MATH- TEST PAPERS

Thursday, 16 April 2026

sony

1) Two non integer roots of the equation

(x²+ 3x)² - (x²+ 3x) -6=0 are

a) (1/2) (-3+ √11), (1/2) (-3- √11)

b) (1/2) (-3+ √7), (1/2) (-3- √7)

c) (1/2) (-3+ √21), (1/2) (-3- √21) d) none

2) Two non integer roots of

{(3x -1)/(2x +3)}⁴ - 5{(3x -1)/(2x +3)}⅖ +4= 0 are

a) -5/7,-2/5 b) -2/4,7/5 c) 5/7,7/5 d) -2/5, 3/5

3) Sum of the roots of the equation

4ˣ - 3. 2ˣ⁺³ + 128=0 are

a) 5 b) 6 c) 7 d) 8

4) The only value of x satisfying the equation is

6√{x/(x +4)} - 2√{(x +4)/x} = 11 where x ∈ R

a) 4/35 b) -4/35 c) 16/3 d) none

5) The number of real values of x satisfying the equation

2(x²+ 1/x²) - 9(x + 1/x) + 14= 0

a) 1 b) 2 c) 3 d) 4

6) The non integer roots of x⁴- 3x³- 2x²+ 3x +1= 0 are

a) (1/2)(3+ √13), (1/2)(3 - √13)

b) (1/2)(3- √13), (1/2)(-3 - √13)

c) (1/2)(3+ √17), (1/2)(3 - √17) d) none

7) The number of real solution of

1/(x +1) + 1/(x +5) = 1/(x +2) + 1/(x +4) is

a) 0 b) 1 c) 2 d) 3

8) Number of real solutions of

(x -1)(x +1)(2x +1)+2x -3)= 15 is

a) 0 b) 2 c) 3 d) 4

9) The number of solutions of the equation

√[2x √(2x +4)]= 4 is

a) 0 b) 1 c) 2 d) 4

10) The number of solutions of

√(3x²+ x +5)= x -3 is

a) 0 b) 1 c) 2 d) 4

11) The number of solutions of

√(4- x) + √(x +9)= 5 is

a) 0 b) 1 c) 2 d) 3

12) The number of real solutions of

√(x²-4x +3) + √(x²-9)= √(4x²- 14x +6) is

a) 0 b) 1 c) 2 d) 4

13) The value of a for which one root of the equation

(a²- 5a +3)x² + (3a -1)x +2=0 is twice as large as other, is

a) -2/3 b) 1/3 c) -1/3 d) 2/3

14) Eange of the function f(x)= (x²+ x +2)/(x²+ x +1), x ∈ R is

a) (1, ∞) b) (1,3/2) c) (1,7/3] d) 1,7/5]

15) If f(x)= x²+ 2bx + 2c² and g(x)= - x²- 2cx + b² are such that minimum f(x)> maximum g(x), then relation between b and c, is

a) no relation b) 0< c<b/2 c) |c|< |b|/√2 d) |c|> √2 |b|

16) if a, b are the roots of x²+ px +1= 0, and c, d are the roots of x²+ qx +1= 0, the value of

E= (a - c)(b - c)(a + d)(b + d) is

a) p²- q² b) q²- p² c) q² + p² d) none

17) If 4ˣ - 3ˣ⁻¹⁾² = 3ˣ⁺¹⁾² - 2²ˣ⁻¹ , then the value of x is

a) 5/2 b) 2 c) 3/2 d) 1

18) For a> 0, a≠ 1, the number of values of x satisfying the equation

2logₓa + logₐₓa + 3 logₐ²ₓ a= 0 is

a) 2 b) 3 c) 4 d) infinite

19) The number of solutions of

√(x+1 - √(x -1)= 1 (x ∈R)

a) 1 b) 2 c) 4 d) infinite

20) If a, b, c are real and a≠ b, then the roots of the equation

2(a - b)x² - 11(a+ b + c)x -3(a - b)= 0 are

a) real and equal

b) real and unequal

c) purely imaginary d) none

21) Let a> 0, b> 0 and c> 0. Then both the roots of the equation

2ax²+ 3bx + 5c= 0

a) are negative

b) have real parts

c) have positive real parts d) none

22) If a, b, c are real, then both the roots of the equation

(x - b)(x - c)+ (x - c)(x - a)+ (x - a)(x - b)= 0 are always

a) positive b) negative c) real d) none

23) The equation

2x - 3/(x -2) = 4 - 3/(x -2) has

a) no root b) one root c) two equal roots d) none

24) If a, b,c are positive real numbers which are in GP , then the equation ax²+ 2bx + c= 0 and dx² + 2ex + f= 0 have common root if a/d, b/e, c/f are in

a) AP b) GP c) HP d) none

25) If P(x)= ax²+ bx + c and Q(x)= - ax²+ dx + c, where ac ≠ 0, then P(x) Q(x)= 0 has

a) no real root

b) exactly two real roots

c) atleast two distinct real roots d) none

26) If the product of the roots of the equation

x²+ 5kx + 2e⁴ˡⁿᵏ -1=0 is 31, then sum of the root is

a) -10 b) 5 c) -8 d) none

27) The number of real roots of

(7+ 4 √3)|ˣ|⁻⁸ + (7- 4 √3)|ˣ|⁻⁸ = 14 is

a) 0 b) 2 c) 4 d) none

28) Sum of all the values of x satisfying the equation

log₁₇log₁₁(√(x +11) + √x)= 0 is

a) 25 b) 36 c) 171 d) 0

29) let α, β be the roots of the equation (x - a)(x - b)= c with c≠ 0. then the roots of the equation (x - α)(x - β)+ c= 0 are

a) a,c b) b,c c) a, b d) a+ c, b+ c

30) If p,q are roots of x²+ px + q= 0, then

a) p=1 b) p= 1 or 0 c) p= -2 d) p= -2 or 0

31) The equation √(x +1) - √(x -1)= √(4x -1), (x ∈R)

a) no solution b) one solution c) two solution d) more than two solutions

32) The sum of all the real roots of the equation

|x -2|²+ |x -2| - 2= 0 is

a) 7 b) 4 c) 1 d) none

33) Let p and q be the roots of x²- 2x + A= 0 and r and s be the roots of x²- 18x + B= 0. If p< q < r< s are in AP, then ordered pair (A, B) is equal to

a) (-3,77) b) (77,-3) c) (-3,-77) d) none

34) In a triangle PQR, angle R= π/2. If tan(P/2) and tan(Q/2) are the roots of the equation ax² + bx + c=0 where a≠ 0, then

a) a+ b= c b) b+ c= a c) a+ c= b d) b= c

35) If α, β (α> β) are the roots of the equation x²+ bx + c= 0 c< 0< b, then

a) 0< α < β b) 0< α < β<|α| c) α < β<0 d) α < 0< |α|< β

36) For the equation 3x²+ px + 3= 0 , p> 0, if one of the roots is square of the other, than p is equals to

a) 1/3 b) 1 c) 3 d) 2/3

37) If the roots are the equation x² - 2ax + a²- 3 = 0 are real and less than 3, then

a) a<2 b) 2≤ a≤3 c) 3<a ≤4 d) a> 4

38) If b> a, then the equation (x - a)+x - b) -1= 0 has

a) both roots in [a, b]

b) both roots in (- ∞, a)

c) both roots in (b, ∞)

d) one root in (-∞,a) and other in (b, ∞).

39) Let α, β be the roots of x² - x + p = 0 and γ, δ be the roots of x² - 4x + q = 0 . If α, β, γ, δ are in GP then the integral value of p and q respectively, are

a) -2,- 32 b) -2,3 c) -6,3 d) -6, -32

40) If a, b, c are not all equal and α and β be the roots of the equation ax² + bx + c = 0, then value of (1+ α+ α²)(1+ β+ β²) is

a) 0 b) positive c) negative d) non negative

41) If a,b,c are in AP and if the equations

(b - c)x²+ (c - a)x + (a - b)= 0 and

2(c + a)x²+ (b + c)x = 0 have a common root, then

a) a², b², c² are in AP

b) a², c², b² are in AP

c) c², a², b² are in AP d) none

42) Value of

x= √[6+ √{6+ √{6+....up to

a) 3 b) 2 c) 1 d) none

43) two complex numbers α and β are such that α + β = 2 and α⁴+ β⁴= 272, then the quadratic equation whose roots are α and β is

a) x²-2x -16= 0

b) x²-2x + 12= 0

c) x²-2x -8= 0 d) none

44) The equation (cos p -1)x² + (cos p)x + sin p = 0 in variable x has real roots, if p belongs to the interval

a) 0,2π) b) (-π,0) c) (-π/2,π/2) d) (0,π)

45) If the roots of the equation

1/(x + a) + 1/(x + b) = 1/c are equal in magnitude but opposite in sign, then their product is

a) (1/2) (a²+ b²) b) - (1/2) (a²+ b²) c) ab/2 d) -ab/2

46) If the quadratic equations x²-11x + a= 0 and x²-14x + 2a= 0 have common root, then the values of a are

a) 0, 24 b) 0,-24 c) 1,-1 d) -2,1

47) If α, β are the roots of the equation ax² + bx + c = 0, then the value of α³+ β³ is

a) (3abc+ b³)/a³

b) (a³+ b³)/3abc

c) (3abc- b³)/a³

d) - (3abc+ b³)/a³

48) If the sum of the roots of the quadratic equations ax² + bx + c = 0 is equal to the sum of the squares of their reciprocals, then

a) ab², ca², bc² are in AP

b) ab², bc², ca² are in AP

c) ab², bc², ca² are in AP d) none

49) If the ratio of the roots of the equation x² + bx + c = 0 is the same as that of the ratio of the roots of x² + qx + r = 0, then

a) br²= qc² b) cq²= rb² c) q²c²= b²r² d) Bpbq= rc

50) If a, b are the non zero distinct roots of x² + ax + b = 0, then the least value of x² + ax + b is

a) 2/3 b) 9/4 c) -9/4 d) 1

51) If a+ b+ c= 0, then the quadratic equation 3ax² + 2bx + c = 0 has

a) at least one root in [0,1]

b) one root in [2,3] and other is [-2, -1]

c) imaginary roots d) none

52) If a< b < c< d, then the equation 3(x - a)(x - c)+ 5(x - b)(x - d)=0

a) real and distinct rootes

b) real and equal roots

c) purely imaginary roots d) none

53) For real x, the function (x - a)(x - c)/(x - b) will assume all real values provided

a) a< b < c b) b< c < a c) c< a < cpb d) none

54) Let a,b,c ∈R and a≠ 0. If α is a root of a²x²+ bx + c= 0, β is a root of a²x²- bx - c= 0 and 0< α < β, then the equation a²x²+ 2bx + 2c= 0 has a root γ that always satisfies

a) γ= (1/2) (α + β) b) γ= α + β/2 c) γ= (α + β) d) α <γ< β

55) Suppose p,q,r,s ∈R and α, β be the roots of x²+ px + q= 0 and α⁴, β⁴ be the roots of x²- rx + s= 0, then the equation x²- 4qx + 2q² - r= 0 has always

a) two imaginary roots

b) two positive roots

c) two negative roots

d) one positive and one negative root

56) The equation

x⁽³/⁴⁾⁽ˡᵒᵍ₂ˣ⁾^²⁺ ˡᵒᵍ₂ˣ ⁻ ⁵/⁴= √2 has

a) exactly two real roots

b) no real root

c) one irrational root d) none

57) Let f(x) be a quadratic expression which is positive for all x, if g(x)= f(x)+ f'(x) then for all real x,

a) g(x)< 0 b) g(x)> 0 c)g(x) = 0 d) g(x)≥ 0

58) If α, β are the roots of ax²+ bx + c= 0, then the quadric equation whose roots are 2α+3 and 2β+3 is

a) 4ax² - 3bx + c= 0

b) 6a¹x² - 4abx + 6c= 0

c) ax² +2(b- 3a)x + 9a+ 2b= 0 d) none

59) If α, β are the roots of the equation ax²+ bx + c= 0, then the equation whose roots are α³, β³ is

a) a³y²+ (b³- 3abc)y+ c³= 0

b) a³y²+ (3abc - b³)y- c³= 0

c) a²y²+ 2aby+ c³= 0 d) none

60) If sinα and cosα are the roots of 25x²+ 5x -12= 0, then value of sin2α is

a) 12/25 b) -12/25 c) -24/25 d) 4/5

61) Let P(x) be a polynomial with integral coefficients . If there exist two integers a and b such that P(a) - P(b)= 1, then

a) both a and b must be even

b) both a and b must be odd

c) a and b are two consecutive integers d) none

62) Let am b, c be non zero real such that

¹₀∫ (1+ cos⁸x)+ax²+ bx+ c) dx

²₀∫ (1+ cos⁸x)+ax²+ bx+ c) dx

Then the quadratic equation has ac¹+ bx + c= 0 has

a) no root in (0,2)

b) at least one root in (1,2)

c) a double root +0,2) d) none

63) If a, b, c are distinct real numbers, then the expression

f(x)= a²{(x - b)(x - c)}/{(a- b)(a - c)} + b²{(x - c)(x - a)}/{(b- c)(b - a)} + c²{(x - a)(x - b)}/{(c- a)(c - b)} is identically equal to

a) x²- (a + b + c)x + abc

b) x² + x - abc c) x² d) none

64) The number of real solutions of the equation

27¹⁾ˣ + 12¹⁾ˣ = 2(8¹⁾ˣ) is

a) 0 b) 1 c) infinite d) none

65) If 0< a < b< c < d, then the quadratic equations ax²+ {1+ a(b + c)}x + abc - d = 0 has

a) real and distinct roots out of which one lies between c and d

b) real and distinct roots out of which one lies between a and b

c) real and distinct roots out of which one lies between b and c

d) nonreal roots

TRIGONOMETRICAL RATIOS AND IDENTITIES

SAP- 1

1) 2(sin⁶x + cos⁶x) - 3(sin⁴x+ cos⁴x)+ 1=0

2) 3[sin⁴(3π/2 - x) + sin⁴(3π+ x)] - 2[sin⁶(π/2+ x) + sin⁶(5π- x)] is equal to

a) 0 b) 1 c) 3 d) sin4x + sin6x e) none

3) sin⁶x + cos⁶x + 3sin²x cos²x = 1

4) 3(sinx - cosx)⁴ + 6(sinx + cosx)² + 4(sin⁶x + cos⁶x) is independent of x.

