TEST PER ON ALGEBRA(XI)

1) Find the smallest positive integer n, for which {(1- i)/(1+ i)}ⁿ is purely real. 

2) If ˣ√a = ʸ√b= ᵖ√c and a,b,c are in GP., show that x,y,p are in AP.        

3) If the roots of ax²+ bx+ c = 0 are both positive, then

A) a>0, b>0, c<0

B) a>0, b<0, c>0

C) a<0, b>0, c<0

D) a<0, b<0, c>0

4) The sum of n terms of a GP is 1/3 (2²ⁿ⁺¹ -2); find its common ratio.

5) Of the numbers 2, x, y, 12 (x>0, y>0), if three are in GP and last three in AP, find x and y.

6) If A be the sum and R the sum of the reciprocals of n terms of a GP, whose first term is 4 and last term 36, find the value of S/R.

7) Find the amplitude of (1+ sinx+ i cosx).

8) Express (1+ ix)(1+ iy)(1- ix)(1- iy) as sum of two squares.

9) If z= x + iy be a complex number, then |z²| equals to

A) z² B) conjugate of z² c) z. conjugate of z d) none

10) For what value of k, does the equation 8x²+ 2x + k=0 have its one root square of the other?

11) In an examination a candidate has to pass in each of the four subjects. The number of ways he can fail in the examination is

A) 1 b) 15 c) 16 d) 24

12) Which term in term in (x + 1/x²)⁹ is independent of x ?

a) 2nd b) 3rd c) 4th d) 5th

13) Show that the sum of the coefficient of xʳ and xʳ⁻¹ in the expansion of (1+ x)ⁿ is equal to the coefficient of xʳ in (1+ x)ⁿ⁺¹.

14) If p, q are the roots of equation ax²+ bx + c=0, evaluate 1/(ap + b) + 1/(aq + b).

15) For what value of p, the expression x²+ 4x +4 and x²+ px + 6 have a common factor?

16) For what values of m, the roots of x²- (m -1)x + m + 1/4= 0 are real and equal?

17) For what value of k, the roots of kx(1- x)= 1 will not be real?

18) In a GP, sum of n terms is 255, the last term is 128 and the common ratio is 2. Find n.

19) Find the square root of 9i.

20) The sum of n terms of a series is n/3 (4n²-1). The 5th term of the series is

a) 81 b) 84 c) 121 d) 165

21) At an election, a voter may vote for any number of candidates not greater than the number to be chosen. There are 6 candidates and 4 of them are to be chosen. Find the number of ways in which a voter may vote.

22) Find the middle terms in the expansion of (1+ x)⁹?

23) The middle term of the progression 3,7,11,......147 is

a) 71 b) 73 c) 75 d) 77

24) If p,q are the roots of x²- 2x +2=0, the least integer n (>0) for which pⁿ/qⁿ =1 , is

a) 2 b) 3  c) 4  d) none

25) A polygon has 44 diagonals; the number of its sides is

a) 8 b) 7 c) 11  d) none

MISCELLANEOUS QUESTIONS -XII(2023/24)

RELATION AND MAPPING

1) For all x∈ z, the function f: z --> is defined by f(x)= 3x+4, Find the function goz --> z  such that (gof=I.         g(x)= (x -4)/3 where x∈z

2) Find the number of equivalence relation on set A={a,b,c} containing elements (b,c) and (c,b).           2

3) If f(x)= sin x, g(x)= x² and h(x)= log x, find the composite function [ho(gof)](x).    2 log(sinx)

4) Let N be the set of all natural numbers and R be the relation on N x N defined by (x,y) R(z,t) => xt(y + z)= yz(x +t). 

 Check whether R is anequivalence relation on N x N.          Yes

5) Let R be the set of real numbers and X={x ∈ R : -1 < x < 1}= Y. Is the mapping f: x --> R defined by f(x)= (2x -1){1 - |2x -1|} bijective ?     Yes

TRIGONOMETRIC FUNCTION

1) PROVE:

a) tan⁻¹{(2 sin2x)/(1+ 2 cos2x)} -1/2  sin⁻¹{(3 sin2x)/(5+ 4 cos2x)}= x.

b) tan⁻¹{(3 sin2x)/(5+ 3 cos2x)} + tan⁻¹{(1/4  tanx)} = x.

c) 2 sin⁻¹(2/√13) + 1/2 cos⁻¹(7/25)+ tan⁻¹(63/16)=π.

2) If cos⁻¹x + cos⁻¹y + cos⁻¹z =π and x + y + z =3/2, then show that x= y= z.

3) If tan⁻¹a + tan⁻¹(1/b)= tan⁻¹3, then find the positive integral values of a and b.    1,2 or 2,7

4) If tan⁻¹[{√(1+ x²) - √(1- x²)}/{√(1+ x² + √(1- x²)}] = θ  ; then show that sin2θ  = x².

5) If (n tanθ)/cos²(α -θ) = {m tan²(α -θ}/cos²θ} then show that 2θ= α - tan⁻¹{(n - m)/(n + m)  tanα}.

6) If φ = tan⁻¹{x √3/(2k - x)} and θ = tan⁻¹{(2x - k)/k √3}, then show that one value of φ - θ is 30°

7) If sin⁻¹(x/a) + sin⁻¹(y/b) = sin⁻¹(c²/ab),  then show that b²x² + 2xy √(a²b² - c⁴) + a²y² = c⁴.

8) Solve: cos⁻¹x - sin⁻¹x = cos⁻¹(x √3).         0, ±1/2

DETERMINANT

By using properties of determinant. Prove that:

1) (x +4)        2x            2x

        2x         x +4          2x  = (5x +4)(x -4)²

        2x          2x           x+4

2) 1        1+ x           1+ x + y

     2        3+2x         1+3x +2y     =1

     3        6+3x         1+6x+3y

3) a         a²        1+ xa³ 

     b         b²        1+ xb³  

     c         c²         1+ xc³

= (1+ abcx)(a - b)(b - c)(c - a)

4) (y + z)²       x²          x² 

         y²       (z + x)²      y²  = 2xyz(x + y+ z)³

         z²            z²     (x + y)²

5)  - yz        y²+ yz        z²+ yz 

   x²+ zx         -zx           z²+ zx 

   x²+ xy      y²+ xy           -xy

= (xy + yz + zx)³

6) (y + z)²        z²           y² 

         z²        (z + x)²       x² 

         y²             x²      (x + y)²

= 2(xy+ yz + zx)³

7) ax - by - cz        ay+ bx              cx+ az

      ay + bx          by - cz- ax          bz+ cy

      cx+ az              bz+ cy          cz - ax- by

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

METRICES

1) If A=  -4       4       3  & B= 1     -1      1

               -7       1       3           1     -2     -2

                5      -3      -1           2      1      3 find AB and use this to solve the following system of equation: x - y + z=4; xx - 2y - 2z=0 ; 2x + y + 3z= 1.       3,-2,-1

2) Let X= 3       2      5

                 4       1      3

                 0       6      7 express X as sum of two matrices such that one is symmetric and other is skew-symmertric.

LIMIT, CONTINUITY AND DIFFERENTIABLE

1) lim ₓ→∞ √x{√(x +3) - √x}.                 3/2

2) lim ₓ→₀ {(x -1 + cosx)/x}¹⁾ˣ .             e⁻¹⁾²

3) lim ₓ→∞ {(x +5)/(x +1)}ˣ.                 e⁴

4) lim ₓ→π/2    (1+ cosx)³ˢᵉᶜˣ.             e³

5) lim ₓ→₀ (1+ 3x)^(x+3)/x.                 e³

6) lim ₓ→₀ {log(x²+ x+1)+ log(x²- x+1)}/(secx - cosx).       1

7) lim ₓ→₀ {xeˣ - log(x +1)}/x².            3/2

8) lim ₓ→₀ {tan(π/4  + x)}¹⁾ˣ.               e²

9) lim ₓ→₀ (sinx + cosx)¹⁾ˣ.                 e

10) lim ₓ→₀ {sin log(1+ x)}/log(sinx +1).        1

1) Prove that the function f(x)= sinπ |x| is continuous at x =0 but not differentiable at the same point.     

2) If g(x) is the universe of f(x) and f'(x)= (1+ x³)⁻¹, show that g'(x)= 1+ {g(x)}³.

3) A function f(x) is defined as follows:

                 - 2sinx,        when -π ≤ x ≤ -π/2

 f(x)=          a sinx + b, when -π/2< x < π/2

                     cosx,        when  π/2≤ x ≤ π

If f(x) is continuous in the interval -π≤ x ≤π. find the value of a and b.     -1,1

DIFFERENTIATION

1) Find dy/dx when

a) xˢᶦⁿ ʸ + yˢᶦⁿˣ =1.          

b) y= tan⁻¹{x/(1+ √(1- x²))}+ sin(2tan⁻¹√{(1- x)/(1+ x)}.

c) (√x)ˣ + (x)^√x.

d) y= tan⁻¹(√x - ⁴√x)/(1+ ⁴√x³).

e) xˡᵒᵍˣ + (sinx)ˣ+ 15x.

2) Find d²y/dx²

a) x= e⁻ᵗ and y = te⁻ᵗ.

b) x = a sin³t and y= a cos³t at t=π/4.

c) x= a(t - sint) and y= a(1+ cost) at t=π/2

d) x=√3(3 sint + sin3t) and y= √3(3cost + cos3t) at t=π/3.

e) if 3px²= y²(p - x⁶), show dy/dx = y³/x³ - 2y/x.

f) If y²(1- x²)= x²+1, show that (1- x⁴)(dy/dx)²= y⁴-1.

g) If √(1- x⁴) + √(1- y⁴)= k(x²- y²), show dy/dx= x√(1- y⁴)/y√(1- x⁴).

h) If f(x)= sin(logx) and y= f{(2x+3)/(3- 2x)} show that dy/dx= 12/(9- 4x²)  cos{log{(2x+3)/(3- 2x)}}.

i) If x= sect - cost, y= secⁿt - cosⁿt, then show that (x²+4)(dy/dx)²= n²(y²+4).

j) 

the decreases for all real values of prove that symmetry with respect if the lion is a normal falling programming problems exactly maximum subject to constraint two Taylor sex 150 200 respectively extensity 6000 problems to minimise the labour cost to produce at least two points two numbers are selected random from 1 to 300 multiply find the probability of the product is divisible by 3% the boys and 10% the girls 60% students and boys

QUICK REVISION (APPLIED MATHS)-XII

28/11/24

1) y= (x -1)eˣ then dy/dx at x=1

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

2) Which one of the following is the value of ∫3ᵃˣ dx ?

a) 3ᵃˣ⁺¹ b)3ᵃˣ c) 3ᵃˣ. logₑ3ᵃ d) 3ᵃˣ/alogₑ3

3) Solve by cramer's rule: x + y-3=0; x + 2y- 5=0.

4) What are the order and degree of d³y/dx³= 4 √(x² dy/dx +7).

5) For what value of x will (x -1)(3- x) have its maximum.

6) Find the area bounded by y= 4x -3, y= 0, x=1 and x=3.

7) The effective rate of interest corresponding a nominal rate of 7% p.a convertible quarterly is

a) 7% b) 7.5% c) 5% d) 7.18%

8) A boat sails a distance of 44km in 4 hours with the current. It takes 4 hours 48 minutes longer to cover the same distance against the current. Find the speed of the boat in still water and the speed of the current.

9) Ravi invested Rs4800 in 6% Rs100 shares quoted at Rs120. Calculate 

a) number of shares held by him.

b) Ravi's income from the investment.

