REVISION TEST CLASS- 12(24/25)

26/11/24

1) y= 4 cos5x then d²y/dx² is 

a) 25y b) -25y c) 25x d) -25x    

2) ∫ (sin2x + cos3x) dx is 

a) sin6x b) -(1/2) cos2x+ (1/3) sin3x c) (1/2) cos2x+ (1/3) sin3x d) -(1/2) cos2x+ (1/3) sin3x + c 

3) The maximum value of 2x³-15x²+24x -15 is

a) -4 b) -31 c) 4 d) 31 

4) ∫ (5 sinx + 2 cosx) dx at (π/2,0) is 

a) 5 b) 2 c) 7 d) none. 

5) The fixed cost of a new product is Rs30000 and the variable cost per unit is Rs800. If the demand function is p(x)= 4500 - 100x, then break-even values is

a) 15 b) 25 c) 35 d) 45 

6) If y=√(3x +2), yd²y/dx² + (dy/dx)² is 

a) 0 b) -1 c) x d) 1 

7) The odds in favour of an event are 3:5. Find the probability of occurrence of this event.   

a) 3/8 b) 2/8 c) 1/8 d) 5/8. 

8) If y= xˣ then dy/dx is 

a) xˣx b) xˣlogx c) xˣ(x +1) d) xˣ(1+ logx) 

9) ∫ cotx dx is 

a) sinx b) log|cosx| c) - log|cosx| d) log|sinx|+ c 

10) The cost function C(x) of a firm is given by C(x)= 2x²- 4x +5. Then marginal cost when x=10.

a) 3 b) 6 c) 36 d) 63 

11) The differential equation of the family of curves y²- 3ay + x³= 0, where a is an arbitrary constant.

a) (x³- y²)dy/dx - 3yx²=0

b) (x³ + y²)dy/dx - 3yx²=0

c) (x³- y²)dy/dx + 3yx²=0

d) (x³+ y²)dy/dx + 3yx²=0 

12) If the matrix 

A= 1       x+ y and B= 1       2 

    x - y      3                 -2     3 with the relation A= B then value of y is 

a) 0 b) 2 c) -2 d) none  

13) ∫ (eˣ - e⁻ˣ)/(eˣ + e⁻ˣ) dx is 

a) log(eˣ - e⁻ˣ) b) log(eˣ + e⁻ˣ) c) (eˣ - e⁻ˣ) d) logx/(eˣ - e⁻ˣ)

14) Find two positive numbers whose sum is 16 and the sum of whose squares is minimum.

a) 8,8 b) 6,8 c) 10,8 d) 12,8 

Test paper -1

Question 1.       

i) If A= 3      1

             7      5, Find x and y so that A²+ xI₂= yA.      (2) 

ii) Evaluate : tan[2 tan⁻¹(1/2) - π/4].   (2)  

iii) State the reason why the relations R ={(a,b): a ≤ b²}

on the set of real number is not reflexive .    (2)

iv) Evaluate: ∫ xeˣ/(x+1)² dx.       (2).   

v) Deepak rolls twice dies and gets a sum more than 9. What is the probability that the number on the first die is even?   (2). 

vi) If y= tan⁻¹ {5x/(1- 6x²)}, -1/√6 < x <1/√6, then show that dy/dx = 2/(1+ 4x²) + 3/(1+ 9x²).    

vij) Solve : (y + xy) dx + y(1- y²) dy=0.      (2) 

Question 2

Let f: [0, ∞)--> R be function defined by f(x)= 9x²+ 6x -5 prove that f is not invertible. Modify only the codomain of f to make f invertible and then find its inverse.    (4)

Question 3

Using properties of diterminants, prove that 

b²+ c²      ab     ac

    bc      c²+a²  bc = 4a²b²c²

    ca        cb    a²+b²                (4)

Question 4

Prove that tan[sin⁻¹(1/√17) + cos⁻¹(9/√85)]= 1/2       (4)

Question 5

Verify Rolle's theorem for f(x)= eˣ(sinx - cosx) on [π/4, 5π/4] and find the point in the interval where the derivativ vanishes.       (4).  

OR

Discuss the differentiability of the function 

          2x -1, x < 1/2

f(x)=                  at x= 1/2

          3-6x, x≥ 1/2.           (4)

Question 6

Using suitable substitution, find 

dy/dx for y = tan⁻¹[{√(1+ x²) -1}/x]        (4). 

Question 7

Evaluate ∫(tanx + tan²x)/(1+ tan²x) dx.        (4) 

Question 8

 Evaluate ³₁∫ (3x²+1) dx.        (2).

Question 9

 Show that the equation of the normal at any point t on the curve x= 3 cost - cos³t and y= 3 sin t - sin³t is 4(y cos³t - x sin³t)= 3 sin4t.      (4)

OR

The side of an equilateral triangle is increasing at the rate of 2 cm/s.  At what rate is its area increasing when the side of the triangle is 20cm ?

Question 10

Solve : (tan⁻¹y - x) dy = (1+ y²) dx.        (4) 

Question 11

Bag I had 2 red and 3 black balls, Bag II has 4 red and one black ball and Bag III has 3 white and two black balls. A bag is selected at random and a ball is drawn at random. What is the probability of drawing a red ball ?    (4). 

OR

A man makes attempts to hit a target. The probability of hitting the target is 3/5. Find the probability that the man hit the target atleast two times in five attempts.    (4). 

Question 12

Using Matrix method solve the following system of linear equations 

x- 2y - 2z -5=0; -x + 3y +4=0; -2x + z -4=0.      (4) 

OR

Find Inverse of

-1      1     2 

 1      2     3

 3      1      1       (4)

Question 13

Show that the right circular cone of atleast curved surface area and given volume has an altitude equals to √2 times the radius of the base.     (4)

OR

 The sum of the surface area of a cuboid with sides x, 2x and x/3 and a sphere is given to be constant. Prove that the sum of their volumes is minimum, if x is equal to three times the radius of the sphere.  Also find the minimum value of the sum of their volumes.       (4)

Question 14

Evaluate ∫ x²/{(x²+4)(x²+9)} dx.        (4)

Question 15

3 persons A, B and C apply for a job of manager in a private company. Chances of their selection (A, B and C) are in the ratio 1:2:4. The probabilities of A, B and C can introduce changes to improve profits of the company are 0.8, 0.5 and 0.3 respectively. if the change does not takes place, find the probability that it is due to the appointment of C.    (4) 

_____

Question 16

a) The vectors from origin to the points A and B are a= 3i - 6j + 2k and b= 2i+ j - 2k respectively. Find the area of the triangle OAB.     (2). 

b) Find the angle between the line (x -6)/3= (y-7)/2= (z -7)/-2 and the plane x + y + 2z =0.     (2)

c) Find the Cartesian equation of the line passing through the points (3,-2,-5) and (1,-4,-7).     (2) 

Question 17

Show that (a x b)=| a.a       a.b|

                                 a.b       b.b       (4)

OR

If the vector ai+ j + k, i+ bj + k and I+ j + ck are coplanar (a,b,c ≠1), then prove that 

1/(1- a) + 1/(1- b) + 1/(1- c)= 1.       (4)

Question 18

Find the shortest distance between the lines whose vector equation are

r= (1- t)i +(t -2)j + (3- 2t)k; and 

r= (s +1)i +(2s -1)j - (2s+1)k.          (4) 

OR

Find the equation of the plane passing through the point (1,1,-1) and  perpendicular to the planes x+ 2y + 3z -7=0 and 2x - 3y + 4z =0.      (4). 