5) (sin⁸x - cos⁸x)= (sin²x - cos²x)(1- 2 sin²x cos²x).

6) (3+ cos4x) cos2x= 4(cos⁸x - sin⁸x).

7) If sinx+ cosx= a, then find the values of|sinx - cosx| and cos⁴x + sin⁴x.

8) If sinx + cosecx = 2, then sin²x + cosec²x is equal to 2. T/F

9) f(x)= cos²x + sec²x≥ 2. T/F

Or minimum value of f(x) is 2.

10) Given A= sin²x + cos⁴x, then for all real x.

a) 1≤ A≤2 b) 3/4≤A ≤1 c) 13/16 ≤A ≤1 d) 3/4 ≤A ≤ 13/16

11) Let A= sin⁸x + cos¹⁴x, then for all real x

a) A≥ 1 b) 0< A ≤1 c) 1/2< A ≤ 3/2 d) none

12) If x, y are acute, sinx= 1/2, cos y= 1/3, then (x + y) belong to

a) (π/3,π/2) b) (π/2,2π/3) c) (2π/3,5π/6) d) (5π/6,π)

13) (tanx + cot x)²= sec²x + cosec²x = sec²x cosec²x.

14) (1+ tan x tan y)² + (tanx - tan y)² = sec²x sec²y.

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

16) 1/(secx - tan x) - 1/cosx = 1/cosx - 1/(secx + tanx).

17) (secx + tan x -1)(secx - tanx +1) - 2 tan x= 0

18) If (secx + tanx)(sec y + tan y)(sec z + tan z)= (secx - tan x)(sec y - tan y)(sec z - tan z) show that each of the side is equal to ±1.

19) If (1+ sinx)(1+ sin y)(1+ sin z)= (1- sin x)(1- sin y)(1- sin z), show that each side is equal to ± cosx cos y cos z.

20) Let f(x)= sinx (sinx + sin3x). Then f(x).

a) ≥ 0 only when x≥ 0

b) ≤ 0 for all real x

c) ≥0 for all real x

d) ≤ 0 only when x ≤ 0

21) The maximum value of (cosx₁). (cosx₂)......(cosxₙ), under the restriction 0≤ x₁, x₂, .....xₙ≤ π/2 and (cotx₁).(Cotx₂).....(cotxₙ)= 1 is

a) 1/2ⁿ⁾² b) 1/2ⁿ c) 1/2n d) 1

22) √{(1- sinx)/(1+ sinx)}= secx - tan x.

23) √{(1+ cosx)/(1 - cosx)}= cosecx + cotx.

24) If sinx + sin²x= 1, then show that cos¹²x + 3 cos¹⁰x + 3 cos⁸x + cos⁶x -1= 0

25) If sinx+ sin²x + sin³x = 1, then cos⁶x - 4cos⁴x + 8cos²x = _____.

26) sec⁴x (1- sin⁴x) - 2 tan²x = 1.

27) tan²x - sin²x = sin⁴x sec²x= tan²x sin²x.

28) (cotx + tant)/(cot y + tanx)= cotx tan y.

29) (sinx + cosx)(tanx + cotx)= secx + cosecx

30) (cosx cosecx - sinx secx)/(cosx + sinx)= cosecx - secx.

31) (1+ cotx - cosecx)(1+ tanx + secx)= 2

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

33) (tanx + secx -1)/(tanx - secx +1)= (1+ sinx)/cosx.

34) cot²x(secx -1)/(1+ sinx) = sec²x. (1- sinx)/(1+ secx).

35) (secx +1- tanx)/(tanx - secx +1)= (1+ cosx)/sinx.

36) cosx/(1- tanx) + sinx/(1- cotx)= sinx + cosx.

37) tₙ= sinⁿx + cosⁿx, then (t₃ - t₅)/t₁ = (t₅ - t₇)/t₃.

38) tanx/(1- cotx) + cotx/(1- tanx)= secx cosecx +1.

39) (sinx + cosecx)²+ (cosx + secx)²= tan²x + cot²x +7.

40) (1+ cotx + tanx)(sinx - cosx)= secx/cosec²x - cosecx/sec²x.

41) (secx - cosecx)(1+ tanx + cotx)= tanx secx - cotx cosecx.

42) {2sinx tanx(1- tanx)+ 2 sinx sec²x}/(1+ tanx)²= 2sinx/(1+ tanx).

43) (tanx + cosec y)²+ (cot y - secx)²= 2 tanx cot y(cosecx + sec y).

44) {(1+ sinx - cosx)/(1+ sinx + cosx)}²= (1- cosx)/(1+ cosx).

45) If 2sinx/(1+ cosx + sinx)= y, then (1- cosx + sinx)/(1+ sinx) is also y.

46) {1/(sec²x - cos²x) + 1/(cosec²x - sin²x)}. sin²x cos²x = (1- sin²x cos²x)/(2+ sin²x cos²x).

47) (cosecx - secx)(cotx - tanx)= (cosecx + secx)(secx cosecx -2).

48) If tanx+ sinx = m and tanx - sinx = n, then show that m²- n² = 4√(mn).

49) Eliminate x from the relations

a secx = 1- b tan x and a² sec²x = 5+ b² tan²x.

50) If cosecx - sinx = m, secx - cosx = n, eliminate x.

51) If cosecx - sinx = a³, secx - cosx= b³, then a²b²(a² + b²)= 1.

52) If cotx + tanx = a, secx - cosx = b eliminate x.

53) If c cos³x + 3c cosx sin²x = m, c sin³x + 3c cos²x sinx = n, then show that (m + n)²⁾³ + (m - n)²⁾³= 2c²⁾³.

54) If cosx + sinx= √2 cosx, show that cosx - sinx =√2 sinx.

55) If 3 sinx + 5 cosx = 5, show that 5 sinx - 3 cosx = ±3.

56) If a cosx + b sin x = p, a sinx - b cosx = q, show that a² + b² = p² + q².

57) If a cosx - b sin x = c, show that a sinx + b cosx = ±√(a² + b² + c²).

58) If a sinx + b cosx = c, then show that (a - b tanx)/(b + a tanx)= ±√(a² + b² + c²)/c.

59) If tan²x = (1- e²), show that secx + tan³x cosecx = (2- e²)³⁾².

60) If ax/cosθ + by/sinθ = (a²- b²) and (ax sinθ)/cos²θ - (by cosθ)/sin²θ = 0, show that (ax)²⁾³ + (by)²⁾³= (a² - b²)²⁾³.

61) If sinθ = (m² - n²)/(m²+ n²), determine the values of tanθ, secθ, cosecθ.

62) If tanθ = 2x(x+1)/(2x +1), determine sinθ and cosθ.

63) If cosθ = 2x/(1+ x²), find the values of tanθ and cosecθ.

64) If secx = p + 1/4p, then secx + tanx = 2p or 1/p

65) If secθ + tanθ = p, obtain the values of secθ, tanθ, sinθ in terms of p.

66) If cosx/cos y = a, sinx/sin y = b, then (a² - b²)sin²y= a² -1

67) If tanθ = p/q, show that (p sinθ - q cosθ)/(p sinθ + q cosθ) = (p² - q²)/(p² + q²).

68) Is the equation sec²θ= 4xy/(x + y)² possible for real values of x and y ?

If not, then find out a relation between x and y so that it may be possible.

69) If m² + m'² + 2mm' cosθ = 1,

n² + n'² + 2nn' cosθ = 1 and mn + m'n' + (mn' + m'n) cosθ = 0 show that m² + n² = cosec²θ.

SAP-2

1) The value of sin⁶θ + cos⁶θ + 3 sin²θ cos²θ is

a) 0 b) 1 c) 2 d) 3

2) The least value of 2sin²θ+ 3 cos²θ is

a) 1 b) 2 c) 3 d) 5

3) The greatest value of sin⁴θ + cos⁴θ is

a) 1/2 b) 1 c) 2 d) 3

4) The value of sin²θ cos²θ(sec²θ+ cosec²θ) is

a) 2 b) 4 c) 1 d) 0

5) If sinθ + cosecθ = 2, then sin²θ + cosec²θ is equal to

a) 1 b) 4 c) 2 d) none

6) For how many values of x between 0 and 2π is the equation

2cosec2x cotx - cot²x = 1 valid ?

a) 0 b) 2 c) 1 d) none

7) Incorrect statement is

a) sinθ= -1/5 b) cosθ= 1 c) secθ= 1/2 d) tanθ= 20

TRUE OR FALSE

8) sec²θ= 4xy/(x + y)² is true if and only if

a) x+ y≠ 0 b) x= y, x≠ 0 c) x= y d) x≠ 0, y≠ 0

9) If x= a cos²θ sinθ and y= a sin²θ cosθ, then (x² + y²)³/(x²y²( is independent of θ.

10) The inequality ₂sin²θ + ₂cos²θ≥ 2√2 holds for all real θ.

11) The equation sinθ = x + 1/x holds true for all real θ.

FILL IN THE BLANK

12) The least value of tan²θ + cot²θ is _____

13) The value of sinθ cosθ(tanθ + cotθ) is ____

14) If for real x, the equation x+ 1/x = 2 cosθ holds, then cosθ= ____

15) If cosecθ - cotθ = q, then the value of cosecθ = _____

SET THEORY

1) Set: A set is a collection of well defined and well distinguished objects of our perception or thought.

The words 'well defined objects ' imply that we must be given a rule with the help of which we should readily be able to say whether a particular object 'belongs to' the set of not. The words 'well distinguished objects ' imply that if the objects of the collection be named, then in doing so, the number of objects will not increase.

The set are usually denoted by capital letters of English alphabet viz, A, B, CA

2) ELEMENTS

The objects, which constitute the set, are said to be elements of the set.

These are also known as members or points of the set. The elements are usually denoted by small letters of English alphabet viz, a, b, c,....

i) If a is an element of the set A, we write it as a ∈A and is ready as " a belongs to A"

ii) If a is not an element of the set A, we write it as a∉ A and is ready as " a does not belong to A"

3) REPRESENTATION OF SETS

There are two methods to represent a set.

a) Roster /Tabulation Method. In this method, the set is represented by listing all its elements, saparating the elements by commas and enclosing them in curvilinear brackets.

b) Defining Property Method. In this method, the set is represented by specifying the common property of the elements.

Thus the set A is represented by A= {a: P(A) is true}.

Here 'a' stands for 'an arbitrary elements' of the set' and (:) stands for 'such that' and P(A) stands for 'common property '

4) FINITE AND INFINITE SETS

a) finite set. If it has finite number of elements.

b) Infinite set. If it has an infinite number of elements.

Order of a finite set is the number of elements it contains.

The order of a finite set A is denoted by O(A).

5) EMPTY SET

A set having no element is....

It is also called Null set or Void set.

6) SINGLETON SET

A set having only one element is....

7) SUB-SETS

a) Subset. Let A and B be two sets. Then the set A is said to be a subset of the set B if each element of A is also an element of B.

Symbolically, we write it as A⊆ B.

Here B is superset of A and is written as B⊇A.

b) Proper Subset. A set A is a proper subset of B if and only if each element of A is in B and there is atleast one element in B, which is not in A.

Symbolically, if A is a proper subset of B, then A ⊂B and A≠ B or A ⊂ B

8) COMPARABLE SETS

Two sets are said to be comparable iff either A ⊂B or B⊃A.

9) EQUAL AND EQUIVALENT SETS

a) Equal Sets. Two sets A and B are said to be equal (written as A= B) iff A ⊂B and B ⊂A.

Two sets A and B are said to be equal if they have exactly same elements.

b) Equivalent Sets , Two sets are said to be equivalent if they have same number of elements.

10) FAMILY OF SETS

A set said to be family of sets if it's elements are also sets.

This is also known as set of sets.

If A= {a,b}, then S={φ, {a}, {b}, {a,b} is the set of sets.

11) POWER SET

The set of all possible subsets of a set A is said to be the power set of A and is denoted by P(A).

If a= P{a,b,c}, then P(A)={φ, {a}{b}, {c}, {a,b}, {b,c}, {a,c}, {a,b,c}}.

12) UNIVERSAL SET

The main set under discussion or the set containing all possible values in the given frame of reference is said to be universal set and is denoted by U or E or X.

13) OPERATIONS IN SETS

a) Union of Sets.

(I) Let A and B be two sets. The union of A and B (denoted by A U B) is the set of all those elements which are either in A or in B or in both.

Symbolically, A∪B= x: x ∈A or x ∈ B}.

II) Let A₁, A₂,..., Aₙ be n(≥2) sets. Then the union of these (denoted by ⁿᵢ₌₁U Aᵢ) is the set of all those elements which are in Aᵢ (1≤ i≤ n) for atleast one value of i.

b) Intersection of sets.

I) Let A and B be two sets. The intersection of A and B (denoted by A∩B) is the set of all those elements which are in both A and B.

II) Let A₁, A₂, .....,Aₙ be n(≥2) sets. Then the intersection of these (denoted by ⁿᵢ₌₁ ∩Aᵢ) is the set of all those elements which are in Aᵢ(1≤ i≤ n) for each i.

14) FUNDAMENTAL RESULTS

i) Identity Laws, A ∪φ, A ∩φ= φ

ii) Idempotent Laws. A ∪A= A, A ∩A = A

iii) Commutative Laws. A ∪ B= B ∪ A, A ∩ B= B∩ A

iv) Associative Laws. A∪(B ∪C)= (A∪B) ∪ C

A∩(B∩C)= (A∩B)∩C

c) Distributive Laws, A∪ +B∩C)= (A∪B)∩(A ∪ C)

A∩(B∪C)= (A∩B)∪(A∩C).

