ b²)x² - 2(ac+ bd)x + (c²+ d²)= 0 are equal, prove that a/b = c/d.

16) If the roots of the equations ax² + 2bx + c= 0 and bx² - 2√(ac)x + b= 0 are simultaneously real, then prove that b² = ac.

CENTRAL TENDENCY (MEAN)

1) If the height of 5 persons are 144cm,152cm,151cm,158cm and 155cm respectively, then mean is

A) 100 B) 122 C) 142 D) 152

2) The Arithmetic mean of first 6 natural numbers is

A) 1.5 B) 2.5 C) 3.5 D) 4.5

3) Arithmetic mean of first 10 odd natural numbers is

A) 10 B) 20 C) 22 D) 42 

4) If the mean of 6,4,7, p and 10 is 8, then p is

A) 10 B) 13 C) 14 D) 15

5) The sum of the deviation of the variate values 3,4, 6, 8, 14 from their mean is

A) 0 B) 1 C) 12 D) 15

6) The mean of 40 observations was 160. It was detected on checking that the value of 165 was wrongly copied as 125 per computation of mean. Then the correct mean is

A) 100 B) 122 C) 142 D) 161 

7) The mean of 100 items was found to be 30. If at the time of calculation 2 items were wrongly taken 32 and 12 instead of 23 and 11, the correct mean is

A) 10.5 B) 22.4 C) 29.9 D) 9.9 

8) The mean monthly salary of 10 members of a group is ₹1445, one more member whose monthly salary is ₹1500 was joined the group. Mean monthly salary of 11 member of the group is

A) 100 B) 1220 C) 1420 D) 1450 

9) The mean of 10 numbers is 25. If 5 is subtracted from every number, what will be the new mean ?

A) 10 B) 12 C) 14 D) 15

10) The mean of 16 numbers is 8. If 2 is added to every number, what will be the new mean?

A) 10 B) 12 C) 14 D) 15

11) The mean of 5 numbers is 18. If one number is excluded, their mean is 16. Then excluded number is

A) 26 B) 22 C) 42 D) 52 

12) If the mean of five observations x, x + 2, x + 4, x + 6, x + 8 is 11, find the mean of first 3 observations.

A) 10 B) 9 C) 7 D) 15

13) The mean of 25 observations is 36. If the mean of first 13 observations is 32 and the mean of the last 13 observations is 39, find the 13th observation.

A) 100 B) 122 C) 142 D) 23 

14) Find the mean of first n natural numbers.

A) n B) n+1 C) n² D) n(n+2)

15) Find the mean of first n odd natural numbers.

A) n B) n+1 C) n² D) n(n+2)

16) If the mean of 1,2,3,....n is 6n/11, find the value of n .

A) n B) 11 C) 21 D) 23

17) If the heights of the five persons are 140,150, 152, 158 cm and 161cm  respectively, find the mean height .

A) 12.2 B) 122.2 C) 132.2 D) 152.2

18) Find the mean of 994, 996, 998,1002 and 1000.

A) 998 B) 1222 C) 1322 D) 1522 

19) mean of first five natural numbers is

A) 2 B) 3 C) 4 D) 5

20) mean of all factors of 10.

A) 2.5 B) 3.5 C) 4.5 D) 5.5

21) find the mean of first 10 even natural numbers.

A) 12 B) 13 C) 14 D) 11

22) find the mean of x, x+ 2, x + 4, x + 6, x + 8.     

A) x+2 B) x+3 C) x+4 D) x+5 

23) find the mean of the first 5 multiples of 3.

A) 2 B) 3 C) 9 D) 10

24) Following are the weights (in kg) of 10 new born babies in a hospital on a particular day :

3.4, 3.6, 4.2, 4.5, 3.9, 4.1, 3.8,  4.5, 4.4, 3.6. find the mean

A) 2 B) 3 C) 4 D) 5

25) The percentage of marks obtained by students of a class in mathematics are: 64, 36, 47, 23, 0, 19, 81, 93, 72, 35, 3, 1. Find their mean.

A) 29.5 B) 39.5 C) 49.5 D) 59.5

26) the number of children in 10 families of a locality are: 2, 4, 3, 4, 2, 0, 3, 5, 1, 1, 5. Find the mean number of children per family.

A) 2 B) 3 C) 4 D) 5 

27) Duration of sunshine(in hours) in Amritsar for first 10 days of August 2022 as reported by the meteorological Department are given below:

 9.6, 5.2, 3.5, 1.5, 1.6, 2.4, 2.6, 8.4, 10.3, 10.9. find mean

A) 2 B) 3 C) 4.6 D) 5.6

28) The mean of marks scored by 100 students was found to be 40. Later on it was discovered that a score of 53 was misread as 83. Find the correct mean.

A) 29.7 B) 39.7 C) 49.7 D) 59.7

29) The traffic police recorded the speed (in km/hr) of 10 motorists as 47, 53, 49, 60, 39, 42, 55, 57, 52, 48. Later on an error in recording instrument was found. Find the correct average speed of the motorists if the instrument recorded 5 km/hr less than in each case.

A) 55.2 B) 35.2 C) 45.2 D) 51.5

30) The mean of the five numbers is 27. If one number is excluded, their mean is 25. Find the excluded number.

