MATHS IIT- JEE

DIFFERENTIATION 

1) y= eˣ tanx + x logₑx.     eˣ(tanx + sec²x) + (log x -1)

2) If y= (logx)/x + eˣ sin2x + log₅x, .         (1- logx)/x²    + eˣ(sin2x + cos2x)  + 1/(logₑ5)

3) If x= exp[tan⁻¹{(y - x²)/x²}].      

a) x[(1+ tan(logx)+ sec²x]

b) 2x[(1+ tan(logx)+ sec²x]

c) 2x[(1+ tan(logx)]+ secx]

d) 2x + x[(1+ tan(logx)]²

4) y= logₑ{tan⁻¹ √(1+ x²)}.    

5) y= (x +1)(x +2)(x +3)

6) y= e⁵ˣ tan(x²+ 2).

7) y= (sinx)ˡᵒᵍˣ.

8) y= (x¹⁾² (1- 2x)²⁾³)/{(2- 3x)³⁾⁴(3- 4x)⁴⁾⁵.

9) y= xˣ.

10) y= ₑx. ₑx². ₑx³.ₑx⁴.

11) xʸ+ yˣ= 2.

12) y= sinx/(1+ cosx)/{1+ sinx/(1+ cosx...

13) x+ y = sin(x - y).

14) x²+ xeʸ + y =0, find y', also find the value of y' at point (0,0).

15) y= a cos t & x= a(t - sin t) find dy/dx at t=π/2.         -1

16) Show that the function represented parametrically by the equations, x= (1+ t)/t³; y= 3/2t² + 2/t satisfies the relationship: x(y')³= 1+ y'.

17) find dy/dx at t=π/4 if y= cos⁴t & x= sin⁴t.        

18) Find the slope of the tangent at a point P(t) on the curve x = at², y= 2at.

19) Differentiate logₑ(tanx) with respect to sin⁻¹(eˣ).      e⁻ˣ√(1- e²ˣ)/(sinx cosx)

) If g is inverse of f and f'(x)= 1/(1+ xⁿ), then g'(c)=?

a) 1+ xⁿ b) 1+ [f(x)]ⁿ c) 1+ [g(x)]ⁿ d) none

) y= xˡᵒᵍˣ with respect to logx

) If g is inverse of f and f(x)= 2x + sinx; then g'(x) equas:

a) -3/x² + 1/√(1- x²)

b) 2+ sin⁻¹x c) 2+ cos g(x)

d) 1/(2+ cos g(x))

) If f(x)= x³+ x² f'(1)+ x f'(2)+ f'''(3) for all x ∈ R. Then find f(x) independent of f'(1), f'(2) and f'''(3).

) If x= a(t + sin t) and y= a(1- cos t) , find d²y/dx².

) y= f(x) and x= g(y) are inverse function of each other then express g'(y) and g"(y) in terms of derivative of f(x).

) If y = ₓₑx² then find y".

) Find y" at x=π/4, if y= x tan x.

) Show that y= eˣ sinx satisfies the relationship y" - 2y' + 2y = 0.

)          (x     x²   x³

If f(x)= 1    2x  3x²

            0     2   6x) find f'(x).

) If f(x)= |eˣ    x²

           log x   sinx | find f'(1).

)           2x          x²         x³

If f(x)= x²+ 2x   1        3x +1

            2x        1- 3x²     5x      then find f'(1).

) d/dx of sin²[cot⁻¹√{(1+ x)/(1- x)}]=

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

) If f(x)= sin⁻¹{2x/(1+ x²)} then find

i) f'(2).

ii) f'(1/2)

iii) f'(1)

) If y= cos⁻¹(4x³- 3x). Then find 

i) f'(-√3/2)

ii) f'(0)

iii) f'(√3/2)

) If √(1- x²)+ √(1- y²)= a(x - y), then show that dy/dx = √{(1- y²)/(1- x²)}

) Find second order derivative of y= sinx with respect to z= eˣ.

) If y= (tan⁻¹x)² then show (1+ x²)²+ 2x (1+ x²) y dy/dx = 2.

) Obtain differential coefficient of tan⁻¹{(√(1+ x²) - 1)/x} with respect to cos⁻¹√[{1+ √(1+ x²)}/2√(1+ x²)].

) y= (secx - tanx)/(secx - tanx) then dy/dx is

a) 2 secx(secx - tanx)

b) - 2 secx(secx - tanx)²

c) 2 secx(secx + tanx)²

d) - 2 secx(secx + tanx)²

) If y= (1+ x² + x⁴)/(1+ x + x²) and dy/dx = ax + b, then the values of a and b are 

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

) Which of the following could be the sketch graph of y= d/dx (x log x)?

Let f(x)= x +3 log(x -2) & g(x)= x +5 log(x -1), then the set of x satisfying the inequality f'x)< g'(x) is

a) (2,7/2) 

b) (1,2) U (7/2, ∞) c) (2, ∞) d) 7/2, ∞)

) Differential coefficient of (x^{(l+ m)/(m - n)})^1/(n - l) . (x^{(m+ n)/(n - l)})^1/(l - m). (x^{(n+ l)/(l - m)})^1/(m - n) w.r.t.x is

a) 1 b) 0 c) -1 d) xˡᵐⁿ

 . 

³²²²⁴ₑᶻˣˣlim ₓ→₀ˢᶦⁿˣⁿ lim ₓ→∞¹⁾ⁿ lim ₓ→₀ˣ²³²⁻¹⁻¹²⁻¹³²²²ˣ⁻¹²²²²²²⁻¹²⁻¹²²²⁴²∞∞∞ₓₓₓⁿᵐᵖᵐᵐⁿᵐᵖⁿᵖₑⁿᵖᵐⁿᵖᵐⁿᵖⁿᵖᵐ⁻¹²²²²¹⁰⁰θ⌈!*/-!#⊂⊂∩∪⨯≺∆∅-*𝔓∫Πₙ₌₁ⁿ⁽¹⁰¹⁻ⁿ⁾ˢᶦⁿˣ¹⁾²ₓ²ₓₓᵐⁿᵐ⁺ⁿ ²ʸˣʸˣʸˣˣʸʸˣˣʸʸˣₑʸ⁺∞θθθθθθ⁻¹⁻¹¹⁰²¹⁰²¹⁰¹⁰²¹⁰²² ₑ ∈

NUMERICAL INEQUALITIES AND INEQUATIONS

EXERCISE - A

1) Solve the following linear inequations:

a) 2x - 4≤ 0.     (-∞,2]

b) -3x +12< 0.      (4, ∞)

c) 4x - 12≥ 0.      [3, ∞)

d) 7x + 9 > 30.    (3, ∞)

e) 5x -3< 3x +1 when 

     i) x is real number.        (- ∞,2)

    ii) x is integer number.       {..... -4,-3,-2,-1,0,1}

   iii) x is a natural number.     {1}

f) 3x + 17 ≤ 2(1- x).        (-∞,-3]

g) 2(2x +3) -10≤ 6(x -2).     [4, ∞)

h) (2x -3)/4  + 9 ≥ 3+ 4x/3.    (- ∞, 63/10]

i) (5x -2)/3 - (6x -3)/5 > x/4.    (4, ∞)

j) (1/2) (3x/5 + 4)≥ (x -6)/3.     (- ∞, 120]

k) 3(x -2)/5 ≥ 5(2- x)/3.     [2, ∞)

l) 1/(x -2) < 0.      (- ∞,2)

m) (x +1)/(x+ 2) ≥ 1.     (- ∞, -2)

n) (x -3)/(x -5) > 0.    (- ∞,3) U (5, ∞)

o) (x -2)/(x+5) > 2.     (-12,-5)

p) (2x +4)/(x -1) ≥ 5.     (1,3]

q) (x +3)/(x -2) ≤ 2.       (- ∞,2) U [7, ∞)

EXERCISE - B

Solve the following linear inequations in R

1) Solve: 12x < 50, when 

    i) x ∈ R.           (-∞, 25/6)

   ii) x ∈ Z.           {...-3,-2,-1,0,1,2,3,4}

   iii) x ∈ N.         {1,2,3,4}

2) Solve: - 4x > 30, when 

  i) x ∈ R.       (- ∞, -15/2)

 ii) x ∈ Z.       {..., -9,-8}

 iii) x ∈ N.         φ

3) Solve: 4x -2 < 8, when 

    i) x ∈ R.       (- ∞, 5/2)

    ii) x ∈ Z.     {....-2,-1,0,1,2}

    iii) x ∈ N.        {1,2}

4) 3x -7 > x +1.      (4, ∞)

5) x +5 > 4x - 10.     (- ∞,5)

6) 3x + 9 ≥ - x + 19.      [5/2, ∞)

7) 2(3- x) ≥ x/5 + 4.       (- ∞, 10/11]

8) (3x -2)/5 ≤ (4x -3)/2.      [11/14, ∞)

9) - (x +3)+4 < 5 - 2x.        (- ∞, -2)

10) x/5 < (3x -2)/4  - (5x -3)/5.        (- ∞, 2/9)

11) 2(x -1)/5 ≤ 3(2+ x)/7.         [44, ∞)

12) 5x/2 + 3x/4 ≥ 39/4.     [3, ∞)

13) (x -1)/3  + 4 < (x -5)/5  - 2.      (- ∞, -50)

14) (2x +3)/4  - 3 < (x -4)/3. - 2.      (- ∞, -13/2)

15) (5- 2x)/3 < x/6  - 5.             (8, ∞)

16) (4+ 2x)/3 ≥ x/2  - 3.           [-26, ∞)

17) (2x +3)/5  - 2 < 3(x -2)/5.      (-1, ∞)

18) x -2 ≤ (5x +8)/3.         [-7, ∞)

19) (6x -5)/(4x +1)  < 0.       (-1/4, 5/6)

20) (2x -3)/(3x -7) > 0.      (- ∞, 3/2) U (7/3, ∞)

21) 3/(x -2) < 1.     (- ∞,2) U (5, ∞)

22) 1/(x -1) ≤ 2.        (- ∞,1) U[3/2, ∞)

23) (5x +8)/(4 - x) < 2.     (-∞,0) U (4, ∞)

24) x/(x -5) > 1/2.        (- ∞, -5) U(5, ∞)

EXERCISE - C

∞ 

∈ φ

MATRIX (APPLICATION)

EXERCISE - A

EXERCISE - B

1) The sum of three numbers is 6, if we multiply the third number by 2 and add the first number to the result, we get 7, by adding second and third number to three times the first number, we get 12, using Matrices find the numbers.       3,1,2