        Ravi sold these shares, when they were quoted at Rs140 and invested the proceeds in 5% Rs10 shares quoted at Rs8. Calculate Ravi's income now (from the newly bought shares) and his percentage return now on his initial investment.        40, 240, 350, 175/24%

10) Ravi invested Rs6840 in buying shares of nominal value Rs30 which are being sold at 20% premium. The dividend on the shares is 12% per annum. Calculate 

a) the market value of shares. 

b) the number of shares bought by Ravi.

c) the dividend Ravi would receive at the end of the year. 36,190,684

27/11/24

1) The probability that a leap year will have 52 Tuesday or Saturday is

a) 2/7 b) 3/7 c) 4/7 d) 1/7

2) If A= 2      -1

            -1       2 and I is the unit matrix of order 2, then find A².

3) Find dy/dx of x³ w.r.t. x³.

4) If 2ˣ + 2ʸ = 2ˣ⁺ʸ, then the value of dy/dx at x= y= 1.

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

5) The maximum value of xy when x + 2y = 8 is

a) 20 b) 16 c) 8 d) 24

6) 1) The fixed cost of new product is ₹18000 and the variable cost per unit is ₹550. If the demand function is p(r)= 4000 - 15x, find the break-even points.

7) 5) A man cover a certain distance from house to office, if he travel at 30 km per hour, then he is late by 10 mins, but if he travels at 40 km/hr then he reaches his office his 5 minutes earlier. Find his distance from home to office. 

8) The value of a machine depreciated by 10% per year during the first two years and 15% per year during the third year. Express the total depreciation of the machine, as percent during 3 years.

9) Find the value(s) of which y=(x²-2x)² is an increasing function.

10) ∫ (e⁵ˣ + e³ˣ)/eˣ dx

Test Paper on Trigonometry - XI

1) If a sin²x + b cos²x = c, b sin²y + a cos²y = d, and a tanx = b tany, show that, 1/a + 1/b = 1/c + 1/d.

2) Evaluate: sin⁴π/8 + sin⁴3π/8 + sin⁴5π/8 + sin⁴7π/8.

3) Show: sin(2 tan⁻¹1/2)= cos(2tan⁻¹1/3).

4) If (sin⁴x)/a + (cos⁴x)/b = 1/(a + b), show (sin¹⁰x)/a⁴ + (cos¹⁰x)/b⁴ = 1/(a+ b)⁴.

5) Evaluate: 1/(2sin10)  - 2 sin 70.    

6) If tanx, tany are the roots of x²+ px + q =0, find the value of sin²(x + y) + p sin(x +y) cos(x + y)+ q cos²(x + y).    

7) If tan(x + y - z)/tan(x - y + z) = tan z/tan y, show that sin(y - z)= 0 or sin2x + sin2y + sin2z =0.

8) Show that cot70 + 4 cos 70= √3.

9) If cos(x - y)+ cos(y - z)+ cos(z - x)= - 3/2, show that, cos(x - y)= cos(y - z)= cos(z - x)= -1/2.

10) If a, b are two values of x satisfying a tanx + b secx = c, show tan(a+ b)= 2ac/(a²- c²).

11) If (tan(a - b))/tana + sin²c/sin²a = 1, show that tana tanb = tan²z.

12) If tanx = (tan a+ tanb)/(1+ tana tanb), show sin2x= (sin2a + sin2b)/(1+ sin2a sin2b).

2) 3/2 5) 1 6) q

REVISION MATHS - IX(2024/25)

5/10/24

1) If cotx = 7/7.5, then cosecx is

a) 7.5/4 b) 8/17 c) 17/15 d) 15/17       

2) If 2x = secA and tanA= 2/x then the value of 2(x²- 1/x²)²= ?

a) 1/2 b) 1/4 c) 1/8 d) 1/16     

3) The value of (sin43° . cos47°+ cos43° sin47°) is 

a) 0 b) 1 c) sin4° d) cos4°     

4) If tanx = 4/5, then cosx =

a) 4/5 b) 3/5 c) 3/4 d) 5/√41    

5) If sinx = 1/√2, then sec2x =

a) 0 b) 1 c) 2 d) none    

6) The value of ( tan35/cot55 + cot78/tan12) is 

a) 0 b) 1 c) 2 d) none      

7) ABC is a triangle. Then sin{(B+ C)/2}=

a) sin(A/2) b) cos(A/2) c) sinA d) cosA     

8) The simplest value of cos53°/sin37° is _____.     

9) if tan35° tan55°= sinx, then lowest positive value of x will be_____.     

10) If cos²x - sin²x = 1/x (x > 1), then cos⁴x - sin⁴x = ____.       

11) The value of (sin 12 . cos 18. sec 78. Cosec72) is___.  

12) The value of tan 15 tan 45 tan 60 tan 75 is ____.     

13) if tanx = 4/5, then x = ____.    

14) If sinx =1/2, then cos2x =_____.    

15) cosx= √3/2, then sin2x=_____. √3/2 

16) The value of (4/sec²x + 1/(1+ cot²x) + 3 sin²x) is ____.    

17) If sinx =1/2, then tan2x =___.       

18) If sin(x - 30°)= 1/2, then the value of cosx is_____.      

   

4/10/24

1) A person deposited Rs100 in a bank and gets the amount Rs121 after 2 years. The rate of compound interest is____%. 

2) If the simple interest for n years at r% p.a. be Rs Pnr/25, then the principle will be Rs____.

3) At same rate percent per annum, the simple interest and compound interest of same principal are same in ____ year. 

4) A person depreciates at a certain rate over time is called____. 

5) The person who gives loan is called____. 

6) Amount of Rs2P per t years at the rate of simple interest r/2% per annum (2P+ ____) Rs.

7) if the ratio of principle and amount for 1 year is 8:9, then the rate of the simple interest per annum is_____.

8) Fixed amount rupee fixed annual interest rate one year compound interest rate and simple interest rate ____. 

9) With the passage of time, someone grows at a certain rate , it is called___. 

2/10/24

1) if a principal becomes twice of it in 10 years, then the rate of a simple interest for annum is 

a) 5% b) 10% c) 15% d) 20%.

2) Interest on Rs a at the simple interest 10% per annum for b months is 

a) ab/100 b) ab/120 c) ab/1200 c) ab/10. 

3) If the ratio of principal and yearly amounts be in the ratio 25:28, then the yearly rate of interest is 

a) 3% b) 12% c) 75/7% d) 8% 

4) If the total interest becomes Rs x for any principal having the rate of simple interest of x% per annum for x years then the principal will be

a) Rsx b) Rs 100x c) Rs 100/x d) Rs 100/x² 

5) The total interest of a principal in n years, at the rate of simple interest of r% per annum is one/109, the principle will be

a) Rs2p b) Rs4p c) Rs3p d) 5p. 

6) If the interest on Rs p at the rate of simple interest of r% per annum in t years is I, then 

a) I= prt b) prt I= 100. I c) prt = 100. I d) none. 

7) A principal becomes twice of its amount in 20 years at a certain rate of simple interest. At that same rate of simple interest, that principal becomes thrice of its amount in 

a) 30 years b) 35 years c) 40 years d) 45 years 

8) A sum of Rs400 amounts to Rs480 in 4 years. What will it amount to if the rate of interest is increased by 2% ?

a) Rs484 b) Rs560 c) Rs512 d) none 

9) At what rate of percent per annum will Rs2304 amount to Rs2500 in 2 years at compound interest ?

a) 9/2% b) 21/5% c) 25/6% d) 13/3% 

10) An amount doubles itself in 5 years with simple interest. What is the amount of interest percent per annum?

a) 10% b) 20% c) 25% d) 30% 

11) A person deposited Rs109 in a bank and got the amount Rs121 for 2 years. The rate of compound interest is

a) 10% b) 20% c) 5% d) 21/2% 

12) In case of compound interest, the rate of compound interest per annum is 

a) equal b) unequal c) both equal or unequal d) none.

13) In case of compound interest 

a) The principals remains unchanged each year 

b) principal changes in each year

c) principal may be equal or unequal in each year d) none 

27/8/24

1) On What sum of money, the difference be

tween the simple interest and compound interest in 2 years at 5% per annum is Rs15 ?

2) A certain sum of money invested at 5% intrest, compounded annually, for 3 years. If the interest computes to Rs2522, determine the principal.

3) In how many years will a sum of Rs800 at 10% per annum compounded semi-annually become Rs926.10 ?

4) Suraj has a fixed deposit in Bank of India of Rs40000 for a period of 3 years. The bank allows a compound interest of 13% compounded half yearly. Find the maturity value.

Day- 8

1) A bar graph is drawn to the scale 1cm= k  units, then a bar of length k cm represents 

a) 1 unit  b) k units c)  2k units  d) k² units 

2) A bar graph is drawn to the scale of 1 cm = x units. If the length of a bar representing a quantity of 702 units is 3.6cm, then x=

a) 165  b) 175 c) 185 d) 195 

3) In figure

bar graph represents sales of two wheelers and four wheelers in a mega city from 2013 to 2016. In which year the difference between the sales of two wheelers and four wheelers is less ?

a) 2013 b) 2014 c) 2015  d) 2016 

4) In the figure, the total number of vehicles (two wheelers and four wheelers ) sold in the year 2013 and 2014 is

a) 26100 b) 28500 c) 25100 d) 27500 

5) In figure, the maximum difference between sales of two wheelers and that of four wheelers, in any year, in the given period is :

a) 1500 b) 1700  c) 1800  d) 2000

6) In figure, the total number of two wheelers sold in four years is

a) 26000 b) 27000 c) 31000 d) 32000

7) in a bar graph, the height of a bar is 5cm and it represent 40 units . The height of the bar representing 56 units is:

a) 11.2cm  b) 5.6cm c) 7cm d) 8cm

8) in a bar graph, length of a bar is 6.4cm and it represent 256 units. The number of units represented by a bar of length 5.3cm is

a) 228  b) 196 c) 212  d) 224

9) In a bar graph, the height of a bar is proportional to the 

a) width of the bar  b) range of the data  c) value of the component d) number of observation in the data.

10) Which one of the following is not the graphical representation of statistical data ?

a) bar graph  b)?histogram c) frequency polygon  d) cumulative frequency distribution

11) In a frequency distribution, ogives are graphical representation of 

a) frequency b) relative frequency  c) cumulative frequency  d) raw data.

12) A frequency polygon is constructed by plotting frequency of the class interval and the

a) upper limit of the class  b) lower limit of the class c) mid value of the class d) any values of the class

13) In a Instagram the area of each rectangle is proportional to

a) the class marks of the corresponding class interval.

b) the class size of the corresponding class interval

c) frequency of the corresponding class interval.

d) cumulative frequency of the corresponding class interval .

14) In the 'less than' type of ogive the cumulative frequency is plotted against 

a) the lower limit of the concerned class interval.

b) the upper limit of the concerned class interval .

c) the mid value of the concerned class interval.

d) any value of the concerned class interval.

15) In a histogram the class interval or the groups are taken along

a)  y-axis  b) x-axis  c) both of x-axis and y-axis  d) in between x and y-axis .

16) A histogram is a pictorial representation of the grouped data in which class intervals and frequency are represpectively taken along

a) vertical axis and horizontal Axis 

b)!vertical access only 

c) horizontal Axis only 

d) horizontal axis and vertical axis.

17) In a histogram, each class rectangle is constructed with base as 

a) frequency  b) class intervals  c) range  d) size of the class

18) Consider the following frequency distribution :

Class interval      Frequency 

5-10                         6

10-15                      12

15-25                      10

25-45                        8

45-75                      15

To draw a histogram to represent the above frequency distribution the adjusted frequency for the class 25-44 is 

a) 6 b) 5 c) 3 d) 2

19) Figure shows the bar graph of number of boys and number of girls in a school from 2014 to 2017.