Question 19

Sketch the graphs of y= x(4- x) and find the area bounded by the curve, x-axis and the lines x= 0 and x= 5.     (4)

_______

Question 20) 

*a) The marginal cost of production of a commodity is 30+ 2x. It is known that the fixed costs are Rs120. Find 

i) Find the total cost of producing 100 units.

ii) Find the cost of increasing output from 100 to 200 units.      (2) 

b) You are given the following two lines regression. Find the regression of Y on X and X on Y and justify your answer.       (2) 

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

i) Find the marginal cost MC

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

Question 21

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

x: 14    12    13     14    12

y: 22    23    22     24    24

Hence, predict the value of y when x= 16.     (4) 

OR

you are given the following data :

                                   x      y

Arithmetic mean     36   85 

standard deviation  11    8 

Correlation coefficient between x and y is 0.66.  find 

i) the two regression coefficients

ii) the two regression equations

iii) the most likely value of y when x= 10.      (4) 

Question 22

Give the total cost function for x units of a commoyas C(x)= x³/3 + 3x²- 7x +16, find 

i) The marginal cost

ii) average cost 

iii) show that marginal average cost is given {xMC - C(x)}/x².     (4) 

OR

Given the price of a commodity is fixed at Rs55 and its cost function is C(x)= 30x +250

i) Determine the break even point.

ii) What is the profit when 12 items are sold.     (4)

Question 23

An aeroplane can carry a maximum 200 passengers. A profit of Rs1000 is made on each executive-class ticket and a profit of Rs600 is made on each economy-class ticr.  The airline reserves at least 20 seats for executive class. However, at least 4 times as  many passengers prefer to travel by economic class than by executive class. Determine how many tickets of each type must be sold in order to maximize the profit for the airline. What is the maximum profit earned?      (6)

Day 11(1/10/14)

1) The function f(x)= (2x²-1)/x⁴, x> 0, decreases in interval_____ .

2) The function g(x)= x + 1/x , x≠ 0 decreases in the closed interval ____ .

3) The largest open interval in which the function f(x)= 1/(1+ x²) decreases___ .

4) The set value of 'a' for which the f(x)= ax + b is strictly increasing for all real x, is ____ 

5) The largest interval in which f(x)= x¹⁾ˣ is strictly increasing is____ .

Day -10 (28/7/24)

1) Show that: 4 tan⁻¹(1/5) - tan⁻¹(1/70) + tan⁻¹(1/99) = π/4.

2) Solve: tan⁻¹(2+ x) + tan⁻¹(2- x) = tan⁻¹(2/3).             ±3

3) Show that sin⁻¹{x/√(1+ x²)} + cos⁻¹{(x+1)/√(x²+ 2x +2)}= tan⁻¹(x²+ x+1).

4) Prove: cot(π/4 - 2 cot⁻¹3)= 7.

5) Find x if sin⁻¹(5/x) + sin⁻¹(12/x) =π/2.        ±13

6) Find the value of sin cot⁻¹ cos(tan⁻¹x). √{(1+ x²)/(2+ x²)}

7) Solve: sin⁻¹6x + sin⁻¹(6 √3 x) =-π/2.           ±1/12

8) Solve: sin⁻¹{2a/(1+ a²)} + sin⁻¹{2b/(1+ b²)} = 2 tan⁻¹x.      (a+ b)/(1- ab)

9) Solve: sin(cos⁻¹cot(2tan⁻¹x))= 0.             ±1, √(3± 2√2)

10) Prove: sin⁻¹(√3/2) + 2tan⁻¹(1/√3) = 2π/3.

11) Show that: sin⁻¹(1/√17) + cos⁻¹(9/√85) = tan⁻¹(1/2).

12) Find the value of cos(2 cos⁻¹x + sin⁻¹x) at x= 1/5.           -2√6/5

13) Solve: cos(sin⁻¹x) = 1/9.            ±4√5/9

14) Show: tan⁻¹(1/3) + tan⁻¹(1/5) + tan⁻¹(1/7) + tan⁻¹(1/8) =π/4.

15) Solve: tan⁻¹2x + tan⁻¹3x =π/4.  1/6

16) Show: tan⁻¹{(1/2) tan2A}+ tan⁻¹(cotA) + tan⁻¹(cot³A) = 0.    

17) Show: 2(tan⁻¹1+ tan⁻¹(1/2) + tan⁻¹(1/3))=π.

18) Show that: tan⁻¹x + cot⁻¹(x +1) = tan⁻¹(x²+ x+1).     

19) Prove: 2 cos⁻¹x = cos⁻¹(2x²-1).

20) Prove: tan⁻¹(1/4) + tan⁻¹(2/9) = (1/2) cos⁻¹(3/5).

21) Evaluate: tan(2tan⁻¹(1/5) - π/4).            -7/17

22) (1/2) tan⁻¹x = cos⁻¹[{1+ √(1+ x²)}/2√(1+ x²)].

23) If tan⁻¹a + tan⁻¹b + tan⁻¹c=π, then show that a+ b+ c= abc.

24) Show: sin⁻¹(4/5) + cos⁻¹(2/√5) = cot⁻¹(2/11).

25) Evaluate:

a) tan(1/2) (cos⁻¹(√5/3)).   (1/2) (3- √5)

b) cos(cos⁻¹(-√3/2) +π/6).          -1

c) sin[π/3 - sin⁻¹(-1/2)].           1

26) If tan⁻¹x + tan⁻¹y + tan⁻¹z =π/2, show that xy+yz+zx= 1.

27) Show that sec²(tan⁻¹2) + cosec²(cot⁻¹3) =15.

28) Show: sin⁻¹(12/13) + cos⁻¹(4/5) + tan⁻¹(63/16) =π.

29) If cos⁻¹(x/2) + cos⁻¹(y/3) = J, show that 9x²- 12xy cosK+ 4y²= 36 sin²K.

30) Show: 

a) tan⁻¹[{√(1+ x²)+ √(1- x²)}/{√(1+ x²) - √(1- x²)]=π/4 + (1/2) cos⁻¹x².

b) cos⁻¹(4/5) + cot⁻¹(5/3) = tan⁻¹(27/11).

c) tan(2tan⁻¹a)= 2tan(tan⁻¹a + tan⁻¹a³).

d) cot⁻¹{(pq +1)/(p - q)} + cot⁻¹{(qr +1)/(q - r)} + cot⁻¹{(rp +1)/(r - p)} = 0.

e) sin[sin⁻¹(1/2) + cos⁻¹(3/5)]= (3+4√3)/10.

f) cos[tan⁻¹(15/8) - sin⁻¹(7/25)]= 297/425.

31) Solve the following:

a) tan⁻¹{1/(2x+1)} + tan⁻¹{1/(4x+1)}) = tan⁻¹(2/x²).        0, -2/3, 3

b) tan⁻¹(x+1) + cospt⁻¹(x-1) = sin⁻¹(4/5) + cot⁻¹(3/4).       ±4√(3/7)

c) tan⁻¹(x -1) + tan⁻¹x + tan⁻¹(x +1) = tan⁻¹(3x).           0, ±1/2

d) sin⁻¹x + sin⁻¹2x=π/3.             1/6

Day- 9 (21/7/24)

1) If A= 1     3 & B= 1     -1

              0     1          0      1

with relation AX = B, then find X

2) A= 1   2   3 B= 1

   0   2 C= 4    4   10

          -1 -3   2       3   4   5      4    2   14

with the relation 2A+ kB = C, then find k.

3) If A= 4      2

              1      1 find (A- 2I)(A - 3I), where I is a unit matrix.

4) Solve:

a) 5x -12y = -9; 7x - 6y = -8.

b) 2x + 3y - 4z = 1, 3x -y +2z = -2, 5x -9y +14z =3.

c) x -2y +3z = 6, x + 4y + z = 12, x -3y + 2z =1.

d) x -2y - 3z = 4, 2x +y -3z = 5, -x +y +2z =3.

5) Construct a 3x4 matrix whose elements are aᵢⱼ= i + j.

6) If x+ y     y - z = 3     -1

        z -2x    y- x     1      1 then find x, y, z

7) If A= 3     1

              7     5, find Matrix x and y so that A²+ xI = yA. Hence find A⁻¹.