2) An amount of Rs 5000 is put into 3 investments at the rate of interest of 6%, 7%, 8% per annum respectively. The total annual income is 358. If the combined income from the first two investments is Rs 70 more than the income from the third, find the amount of each investment by Matrix method.    1000,2200,1800

3). A mixture is to be made of three foods A, B, C. The three foods A, B, C contain nutrients P, Q, R as shown below 

           Ounces per pound of Nutrients 

Food     P     Q    R 

A           1      2    5

B           3      1    1 

C           4      2    1

How to form a mixture which will have 8 ounces of P, 5 ounces of Q and 7 ounces of R ?   1,1,1

4) A transport company uses 3 types of trucks A, B , C to transport three types of vehicles M,N, P. The capacity of each truck in terms of 3 types of vehicle is given as follows:

      M    N     P

A    1     3     2 

B    2     2     3 

C    3     2     2

Using Matrix method, find the number of trucks of each type required to transport 85, 105, 110 vehicles of M,N, P types respectively.     15,20,10

5) In an engineering workshop there are 10 machines for drilling, 8 machines for turning and 7 machines for grinding. Three types of brackets are made. Type I required 0 minutes for drilling, 5 minutes for turning and 4 minutes for grinding. The corresponding times for type II and III brackets are 3, 3, 2 and 3, 2, 2, minutes respectively. How many brackets of each types should be produced per hour so that all the machines remains fully occupied during an hour ? Solve by using Matrix method .      5,55,145

6) Given the following National Income model :

C= a+ bY (a> 0, 0 < b < 1)

I= d + eY  (d> 0, 0< e < 1

Y= I + C

Solve the indigenous variables C, I and Y using Matrix method.       (a- ac+ bd)/(1- b - c), (d+ ac - bd)/(1- b - c), (a+ d)/(1- b - c)

7) The sum of three numbers is 2. If twice the second number is added to the sum of first and third, the sum is 1, By adding second and third number to five times the first number, we get 6 p. Find the three numbers by using metrix .   1,-1?2

8) An amount of Rs 10000 is put into 3 investments at the rate of 10, 12 and 15% per annum. The combined income is Rs 1310 and the combined income of first and second investment is Rs 190 short of the income from the third. Find the investment in each using Matrix .     2000,3000,5000

9) A company produces three products everyday. Their production on a certain day is 45 tons. It is found that the production of third product exceeds the production of first product by 8 tons while the total production of first and third product is twice the production of second product. Determine the production level of each product using Matrix method.      11,15,19

10) The price of 3 commodities P, Q and R are Rs x, y, z per unit respectively. A purchases 4 units of R and sells 3 units of P and 5 units of Q, B purchases 3 units of Q and sells 2 units of P 1 units of R. C purchases 1 unit of P and sells 4 units of Q and 6 units of R. In the process A, B and C earns Rs 6000, Rs 5000 and Rs 13000 respectively. If selling the unit is positive earnings and buying the units is negative earnings , find the price per unit three commodities by using metric method.    3000, 1000, 2000

11) Two factories decided to award their employees for three values of (a) adaptable to new techniques, (b) careful and alert in difficult situations and (c) keeping calm in tense situations at the rate of Rs x, y, z per person respectively. The first factory decided to honour respectively 2,4 and 3 employees with a total prize money of Rs 29000. The second factory decided to honour respectively 5,2 and 3 employees with the prize money of Rs 30500. If the three prizes per person together cost Rs 9500, then 

i) represent the above situation by Matrix equation and form linear equations using matrix multiplication.

ii) solve these equations using matrices.     2500,3000,4000

12) A total amount of Rs 7000 depreciated in three different saving bank accounts with annual interest rates 5%, 8% and 17/2% respectively. The total annual interest from these 3 account is Rs 550. Equal amounts have been deposited in the 5%, 8% saving accounts. Find the amount deposited in each of the three accounts, with the help of matrices.      1125,1125,4750

13)  A shopkeeper has 3 varieties of pens A, B and C. Meenu purchased 1 pen of each variety for a total of Rs 21. Jeen purchased 4 pens A veriety, 3 pens of B variety and 2 penc of C variety for Rs 60. While Shikha purchased 6 pens of A variety, 2 pens of B variety and 3 pens of C variety for Rs 70. using Matrix method find the cost of each pen.    5,8,8

14) Purvi has invested a part of his investment in 10% bond A and a part in 15% bond B. His interest interest during first year is Rs 4000. if he invests 20% more in 10% Bond A and 10% more in 15% bond B his income during 2nd year increases by Rs 500. Find his initial investments and new investment in bonds A and B using Matrix method.   10000 in A, 20000 in B, new Rs 12000 in A Rs 22000 in B

15) To control a crop disease it is necessary to use 8 units of chemical A, 14 units of chemical B and 13 units of chemical C. One barrel of spray P contains one units of A, 2 units of B and 3 units of C. One barrel of spray Q contains 2 units of A, 3 units of B and 2 units of C. One barrel of spray R contains 1 unit of A, 2 units of B and 2 units of C. Using Matrix method, find how many barrel of each spray be used to just meet the requirement ?      P: 1 barrel ; Q: 2 barrel; R: 3 barrel 

16) A firm produces two products And B passing through two machines X and Y before completion. X can produce either 10 units of A or 15 units of B per hour. Y can produce 15 units of either product per hour. Find daily production of A and B if time available is 12 hours on machine X and 10 hours on Y per day using Matrix inversion method.    60,90

17) The equilibrium condition for three related markets is given by 

11p₁ - p₂ - p₃ = 31

- p₁ + 6p₂ - 2p₃ = 26

- p₁ - 2p₂ + 7p₃ = 24

Using Matrix inversion method, find the equilibrium prices of each market.   4,7,6

EXERCISE - C

FUNCTION, DIFFERENTIATION, APPLICATION OF DER, MATRIX, DETERMINANTS, COMPOUND INT, ANNUITY, UNIVERIATE DATA, BIVERTIA DATA, INDEX NUMBER, TIME SERIES

FUNCTION

1) If f(x)= (x²- 5x +6)/ (x²- 8x +12) show that f(2) is not defined and also find f(-5).       8/11

2) If f(x)=eᵃˣ⁺ᵇ,. show that eᵇ. f(x + y)= f(x).f(y).

3) If f(x)= |x| - |-x|. Where  |x| is the greatest integer not exceeding x, find the value of f(3.5) and f(-3.5).    0.5,7.5

4) Find the domain of definition of the function 5/√{(x +1)(x -3)}.     - ∞<x <-1 and 3<x < ∞

5) If f(x)= |x| - 2x, find f(-1), f(1).     3,-1

6) If f(x)= (ax - b)/(bx - a), show that f(a). f(1/a) - f(b) f(1/b)= 0.

7) Show that the function f(x, y)= (5x²- 7y²)/3y² is a homogeneous function.

8) If f(x)= (eˣ -1)/(eˣ +1) and f(a)= (1+ f(x))/(1- f(x)), then show that f(x + a)= f(x). f(a)

9) If f(x)= (1+ eˣ)/(1- eˣ) , then show that f(x) is an odd function.

10) If f(x)=x²- x, then show that f(h +1)= f(-h)

11) If f(x)= (ax + b)/(bx + a) show that f(x). f(1/x)= 1

12) If y= f(x)= (x + 1)/(x + 2), find f(y) and f{f(1/x)}.      (2x+3)/(3x +5), (2+ 3x)/(3+ 5x)

13) If f(x)= (x -1)/(x +1), then show that {f(a) - f(b)}/(1+ f(a)f(b))= (a - b)/(1+ ab)

14) If f(x)= (2x +1)/(2x²+1) , g(x)= 2f(2x), then find g(24).     22/51

15) If f(x +3)= 3x²- 2x +5, find f(x -1).   3x²- 26x +61

16) Find {f(x+ h) - f(x)} when f(x)= (1- x)/(1+ x).

17) A function is defined as follows 

f(x) = x when x > 0, = - x when x=0 obtain lim ₕ→₀₊ (f(h) - f(0))/h and lim ₕ→₀₋ f(h) - f(0)/h

What can you say about the derivative of f(x) at x=0.        The function does not exist.

18) Find the domain of definition of the following function:

a) (x +2)/√(x²- x -2).       ∞ ≤ x < -1, 2≤ x < ∞

b) (4x -5)/√(x²- 7x +12).       -∞ ≤ x < 3, 4≤ x < ∞

c) log((x²- 5x +6).            x> 3, x> 2 or -∞ ≤ x < 2, 3≤ x < ∞

d) (x²- -5x +6)/(x²- 8x +12).       Domain of the function is all values of x except 6.

19) Find the range of the function 2x/(4 + x²) where x is real     -1/2 ≤ y < 0, 0≤ y ≤ 1/2

20) If y= f(x)= (ax + b)/(CX + a), then show that f(y)= x.

21) If x is real, Find the range of the function x/(1+ x²).   -1/2≤ y ≤1/2

22) If f(x)= (ax - b)/(bx - a), show that f(a) - f(b)f(1/b)= 0

LIMITS 

1) im ₓ→₁ (x²- 2x +3)/(x +4).     2/5

2) lim ₓ→₀ (x²+ x -12)/(x -3).      7

3) lim ₓ→₃ (x²- 9)/(x -3).      6

4) lim ₓ→₂ (x²+ x -6)/(x²- x -2).   5/3

5) lim ₓ→ₐ (√x - √a)/(x - a).       1/2√a

6) lim ₓ→ₐ (x¹⁾³ - a¹⁾³)/(x - a).    1/3³√a²

7)  lim ₓ→∞(1- √x)/(1+ √x).       -1

8) lim ₓ→₋₄ [1/(x +4)  + 1/(x²-4)].    Does not exist 

9) lim ₓ→₂ (2x²- 7x +6)/(3x² -7x +2).    1/5

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

11) lim ₓ→₂ (x²+ x -6)/(x²- x -2).    5/3

12) lim ₓ→₀ {√(a+ x) - √a}/x.       1/2√a

13) If lim ₓ→₂ (ax²- b)/(x -2)= 4, find the values of a and b.     1,4

14) lim ₓ→₁(x²+ 5x -6)/(x²- 3x +2).    -7

15) lim ₓ→₂ {√(x +7) -3}/(x -2).    1/6

16) lim ₓ→₀ {√(1+ 2x) - √(1- 3x)}/x.   5/2

17) lim ₓ→₃ (x²+ x - 12)/(x -3).    7

18) lim ₓ→₀ {√(x + h - √x}/h.    1/2√x

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

20) lim ₓ→₃ (x -3)/{√(x -2) - √(4- x)}.    1

21) lim ₓ→₂(x²- 5x +6)/(x²- 3x +2).   -1

22) lim ₓ→ ₋ ₁ (2x² - x -3)/(x²- 2x -3).   5/4

23) lim ₓ→₀ {√(1+ x) -.√(1- x)}/x.     1

24) lim ₓ→₁ (x²- 1)/{√(3x -1) -.√(5x -1)}.   -4

25) lim ₓ→₁(x² -3x +2)/(x²- 4x +3).    1/2

26) lim ₓ→₋₁(2x²- x -3)/(x²- 2x -3).    -1

27) lim ₓ→₀ (eᵃˣ - eᵇˣ)/x.      (a- b)