In which year the difference between the number of boys and the number of girls was more ?

a) 2014  b) 2015  c) 2016 d) 2017 

20) In figure, total number of students in the year 2015 was

a) 1160 b)  1270 c) 1380 d)  1490

21) In figure , the minimum difference between the number of boys and girls in any year in the given period was

a) 90 b) 70 c) 50 d) 30 

22) In figure, in which year the number of girls more than the number of boys?

a) 2014 b) 2015 c) 2016 d) 2017

23) In figure , the ratio between the number of student in the year 2016 and 2017 was

a) 107 :145 b) 127 : 145  c) 29 :36 d) 107: 127

CASE STUDY 

1) Following bar graph represents the sales of the cold drinks of two companies A and B from 2015 to 2018.

Read the above bar graph and answer the following questions :

i) The year in which the difference between the sells of two companies was highest, was 

a) 2018  b) 2015 c) 2016  d) 2017 

ii) Total sales of A and B in the year 2016 was 

a) 1160000 b) 1270000 c) 1380000 d) 1490000

iii) The minimum difference between the sales of company A and B in any year in the given period was 

a) 90000 b) 70000 c) 50000 d) 300000

iv) In which year was the sales of company B more than the sales of company A?

a) 2015 b) 2016  c) 2017  d) 2018

v)  The ratio of the total sales in the year 2017 and that in 2018 was 

a) 107 :145 b) 29:36  c) 127: 145 d) 107: 127

2) Read the following bar graph and answer the following questions:

i) In the which year was the difference between sales of the scooters and the sales of cars the least ?

a) 2015  b) 2016 c) 2017 d) 2018

ii) Total number of the vehicles (scooters and cars) sold in the year 2015 and 2016 was

a) 26100 b) 28500 c) 25100 d) 27500

iii) The maximum difference between the sales of scooters and cars , in the given period was

a) 1500  b) 1700 c) 1800 d) 2000

iv) The total number of scooters sold in the 4 years was

a)  26000 b)  27000 c)!31000 d) 32000

v) The ratio between the total number of vehicles sold (scooters and cars) in the year 2016 that in the year 2018.

a) 41: 46  b) 69: 91 c) 147 :182 d) 46: 49 

3) Population census in India is conducted every 10 years. The first complete census was taken in 1881 and 15th decennial census taken in 2011. The 16th decennial census was to be conducted in 2021 but due to the COVID it will be taken in 2022. The data obtained from the census of a town has been represented by a bar graph shown in figure. It represents the number of persons living in various age groups in the town. Observe the bar graph and answer the following questions:

i) What is the total of persons living in the town in the age-groups 10-15 and 60-65 ?

a) 2000 b) 2200 c) 2100 d) 1900

ii) How many persons are more in the age group 10 to 15 than in the age group 30 to 35 ?

a) 200 b) 250 c) 300 d) 350

iii) What is the total population of the town ?

a) 6700 b) 6400 c) 7700 d) 6600

iv) What is the number of persons in the age-group of 60-65 ?

a) 900 b) 750 c) 850 d) 800

v) What is the age group of exactly 1200 persons living in the town ?

a) 10 to 15 b) 20-25 c) 30-35 d) 40-45

4) A healthcare survey was done by the State Health and Family Welfare Care Board of the State of Punjab. The data is collected by forming age groups i.e.,10 - 15, 20 -25, 30 -35, 40 -45, 50 -55, 60- 65, 70-75. The overall data from a town is the given below in the form of a bar graph. Read the data carefully and answer the question that follow :

i) How many persons are more in the age group 10 - 15 than the age group 30-35?

ii) What is the age group of exactly 1200 persons living in the town?

iii) What is the percentage of the youngest age group persons over those in the oldest age group?

iv) What is the total population of the town ?

Assertion- Reason 

Each of the following examples contains STATEMENT-1 (Assertion) and STATEMENT-2 (Reason) and has following four choices (a), (b), (c), and (d) only one of which is the correct answer. Mark the correct choice.

a) Statement -1 and Statement -2 are true Statement-2 is a correct explanation for statement-1.

b) Statement-1 and statement-2 are true; Statement-2 is not a currect explanation for statement-1.

c) Statement -1 is true, Statement -2 is false.

d) Statement -1 is false, Statement -2 is true.

1) Statement -1(A): The graphical representation of the frequency distribution

 Marks: 0-20 20-40 40-60 60-100 

No. of Student: 10 15 20. 25

is as given figure 

statement-2 (R): In a histogram, the area of the rectangles are proportional to the frequencies. c

2) Statement -1(A): A bar graph is a pictorial representation of the numerical data by a number of rectangles of uniform width erected horizontally or vertically with equal spacing between them.

Statement-2 (R): In order to draw the histogram when mid-points of class intervals are given, it is assumed that the frequency corresponding to the variate value a (say) is spread over the interval a - h/2 to a + h/2 , where h is the jump from one value to other. b

3) Statement -1(A): In a histogram, the areas of each rectangle is proportional to the frequencies of its classes.

Statement-2(R): in a histogram the lengths or height of rectangles are proportional to the frequencies .

4) Statement -1(A): in a histogram, the area of each rectangle is proportional to the class size of the corresponding interval.

Statement-2 (R): To draw the histogram of a continuous frequency distribution with unequal class intervals, the frequencies of classes are adjusted by using the formula:

Adjusted frequency of a class= minimum class size/ class size   x frequency of the class.

5) Statement -1(A): To draw the histogram of a continuous frequency distribution when class marks of class intervals are given, it is asumed that the frequency corresponding to the class mark a is spread over the interval a - h/2 to a + h/2, where h is the jump from one value to the other.

Statement-2 (R): The class marks of a continuous frequency distribution are:

1.04, 1.14, 1.24, 1.34, 1.44, 1.54, 1.64, then the last interval is 1.55-1.73.

Day- 7(19/6/24)

1) (a - b)³+ (b - c)³+ (c - a)³ is equal to 

a) 2a³+ 2b³+ 2c³ 

b) (a - b) (b - c)(c - a)

c) 0

d) 3(a - b)(b - c)(c - a)

2) If x + y=12 and xy = 27, then x³+ y³=

a) 765 b) 756 c) 657 d) 675

3) If x+ y= -4 then x³+ y²- 12xy +64=

a) -64 b) 128 c) 0 d) none

4) If x = 2y+6, then x³- 8y³- 36xy= 

a) 216 b) -216 c) 36 d) -36

5) (a+ b+ c){(c - b)²+ (b - c)²+ (c - a)²}=

a) a³+ b³+ c³- 3abc b) a³+ b³+ c³ c) 2(a³+ b³+ c³- 3abc) d) 3abc

6) If a³+ b³= 5 and a+ b=1, then ab= 

a) -4/3 b) 4/3 c) -3/4 d) 3/4

7) If a³+ (b - a)³ - b³ = k(a - b), then k= 

a) ab b) 3ab c) -3ab d) 3

8) If a+ b+ c= 0, then a²/bc + b²/ca + c²/ab= 

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

9) The factor of x³ - x²y - xy²+ y³, are 

a) (x+y)(x²- xy+ y²) 

b) (x+y)(x²+ xy+ y²) 

c) (x+y)²(x- y) 

d) (x-y)²(x + y) 

10) The factor of x³ - 1 +y³ + 3xy, are 

a) (x-1+y)(x²+1+ y²+ x + y - xy) 

b) (x+1+y)(x²+1+ y²+ 1- x - y - xy) 

c) (x-1+y)(x²-1- y²+ x + y - xy) 

d) 3(x-1+y)(x²-1+ y²) 

11) The factor of 8a³+ b³- 6ab +1 are 

a) (2a+ b-1)(4a²+ b²+1- 3ab- 2a)

b) (2a- b+1)(4a²+ b²+1- 4ab- 2a+ b)

c) (2a+ b+1)(4a²+ b²+1- 2ab- b -2a)

d) (2a+ b-1)(4a²+1- 2ab- b- 4a)

12) (x + y)³ -(x - y)³ can be Factorized as

a) 2y(3x²+ y²) b) 2x(3x²+ y²) c) 2y(3y²+ x²) d) 2x(x²+ 3y²)

13) The expression (a - b)³ + (b - c)³+ (c - a)³ can be Factorized as 

a) (a- b)(b - c)(c - a)

b) 3(a- b)(b - c)(c - a)

c) -3(a- b)(b - c)(c - a)

d) (a+ b+c)(a²+b² + c²- ab - bc - ca)

14) The value of {(2.3)³ - 0.027}/{(2.3)²+ 0.69+ 0.09}, is 

a) 2 b) 3 c) 2.327 d) 2.273

15) The value of {(0.013)³ +(0.007)³}/{(0.013)² - 0.013 x 0.007+ (0.007)²} is 

a) 0.006 b) 0.02 c) 0.0091 d) 0.00185

16) The factors of a² - 1 - 2x - x², are

a) (a - x +1)(a - x -1) 

b) (a + x +1)(a - x +1) 

c) (a + x +1)(a - x -1) d) none

17) The factors of x⁴+ x²+ 25, are

a) (x²+ 3x +5)(x²- 3x +5)

b) (x²+ 3x +5)(x²+ 3x -5)

c) (x²+ x +5)(x²- x +5) d) none

18) The factors of x²+ 4y²+ 4y - 4xy - 2x - 8, are

a) (x - 2y -4)(x - 2y +2)

b) (x - 2y +2)(x - 4y -4)

c) (x + 2y -4)(x + 2y +2) d) none

19) The factors of x³- 7x + 6, are

a) x(x -6)(x -1)

b) (x² -6)(x -1)

c) (x +1)(x +2)(x -3)

d) (x +3)(x -2)(x -1)

20) The expression x⁴+ 4 can be Factorized as 

a) (x²+ 2x +2)(x²- 2x +2)

b) (x²+ 2x +2)(x²+ 2x -2)

c) (x²- 2x -2)(x²- 2x +2)

d) (x²+2)(x²- 2)

21) If 3x = a+ b + c, then the value of (x - a)³+ (x - b)³+(x - c)³ -3(x - a)(x - b)(x - c), is 

a) a+ b + c b) (a - b)(b - c)(c - a) c) 0 d) none

22) If (x + y)³ - (x - y)³ - 6y(x²- y²)= ky³, then k=

a) 1 b) 2 c) 4 d) 8

23) If x³- 3x²+ 3x +7= (x +1)(ax²+ bx + c), then a+ b + c= 

a) 4 b) 12 c) -10 d) 3

24) If x/y + y/x = -1 (x,y ≠ 0), then the value of x³- y³ is 

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

25) Which of the following is a factor of (x + y)³ - (x + y³)?

a) x²+ y²+ 2xy 

b) x²+ y²- xy 

c) xy² d) 3xy

Assertion- Reason based 

Each of the following examples contains STATEMENT-1(Assertion ) and STATEMENT-2( (Reason) and has following four choices (a), (b), (c) and (d ), only one of which is the correct choice.

a) Statement-1 and Statement -2 are True; statement-2 is a correct explanation for statement-1

b) Statement-1 and statement-2 are True ; Statement -2 is not a correct explanation for Statement-1.

c) Statement -1 is True , Statement -2 is False .

d) Statement -1 is False , Statement -2 is True .

1) Statement -1 (A): The value 1000³ - 900³ - 100³ is 270000000

    Statement -2 (R): If a+ b + c= 0, then a³+ b³+ c³= 3abc. a

2) Statement -1 (A): The value of (0.093³+ 0.007³)/(0093² - 0.093 x 0.007 + 0.007²) is 0.1.