15) Disjoit Sets

If and only if they have no common element.

Let A and B be two sets

Here A ∩B= φ.

16) DIFFERENCE OF SETS

Let A and B be two sets. Then (A - B) is the set of those elements of the set A which are not in the set B.

Symbolically. A - B= {x: x ∈A and x ∉B}

Similarly, B - A ={x: x ∈B and x ∉A}

17) SYMMETRIC DIFFERENCE OF SETS

Let A and B be two sets. Then their symmetric difference is the union of the sets A- B and B- A. This is denoted by A ∆B.

Symbolically

A∆B ={x: x ∈A - B or x ∈B - A}

18) COMPLEMENT OF A SET

Let X be the universal set and A be any set. Then the complement of the set A is the set of all those elements of X, which are not in the set A.

This is denoted by Aᶜ or A' or X - A

Symbolically, Aᶜ={x: x∈X and x ∉A}.

19) FUNDAMENTAL RESULTS

a) Xᶜ= φ, φᶜ= X

b) (Aᶜ)ᶜ= A

c) If A ⊆ B, then B⊆Aᶜ

d) A U Aᶜ= X and A∩Aᶜ = φ

e) De-morgan's Laws. (AUB)ᶜ = Aᶜ∩ Bᶜ

(A∩ B)ᶜ= AᶜU Bᶜ

20) USE OF SETS IN PRACTICAL PROBLEMS

If A, B, C are finite sets of n elements each, then

a) n(AUB)= n(A)+ n(B); if A, B are disjoint

b) n(AUB)= n(A)+ n(B) - n(A∩B); if A, B are not disjoint

c) n(A∩Bᶜ)= n(A) - n(A∩B)

d) n(B∩Aᶜ)= n(B)- n(A∩B)

e) n(AUB)= n(A∩Bᶜ)+ n(B∩Aᶜ)+ n(A∩B).

f) n(AUBUC)= n(A)+ n(B)+ n(C) - n(A∩B) - n(B ∩C) - n(A∩C)+ n(A∩B∩C).

21) ORDERED PAIR

An ordered pair is a pair of entries in the specified order.

In the ordered pair (a,b), a is the first element and b the second element.

22) CARTESIAN PRODUCT/DIRECT PRODUCT OF SETS

a) The set of all ordered pairs of elements (a,b); a ∈A, b ∈ B is called Cartesian product of two sets A and B and is denoted by A x B

Symbolically, A x B={(a,b): a ∈A, b∈B}.

b) The Cartesian product of n(>2) sets A₁, A₂,......Aₙ is the set of all ordered n-triples (a₁, a₂, .......aₙᵢ), where aᵢ∈ Aᵢ (1≤ i≤ a) and is denoted by A₁x A₂ x......x Aₙ or ⁿᵢ₌₁ΠAᵢ.

Symbolically, ⁿᵢ₌₁ΠAᵢ= {(a₁, a₂,......,aₙ): aₙ ∈ Aₙ, 1≤ i≤ n}.

23) FUNDAMENTAL RESULTS

a) Ax B≠ B x A

b) A x φ= φ x = φ

c) n(Ax B)= n(B x A)= n(A) x n(B)

d) If A ⊆B, C ⊆ D, then A x C ⊆ B x D

e) n(A₁x A₂ x......x Aₙ)= n(A₁) x n(A₂)......xXₙ (Aₙ)

IN FOCUS

ⱼ

φΠ ᵢ

φφ

ᶜᶜᶜᶜφ

∪∪∪∪

ⱼ∉∆

₁₂ₙᵢⱼ∉∆

⊆⊇⊂∩∪∀≺∩⊃⊄⊅¹²₁₂ₙⁿᵢ₌₁ᵢ₁₂₌₁ᵢ₁₂ₙᵢⱼ∉∆ₙⁿᵢ

1) Let N be the set of non-negative integers, I the set of integers Nₚ the set of non positive integers, E the set of even integers and P the set of prime numbers. Then

a) I - N= Nₚ

b) N ₚ∩ Nₚ = φ

c) E ∩C = φ

d) N ∆Nₚ= I - {0}

2) Let A and B be two sets, then (AUB)ᶜ U(Aᶜ∩B) equals

a) Aᶜ b) Bᶜ c) A d) none

3) If A and B are two sets , then A∩(AUB)ᶜ equals

a) A B) B C) φ d) none

4) The set (A∩Bᶜ)ᶜ U (B∩C) equals

a) Aᶜ U Bᶜ b) Aᶜ U Cᶜ c) Aᶜ U B U C d) none

5) Let U be the universal set and AU BU C= U Then [(A - B) U(B - C) U(C A)]ᶜ equals

a) AUBUC b) A∩B∩C c) AU(B∩C) d) A∩(B UC)

6) The set (AUBUC)∩(A∩Bᶜ∩Cᶜ)∩Cᶜ equals

a) A∩C b) B U Cᶜ c) B ∩Cᶜ d) none

7) If A and B are two sets, then A ∩(AUB) equal

a) A b) B c) Aᶜ d) Bᶜ

8) If A={1,3,5,7,9,11,13,15,17}, B={2,3,.....18} and N is the universal set, then Aᶜ U ((A U B)∩ Bᶜ is

a) A B) N c) B d) none

9) If A and B are disjoint nonempty sets, then A - (A - B) equals

a) A B) B C) φ d) A U B

10) Which of the following is empty set?

a) {x: x is a real number and x² -1=0}

b) {x: x is a real number and x² +1=0}

c) {x: x is a real number and x² -9 =0}

d) {x: x is a real number and x²= x +2}

11) Which of the following is a singleton set?

a) {x: |x|=5, x ∈ I}

b) {x: |x|=6, x ∈ N}

c) {x: x²=5, x ∈ N}

d) {x: x²+ 3x +2=0, x ∈ N}

12) Which of the following does not have a proper subset:

a) {x: x ∈ N, 4< x<5}

b) {x: x ∈ Q}

c) {x: x ∈ WpQ, 4< x<5} d) none

13) If A, B and C are any three sets, then A - (BUC) equals

a) (A-- B) U(A - C)

b) (A- B) ∩(A - C)

c) (A-- B) U C

d) (A - B) ∩ C

14) If A, B, C are three sets, then A∩(B U C) equals

a) (AUB) ∩(AUC)

b) (A∩B) U(A∩C)

c) (AUB) U (AU C) d) none

15) If Q={x: x= 1/y, where x ∈N} then:

a) 0∈Q b) I∈ Q c) 2∈ Q d) 2/3 ∈Q

16) If A={x: x∈I, -2≤ x ≤ 2}, B==x: x∈ I, 0≤ x ≤ 3}

C={x: x ∈N, 1≤ x ≤ 2}

D={(x,y): (x,y) ∈N x N, x+ y= 8}, then

a) n(BUC)= 5

b) n(D)= 6

c) n(AU(BUC)(= 5 d) none

17) If for α∈ N, αN= {αx: x ∈ N}, then the set 8N ∩6N is

a) 8N b) 12N c) 24N d) 48N

18) Let n(A)= 3 and n(B)= 6 and A B. Then the number of elements in A ∩B is

a) 3 b) 9 c) 6 d) none

19) Sets A and B have 3 and 6 elements respectively. What can be the minimum number of elements in AUB ?

a) 3 b) 6 c) 9 d) 18

20) Two finite sets have m and n elements. The total number of subsets of the first set is 56 more than the total number of subsets of second set. The values of m and n are

a) 7,6 b) 6,3 c) 5,1 d) 8,7

21) If A, B and C are any three sets, then Ax (BUC) is

a) (AxB) U (Ax C)

b) (A UB) x (AU C)

c) (Ax B) ∩(Ax C)

d) none

22) If A, B and C are any three sets, then A x (B ∩C) is

a) (Ax B) U(Ax C)

b) (Ax B)∩(Ax C)

c) (AUB) x (AUC)

d) (A∩B) x (A∩C)

23) If S₁={1,2,3,.....20}, S₂= {a,b,c,d}, S₃={b,d,e,f}. The number of elements of (S₁x S₂) U(S₁x S₃) is

a) 100 b) 120 c) 100 d) 40

24) If A={1,2,3,6,11,18,21}, B={5,7,9} and N is the universal set, then Aᶜ U(A U B) ∩ Bᶜ equals

a) A b) B C) N d) N - A

25) The set (AUB)∩(A ∩Bᶜ∩Cᶜ)ᶜ ∩Cᶜ equals

a) A∩C b) B∩Cᶜ c) Bᶜ∩Cᶜ d) none

26) Suppose A₁, A₂, ......A₃₀ are thirty sets, each having 5 elements and B₁, B₂, .....Bₙ are n sets, each element of S belongs to exactly 10 if the Aᵢ's and exactly 9 of Bᵢ's. Then n is equal to

a) 15 b) 3 c) 45 d) none

27) Consider the set of all determinants of order 3 with entries 0 or 1 only. Let B be the subset of A consisting of all determinants with value 1. Let C be the subset of the set of all determinants with value -1. Then

a) C is empty

b) B has as many elements as C has

c) A= B U C

d) B has twice as many elements as C has

28) If (1,3),(2,5) and (3,3) are the elements of Ax B and the total number of elements in A x B is 6, then the remaining elements of Ax B are

a) (1,5),(2,3),(3,5)

b) (5,1),(3,2),(5,3)

c) (1,5),(2,3),(5,3) d) none

29) If A, B, C be three sets such that AUB = AUC and A∩B= A∩C, then

a) A= B b) B= c c) A= C d) A= B = C

30) Let A={x,y): y= eˣ, x∈ R}, B= [(x,y): y= e⁻ˣ, x ∈ R}. Then

a) A∩B= φ b) A∩B≠ φ c) AUB= R d) none

31) Let A={(x,y): y= eˣ, x ∈R}, B={(x,y): y= x, x∈ R}. Then

a) B⊆ A B) A ⊆ B C) A∩B= ∪φ d) AU B = A

32) If X={4ⁿ - 3n -1: n∈ N} and Y={9(n -1); n∈ N} then X U Y is

a) X b) Y c) N d) none

33) If X= {8ⁿ - 7n -1: n∈ N} and Y= {49(n -1)| n ∈ N}, thn

a) X⊂ Y b) Y⊂ X c) X= Y d) none

34) If the sets A and B are defined as:

A={(x,y): y= 1/x, 0 ≠ x R}

B={(x,y): y= - x, x ∈ R}, then:

a) A∩B= A b) A∩B= B c) A∩B= φ d) none

35) If A={φ, {φ}}, then the power set of A is

a) A B) {φ, {φ}, A} c) {φ, {φ}, {{φ}}A} d) none

36) In a class of 100 students, 55 students have passed in mathematics and 67 students have passed in Physics. Then the number of students who have passed in Physics only is

a) 22 b) 33 c) 10 d) 45

37) Out of 800 boys in a school, 224 played cricket, 240 played hockey and 336 played basketball. Of the total, 64 played both basketball and hockey, 80 played cricket and hockey; 24 played all the three games. The number of boys which did not play any game is

a) 128 b) 216 c) 240 d) 160

38) From 50 students taking examinations in mathematics, physics and chemistry, 37 passed mathematics, 24 physics and 43 chemistry. Atmost 19 passed mathematics and physics, atmost 29 mathematics and chemistry and atmost 20 physics and chemistry. The largest possible number that could have passed all three examination is

a) 9 b) 10 c) 12 d) none

39) Of the members of three athletic teams in a school 21 are in the cricket team, 26 are in the hockey team and 29 are in the football team. Among them, 14 play hockey and cricket, 15 play hockey and football, and 12 play football and cricket. Eight play all the three games. The total number of members in the three athletic teams is

a) 43 b) 49 c) 76 d) none

40) In a college of 300 students, every student reads 5 newspaper and every newspaper is ready by 60 students. The number of newspaper is

a) atleast 30

b) atmost 20

c) exactly 25 d) none

Wednesday, 25 March 2026

REVISION - X (26/27)

LINEAR INEQUATIONS

1) If 2x - 7 < 4, where x is a natural number less than 8, than the solution set is:

a) {0,1,2,3,4} b) {1,2,3,4,5} c) {1,2,3,4,5,6} d) {0,1, 2,3,4,5,6}

2) If - x ≥ -3 then:

a) x≤ -3 b) x ≥ 3 c) x = 3 d) x ≤ 3

3) if 2 + 4 x< 2 x - 5 ≤ 3x ∈Z, then the solution set is :

a) {5,4} b) {- 5,-4} c) {- 5, -4, -3} d) {- 4,-3,-2,-1}

4) if 2≤2x - 3 ≤ 5, x∈ R, then the solution set is:

a) {2.5≤ x ≤ 4, x∈ R} b) {2≤ x ≤ 5, x∈ R} c) {3≤ x ≤ 5, x∈ R} d) {2< x < 4, x∈ R}

5) If a> b, then:

a) a - c ≤ b - c b) a - c≥ b - c c) a - c = b - c d) a - c > b - c

6) If x≥ 5 and- ax ≥ 5a, then :

a) a > 0 b) a < 0 c) both a and b d) neither a nor b

7) If x+1≥ 13 - 5x, x ∈{1,2,3,4.....10}, then the solution set is:

a) {1,2,3,4,5,6} b) {6,7,8,9,10} c) {7,8,9,10} d) {6,7,8.....}

8) If 7 - 5x ≥ 3x -1, then the solution set, when x ∈ W is:

a) {0,1} b) {0} c) {1} d) {0,1,2}

9) Given a >0, b >0, c >0 and d <0, then a < b implies :

a) a+ d> b + d b) a - d < b - d c) a - d > b - d d) a + d = b + d

10) Given 2x - 5≤ 5x +4 < 11. If x ∈ I, the solution set is:

a) {-2,-1,0,1} b) {-3,-2,-1,0,1} c) {-3,-2,-1,0} d) {-2,-1,0,1}

11) For the inequation -12< 3 - 4x ≤ 11, x ∈ N, the solution set on the number line can be shown as:

12) If 23> 3 + 4x ≥ -1, x ∈ R, then the greatest integer value of x is:

a) 5 b) 4 c) 3 d) 2

13) If 2x - 5≤ 5x + 4 < 11, x ∈ I, then the solution set can be represented as:

14) If 2x -3< x +1 ≤ 4x +7, x ∈ R, then the smallest integer value of x is:

a) -2 b) -1 c) 0 d) 1

15) If -9(x -7)≥ 45 - 21x > x +1, x ∈ R, then the solution set is:

a) {-3/2≤ x < 2, x ∈R}

b) {-3/2 < x < 2, x ∈R}

c) {-2/3 ≤ x ≤ 1, x ∈R}

d) {-1/3 ≤ x ≤ 2, x ∈R}

16) If 2x - 5 ≤ 5x + 4 < 11, x ∈ I, then the smallest whole number for x is:

a) 0 b) 1 c) -3 d) 2

17) If 5 - 3x < 11, x ∈ R, then the solution set is:

a) {x> -2, x∈R} b) {x≥ -2, x∈R} c) {x< 2, x∈R} d) {x< -2, x∈R}

18) Given 3x -1 ≤ x +5, x ∈N, then the solution set is:

a) {1,2,3} b) {1,2,3,4} c) {1,2} d) {0,1,2,3}

19) If 8 < 5(x +1) -2 ≤ 18, x ∈R, then the smallest integer value of x is:

a) 1 b) 0 c) -1 d) 2

20) Given a >0, b >0, c >0 and d <0. Then a > b implies:

a) ad >bd b) ad = bd c) ad < bd d) none

SHORT ANSWER TYPE QUESTIONS

1) Find the value of x, which satisfies the inquation -2≤ 1/2 - 2x/3 ≤ 11/6, x ∈N.