8) If A= 1 & B= 3     1     -2

              5

              7

Verify (AB)'= B'A'.

9) Find the inverse of a       b

                                      c       d given that ad - bc ≠ 0.

10) If A= 1      1

                 0     1 show that A³= 1    3

                                                     0    1 

11) Find A= 1    1    2 B= 1    2 

                     2    1    0       2    0

                                         -1    1

Verify B'A' = (AB)'.

12) If A= x²  B= x   C= -2

                y²       2y        9 with the relation A - 3B= C then find x and y.

13) If A= 1   x   1 B= 1   3   2 & C= 1

                                   2   5   1           2

                                  15  3   2           x with the relation ABC= 0, then find x.

14) Find the inverse of 

A= -1     1     2 

       3    -1     1

      -1     3     4 

15) Find the adjoint of the matrix 

A= 1      0       -1

      3      4        5

      0     -6       -7 and hence find the inverse matrix.

16) If A= 2       3 

                5      -2 then show that A⁻¹= A/19

17) A= x²  B= 2x  C= 7

             y²        3y      -3 with the relation A+ 2B = 3C, then find x and y.

18) Construct a matrix of order 3x2 whose element in iᵗʰ row and jᵗʰ column is given by aᵢⱼ = (3i+ j)/2.

19) If A= x²   3   4  B= -3x 1  -5 C= 4  4   -1

                1    9   8        -3   -2   -6   -2  7    2 with the relation A+ B = C find the value of x.

20) Let a be a square matrix, show that (1/2) (A+ A') is symmetric matrix and that (1/2)(A- A') is a skew symmetric matrix. Hence show that every square matrix may be expressed as the sum of symmetric and skew symmetric matrices.

21) If x+ y= 7    0 & x -y= 3   0

                    2    5              0   3 then find Matrix x and y.

22) If 2x+ 3y=2   3 & 3x +2y=-2   2

                        4   0                   1  -5 then find Matrix x and y.

23) If A= 1    -1 B= 2     1

                2     3       1     0 then show (A+ B)²≠ A²+ 2AB+ B²

24) If A= 3    -2

                4    -2 with the relation A²= kA - 2I, I is the unit matrix, then find k

25) If A= x   y B= -2   0 C= 1     2 

               -1   5      -1   5       0    0 with the relation 2A = B - C, then find x and y.

26) If (x y z) -(-5 3 0)= (-5 6 7), determine x, y, z.

27) If A= 2   -3 B= a   b C= -3     4

                4    0       c   d        5    -1 determine a, b, c, d.

28) Express in a single matrix 

A= 1    3  B= 8    4

      1   -4       4    8 as relation 4A - (1/2) B

29) If A= 1   2   3 B= 1   0   0

                0   1   2       0   0   0

                0   0   1       0   0   1 with the relation -2(X + A)= 3X + B, then find Matrix X.

30) If A= 3     5

               -4     2 find A²- 5A - 14I, then find inverse of A

31) A men invests Rs50000 into two types of bonds . The first bond pays 5% interest per year and the second bond pays 6% intrest per year. Use matrix multiplication to determine how to divide Rs50000 among two types of bonds so as to obtain an annual total interest of Rs2780.

Day- 8 (16/7/24)

1) ∫ (x³+ 5x²+ 6x +9) dx.

2) ∫ (1+ x)³/x dx

3) ∫ √x(x⁵ + 3/x) dx.

4) ∫ (x √x - (1/3) √x + 11/√x) dx 

5) ∫ (8²⁺ˣ +2)/2ˣ⁺³ dx

6) ∫ (e⁵ˣ + e³ˣ)/(eˣ + e⁻ˣ) dx.

7) ∫ (e⁵ˡᵒᵍˣ - e⁴ˡᵒᵍˣ)/(e³ˡᵒᵍˣ - e²ˡᵒᵍˣ). dx

8) ∫(x³- 4x²+ 5x -2)/(x²- 2x +1) dx 

Day- 7 (30/6/24)

1) The value of sin{2 tan⁻¹(1/2)}+ cos( tan⁻¹ (2√2)) is

a) 12/13 b)13/14 c) 14/15  d) none 

2) cot⁻¹(-3/4)=

a) π+ sin⁻¹(-4/5)

b) π+ tan⁻¹(-4/3)

c) tan⁻¹(-4/3)

d) tan⁻¹(-4/3) -π

3) Find dy/dx of tan2x tan3x tan5x

4) If xy = e - eʸ, then find dy/dx at x= 0.

5) If x = a cos³k, y= a sin³k then √{1+ (dy/dx)²} equal to 

a) cosk b) sink c) seck d) coseck

6) If A= 1     0     0

              1     1     2

              3    -1     9 then the value of det(adj(adjA))=

a) 11 b) 121 c) 1331 d) 14642

7) For the set of linear equations Kx + 3y + z= 0, x + Ky + 3z = 1 and 3x + y + 5z = 2

The value/s of K, for which the equations does not have unique solution 

8) From the Matrix equation AB= AC, we say B= C provided

a) A is singular matrix 

b) A is a square matrix 

c) A is a skew symmetric Matrix 

d) A is non singular matrix 

9) If y= (1+ 1/x)ˣ, then find dy/dx

10) If y= cos⁻¹√[{√(1+ x²) +1}/2√(1+ x²)}, then dy/dx is

a) 1/(1+ x²)  b) 1/(1- x²) c) 1/2(1+ x²) d) n

11) If A= -4   -3    -3

                 1     0     1

                  4    4     3 then A⁻¹ is

a) Aᵀ b) A c) A²+ A - I d) A²- A - I

Day 6 (2/6/24)

1) Find the interval in which the function f(x)= x³- 12x²+ 36x +17 is

a) increasing. (-∞, 2) U (6,∞)

b) Decreasing. (2,6).         

2) Fill the Gap

3       1975        1978

4       1982        1986 = _______

5       1995        2000 

3) The product of two non-zero matrices must be a non-zero matrix. T/F

4) If y= logₑlogₑx, x > 1 which one of the following answers is true.

a) x dy/dx = 1 b) (x logₑx) dy/dx = 1 c) (logₑx) dy/dx = 1 d) (logₑx) dy/dx = x.

5) If x= a(k - sin k) and y= a(1+ cos k) then which one of the following is the value of dy/dx ?

a) - cot(k/2) b) - cot k c) - tan(k/2) d) cot(k/2)

6) Examine whether AB= BA for the two matrices 

A= 1      5 & B= 0       0

      1      3          0        1.

7) If y= (secx)ᵗᵃⁿˣ, then find dy/dx.

8) If x³+ y³= 2xy, then find the value of dy/dx at the point (1,1).

9) if a, b, c are three non-zero real numbers , then show that 

1+ a     1       1

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

  1        1      1+c

10) If eˣʸ - 4xy = 4, find the value of dy/dx.

11) If f(x)= {(1+ x)/(2+ x)}³⁺²ˣ, find the value of f'(0).

5/5/24 Day- 6

      (2 Marks for each question)

1) Use matrix rule to solve 2x+ 3y =10 and x + 6y= 4. 

        

2) Find dy/dx if y+ siny = x².          

3) Solve for x the determinants if 

x²     x     1

0      2     1 = 28

3      1     4.                          

4) Find the inverse of -2       5

                                       3       4 

5) Using elementary row transformation, find the inverse of the matrix 

1      2

3      7

6) If y= √{(1- cos2x)/(1+ cos2x)} find dy/dx.