    Statement -2(R): a³+ b³= (a+ b)(a²- ab + b²). a

3) Statement -1(A): a³(b - c)³+ b³(c - a)³+ c³(a - b)³= 3(a - b)(b - c)(c - a)

Statement -2(R): if a+ b + c = 0, then a³+ b³+ c³= 3abc. d

4) Statement -1(A): (a+ b + c){(a - b)²+ (b - c)²+ (c - a)²}= 2(a³+ b³+ c³ - 3abc)

Statement -2(R) If a+ b + c = 0 then (a+ b)³+ (b + c)³+ (c + a)³= - 3abc. b

5) Statement -1(A): The product of (x²+ 4y²+ z²+ 2xy + xyz - 2yz) and (-z + x - 2y) is x³- 8y³- z³ - 6xyz

Statement -2(R): a³+ b³+ c³ - 3abc = (a + b + c)(a²+ b²+ c² - ab - bc - ca). a

Statement -1(A): a²+ b²+ c²- ab - bc - ca = 0 if and only if a= b= c.

Statement -2(R): a³+ b³+ c³ - 3abc = (a+ b + c)(a²+ b²+ c²- ab - bc - ca). b

6) statement-1(A): (a - b)³+ (b - c)³+ (c - a)³= 3(a - b)(b - c)(c - a)

Statement -2:(R): if a+ b + c = 0, then a³+ b³+ c³= 3abc. a

7) Statement-1(A): if 3x= a+ b + c, then (x - a)³+ (x - b)³+ (x - c)³= 3(x - a)(x - b)(x - c)

Statement -2(R): if a+ b + c= 0, then a³+ b³+ c³= 3abc. a

8) Statement -1(A): if a+ b + c = 5 and ab + bc+ ca= 10, then a³+ b³+ c³ - 3abc = 25

Statement-2(R): a³+ b³+ c³ - 3abc = (a+ b + c){(a+ b + c)² -3(ab + bc+ ca)}. d

9) Statement -1(A): If a,b,c are all non-zero such that a+ b + c = 0, then a²/bc + b²/ca + c²/ab = 3

Statement-2 (R): If a+ b + c = 9 and a²+ b² + c²= 35, then ab + bc+ ca= 23. b

10) Statement -1(A): The value of (0.027³+ 0.023³)/(0.027²- 0.027 x 0.023 + 0.023²) is 0.05

Statement-2 (R): a³- b³= (a- b)(a²- ab + b²). c

Day- 6 ( 14/6/24)

1) The square root of a²+ 1/a²+ 2 is

a) a+ 1/a b) a- 1/a c) a²+ 1/a² d) a²- 1/a²

2) The square root of a+ 1/a - 2 is 

a) a -1/a b) √a+ 1/√a c) ±(√a- 1/√a) d) a+ 1/a

3) The value of (a+ b)²/{(b -c)(c - a)} + (b + c)²/{(a - b)(c - a) + (c + a)²/{(a - b)(b - c)} is 

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

4) The square root of the expression (1/abc) (a²+ b²+ c²) +2(1/a + 1/b+ 1/c) is 

a) (a+ b+ c)/abc b) √a + √b + √c c) √(bc/a) + √(ca/b) + √(ab/c) d) √(a/bc)+ √(b/ca) + √c/ab)

5) The square root of x²/9 + 9/4x² - x/3 - 3/2x + 5/4 is

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

6) The square root of the expression is (xy + xz - yz)² - 4xyz(x - y) is 

a) xy + yz - 2xyz b) x + y - 2xyz c) xy + z - y d) xy + yz - xz

7) The square root of a²/4 + 1/a² - 1/a + a/2 - 3/4 is

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

8) The expression (4a + 5b+ 5c)² - (5a + 4b+ 4c)² + 9a² is a perfect square of the expression

a) √3(b + c) b) 3(a+ b + c) c) 3(b+ c) d) 3(-b + c - a)

9) The expression (3a + 2b+ 3c)² - (2a + 3b+ 2c)² + 5b² is a perfect square of the expression √(a + b+ c) b) √((a + b) c) √5(a +c) d) √5(a - b+ c)

10) If a/b + b/a =2, then (a/b)¹⁰ - (b/a)¹⁰ is equal to 

a) (2¹⁰-1)/2¹⁰ b) 2 c) 0 d) (2²⁰+1)/2¹⁰

11) If ab c = 6 and a + b + 6 = 6, then 1/ac + 1/ab + 1/bc =

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

12) √{(a + b+ c)²+ (a + b- c)²+ 2(c² -b²- a²- 2ab) is equal to 

a) 2c b) 2a c) 2b d) a + b+ c

13) If a/b + b/a = -1, then a³- b³=

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

14) If a+ b=8 and ab= 12, then a³+ b³=

a) 244 b) 224 c) 144 d) 284

15) If (a + 1/a +2)²=4, then a²+ 1/a²=

a) 12 b) 13 c) 14 d) -14

16) If x+ 1/x =7, then x³- 1/x³=

a) 9√5 b) 144√5 c) 135√5 d) √5

17) {(a - b)³ - (a + b)³}/2 + a(a²+ 3b²)=

a) a³- b³ b) (a + b)³ c) a³+ b³ d) (a - b)³

18) If x+ 1/x =5, then x²+ 1/x²=

a) 25 b) 10 c) 23 d) 27

19) If x+ 1/x =2, then x³ + 1/x³=

a) 64 b) 14 c) 8 d) 2

20) If x+ 1/x =4, then x⁴ + 1/x⁴=

a) 196 b) 194 c) 192 d) 190

21) If x+ 1/x =3, then x⁶ 1/x⁶=

a) 927 b) 414 c) 364 d) 322

22) If x² + 1/x² =102, then x- 1/x=

a) 8 b) 10 c) 12 d) 13

23) If x³+ 1/x³ =110, then x + 1/x =

a) 5 b) 10 c) 15 d) none

24) If x³- 1/x³ =14, then x- 1/x=

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

25) If a+ b+ c= 9 and ab+ bc+ ca= 23, then a²+ b²+ c²=

a) 35 b) 58 c) 127 d) none 

26) (a - b)³+ (b - c)³+ (c - a)³=

a) (a+ b+ c)(a²+ b²+ c²- ab - bc - ca)

b) (a - b)(b - c)(c - a)

c) 3(a - b)(b - c)(c - a) d) none 

27) a+ b= 3 and ab = 2, then a³+ b³=

a) 6 b) 4 c) 9 d) 12

28) If a- b =-8 auab = -12, then a³- b³=

a) -244 b) -240 c) -224 d) -260

29) if the volume of a cuboid is 3x²- 27, then its possible dimensions are 

a) 3, x, -27x b) 3, x -3, x+3 c) 3, x², 27x d) 3,3,3

30) 75 x 75 +2 x 75 x 25+25 x 25 is equal to 

a) 10000 b) 6250 c) 7500 d) 3750

31) (x - y)(x+ y)(x²+ y²)(x⁴+ y⁴) is equal to 

a) x¹⁶- y¹⁶ b) x⁸- y⁸ c) x⁸+ y⁸ d) x¹⁶+ y¹⁶ 

32) If x⁴+ 1/x⁴ =623, then x + 1/x=

a) 27 b) 25 c) 3√3 d) -3√3

33) If x- 1/x = 15/4, then x + 1/x =

a) 4 b) 17/4 c) 13/4 d) 1/4

34) If 3x+ 2/x = 7, then 9x² - 4/x² =

a) 25 b) 35 c) 49 d) 30

35) If a²+ b²+ c²- ab - bc - ca = 0, then 

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

36) If a+ b + c = 0, then a²/bc + b²/ca + c²/ab is

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

37) If a¹⁾³ + b¹⁾³ + c¹⁾³= 0, then

a) a+ b+ c= 0 b) (a+ b + c)³= 27abc c) a+ b + c = 3abc d) a³+ b³+ c³= 0

38) If a+ b + c = 9, then ab+ bc + ca =23, then a³+ b³+ c³- 3abc= 

a) 108 b) 207 c) 669 d) 729

39) {(a² - b²)³+ (b²- c²)³+ (c²- a²)³}/{(a - b)+ (b - c)+(c - a)}=

a) 3(a + b) (b +c)(c +a) b) 3(a - b)(b - c)(c - a)} c) (a - b) (b - c)(c - a) d) (a + b) (b +c)(c+ a)

40) The product (a + b)(a - b)(a²- ab+ b²)(a²+ ab+ b²) =

a) a⁶+ b⁶ b) a⁶- b⁶ c) a³- b³ d) a³+ b³

41) The product (x²-1)(x⁴+ x²+1) is equal to 

a) x⁸-1 b) x⁸+1 c) x⁶-1 d) x⁶+1

42) If a/b + b/a = 1, then a³+ b³=

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

43) If 49a²- b = (7a + 1/2)(7a - 1/2), then the value of b is 

a) 0 b) 1/4 c) 1/√2 d) 1/2

44) One of the factors of (5x +1)² -(5x -1)² is 

a) 5 + x b) 5- x c) 5x -1 d) 20x

45) If 9x² - b =(3x + 1/2)(3x - 1/2), then the value of b is 

a) 0 b) 1/√2 c) 1/4 d) 1/2

46) The Coefficient of x in (x +3)³ is 

a) 1 b) 9 c) 18 d) 27

47) The value of 249²- 248² is 

a) 1 b) 477 c) 487 d) 497

48) Which of the following is a factor of (x + y)³-(x³+ y³)?

a) x²+ 2xy + y² b) x² - xy + y² c) xy² d) 3xy

49) If x/y + y/x = -1 (x,y ≠ 0), the value of x³- y³ is 

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

50) If x + y=2 and xy = 1, then x⁴+ y⁴=

a) 6 b) 4 c) 8 d) 2

51) If x² + y²+ xy =1 and x + y = 2, then xy=

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

52) If a, b, c are natural numbers such that a²+ b²+ c²= 29 and ab + bc + ca = 26, and a+ b + c=

a) 9 b) 6 c) 7 d) 10

53) If 2x + y/3= 12 and xy = 30, then 8x³+ y³/27=

a) 1008 b) 168 c) 106 d) none

ASSERTION- REASON 

Each of the following examples contains STATEMENT-1 (Assertion) and STATEMENT-2 (Reason) and has following four choices (a), (b), (c) and (d), only one of which is the correct answer. Mark the correct choice.

a) Statement-1 and statement-2 are true; Statement-2 is a correct explanation for statement-1 .

b) Statement -1 and Statement-2 are true; Statement -2 is not a correct explanation for statement-1.

c) Statement -1 is true, statement -2 is false.

d) Statement -1 is False, Statement -2 is true.

1) Statement -1(A): √{(a+ b + c)+ (a - b + c)+2(b²- a²- c²- 2ac)}= 2b

Statement-2 (R): (x + y+ z)²= x²+ y²+ z²+ 2(xy + yz + zx). a

2) Statement -1(A): a³+ b³+ 3ab -1= (a+ b -1)(a²+ b²+ a+ b - ab +1)

Statement-2 (R): a³+ b³+ c³- 3abc= (a+ b + c)(a²+ b²+ c²+ ab + bc + ca). c

3) Statement -1(A): (a - b)³+(b - c)³+(c - a)³= 3(a - b)(b - c)(c - a)

Statement-2 (R): If a+ b + c = 0, then a³+ b³+ c³= 3abc. a

4) Statement -1(A): a²+ b²+ c²- ab - bc - ca = 0 if and only if a= b = c.

Statement-2 (R): (a+ b + c)²= a²+ b²+ c²+ 2ab + 2bc + 2ca. b

5) Statement -1(A): a+ b + c = 6 and 1/a + 1/b + 1/c = 3/2, then a/b + a/c + b/a + b/c + c/a + c/b = 6

Statement-2 (R): (a + b + c)²= a²+ b²+ c²+ 2(ab + bc + ca). b

6) Statement -1(A): if a+ b + c = 0, then a³+ b³+ c³= 3abc

Statement-2 (R): a³+ b³+ c³ - 3abc = (a + b + c)(a²+ b²+ c²-ab - bc - ca).     