A) 25 B) 35 C) 54 D) 55

31) The mean weight per student in a group of seven students is 55 kg. The individual weight of 6 of them (in kg) are 52, 54, 55, 53, 56 and 54. Find the weight of the 7th student.

A) 21 B) 31 C) 41 D) 61

32) The mean weight of 8 numbers is 15. If each number is multiplied by 2, what will be the new mean ?

A) 20 B) 30 C) 40 D) 50

33) The mean of 5 numbers is 18. If one number is excluded, their mean is 16. Find the excluded number.

A) 26 B) 36 C) 46 D) 56

34) The mean of 200 items was 50. Later on it was discovered that the two items were misread as 92 and 8 instead of 192 and 88. Find the correct mean.

A) 20.9 B) 30.9 C) 40.9 D) 50.9

35) Find the sum of the deviations of the variate values 3, 4, 6, 7, 8, 14 from their mean.

A) 0 B) 3 C) 4 D) 5 

36) The mean of 11 numbers is 35. If the mean of first 6 number is 32 and that of last 6 number is 37. Find the 6th number.

A) 29 B) 39 C) 49 D) 59

37) Find the mean of n terms of an AP with first term a and common difference d.

A) 2a B) 3d C) 4ad D) a+ (n-1)/2

38) Find the mean of n terms of the AP 1, 4, 7, 10,.....

A) 2n B) 2n -1 C) (3n-1)/2 D) 2n+1

39) There are 40 numbers in a group.  If the mean of first 10 is 4.5 and that of the remaining 30 is 3.5, find the mean of the whole group.

A) 2.75 B) 3.75 C) 4.75 D) 5.75

40) A group of 10 items has mean 6. If the mean of 4 of these items is 7.5, find the mean of remaining items.

A) 2 B) 3 C) 4 D) 5

41) The AM of a set of 50 numbers is 38. If two numbers of the set, namely 55 and 45 are discarded, find the AM of the remaining set of numbers.

A) 27.5 B) 37.5 C) 47.5 D) 57.5

42) find the AM of a, a+ d, a+ 2d, ....., a+ 2nd.

A) an B) and  C) a- nd D) a+ nd

Short Question for XII

FUNCTION

EXERCISE -1

1) If A{1,2,3} and B ={a,b}, Write total number of functions from A to B. 8

2) If A{a,b,c} and B ={-2,-1,0,1,2}, Write total number of one-one functions from A to B. 60

3) Write total number of one-one functions from set A I={1,2,3,4} to set B ={a,b,c}. 0

4) If f: R--> R is defined by f(x)= x², Write f⁻¹(25). {-5,5}

5) If f: C--> C is defined by f(x)= x², Write f⁻¹(-4). Here C denotes the set of all complex numbers. {2i, -2i}

6) If f: R--> R is defined by f(x)= x³, Write f⁻¹(1). {1}

7) Let C denote the set of all complex numbers. A function f: C--> C is defined by f(x)= x³, Write f⁻¹(-1). {1, w,w²}

8) Let f be a function from C(set of all complex numbers) to itself defined by f(x)= x³, Write f⁻¹(-1). {-1,- w,-w²}

9) Let f: R--> R is defined by f(x)= x⁴, Write f⁻¹(1). {-1,1}

10) Let f: C--> C is defined by f(x)= x⁴, Write f⁻¹(1). {-1,-i, 1,i}

11) If f: R--> R is defined by f(x)= x², Write f⁻¹(-25). ¢

12) If f: C--> C is defined by f(x)= (x-2)³, Write f⁻¹(-1). {1,2,-w,2-w²}

13) If f: R--> R is defined by f(x)= 10x -7,, Write f⁻¹(x). (x+7)/10

14) Let f: (- π/2, π/2) --> R be a function is defined by f(x)= [cos x], Write range of (f) . {1, cos 1, cos 2}

15) If f: R--> R is defined by f(x)= 3x -4 Write f⁻¹(x). (x+4)/3

16) If f: R--> R g: R--> R are given by f(x)= (x +1)² and g(x)= x²+1, then write the value of gog(-3). 121

17) Let A= {x belongs to R: -4≤x ≤ 4 and x ≠ 0} and f: A--> R be defined by f (x)= |x|/x. Write the range of f. {-1,1}

18) Let f: (- π/2, π/2) --> A be defined by f(x)= sin x. If f is a bijection, write set A. A[-1,1]

19) If f: R--> R⁺ is defined by f(x)=a ˣ, a> 0 and a≠ 1. Write f⁻¹(x). Logₐ x

20) Let f: R--{-1}--> R--{1} is defined by f(x)= x/(x+1). Write f⁻¹(x). x/(1-x)

21) Let f: R--{-3/5}--> R--{1} be a function defined by f(x)= 2x/(5x+3). Write f⁻¹: Range of f --> R --{-3/5}. 3x/(2 - 5x)

22) Let f: R--> R, g: R--> R be two functions defined by f(x)= x²+x+1 and g(x)= 1 - x². Write fog(-2). 7