7) without expanding the determinants show that 

1       a      b+ c

1       b      c+ a = 0

1       c      a + b

8) If eˣ + eʸ = eˣ⁺ʸ, show dy/dx + eʸ⁻ˣ = 0       (3)

9) Find dy/dx if y tanx - y² cosx + 2x = 0

10) Without expanding the determinants show that 

x+ y      y+ z      z+ x

   z          x            y = 0

   1          1           1

11) If A= x  4   1 & B= 2  1  1 & C= x

                                     1   0  2         4

                                     0   2 -4        -1 with the relation ABC= 0, find x.

12) Find the value of k if the function defined by 

f(x)= 2x- 1, x < 2

             k,   x= 2

          x+1,  x> 2 is continuous x= 2

Day- 5

1) Find the value of x, y, z and w from:

x+ 3     2y + x = - x -1     0

z -1      4w -8        3       2w

2) Find 2 x 3 matrix [aᵢⱼ]₂ₓ₃ , where aᵢⱼ = i + 2j.

3) If A= 1   3 & B= -1    -2

              3   2           0     5

              2   5           3     1 and A+ B - D = 0, then Find the Matrix D.

4) If A= 8     9

            7/2   8

             1    -1 then find the additive inverse of A.

5) If A+ B= 4   4 & A- B= 3       2

                   2   4              -2       0 then find AB.

Day- 4

1) If A= [aᵢⱼ] is a square Matrix of order 2 such that aᵢⱼ = { 1, when i≠ j

                            0, when I= j

Then A² is

a) 1   0 b) 1   1 c) 1    1 d) 1    0

     1   0     0   0     1     0     0    1       (1)

2) If A and B are invertible square matrices of the same order, then which of the following is not correct?

a) adj A= |A| A ⁻¹ 

b) det (A)⁻¹= [det(A)]⁻¹

c) (AB)⁻¹= B⁻¹A⁻¹

d) (A+ B)⁻¹= B⁻¹+ A⁻¹.                   (1)

3) If f(x)= { kx/|x|, if x < 0

                     3,      if x ≥ 0 is continuous at x= 0, then the value of k is 

a) -3 b) 0 c) 3 d) any real number.     (1)

4) The value of |A|, if

A= 0      2x -1     √x

    1- 2x     0       2√x

   - √x    - 2 √x     0 where x ∈ R⁺ is

a) (2x +1)² b) 0 c) (2x+1)³ d) none.    (1)

5) Given that A is a square Matrix of order 3 and |A| = -2, then | adj (2A)| is equal to 

a) -2⁶  b) 4 c) -2⁸ d) 2⁸.          (1)

6) Using the Matrix method, solve the following system of linear equations.

2/x + 3/y + 10/z = 4

4/x - 6/y + 5/z = 1

6/x + 9/y - 20/z = 2.                (5)

Day - 3

1) If A= 0    0      -1

              0   -1       0

             -1    0       0 , then the only correct statement about the Matrix A is

a) A⁻¹ does not exist b) A= (-1)I c) A is a zero Matrix d) A²= I.             (3)

2) If A= 1   -1    1 and 10 B= 4    2     2

              2    1    3                  -5.   0     x

              1    1    1                   1   -2     3 if B is the inverse of the Matrix A, then the value of x is.                      (3)

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

3) The Matrix 0      7      4

                       -7      0     -5

                       -4      5      0 is

a) symmetric b) skew symmetric c) none singular d) orthogonal.              (1)

4) Let A= a.    0 and B= 1        0

                 1     1              5        1 if A²= B, then the value of a is

a) 1 b) -1 c) 4  d) no real value of a     (2)

5) Let A and B are two square Matrices such that AB= A and BA = B. Then A² is equal to

a) O b) I c) A.  d) B.              (1)

6) If A= 3      2 and AC= 19    24

             4       5                 37    46, then find the Matrix C.         (2)

7) If A= 7    1    2 B= 3       C= 4

             9     2    1      4             2

                                   5

Simplify With the relation AB + C.    (2)

8) For how many values of x in the closed interval [-4, -1], the Matrix

3       -1+ x       2

3          -1       x+2 is singular?

x+3      -1         2                                (1)

Day- 2

1) The product of two non-zero matrices must be a non-zero Matrix. T/F.       (1)

2) Examine AB= BA for the two Matrix

A= 1        5 B= 0         0

      1        3       0         1               (2)

3) If X= 1         -1 & Y= x         1

              2         -1          y        -1 and 

(X + Y)²= X²+ Y², find x, y.             (3)

4) If A= 2       -1 & B= -1         -8       -10

              1        0           1         -2        -5

             -3        4           9         22      15 with the relation AP= B, find matrix P.    (2)

5) 2) A 3x3 matrix A=(aᵢⱼ) whose elements are given aᵢⱼ= i + j.          (1)

A) 2     3    4    B) 3     3      4

     3     4    5         4     5      6

     4     5    6         5     6     7

C) 0    1      2 D) none

     3    4      5

     5    6      7

Day 1

1) If A= 2      -1

            -1        2 and I is the unit Matrix of order 2, then A²=?

a) 4A- 3I b) 3A -

 4I c) A- I d) A+ I

2) If A= - x    - y

                z     t then the transpose of adjA is

a) t.   z   b) t     y       c) t     -z

    -y  -x      -z   -x            y    -x  d) none

3) If A= 3.     5  B= 1      17

              2      0        0     -10 then |AB|=?

a) 80    b) 100  c) -110  d) 92

4) If A= 5     6    -3

            -4      3     2

            -4     -7     3 then the cofactor of the element of second row are-

a) 3,3,11 b) 3,-3,11 c) -39,3,-11 d) 39,-3,11

5) If A= a     b and A²= m      n

              b.    a                n      m then

a) m= 2ab, n= a²+ b²

b) m= a²+ b², n= ab

c) m= a²+ b², n= 2ab

d) m= a²+ b², n= a²- b²

Test paper For ENGINEERING

MARKS- 30 Time: 45 minutes 1x24= 24

1) If the determinant 

P= 1    α     3 

      1    3     3

      2   4     4 is adjoint of a 3x3 Matrix A and |A|= 4 then α is equals to 

a) 11   b) 5   c) 0   d) 4 

2) The term endependent of x in expansion of 

{(x +1)/(x²⁾³ - x¹⁾³+1) - (x -1)/(x - x¹⁾²)}¹⁰ is 

a) 120  b) 210  c) 310  d) 4 

3) If the determinants 

     x     1    1 & B= x     1

A=1     x    1          1     x then dA/dx=

a) 3B+1  b) 3B  c) -3B  d) 1- 3B

4) The value of C₀ + 2C₁ + 3C₂ +....+ (n+1) Cₙ = 576, then n is 

a) 7 b) 5 c) 6 d) 9

5) The remainder when, (10¹⁰+ 1)(10¹⁰+ 2) is divided by 6 is

a) 2  b) 4  c) 0  d) 6 

6) If (1+ x + x²)ⁿ = 1+ a₁x + a₂x²+ ....+ a₂ₙx²ⁿ, then 2a₁ - 3a₂+ ... -(2n +1)a₂ₙ =

a) n  b) -nl  c) n +1  d) -nl -1

7) The value of x satisfying the equation of determinant

Cos2x      sin2x      sin2x

Sin2x       cos2x     sin2x= 0

Sin2x       sin2x      cos2x

And x ∈[0,π/4] is 

a) π/4  b) π/2  c) π/16  d) π/8

8) If t₅, t₁₀, t₂₅ are 5ᵗʰ, 10ᵗʰ, and 25ᵗʰ terms of an AP respectively, then the value of determinant

t₅       t₁₀        t₂₅ 

5       10        25

1        1          1 is equal to 

a) -40  b) 1  c) -1  d) 0

9) Five dice are tossed. What is the probability that five numbers shown will be different ?