7) Statement -1(A): (a+ b + c)² = a²+ b²+ c²-2(ab+ bc + ca)

Statement-2 (R): a³+ b³+ c³ - 3abc = (a + b + c)(a²+ b²+ c²-ab - bc - ca).     

8) Statement -1(A): a³ + 3ax/8 + cpx³/64 - 1/8 = (a + x/4 - 1/2)(a²+ x²/16 + 1/4 - ax/4 + x/8 + a/2)

Statement-2 (R): a³+ b³+ c³ + 3abc = (a + b + c)(a²+ b²+ c²+ ab + bc + ca).     

9) Statement -1(A): If a+ b + c =0, ab + bc+ ca = 11, then a²+ b² + c²= 14

Statement-2 (R): (a+ b+ c)³ = a²+ b²+ c²+ 2(ab + bc + ca).  

10) Statement -1(A): {(x²- y²)³+(y²- z²)³+(z³- x²)³}/{(x - y)³+(y - z)³+ (z - x)³}= (x + y)(y+ z)(z + x).

Statement-2 (R): If a + b + c= 0, then a³+ b³+ c³= 3abc.

11) Statement -1(A): (1/abc) (a²+ b² + c²)+ 2(1/a + 1/b+ 1/c) is √(a/bc) + √(b/ca) + √(c/ab).

Statement-2 (R): a³+ b³+ c³ - 3abc = (a + b + c)(a²+ b²+ c²-ab - bc - ca).     

Day- 5 (11/5/24)

1) simplify 

a) (a²+ 3b³)(a³- 2b²)

b)

 (ab - 3ad/2)(2ab + 3cd)

c) (2/5 + x)(2/5 - x)(4/25 + x²)

2) Simplify with the help of formula 

a) 88 x 112

b) 10.8 x 9.2

c) 10/3 x 14/3

3) Expand:

a) (8+ 3p)²

b) (4+ √5 y)²

c) (3a + 5b/3)²

d) (2/a - 3/b)²

e) (3x - 1/3x)²

4) If x+ 1/x = 4, find the value of 

a) x - 1/x b) x²+ 1/x² c) x⁴+ 1/x⁴

5) If x²+ 1/x²= 102, find the value of 

a) x - 1/x b) x + 1/x c) x²- 1/x² d) x⁴ - 1/x⁴

6) If a²+ b²= 13 and ab = 6, find the value of a+ b and a- b

7) If a²+ b²= 52 and ab = 24, find the value of a - b

8) Find the value of 36x²+ 49y²+ 84xy, when x= 3, y= 6.

Day -4 7/5/24

1) Factorize the following:

a) x²- 1 - 2a - a²

b) 2 - 50x²

c) 20x² - 45

d) a - b - a²+ b²

e) a²+ b - ab - a

f) x - 64x³

g) a(a + b - c) - bc

h) 1+ 2ab - a²- b²

i) a²+ b²+ 2ab - c²

j) ab(c²+ 1) + c(a²+ b²)

Day -3

1) The value of (0.538 x 0.538 - 0.462 x 0.462)/(1- 0.924)

2) The approximate value of of

(6.385x 6.385 - 5.385 x 5.385)/(6.385 x 6.385 + 2x 6.385 x 5.385 + 5.385 x 5.385)

3) The value of {856+ 167)²+ (856 - 167)²}/(856 x 856 + 167 x 167)

4) If a+ b + c= 0 then the value of (a + b - c)³+ (b + c - a)³+ (c + a - b)³

5) If (x + 1/x)= 3 then find the value of x²+ 1/x².

6) If x- 1/x = 1/2 then value of 4x²+ 4/x²

7) If 2x - 3/x then the value of 4x²- 9/x²

8) Factorize:

a) t⁴ - 16

b) x² - (y+1)²

c) x² + 1/x² - 3.

d) (x² - 2a - 1 - a²)

e) 2 - 50x²

Day- 2 

1) Find the value of (a+ b)² - (a - b)².

2) If a²bc²= 5³ and ab²= 5⁶ then find the value of abc.

3) (a+ b)²= a²+ 2ab + b² is true for all

a) natural numbers only b) integers only c) real numbers d) cannot say

4) If x+ y= 17 and x²+ y²= 167 then find the value of xy.

5) If m- n = 16 and m²+ n²= 400 then find the value of mn.

6) (a²/5 - b²/3)(a²/5 + b²/3) = ?

7) If (5x + 1/2)(5x - 1/2)= 25x² - p then find the value of p.

8) The square of (4x - 5y) is...

9) Expand (11x - 9xy)².

10) If x- y= 1 and x²+ y²= 41 then find the value of x + y

11) What is (a+ b)(a - b)(a⁴+ b⁴) equal to?

Day- 1

1) If (3x -4)(5x +7)= 15x²- ax - 28 then find the value of a.

2) The product of (x²+ 3x +5) and (x²-1) is.

3) In the product (2- y)(5- 3y)(1- 7y), the co-efficient of y² is...

4) Given that (3x -1)(x + p)= 3x²+ qx -2, find the value of p+ q - pq.

5) The value of (a+ b)²+ (a - b)² is 

6) If a²+ b²= 47 and ab = 19/2 then find the value of 2(a+ b)²+ (a - b)².

7) If ab= 6 and a+ b = 5 then find the value of a²+ b².

8) If x+ y =17 and x²+ y²= 167 then find the value of xy.

9) If a²+ b²= 74 and ab = 35 then find the value of a+ b.

10) If xy = b and 1/x² + 1/y²= a then find the value of (x + y)².

28/11/23

1) Find the difference between compound and simple intrest at 5% per annum for 4 years on Rs20000.

2) Solve: z + √z = 6/25.

3) (6x+2)/4 + (2x²-1)/(2x²+2) = (10x -1)/4x.

4) Factorize:

a) a⁴- 2a²b² + b⁴

b) x⁴- (y+ z)⁴

5) Evaluate:cos²60. Cos²45. Cos²30.

6) If tanx =a/b, find the value of (a sinx + b cosx)/(asinx - b cosx).

7) A diameter of a circle has the extreme points (7,9) and (-1,-3). Then find coordinates of the centre.

8) 4ˣ⁻¹ = 3. 2ˣ - 8. Find x.

9) If (5⁵ + 0.01)² - (5⁵ - 0.01)²= 5ˣ then x is

10) If log2= 0.3010, log3= 0.4771, log7= 0.8451, find the value of log294.

25/11/23

1) If 4ˣ = 8ʸ then find x/y - 1.

2) Evaluate: (2ᑫ. 6ᵖ⁺¹. 10ᵖ⁻ᑫ . 15ᵖ⁺ᑫ⁻²)/(4ᵖ. 3²ᵖ⁺ᑫ. 25ᵖ⁻¹).        

3) If log2= x and log3= y, then find the value for log60.

4) Evaluate: 4 log(8/25) - 3 log(16/125) - log 5.

5) Solve: 3x²- 14x +16=0.

6) Solve: x²- (a+ b)x + ab=0.

7) In ∆ABC, AB=26cm, BC=28cm and the altitude AD=24cm. Calculate AC.

8) The area between two concentric circles is 3168cm². Find the radii of the two circles if

A) their sum is 42cm

B) their difference is 28cm

23/11/23

1)  (3+ √5)/(3- √5)= a + b √5, find the value of a and b.

2) If x= 4 +√15. Then find the value of x² + 1/x²

3) Simplify (√11- √7)(√11+ √7)

4) Rationalise: 1/(7+ 3 √2)

5) Rationalise: 5/(3√3+ 2√2).

6) If (2/3)⁶(9/4)⁵=(3/2)ᵐ⁺² then find the value of m

7) simplify: x³ +3 √x³/√x

8) simplify: (√3+ 1)(1- √12)+ 9/(√3+ √12)

9) 12x² - 7x +1

10) 2x² + 7x +3.

11) 6x² + 5x -6.

12) 3x² - x - 4

TEST PAPER(1)- XII (2023/24)

1) Choose the correct answer from the given alternatives: 1 x 10=10

i) The domain for which the function f(x)= 3x²- 2x and g(x)= 3(3x -2) are equal, will be

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

ii) The value of tan {π/2 - tan⁻¹(1/3)} is equals to 

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

iii) If two rows or two columns of a determinant are identical then value of the determinant is 

a) 0 b) 2 c) - 1 d) 1 

iv) if f(x)= - f(x), then the value of ᵃ₋ₐ∫ f(x) dx is equals to

a) 2a b) a c) a/2 d) 0 

v) If y= tan⁻¹{5 - x).(1+ 5x)}, then value of dy/dx is

a) -1/(1+ x²) b) 1/(1+ x²) c) 5 d) 5/(1+ x²) 

vi) If P(A)= 3/7, P(B)= 4/7 and P(A∩B) =2/9, then the value of P(A/B) is equals to 

a) 7/18 b) 14/27 c) 5/18 d) 4/9

vii) A coin is tossed 10 times . The probability of getting had six times is

a) ¹⁰C₅. 1/2¹⁰ b) ¹⁰C₃. 1/2¹⁰ c) ¹⁰C₄. 1/2¹⁰ d) ¹⁰C₈. 1/2¹⁰

viii) If xy =1 then the value of d²y/dx² is

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

ix) The integrating factor of the differential equation (x + y+ z) dy/dx =1 is

a) e⁻ʸ b) eˣ c) e⁻ˣ d) eʸ

x) Form the differential equation of the family of curves y= A cos(x + B), where A and B are arbitrary constants.

2)a) Find the maximum value of 1      1          1

                                                         1   1+sinx   1

                                                         1       1  1 + cosx             (1)

b) Solve the equation for x: 

 cos(tan⁻¹x)= sin(cot⁻¹(3/4)).               (1)

c) Find the intervals in which the function f(x)= 3x⁴- 4x³- 12x²+ 5 is strictly decreasing.    (1)

4) Given that event A and B are such that P(A)= 1/2, P(B)= p, P(A U B)= 3/5. Find p if A and B are

a) mutually exclusive.

b) independent.                            (2)

5) Evaluate ∫ (x sinx)/(1+ sinx) dx at (π,0).                 (2)

6) For what value of x is the given matrix

    2x+4            4

     x+ 5            3 is a singular matrix ?          (2)

7) If y = xʸ, prove that x dy/dx = y²/(1- y logx).    (2)

8) The probabilities of A, B, and C solving a problem are 1/2, 1/3 and 1/4 respectively . Find the probability that the problem will be solved.    (3)

9) Prove: 1+ a      1        1

                   1      1+b      1 = abc(1/a+1/b+1/c).        (3)

                   1         1     1+c 

10) If the following function is differentiable at x=2, then find the value of a and b.

     f(x)= x²,          if x ≤ 2

             ax + b,    if x > 2.                         (3)

OR

Find the value of k. So that the function f defined by 

f(x)= kx +1,       if x ≤π

          cosx,       if x > π

is continuous at x =π

11) if cos⁻¹(x/a)+ cos⁻¹(y/b)= K, Show that x²/a² - (2xy cosK)/ab + y²/b² = sin²K.    (4)

12) If y= {x + √(1+ x²)}ⁿ, then show that (1+ x²) d²y/dx² + x dy/dx = n²y.       (4)

13) Two balls are drawn one after another (without replacement) from a bag containing 2 white, 3 red and 5 blue balls. What is the probability that atleast one ball is red ?   (4)

14) Evaluate ∫ (3x +1)/√(5 - 2x - x²) dx.            (4)

OR

Prove ³₁∫ dx/{x²(x +1} = 2/3 + log(2/3).