Graph the solution set on the number line.

2) Solve the following inequation, write the solution set and represent it on the number line.

-3(x -7)≥ 15 - 7x > (x +1)/3, x ∈R

3) Solve the following inequation, write down the solution set and represent it on the real number line:

-2+ 10x ≤ 13x +10 < 24+ 10x, x ∈ Z

4) Solve the following inequation and write down the solution set:

11x - 4 < 15x +4 ≤ 13x + 14, x ∈ W

Represent the solution on a real number line.

5) Solve the given inequation and graph the solution set on the number line:

2y - 3 < y +1 ≤ 4y +7, y ∈ R

6) Solve the following inequation and represent the solution set on the number line:

2x -5 ≤ 5x +4 < 11, x ∈I

7) Solve the following inequation and write the solution set:

13x -5 < 15x +4 < 7x +12, x∈R

1) Find the mean of the following distribution:

x: 4 6 9 10 15

f: 5 10 10 7 8

2) Following table shows the weights of 12 students:

Weight (in kgs): 67 70 72 73 75

No of students: 4 3 2 2 1

Find the mean weight.

3) Find the mean of the following distribution:

X: 10 30 50 70 89

F: 7 8 10 15 10.

4) If the mean of the following distribution is 6, find the value of p.

X: 2 4 6 10 p+5

F: 3 2 3 1 2

5) Find the value of p, if the mean of the following distribution is 7.5.

X: 3 5 7 9 11 13

F: 6 8 15 p 8 4

6) Find the missing frequencies in the following frequency distribution if it is known that the mean of the distribution is 1.46.

No of accidents(x): 0 1 2 3 4 5 total

Frequency (f): 46 ? ? 25 10 5 200

BOOSTER - C

1) The following table shows the weight of 12 students:

Weight (in kg): 67 70 72 73 75

No of students: 4 3 2 2 1

Find the mean weight. 70.25 kg

2) Find the mean wage from the data given below:

Wages: 800 820 860 900 920 980 1000

F: 7 14 19 25 20 10 5 891.2

3) Apply step-deviation method to find the AM of the distribution:

Variate (x) Frequency(f)

5 20

10 43

15 75

20 67

25 72

30 45

35 39

40 9

45 8

50 6 22.214

3) The weight in kilograms of 60 workers in a factory are given in the following frequency table. Find the mean weight of a worker.

Weight (in kg): 60. 61 62 63. 64 65

No of workers: 5 8 14 16 10 7 62.65

4) The table given the distribution of villages under different heights from sea level in a certain region. Compute the mean height of the region:

Height: 200 600 1000 1400 1800 2200

F: 142 265 560 271 89 16 984.51

CENTRAL TENDENCY

1) The weight of 6 persons in a firm are 64, 66,63,69,75,68 kg respectively. What is their mean weight?

a) 56.7 b) 67.5 c) 76.5 d) 65.7

2) The profit (in Rs) of a small shopkeeper of a week is 207,205,210,221,230,204,218. What is his mean profit per day?

a) 115 b) 225 c) 215 d) 125

3) The AM of 1,3,5,.......,29 is

a) 13 b) 15 c) 14 d) 16

4) The word length in each of the 40 words are as follows:

X: 2 3 4 5 7

F: 6 8 12 10 4

What is the mean of the above distribution?

a) 4.35 b) 4.05 c) 4.25 d) 4.15

5) The AM of a variable x is 40. What will be the mean of the variable y, when y= 4x - 10.

a) 160 b) 165 c) 155 d) 150

6) The mean age of a group of 30 girls is 20 years, and that of a group of 20 boys is 30 years. If the two groups are taken together to form a new group, what is the mean of this group?

a) 22 b) 24 c) 23 d) 25

7) The mean marks of 170 students in a certain class is 75. The mean mark of boys in the class is 85, and of the girls is 65. Find the number of boys and girls in the class.

a) 85,85 b) 65,105 c) 75,95 d) 80,90

8) The mean marks obtained in an examination by two gr of students were found to be 75, 85 respectively. What will be the ratio of students in the two groups, if the mean marks for all students was 100.

9) Suppose x takes the value 2,4,6,8,10,12. Then what will be the value of ∑(xᵢ - mean x)?

a) 0 b) 1 c) 2 d) 3

10) Sum of the deviation from mean is

a) 0 b) 1 c) mean of the number d) can't determine

11) For a set of 10 observations, what will be the mean for the values of x: 10,10,10,.....10.

a) 8 b) 9 c) 10 d) 11

12) The AM of 7 items is 10. If one more item is added to the series, then the AM becomes 12. Find the value of 8th item.

a) 24 b) 26 c) 28 d) none

13) What will be the weighted AM of the first n natural numbers when the numbers are weighted by the corresponding numbers?

a) (n+1)/3 b) (2n+1)/2 c) (2n+1)/3 d) (2n+3)/2

14) The AM of a set of values is 15. If 5 is added to each value, the new AM will be

a) 10 b) 15 c) 20 d) none

15) The AM of 1, 2,2²,......2⁹ is

a) 18.341 b) 18.431 c) 18.143 d) 18.413

16) The weighted AM of first 11 natural numbers whose weights are equal to the corresponding numbers, is

a) 6.77 b) 8.55 c) 7.66 d) none

17) The marks of 7 students in a test in statistics are 0,100,12,18,17,10,32. A suitable average of these marks is

a) Mean b) median c) mode d) none

18) If 3,9,6,7,10,8,4,1,5,2 are the observations, then find the median

a) 5 b) 5.5 c) 6.5 d) 6

19) Which quartile is the median

a) 1st quartile b) 3rd quartile c) 2nd quartile d) none

20) If the relation between two variables x and y be 3x + 7y=50, and median of y is 5. Then median of x is

a) 4 b) 5 c) 6 d) 7

21) If the mean and median are 24 and 26 respectively. Find mode

a) 27 b) 28 c) 29 d) 30

22) In usual symbols, mode= 3 median - a x mean, where a is

a) 1 b) 2 c) 3 d) none

23) How many quartiles are there

a) 1 b) 2 c) 3 d) 4

Matrix - Test

) If A= 2 0 & B= 1 2

1 2 2 1 find A+ B

2) If A= 1 2 3 & B= 1 2

4 5 6 3 4

find A+ B

3) If A= 0 2 & B= 7 8

2 1 1 4 find

a) 2A + 3B

b) 3A - B

c) AB

4) If A= 1 3

3 4 and A²- kA - 5=0, then find k.

5) If A= 1 0 & B= 0 1

0 1 -1 0

Then show that (aA+ bB)(cA+ dB)= (ac - bd)A+ (ad + bc)B.

BANKING

Sap-1

1) Amit deposited Rs 150 per month in a bank for 8 months under the recurring deposits. If the rate of interest is 8% p.a. and intrest is calculated at end of every month?

2) Laxmi took a cumulative time deposit account of Rs 240 per month at 10% p.a. she received Rs 3840 on maturity. Find the period for this account.

3) Manoj opened Recurring Deposit Account in a bank and deposited Rs 500 per month for 3 years. The bank on Rs 20220 on maturity. Find the rate of interest paid by the bank.

4) Raju opened a Recurring Deposit Account with the Bank of Rajasthan and deposits Rs 600 per month for 20 months. Calculate the maturity value of this account, if the bank pays interest at the rate of 10% per annum.

5) Miss Anshu Gupta deposited Rs 350 per month for 20 months under Recurring Deposit Scheme. Find the total amount payable by the bank on maturity of the account if the rate of interest is 11% per annum.

6) Mrs. Mathew opened Recurring Deposit Account in a bank with Rs 500 per month for 5/2 years. Find the amount she will get on the maturity. If the interest is paid on monthly balance at 12.5% per annum .

7) Calculate the amount received on maturity of a recurring deposit of Rs 150 per month for 1 year 6 months. if the rate of interest is 11% per annum .

8) Amar deposits Rs 1600 per month in a Recurring Deposit for 3 years at the rate of 9% p.a simple interest. Find the amount Amar will get at the time of maturity.

9) A Recurring Deposit Account of Rs 1200 per month has a maturity value of Rs 12440. If the rate of interest is 8% and the interest is calculated at the end of every month, find the time (in months) of this Recurring Deposit Account .

10) Sujata deposited , a certain sum of money, every months, for 5/2 years in a cumulative Time Deposit Account. At the time of maturity she collected Rs 4965. if the rate of interest was 8% p.a., find the monthly deposit.

Sap-2

1) Sunita paid Rs 300 per month for 2 years. He received Rs 7875 as the maturity amount. Find the rate of interest.

2) Meena has a cumulative Time Deposit Account of Rs 340 per month at 6% per annum. If she get Rs 7157 at the time of maturity, find the time for which the account was held.

3) On depositing Rs200, every month paying 9% p.a interest, a person collected Rs 2517 at maturity. Find the period.

4) Mamta has a cumulative Time Deposit Account in a bank. She deposits Rs 800 per month and gets Rs 15198 as maturity value. If the rate of interest be 7% p.a., find the total time for which the account was held.

5) Karim has a recurring deposit account for 2 years at 10%. If he receives Rs 1900 as interest, find the value of monthly installment paid by him.

6) Saloni has a cumulative time deposit account of Rs 340 per month at 6% p.a., if she get 7157 at the time of maturity, find the total time for which the account was held.

7) A man deposited Rs 150, every month in a bank for 8 months under the recurring deposit scheme. Find the maturity value of his deposits, if the interest is calculated every month and the rate of interest is 8% p.a.

8) Calculate the amount receivable on maturity of

a) recurring deposit of Rs 1200, deposited every month for 24 months at 10% p.a.

b) recurring deposit of Rs 100, every month for 5 years at 11% p.a.

c) recurring deposit of Rs 500, every month for 27 months at 10.5% p.a.

Saturday, 21 March 2026

Test-12 (2026/27)

H. W -1

SAP- 1

1) If A= 2 0 3 & B= 1 2 3

1 2 0 2 1 4 find A+ B

2) If A= 1 2 3 & B= 1 2

4 5 6 3 4

6 8 9 5 6 find A+ B

3) If A= 0 2 3 & B= 7 8 3

2 1 4 1 4 3 find

a) 2A + 3B

b) 3A - B

c) AB

4) If A= 1 3

3 4 and A²- kA - 5=0, then find k.

5) If A= x y z & B= a h g & C= x

h b f y

g f c z

Then show ABC= ax²+ by²+ cz²+ 2fyz + 2gzx+ 2hxy.

6) If A= 1 0 & B= 0 1

0 1 -1 0

Then show that (aA+ bB)(cA+ dB)= (ac - bd)A+ (ad + bc)B.

7) If A= 1 0 0 & B= x₁ y₁ z₁

0 1 0 x₂ y₂ z₂

0 0 1 x₃ y₃ z₃

Then show that AB= BA = B

Sap-2 (H. W)

1) If A= 1 -1 & B= a 1

2 -1 b -1 and (A+ B)¹= A²+ B², find a, b, using the value of a, b, verify whether AB= BA.

2) If A= 1 -1

1 1 show A/√2 is a orthogonal matrix. A. A'= I is orthogonal matrix

3) Find the inverse of A

A= 1 -3 2

2 5 -1

-3 1 4

4) A= 20 10

10 20 find inverse of A

5) Solve: x + 2y - z= 9, 2x - y - 4z= -7; 3x + 2y - 3z= 2.

SAP-1

1) If A= 2 -1

-1 2 and I is the unit matrix of order 2, then A² is equal to

a) 4A - 3I b) 3A - 4I c) A - I d) A + I

2) The multiplicative inverse of

2 1

7 4 is

a) 4 -1 b) 4 -1 c) 4 -7 d) -4 -1

-7 -2 -7 2 7 2 7 -2

3) Assuming that the sums and products given below are defined, which of the following is not true for matrices?

a) AB= AC does not imply B= C

b) A+ B= B+ A

c) (AB)'= B'A'

d) AB= O implies A= O or B= O.