23) Let f: R---> R be defined by f(x)= (2x- 3)/4. Write fof⁻¹(1). 1

24) Let f be an invertible real function. Write (f⁻¹of)(1)+ (f⁻¹of)(2)+....(f⁻¹of)(100). 5050

25) Let A={1,2,3,4} and B={a,b} be two sets. Write total number of onto functions from A to B. 14

26) Write the domain of the real function f(x)= √{x - [x]}. R

27) Write the domain of the real function f(x)= √{[x] - x}. ¢

28) Write the domain of the real function f(x)= 1/√{|x| - x}. (- ∞ ,0)

29) Write whether f: R---> R given by f(x)= x + √(x²) is one-one, many-one, onto or into. Many one into 

30) If f(x)= x+7 and g(x) =x-7, x belongs to R, write fog(7). 7

EXERCISE -2

Short Question for XI

TRIGONOMETRY

1) 25° when measured in radians is

A) 5π/18 B) 5π/24 C) 6π/36 D) none

2) 162° when measured in radians is

A) 5π/10 B) 9π/10 C) 4π/3 D) 5π/4

3) (8π/5)ᶜ = ?

A) 272° B) 302°  C) 288°  D) 316°

4) 11ᶜ= ?

A) 315° B) 372°  C) 418°  D) 630°

5) 1ᶜ= ?

A) 56°27'22" B) 57°16'22"  C) 55°18'32"  D) 57°26'32"

6) 3°45' expressed in radians is

A) (π/36)ᶜ B) (π/54)ᶜ C) A) (π/48)ᶜ B) (5π/96)ᶜ

7) 50°37'30" =?

A) (5π/16)ᶜ B) (7π/18)ᶜ C) A) (9π/32)ᶜ B) (11π/36)ᶜ

8) In a right angled triangle, the difference between two angles is (π/15)ᶜ. The measure of the smallest angle is

A) 40° B) 45°  C) 36°  D) 39°

9) The angles of a triangle are in AP and the greatest angle is double the least. The largest angle measures?

A) 60° B) 80°  C) 75°  D) 90°

10) The angles of a triangle are in AP and the ratio of the number of degrees in the least two the number of radians in the greatest is 60:π. The smallest angle is

A) 15° B) 30°  C) 45°  D) 60°

11) In a circle, the central angle of 45° intercepts an arc of length 33 cm. The radius of the circle is 

A) 21cm B) 35cm  C) 42cm  D) 14cm

12) In a circle of radius 14cm an arc subtends an angle of 36° at the centre.. The length of the arc is 

A) 6.6cm B) 7.7cm  C) 8.8cm  D) 9.2cm

13) The minute hand of a watch is 1.4cm long. How far does its tip move in 45 minutes?

A) 6cm B) 6.3cm C) 6.6cm D) 7cm

14)  If the arcs of the same length in two circles subtend angles of 60° and 75° at their respective centres, the ratio of their radii is

A) 4:5 B) 5:4  C) 3:5 D) 5:3 

15) A wire of length 121cm is bent to form an arc of a circle of radius 180cm. The angle subtends at the centre by the arc is

A) 36°20' B) 34°49' C) 38°30' D) 39°10'

16) A horse is tied to a post by a rope. If the horse moves along a circular path, always keeping the rope tight and describes 88m when it traces 72° at the centre, then the length of the rope is

A) 35m B) 70m C) 17.5m D) 22m

17) A pendulum swings through an angle of 42° in discribing an arc of length 55cm. The length of the pendulum is

A) 56cm B) 60cm C) 75cm D) 88cm

18) The radius of a circle is 30 cm. The length of the arc of the circle whose chood is 30 cm long, is

A) 9πcm B) 10πcm C) 12πcm 13.6πcm

19) A wheel makes 180 revolutions in 1 minute. How many radians does it turn in 1 second?

A) 3π B) 4π C) 6π D) 12π

20) A railway train is moving on a circular curve of radius 1500m at a speed of 90 km/hr. Through what angle has it turned in 11 seconds?

A) 10°30' B) 11°40' C) 12° D) 16°30'

21) When a clock shows the time 7:20, what is the angle between its minute hand and the hour hand ?

A) 60° B) 80° C) 100° D) 120°

22) the angle between the hour and the minute hand of a clock at half past three is

A) 54° B) 63° C) 72° D) 75° 

23) The angle between the minute hand and the hour hand of a clock when the time is 8:25 a.m. is 

A) 107°15' B) 105° C) 102°30' D) 92°45

24) The length of a pendulum is 60cm. The angle through which its swings when its tip describes an arc of length 16.5 cm is 

A) 15°30' B) 15°45' C) 16°15' D) 16°30'

25) the angles of a quadrilateral in degrees are in AP and the greatest angle is 120°. The smallest angle is 

A) π/4 B) π/3 C) π/5 D) π/6

26) The perimeter of a sector of a circle is equal to half the circumference of the circle. The angle of the sector is

A) π/4 B) π/2 C) π-2 D) π+2

27) sin (25π/3)=

A) 1/2 B) 1/√2 C) √3/2 D) -√3/2

28) cos (41π/4)=?

A) 1/√2 B) -1/√2 C) √3/2 D) -√3/2

29) tan(-16π/3)=?