14) Find the equations of the tangent to the curve y= x²- 2x +7 which is

a) parallel to the line 2x - y +9=0.

b) perpendicular to the line 5y - 15x =13.           (4)

OR

Find the intervals in which the function f given by f(x)= sinx - cosx, 0 ≤ x ≤ 2π is strictly increasing or strictly decreasing.

15) a) Solve : dy/dx + y/x = x².           (2)

b) Solve : dy/dx + 1 = eˣ⁺ʸ.                 (2)

16) Let f: N --> N be a function defined as f(x)= 4x²+ 12x +15. Show that f: N --> S is invertible (where S is the range of f). Find the inverse of f and hence f⁻¹(31) and f⁻¹(87).    (4)

17) Using matrices , solve the following equations: 5x+ 3y + z =16, 2x+ y + 3z =19, x+ 2y + 4z =16pp25.             (6)

OR

If A= 1       -1       0

         2        5       3

         0        2       1   Find A⁻¹.          

18) Prove that the area of a right angle triangle of given hypotenuse is maximum, when the triangle is isosceles.         (6)

OR

Show that of all the rectangle inscribed in a given fixed circle, the square has the maximum area.

19) Evaluate 

a) ∫ x² sin⁻¹x dx. (3)

b) ∫ x/(x²+ 4x +3) dx.       (3) 

20) A pair of dice is thrown 4 times. If getting a double is considered a success, find the probability distribution of the number of successes and show that its mean is 2/3.   (3)

21) a) One bag contains 4 white and 5 black balls. Another bag contains 6 white and 7 black balls . A ball is transferred from the first bag to the second and then a ball is drawn from the second. Find the probability that the ball drawn is white.          (3)

++++++++++++++++++++++ +++++++++

22) i) If a= i+ 3j - k and b = 2i + 6j + μk. If a and b vectors are parallel, then the value of μ is

a) 3 b) -6 c) -3 d) -2 (1)

ii) The angle between the two planes x - y +2z =9 and 2x + y +z =7 is

a) 30° b) 45° c) 60° d) 90° (1)

iii) If a= 2i + j + 3k and b = 3i + 5j - 2k represent two adjacent sides of a triangle, then find the area of the triangle.          (1)

iv) If A(8,2,0), B(4,6,-7), C(-3,1,2) and D(-9,-2,4) are four given points , then find the angle between AB and CD.           (1)

23) If a and b are two vectors, then show that

 (a x b)²= a.a           a.b

                a. b           b.b                 (4)

24) If a= 7i - 2j + 3k, b= i - j +2k, c= 2i + 8j are three vectors then find a.(b x c) and (a x b).c.              (4)

25) Find the equation of plane passing through the line of intersection number planes x + 2y + 3z -5=0 and 3x - 2y - z +1=0 and cutting off equal intercepts on the OX and OZ axes.              (4)

OR

Find the coordinates of the points in the line (x -1)/2 = (y +2)/3= (z -3)/6 which are at a distance of 3 units from the point (1,-2,3).

26) Find the area of the region included between the parabola y= 3x²/4 and the line 3x - 2y +12=0.           (4)

++++++++++++. ++++++++++

27)a) The fixed cost of a product is Rs18000 and the variable cost per unit is Rs550. If the demand function is p(x)= 4000 - 150x, find the break-even values.      (2)

b) Given x + 4y =4 and 3x + y =16/3 are regression lines. Find the line of regression of x and y.                 (2)

c) The cost function for a commodity is C(x)= Rs(200+20x - x²/2)

i) Find the marginal cost (MC).

ii) Calculate the marginal cost when x= 4 and interprete it.       (2)

28) Two regression lines are represented by 2x + 3y -10=0 and 4x + y -5=0. Find the lines of regression of y on x.              (4)

OR

Fit a straight line to the following data, treating y as the dependent variable.

X: 1       2        3        4         5

Y: 7       6        5        4         3 

Hence, estimate the value of y when x=3.5.

29) The marginal cost function of a firm is MC= 33 logx. Find the total cost function when the cost producing one unit is Rs11.         (4)

30) A manufacturer produces two products A and B. Both the products are processed on two different machines. The available capacity of the first machine is 12 hours and that of the second machine is 9 hours per day. Each unit of product A requires 3 hours and both machines and each unit product B requires 2 hours on the first machine and 1 hour on the second machine. Each unit of product A is sold at profit of Rs7 and that of B at a profit of Rs4. Find graphically the production level per day for maximum profit.

 (6)

Quick Revision Test -X(2023/24)

1) tanx=4/3, find without using tables ,the value of sinx + cosx(both sinx and cosx are positive).       1.4

2) Given sinx= p/q, find in terms of p and q, cosx + tanx.     √(q²- p²)/q  + p/(q²- p²).

3) Given tanA=3/4 that find the value of 

a) sinA.        3/5

b) cosA. (A is an acute angle).            4/5

4) In the adjoining figure, 

∆ ABC is right angled triangle at B, ∆ BSC is right angled at S, and ∆ BRS is right angled at R, AB= 18cm, BC =7.5cm, RS= 5cm, angle BSR = x° and angle SAB= y°. Find

a) tan x°.                6/5

b) sin y°.               5/13

c) cos y°.            12/13

5) Using the measurements given in the figure.

a) Find the value of sinφ and tanθ.       5/13,4/3

b) Write an expression for AD in terms of θ.          9/cos θ or 12/sinθ

6) In the triangle given alongside, find θ,

 if B=90°, AB= 20cm and AC= 40cm.  30°

7) Find the value of (sin² 45 +cos² 45)/tan²60.      1/3

8) find the value of (sin² 30+ cos² 45+ sin² 60)/ Tan2 30.        9/4

9) Evaluate : cos 90°+ cos² 45 sin 30° tan45°.        1/4

10) Evaluate : 4/3  tan²30°+ sin²30° - 3 cos² 60° +3/4 tan² 60° - 2 tan²45.   25/36

11) If 4 sin² θ -1=0 and angle θ is less than 90°, find the value of θ and hence the value of cos²θ+ tan²θ.       30°, 13/12

12) if x= 30°,  verify, tan2x= 2tanx/(1- tan²x).

13) if 0≤ x ≤90°, state the numerical value of x for which sin x°= cos x°.    45°

14) ABC is a right angled triangle, right angled B. 

Given that angle ACB= θ, side AB= 2 units and side BC= 1 unit, find the value of sin²θ + tan²θ.      4.8

15) If the length of a shadow cast by a pole be √3 times the length of the pole, find the angle of elevation of the Sun.   30°

16) In the given figure:

 find

a) AD.              35.32cm

b) the perpendicular distance between BC and AD.       5cm

17) The shadow of a tower on level ground increases in length by x metres when the altitude of the sun changes from 45° to 30°. Calculate the value of x, given that the height of the tower is 25 m.           18.3cm

18) AD is drawn perpendicular to BC, the base of an equilateral triangle ABC. Given BC=10cm, find the length of AD correct to 1 place of decimals.     8.7cm

19) From a light house the angles of depression of two ships on opposite sides of the light house were observed to be 30° and 45°. If the height of the lighthouse is 90 m and the line joining the two ships passes through the foot of the light house , find the distance between the two ships . Give your answer correct to 2 decimal places.     245.88 m

20) An observer standing in 60m away from a building notices that the angles of elevation of the top and the bottom of a flag-staff on the building are respectively 60° and 45°. Find the height of the flG-staff.         43.92

21) Two men are on the opposite sides of a tower. They measue the angle of elevation of the top of tower as 45° and 30° respectively. If the height of the tower is 40m find the distance between the men.      109.28

22) The shadow of a tower, when the angle of elevation of the sun is 45°, is found to be 10m longer than when it is 60°. Find the height of the tower.      23.66m

23) A ladder of length of 4 m makes an angle of 30° with the floor while leaning against the wall of a room. If the foot of the ladder is kept fixed on the floor and it is made to lean against the opposite wall of the room, it makes an angle of 60° with the floor. Find the distance between these two walls of the room.        5.464m

24) Two men are on the opposites sides of a tower. they measure the angles of elevation of the top of the tower 45° and 60° respectively. if the height of the tower is 40 m, find the distance between the men.       63.09m

25) The angle of elevation of the top of the tower from a point on the ground is found to be 45°. By going 50 m further away from the tower, it is found to be 30°. Find the height of the tower.        68.3

26) From the top of a building 60 m 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.    40cm

27) Two vertical poles are fixed 60m apart. The angle of depression of the top of the first as seen from the top of the second, which is 150m high is 30°. Find the height of the first pole.         120m

28) An artist climbs a rope stretched from the top of a pole and fixed on the ground. The height of the pole is 10 m and the angle made by the rope with the ground is 30°. Find the length of the rope.      20m

Quick Revision- Class- VIII(2023/24)

28/11/23

1) Evaluate: -4/5 x 3/7 x 15/16 x (-14/9).

2) solve for y15(y -4)- 2(y -9) +5(y+6)=0

3) A bag has 4 red balls and 2 red yellow balls. A ball is drawn from the bag without looking into the bag. What is probability of getting a red ball?

4) Find the smallest square number that is divisible by each of the numbers 4, 9 and 10.

5) An article was purchased for Rs1239 including GST of 18%. Find the price of the article before GST.

6) A farmer has enough food to feed 20 animals in his cattle for 6 days. How long would the food last if there were 10 more animals in his cattle?

7) The probability of getting a number '0' is one throw of a die is

A) 0 B) 1/6 C) 1

8) The probability of getting a number 6 is one throw of a die is

A) 0 B) 1/6 C) 1

9) (x+7)/3 - (3x -2)/5 = 3.

10) (1- x)/6 + 2x/3 - (1- 7x)/4 = 13/6

11) Factorize the following:

a) 12x² - 7x +1

b) 2x² + 7x +3.

Test - XII(2023/24)

1) Prove : 1     x    x²

                1     y      y² 

                1     z      z²

= (x - y)( y - z)(z - x).

2) For the function f(x)= 2x³ - 24x + 5, find

a) the intervals where it is increasing.

b) the interval where it is decreasing. 

3) Evaluate : dx/(5+ 2cosx) at (π, 0). 

4) Evaluate : ∫ √tanx dx.

5) Find the value ¹₀∫(2x²+ 5x).

6) The two lines of regression for a biveriate distribution (x,y) are 3x + 2y = 7 and x + 4y =9. Find the regression coefficients bᵧₓ and bₓᵧ.

TEST - XI- 2025

TEST PAPER - 1

1) If α, β be the roots of the equation ax²+ bx + c = 0 and γ, δ those of the equation px²+ qx +r=0, show that, ac/pr = b²/q², if αδ = βγ..

2) If the sum of the first 2n terms of a GP  is twice the sum of the reciprocal of the terms , then show that continued product of the terms is equal to 2ⁿ.

3) If 9α=π, find the value of sinα sin2α sin3α sin4α.

4) If tanx = (tanα - tanβ)(1- tanα tanβ), then show that 

Sin2x= (sin2α - sin2β)/(1- sin2α sin2β).

5) If m(tan(α- β)/cos²β = ntanβ/cos²(α- β), show that 

β = (1/2) [α - tan⁻¹{(n - m)/(n + m)} tanα].

6) If lx + my = 0 be the perpendicular bisector the statement joining the points (a,b) and (c,d), then show that 

(c - a)/I = (d - b)/m = 2(la + mb)/(l²+ m²)

7) Determine the sign of the expression 

(x -1)(x -2)(x -3)(x - 4)+ 5 for real values of x.

8) If cotx = 2 and cot y = 3, then find (x + y).

9) Find square root of 4ab - 2i(a²- b²).

10) If θ{x}= (x -1) eˣ +1, show that θ{x} is positive for all the values of x > 0.