4) If A=1. 0 2 & Adj A= 5 a -2

-1 1 -2 1 1 0

0 2 1 -2 -2 b

Then the values of a and b are

a) a= -4, b= 1 b) a= -4, b= -1 c) a= 4, b= 1 d) a= 4, b= -1

5) If A= -1 0

0 2 then the value of A³- A² is

a) I b) A c) 2A d) 2I.

6) If A= -x - y

z t, then transpose of adj A

a.) t z b) t y c) t -z d) none

-y -x -z -x y -x.

7) If A square metrix of order 3x3 and λ is a scalar, then adj(λA) is equal to

a) λ adj A B) λ² adj A c) λ³ adj A d) λ⁴ adj A.

8) The inverse of 5 -2

3 1

a) -2/13 5/13 b) 1 2 c) 1/11 2/11 d) 1 3

1/13 3/13 -3 5 -3/11 5/11 -2 5

Tuesday, 20 January 2026

Mixed b/c/p/m

MIXED CHEMISTRY

1) What are 'groups' of modern periodic table ? What does the 'group number' ?

2) Name and state the following with the reference to the elements of the periodic table.

a) The novel gas having an electronic configuration 2, 8, 8 .

b) The non-metal in period three having a valency one.

c) The formula of the hydride of the halogen in period 3.

d) The element from the elements C, O, N, F having the maximum nuclear charge.

e) The element with the largest atomic size from the elements of period -- 1, 2 and 3.

3) Fill up the blanks with appropriate words.

a) Elements of the extreme left of the modern periodic table are ___ reactive, while elements of the extreme right [group -189(0)] are ____reactive. [least/at most]

b) Atomic size of neon is ____( more/less) than the atomic size of fluorine .

c) An atom is said to be non metal if it ____(gains /loses) one or more electrons.

d) nuclear charge of an atom is the______( negative /positive) charge on the molecules of an atom, equivalent to the atomic _____(number/ mass) of an atom.

e) covalent compounds are formed by sharing electron pairs between non metallic atoms. non metallic ions having _____valence electrons (4, 5, 6, 7) share 1, 2 or 3 pairs of electrons respectively.

f) In the reaction of Cl₂ + 2KI ---> 2KCl + I₂.

The conversion of 2I to I₂ is deemed as ____(oxidation/ reduction).

g) An example of a base which is not alkali is ____(caustic soda/ zinc hydroxide) solution.

h) An insoluble salt prepared by direct combination or synthesis is. ___(FeCl₃/FeSO₄/FeS/ Fe(NO₃)₂).

i) The hydroxide which is soluble in excess of NaOH is ____(Zn(OH)₂/ Fe(OH)/ Fe(OH₂))

j) To distinguish soluble salts of zinc and lead ____(NaOH/ NH₄OH) can be used.

k) On electrode at which anions donates excess electrons and are oxidised to neutral atmoss is the ____(anode/cathode).

l) On electrolysis Ag¹⁺ and H ions migrate to the ___(cathode/anode) and ____(Ag¹⁺/HI⁻) are discharged.

m) MA²⁺ forms M, a natural atom by electron ____(less/gain).

n) The metal whose hydroxide does not decompose on heating but its nitrate decomposes is ____(Ca/Al/Na/Fe).

o) During electrolytic reduction of alumina the insert electrode is___( reduced/oxidised) to a neutral gas.

p) aluminium powder a constituent of points , prevents ___(heat radiation/ formation of rust/conduction of electric current).

q) German silver contains ____(Cu-Zn-Sn/Cu-Zn-Ni/Cu-Ph/Ni)

r) the non metallic components of stainless steel is ___(Sulphur/Phosphorus/ carbon).

s) hydrogen chloride gas on heating above 500°C gives hydrogen and chlorine. The action/ thermal dissociation.

t) Addition of ___(sodium nitrate/zinc nitrate/silver nitrate) to hydrochloric acid, gives an insoluble precipitate of the respective chloride.

u) The gas/es which is/are heavier than air and highly soluble in water (NH₃/HCl/CO₂/H₂S)

v) Ammonia in the liquefied from is___ (acidic/ basic/ neutral)

w) the oxidised product obtained on reaction of H₂S gas with dilute HNO₃ is ___(Sulphur /sulphuric acid).

x) A mineral acid obtained from concentric nitric acid on reaction with a non metal is ___( hydrochloric acid/ sulphuric acid/ carbonic acid ).

y) The dehydrated product oobtained when cane sugar reacts with concentric H₂SO₄. (CO/C/CO₂).

z) The reduced produced obtained from hydrogen supplied reacts with concentric H₂SO₄/ SO₂/S/H₂O).

a) The IUPAC name of methyl acetylene (1- butane/propyne/ethyne).

b) State the factor which affect

a) electro affinity

b) electro negative of elements in a periodic table.

3) What is meant by the 'chemical bond' and 'chemical bounding'.

4) State why noble gases are unreactive while atoms of elements other than noble gases are chemically reactive.

5) State the reason why ammonia is evolved when ammonia is evolved when an ammonium salt and alkalis are heated.

6) Give balanced equation for the preparation of the following salts.

i) CUSO₄ ii) NaHSO₄ iii) FeSO₄ iv) PbSO₄ using dilute H₂SO₄ v) Na₂SO₄

7) State giiving reasons , in what state of medium does

a) NaCl

b) HCl

c) NH₃ gas conduct electricity.

8) Name three organic compounds and one neutral liquid which are non electrolytes.

9) Give a reason why metals ---copper, silver and lead are electro refined but K, Na and Ca are not.

10) Explain the term 'dectrometallurgy'. At which electrode metal always deposited.

11) State the electrode reaction at the respective electrodes at the respective electrode reaction at the respective electrode reaction at the respective electrodes during extraction of Al from Al₂O₃.

12) State the basis on which the general characteristic properties of metals and non-metal are associated.

13) State why alkali metals are strong agents, while halogens( non metal) are strong oxidizing agents.

14) Explain in brief the electrolyte method of further purification of aluminium metal.

15) state reason why zinc is used in used in

a) galvanization

b) dry cells

c) alloys

16) Give the equation for preparation of HCl gas by synthesis. State two conditions involved in the synthesis.

17) Give four different word equations relating to acidic properties of an aqu. solution of HCl gas.

18) convert hydrochloric acid to nascent chlorine.

19) Give three test for hydrochloric acid. Convert silver nitrate to a soluble salt of silver using hydrochloric acid and alkali.

20) State by nitrogenous matter produces ammonia. State a liquid source of Ammonia .

21) Give balanced equation with all conditions to option with all conditions to obtain NH₃ from N₂ and H₂.

22) convert ammonia to nitric oxide by catalytic oxidation of ammonia . State all conditions .

23) State a reason why reaction of liquor ammonia. State all conditions.

24) State five tests for ammonia where a colour change involved.

25) State the colour of

a) i) pure nitric acid ii) nitric acid obtained in the laboratory iii) nitric acid from laboratory preparation after passage of air addition of water to it.

b) State which reaction of ammonia forms the first step of Oswald's process.

c) state how addition of nitric acid to acidified FeSO₄ serves as a from former.

d) Name two naturally occurring nitrates . Give equations for three different methods of preparing nitrates.

e) give equations for the heat on i) two different nitrates types which evolve only one gas. ii) 3 different nitrates which leave a coloured metallic residue . iii) A nitrate which leaves no residue.

26) a) State how SO₂ is obtained from Cu and conc. H₂SO₄.

b) Give the reaction of SO₂ with chlorine. What type of reaction of SO₂ is it.

c) Give the reaction reaction for the use SO₂ in i) bleaching ii) food preservation.

d) State how addition of a) copper b) NaCl to hot con H₂SO₄.

27) a) Draw the structural formula and give the name of two isomers of butane and 3 isomers of pentane .

b) State what are alcohols including denatured alcohol. Draw the structural formula for method and ethanol.

c) state what are alkenes. Give a reason why alkanes are called olefins.

d) State what are functional groups ? Name the following functional groups.

-OH, -CHO; -COOH; X= - F, -Cl, Br, I; -Cl=O; -C-O-C.

28) a) calculate the percentage of iron in K₃Fe(CN)₆ (K=39, Fe= 56, C= 12, N= 14(. 17.02% of Fe

b) calculate the percentage of water of crystallization in CuSO₄. 5H₂O. (Cu= 63.5, S= 32, O= 16, H= 1). 36.07%

c) Two the organic compounds x and y containing carbon and hydrogen only have vapour densities 13 and 36 respectively. State the molecular formula of x and y. (C=12, H=1). C₂H₂; C₆H₆

d) A gaseous hydrocarbon weight 0.7g and contains 0.60g of carbon. Find the molecular weight is 70. (C=12, H=1). C₅H₁₀

e) What mass of silver chloride will be obtained by adding an excess of hydrochloric acid to a solution of silver nitrate. (Cl=35.5, Ag=108, N=14, O=16, H=1). 0.287g

f) Zinc blend is roasted in air. Calculate

a) The number of mole of sulphur to produce 22.4 lits of SO₂ at s.t.p (S=32, Zn= 35, O= 16) 8 moles, 97g

g) Calculate the ammonia gas obtained when 32.6g. ammonium chloride reacts with calcium hydroxide during the laboratory preparation of ammonia (2NH₃). (N=14, H=1, O=16, S= 32, Cl= 35.5). 1036g

h) What volume of oxygen would be required for the complete combustion of 100 lits of ethane according to the following equation

2C₆H₆ + 7O₂ --> 4CO₂ + 6H₂O. 350 lits

i) 4NO+ CH₄ ---> CO₂ + 2H₂O + 4N₂. If all volume are measured at the same temperature and pressure calculate the volume of N₂O required to give 150 cc of steam. 300 cc

₂₂₂₃₂₂₂₄₂₂₄₄₄₄₂₄₂₄₃₂₃₃₂₂₄₂₂₄₂₂₂₂₃₂₄ ⁻₂₅₂₂₃₂₂₂₂₃₂₂₂₃₂₂₄₄₂₄₄₂₂₂₂₇₂₄₂₂₃₂₅₂₂₇₂₂₇₃ ₆ ₄₂₂₂₆₆₅₁₀₃₆₆₂₂₂₄₂₂₂₂

MATHS

1) Girija buys a TV set for Rs 13300 and gets 6% rebate on it. He has to pay a GST of 10%. Find the total amount he has to pay.

2) The listed price of a refrigerator is Rs 15000. The customer is allowed a rebate of 50/3%. If the customer pays Rs 13750 for it, calculate the rate of GST.

3) Leena wants to buy an electric iron which is listed Rs 1308. She has to pay a GST of 9%. The shopkeeper reduces the price so that Leena has to pay Rs 1308 inclusive of GST. Find the reduced price.

4) Sita deposits Rs 400 per month in a bank in recurring deposit. On maturity she gets Rs 97854.40. find the period of which she had deposited.

5) Which is the better investment 8% Rs 100 shares at Rs 20 premium or 6% Rs 100 shares at Rs 20 discount?

6) A company declares a semi annual dividend of 5%. Sanjay owns 25 shares of par value Rs 12.50 each. How much annual dividend must be receive ?

7) A company having a capital stock of Rs 450000 declares a dividend of 4% semiannually.

a) What is the annual income of a stock holder owning 135 shares at par of Rs 10?

b) What is the total amount of dividend paid annually by the company.

8) Vinay owns 150 Rs 25 shares of a company which declares a dividend of 12%. What is Vinay's dividend income? If he sells the shares at Rs 40 and invests the proceeds in 7% stock (par value Rs 100) at Rs 80, what is the change in his dividend income?

9) Mr. Gupta purchased 360 Rs 50 shares at Rs 20 premium. The company declares an annual dividend of 12%.

a) Find his dividend income from the shares.

b) Find his total investment in the shares.

10) A man invests Rs 1426 in 5% stock at Rs 115. He sells this stock at Rs 125 and invests the proceeds in 3% stock at 93. Find the change in his income.

11) Solve: (11- 2x)/(9-3x) ≥ 5/8, x belongs to R, x< 3.

12) 8/3≤ x + 1/3 < 10/3, x belongs to R. Hence represent the solution on a number line.

13) A={ x: -1< x ≤ 5, x belongs to R}

B={x: -4≤ x < 3, x belongs to R}

Represent a) A ∩ B b) A' ∩ B on different number lines, where universal set is R.

14) Solve the equation 3x½- x -7= 0 and give your answer correct to 2 decimal places.

15) Find the roots of x½- 6x +2= 0, using formula.

16) Find the roots of √3 x²- 9x + 6√3=0 using formula.

17) Solve for x, (4x²-1) -3(2x +1)+ x(2x +1)= 0.

18) Solve for x, using formula, x²- 1/x²= (29/10) (x - 1/x).

19) Solve: √(x +15)= x+ 3, x belongs to N.

20) Solve: √{x(x-3)}=√10. State the sum of the roots.

21) Solve: √(6x -5) - √(3x -2)= 2.

22) A two digit number is four times the sum and two times the product of its digits. Find the number.

23) A says to B, I am twice as old as you were when I was as old as you are. If the product of their ages is 588. Find their present ages.

24) A number consists of two digits such that the square of the digit in the ten's place exceeds the digit in the unit place by 11. If the number is five times the sum of the digits, find the number.

25) Ten years ago, the sum of the ages of two sons was half of their father's age. The ratio of the present ages of the two sons is 4:3 and the sum of the present ages of all the three is 117 years. Find the present ages of the father and each of the two sons.

26) A party of students arranged an excursion costing Rs 540, whose amount was to be shared equally by all of them. But later, it was found that three of the students, could not pay, though they had joined the excursion. As a result, the rest of the students had each to pay Rs 2 more. Find the total number of students in the party.

27) In a certain examination, the those number of candidates passed was four times the number of those who failed. If the number of candidates that appeared had been 35 less, and the number that failed had been 9 more, the number passing would have been twice the number failing. Find the total number of candidates that appeared at the examination.

28) In a certain year, the incomes of Nitesh and Mitesh are in the ratio 5:6; their expenses are in the ratio 6:7, and their savings are in the ratio 4:5. If the sum of their expenses during that year is Rs 3900, find the income of each.