28) lim ₓ→ₐ (3x⁴+ 2x²+ 1)/(x⁴+ 2x²+1).   3

29) lim ₓ→₁ (x²+ 4x -5)/(x -1).      6

30) Lim ₓ→₃ (x -3)/{√(x -2) - √(4- x)}.    1

31)  lim ₓ→∞ (4x²- 3x +2)/(5x⁴+ 2x²+3).      0

32)  lim ₓ→₀ x/{√(1+ x) - √(1- x)}..      1

33) lim ₓ→∞ (3x³ + 2x -1)/(4x³+ 3x²-2) .     3/4

34) lim ₕ→₀ {f(2+ h) - f(2)}/h where f(x)= 2x²- x +1.      7

35) lim ₓ→₂₅ {(√x -5)(x +1)}/(x²- 24x -25).       1/10

36) lim ₓ→∞ (3h⁴- 2h² +1)/(h⁴- 2h²+3).       3

37) lim ᵧ→₀ (1/y) {√(1+ 2y) - √(1- 2y)}.       2

38)  lim ₓ→₀ {(√1+ x²) - √(1+ x)}/{√(1+ x³) - √(1+ x)}.     1

39) lim ₓ→₅ (x³-125)/(x⁴- 625).       3/20

40) lim ₓ→∞ (x² + 3x +2)/(x³ + x -4).     0

41) lim ₓ→₀ {√(x²+ a) - √(a - x²)}/x².      1/√a

41) lim ₓ→₃ (x² + x -12)/(x²+2x - 15).     7/8

42) lim ₓ→₀ {√(1+2x) - √(1- 3x)}/x.       5/2

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

44) lim ₓ→₋₄ [1/(x +4) + 8/(x²-4)].   Does not exist 

45) lim ₓ→₂ (2x²- 7x +6)/(3x²- 7x +2).     1/5

46) lim ₓ→₀ {f(x +h) - f(x)}/h, where f(x)= 1/√x. (x > 0).     -1/2√x³

47) lim ₓ→∞ (15x⁷ + 12x +17)/(5x⁷+ 9x²+12).     3

48) lim ₓ→₀  {√(a+ x) - √a}/2x.        1/4√a

49) lim ₓ→∞ (3x²- 4x +6)/(x²+ 6x -7).     3

50) limₓ→₀ (e³ˣ - e²ˣ + 2x)/x.     3

51)  lim ₓ→₀ (14ˣ - 7ˣ - 2ˣ  +1)/x².     Logₑ7 logₑ2

52) lim ₕ→₀ {f(2+ h) - f(2)}/h, where f(x)= 2x²- 7x +1.       1

53) lim ₓ→₀ ∈ ∞ lim ₓ→ₐ lim ₙ→∞

54) lim ₓ→₃ {x - √(x-a)(x -b)/(x²+2x - 15).         (b + a)/2

55) lim ₓ→₃ (√x - √3)/(x²- 9).          1/12√3

56) lim ₓ→∞{1/(1+ n) + 1/(2+ n) + 1/3- n) + ....+ 1/2n}.       0

CONTINUITY 

1) f(x)= (x²- 9)/(x -3). When x≠ 3. State the value of f(3) so that f(x) is continuous at x=3.

2) Find f(2) so that f(x)= (x²- 4)/(x -2) may be continuous at x=2.

3) For what value of f(3), f(x)= (x²-9)/(x -3) will be continuous at x= 3 ?

4) Draw the graph of f(x)= x²/x and g(x)= x. Rough sketches only are to be given. From the graphs so drawn state which of the two functions is not continuous? What is the point of discontinuty ? Indicate the point of discontinuty of the function (2x²+ 6x -5)/(12x²+ x -20).

5) Sketch the graph of f(x)= |x|/x . From the graph, examine continuity of f(x) at x=0.

6) Sketch the graph of f(x)= |x|. From the graph, examine continuity of f(x) at x=0.

7) Sketch the graph of f(x)= 1 for x ≥ 0

                                            = -1 for x ≤ 0

From the graph discuss whether lim ₓ→₀ f(x) exist or not 

8) A function f(x) is defined as follows 

f(x)= 3+ 2x for -3/2≤ x < 0

      = 3- 2x for 0≤ x < 3/2

      = -3 - 2x for x ≥ 3/2

Show that f(x) is continuous at x=0 and discontinuous at x= 3/2.

9) Sketch the graph of 

f(x)= 2x +1 when x≥ 1

      = 2x -1 when x < 1

From the graph examine whether f(x) is continuous at x= 1 or not 

10) Draw the graph of the following function 

f(x)= 1 when x > 0

      = 0 when x = 0

      = -1 when x < 0

Examine the continuity of f(x) at x=0 from the graph.

11) Sketch the graph of 

f(x)= 3x +1 when x ≥ 1

      = 3x -1 when x < 1

From the graph examine whether f(x) is continuous at x= 1 or not 

12) f(x)= (x²-9)/(x -3), when x≠ 3. State value of f(3) so that f(x) is continuous at x= 3.

13) Sketch the graph of the function 

f(x)= - x when x ≤ 0

      = x when 0< x.

From the graph examine the continuity of f(x) at x= 0.         Con

14) Sketch the graph of the function

 f(x)= 3+ 2x when x ≤ 0

          3 -2x when x> 0

From the graph examine the continuity of f(x) at x=0.     C

15) Sketch the graph of the function defined by 

f(x)= x -1 when x> 0

      = 1/2 when x =0

      = x +1 when x< 0

From the graph examine continuity of f(x) at x = 0.      D

16) Draw the graph of the following function 

f(x)= 2x -1.  0≤ x ≤ 4

      = 2- x²    -4< x < 0

State from the graph whether f(x) is continuous at x = 0.

17) Draw a rough sketch of the function f(x)= x/|x| and discuss its continuity at x=0.     D

18) Examine the continuity of the function defined by 

f(x)= x -1 when x> 0

          =1/2 when x= 0

        = x +1 when x < 0

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

f(x)= |x -3|/(x -3), if x ≠3

      =.   1               if x= 3

Discuss the continuity of f(x) at x = 3.    D

20) Given f(x)= (x²-4)/(x -2), if x≠ 2. Find the value of f(2) and show that  f(x) is continuous at x = 2.

21) Discuss continuity of f(x) at x = -2, where 

f(x)= {x + (x +2)/|x +2|, if x≠ -2

     = -1, x = -2.                          C

22) Examine the continuity of the function 

f(x)= 2- 3x when x > 0

      = 2       when x = 0

      = 2+ 3x when x< 0 at x = 0.      C

23) If f(x)= (6- 4x)/(1+ 2x + 2x²), find f(0). Is the function continuous at x= 0 ?     Y

DIFFERENTIATION 

Find the first principle the derivative of 

1) 1/x³.       -2/x³

2) x²- 2x.             2x - 2

3) √x at x = 4.        1/4

4) 2x³+ 3.             6x²

5) 3x³ + 7.       9x²

6) 5x²+ 2.         10x

7) x³+ 4 at x = 1.       3

8) Evaluate lim ₕ→₀ {f(x+ h) - f(x)}/h where f(x)= 2x²+ 3x - 4.      4x +3

9) (x -1)³ at x=1.        0

10) x³.        3x²

11) (3- 5x)¹⁾².      (-15/2)+ √(3- 5x)

12) 2ˣ logx.           2ˣ(1+ x logx log2)/x.

13) 

DIFFERENTIATE

1) y= ₑax²+ bx + c.       (2ax + b)ₑax²+ bx + c.

2) x/(eˣ -1).        (eˣ(1- x) -1)/(eˣ - 1)².

3) 2ˣ. x⁵ .        2ˣx⁴(5+ x log 2)

4) (x² - 3x -5)¹⁾².      (3(2x -3)√(x²- 3x -5))/2

5) (2- 5x)¹⁾².        (15/2) √(2- 5x).

6) x⁵⁾² logx.            x³⁾²(1+ (5/2) logx)

7) If y= x/√(1- x²) then show that (1- x²) dy/dx = y/x.

8) x= at², y= 2at..        1/t

9) x²+ y²= 2a².         -x/y

10) 3⁴ˣ + 3/³√x.         3⁴ˣ. 4 logè - 1/³√x⁴

11) (5- 4x)/(5+ 4x).      -40/(5+ 4x)²

12) log(x + √(x²+ a²)).            1/√(x²+ a²)

13) If xᵐ yⁿ = (x + y)ᵐ⁺ⁿ show that dy/dx = y/x.

14) x². 5³ˣ.       x. 3³ˣ(2+ 3x log5)

15) (x²- 2x -3)/(x - 1).     (x²- 2x +5)/(x - 1)²

16)  xʸ. yˣ= 1.     -(y + x logy)y/(y logx + x)x

17) (x²+1)/(x -1).       (x²- 2x - )/(x - 1)²

18) xˣ.     (1+ log x)xˣ

19) xʸ + y = 1.        yxʸ⁻¹/(1+ xʸ log x)

20) 10ˣ. x¹⁰.         10ˣ. x⁹(x log 10 + 10)

21) x²/a² + y²/b² = 1.     - b²x/a²y

22) xʸ + xy = 8.      -y(1+ xʸ⁻¹)/x(xʸ⁻¹ log x +1)

23) xˣ + x².       xˣ + xˣ log x + 2x

24) √(x²+ a²).       x/√(x²+ a²)

25) (x²+ 1)eˣ.        eˣ(x +1)²

26) (1+ x)ˣ.          x(1+ x)ˣ⁻¹ + (1+ x)ˣ log(1+ x)

27) x³+ 3x²y + y³ = a³.         -(x²+ 2xy)/(x²+ y²)

28) 3x²- x²y + 2y³= 0.         (2xy - 6x)/(6y²- x²)

29) 7²ˣ+ 2ˣ.     7²ˣ2 log 7 + 2ˣ log 2.