11) If y= f{x}= (x +1)/(x +2), Show that, f(y)= (2x +3)/(3x +5).

12) If f(x)= tan(x - π/4), find f(x) . f(-x)

COMPLEX NUMBERS 

1) Simplify: 1+ i²+ i⁴+ i⁶.    

2) Write in the form of a + ib where √-1= i

a) √-144 + √441.         

b) √-27 x √12 - √-125 x √-5.     

3) Find the conjugate of (2+ 3i)².      

4) Find x and y if (3x -7) + 5iy = 2y +3 - 4(1- x)i.     

5) Find the modulus of the complex number -12 + 5i.    

6) Express the reciprocal of the complex number 3+ i √5 in the form a+ ib. 

19/825

SET THEORY 

BOOSTER - A

1) The number of proper subset in a set consisting of four distinct elements is

a) 4 b) 8 c) 16 d) 64

2) The number of proper subsets in a set consisting of five distinct elements is 

a) 5 b) 10 c) 32 d) 31

3) If x ∈A=> x ∈ B then 

a) A= B b) A ⊂B C) A ⊆B d) B ⊆A

4) If A ⊆ B and B ⊆ A then 

a) A= ∅ b) A ∩B = ∅ c) A= B d) none 

5) For two sets if A ∪B = A ∩B then 

a) A ⊆B b) B ⊆ A c) A= B d) none

6) A - B = ∅ iff

a) A≠ B b) A ⊂B c) B ⊂ A d) A ∩B = ∅

7) If A ∩ B = B then 

a) A ⊆B b) B ⊆ A c) A= B d) A= ∅

8) If A and B are two disjoint sets then n(A ∪B)=

a) n(A)+ n(B) b) n(A) - n(B) c) 0 d) none

9) For any two set A and B, n(A)+ n(B) - n(A ∩B)=

a) n(A ∪B) b) n(A) - n(B) c) ∅ d) none 

10) The dual of A ∪ U= U is

a) A ∪ U= U b) A ∪∅= ∅ c) A ∪∅ = A d) A ∩∅= ∅

11) The dual of A ∪(B ∩C) = (A ∪B) ∩ (A ∪ C) is 

a) (A ∩ B) ∪ (A∩ C)

b) (A ∪B) ∪(A ∪C)

c) (A∩B) ∩ (A ∩ C)

d) (A∪B)∪ (A ∪C)

12) State which of the following statements is true?

a) Subset of an infinite set is so an infinite set 

b) The set of even integers greater that 889 is an infinite set.

c) The set of odd negative integers greater than (-150) is an infinite set.

d) A={x : x is real and 0< x ≤1) is a singleton set.

13) State which of the following statements is not true?

a) If a ∈ A and a ∈ B then A ⊆B.

b) If A⊆B and B ⊆C then A ⊆ C.

c) If A ⊆ B and B ⊆A, then A= B.

d) For any set A, if A ∪∅ = ∅(∅ being the null set) then A= ∅.

14) State which of the following is the set of factors of the number 12

a) {2,3,4,6} b) {2,3,4,6,12} c) {2,3,4,8,6} d) {1,2,3,4,6,12}

15) State which of the following is a null set?

a) {0} b) {∅}

c) {x: x is an integer and 1< x <2}

d) {x: x is a real number and 1< x <2}

16) If B be power set of A, state which of the following is true?

a) A ⊃B b) B ⊃A c) A ∈B d) A= B

17) If x ∈ A ∪B, State which of the following is true?

a) x ∈A b) x ∈B c) x ∈ A∀ x ∈B d) x ∈A ∧ x ∈B

18) If x ∈ A ∩B, state which of the following is true?

a) x ∈ A ∧ x ∈B b) x ∈B c) x ∈A ∨ x ∈B d) x ∉ A

19) If A= {2,4,6,8}, state which of the following is true?

a) {2,4} ∈ A b) {2,4} ⊆A c) {2,4} ⊂ A d) {2,4} ∈ Aᶜ

20) State which of the following statements is true?

a) {a} ∈ {a, b,c}

b) a ∉ {a,b,c}

c) a ⊂ {a,b,c}

d) {a} ⊂ {a,b,c}

21) State which of the following four sets are equal?

a) A={0} b) B={∅} 

c) C={x : x is a perfect square and 2≤ x ≤6}

d) D={x : x is an integer and -1< x < 1}

22) Some well defined sets are given below. Identify the null set:

a) A==x: x is the cube of an integer and 2≤ x≤7}

b) B={0} c) C={∅} d) D={x: x is an integer and 2< x ≤3}

23) State which of the following sets is an infinite set?

a) A={x : x is an integer and -1≤ x < 1}

b) B= set of negative even integers greater than (-100)

c) C= set of positive integers less than 100

d) D= {x: x is real and -1≤ x <1.

TEST PAPER -5

1) Answer the following questions (alternatives are to be noted): 1x10

a) Fill in the gap :

The quadratic equation with real co-efficients having 5i as one of roots is ____.

b) If A and B are two sets containing m and n elements respectively, how many different relations can be defined from A to B?

a) m+ n b) mn c) 2ᵐⁿ d) 2ᵐ⁺ⁿ

OR

Which one of the following is the value of n when ⁿC₄ = ⁿC₃ ?

a) 0 b) 2 c) 5 d) 7

c) If sinθ= 3/5 and θ is in the second quardant, the value of sin2θ is

a) 24/25  b) -24/25  c) 7/25  d) -7/25

Which one is correct?

d) Fill in the gap:

(Sin²60 + sin²30)/(Sec²50- tan²50)= ____

OR

Value of cos(-2220°)

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

e) Value of cos(π/4 + x) + cos(π/4 - x) is 

a) √2 b) √2 cosx c) cosx d) -√2 cosx

f) The gradient of a line parallel to x-axis is 

a) -1 b) 0 c) 1 d) undefined 

OR

 The radius of the circle x = 2 cost +3, y= 2sin t + 5, t being the parameter is 

a) 2  b) 3  c) 4  d) 5 

g) 3 points (-2,-5)(2,-2) and (8,a) are collinear. The value of a is

a) 2.5 b) 1.5 c) -2.5 d) -1.5

h) Which one of the following is the value of 

lim ₓ→ₐ (√x - √a)/(x - a) ?

a) √a/2 b) 2/√a c) 1/2√a d) 2√a

i) State whether the following relation is true or false :

(A U A's)' = ∅ (A' denotes the complement of the set A)

j) State whether the following statement is true or false :

The set A is a null set, when A={x: x is a real and x²+9=0}

OR

Which one of the following is the set of odd integer divisible by 2 ?

a) ∅ b) U c) {0} d) {∅}

2) a) Answer any two questions (2 x2)

i) If the roots of the equation x²- px +q=0 are in the ratio 2:1, Express q as a fraction of p.

ii) If z₁ = (-i +1)/√2 and z₂ = (1+ i√3)/2, find the principal argument of z₁z₂.

iii) Out of 14 footballers, two are goalkeepers, in how many ways a team of 11 footballers, containing only one goalkeeper, can be selected ?

b) Answer any two questions (2x2)

i) Find x if cosèx = cos2x.

ii)  What is the sum of the series:

sinx + sin(π+ x)+ sin(2π+ x)+ ......+ Sin(nπ+ x)? n being a positive integer.

iii) prove that : sin(7π/12) = (1/4) (√6+ √2).

c) Answer any two questions (2x2)

i) If two opposite angular points of a square are (3,5) and (1,-3). Find the area of the square.

ii) If the straight lines ax - 3y + 5 = 0 and 3x + 2y - 7 = 0 intersects at (1,2), find the value of a.

iii) if (a,0) and (0,b) are two vertices of a triangle, find the third vertex of it so that the centroid is at origin.

d) Answer any one question: (2x1)

i) if a = 5i - 2j, b= i + 3j, find the magnitude of 2a - b.

ii) The position vector of the points A and B are the respectively i+ 2j and -3i + 6j. Find the position vector of midpoint of AB.

e) Answer any two questions (2x2)

i) Find the limit: lim ₓ→₀ (eˢᶦⁿˣ -1)/x.

ii) Find the domain of the real valued function f(x)= (x²+ 2x +3)/(x²- 5x +6).

iii) If y= xeˣ, prove x(dy/dx)= (1+ x)y.

3) a) Answer any three questions: (4x3)

i) If a,b,c are in GP such that a + b + c = bx, then show that either x< -1 or x = 3.

ii) In how many ways can the letters of the word PENCIL be arranged so that N is always next to E?

iii) Find the square root of √(-8i).

iv) If the roots of the equation ax²+ bx + c =0  are in the ratio m: n then show that √(m/n) + √(n/m) = √(b²/ac).

v) Find the 4th term from the end in the expansion of (4x/5 - 5/2x)⁹.

b) Answer any three questions (4x3)

i) prove that: tan(π/3 + θ) tan(π/3 - θ)= (2 cos2θ+1)/(2cos2θ -1).

ii) show that the solution of the equation tanax = tanbx, a²+ b²≠ 0 are in AP. Find the command difference of the series.

iii) Prove : cosx/(1- sinx)= tan(π/4 + x/2).

iv) Solve: 2cos²x + 3sinx =0.

v) (sinx - siny)/(cosx + cosy)= tan{(x - y)/2}.

c) Answer any two questions (4x2)

i) A(1,2) and B(5,-2) are two points P is a point moving in such a way that area of the triangle ABP is 12 sq units . Find the locus of P.

ii) A(a,0) and A'(-a,0) are two points. P is a point such that AP makes an angle 45° with A'P. Find the locus of P.

iii) Find the equation the circle passing through the points of intersection of the circles x²+ y²= 9 and x²+ y² - 4x + y - 6 = 0 and through the origin.

d) Answer any two questions (4x2)

i) A, B, C are three sets such that A U B = A U C and A' U B = A' U C. prove that B = C, (A' denote the complement of A).

ii) If f(x)= x -1 and g(x)= x²+1, find 

(f+ g), (f - g), f/g

iii) Let A={1,2,3,4,6} and let R={(a,b): a, b ∈ A and a divides b}

A) write R in the roaster form 

B) Find domain(R) and range (R).

e) Answer any one question (4x1)

i) A card is drawn from a well shuffled pack of 52 cards. Find

A) the odds in favour of getting a face card.

B) the odds against getting a spade.

ii) Find the variance and standard deviations for the following data:

X: 10   15    18     20     25

f:   3     2       5      8        2

f) Answer any one question 4x1)

i) Find the mean deviation about mean for the following data:

Height(in cm)  Number of boys 

95-105                 9

105-115              13

115-125              25

125-135              30

135-145              13

145-155              10

If is a relation on the set of natural number given by the function defined by let the function define continuous at find the value of exam in the existence 

Find out the 99 term of the series 27 14 23 34 the sum of the three numbers in GP 7 some other square is 21 what is the sum of their series are positive numbers are in GP so that they are log exams to a fixed base are in AP prove that complex cube root of unity if one root of a equation square of the other show that how many numbers line between 101 can be found with the digit 034689 if x y z 3 real number show that prove that solve find the range of the function find the limit find out the

4/11/24

A. P

1) If the third and the 6th terms of an AP are 7 and 13 respectively, find the first term and the common difference. 

2) Find the sum of all natural numbers between 100 and 1000 which are multiple of 5.

3) How many terms of the AP -6, -11/2, -5, .... are needed to give the sum -25? 

4) Determine the sum of the first 25 terms of an AP if a₂= 2 and a₇= 22. 

5) If the first term of an AP is 2 and the sum of first five terms is equals to the one fourth of the sum of the next 5 terms , show that the 20th term is -112.   

6) Insert 3 Arithmetic mean between 2 and 10. 