a) 5/24  b) 5/18  c) 5/27  d) 5/81

10) if the events A and B are independent and if P(A')= 2/3, P(B)= 2/7 then P(A∩B) is equals to 

a) 4/21   b) 3/21  c) 5/21  d) 1/21

11) Let P= [aᵢⱼ] be a 3 x 3 matrix and let Q= [bᵢⱼ], where bᵢⱼ = 2 ᶦ⁺ʲ aᵢⱼ for 1 ≤ i, j ≤ 3. If the determinant of P is 2, then the determinant of the matrix Q is 

a) 2¹⁰ b) 2¹¹ c) 2¹² d) 2¹³

12) x(xⁿ⁻¹ - nαⁿ⁻¹) + αⁿ (n -1) is divisible by (x - α)² for 

a) n> 1 b) n > 2 n ∈ N d) none 

13) The sum of the series 1+ 3²/2! + 3⁴/4! + 3⁶/6!+....to ∞ is

a) e⁻³ b) e³ c) (1/2)(e³ - e⁻³) d) (1/2) (e³ + e⁻³)

14) Value of the series x/1.2 + x²/2.3 + x³/3.4 +.... is

a) 1- {(1- x)/x} log(1- x)

b) 1- {(1- x)/x} log(1+ x)

c) 1 + {(1- x)/x} log(1- x) d) none 

15) Let the coefficient of powers of x in the 2ⁿᵈ, 3ʳᵈ and 4ᵗʰ terms in the expansion of (1+ x)ⁿ, where n is a positive integer, be in Arithmetic progression. The sum of the co-efficients of odd powers of x in the expansion is

a) 32 b) 64 c) 128 d) 256

16) The sum of the infinite series 

1+ 1/3 + (1.3)/(3.6) + (1.3.5)/(3.6.9) + (1.3.5.7)/(3.6.9.12) + .....is equal to 

a) √2  b) √3  c) √(3/2)  d) √(1/√3))

17) The number of real values of K for which the system of equations 

x+ 3y + 5z = Kx

5x+ y + 3z = Ky

3x+ 5y + z = Kz

has infinity number of solution is

a) 1  b) 2  c) 4  d) 6

18) Let Sₖ be the sum of an infinite GP series whose first term is K and common ratio is K/(K +1) (K> 0). Then the value of ∞ₖ₌₁∑ (-1)ᴷ/Sₖ is equal to 

a) log4 b) log2 -1 c) 1- log2 d) 1- log4

19) Let A and B be two events with P(A')= 0.3, P(B)= 0.4 and P(A ∩B) = 0.5 then P(B/A U B') is equal to 

a) 1/4 b) 1/3 c) 1/2 d) 2/3 

20) Three numbers are choosen at random without replacement from {1, 2, 3, .....,8}. The probability that their minimum is 3, given that their maximum is 6, is

a) 1/4 b) 2/5 c) 3/8 d) 1/5

21) If C₀, C₁, C₂, C₃,.... are binomial coefficients in the expansion of (1+ x)ⁿ. Then C₀/3 - C₁/4 + C₂/5 - C₃/6+.... is equal to 

a) 1/(n +1) - 2/(n+2) + 1/(n +3)

b) 1/(n +1) + 2/(n+2) - 1/(n +3)

c) 1/(n +1) - 1/(n+2) + 1/(n +3)

d) 2/(n +1) - 1/(n+2) + 2/(n +3)

22) If the matrix 

A= a       x

      y       a and xy=1. Then det(AA') is equal to 

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

23) Let A and B any two events. Which one of the following statements is always true ?

a) P(A'/B) = P(A/B)

b) P(A'/B) = P(B'/A)

c) P(A'/B) = 1- P(A/B)

d) P(A'/B) = 1- P(A/B')

24) The inverse of a symmetric matrix is

a) skew symmetric 

b) symmetric

c) diagonal matrix d) none 

AIEEE-21

1) Let X be the universal set for sets A and B. If n(A)= 200, n(B)= 300 and n(A∩B)= 100, then n(A' ∩ B') is equal to 300 provided n(X) is equal to

a) 600 b) 700 c) 800 d) 900

2) The sum of n terms of the infinite series 1.3²+ 2.5²+ 3.7²+...... ∞ is

a) (n/6)(n+1)(6n²+14n+7) b) (n/6)(2n+1)(3n+1) c) 4n³+ 4n²+ n d) none

3) If x²ᵏ occurs in the expansion of (x + 1/x²)ⁿ⁻³, then

a) n - 2k is a multiple of 2.

b) n - 2k is a multiple of 3.

c) n = 0 d) none

4) If y= 1- x + x²/2! - x³/3! + x⁴/4! - ........, then d²y/dx² is

a) x  b) - x  c) - y  d) y

5) The value of ∫ secx dx/√{sin(2x+θ)+ sinθ} is

a) √{(tanx + tanθ) secθ}+ c

b) √{2(tanx + tanθ) secθ}+ c

c) √{2(sinx + tanθ) secθ}+ c d) none

6) The solution of the equation x² d²y/dx² = log x when x=1, y=0 and dy/dx = -1 is

a) y= (1/2)(logx)²+ logx

b) y= (1/2)(logx)²- logx

c) y= -(1/2)(logx)²+ logx

d) y= - (1/2)(logx)²- logx

7) If C²+ S²= 1 then (1+ C + iS)/(1+ C - iS) is equal to

a) C+ iS b) C- iS c) S+ iC d) S- iC 

8) The number of real roots of (x + 1/x)³+ (x + 1/x)= 0 is

a) 4 b) 6 c) 2 d) 0

9) If Tₚ , Tq , Tᵣ are pᵗʰ , qᵗʰ and rᵗʰ terms of an AP, then

Tₚ      Tq     Tᵣ

 p       q       r

 1        1      1 is equal to

a) p+ q+ r b) 0 c) 1 d) -1

10) If A is a square Matrix of order n x n and k is a scalar then adj.(kA) is equal to

a) kⁿ⁻¹ adj. A b) kⁿ⁺¹ adj. A c) kⁿ adj. A d) k adj. A

11) 6 persons A, B, C, D, E and F are to be seated at a circular table. The number of ways this can be done if A must have either B or C on is right and B must have either C or D on his right is 

a) 24  b) 12 c) 18 d) 36

12) For what values of α

lim ₓ→∞ √(2α²x²+ αx+7) - √(2α²x²+7) will be 1/2√2

a) any value of α  b) α ≠ 0 c)α = 1 d) α = -1

13) A window is in the form of a rectangle with the semicircular bend on the top. If the perimeter of the window is 10m, the radius in metres of the semicircular bend that maximize the amount of light admitted is

a) 20/(4+π) b) 10/(4+π) c) (10 -π) d) none

14) There is line with positive slop λ through origin which cuts off a segment of length √10 between the parallel line 2x - y + 5 = 0 and 2 x - y + 10 = 0. Then λ should be

a)  1/2  b) 1/3  c) 1/5 d) none

15) Let C₁ and C₂ be the circles given by equations x²+ y²- 4x -5= 0 and x²+ y² + 8y +7 = 0. Then the circle having the common chord of C₁ and C₂ as its diameter has

a) centre at (- 1, - 1) and radius 2

b) centre at (1, - 2) and radius √5

c) centre at ( 1, - 2) and radius 2

16) The equation of common tangent to the parabola y²= 16x and the circle x²+ y²= 8 are

a) y= x+ 4; y= - x -4 b) y= 2x+ 4; 2y= - x +9 c) y= x+ 9; y= - x -4 d) none

17) An ellipse has OB as semi minor axis. F and F' its foci and the angle FBF' is a right angle. Then the eccentricity of the ellipse is 

a) 1/√2  b) 1/2 c) 1/4  d) 1/√3

18) The angle between the lines 2 x= 3y= - z and 6x = - y = -4z is 

a) 0° b) 30° c) 45° d) 90°

19) A vector parapendicular to the plane containing the vectors i - 2j - k and 3i - 2j - k is inclined to the vector i + j + k at an angle

a) tan⁻¹√14 b) sec⁻¹√14 c) tan⁻¹√15 d) none

20) Three integers are chosen at random from the first 20 integers. The probability that their product is even, is 

a) 2/19 b) 3/29  c) 17/19 d) 4/29

21) If cotθ + tanθ = m and secθ - cosθ = n, then which of the following is correct?