A) √3 B) -√3 C) 1/√3 D) -1/√3

30) cot(29π/4)=?

A) -1 B) 1 C) √3 D) 1/√3

31) sec(-19π/3)=?

A) -2 B) -1/2 C) 2 D) 1/2 

32) Cosec(-33π/4)=?

A) -√2 B) √2 C) 1/√2 D) -1/√2 

33)cos 15π=?

A) 1 B) -1 C) 0 D) none

34) sec 6π

A) 1 B) -1 C) √2 D) -√2  

35) tan 5π/4=?

A) √3 B) 1/√3 C) 1 D) -1

36) sin(765°)=?

A) √3/2 B) -√3/2 C) 1/√2 D) -1/√2

37) cot(-600°)=?

A) -1 B) -√3  C) -1/√3 D) none

38) If sinx =-2√6/5 and x lies in quadrant III, then cot x =?

A) 1/2√6 B) -1/2√6 C) 3/2√6 D) -3/2√6

39) If cos x =- √15/4 and π/2 < x< π, then sin x= ? 

A) 3/4 B) -3/4 C) 1/4 D) -1/4

40) If sec x =- 2 and π < x< 3π/2, then sin x= ? 

A) √3/2 B) -√3/2 C) 1/2 D) -1/2

41) If cosec x =- 2/√3 and x lies in quadrant IV,  then tan x= ? 

A) 1/√3 B) -1/√3 C) √3 D) -√3

42) if cot x=√5 and x does not lie in the quadrant I, then the value of cosec x and sec x are respectively.

A) √6,(√6/5) B) -√6,√(6/5) C) -√6,-(√6/5) D) -√6,(√6/5)

43) If cos x =- 1/2 and x lies in quadrant II, and then (2sin x+ tan x) = ? 

A) 0 B) 3√3/2 C) -√3/2 D) none

44) If cos x =- 3/5 and π < x< 3π/2, then (cosec x + cot x)/(sec x - tan x) = ? 

A) 1/6 B) 2/3 C) 3/2 D) 1/3

45) If sin x = 3/5 and π/2 < x< π, then (2 sec x- 3 cot x)= ? 

A) 3/2 B) -3/2 C) 13/2 D) -13/2

46) If sec x =√2 and 3π/2 < x< 2π, then (1+ tan x + cosec x)/(1- cot x- cosec x)= ? 

A) √3/2 B) -1 C) -3/8 D) 3/4

47) If cos x =- 12/13 and π < x< 3π/2, then (cot x + cosec x) = ? 

A) 1/5 B) -1/5 C) 3/5 D) -3/5

48) If cot x =- 12/5 and π/2 < x< π, then (1+ sin x - cos x)/(1- sin x + cos x)= ? 

A) 13/4 B) -13/4 C) 15/2 D) -15/2

49) If sin x =4/5 and π/2 < x< π, then (5 cos x+ 4 cosec x + 3 tan x)/(4 cotx + 3 sec x + 5 sin x) = ? 

A) -1/2 B) 1 C) 1/2 D) -1

50) If sec x = 13/5 and x is acute, then (4- 3cot x)/(3+ 4 tanx) = ? 

A) 55/252 B) 44/305 C) 54/255 D) 33/215

51) cos 135 =  ? 

A) 1/√2 B) -1/√2 C) 1/2 D) -1/2

52) sec 120 =  ? 

A) √2 B) -√2 C) 2 D) -2

53) cosec 150 =  ? 

A) -2 B) 2 C) √2 D) -√2

54) sin 315 =  ? 

A) 1/√2 B) -1/√2 C) √2 D) -√2

55) cos 405 =  ? 

A) √2 B) 1/√2 C) -√2 D) -1/√2

56) tan(11π/6) =  ? 

A) -1/√3 B) 1/√3 C) √3 D) -√3

57) cot 675 =  ? 

A) -1 B)  1 C) -√3 D) √3

58) sin(31π/3) =  ? 

A) 1/2  B) -1/2 C) √3/2 D) -√3/2

59) cot (-600) =  ? 

A) -√3 B)  1/√3 C) -1/√3 D) √3

60) cosec(-1100) =  ? 

A) 2/√3 B)  -2/√3 C) 2 D) -2

61)  sec(-33π/4) =  ? 

A) -√2  B) √2 C) -√3/2 D) √3/2

62) Tan (-25π/3) =  ? 

A) -√3 B) √3 C) -1/√3 D) 1/√3

63) The value of Cot (π/3),  Cot (π/4), Cot (π/6) are in 

A) AP  B) GP C) HP D) none

64) Which is smaller, sin 64° or cos 64°?

A) sin 64° B) cos 64° C) both are equal D) can not be compared

65) Which is larger, sin 24° or cos 24°?

A) sin 24° B) cos 24° C) both are equal D) can not be compared

66) The extremum values of sin x are

A) 0 and 1 B) -1 and 0 C) -1 and 1 D) -√3/2 and 1/√2

67) The extremum values of cos x are

A) 0 and 1 B) -1 and 0 C) -1 and 1 D) -√3/2 and √2

68) The values of sec x can 

A) never be greater than 1 B) never be less than 1  C) never be equal to 1 D) never lie between -1 and 1

69) tan(150) =  ? 