30) x= y log(x²y²).      (x - 2y)/(2(log xy +1).

31) y= xˣ.        (1+ logx)xˣ

32) x³+ y³= 3axy.       (ay - x²)/y²- ax)

33) x²/a²+ 2xy/h + y²/b²= 1.       -(a²y + hx)b²/(b²x + hy)a².

34) x= ct, y= c/t.        - 1/t²

35)  eˣʸ = 4(1+ xy) and eˣʸ ≠ 4 then show dy/dx = -y/x.

36) xʸ = yˣ.       (x logy - y)y/(y logx - x)x.

37) y=√{(1+ x)/(1- x)}.       1/{(1- x)√(1- x²)}

38) x= t/(1+ t), y= t/(1- t).       {(1+ t)/(1- t)}²

39) x⁴ₑ3x².       2x³(2+ 3x²)ₑ3x²

40) 3x²+ 2xy - y²= 4.        -(y + 3x)/(x - y)

41) aˣ + xʸ = 4.        -(aˣ log a + yxʸ⁻¹)/(xʸ logx)

42) xˣ +2.      (1+ logx)xˣ + 2ˣ log

43) x= log(xy).       (x - y)/x(1+ log(xy))

44) x= ct³, y= c/t².         -1/t⁶

45) s= t¹⁻ᵗ + t⅖ find ds/dt.       t¹⁻ᵗ{(1- t)/t   logt)} + 2t

46)  (eˣ +1)y= eˣ -1.     e(1- y)/(eˣ +1)

47) y= (xˣ)ˣ.                   ₓxˣ⁺¹(1+ 2 logx)

48) yˣ = eˣ.        (1/e  - logy)y/e

49) x= √(1+ t); y= √(1- t) at t= 1/2.        - √{(1+ t)/(1- t)}

50) xy = eˣ⁻ʸ show that dy/dx = logx/(1+ logx)²

51) log{√(x - a) + √(x - b)}.     1/2√{(x - a)(x - b)}

52) x⁴+ x²y²+ y⁴= 0.      -x(2x² + y²)/y(x²+ 2y²)

53) y= (1+ 2x)ˣ .        (1+ 2x)ˣ .{2x/(1+ 2x) + log(1+ 2x}

54) y= log(ax²+ bx + c).     (2ax + b)/(ax²+ bx + c).

55) x= 1/(1+ t), y= 1/(1- t).           {(1+ t)/(1- t)}²

56) y= xˣ+ log(3x²+ 4x+ 5).         xˣ(1+ logx). (6x+4)/(3x²+ 4x +5)

57) x= 3at/(1+ t³), y= 3at²/(1+ t³).      (2t - t⁴)/(1- 2t³)

58) y= x²/(a²- x²) at x=1.        2a²/(a²- 1)²

59) y= e²ᵐˣ + e⁻²ᵐˣ show that d²y/dx²- 4m²y =0

60) ₓxˣ.          ₓxˣ{xʸ/x + log xˣ(1+ logx)}

61) x²ᵖ yᑫ = (x + y)²ᵖ⁺ᑫ.       y/x

62) x= 2at/(1+ t)², y= (1- t²)/(1+ t²).         -2t/a(1- t²)

63) xˣ + ₑax²+ bx + c.          xˣ(1+ logx) + ₑax²+ bx + c(2ax + b)

64)  If y= (x + √(1+ x²))ᵐ, show that (1+ x²)y₂ + xy₁ - m²y=0

65) If y= (x -2)/(x +2), show that 2x dy/dx = 1- y².

66) If yˣ = eʸ⁻ˣ, show that dy/dx = (logₑy)²/logy.

67) Given x= t + 1/t and y= t - 1/t, find value of d²y/dx⅖ at the point t=20.         

68) If x³- 2x²y²+ 5x + y -5=0, then find d²y/dx² at x= 1 , y= 1

69) If f(x)= {(a + x)/(b + x)}ˣ + 2x, show that f'(0)= 2 + log(a/b).

70) If x⁴ + x³y³+ y⁴= 0

71) f(x)= {(a+ x)/(b+ x)}ᵃ⁺ᵇ⁺²ˣ show that f'0)= (2 log(a/b) + (b²- a²)/ab)(a/b)ᵃ⁺ᵇ

EULER'S THEOREM 

1) State Euler's theorem for homogeneous function of 2 variables of degree 'n'.    

MAXIMUM AND MINIMUM 

1) Find maximum and minimum values of the function y= x³- 3x -1.      3,-1

2) Find the maximum or minimum point of the function y= 5- x - x²

3) Show that the maximum value of the function 2x + 1/2x is less then its minimum value.       

4) Examine x³- 9x²+ 24x -12 of maximum and minimum values of x³- 9x²+ 24x -12.

5) Show that the function f(x)= 12 -24x - 15x²- 2x³ has a maximum at x= -1, minimum at x= -4 and point of inflexion at x= 5/2.

6) Show that the the maximum value of x³+ 1/x³ is less than its minimum value.

6) Find the maximum and minimum values of y= x/{(x -1)(x -4)}.      -2

7) Determine whether the following 2x³/3 - 6x²+ 20x - 5 has a maximum or a minimum.

8) Show that maximum value of f(x)= x +1/x is less than its minimum value.

9) Examine whether the curve y -3 = 6(x - 2)⁵ has a point of inflexion at (2,3).

10) Show that maximum value of 2x + 1/x is less than its minimum value.

11) Find the point of inflexion of the curve y+ 5= x³- 3x²+ 9x.

12) Find the maximum and minimum value of the function 2x³- 15x²+ 36x.     28,27

13) Divide the number 20 in two parts in such a way that their product will be maximum.   10,10

14) If x+y= 2, show that the maximum value of (4/x + 36/y is less than its minimum value.

APPLICATION OF DERIVATIVES (COMMERCE)

1) A firm produces x tons of a valuable metal per month at a total cost c given by:

C= Rs (x³/3 - 5x² + 75x +10).

Find at what level of output the marginal cost attains its minimum.     5

2) A firm produces x units of output per week at a total cost of Rs (x³/2 - x² + 5x +3).

Find at what level of output the marginal cost and the average variable cost attains their respective minima.        2/3

3) A radio manufacturer finds that he can sell x radios per week at Rs p each where p= 2(100 - x/4). His cost of production of x radios per week is Rs (120x + x²/2). Show that his profit is maximum when the production is 40 radios per week. Find also his maximum profit per week.     40, Rs1600

4) A radio manufacturerproduces  x sets per week at total cost of Rs x²+ 78x + 2500. He is monopolist and the demand function for his product is x= (600- p)/3 when the price is Rs p per set. Show that the maximum net revenue (profit) is obtained when 29 sets are produced per week. What is the monopoly price?       Rs405

5) A manufacturer can sell x items per month at a price p= 300 -2x rupees. Manufacturer's cost of production y rupees of x items is given by y= 2x + 1000. Find the number of items to be produced per month to yield the maximum profit.     75

6) The cost of manufacturing a certain article is given by the formula C= 5+ 48/(x + 3x²) where x is the number of articles manufactured. Find the minimum value of C.      41

7) The total cost function C for producing x units of an article per day is given by C= Rs (400 -16x + x²). Find the average cost function and the level of output at which their function is minimum.      20 

8) If the cost function for x units is given by C= Rs (400- 15x - x²) obtain the 

a) average cost 

b) average variable cost.     400/x - 16 - x, (-6- x)

9) The total cost C, of making x units of a product is C= axⁿ + b, where a,b,n are positive constants. Find marginal cost and marginal average cost.     naxⁿ⁻¹, axⁿ⁻¹ + b/x, a(n -1)xⁿ⁻² - b/x²

INTEGRATION 

1) ∫ x/(x +1) dx.       x - log(x +1)

2) ∫ √x(x½+ 3x +4)dx.      (2/7) √x⁷ + (6/7) √x⁵+ (8/3) √x

3) ∫ (x²+ 1)²/x³ dx.      x²/2 + 2 logx - (1/2x²)

4) ∫ 3dx/(x²-1).       (3/2) Log{x -1)/(x +1)

5) ∫ √{(x +1)/(x -1)} dx.       √(x²-1) + log|x + √(x²-1)|

6) ∫ x²/³√(3x +5) dx.      (1/8) ³√(3x +5)⁸ - 2 ³√(3x +5)⁵ + (25/2) ³√(3x +5)²

7) ∫ logx/(x +1)² dx.        - log(x +1) + log|x/(x +1|

8) ∫ x⁴/(³√(2x⁵+3) dx.     (3/20)³√(2x⁵+ 3)²

9) ∫ (x -2)eˣ/(x -1)² dx.     eˣ/(x -1)

10) ∫ dx/(eˣ+1).      eˣ/(eˣ +1)

11) ∫ x √(x²-1) dx.     (1/4) (x⁴- 2x²+1)

12) ∫ (1+ 2/√x +3) dx.       4√x(√x + 1)

13) ∫ x³/(x -1) dx. .       (2x³+ 3x²+ 6x -11 + 6 log(x -1)/6

14) ∫ (4x -3)/x²  dx.     32x²- 144x + 27/x + 108 logx

15)  ∫ (x -2) dx/³√(x²- 4x -5).      (3/4) ³√(x²- 4x -5)²

16) ∫ dt/(2t²+ 3t +1).         (1/2) Log|(2t+1)/(t+1)|

17) ∫ (x +1)²/√x dx.          2∫∫ ∫∫∫∫∫∫∫∫∫∫∫∫∫∫  ∫∫∫∫∫ ∫∫∫∫∫∫∫∫∫∫∫∫∫∫  ∫∫∫∫∫ ∫∫∫∫∫∫∫∫∫∫∫∫∫∫  ∫∫∫∫∫ ∫∫∫∫∫∫∫∫∫∫∫∫∫∫  ∫∫∫∫∫ ∫∫∫∫∫∫∫∫∫∫∫∫∫∫  ∫∫∫∫∫ ∫∫∫∫∫∫∫∫∫∫∫∫∫∫  ∫∫∫∫∫ ∫∫∫∫∫∫∫∫∫∫∫∫∫∫  ∫∫∫∫∫ ∫∫∫∫∫∫∫∫∫∫∫∫∫∫  ∫∫∫∫∫ ∫∫∫∫∫∫∫∫∫∫∫∫∫∫  ∫∫∫∫∫ ∫∫∫∫∫∫∫∫∫∫∫∫∫∫  ∫∫∫∫∫ ∫∫∫∫∫∫∫∫∫∫∫∫∫∫  ∫∫∫∫∫ ∫∫∫∫∫∫∫∫∫∫∫∫∫  ∫∫∫∫∫ ∫∫∫∫∫∫∫∫∫∫∫∫∫∫  ∫∫∫∫∫ ∫∫∫∫∫∫∫∫∫∫∫∫∫∫  ∫∫∫∫∫ ∫∫∫∫∫∫∫∫∫∫∫∫∫∫  ∫∫∫∫∫ ∫∫∫∫∫∫∫∫∫∫∫∫∫∫  ∫∫∫∫∫ ∫∫∫∫∫∫∫∫∫∫∫∫∫∫  ∫∫∫∫∫ ∫∫∫∫∫∫∫∫∫∫∫∫∫∫  ∫∫∫∫∫ ∫∫∫∫∫∫∫∫∫∫∫∫∫∫  ∫∫∫∫∫ ∫∫∫∫∫∫∫∫∫∫∫∫∫∫  ∫∫∫∫∫ ∫∫∫∫∫∫∫∫∫∫∫∫∫∫  ∫∫∫∫∫ ∫∫∫∫∫∫∫∫∫∫∫∫∫∫  ∫∫∫∫∫ ∫∫∫∫∫∫∫∫∫∫∫∫∫∫  ∫∫∫∫∫ 