a) m(mn²)¹⁾³ - n(nm²)¹⁾³ = 1

b) m(m²n)¹⁾³ - n(mn²)¹⁾³ = 1

c) n(mn²)¹⁾³ - m(nm²)¹⁾³ = 1

d) n(m²n)¹⁾³ - m(mn²)¹⁾³ = 1

22) If 3 sin⁻¹{2x/(1+ x²)} - 4 cos⁻¹{(1- x²)/(1+ x²)} + 2 tan⁻¹{2x/(1- x²)} =π/3, then value of x is

a) √3 b) 1/√3 c) 1 d) none

23) The A. M of ²ⁿ⁺¹C₀ , ²ⁿ⁺¹C₁ , ²ⁿ⁺¹C₂ , .... ²ⁿ⁺¹Cₙ is

a) 2ⁿ/n b) 2ⁿ/(n +1) c) 2²ⁿ/n d) 2²ⁿ/(n +1)

24) Let p and q be two statements, then (p∪q) ∪ - p is 

a) tautology b)  contradiction c) Both a and b d) none

25) Let f(x)= x - [x], for every real number of x, where [x] is the integral parts of x. Then  ¹₋₁∫ f(x) dx is equal to

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

Directions (26-30): This section contains 5 questions numbered 26 to 30. Each question contains statement-1 (Assertion ) and statement-2 (Reason). Each question has four choices (a), (b), (c) and (d) out of which ONLY ONE is correct.

a) Statement-1 is True , Statement-2 is True , Statement-2 is a correct explanation for Statement -1.

b) statement-1 is true , statement-2 is 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.

26) statement -1: 5/3 and 5/4 are the eccentricity of two conjugate hyperbolas.

      Statement -2: If e and e₁ are the eccentricities of two conjugate hyperbolas, then ee₁ > 1.

27) Statement -1: The maximum area of triangle formed by the point (0,0), (a cosθ , b sinθ), (a cosθ , - b sinθ) is (1/2) |ab|.

       Statement -2: Maximum value of sinθ is 2.

28) Statement -1: The Coefficient of xⁿ in the binomial expansion of (1- x)⁻² is (n +1).

       Statement-2: The Coefficient of xʳ in (1- x)⁻ⁿ when n ∈N is ⁿ⁺ʳ⁻¹C ᵣ.

29) Statement -1: 20 persons are sitting in a row. Two of these persons are selected at random, The probability that two selected person are not together is 0.7.

      Statement-2: if A is an event, then

P(not A)= 1- P(A).

30) Statement -1:  A flagstaff of length 100m stands on tower of height h. if at a point on the ground the angle of elevation of the tower and top of the flagstaff be 30°, 45°, then h= 50(√3 +1)m.

     Statement -2: A flagstaff of length 'd' stands on tower of height h. If at a point on the ground the angle of elevation of the tower and top of the flagstaff be α, β then h= d cotβ/(cot α - cotβ).

REVISION ICSE X- TEST(2023/24)

30/9/24

1) The rth term of an AP is n and it's nth term is r; show that its mth term is r+ n- m.

2) If the 1st term of an AP is 34 and 6th term is 48, then find 2nd, 3rd, 4th and 5th term of the AP. 

3) Find the middle term (or terms) and the sum of the following arithmetic series:

3+7+11+15+....+95.

4) The fifth term of an AP is 30 and its twelfth term is 65, find the sum of its 20 terms.

5) The sum of n terms of an AP is 3n²+ 5n. Find the number of the term which is equal to 152. 25

6) How many terms of the series {22+ 18+ 14+ 10 +....} must be added to get the sum 64 ? 

7) Insert 7 arithmetic mean between 1 and 41.

8) Find the sum of all natural number between 500 and 1000 divisible by 13. 

25/8/24

1) Show: 

a) (1+ secx- tanx)/(1+ secx + tanx) = (1- sinx)/cosx

b) (sinx - cosx +1)/(sinx + cosx -1)= 1/(secx - tanx)

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

d) (sinx - siny)/(cosx + cosy) + (cosx - cosy)/(sinx + siny)= 0

f) cos⁴x - cos²x = sin⁴x - sin²x

18/8/24

TEST PAPER -1

Time allowed=5/2 hours  max. marks :80

General instructions 

       Attempt all questions from section A and any 4 questions from section B.

All working including rough work, must be clearly shown, and must be done on the same sheet as the rest of the answer.

      Omission of essential working will result in loss of marks.

The intended marks for questions or parts of questions are given in bracket [ ]

       Mathematical tables and graph paper are provided 

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

       Section A (40 Marks)

(Attempt all questions from this Section)

______________::::::________________

Question 1: Choose the correct answer to the questions from the given options:  [15]

 (Do not copy the questions , write the correct answers only)

i) If A= 2   0 B= x & C= 2

            0    4      y         -8, with the relation AB= C then the value of x and y respectively are

a) 1,-2 b) -2, 1  c) 1,2 d) - 2, -1 e) none

ii) If x -2 is a factor x³- Kx - 12, then the value of k is:

a) 3 b) 2 c) -2 d) -3 

iii) In the given,

RT is a tangent touching the at S. If angle PST= 30° and angle SPQ= 60° Then angle PSQ is :

a) 40° b) 30° c) 60° d) 90°

iv) A letter is chosen random from all the letters of the English alphabets, The probability that the letter chosen is a vowel, is 

a) 4/26  b) 5/26  c) 21/26  d) 5/24 

v) If 3 is a root of the quadric equation x²- px +3= 0, then p is equal to:

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

vi) In the given figure

angle BAP= angle DCP= 70°, PC= 6cm and CA= 4cm, then PD: DB

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

vii) The printed price of an article is Rs3080. If the rate of GST is 10%, then the GST charged is:

a) Rs154  b) Ra308 c) Rs30.80 d) Rs15.40

viii) (1+ sinA)(1- sinA) is equal to:

a) cosec²A b) sin²A c) sec²A d) cos²A

ix) The coordinates of the vertices of ∆ ABC are respectively (-4,-2), (6,2) and (4,6). The centroid G of ∆ ABC is:

a) (2,2) b) (2,3) c) (3,3) d) (0,-1) 

x) The n-th term of an arithmetic progression is 3n+ 5. The 10th term is:

a) 15  b) 25  c) 35  d) 45 

xi) The mean proportional between 4 and 9 is:

a) 4 b) 6 c) 9 d) 36 

xii) Which of the following cannot be determined graphically for a grouped frequency distribution ?

a) median b) mode c) quartiles d) mean 

xiii) Volume of a cylinder of height 3 cm is 48 π cm³, Radius of the cylinder is:

a) 48cm b) 16 cm c) 4cm d) 24cm

xiv) Naveen deposits Rs800 every month in a recurring deposit account for 6 months. If he receives Rs4884 at the time of maturity, then the interest he earns is:

a) Rs84  b) Rs42  c) Rs24 d) Rs284

xv) The solution set for the equation 2x + 4 ≤ 14, x ∈ W is:

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

Question 2:          

i) Find the value of 'a' if x - a is a factor of the polynomial 3x³+ x²- ax - 81.   [4] 

ii) Salman deposits Rs100 every month in a recurring deposit account for 2 years. If he receives Rs2600 on maturity, find:

a) the total intrest Salman earns.

b) the rate of interest.           [4]

iii) In the given figure,

O is the centre of the circle. CE is tangent to the circle at A. If angle ABD= 26°, then find:

a) angle BDA

b) angle BAD

c) Angle CAD

d) Angle ODB.           [4]