A) 1/√3 B)  -1/√3 C) √3 D) -√3

70) sec(150) =  ? 

A) 2/√3 B)  -2/√3 C) 2 D) -2

71) cot(120) =  ? 

A) -1/√3 B)  1/√3 C) -√3 D) √3

72) sin 105 + cos 105 =  ? 

A) √2 B)  1/√2 C) 1/√3 D) 2/√3

73) sin 15 =  ? 

A) √3/2√2 B) (√3+1)/2√2 C) (√3-1)/2√2 D) (√2-1)/√2

74) cos 15 =  ? 

A) (√3+1)/√2 B) (√3+1)/2√2 C) (√3-1)/2√2 D) (√3-1)/√2

75) tan 15 =  ? 

A) (√3+1)/(√3-1) B) (√3-1)/(√3+1) C) (√2+1)/(√2-1)  D) (√2-1)/(√2+1)

76) sin 75 =  ? 

A) (√3+1)/2√2  B) (√3-1)/2√2  C) (√2+1)/2√2 D) (√2-1)/2√2

77) cos 75=?

A) (√3+1)/2√2  B) (√3-1)/2√2  C) (√2-1)/2√2 D) (√2+1)/2√2

78) tan(13π/12)=?

A) (2+ √3) B) (1+ √2) C) (2 - √3) D) (√2-1)

78) (sin70 cos 10 - cos 70 sin 10)=?

A) 1/√2 B) 1/2 C) √3/2  D) -√3/2

80) (sin36 cos 9 - cos 36 sin 9)=?

A) 1/√2 B) 1/2 C) √3/2  D) 1

81) (cos 80 cos 20 + sin 80 sin 20)=? 

A) 1/√2 B) 1/2 C) √3/2 D) 1

82) (cos 50 cos 10 - sin 50 sin 10)=? 

A) 1/√2 B) 1/2 C) √3/2 D) 1

83) sin(40 +x) cos(10 + x) - cos(40 +x) sin (10 + x) =? 

A) √2 B) 1/2 C) √3/2 D) none

84) sin 7π/12 cos π/4 - cos 7π/12 sin π/4 =? 

A) 1/√2 B) √2 C) √3/2 D) none

85) sin π/4  cos π/12 - cos π/4 sin π/12 =? 

A) 1/2 B) √2 C) √3/2 D) none

86) cos 2π/3 cos π/4 - sin 2π/3 sin π/4 =? 

A) (√3+1)/√2 B) -(√3+1)/√2 C) (√3+1)/2√2 D) -(√3+1)/2√2

87) sin π/12 =? 

A) (√3-1)/2√2 B) -(√3+1)/2√2 C) (√3+1)/2√2 D) none

88) sin 5π/12 =? 

A) (√3-1)/2√2 B) (√3+1)/2√2 C) - (√3-1)/2√2 D) -(√3+1)/2√2

89) If sin x =15/17 and cos y= 12/13, where x and y both lie in quadrant I, then sin(x+ y)=?

A) 171/221 B) 180/221 C) 220/221 D) 181/221

90) If sin x =3/5 and cos y= -12/13, where x and y both lie in quadrant II, then sin(x- y)=?

A) 16/65  B) -16/65 C) 33/65 D) -33/65

91) If cos x =4/5 and cos y= 12/13, where x and y both lie in quadrant IV, then cos(x+ y)=?

A) 33/65 B) -33/65 C) 16/65 D) -16/65

92) If cot x =1/2 and sec y= -5/3, where x lie in quadrant III, and y lies quadrant II then tan(x+ y)=?

A) 5/11 B) 211 C) -6/11 D) 10/11

93) cos 15 - sin 15 =  ? 

A) 1/2 B)  1/√2 C) (√2-1)/√2 D) (√2+1)/√2

94) cot 105 - tan 105 =  ? 

A) √3 B)  2√3 C) √3/2 D) (√3+1)/(√3-1)

95) 2 sin 5π/12 sin π/12 =  ? 

A) √2 B)  1/√2 C) 1/2 D) √3/2

96) 2 cos 5π/12 cos π/12 =  ? 

A) √2 B)  1/√2 C) 1/2 D) √3/2

97) 2 sin 5π/12 cos π/12 =  ? 

A) √3/2 B)  (2+√3)/2 C) 1/2 D) (√3+1)/2

98) {sin(180+x) cos(90+x) tan(270-x) cot(360- x)}/{sin(360-x) cos(360+x) cosec(-x) sin(270+x)} =  ? 

A) √2 B)  √3/2 C) 3/2 D) 1

99) {sin(40+x) cos(10+x) - cos(40 +x) sin(10-+x)} =  ? 

A) 2 B) √3/2 C) 1/2 D) none

100) {cos(90+x) sec(270+x) sin(180+x)}/{cosec(-x) cos(270- x) tan(180+x)} =  ? 

A) cos x B) sec x C) cot x D) none

101) cosx+ sin(270+x) -  sin(270-x)+ cos(180+x) =  ? 

A) 2cos x B) 2sin x C) 0 D) none

102) (cos8 - sin 8)/(cos8 + sin 8) =  ? 