) Find the slope of the curve at the point t=2 when x= t²- 3., y= 2t +1.      1/2

) Find the gradient of the curve log(xy)= x²+ y² at the point (1,1).      -1

) State Rolle's Theorem.

) 

DIFFERENTIATION 

WB MATHS - X

INTREST 

1) If a principal becomes twice of it n 10 years, then the rate of simple interest per annum is 

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

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

a) Rs ab/100 b) Rs ab/120 c) Rs ab/1200 d) Rs ab/10

3) If the ratio of principal and yearly amount 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 intrest 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) Rs x 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 pnr/25, The principal will be 

a) Rs 2p b) Rs 4p c) Rs p/2 d) none

6) If the intrest 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 x I c) prt= 100 x 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 Rs 400 amounts to Rs 480 in 4 years. What will it amount to if the rate of interest is increased by 2%?

a) Rs 484 b) Rs 560 c) Rs 512 d) none 

9) At what rate of percent per annum will Rs 2304 amount to Rs 2500 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 intrest percent per annum?

a) 10% b) 20% c) 25% d) 30%

11) A person deposited Rs 100 in a bank and got the amount Rs 121 for two 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 principal remains unchanged each year 

b) Principal changes in each year 

c) Principal may be equal or unequal in each year 

d) none 

1b 2b 3b 4c 5b 6c 7c 8c 9c 10b 11a 12c 13b 

1) A person deposited Rs 100 in a bank and gets the amount Rs 121 after two years. The rate of compound interest is ___%.

2) If the simple interest of a principal for n years at r% per annum be Rs (pnr)/25, then the principle will be Rs ____

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

4) A person depreciated certain rate over time its called ____.

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

6) If the ratio principal and amount for 1 year is 8:9, then the rate of the simple interest per annum is ____.

7) Fixed amount rupees fixed annual interest rate 1 year compound interest rate and simple interest rate ____.

8) With the passage of time, someone grows at a certain rate, it is called ___.

1) 10% 2) 4p 3) one 4) homogeneous depreciation 5) Rs (Prt)/100 6) 25/2% 7) equals 8) Identical 

1) The amount of Rs 2p in t years at the rate of simple interest of r/2% per annum is Rs (2p + prt/100).

2) The difference between the simple interest and the compound interest of Rs 100 in 1 year at the rate of 10% p.a is Rs 1.

3) At same rate of interest the simple interest for 2 years is more than the compound interest on the same principle.

4)  The present value of an article is Rs 100. The value of the article depreciates by 10% per annum. After 29 years the cost of the article be Rs 81.

5)  The present price of an article is Rs x. The rate of increase in price per year is y%. After z years the cost of the article will be Rs (1+ y/100)².

6) Since the rate of interest on the specified principal remains the same, the simple interest for 2 years will be higher than the compound interest.

7) When time and rate of interest are equal, total interest is proportional to principal.

8) If the principal and the rate of the simple interest per annum are the same, the total interest is inversely related to time.

9)  Any principal amount will double in 20 years at the rate of 5% simple interest per annum.

10) The compound interest on a principal at the same rate of interest per annum for the same period of time will be less than the simple interest.

11) In the compoun intrest, the interest is the sum of the principal amount over a specified period of time. For this reason the amount of principal keeps on increasing gradually. 

1) t 2) f 3) f 4) t 5) t 6) f 7) t 8) f 9) t 10) f 11) t

2 MARKS

1) If the ratio of a principal and its compound for 5 years is 5:6, decide the annual simple rate of interest.

2) What will be the monthly interest of Rs 1 at the rate of 5% simple interest per annum.

3) The rate of simple interest per r reduces from 4% to 15/4% and for this, a person's annual income decreases by Rs 60. Determine the principal of that person .

4) A sum of money is doubled in 8 years at r% rate of compound interest per annum. At the same rate in how many years it will be four times of the Sum ?

5) Calculate Whats sum of money will amount to Rs 3528 after 2 years at the rate of 5% compound interest per annum.

6) For how many years will the principal amount double at the rate of 25/4% simple interest per annum.

7) At what rate of annual simple interest is the interest of a principal 8/25 part of the principle amount in 4 years ?

8)  What is the rate of simple interest per annum, the interest of 2/5 a rupee for 10 years will be part of the compound ?

9) If the compounded compound interest of Rs 400 for 2 years is Rs 441, then what is the compound interest rate per annum ?

10) 5% compound interest of a few rupees in 2 years at the rate of compound interest per annum is Rs 615, then find the principal.

11) What is the rate of simple interest per annum, when the interest on some money in 10 years will be 2/5 parts of its amount (principle along with interest).

12) if the rate of compound interest for the first and second year are 5% and 6% respectively, let us calculate the compound interest on Rs 5000 for 2 years.

1) 4% 2) Rs240 3) Rs 24000 4) 16 5) Rs 3200 6) 16 year 7) 8 8) 20/3% 9) 5% 10) Rs6000  11) 20/3% 12) Rs 565

5 MARKS 

1) The price of a machine in a factory of your uncle depreciates at the rate of 10% every year. If its present price is Rs 6000 then what will be its price after 3 years ?

2)  If interest is compounded half yearly what will be the compound interest and amount on Rs 8000 at the rate of 10% compound interest per annum for 3/2 years ?

3) Khawaish has taken a loan of rse64000 from a bank, if the rate of interest be 2.5 paise per annum, calculate the compound interest payable after 2 years.

4)  The difference between simple interest and compound interest for 2 years of a sum of money becomes Rs 80 at 4% interest per annum. Calculate the sum of money.

5) As a result of Sarba Siksha Abhiyan, the students leaving the school before completion, the students are readmitted, so the number of students in a year is increased by 5% in comparison to its previous year. If the number of such readmitted students in a district be 3528 in the present year, calculate the number of students readmitted 2 years before in this manner.

6) Calculate in how many years Rs 50000 will amount to Rs 60500 the rate of 10% compound interest per annum .   

7) Calculate the compound interest and amount on Rs 1600 for 3/2 years at the rate of 10% compound interest per annum, compounded at an interval of 6 months .

8) At present the sum of the number of the student in Magnus Academy is 3993. if the increase in the number of students in a year was 10% of its previous year, Calculate the sum of the number of Students 3 years before in all in Magnus Academy.

9)  Calculate the sum of the money, if the difference between compound interest and simple interest for 2 years at the rate of 9% interest per annum is Rs 129.60.

10) As a result publicity on smoking, the number of smokers is decreased by 25/4% every year in comparison to its previous year. If the number of smokers at present in a city is 33750, calculate the number of smokers in that city 3 years before.

11)  A bank gives 5% simple interest per annum, In that Bank Archana deposits Rs 15000 at the beginning of the year, but withdraws Rs 3000 after 3 months and then again, after 3 months, she deposits Rs 8000. Find the amount (principal along with interest) Archana will get at the end of the year.

12) Kushagra deposity the money for each of his two daughters in such a way that when the ages of each of his daughters will be 18 years, each one will get Rs 120000. The rate of simple interest per annum in the bank is 10% and the present ages of his daughters are 13 years in 8 years respectively. Find the money he had deposited separately in the bank for each of his daughters.

13) if simple interest and compound interest of a certain sum of money for 2 years are Rs 8400 and Rs 8652, then find the sum of money and the rate of the interest.

14) Raju got Rs 10000 when retired from his service. He deposited some of that money in the post office and get Rs 5400 in total per year as interest. If the rate of simple interest per annual in the bank and in the post office at 5% and 6% respectively, then find the money he had deposited in the bank and the post office.

15) Robin Sir takes a loan amount of Rs240000 from a bank for constructing a building at the rate of simple interest of 12% per annum . After 1 year of taking the loan he rents the house at the rate of Rs5200 per month. Determine the number of years he would take to repay his loan along with interest from the income of the house rent.

1) Rs 4374 2) Rs 9261, Rs 1261 3) Rs 3240 4) Rs 50000 5) 3200 6) 2 years 7) Rs 252.20  8) 3000 9) Rs 16000 10) 40960 11) Rs 20837.50 12) 80000, 60000 13) 6%, 70000 14) 60000 in bank, 40000 in P. O 15) 9 years 

QUADRATIC EQUATION 

1) The product of two roots of the equation x²- 7x +3=0 is

a) 7 b) -7 c) 3 d) -3

2) Under what condition one root of the equation ax²+ bx+ c=0 is zero 

a) a= 0 b) b= 0 c) c= 0  d) none

3) 2x²- 3x + k +2 =0 one root of the equation is zero. The product. The value of k is 

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

4) If two roots of equation x² + 4x +k =0 are equal, then the value of k is 

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

5)  If two roots of equation x² - 6x +k =0 are real and unequal , then the value of k is 

a) more than 6 b) less than 9 c) more than 9 d) less than 9

6) The sum of two roots of the equation x² - 6x +2 =0 

a) 2  b) -2 c) 6 d) -6

7) If the product of two roots of the equation x² - 3x +k = 10 is -2, what is the value of k?

a) -2 b) -8 c) 8 d) 12

8) If two roots of the equation ax² + bx + c=0 (a≠ 0) be equal, then 

a) c= - b/2a b) c= b/2a c) c= - b²/4a d) c= b²/4a

9) The root/s of equation x²/x = 6 is/are

a) 0 b) 6 c) 0 and 6 d) -6

10) If two roots of the equation (k +1)x² + 2kx + (k +2) =0 are equal and negative then value of k is 

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

11) If roots of the equation ax² + bx + c=0 (c≠ 0) are real and unequal then b²- 4ac will be 

a) > 0 b) =0 c) < 0 d) none 

12) The number of roots of quadratic equation is 

a) one b) two c) three d) none 

13) If ax² + bx + c is a quadratic equation then

a) b≠ 0 b) c≠ 0 c) a ≠ 0 d) none 

14) The highest power of the variable of a quadratic equation is 

a) 1 b) 2 c) 3 d) none 

15) The equation 4(5x² -7x + 2)= 5(4x²- 6x +3) is 

a) linear b) quadratic c) 3rd degree d) none 

1c 2c 3a 4d 5d 6c 7c 8d 9c 10c 11a 12b 13c 14b 15a 

16) The roots of the equation ax² + 2bx + c= 0 (a≠ 0) are real and equal, then b²= ____

17) If the product and sum of two quadratic surds is a rational number, then the surds are ____ surd.