Question 3

i) Solve the following quadratic equation : x²+ 4x - 8= 0. Give your answer correct to one decimal place. ( use mathematical tables if necessary).      [4]

ii) Prove the following identity: (sin²x -1)(tan²x +1)+ 1= 0.       [4]

iii) Use graph sheet to answer this question. Take 2cm = 1 unit along both the axes.

a) plot A, B, C, where A(0,4), B(1,1) and C(4,0)

b) Reflect A and B on the x-axis and name them as E and D respectively.

c) Reflect B through the origin and name it F. Write down the coordinates of F.

d) Reflect B and C on the y-axis and name them as H and G respectively.

e) join points A, B, C, D, E, F, G, H and A in order and name the closed figure formed.   [5]

     Section - B (40 Marks)

(Attempt any four questions from this section)

_______________________________________

Question 4

i) If A= 1 3 B= 1  2 C= 4   1 & D= 1  0

             2 4      2  4       1   5          0  1

Find A(B+ C)- 14I.               [3]

ii) ABC is a triangle whose vertices are A(1,-1), B(0,4) and C(-6,4). D is the midpoint of BC. Find the:

a) coordinates of D

b) the equation of the median AD.       [3]

iii) In the given figure, O is the centre of the circle. PQ is a tangent to the circle at T. chord AB produced meets the tangent at P. 

AB= 9cm, BP=16cm, angle PTB= 50°, angle OBA= 45°

Find :

a) length of PT

b) angle BAT

c) angle BOT

d) angle ABT.         [4]

Question 5:

i) Mrs Arora bought the following articles from a departmental store:

S. No  Items  Price     rate of GST  discount 

1.      hair oil  Rs1200      18%         Rs100

2. cashew nuts Rs600    12%            --

Find the:

a) total GST paid.

b) total bill amount including GST.     [3]

ii) Solve the following inequation. Write down the solution set and represent it on the real number.

-5(x -9)≥ 17 -9x > x+2, x ∈ R.       [3]

iii) In the given figure,

AC || DE || BF.

If AC =24cm, EG= 8cm, GB= 16cm, BF= 30cm.

a) prove ∆ GED ~ ∆ GBF

b) Find DE

c) DB: AB.       [4]

Question 6:

i) The following distribution gives the daily wages of 60 workers of a factory.

Daily income in Rs     frequency 

200-300                         6

300-400                        10

400-500                        14

500-600                        16

600-700                        10

700-800                         4

Use graph paper to answer this question.

Take 2 cm= Rs100 along one axis and 2cm =2workers along the other axis.

Draw a histogram and hence find the mode of the given distribution.    [3]

ii) The 5th and 9th term of an Arithmetic progression are 4 and -12 respectively. Find :

a) the first term 

b) common difference

c) sum of 16 terms of an AP .       [3]

iii) A and B are the two points on the x-axis and y-axis respectively.

a) Write down the coordinates of A and B.

b) P is a point on AB such that AP: PB = 3:1. Using section formula find the coordinanates of point P.

c) Find the equation of a line passing through P and perpendicular to AB.     [4]

Question 7:

i) A bag contains 25 cards numbered through 1 to 25. A card is drawn at random. What is the probability that the number on the card drawn is:

a) a multiple of 5.

b) a perfect square 

c) A prime number ?      [3]

ii) A man covers a distance of 100km, travelling with a uniform speed of x kmph. Had the speed bean 5 kmph more it would taken 1 hour less. Find x, the original speed.   [3]

iii) A solid is in the shape of hemisphere of radius 7cm, surmounted by a cone of height 4 cm. The solid is immersed completly in a cylindrical container filled with water to a certain height. If the radius of the cylinder is 14 cm, find the rise in the water.  [4]

Question 8:

i) The following table gives the marks scored by a set of students in an examination. Calculate the mean of the distribution by using the shortcut method.

 Marks   number of students

 00-10        3 

10-20         8 

20-30        14

30-40         9

40-50         4

50-60         2        [3]

ii) What number must be added to each of the numbers 4, 6, 8, 11 in order to get the four numbers in proportion ?     [3]

iii) Using ruler and compass, construct a triangle ABC in which AB= 6cm, angle BAC= 120° and AC= 5cm. Construct a circle passing through A, B and C. Measure and write down the radius of the circle.     [4]

Question 9:

i) Using componendo and dividendo solve for x:

{√(2x+2) + √(2x -1)}/{√(2x +2) - √(2x -1)}= 3.        [3]

ii) Which term of the arithmetic progression(AP) 15, 30, 45, 60....is 300 ? Hence find the sum of all the terms of the arithmetic progression.   [3]

iii) From the top of the tower 100m high a man observes the angles of depression of two ships A and B , on opposite sides of the tower as 45° and 30° respectively. if the foot of the tower and the ships are in the same horizontal line find the distance between the two ships A and B to the nearest metre.     [4]

Question 10:

i) Factorise completely using factor theorem:

2x³- x²- 13x -6.          [4]

ii) Use graph paper to answer this question.

 During a medical checkup of 60 students in a school, weights were are recorded as follows:        [6]

Weight (in kg)    no of students 

28-32                    2

30-32                    4 

32-34                  10 

34-36                  13 

36-38                  15

38-40                   9

40-42                   5

42-44                   2

 taking 2 cm 2 kg along one axis and 2cm= 10 students along the other axis, draw an ogive. Use your graph to find the 

a) median 

b) upper quartile 

c) number Students whose weight is above 37 kg.       [6]

Day- 11(14/8/24)

1) On What sum of money, the difference between 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.

5) Sita deposit Rs400 per month in a bank in recurring deposit. On maturity she gets Rs97854.40. Find the period of which she had deposited .

6) Which is the better investment 8% Rs100 shares at Rs 20 premium or 6% Rs100 shares at 20 discount?

7) A company declares a semi-annual dividend of 5%. Sanjay owns 25 shares of par value Rs12.50 each . How much annual dividend must he receive ?

8) A company having a capital stock of Rs 450000 declares a dividend of 4% semi-annually.

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

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

9) Vinoy owns 150 Rs 25 shares of a company which declares a dividend of 12%. What is Vinay's dividend income? If he sells the shares at Rs40 and invests the proceeds in 7% stock (par value Rs100) at Rs80, what is the change in his dividend income ?

10) Mr Gupta purchased 360 Rs50 shares at Rs20 premium. The company declares an annual dividend 12%.

a) Find his dividend income from the shares.

b) Find his total investment in the shares.

11) A man invests Rs1426 in 5% stock at Rs115. He sells this stock at Rs125 and invests the proceeds in 3% stock at 93. Find the change in his income.

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

13) Solve: 8/3 ≤ x + 1/3 < 10/3, x ∈ R. Hence represent the solution on a number lines.

14) A= {x: -1< x ≤ 5, x ∈ R} 

B={x : -4 ≤ x < 3, x ∈ R}

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

15) Solve the equation 3x²- x - 7= 0 and give your answer correct to 2decimal places.

16) Find the roots of x²- 6x +2= 0, using formula method.

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

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

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

20) Solve the following quadratic equation: √(x +15) = x + 3, x ∈ N.

21) Solve: √(x(x -3))= √10, State sum of the roots.

22) Solve : √(6x -5) - √(3x -2)= 2 using formula method.

23) A two-digit number is 4 times the sum and two times the product of its digits . Find the number.

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

25) A number consists of two digits such that the square of the digit in the ten's place exceeds the digits in the unit place by 11. If the number is 5 times the sum of the digits, find the number.

26) 10 years ago, the sum of the ages of two sons was half of their father's age. The ratio of the present ages of the two sons is 4:3 and the sum of the present ages of all the three is 117 years. Find the present ages of the father and each of the two sons.

27) A party of students arranged an excursion costing Rs540, whose amount was to be shared equally by all of them. But later, it was found that three of the students, could not pay, though they had joined the excursion . As a result , the rest of the students had each to pay Rs2 more. Find the total number of students in the party.