7) The sum of all odd numbers between 1 and 100 divisible by 3, is

a) 83667 b) 90000 c) 83660 d) none 

   

2/11/24

1) lim ₓ→₁/₂ (4x²-1)/(2x -1).

2) lim ₓ→₀ sin3x/sin2x. 

3) lim ₓ→₀ (sin2x + sin6x)/(sin5x - sin3x). 

4)

 lim ₓ→₀ (tan3x - 2x)/(3x - sin²x). 

5) lim ₓ→∞ (2x²+ 7x +5)/(4x²+ 3x -1). 

6) lim ₓ→₀ (2sinx - sin2x)/x³.

7) lim ₓ→ₐ {√(1+ ax) - √(1- ax)}/x.

8) lim ₓ→π sinx/(π - x). 

14/11/23

Find dy/dx of following:

1) y= (Secx+ tanx)/

(Secx - tanx).

2) y= sinx°

3) x = sint, y= cospt.

4) √(sin√x).

5) If f(x)= logₓ(logₑx), then find f'(e).

11/11/23

2) out of 20 members of a family 15 prefer tea and 9 prefer tea but not cofee. Each member of the family prefers atleast one drink. Find the number of members who

A) prefer both drinks. 6

B) prefer cofee but not tea. 5

3) out of 50 students of a school 35 students prefer Bengali and 25 students prefer both Bengali and English. Each student prefer atleast one subject. Find the number of students who

A) prefer English. 40, 15

B) prefer English but not Bengali.

4) Out of 1600 students in a school, 390 played cricket, 580 played football, 450 played hockey, 90 played cricket and hockey, 125 played hockey and football and 155 played cricket and football, 50 played all three games. How many students did not play any game? 500

7/11/23

1) If A={5,6,9,10}, B={7,8,9,11} and C=

{9,10,11,12} then show that AU(BUC)= (AUB)UC

3) With the help of set operation find the H.C.F. of 35, 77, 119. 7

4) In an examination, out of 100 students 70 passed in Mathematics, 65 passed in Physics and 55 passed in chemistry. Of these students 50 passed in Mathematics and physics. 45 passed in Mathematics and Chemistry, 40 passed in Physics and chemistry and 35 passed in all three subjects. 

A) how many students passed in exactly two of the three subjects? 

B) how many students passed in exactly one of the three subjects? 

C) how many students failed in three subjects? 30, 25, 10

5/11/23

1) Write down the first five terms of the sequence, whose nth term is (-1)ⁿ⁻¹. 5ⁿ⁺¹. 25,-125,625,-3125, 15625

2) If the 3rd and 6th terms of an AP are 7 and 13 respectively, find the first term and the common difference. 3 and 2

3) find the sum of all natural numbers between 100 and 1000 which are multiple of 5. 98450

4) how many terms of the AP -6, -11/2, -5,.... are needed to give the sum -25 ? 5 or 20.

6) If the first term of an AP is 2 and the sum of first five terms is equal to one fourth of the sum of the next five terms, show that the 20th term is --112

7) Insert 3 arithmetic mean between 2 and 10. 4,6,8

8) The sum of three decreasing numbers in AP is 27. If -1, -1, 3 are added to them respectively, the resulting series is in GP. The numbers are 

A) 5,8,13 B)15,9,3 C)13,9,5 D) 17,9,1

9) The sum of all odd numbers between 1 and 100 which are divisible by 3, is..

A) 83667 B) 90000 C) 83660 D) n 

10) If 7th and 13th terms of an AP be 34 and 64 respectively, then its 18th term is.

A) 87 B) 88 C) 89 D) 90 

11) If the sum of p terms of an AP is q and the sum of q terms is q, then the sum of the p + q terms will be..

A) 0 B) p-q C) p+q D) -(p-q)

12) If the sum of n terms of AP be n² - n and its common difference is 6, then its first term is..

A) 2 B) 3 C) 1 D) 4 

13) Sum of all two digit numbers which when divided by 4 yield Unity as reminder is..

A) 1200 B)1210. C)1250. D) n

14) In n AM's introduced between 3 and 17 such that the ratio of the last mean to the first mean is 3:1, then the value of n is..

A) 6 B) 8 C) 4 D) n 

15) The 1st and last terms of an AP are 1 and 11. If the sum of its terms is 36, then the number of terms will be.

A) 5 B) 6 C) 7 D) 8 

16) Find the sum of all odd integers from 1 to 1001. 251001

17) If the ratio between the sums of n terms of two AP is (7n+1):(4n+27) find the ratio of their 11th term. 148: 111

18) If the sum of m terms of an AP be n and the sum of n terms be m, show that the sum of m+n terms is -(m+n).

19) If the sum of n terms of an AP is (pn+ qn²), where p and q are constants, find the common difference. 2q

20) In an AP, the first term is 2 and the sum of first five terms is one-fourth of the sum of next terms. Show that the 20th term is - 112 and the sum of first 20 term is -1100.

21) If the sum of n terms of an AP is given by (3n²+ 4n), find its rth term. 6r +1

22) The digits of a three-digit numbers are in AP and their sum is 15. The number obtained by reversing the digits is 594 less than the original number. Find the number. 852

23) Between 1 and 31, m numbers have been inserted in such a way that the ratio of 7th and (m-1)th numbers is 5:9. Find the value of m. 14

24) In the arithmetic progression whose common difference is non zero, the sum of the first 3n terms is equal to the sum of next n terms. Then the ratio of the sum of the first 2n terms the next to 2n terms is 

A) 1/5. B) 2/3 C) 3/4 D) none

25) If four numbers in AP are such that their sum is 50 and the greatest number is 4 times the least, then the numbers are:

A) 5,10,15,20 B) 4,10,16,22

C) 3,7,11,15 D) none

26) The first and the last term of an AP are a and l respectively. if S is the sum of all the terms of the AP. and the common difference is given by (l²-a²)/{k -(l+a)}, then k is

A) S B) 2S C) 3S D) none

27) If the sum of the first n even natural number is equal to K times the sum of the first n odd natural numbers, then k is..

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

28) If the first, second and last term of an AP are a,b and 2a respectively, then its sum is 

A) ab/{2(b-a)} B) ab/(b-a)

C) 3ab/{2(b-a)} D) none

29) If x is the sum of an arithmetic progression of n odd number of terms and y the sum of the terms of the series in odd places, then x/y is

A) 2n/(n+1) B) n/(n+1)

C) (n+1)/2n D) (n+1)/n 

30) If the first term of an AP is 2 and common difference is 4, then the sum of its 40 terms is

A) 3200 B) 1600 C) 200 D) 2800

31) The number of terms of the AP 3, 7, 11, 15, ... to be so that the sum is 406 is...

A) 5 B) 10 C) 12 D) 14 E) 20

32) If a(1/b+ 1/c), b(1/c + 1/a), c(1/a + 1/b) are in AP , then

A) a, b, c are in AP

B) 1/a, 1/b, 1/c are in AP

C) a, b, c are in HP

D) 1/a, 1/b, 1/c are in GP. 

33) If the sum of the three numbers in AP be 18 then what is the middle term ? 6

34) The fifth term and the 11th term of an AP are 41 and 20 respectively. Find the first term. What will be the sum of first 11 terms of the AP. ? 425/2

36) The middle term, of an AP having 11th term is 12. Find the sum of the 11 terms of that progression. 132

37) There are n arithmetic means between 4 and 31. If the second mean : last mean=5: 14 then find the value of n. 8

38) If the sum of the first P terms of an AP be equal to the sum of the first Q terms then show that the sum of the first P +Q terms is zero.

) Find the sum upto n terms of the series 1²- 2²+ 3²- 4²+ 5²- 6²+.. .. -n/2 (n+1) (n= 2r)

39) if the sum of p terms of an AP is to the sum of q terms as p²:q², show that (pth term)/(qth term)= (2p-1)/(2q-1).

40) The first term of an AP is a, the second term is b and the last term is c. Show that the sum is {(a+c)(b+c-2a)}/{2(b-a)}.

41) The sides of a right angled triangle are in AP. if the smallest side is 5cm then find the largest side. 25/3

42) find the sum of natural numbers from 1 to 200 excluding those divisible by 5. 16000 

43) Show that the sum of all odd numbers between 2 and 1000 which are divisible by 3 is 83667 and of those not divisible by 3 is 166332.

44) Find the 14 A. M which can be inserted between 5 and 8 and show that their sum is 14 times the Arithmetic mean between 5 and 8.

45) Divide 25/2 into five parts in AP, such that the first and the last parts are in the ratio 2: 3. 2,9/4,5/2, 11/3, 3.

46) For what value of m, the sequence 2(4m+7), 6m + 1/2, 12m-7 forms an AP. -3/4 

47) Find the 20th term of the AP 80, 75, 70,... Calculate the number of terms required to make the sum equal to zero. 35 

48) Prove that if unity is added to the sum of any number of terms of the AP 3, 5,7,9...the resulting sum is a perfect square.

49) The sum of n terms of the series 25, 22, 19, 16,.. is 116. Find the number of terms and the last term. The given series is AP. 18405

50) Find the sum of all natural numbers from 100 to 300:

a) which is divisible by 4. 10200

b) excluding those which are divisible by 4. 30000

c) which are exactly divisible by 5. 

d) which are exactly divisible by 4 and 5. 8200, 2200

e) which are exactly divisible by 4 or 5. 16200

4/11/23

1) If x= 2+ 3i and y= 2- 3i, find the value of (x³- y³)/(x³+ y³).

2) If w be an imaginary cube root of 1, show that (1- w²)(1- w⁴)(1- w⁸)(1- w¹⁰)= 9.

3) If 7 cosx + 5 Sinx =5, find the value of 5 cosx - 7 Sinx.

4) If x =π/19, show that (sin 23x -sin 3x)/(sin 16x + sin 4x) =-1.

5) Evaluate: {cot570 + sin(-330)}/{tan(-210)+ cosec(-750)}.

31/10/23

1) Find the value of:

a) cos(-1170°). 

b) cos(-870°). 

c) tan(-1755°). 

d) cot 660° + tan(-1050°). 

e) sec(-945°). 

f) cosec(-840°). 

28/10/23

Find dy/dx of the following: 

1)y=  x°

a) 1 b) x°. x c) x d) none

b) y= f(x)

a) f(x) b) f'(x) c) x d) 1 e) none

3) y= c

a) cx b) c c) 1 d) 0 e) 

none

4) dy/dx is the the derivative of 

a) y respect to x

b) x respect to y

c) y respect to c

d) x respect to c

e) none

5) y= 2ˡᵒᵍ ˣ

a) 2ˡᵒᵍ ˣ b) 2ˡᵒᵍ ˣ. 2 c) log 2 d) none

6) y=  logₑx 

a) 1/x b) 1/logₑx c) 1/x logₑe  d) none

7) y= logₐx

a) 1 b) 0 c) 1/x d) none

8) y= logₓx.

a) 1 b) 0 c) 1/x d) none

9) y= log ₓe.

a) x b) 1/e c) 1/x d) none

10) log₇7.

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

11) y= xᵐ

a) mx b) mxᵐ⁻¹ c) 0 d) none

12) y= mˣ.

a) mˣlog m b) mˣ c) m logx d) none

12) y= mˣ + xᵐ + mᵐ + mxᵐ

13) y= (x²+3)⁵.

14) ₂3x²

15) log(x²-5)

16) (2x³+5)¹⁰

17) √(2x²+ 9).

18) √(x²+ a²).

19) The derivative of an even function is always an odd function.    T/F

20) The derivative of an odd function is always an even function. T/F

21) y= √(x²-1) at x=1.

22) y= 1/(3x -1) at x=0

23) log(ax + b)³

24) (2x²+3)⁴.

25) log(logx).

26) √(logx)³.