18) The equation (a-2)x²+ 3x +5=0 will not be a quadratic equation for a = ____

19) If in a quadratic equation ax² + bx + c=0 (a≠ 0), b²= 4ac, then the roots of the equation will be real and ____.

20) If the sum and product of two quadratic equation is a fundamental number, then both the conditions are ____.

21) 7x² - 12x + 18=0. The ratio of the sum and the product of the roots of the equations_____.

22) ax² + bx + c= 0 (a≠ 0). If both the roots of the equation are mutually inverse (reciprocal), then c= _____.

23) ax² + bx + c= 0 (a≠ 0) if both the roots of the equation are mutually inverse and negative, then a+ c =____'

16) ac 17) pure 18) 2 19) equal 20) supplement 21) 2:3 22) a 23) 0 24) 

24) √π is a quadratic.    T/F

25) x² + x + 1= 0 both the roots of the equation are real.    T/F

26) x² + x + 2 = 0 both the roots of the equation are not real.    T/F

27) x²= 25 the only root of the equation is 5.     T/F

28) 3x² - 2 √6x +2=0 both of the equation will be real and equal.    T/F

24) f 25) f 26) t 27) f 28) t

29) If the roots of the equation 5x² - 2x + 3 = 0 be a and b, find the value of (1/α + 1/β).

30) If the roots of the equation be 2 and -3, then write the equation.

31) Without solving, find the values of p for which the equation x²+ (p -3)x + p=0 has real and equal roots.

32) If kx²+ 2x + 3k =0 (k≠ 0) the sum and product of the roots of the equation are the same, write the value of k.

33) If x² - 22x + 105 =0 the equation has two roots α and β then determine the value of 

(α - β).

29) 2/3  30) x²+ x - 6=0 31) 9,1 32) -2/3 33) ±8

34) Solve: 1/(x -3) - 1/(x +5) = 1/6.        7,-9

35) The product of two consecutive positive odd numbers is 143. Determine the numbers.      11,13

36) Determine the equation whose roots are the square of the roots of the equation x²+ x +1=0.       

37) If the price of 1 dozen pens is reduced by Rs 6, then 3 more pens can be bought in Rs 30. Find the price of 1 dozen pens before the reduction of price.     Rs 38

38) Solve: {(x+4)/(x -4)}² - 5(x+4)/(x -4)+ 6 =0, x ≠ 4.       8,12

39) The digit in the unit's place of a two digited number is 6 more than that at the ten's place. The product of the digits is 12 less than the number. Find the possible values of the digits in the unit's place.      8 or 9

40) If 5 times of a positive whole number is less by 3 then twice of its square, then find the number.        3

41) solve: 1/x  - 1/(x - b) = 1/a - 1/(a + b).        a, -(a+ b)

42) Solve: x²- (√3+ 2) x + 2√3=0.      √3, 2

43) (x+3)/(x -3) + 6{(x-3)/(x +3)}= 5.        9,6

44) x/(x +1)  + (x +1)/x   = 25/12, x≠0,-1.       3,-4

45) (2x +1)  + 3/(2x +1) = 4.      0,1

46) (x +1)/2  + 2/(x +1)  = (x +1)/3 + 3/(x +1)  - 5/6.      0,-7

47) {(x+a)/(x -a)}² - 5{(x+a)/(x -a)} + 6 =0, x≠ a.      2a, 3a

48) The tens digit of a two digit number is 3 less than the unit. Substracting their product from the number gives the disjunction 15. Find the unit's digit of the number.    5 or 9

49) There are two in our school's cistern. A cistern is filled in a 1/x minutes by both of drains. The second tap takes 5 minutes longer than the first. In how much time will each tap fill the cistern separately?      20,25 minutes 

50) When two digits positive numbers is multiplied by its unit digit, the product is 189. If the tens digit's is twice the unit digr, then what is the unit's digit?      3

51) (b - c)x² + (c - a)x + (a - b)=0 if both the roots of a quadratic equation are equal, show that 2b= a+ c.

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

53) ax²+ bx + c=0 if one root is twice the other, then show that 2b²= 9ac.

54) There is a square park in our locality. The area of a rectangular Park of length 5m more than the length of one side of this park and 3m less in width is equal to that of a square Park. then find the length of the side of the square Park.      9m

55) Pralay Babu of our village decided to plant 350 red pepper plants in one of his rectangular plots of land . After planting the sapling in rows, he saw that after planting in 24 more rows than the number of saplings planted in each row, 10 saplings were left, Then how many saplings have been planted by them in each row?      10

56) The length of the base of a triangle is 18m more than twice its height. If the area of the triangle is 360 square metres. Find the height ?       15m

57) The distance between two places is 200 km. The time taken by motor car to travel from one place to another is 2 hours less than it takes to travel by jeep. If the speed of the motor car is 5km per hour less than that of the Jeep, then find the speed of the motor car.         20

58) The area of Anita's rectangular land is 2000 square meters and its boundary is 180 meters , then find the length and the breadth of the Anita's rectangular land.     50,40

59) The velocity of a boat in still water is 8 km/hr. A boat takes 5 hours to go 15km downstream and 22km upstream, then what is the velocity of the stream?     8/5

SHORT QUESTION VII/ VIII/IX

RATIONAL NUMBERS

BOOSTER - A

1) (-5/16 + 7/12)= ?

a) -7/48 b) 1/24 c) 13/48 d) 1/3

2) (8/-15 + 4/-3)= ?

a) 28/15 b) -28/15 c) -4/5 d) -4/15

3) 7/-26  + 16/39= ?

a) 11/78 b) -11/78 c) 11/39 d) -11/39

4) 3+ 5/-7= ?

a) -16/7 b) 16/7 c) -26/7 d) -8/7

5) 31/-4  + -5/8= ?

a) 67/8 b) 57/8 c) -57/8 d) -67/8

6) What should be added to 7/11 to get -4/15 ?

a) 17/20 b) -17/20 c) 7/20 d) -7/20

7) 2/3 + -4/4 + 7/15 + -11/20= ?

a) -1/5 b) -4/15 c) -13/60 d) -7/30

8) The sum of two numbers is -4/3 if one of the numbers is -5, what is the other?

a) -11/3 b) 11/3 c) -19/3 d) 19/3

9) What should be added to -5/7 to get -2/3 ?

a) -29/21 b) 29/21 c) 1/21 d) -1/21

10) What should be substracted from -5/3 to get 5/6 ?

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

11) (-3/7)⁻¹= ?

a) 7/3 b) -7/3 c) 3/7 d) none 

12) The product of two rational numbers is -28/81. If one of the numbers is 14/27 then the other one is 

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

13) The product of two numbers is -16/35. If one of the numbers is -15/14, the other is

a) -2/5 b) 8/15 c) 32/75 d) -8/3

14) What should be substracted from -3/5 to get -2?

a) -7/5 b) -13/5 c) 13/5 d) 7/5

15) The sum of two rational numbers is -3. If one of them is -10/3 then the other one is 

a) -13/3 b) -19/3 c) 1/3 d) 13/3

16) Which of the following numbers is in standard form?

a) -12/26 b) -49/70 c) -9/16 d) 28/-105

17) -9/16 x 8/15=?

a) -3/10 b) -4/10 c) -9/25 d) -2/5

18) -5/9 ÷ 2/3=?

a) -5/2 b) -5/6 c) -10/27 d) -6/5

19) 4/9 ÷ ? = -8/15

a) -32/45 b) -8/5 c) -9/10 d) -5/6

20) Additive inverse of -5/9 is

a) -9/5 b) 0 c) 5/9 d) 9/5

21) Reciprocal of -3/4 is

a) 4/3 b) 3/4 c) -4/3 d) 0

22) A rational number between -2/3 and 1/4 is

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

23) The reciprocal of a negative rational number 

a) is a positive rational number 

b) is a negative rational number 

c) can be either a positive or a negative rational number 

d) does not exist.

BOOSTER - B

1) Find the additive inverse of: 

a) 7/-10

b) 8/5

2) The sum of two rational numbers is -4. If one of them is -11/5, find the other.

3) What number should be added to -3/5 to get 2/3?

4) What number should be substracted from -3/4 to get -1/2 ?

5) Find the multiple inverse of:

a) -3/4

b) 11/4

6) The product of two numbers is -8. If one of them is -12, find the other.

7) Evaluate: 

a) -3/5 x 10/7

b) (-5/8)⁻¹

c) (-6)⁻¹

8) Name the property of multiplication shown by each of the following statements:

a) -12/5 x 3/4= 3/4 x -12/5

b) -8/15 x 1= -8/15

c) (-2/3 x 7/8) x -5/7= -2/3 x (7/8 x -5/7)

d) -2/3 x 0 = 0

e) 2/5 x (-4/5 + -3/10)= (2/5 x -4/5) ÷ (2/5 x -3/10)

9) Find two rational numbers lying between -1/3 and 1/2.