A) tan 8 B) tan 37 C) tan 52 D) none

103) {cos(π+x) cos(-x)}/{ cos(π-x) cos(π/2 + x)  =  ? 

A) -cot x B) cot x C) -tan x D) tan x

104) cos(π/4 +x) +  cos(π/4 -x) =  ? 

A) 2cos x B) √2 cos x C) 2 sin x D) √2 sin x

105) cos(3π/4 +x) - cos(3π/4 - x) =  ? 

A) √2 sin x B) 2 sin x C) -√2 sin x D) 1/√2 sin x

106) (sin 3x - sin x)/(cosx - cos 3x) =  ? 

A) tan 2x B) cot 2x C) - tan 2x D) - cot 2x

107) (cos 6x + cos 4x)/(sin6x - sin4x)= ? 

A) cot x B) - tan x C) - tan x D) - cot x

108) (cos4x +cos3x+ cos2x)/(sin4x + sin 3x + sin2x) =  ? 

A) tan 2x B) cot 2x C)  tan 3x D) cot 3x

109) (sin 7x - sin 5x)/(cos7x + cos5x)= ? 

A) tan x B) cot x C)  tan 2x D)  cot 2x

110) (sin²6x - sin²4x)=  ? 

A) sin10x B) sin 2x C) sin10x sin 2x D) n

111) cos20 cos40 cos80 =  ? 

A) 1/16 B) 1/8 C) √3/8 D) √3/16

112) sin 10 sin 50 sin 70 =  ? 

A) 1/16 B) 1/8 C) √3/8 D) √3/4

113) 2cos 45 cos 15  =  ? 

A) 3/2 B) (√3+1)/2 C) (√3-2)/2 D) √3/2

114) 2 sin 75 sin 15  =  ? 

A) 1/2  B) (√3+1)/2 C) √3/2 D) n

115) cos 15 - sin 15 =  ? 

A) 1/2 B) 1/√2 C) 1/2√2 D) √3/2

116) If sinx =-1/2 and x lies in quadrant III, then sin 2x =  ? 

A) 1/2 B) 1/2√3 C) √3 D) √3/2

117) If sec x =-13/12 and π/2< x<π, then cos 2x =  ? 

A) -120/169 B) 119/169 C) -120/119 D)n

118) If tan x =-3/4 and 3π/2< x<2π, then tan 2x =  ? 

A) -24/25 B) 7/25 C) 24/7 D) -24/7

119) If sin x = 1/3  then sin 3x =  ? 

A) 1 B) 1/9 C) 7/9 D) 23/27

120) If cos x = 1/2 then cos 3x =  ? 

A) 3/2 B) 1/6 C) -1  D) 2/3

121) cos² π/12 - sin²π/12 =  ? 

A) 1/2 B) √3/2 C) 3/2  D) -1/2

122) (1+sin 2x - cos2x)/(1+ sin2x+ cos2x) =  ? 

A) tan 2x B) tan x C) cot2x D) cot x

123) If cos x =-3/5 and π/2< x<π, then sin(x/2) =  ? 

A) -2/√5 B) 2/√5 C) -2/√5 D) 1/√5

124) If cos x =-3/5 and π/2< x<π, then cos(x/2) =  ? 

A) 1/√5 B) -1/√5 C) 2/√5 D) -2/√5

125) If cos x =-4/5 and π< x<3π/2,  then cos(x/2) =  ? 

A) 1/√10 B) -1/√10 C) 3/√10 D) -3/√10

126) If tan x =3/4 and π < x<3π/3, then tan(x/2) =  ? 

A) 3  B) 1/3 C) -3 D) -1/3

127) If cos x =-1/3 and x lies in quadrant III, then tan(x/2) =  ? 

A) √2 B) -√2 C) √3 D) -√3

128) If sin x =-1/2 and x lies in quadrant IV then sin(x/2) =  ? 

A) √{2 +√3}/2 B) {√3+ √2}/2 C) √(2- √3)/2 D) none

129) (1+ cosx)/((1- cosx)=?

A) tan²(x/2) B) cot²(x/2) C) sec²(x/2) D) cosec²(x/2)

130) √{(1+ sinx)/((1- sinx)} =?

A) tan(x/2) B) cot(x/2) C) tan(π/4+ x/2) D) cot(π/4 + x/2)

131) sinx/((1+ cos x) =?

A) tan(x/2) B) cot(x/2) C) tan(π/4+ x/2) D) n

132) cot(x/2) - tan(x/2) =?

A) 2tanx B) 2cot x C) 2 sin x D) 2cos x

133) sin 18°=?

A) (√5-1)/4 B)  (√5+1)/4 C)  (√3+ 1)/2 D)  (√3 -1)/2

134) cos 18°=?

A) √(10- 2√5)/4 B) √(10+2√5)/4  C)  (√5+ 1)/4 D) none

135) cos 36°=?

A) (√5-1)/4 B) (√5+1)/4  C)  (√3+ 1)/2 D) 

(√3- 1)/2

136) sin 36°=?

A) (√5 -1)/4 B) √(10+2√5)/4  C) √(10- 2√5)/4 D) none

137) sin 54°=?