29) The length of a rectangle exceeds its breadth by 5m. If the breadth were doubled and the length reduced by 9m, the area of the rectangle would have increased by 140 square.m, find its original area.

30) If the usual speed of a train is reduced by 4km per hour, it takes one hour more to cover a distance of 360 km than it usually does. Find the usual speed of the train in km per hour.

31) The area of a right angle triangle is 60 square. units. Determine its base and the altitude if the latter exceeds the former by 7 units.

32) The points A((2,1), B(0,3) and C(-3,-2) are the vertices of a triangle.

a) plot the points on the graph paper.

b) Draw the triangle formed by reflecting these points in the x-axis.

c) Are the two triangles congruent?

33) P,Q, R are the points (0,0), (2,6) and (10,2) and PM is the median of ∆ PQR. Find the coordinates of the image?

a) A' of A under reflection in the x-axis

b) B' of B under reflection in the line AA'.

c) A" of A under reflection in the y-axis

d) B" of B under reflection in the line AA'.

34) If a: b= c: d, show (a²+ ac + c²)/(a²- ac + c²)= (b²+ bd + d²)/ (b²- bd + d²).

35) If a/(b + c) = b/(c + a)= c/(a + b), show that each ratio 1/2 or -1.

36) Using properties of proportion, solve for x:

(2x²+ 6x)/(12x²+ 4)= 682/364.

37) If a,b,c are in continued proportion, show that (ba+ bc) is the mean proportion between (a²+ b²) and (b²+ c²).

38) If x+2 is a factor of x²(x +6)+ Kx + 6, find the value of k.

39) If x -3 is a factor of x³- 3x⅖+ 4x + k, find the value of k. Hence find the other factors.

40) If x-1 is a factor of x³(x +3)+ x(x -3) - 2k, find the value of k. Hence find the other factors, if x+1 is also one of the factors.

41) Show that x-1 is a factor of x³- 7x²+ 14x -8. Hence, completely factorise.

42) If A= (k 3 & B= (2 & C= 1

0 1) -3) -3) with the relation AB = C, then find k.

43) If A= a 1 & B= 4 3 & C= b 11

1 0 3 2 4 c with the relation AB= C, then find a, b, c.

44) If A= 4 1

-1 2 show that A²- 6A + 9I= 0.

45) If A= -3 2 & B= 1 a

-2 -4 b 0 and (A+ B)(A - B)= A²- B², find the values of a and b.

46) If A=2 -1 & B= 2

4 3 -3 find matrix X such that AX = B.

47) If A= 1 1

1 1 find matrices B and C of order 2x 2 such that AB= AC but B≠ C

48) If A= 2 5 & B= 4 -1

0 1 2 0 and equation

5A+ 3B + 2X= 0 42

4 9, find the matrix X.

49) Find the coordinates of the point dividing the join of (1,2) and (2,1) in the ratio of 3:4.

50) C divides AB in the ratio 5:4. If A is (2,7) and C is (-3,-8), find the coordinates of B.

51) A is (6,10), B is a point such that the origin divides AB internally in the ratio 1:2. Find the coordinates of B.

52) Find the ratio in which the point (7, a) divides the join of (-5,2) and (3,6) hence find a.

53) Obtain the ratio in which the x-axis divides the segment joining A(-2,3) and B (4,-6). Hence or otherwise find the coordinates of the point of intersection of the x-axis and AB.

54) Find the equation of lines whose y-intercept is 2 and slope is 3.

55) find the y-intercept and slope of

a) 3y - 6x= 2

b) 2x - 3y = 6.

56) Find the equation of the line through (2,0),(0,2).

57) Find the equation of a line through (0,3) and parallel to 2y - 4x =1.

58) Find the equation of a line through (0,-5) and parallel to x/2 + y/3= 1.

59) Find the equation of a line through (1,2) and perpendicular to y= 2x +1.

60) Find the equation of a line through the origin and perpendicular to 3x + 5y=7.

61) A(2,4) and C(8,10) are opposite vertex of a rhombus ABCD. Find

a) the midpoint of AC

b) the slope of AC

c) the equation of the diagonal BD.

62) In the given figure angle XZS= YZR= SRZ, XY= 5.2cm and RS= 3.9 cm. Find the ratio of area (∆ XYZ): area (∆ ZSR).

63) A model of a ship is made to a scale 1:300.

a) The length of the model is 6m. Find the length of the ship.

b) The volume of the model is 250 litres. Find the volume of the ship in m³.

c) The area of the deck of the ship is 18000 m². Find the area of the deck of the model.

64) Construct an equilateral ABC in which each side is 5cm long. Construct a point O which is equidistant from the three vertices. Construct perpendicular from O to each side. Is equidistant from each side?

65) Draw a circle of radius 4cm and mark two chords AB and AC of the circle of length 6cm and 5cm respectively.

a) construct the locus of points, inside the circle that are equidistant from A and C.

b) Construct the locus of points inside the circle that are equidistant from AB and AC

66) From the adjoining diagram

a) Show BM/AL = MC/LQ

b) If BM= (1/2) MC, find LM/QR.

67) PQRS is a parallelogram, SL= SR. show that

a) QM= QR

b) ∆ LSR= ∆ ||gm PQRS

If PQ= 2 PS, show that ∆ QRM= (1/2) ∆ PQR

68) PQRS is a parallelogram, PZ: ZQ= 2:3, Find

a) QX/XS

b) QX/QS

c) QY/YS

d) QY/QS

69) From the diagram. Find the value of x. Name the isosceles triangle in the figure.

70) If O is the centre of the circle and angle ROS=42, find angle RTS.

71) If C is the centre of the circle, PQRS is a parallelogram and angle PQR= 42, find angle PXR, SCX.

72) A circus tent is cylindrical to a height 3m and conical above it. If its diameter is 105m and slant height of the cone is 53m, find the total area of the canvas required. (π=22/7).

73) A cone is 8.4cm high and the radius of its base is 2.1cm. it is melted and recast into a sphere. Determine the radius of the sphere.

74) The diameter of a sphere is 6cm. It is melted and recasted into a sphere. Determine the radius of the sphere.

75) The diameter of a sphere is 6cm. It is melted and drawn into a wire of diameter 0.2m. find the length of the wire.

76) Prove:

a) 1/(cosx + sin x -1). + 1/(cosx + sinx +1) = cosecx + secx.

b) tan²x - tan²y = (cos² y - cos²x)/(cos² y cos² x) = sin²x - sin²y)/(cos²x cos²y).

c) If x= a sec m cos n, y= b secm sec n, z= c tan m , show that x²/a² + y²/b² - z²/c²= 1.

d) If sinx + sin²x = 1, show cos²x + cos⁴x = 1.

e) show: cos⁶x + sin⁶x = 1- 3 cos²x. sin²x.

f) If 6 sin²x + 2 cos²x = 3 find x.

g) Show: (tanx + secx -1)/(tanx - secx +1)= (1+ sinx)/cosx.

h) show: (1+ cot x - cosecx) (1+ tanx + secx)= 0

i) show (sinx + cosecx)²+ (cosx + secx)²= tan²x + cot²x +7.

76) A man standing 20m from a tower measure the angles of elevation of the top and bottom of a flagstaff on the tower as 30° and 60°. Calculate the height of the flagstaff to the nearest 0.1m.

78) A man on the deck of a ship is 20m above water level. He observes that the angle of a elevation of the top of a cliff is 60° and the angle of depression of the base is 30°. Find the distance of the cliff from the ship and the height of the cliff.

79) For the following distribution, draw a histogram and find mode.

Mass (kg) frequency

44-47 13

48-51 27

52-55 43

56-59 36

60-63 23

80) Calculate the mean for the following distribution:

Class frequency

10-16 1

16-22 10

22-28 5

28-34 3

35-40 6

81) Find mean for the following distribution

Marks students

05-10 5

10-15 6

15-20 15

20-25 10

25-30 5

30-35 4

35-40 3

40-45 2

82) Following are scores made by a cricketer:

12,24,48,27,0,91,40,13,6,32.

Calculate a) the mean b) the median score

83) For the following distribution, draw the ogive, estimate the median

Class frequency

0-2 17

3-5 22

6-8 29

9-11 18

12-14 9

15-17 5

Also find (i) lower quartile ii) the upper quartile iii) the semi-inter quartile

84) Following are the marks of 100 candidates

Marks candidate

00-10 5

10-20 10

20-30 22

30-40 40

40-50 15

50-60 8

a) state the pass mark if 75% of the candidates passed.

b) State the mark which 40% of the candidates exceeds

What percentage of candidates get less than 20 marks.

85) The median of the following observations 11,12,14, 18, (x +14),30,32,35,41, arranged in ascending order is 24. Find x.

86) Two fair dice are thrown. Find the probability of getting

a) the same score on the first die as on the second.

b) the score of second die is greater than the score of first.

Friday, 16 January 2026

b. aom

Type-1)

1) lim ₓ→₀ (7x²-5x+1). 1

2) limₓ→₀ (2x³+3x+4)/(x²+3x+2). 2

3) limₓ→₃√(2x+3)/(x+3). 1/2

4) limₓ→₁ √(x+8)/√x. 3

5) lim ₓ→₁(x²+1)/(x+1). 1

6) lim ₓ→ₐ (√a+√x)/(a+x). 1/√a

7) limₓ→₁{1+(x-1)²}/(1+x²). 1/2

8) lim ₓ→₂ (3x²-x+1)/(x-1). 11

9) limₓ→₁ (4-x). 3

10) lim ₓ→₀(ax²+b)/(cx+d). b/d

11) limₓ→_₁(x³ - 3x +1)/(x-1) -3/2

12) limₓ→₀ (3x+1)/(x+3). 1/3

Type: 2. *******

1) lim ₓ→₁ (x²-1)/(x-1). 2

2) limₓ→₋₅ (2x²+9x-5)/(x+5). -11

3) limₓ→₃(x²-4x+3)/(x²-2x-3). 1/2

4) limₓ→₄ (x²-16)/(√x -2). 32

5) limₓ→₀ {(a+x)³-a³}/x. 3a²

6) lim ₓ→₁(x-1)/(2x²-7x+5). -1/3

7) lim ₓ→₁ (x²-√x)/(√x-1). 3

8) limₓ→₃(x²-9)/{1/(x-3)+1/(x+3)}. 6

9) limₓ→₁ (x-1)/(2x²-7x+5). -1/3

10) limₓ→₃ (x²-7x+12)/(x²-9). -1/6

11) limₓ→₂ (7x²-11x-6)/(3x²-x-10). 17/11

12) limₓ→₂ (x³-8)/(x-2). 12

13) limₓ→_₁(2x²+5x+3)/(x³+1). 1/3

14) limₓ→₂ x²(x²-4)/(x-2). 16

15) limₓ→₂{(x⁸-16)/(x⁴-4)+(x²-9)/(x-3)}. 25

16) limₓ→₂ (x-2)/(√x -√2). 2√2

17) limₓ→₀ {(1+x)²-(1-x)²}/2x. 2

18) limₓ→₁ (x²+5x-6)/(x²-3x+2). -7

19) lim ₓ→₁/₂{(8x-3)/(2x-1) - (4x²+1)/(4x²-1)}. 7/2

20) limₓ→₃(x²+x-12)/(x-3). 7

21) limₓ→₁(x²+4x-5)/(x-1). 6

22) limₓ→₀ {(1+x)²-1}/x. 2

23) limₓ→₂(x²-5x+6)/(x²-3x+2). -1

24) limₓ→₂(x²+x-6)/(x²-x-2). 5/3

25) limₓ→₂(x²-5x+6)/(x²-7x+10) 1/3

26) limₓ→₃(x²+2x-15)/(x²-2x-3). 2

27) limₓ→₁(x³-1)/(x²-1). 3/2

28) limₓ→₂{1/(x-2) - 1/(x²-3x+2)}. 1

29) limₓ→₁(x²-3x+2)/(x³-4x+3). 1

30) limₓ→₀{(4+3x)³-8x²}/{4(4-x)²}.1

31) limₓ→₂(2x²-3x+7)/(x³+5x+1) 9/19

32) limₓ→_₁(2x²+5x+3)/(x³+1). 1/3

33) limₓ→₁(2x⁴-3x+1)/(x³-5x²+4x). -5/3

34) limₓ→₃(x³-8x²+45))(2x²-3x-9). -7/3

35) limₓ→₃(x³-6x-9)/(x⁴-81). 7/36

36) limₓ→√₂ (x⁴-4)/(x²+3x√2-8). 8/5

37) limₓ→₁(x⁴-3x³+2)/(x³-5x²+3x+1) 5/4

38) limₓ→₁{(2x-3)(√x -1)}/(2x²+x-3) -1/10

39) limₓ→₃(x²-9){1/(x+3) + 1/(x-3)} 6

40) limₓ→₂(x³-6x²+11x-6)/(x²-6x+8) 1/2

41) limₓ→₁/₂ (8x³-1)/(16x⁴-1) 3/4

42)limₓ→₄(x²-x-12)¹⁸/(x³-8x²+16x)⁹ 7¹⁸/4⁹

43) limₓ→₁{1/(x²+x-2) - x/(x³-1)}. -1/9

44) limₓ_₃(x³-7x²+15x-9)/ (x⁴-5x³+27x-27) 2/9

45) limₓ→√₂. (x⁹- 3x⁸+ x⁶- 9x⁴- 4x²- 16x+84)/(x⁵-3x⁴-4x+12). (8√2-31)/(√2-3)

46) limₓ→₃ (x⁴- 81)/(x²-9). 18

47) limₓ→₃(x²-x-6)/(x³-3x²+x-3). 1/2

48) limₓ→₋₂ (x³+x²+4x+12)/ (x³-3x+2). 4/3

49) limₓ→₁(x³+3x²-6x+2)/ (x³+3x³-3x-1). 1/2

50) limₓ→₁(x⁴-3x³+2)/(x³-5x²+3x+1) 5/4

51) limₓ→₂ (x³+3x²-8x-2)/(x³-x-6). 15/11

52) limₓ→₂ (x⁴ -16)/(x-2). 32

53) limₓ→₁{(x-2)/(x²-x) - 1/(x³ -3x²+2x)}. 2

54) limₓ→₂ {1/(x-2) - 2(2x-3)/(x³- 3x² +2x)}. -1/2

55) lim ₕ→₀ {f(1+h)-f(1)}/h, when f(x)= 1/x. -1

Continue.........