BOOSTER - C

1) What should be added to -3/5 to get -1/3?

a) 4/5 b) 8/15 c) 4/15 d) 2/5

2) What should be substracted from -2/3 to get 3/4?

a) -11/12 b) -13/12 c) -5/4 d) -17/12

3) (-5/4)⁻¹=?

a) 4/5 b) -4/5 c) 5/4 d) 3/5

4) The product of two numbers is -1/4. If one of them is -3/10, then the other is

a) 5/6 b) -5/6 c) 4/3 d) -8/5

5) -5/6 ÷ -2/3=?

a) -5/4 b) 5/4 c) -4/5 d) 4/5

6) 4/3 ÷ ? = -5/2

a) -8/5 b) 8/5 c) -8/15 d) 8/15

7) Reciprocal of -7/9 is

a) 9/7 b) -9/7 c) 7/9 d) none 

8) A rational number between -2/3 and 1/2 is

a) -1/6 b) -1/12 c) -5/6 d) 5/6

9) Fill in the blanks 

a) 25/8 × ___= - 10.

b) -8/9 x ___= -2/3.

c) (-1) + ___= -2/9

d) 2/3 -  ____= 1/15.

10) WRITE TRUE/FALSE 

a) Rational numbers are always closed under substraction.

b) Rational numbers are always closed under division.

c) 1 ÷ 0= 0.

d) Subtraction is commutative on rational numbers.

e) -(-7/8) = 7/8

EXPONENTS 

BOOSTER - A

1) The value of (2/5)⁻³ is 

a) -8/125 b) 25/4 c) 125/8 d) -2/5

2) The value of (-3)⁻⁴ is 

a) 12 b) 81 c) -1/12 d) 1/81

3) The value of (-2)⁻⁵ is 

a) -32 b) -1/32 c) 32 d) 1/32

4) (2⁻⁵ ÷ 2⁻²)= ?

a) 1/128 b) -1/128 c) -1/8 d) 1/8

5) The value of (3⁻¹ + 4⁻¹)⁻¹ ÷ 5⁻¹ is 

a) 7/10 b) 60/7 c) 7/5 d) 7/15

6) (1/2)⁻² + (1/3)⁻² + (1/4)⁻² = ?

a) 61/144 b) 144/61 c) 29 d) 1/29

7) {(1/3)⁻³ - (1/2)⁻³} ÷ (1/4)⁻³= ?

a) 19/64 b) 27/16 c) 64/19 d) 16/25

8) [{(-1/2)²}⁻²]⁻¹= ?

a) 1/16 b) 16 c) -1/16 d) -16

9) The value of x for which (7/12)⁻⁴ x (7/12)³ˣ = (7/12)⁵, is 

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

10) If (2³ˣ⁻¹ + 10) ÷ 7= 6, then x is equal to 

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

11) (2/3)⁰= ?

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

12) (-5/3)⁻¹= ?

A) 5/3 b) 3/5 c) -3/5 d) none 

13) (-1/2)³= ?

a) -1/6 b) 1/6 c) 1/8 d) -1/8

14) (-3/2)² = ?

a) -9/16 b) 9/16 c) 16/9 d) -16/9

15) 3670000 in standard form is

a) 367 x 10⁴ b) 36.7 x 10⁵ c) 3.67 x 10⁶ d) none 

16) 0.0000463 in standard form is

a) 4.63 x10⁻⁷ b) 4.63 x10⁻⁵ c) 4.63x 10⁻⁹ d) 46.3 x 10⁻⁶

17) 0.000367x 10⁴ in usual fom is 

a) 3.67 b) 36.7 c) 0.367 d) 0.0367

BOOSTER- B

1) Evaluate:

a) 3⁻⁴

b) (-4)³

c) (3/4)⁻²

d) (-2/3)⁻⁵

e) (5/7)⁰

2) Evaluate: {(-2/3)³}⁻².

3) Simplify : {3⁻¹ + 6⁻¹} ÷ (3/4)⁻¹.

4) By what number should (-2/3)⁻³ be divided so that the quotient is (4/9)⁻² ?

5) By what number should (-3)⁻¹ be multiplied so that the product becomes 6⁻¹ ?

6) Express each of the following in standard fom:

a) 345

b) 180000

c) 0.000003

d) 0.000027

BOOSTER- C

1) The value of (-3)⁻³ is 

a) -27 b) 9 c) -1/27 d) 1/27

2) The value of (3/4)⁻³ is 

a) -27/64 b) 64/27 c) -9/4 d) 27/64

3) (3⁻⁶ ÷ 3⁴)= ?

a) 3⁻² b) 3² c) 3⁻¹⁰ d) 3¹⁰

4) If (5/12)⁻⁴ x (5/12)³ˣ= (5/12)⁵, then x= ?

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

5) (3/5)⁰= ?

a) 5/3 b) 3/5 c) 1 d) 0

6) (-6/5)⁻¹= ?

a) 6/5 b) -6/5 c) 5/6 d) -5/6

7) (-1/3)³= ?

a) -1/9 b) 1/9 c) -1/27 d) 1/27

Fill in the blanks:

a) 360000 written in standard fom is____

b) 0.0000123 written in standard form is _____

c) (-2/3)⁻² = _____

d) 3 x 10⁻³ in usual form is_____

e) 5.32 x 10⁻⁴ in usual form is ____

SQUARES AND SQUARE ROOTS 

BOOSTER - A

1) Which of the following number is not a perfect square ?

a) 7056 b) 3969  c) 5478  d) 4624 

2) Which of the following numbers is not a perfect square ?

a) 1444 b) 3136 c) 961 d) 2222

3) Which of the following numbers is not a perfect square ?

a) 1843 b) 3721 c) 1024 d) 1296

4) Which of the following numbers is not a perfect square ?

a) 1156 b) 4787 c) 2704 d) 3969

5) Which of the following numbers is not a perfect square ?

a) 3600 b) 6400 c) 81000 d) 2500

6) Which of the following cannot be the unit digit of a perfect square number ?

a) 6 b) 1 c) 9 d) 8

7) The square of the proper fraction is 

a) larger than the fraction 

b) smaller than the fraction 

c) equal to the fraction  d) none 

8) if n is odd, then 1+ 3 + 5+ 7+.... to n terms is equals to

a) n²+1 b) n² - 1 c) n² d) 2n²+1

 

9) Which of the following is a Pythagorean triplet ?

a) (2,3,5) b) (5,7,9) c) ( 6,9,11) d) (8,15,17)

10) What least number must be subtracted from 176 to make a perfect square ?

a) 16 b) 10 c) 7 d) 4

11)  What least number must be added to 526 to make it a perfect square?

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

12) What least number must be added to 15370 to make a perfect square?

a) 4 b) 6 c) 8 d) 9

13) √0.9= ?

a) 0.3 b) 0.03 c) 0.33 d) 0.94

14) √0.1=?

a) 0.1 b) 0.01 c) 0.316 d) none 

15) √0.9 x √1.6= ?

a) 0.12 b) 1.2 c) 0.75 d) 12

16) √288/√128= ?

a) √3/2 b) 3/√2 c) 3/2 d) 1.49

17) √9/4= ?

a) 5/2 b) 3/2 c) 5/4 d) none 

18) Which of the following is the square of an even number ?

a) 196 b) 441 c) 626 d) 529

19) Which of the following is the square of an odd number ?

a) 2116 b) 3844 c) 1369 d) 2500

BOOSTER - B

1) Evaluate : √11236

2) Find the greatest number of the 5 digit which is a perfect square. What is the square root of this number ?

3) Find the least number of 4 digits which is a perfect square. What is the square root of this number ?

4) Evaluate : √0.2809.

5) Evaluate: √3 correct up to two places of decimal.

6) Evaluate: √48/√243.

BOOSTER - C

1) Which of the following numbers is not a perfect square ?

a) 529 b) 961 c) 1024 d) 1222

2) √9/4=?

a) 5/2 b) 5/4 c) 3/2 d) none 

3) Which of the following is the square of an even number ?

a) 529 b) 961 c) 1764 d) 2809

4) What least number must be added to 521 to make it a perfect square?

a) 3 b) 4 c) 5 d) 8

5) What least number must be subtracted from 178 to make it a perfect square ?

a) 6 b) 8 c) 9 d) 7

6) √72 x √98= ?

a) 42 b)  84 c) 64 d) 77

Fill in the blanks:

a) 1+ 3+5+7+9+11+13= ____

b) √1682= ___

c) The smallest square number exactly divisible by 2, 4, 6 is ___

d) A given number is a perfect square having n digits, where n is odd. Then its square root will have____ digits.

CUBES AND CUBE ROOTS 

BOOSTER - A

1) Which of the following numbers is a perfect cube ?

a) 141 b) 294  c) 216  d) 496 

2) Which of the following numbers is a perfect cube?

a) 1152 b) 1331 c) 2016 d) 739 

3) ³√512=?

a) 6 b) 7 c) 8 d) 9 

4) ³√(125 x 64)= ?

a) 100 b) 40 c) 20 d) 30

5) ³√(64/343)= ?

a) 4/9  b) 4/7  c) 8/7  d) 8/21 

6) ³√(-512/729)=?

a) -7/9 b) -8/9 c) 7/9 d) 8/9

7) By what least number should 648 be multiplied to get a perfect cube ?

a) 3 b) 6 c) 9 d) 8 

8) By what least number should 1536 be divided to get a perfect cube ?

a) 3 b) 4 c) 6 d) 8 

9) (13 0/10)³=?

a) 1027/1000 b) 2027/1000 c) 2197/1000 d) none 

10) (0.8)³=?

a) 51.2 b) 5.12 c) 0.512  d) none 

BOOSTER - B

1) Evaluate: (7/5)⅔.

2) Evaluate ³√4096.

3) Evaluate: ³√(216 x 343).

4) Evaluate: ³√(64/125).

BOOSTER - C

1) (7/4)³= ?

a) 91/64 b) 155/64 c) 343/64 d) none 

2) Which of the following numbers is a perfect cube ?

a) 121 b) 169  c) 196 d) 216 

3) ³√(216x64)= ?

a) 64 b) 32  c) 24  d) 36

4) ³√(- 343/729)=?

a) 7/9 b) -7/9  c) -9/7  d) 9/7

5) By what least number should 324 be multiplied to get a perfect cube ?

a) 12  b) 14  c) 16 d) 18 

6) ³√(128/³√250)=?

a) 3/5 b) 4/5  c) 2/5 d) none 

7) Which of the following is a cube of an odd number?

a) 216 b) 512 c) 343 d) 1000

Fill in the blanks 

a) ³√(ab)= ³√a x ___

b) ³√(a/b)= ___

c) ³√-x = ____

d) (0.5)³= ____

PLAYING WITH NUMBERS 

BOOSTER - A

1) If 5x6 is exactly divisible by 3, then the least value of x is

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

2) If 64y8 is exactly divisible by 3, then the least value of y is 

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

3) If 7x8 is exactly divisible by 9, then the least value of x is 

a) 0 b) 2 c) 3 d) 5

4) If 37y4 is exactly divisible by 9, then the least value of y is 

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

5) If 4xy7 is exactly divisible by 3, then the least value of x+ y is 

a) 1 b) 4 c) 5  d) 7

6) If x7y5 is exactly divisible by 3, then the least value of x+ y is 

a) 6 b) 0 c) 4 d) 3

7) If x4y5 is exactly divisible by 9, then the least value of x+ y + z is 

a) 3 b) 6 c) 9 d) 0

8) If 1A2B5 is exactly divisible by 9, then the least value of A+ B is 

a) 0 b) 1 c) 2 d) 10

9) If the 4digit number x27y is exactly divisible by 9, then the least value of x+ y is 

a) 0 b) 3 c) 6 d) 9

BOOSTER - B

1) Find all possible values of x for which the 4digit number 320x is divisible by 3. Also, find the numbers.