A) (√5+1)/4 B) (√5-1)/4  C)  √(10+2√5)/4 D) √(10-2√5)/4

138) cos 72°=?

A) (√5+1)/4 B) (√5-1)/4  C)  (2+√5)/3 D) (2 - √5)/3

139) cos 54°=?

A) (√5+1)/4 B) (√5-1)/4  C)  √(10+2√5)/4 D) √(10-2√5)/4

140) sin 72°=?

A) (√5+1)/4 B) (√5-1)/4  C)  √(10+2√5)/4 D) √(10-2√5)/4

141) 2sin 45/2 cos 45/2 =?

A) 1 B) 1/2  C) 1/√2 D) √2

142) (2 cos² 15 - 1) =?

A) 3/2 B) √3/2  C) 2√3 D) √3/√2

143) 3sin 40- 4 sin² 40 =?

A) 3√3 B) 3/2 C) none D) √3/2

144) 8cos³20- 6 cos 20 =?

A) 5/2 B) 5/3  C) 1 D) √3/2

145) If tan x= a/b, then a sin2x+ b cos2x = ?

A) a B) b  C) a+ b D) a-- b

146) cotx - 2 cot 2x =?

A) tanx B) cosx  C) sinx D) cos2x

147) cos 2x+ 2 sin²x =?

A) 2 B) 1/2  C) 1 D) 3/2

148) sin2x/(1-  cos 2x) =?

A) tanx B) cot x  C) sec x D) cosec x

149) sin2x/(1+ cos 2x) =?

A) tanx B) cot x  C) sec x D) cosec x

150) tan 2x/(1+ sec 2x) =?

A) tanx B) cot x  C) sin x D) cos x

151) (1- cos 2x+ sinx)/(sin2x+ cosx) =?

A) tanx B) cot x  C) sec x D) cosec x

152) cosx/(1-  sinx) =?

A) tan(x/2) B) tan(π/4+ x/2)  C) tan(π/4- x/2) D) none

153) 

PERMUTATION AND COMBINATIONS 

EXERCISE -1

1) If 1/6! + 1/7! = x/8!, Then x=?

A) 32 B) 48 C) 56 D) 64

2) If ⁿ⁻¹P₃ :  ⁿP₄ = 1:9, then n=?

A) 12 B) 11 C) 9 D) 10 

3) ⁿPₙ/ⁿPₙ₋₂ =?

A) 1/2 B) 2 C) 1/(n-2) D) n(n-1)

4) ¹⁵Pₙ =2730, then n=?

A) 3 B) 4 C) 5 D) 6

5) ⁷P₃ =?

A) 105 B) 140 C) 210 D) 175

6) ⁿP₅ = 20. ⁿP₃, then n=?

A) 8 B) 9 C) 10 D) 11

7) ¹⁵Pₙ₋₁ : ¹⁶Pₙ₋₂=  3:4, then n= ?

A) 8 B) 10 C) 12 D) 14

8) ⁿC₁₀ =ⁿC₁₄ , then n=?

A) 4 B) 24 C) 14 D) 10

9) If ⁿC₃ = 220, then n=?

A) 9 B) 10 C) 11 D) 12

10) ⁿCₐ + ⁿCₐ₊₁ = ⁿ⁺¹Cₓ , then x=?

A) a-1 B) a C) a+1 D) n

11) ³⁶C₃₄ = ?

A) 1224 B) 612 C) 630 D) none

12) ⁿCₐ/ⁿCₐ₋₁ =?

A) (n-r)/r B) (n-r -1)/r  C) (n-r +1)/r D) none

13) ⁿC₁₈= ⁿC₁₂ , then ¹²Cₙ=?

A) 248 B) 496 C) 992 D) none

14) ⁶⁰C₆₀ =?

A) 60! B) 1 C) 1/60 D) none

15) In how many ways can 5 persons occupy 3 seats ?

A) 15 B) 20 C) 30 D) 60 

16) In how many ways can 5 children stand in a queue ?

A) 5 B) 25 C) 60 D) 120 

17) In how many ways can 4 different books be arranged in a self ?

A) 4 B) 8 C) 24 D) 16 

18) 10 students are participating in a race. In how many different ways can the first prize be own ?

A) 30 B) 60 B) 120 D) 720

19) three different rings are to be worn in four fingers. In how many ways can this be done ?

A) 12 B) 24 C) 64 D) 81 

20) 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.

A) 360 B) 720 C) 3600 D) 1800

21) How many words with or without meaning can be formed by using all the letters of the word, DELHI, using each letter exactly once ?

A) 20 B) 60 C) 120 D) 5

22) It is required to seat 5 man and 4 Women in a row so that the women occupied the even places. How many such arrangements are possible ?

A) 24 B)120 C) 720 D) 2880

23) How many three-digit numbers are there ?

A) 648 B) 729 C) 900 D) 1000

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

A) 648 B) 720 C) 729 D) none

25) How many four-digit numbers can be formed with no digital repeated by using the digit 3,4,5,6,7,8 and 0?

A) 280 B) 560 C) 720 D) 840

26) How many 3-digits even numbers can be formed with no digit repeated by using the digit 0, 1, 2, 3, 4 and 5 ?