Type: 3. ------------

1) limₙ→₀{√(x+n) -√(x)}/n. 1/2√x

2)limₓ→₀ {√(1+x) - √(1+x²)}/x. 1/2

3) ltₓ→₀{√(1+x) -√(1+x²}/{√(1-x²)-√(1-x)}. 1

4) limₓ→ₐ{√(a+2x)-√(3x)}/ {√(3a+x) -2√(x)}, a≠ 0. 2/3√3

5) limₙ→₀ 1/n{1/√(x+n) - 1/√(x)}. -1/(2x√x)

6) limₓ→₀ {√(1+x) - 1}/x. 1/2

7) limₓ→ₐ{√(x) - √(a)}/(x-a). 1/2√a

8) limₓ→₄ {3 -√(5+x)}/(x-4). -1/6

9) limₓ→₀{√(x+2) - √(2)}/x. 1/2√2

10) limₓ→₀ x/{√(1-x)- 1}. 2

11) limₓ→₀ {√(1+x) -√(1-x)}/2x. 1/2

12)

13) limₓ→₀ {√(1+x+x²) -1 }/x. 1/2

14) limₓ→₀{√(1-x³) -√(1+x³)}/x². 0

15) limₓ→₄ {3-√(5+x)}/{3-√(5-x). 1/3

16) limₓ→₃{3-√(6+x)}/{√3 -√(6-x). -1/√3

17) limₓ→₀{√(1+x) -√(1+x²)}/{√(1-x²) - √(1-x)}. 1

18) limₓ→₂(x²-4)/{√(3x-2)-√(x+2)}. 8

19) limₓ→₃{√(3x+7)-√(7x-5)}/{√(5x-6) - √(2x+3)}. - 1

20) limₓ→₁{√(x+8)-√(8x+1)}/{√(5-x)- √(7x-3)}. 7/12

21) limₓ→₁ {³√(x+7)- ³√(7x+1)}/(x-1) 1 - ³√7

22) limₓ→₂{2-√(2+x)}/{³√2- ³√(4-x)} - 3/³√16

23) limₓ→₀ x/{√(a+x)-√(a-x)}. √a

24) limₓ→₄ (x²-16)/{√(x²+9) -5} 10

25) limₓ→ₐ{√(a+2x)-√(3x)}/{√(3a+x)- 2√x} 2/3√3

26) limₓ→₁{(2x-3)(√x -1)}/(2x²+x-3) -1/10

27) limₓ→√₁₀ {√(7-2x)-(√5-√2)}/(x²-10). (√5+√2)/6√10

28) limₓ→₂ {√(x²+1)-√5}/(x-2) 2/√5

29) limₓ→₂ (2-√x)/(4-x). 1/4

30) limₓ→ₐ (x-a)/(√x - √a). 2√a

31) limₓ→₂ (x-2)/(√x-√2). 2√2

32) limₓ→₃ {√(x-3)+√x -√3}/√(x² -9) -1

Continue.......

Type : 4

1) limₓ→₂(x¹⁰ - 1024)/(x-2). 5120

2) limₓ→₁ (xᵐ -1)/(x-1) m

3) limₓ→₃ (x⁵-243)/(x²-9). 135/2

4)limₓ→ₐ(x⁵-a⁵)/(x³-a³). 5a²/3

5) limₓ→₅ (x⁴-625)/(x³-125). 20/3

6) limₓ→₂ (x¹⁰ -1024)/(x⁵ -32). 64

7) limₓ→₉ (x³/² -27)/(x-9). 9/2

8) limₓ→ₐ(x³/⁵-a³/⁵)/(x¹/³-a¹/³). 9/

a⁴/¹⁵/5

10) limₓ→₁(xᵐ -1)/(xⁿ -1) m/n

11) limₓ→ₐ (x√x- a√a)/(x-a) 3√a/2

12) limₓ→₂ (x⁷-2⁷)/(x³-2³). 112/3

13) limₓ→₀ {(1+x)ⁿ -1}/x. - n

14)limₓ→¹ {(1+x)⁶ -1}/{(1+x)² -1}. 3

15) lim ₓ→ₐ(x²⁾⁷- a²⁾⁷)/(x-a) 2/7a⁵⁾⁷

16) lim ₓ→₋₁/₂ (8x³+1)/(2x+1). 3

17) limₓ→ₐ{(x+2)⁵/² -(a+2)⁵/²}/(x-a) 5/2 √(a+2)³

18) lim ₓ→₂ (x-2)/(³√x - ³√2). 3(2²⁾³)

19) If lim ₓ→₂ (xⁿ - 2ⁿ)/(x-2)= 80 and n∈ ℕ find n 5

20) If lim ₓ→₁(x⁴-1)/((x-1)= lim ₓ→k (x³ - k³)/(x² - k²). Find k 8/3

21) If limₓ→_ₐ (x⁹+a⁹)/(x+a) = 9, then find the value of a. ±1

22) limₓ→³ (xⁿ-3ⁿ)/(x-3) =108 and if n is positive integer find n. 4

Continue........

Type : 5

1)lim→₀ (e⁻ˣ -1)/x. -1

2) limₓ→₀ (eᵃˣ-1)/ax. 1

3) limₓ→₀ (eᵃˣ-1)/mx. a/m

4) limₓ→₀ (e⁵ˣ -1)/3x. 5/3

5) limₓ→₀ (eᵃˣ - eᵇˣ)/x. a-b

6) limₓ→₀ (e⁷ˣ - e³ˣ -e⁴ˣ +1)/x². 12

7) limₙ→₀{ ₑ(x+n)² - ₑx²)}/n. 2x

8) lim ₓ→⁰ (eˣ- e)/(x-1). e

9) limₓ→₀ (ₑlog x ₋ ₁)/ₑˣ⁻¹ ₋ ₁) 1

10) lim ₓ→₀ (eˣ - e²)/(x-2). e²

11) limₓ→₀ (e⁷ˣ - 1)/9x 7/9

12) lim ₓ→₀ (eˣ - e⁻ˣ)/x. 2

13) limₓ→₀ (e¹⁵ˣ - e⁷ˣ)/x. 8

14) limₓ→₀ (e⁷ˣ + e⁵ˣ -2)/x. 12

Type : 6

1) limₓ→₀ (3⁵ˣ - 1)/x. 5 log 3

2) limₓ→₀ (2³ˣ -1)/x. 3 log 2

3) limₓ→₀ (2ᵃˣ - 3 ᵇˣ)/x. alog 2-b log 3

4) limₓ→₀ (12ˣ -3ˣ- 4ˣ +1)/x² log 3. Log 4

5) limₓ→₀(aˣ - bˣ)/x. log(a/b)

6) limₓ→₀ (10ˣ -2ˣ- 5ˣ +1)/x². log 5. log 2

Continue......

Type : 7

1) lim ₓ→₀ {log(1+7x)}/x. 7

2) limₓ→₁ (log x)/(x-1). 1

3) limₓ→₀ {log(6+x)- log(6)}/x. 1/6

4) limₓ→₂ {log(x) - log(2)}/(x-2). 1/2

5) limₓ→ₑ (logx -1)/(x-e). 1/e

6) limₓ→₁(x²- x log x+ log x -1)/(x-1) 6

7) limₓ→₀ x{log(x+a) - log x}. a

8) limₓ→⁰ x{log(x+5) - log x}. 5

9) limₓ→₄ (x⁷/²- 4⁷/²)/{ log(x-3)} 112

Continue.......

Type: 8

1) limₓ→∞ (4x-3)/(2x+7). 2

2) limₓ→∞(3x²+2x-5)/(x²+5x+1). 3

3) limₓ→∞ (x³+6x²+1)/(x⁴+3). 0

4) limₓ→∞(3x³+x²-1)/(x²-x+7). ∞

5) limₓ→∞ (5x-6)/√(4x²+9). 5/2

6) limₓ→∞{√(3x²-1)-√(2x²-1)}/ (4x+3). (√3-√2)/4

7) limₓ→∞{√x √(x+c) -√x). c/2

8) limₓ→∞{√(x²+x+1) - √(x²+1)}. 1/2

9) limₓ→∞{(x+1)(2x+3)}/{(x+2)(3x+4)}. 2/3

10) limₓ→∞ {x - √(x² - x)}. 1/2

11) limₓ→∞{√(x²+5x+4)- √(x²-3x+4)}. 4

12) limₓ→∞ 2x{√(x²+1)-x}. 1

13) limₓ→∞{√(x²+1)-³√(x²-1)}/{ ⁴√(x⁴+1)- ⁵√(x⁴+1)}. 1

14) limₓ→∞(5x³-3x+1)/(7x³+2x²-2). 5/7

15) limₓ→_∞(5-6x²)/(1++2x-3x²) 2

16) limₓ→∞(x√x+√x -1)/(5√x+1) ∞

17) limₓ→∞ {(x+1)(2x+1)(3x+1)}/ {(x²+1)(5x-3)}. 6/5

18) limₓ→∞{1²+2²+...+x²}/{(x-2)(x+3)(x-4)}. 1/3

19) limₓ→∞{1+ 1/2 + 1/2²+.... to n terms}. 2

20) limₓ→∞{1+3+5+... to n terms}/(n² -1). 1

21) limₓ→∞{2+5+8... to(2n+1)}/{1+2+3+.... to n terms} 12

22) limₓ→∞(1.2+2.3+3.4+...to n terms)/{(3-n)(n+1)(n+2). -1/3

23) limₓ→∞{√(x²-2x+1) - √(x²-5x-3)}. 3/2

24) limₓ→∞ [³√x²{³√(x+1)- ³√x}] 1/3

25) limₓ→∞{(2x-1)³(x²+1)}²/{(x³-2x+1)(3x+1)}. 8/3

26) limₓ→∞{(2x³-x+1)²(x²-1)³}/ {(3x+1)⁴(2x⁴-3x+1)²}. 1/81

27) limₙ→∞ (1+3+....+n)/n². Or limₙ→∞ ∑n/n² 1/2

28) limₙ→∞ ∑n³/n⁴ 1/4

EXERCISE -1

1) A manufacturer finds that the production cost of each article produced by his firm is ₹25 and the other fixed costs are ₹25000. If each article is sold for ₹35, Find

A) cost function. C(x)= 25000+ 25x

B) Revenue function. R(x)= 35x

C) break even point. 2500

2) The fixed cost of a new product is ₹35000 and the variable cost per unit ₹500. If the demand function p(x)= 5000 - 10x, find the break-even values. 10, 35

3) The fixed cost in the variable cost of x units of a product of a company are ₹ 30000 and ₹75x respectively. If each unit is sold for ₹125, find break-even point. 600

4) A company decides to set up a small production plant for manufacturing clocks. The total cost of initial set up(fixed cost) is ₹9 lakhs. The additional cost (variable cost) for producing each clock is ₹300. Each clock is sold at ₹750. During the first month 1500 clocks are produced and sold.

A) determine the cost function for C(x) for producing x clocks. 300x + 900000

B) determine the revenue function R(x) for the sale of x clocks. 750x

C) determine the profit function p(x) for the sale of x clocks. 450x - 900000

D) what profit or loss the company incurs during the first month when all 1500 clocks are sold ? 225000

E) determine the break even point. 2000

5) The printing cost of 1900 and 1300 copies of a book are 5100 and 3700 respectively. Find the equation of the cost curve of printing assuming it to be linear. If the selling price is ₹3 per copy, find the number of copies that must be printed so that

A) there is no profit or loss. y= 7x/3 + 2000/3, 1000 copies

B) a profit of ₹ 40. 1600 copies

6) A publishing house finds that the production cost directly attributed to each book is ₹30 and that the fixed costs are ₹15000. If each book can be sold for ₹45 then find

A) cost function. 30x+15000

B) the revenue function. 45x

C) break-even point. 1000

7) A firm knows that the demand function for one of its products is linear. It also knows that it can sell 1000 units when the price is ₹4 per u6, and it can sell 1500 units when the price is ₹2 per unit. Find

A) demand function. 8 - x/250

B) total revenue function. 8x -x²/250

C) marginal revenue function. 8 - x/125

8) ABC Co. Ltd. is planning to market a new model of shaving razor. Rather than set the selling price of the razor based only on production cost estimates, management pulls the retailer of the razors to see how many razors they would buy for various prizes. From this survey it is determined that the unit demand function (the relationship between the amount x each retailer would buy and the price p he is willing to pay) is

x= 1500 p + 30000

The fixed costs to the company for production of the razors are found to be ₹28000 and the cost for material and labour to produce each razor is estimated to be ₹8.00 per unit. What price should the company charge retailer in order to obtain a maximum profit? at 14, ₹26000, 9000

EXERCISE -2

1) The unit demand function is x= (25- 2p)/3, where x is the number of units and p is the price. Let the average cost per unit be ₹40. Find

A) the revenue function R in terms of price p. (25 - 2p)/3

B) the cost function C. 40(25- 2p)/3

C) the profit function P. (-2p² + 105p - 1000)/3

D) the price per unit that maximizes the profit function. When p= 105/4

E) the maximum profit.

2) The demand function faced by a firm is p= 500 - 0.2x and its cost function is C= 25x + 10000 (p= price, x= output and C= cost). Find

A) the output at which the profits of the firm are maximum. 1187.50

B) the price it will charge. 262.50

7) For a manufacturer of dry cells, the daily cost of production C for x cells is given by C(x)= ₹(2.05x + 550). If the price of a cell is ₹ 3. Determine the minimum number of cells those must be produced and Sold daily to ensure no loss. 579

9) The daily cost of production C for x units of an assembly is given by C(x)= ₹(12.5 x + 6400).