2) Find all possible values of y for which the 4digit number 64y3 is divisible by 9. Also, find the numbers.

3) The sum of the digits of a 2-digit number is 6. The number obtained by interchanging its digits is 18 more than the original number. Find the original number.

4) Which of the following numbers are divisible by 9 ?

a) 524618. b) 7345845  c) 8977148

5) Replace A, B and C by suitable numerals 

  5   7   A

- C   B  8

----------------

  2   9   3

6) Replace A, B and C by suitable numerals 

  7| 6 A B| 8C

     -5 6

  ----------

        6  B

      -6   3

  --------------

        ×

7) Replace A, B and C by suitable numerals 

    A     B

×  B     A

--------------

B  C    B

BOOSTER - C

1) If 7x8 is exactly divisible by 3, then the least value of x is 

a) 3 b) 0 c) 6  d) 9

2) If 6x5 is exactly divisible by 9, then the least value of x is 

a) 1 b) 4 c) 7 d) 0

3) If xé8y is exactly divisible by 9, then the least value of x + y is 

a) 4 b) 0 c) 6  d) 7

4) If 486*7 is exactly divisible by 9, then the least value of * is 

a) 0  b) 1 c) 3  d) 2

OPERATION ON ALGEBRAIC EXPRESSION 

BOOSTER - A

The sum of (6a + 4b - c +3), (2b - 3c +4),(11b - 7a + 2c -1) and (2c -5a -6) is 

a) (4a - 6b +2) 

b) (-3a +14b -3c +2) 

c) (-6a +17b) 

d) (-6a +6b +c -4)

2) (3q + 7p²- 2r³ +4) - (-2q + 4p²+ 7r³ -3)=?

a) 2q + p² + 5r³ +1

b) 5q + 11p²- 9r³ +7

c) -5q - 3p² + 9r³ -7

d) 5q + 3p²- 9r³ +7

3) (x +5)(x -3)=?

a) x²+ 5x - 15

b) x² -3x - 15

c) x²+ 2x +15

d) x²+ 2x - 15

4) (2x +3)(3x -1)=?

a) 6x²+ 8x - 3

b) 6x² +7x - 3

c) 6x² -7x -3

d) 6x² -7x +3

5) (x +4)(x - 4)=?

a) x²+ 16

b) x² +4x + 16

c) x²+ 8x +16

d) x²+ 16x 

6) (x -6)(x -6)=?

a) x² - 36

b) x² +36

c) x²-6x +36

d) x²-12x +36

7) (2x +5)(2x -5)=?

a) 4x²+ 25x 

b) 4x² - 25

c) 4x²-10x +25

d) 4x²+ 10x - 25

8) 8a²b³ ÷ (-2a)=?

a) 4ab² b) 4a²b c) -4ab² d) - 4a²b

9) (2x²+ 3x +1)÷ (x +1)=?

a) x+1 b) 2x+1 c) x+3 d) 2x+3

10) (x² -4x +4)÷ (x -2)=?

a) x-2 b) x+2 c) 2- x d) 2+ x+x²

11) (a+1)(a -1)(a²+1)=?

a) (a⁴- 2a²-1) b) (a⁴- a²-1) c) (a⁴ -1)  d) (a⁴ + 1) 

12) (1/x + 1/y)(1/x - 1/y)=?

a) (1/x²- 1/y²) b) (1/x² + 1/y²) c) (1/x² +  1/y² - 1/xy) d)  (1/x² - 1/y² + 1/xy)

13) If x+ 1/x = 5, then  (x² +  1/x²)=?

a) 25 b) 27 c) 23 d) 626/25

14) If x - 1/x = 6, then (x² + 1/x²)=?

a) 36 b) 38 c) 32 d) 1397/36 

15) (82)¹- (18)²= ?

a) 8218 b) 6418 c) 6400 d) 7204

16) 197 x 203=?

a) 39991 b) 39999 c) 40009 d) 40001

17) If a+ b=12 and ab= 14 then a²+ b² is 

a) 172 b) 116 c) 162 d) 126

18) If a+ b=7 and ab= 9 then a²+ b² is 

a) 67 b) 31 c) 40 d) 58

19) If x=10 then the value of 4x²+ 20x +25 is

a) 256 b) 425 c) 625 d) 575

FACTORIZATION 

BOOSTER - A

1) 7a²- 63b²=?

a) (7a -9b)(9a + 7b)

b) (7a -9b)(7a + 9b)

c) 9(a - 3b)(a + 3b)

d) 7(a - 3b)(a + 3b)

2) 2x - 32x³=?

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

b) 2x(1- 2x)²

c) 2x(1+2x)²

d) 2x(1 - 4x)(1+ 4x)

3) x³- 144x=?

a) x(x -12)² b) x(x +12)² c) x(x -12)(x +12) d) none 

4) 2- 50x²=?

a) 2(1- 5x)²  b) 2(1+ 5x)²  c) (2 - 5x)(2+ 5x) d) 2(1- 5x)(1+ 5x)

5) a²+ bc+ ab+ ac=?

a) (a+ b)(a+ c) b) (a+ b)(b+ c) c) (b+ c)(a+ c) d) a(a+ b+ c)

6) pq²+ q(p -1) -1=?

a) (pq +1)(pq -1)   b) p(q +1)(q -1)

c) q(q +1)(p -1)     d) (pq -1)(q +1)

7) ab - mn + an - bm=?

a) (a - b)(m - n)

b) (a - m)(b+ n)

c) (a - n)(m + b)

d) (m - a)(n - b)

8) ab - a - b +1=?

a) (a - 1)(b - 1)     b) (1 - a)(1 - b)

c) (a - 1)(1 - b)     d) (1 - a)(b - 1)

9) x²- xz + xy - yz=?

a) (x - y)(x + z) b) (x - y)(x - z) c) (x + y)(x - z) d) (x - y)(z - x)

10) 12m¹- 27=?

a) (2m -3)(3m -9)

b) 2(2m -9)(3m -1)

c) 3(2m -3)(2m +3) d) none 

11) x³- x =?

a) x(x²- x) b) x(x- x²) c) x(1+x)(1- x) d) x(x+1)(x -1)

12) 1- 2ab - (a²+ b²)=?

a) (1+ a - b)(1+ a+ b)

b) (1+ a + b)(1- a+ b)

c) (1+ a + b)(1+0- a- b)

d) (1+ a - b)(1- a+ b)

13) x²+ 6x +8=?

a) (x +3)(x +5) b) (x +3)(x +4) c) (x +2)(x +4) d) (x +1)(x +8) 

14) x²+ 4x -21 =?

a) (x - 7)(x +3) b) (x +7)(x -3) c) (x -7)(x -3) d) (x +7)(x +3) 

15) y²+ 2y - 3 =?

a) (y - 1)(y +3) b) (y +1)(y -3) c) (y -1)(y -3) d) (y +2)(y -3) 

16) 40+ 3x - x²=?

a) (x +5)(x -8) b) (5- x)(8+x) c) (5+ x)(8 - x) d) (5 - x)(8- x) 

17) 2x²+ 5x +3 =?

a) (x +3)(2x +1) b) (x +1)(2x +3) c) (2x +5)(x -3) d) none 

18) 6x² - 13x +6 =?

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

19) 4x² - 8x +3 =?

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

20) 3+ 23y - 8y²=?

a) (1- 8y)(3+ y) b) (1 + 8y)(3- y) c) (1- 8y)(y -3) d) (8y -1)(y+ 3) 

LINEAR EQUATIONS

BOOSTER - A

1) If 2x -3= x +2, then x is

a) 1 b) 3 c) 5 d) 7

b) If 5x + 7/2 = 3x/2 - 14, then x is 

a) 5 b) -5 c) 6 d) -6

3) If z= 4(z+10)/5 then z is 

a) 40 b) 20 c) 10 d) 60

4) If 3m= 5m - 8/5, then m is 

a) 2/5 b) 3/5 c) 4/5 d) 1/5

5) If 5t - 3 = 3t -5, then t is 

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

6) If 2y + 5/3= 26/3 - y, then y is

a) 1 b) 2/3 c) 6/5 d) 7/3

7) If (6x +1)/3 +1 = (x -3)/6, then x is 

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

8) If n/2 - 3n/4 + 5n/6= 21, then n is 

a) 30 b) 42 c) 36 d) 28

9) If (x +1)/(2x +3) = 3/8, then x is 

a) 1/4 b) 1/3 c) 1/6 d) 1/2

10) If (4x +8)/(5x +8) = 5/6, then x is 

a) 4 b) 6 c) 8 d) 12

11) If n/(n +15) = 4/9, then n is 

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

12) If 3(t -3)= 5(2t +1), then t is 

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

13) Four -fifths of a number is greater than three-fourth of the number by 4. The number is 

a) 12 b) 64 c) 80 d) 102

14) The ages of A and B are in the ratio 5:7. Four years from now the ratio of their ages will be 3:4. The present age of B is 

a) 20 b) 28 c) 15 d) 21

15) The base of an isosceles triangle is 6cm and Its perimeter is 16cm. Length of each of the equal sides is

a) 4cm b) 5cm c) 3cm d) 6cm

16) Sum of three consecutive integers is 51. The middle one is 

a) 14 b) 15 c) 16 d) 17

17) The sum of two numbers is 95. If one exceeds the other by 15, the smaller of the two is

a) 40 b) 35 c) 45 d) 55

18) Number of boys and girls in a class are in the ratio 7:5. The number of boys is 8 more than the number of girls. The total class strength is

a) 56 b) 52 c) 48 d) 36

TEST PAPER