A) 50 B) 52 C) 54 D) 56 

27) The number of positive integers greater than 6000 less than 7000 which are divisible by 5 with no digit repeated is

A) 28 B) 56 C) 84 D) 112

28) How many 10 digit numbers can be formed by using the digit 1 and 2 ?

A) ¹⁰P₂  B) ¹⁰C₂ C) 2¹⁰ D) 10!

29) How many words beginning with T and ending with E can be made with no letter repeated out of the letters of the word TRIANGLE?

A) ⁸P₆ B) 720 C) 722 D) 1440 

30) How many words can be formed from the letters of the word DAUGHTER so that the vowels always come together ?

A) 720 B) 2160 C) 4320 D) none

31) How many words can be found from the letters of the word LAUGHTER so that the vowels are never together ?

A) 3600 B) 4320 C) 36000 D) 40320

32) In how many ways can the letters of the word MACHINE be arranged so that the vowels may occupy only odd positions ?

A) 288 B) 576 C) 5040 D) none

33) in how many ways can the letter of the word PENCIL be arranged so that N is always next to E?

A) 120 B) 240 C) 720 D) 1440 

34) In how many ways can the letters of the word APPLE be arranged?

A) 6 B) 60 C) 90 D) 120

35) How many words can be formed by using all the letter of the word ALLAHABAD?

A) 9! B) 1890 C) 3780 D) 7560

36) How many words can be formed using the letter A thrice, the letter B twice and the letter C once ?

A) 6 B) 60 C) 90 D) 120 

37) In how many ways can the 10 books be arranged in a self so that a particular pair of books shall be always together ?

A) 8! B) 9! C) 2 . 8! D) 2 . 9!

38) In how many ways can 10 books be arranged in a shelf so that a particular pair of books shall be never together ?

A) 8! B) 9! C) 2. 9! D) 8.9.

39) How many 4 digit numbers are there where a digit may be repeated any number of times in each number ?

A) 5040 B) 9000 C) 10000 D) 4500

40) How many ways can 6 boys be arranged in a row?

A) 5! B) 6! C) 6 D) 2.6! 

41) In how many ways can 6 girls be seated in a circle ?

A) 6 B) 6! C) 5! D) 2.5!

42) How many diagonals are there in a polygon of n sides?

A) n(n-1)/2 B) n(n-2)/2 C) n(n-3)/2 D) n(n+1)/2  

43) How many diagonals are there in an octagon ?

A) 28 B) 24 C) 20  D) 36 

44) A polygon as 54 diagonals. Number of sides in this polygon is.

A) 9  B) 12 C) 15 D) 16 

45) There are 10 points in a plane, out of which 4 points are collinear. The number of line segments obtained from the pairs of these points is

A) 39 B) 40 C) 41 D) 45 

46) There are 10 points in a plane, out of which 4 points are collinear. The number of triangles formed with the vertices as these points is 

A) 20 B) 120 C) 116  D) none

47) Out of 7 consonants and 4 vowels, how many words of 3 consonants and 2 vowels can be formed?

A) 330 B) 1050 C) 6300 D) 25200

48) In how many ways can a committee of 5 member be selected from 6 men and 5 ladies, consisting of 3 men and 2 ladies?

A) 25 B) 50 C) 100  D) 200

49) Out of 5 men and 2 women, a committee of 3 is to be formed. In how many ways it be formed if at least one woman is included in each committee ?

A) 21 B) 25 C) 32 D) 50 

50) A committee of five is to be formed out of six gents and four ladies. In how many ways can this be done when each committee may have atmost 2 ladies ?

A) 120 B) 160 C) 180 D) 186 

51) How many different teams of 7 players can be chosen out of 10 players?

A) 720 B) 70 C) 120 D) none 

52) 12 persons meet in a room and each shakes hands with all the others. How many handshakes are there ?

A) 144 B) 132 C) 72 D) 66 

53) In how many ways can we select 9 balls out of 6 red balls, 5 white balls and five blue balls if 3 balls if each colour are selected 

A) 40 B) 200 C) 2000 D) 400

54) How many ways can a cricket team be chosen out of a batch of 15 players, if a particular player is always chosen?

A) 1364 B) 364 C) 1001 D) none

55) In how many ways can a cricket team be chosen out of a batch of 15 players, if a particular player is never chosen ?

A) 364  B) 1001 C) 1364 D) none 

56) For the post of 5 teachers, there are 23 applicants. 2 posts are reserved for SC candidates and there are 7 SC candidates among the applicants. In how many ways can the selection be made?

A) 5880 B) 11760 C) 3920 D) none

57) In how many ways can 5 white balls and 3 black balls be arranged in a row so that no two black balls are together 

A) 192 B) 40 C) 20 D) 120 

58) In an examination a candidate has to pass in each of the five subjects. In how many ways he fail ?

A) 5 B) 10 C) 21 D) 31

59)  An examination paper content 12 questions consisting of two parts, A and B. Part A contains 7 questions and part B contains 5 questions. A candidate is required to attempt 8 questions, selecting at least three from each part. In how many ways can the candidate select the questions   ?

A)210 B) 175 C) 420  D) none