A) if each units is sold for ₹25, determine the minimum number of units that should be produced and Sold to ensure no loss. 513

B) if the selling price is reduced by ₹2.50 per unit, what would be the break-even point. 640

C) if it is known that 500 units can be sold daily, what price per unit should be charged to guarantee no loss ? 25.30

10) A firm produces x tonnes of output per week at a total cost of ₹(x³/10 - 5x² + 60x +100). Find

a) average cost. x²/10- 5x + 60 + 100/x

b) average variable cost. x²/10 - 5x + 60

c) marginal cost. 3x²/10- 10x+60

11) A calculator manufacturing company introduces production bonus to the workers that increases the cost of a calculator. The daily cost of production C for y calculators is given by C(y)= ₹ 2.05y + ₹ 550.

A) If each calculator is sold for ₹3, determine the minimum number that must be produced and sold daily to ensure no loss. 579

B) if the selling price is increased by 30 paise per piece, what would be the break even point ? 440

C) if it is known that at least 500 calculator can be sold daily, what price the company should charge per piece of calculator to guarantee no loss? 3.15

12) The total cost and the total revenue of a company that produces and sales x units of a product are respectively C(x)= 10x + 400 and R(x)= 60x - x³. Find

A) the break-even values. 10 & 40

B) the values of x that gives a profit. 10< x < 40

C) the values of x that results in a loss. x< 10 and x> 40

13) The total cost and the total revenue functions of a company that produces and sales x units of a particular product are given by C(x)= 5x + 350 and R(x)= 50x - x² respectively. Find

A) the break even values of x. 10 & 35

B) the values of x that produces a profit. 10< x < 35

C) the values of x that result in a loss. x< 10 and x > 35

14) The total cost and the total revenue of a company that produces and sells x units of a product are respectively C(x)= 5x +350 and R(x)= 50x - x². Find

A) the break-even values. 10 or 35

B) the value of x that produces a profit. 10< x < 35

C) the value of x that results in a loss. x< 10 or x> 35

15) The cost function C(x) of a firm is given by C(x)= 2x² - 4x + 5. Find

A) The average cost.

B) the marginal cost when x= 10. 16.5 units, 36 units

16) A firm produces x units of a article at a total cost of ₹(5+ 48/x + 3x²). Find the minimum value of the total cost. At x= 2, ₹ 41

17) A firm produces x units of output at a total cost of ₹(2x/3 + 35/2). Find the cost when the output is 4 units, the average cost of output of 10 units, and the marginal cost when output is 3 units. ₹20.16, ₹2.42, ₹0.67

18) A firm produces x units of output per week at a total cost of ₹(x³/3 - x² + 5x + 3). Find the output levels at which the marginal cost and the average variable cost attains their respective minima. 1, 1.5

19) A firm produces x tons of a valuable metal per month at a total cost C given by C= ₹(x³/3 - 5x² + 75x +10). Find at what level of output the marginal cost attains it's minimum. 5 tons.

20) Let the cost function of a firm be given by the equation: C(x)= 300x - 10x² + x³/3, where C(x) stands for cost function and x for output. Calculate the output at which

A) the marginal cost is minimum. 10

B) the average cost is minimum. 15

C) average cost is equals to Marginal cost. 15

21) The efficiency E of a small manufacturing concern depends on the number of workers w and is given by 10E = - w³/40 + 39w - 392. Find the strength of the workers which gives maximum efficiency. 20

22) A company after examining its cost structure and revenue structure has determined that the following functions approximately describe its cost and revenues:

C= 100 + 0.015x² and R= 2x where C= total cost, R= total revenue and x= number of units produced and Sold. Find the output rate which will maximum profits for the firm. 66.67

23) A firm can sell x items per week at a price p= (300 - 2x) rupees. Producing items cost the firm y rupees where y= 2x + 1000. How much production will yield maximum profits ? 74

24) The total revenue function and the total cost function of a company are given by R= 21q - q² and C= q³/3 - 3q² - 7q + 16 respectively, where q is the output of the company. Find the output at which the total revenue is maximum and the output at which the total cost is minimum. 10.5, 7

25) The demand function of a firm is p= 500 - 0.2x and its cost function is c= 25x - 10000, where p is the price and x is the output. Find the output at which the profit of the firm will maximum. Also find the price it will charge. 1187.5, 262.5

26) The demand function of a producer is 3q= 98 - 4p and its average cost is 3q +2, where q is the output and p is the price. Find the maximum profit of the producer. 33.75 units

27) A radio manufacturer finds that he can sell x radio per week at ₹p each, where p= 2(100 - x/4). His cost of production of x radios per week is ₹ (120x + x²/2). Show that his profit is maximum when the production is 40 radios per week. Find also his maximum profit per week. 1600

28) A manufacturer produces x units per month at a total cost of ₹(x²/25 + 3x + 100). There is no competition in the market and the demand follows the rule x= 75 - 3p, where p is the selling price per article. Find x such that the net revenue is maximum,. also find the monopoly price. 30, 15

29) A firm produces x units of output at a total cost of ₹(300x - 10x² + x³/3). Find

A) output at which marginal cost is minimum. 10

B) output at which average cost is minimum. 15

C) output at which average cost is equal to marginal cost. 0, 15

30) The demand function of a monopolist is given by p= 1500 - 2x - x². Find the marginal revenue for any level of output x. Also, find marginal revenue when x= 0. 1160

31) A firm produces x tons of output per week of a total cost of ₹(x³/8 - 4x² + 12x +3). Find the level of output at which average variable cost attains minimum value. 16

32) The manufacturing cost of an item consists of ₹900 as overheads, the material cost is ₹3.00 per item and the labour cost is ₹ x²/100 for x items produced. How many items must be produced to have average cost minimum. 300

33) The total cost function of a firm is C= x³/3 - 5x² + 28x +10. Where C is total cost and x is output. A tax at the rate of ₹2 per unit of output is imposed and the producer adds it to his cost. If the market demand function is given by

p= 2530 - 5x.

Where ₹ p is the price per unit of output, find the profit maximizing output and price. 50, 2280

34) Given the demand and cost functions:

p= 10 - 4x

C= 4x

Find

A) the maximum quantity, price and the profit on this level. 12

B) what will be the new equilibrium after a tax of ₹0.50 is imposed ? 12.25

*C) the tax rate that will maximizing tax revenue and determine that tax revenue. 8

EXERCISE-A

1) If f(x)= 2x² - √x +1, find

a) f(4). 31

b) f(0). 1

c) f(1/4). 5/8

2) If f: x --> (x²-4)/(x-2) then find

a) f(1). 3

b) f(2). Undefined

3) If the function f: N --> N is defined by f(x)=√x, then find f(25)/{f(16)+f(1)}. 1

4) If f(x)= x³/3 - x²/2 + x -16, then find f(1/2). -187/12

5) if f(x)= 7x⁴- 2x³- 8x -5, find f(-1). 12

5) If f(x)=2x²- 3 √x +2, find

A) f(0). 1

B) f(4). 27

C) f(h+2). 2h+3 - 3 √(h+1)

6) Find {f(x+h) - f(x)}/h when

a) f(x)= 4x²+ 2x -3. 2(4x+ 2h+1)

b) f(x)= (1-x)/(1+x). -2/{(1+x)(1+x+h)}

7) f(x)= (x²- 5x+6)/(x²- 8x +12), find

A) f(2). 0

B) f(-5).

C) f(0).

D) f(a).

E) f(h+1).

F) f(h-1)

G) f(2+ h).

H) f(2/h).

8) If f(x)= (ax + b)/(bx + a), find f(x) f(1/x).

9) if g(x)= (x - a)/x + x/(x - b), find the value of g{(a+b)/2}.

EXERCISE -B

1) 2x -1 when x≤2

If f(x)= x² -1 when 2 < x < 3

2x +2 when x≥ 3

Find

a) f(-1). -3

b) f(2). 1

c) f(2.5). 5.25

d) f(3). 8

e) f(3.5). 8

2) f(x) = 2x-1, when x≤ 0

x², when x > 0.

Find

a) f(1/2). 1/4

b) f(-1/2). -2

3) 1+x, -1≤ x <0

If f(x)= x² - 1, 0< x < 2

2x, 2 ≤ x

Find

a) f(3). 6

b) f(-2). -1

c) f(1/2). 3/4

d) f(2-h) h²- 4h+3

e) f(-1+ h). h

4) 3x-2 when x ≤ 0

If f(x)= x+ 1 when x > 0

Find

A) f(-1). -2

B) f(0). -5

5) 2x²+1 ; x ≤ 2

If f(x)= 1/(x -2) ; 2< x ≤ 3

2x -5 ; x > 3

Then find the value of

A) f(-1)

B) f(0)

C) f(√2)

D) f(-2)

E) f(4)

F) f(2.5)

EXERCISE - C

1) If f(x)= log{(1-x)/(1+x), then f(p) + f(q)= f{(p+q)/(1+pq)}.

2) If f(x)= 2x √(1- x²), then f(sin (x/2))= sin x

3) If f(x)= x(x-a)/(b-a) + x(x-b)/(a-b), then f(a) + f(b)= f(a+b).

4) If f(x)= (x-1)/(x+1), then {f(x) - f(y)}/{1+ f(x). f(y)}= (x-y)/(x+y)

5) If f(x) = (x-p)/x + x/(x-q) then f{(p+q)/2}= 4pq/(p² - q²).

7) If f(x)= (ax+b)/(bx+a), then f(x).

f(1/x)= 1.

EXERCISE - D

1) If f(x)= (ax + b)/(bx - a), find f{f(1/x)}. 1/x

2) If f(x)= (1- x)/(1+ x), then show that f{f(x)} = x.

3) f(x)= (x+1)/(x+2), find f{f(1/x)}. (3x+2)/(5x+3)

4) if f(x) = (4x -5)/(3x -4), find the value of f{f(x)}.

5) if f(x)= 1/(1+ x) then find f[f{f(x)}].

EXERCISE - E

1) If f(2x -1)= (3x+1)/(x-1) then find f(2- x). (11-3x)/(1-x)

2) f{(x-1)/(2x+1)= 2- x , then find f(3x-1). (2-5x)/(1-2x)

3) If f(2x-1)= (3x-1)/(x+1) find

a) f{f(4)}. 23/17

b) f{f(1-3x)}. (8-15x)/(8-9x)

4) If g(x -1)= 7x -5 then find the value of

g(x +2).

5) if g(2x -1)= (x+1)/(x+2). Then find

A) g(x).

B) g(2)

C) g(0)

D) g(h+1)

E) g(2+ h).

F) g(2/h)

G) g(- x)

6) Given f{(x-2)/(x+3)= (x -1)/(2x+1), find the value of f(5 -2x)

7) f(x+3)= 3x²- 2x +5, then find the value of f(x -1).

8) If f(x +1)= x³ then find f(1- x).

EXERCISE - F

1) y = f(x)= (x+1)/(x+2). Then find

A) f(y).

B) f(1/x).

C) f{f(x)}.

2) y= f(x) =(ax + b)/(ax - a), then find the value of f(y).

3) If y= f(x)= (2-x)/(5+3x) and z= f(y), express z in terms of x. (7x+8)/ (12x+31)

4) If y= (x -3)/(2x+1) and z= f(y), express z = f(x).

5) If y= f(x)= (5x+3)/(4x-5) then show that f(y)= x

6) If y= f(x)= (3x+1)/(3x-m) and f(y)= x, find m. 2

7) If y= f(x)= (3x+4)/(5x-m) and f(y)= x, find m. 3

8) If y= f(x)= (3x -5)/(2x-m) and f(y)= x, find m.

EXERCISE - G

1) If f(x) = 2x² - 5x +4, for w

hat value of x is 2f(x) = f(2x)

2) Let f(x) =10x²- 13x +13. Find the value of x for which f(x) =16.

3) If f(x)= ax²+ bx + c and f(1)= 3, f(2)=7,

f(3)=13, find the value of a, b, c.

4) If f(x)= a/x + b + cx and f(1)=5, f(-2)=2, f(-1)=-3. Then find the value of f(-3).

EXERCISE - H

E) Show that the following are even

1) 5ˣ + 5⁻ˣ

2) x(eˣ + 1)/(eˣ-1)

3) 1/x log √{x + √(x²+1)}.

EXERCISE - I

F) Show that following are odd

1) x + x³

2) 5ˣ - 5⁻ˣ

3) (eˣ + 1)/(eˣ -1)

4) log {(1+x)/(1-x).

5) log {√(1+x²)+ x}

6) log {√(1+x²) - x}

7) If f(x)= x³ -(k-2)x²+ 2x, for all x and if it is an odd function, find k. 2

EXERCISE - J

G) Find the domain of the following

1) 1/(x²- 3x +2).

2) x²/(x²- 5x +6).

3) (x+3)/(x²- x -2).

4) (x²- 5x +6)/(x²- 8x + 12).

5) 1/(x²- 4).

6) x²/(x²- 25).

7) 1/(x²+1).

8) √(x -10)

9) √(x -a)

10) √(6 -x)

11) √(k -x)

12) √(x² - 5x +6)

13) √(x²- 3x +2)

14) √(x²- 7x +12)

15) √(x²- x - 2)

16) √(x²- 8x +12)

17) √(x²- 4)

18) √(x²- 25)

19) 1/√(x²- 5x +6)

20) 1/√(x²- 7x +12)

21) 1/√(x²- 3x +2)

22) 1/√(x²- 9)

23) 1/√(21-x)

24) 1/√(9- x²)

25) 1/√(16-x²)

EXERCISE- K

Find the range of the following:

1) x/(1+ x²).

2) x²/(1+ x²).

3) x/(x² - 5x +9).

4) (3x-5)/(x² -1).

5) √(4- x²).