28) In a certain examination, the those number of candidates passed was 4 times the number of those who failed. If the number of candidates that appeared had been 35 less, and the number been twice the number failing. Find the total number of candidates that appeared at the examination

Day-10(11/8/24)

PROBABILITY - TEST

1) A coin is tossed 500 times. We obtain a head 260 times. On tossing the coin at random, find the probability of getting 

i)A head

a) 12/25 b) 13/25 c) 14/25 d) 16/25

ii) A tail 

a) 12/25 b) 13/25 c) 14/25 d) 16/2

2) A die is rolled 50 times and the number 6 is obtained 8 times. Now, if the die is rolled at random, find the probability of getting the number 6.

a) a) 4/25 b) 13/25 c) 14/25 d) 16/25

3) There are 10 cards in a bag bearing numbers from 1 to 10. A card is drawn from the bag 60 times. Each time, the number obtained is noted and the card drawn is replaced. It was found that the card number bearing number 7 was drawn 4 times. Now, if a card is drawn at random, find the probability of getting the card bearing 7.

a) 1/15 b) 2/15 c) 4/15 d) 7/15

4) A die is rolled 100 times and the outcomes are noted and tabulated as shown :

Outcomes:  1    2   3   4     5     6 

Frequency:  9  15 19  21   24  12 

When a die is rolled at random, find the probability of getting the number 

i) 2

a) 3/20 b) 21/100 c) 3/25 d) 6/25

ii) 4

a) 3/20 b) 21/100 c) 3/25 d) 6/25

iii) 6

a) 3/20 b) 21/100 c) 3/25 d) 6/25

ASSASSIN REASONS QUESTIONS:

DIRECTIONS: In the following questions , a statement of Assertion(A) is followed by a statement of Reason(R). Choose the correct option:

1) Assertion (A): A coin is tossed 16 times and the outcomes are recorded as below:

H  T  T  H  T  H  H  H  T  T  H  T  H  T  T  H

 The probability of occurrence of a both is 50% .

Reason : When a coin is tossed, there are two possible outcomes-- Head and Tail.

a) both Assertion (A) and Reason(R) are true and Reason(R) is the correct explanation of Assertion (A).

b) both Assertion (A) and Reason(R) are true and Reason(R) is not the correct explanation of Assertion (A).

c) Assertion (A) is true but Reason(R) is false.

d) Assertion (A) is false but Reason (R) is true.

2) Assertion (A): When a spinner with 3 colour Red(R), Green (G) and Black (B) as shown is rotated , red and green colours have equal probability to show up with arrow. 

Reason (R): The probability of occurrence of an event E is given by

P(E)=  total number of trials/number of trials in which occurs

a) both Assertion (A) and Reason(R) are true and Reason(R) is not the correct explanation of Assertion (A).

b) both Assertion (A) and Reason(R) are true and Reason (R) is not correct explanation of Assertion (A).

c) Assertion (A) is true but Reason (R) is false

d) Assertion (A) is false but Reason (R) is true

Day- 9 (8/7/24)

1) From a pack of cards of 52 cards two are drawn at random. Find the chance that one is a knave and the other a queen.

2) Show that the chances of throwing six with 4,3 or 2 dice respectively are as 1:6:18

3) What is the chance of throwing a number greater than 4 with an ordinary die whose faces are numbered from 1 to 6 ?

4) Find the chance of throwing 9 atleast in a single throw with two dice.

5) From 20 tickets marked with the first 20 numerals, one is drawn at random, find the chance that it is a multiple of 3 or of 7 ?

Day- 8(24/6/24)

1) If x²= 3x, then 

a) x= 0 b) x=0 or x=3 c) x=3 d) x=0 and x= 3

2) If 3x²+ 8= 10x, then 

a) x= 2 or 4/3 b) x=2 or x=3 c) x=3 or 4/3 d) x=1 and x= 1/3

3) The quadratic equation whose solution set is {-2, 3} is 

a) x²- 3x -6=0 b) x²- x + 6=0 c) x² +x -6=0 d) x²- x -6=0 

4) One of the roots of 2x²- 7x +6=0, is

a) -2 b)2 c)!-3/2 d) 4 

5) The discriminant of the quadratic equation 2x²- x +3 =0 is:

a) 24 b) 25 c) -23 d) -20

6) On solving x² +4x - 21=0, we get 

a) x= -3, or -7 b) x=7 or -3 c) x =7 or 3 d) x =-7 or 3

7) The roots of the quadratic equation 2x² + x x - 1 =0 are

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

8) The discriminant of x²- 4x - 7 =0 is:

a) 44 b) √44 c) 12 d) -12

9) The quadratic equation whose roots are -1, -5 is:

a) x² +6x +5 =0 b) x²- 6x +5 =0 c) x²+ 6x - 5=0 d) x²- 6x - 5=0 

10) For the equation 3x²- 4x - 2 =0, the roots are:

a) real and equal b) real and unequal c) imaginary d) both (a) and (b)

11) For the quadratic equation 2x²- 4x + 1 =0, the discriminant is:

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

12) For the quadratic equation 2x²- 3x + 1 =0, the discriminant is:

a) 1 b) -1 c) 0 d) 9

13) For the quadratic equation 9x²+ 6x +1 =0, the discriminant is :

a) +ve b) -ve c) 0 d) imaginary 

14) For the quadratic equation 2x² + ax - a²=0, the sum of the roots is:

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

15) Given a quadratic equation mx²+ 8x -2 =0, m≠ 0. For this quadratic equation the value of discriminant is:

a) 64- 8m b) √(64+ 8m) c) 64+ 8m d) √(8m - 64)

16) For the quadratic equation 3x²+ 7x + 8 =0, the roots are:

a) real and distinct b) real and equal c) imaginary d) both (a) and (b)

17) The roots of the quadratic equation 2x²- kx + k =0 are equal . If k ∈N, then k is 

a) 0 b) 8 c) 6 d) -6

18) Given the quadratic equation x² + 2√2x +1 =0. The roots of the quadratic equation are:

a) (-√2±1)/2 b) (√2±1)/2 c) (√2±1) d) (-√2±1)

19) The quadric equations (m +1)x² + 2(m +3)x +(m +8) =0. has equal roots. The value of m is :

a) 1/3 b) 1/4 c) 3 d) -3 

20) The quadratic equation x² + m(2x + m -1)+ 2 =0 has equal roots . The value of m is

a) 1 b) 2 c) -2 d) 0 

5/5/24 day- 7

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

2) Solve: 21x²- 8x - 4 = 0.      (2)

3) Find the coordinates of the image of (5,-4) after reflection in

a) x= 0

b) y= 2.             (2)

4) List the solution set of the following inequation and graph the solution set:

1/2 + 8x > 5x - 3/2 p, x belongs to Z.        (3)

5) Calculate the ratio in which the line joining A(6,5) and B(4,-3) is divided by the line y= 2.     (2)

6) Given A= 1      1

                     8      3 evaluate A²- 3A.       (3)

7) Show that: √{(1- cosx)/(1+ cosx)= sinx/(1+ cosx).      (3)

8) Calculate the amount receivable on maturity of a recurring deposit of Rs800 every month for 5 years at 11% per annum.        (3)

9) A and B are points on positive side of the x-axis and y axis respectively. The point P(4,3) divides the join of AB in the ratio 3:1. Find the equation the line AB.    (2)

Day- 6

1) When 7x²- 3x +8 is divided by x -4, find the remainder (using remainder theorem (2)

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

3) Find p and q if g(x)= x +2 is a factor of f(x)= x³- 0x + x + q and f(2)= 4.      (3)

4) Find the equation of a line that passes through (1,3) and slope is 3.    (3)

5) The midpoint of the line joining A(2,p) and B(q,4) is (3,5). Find the numerical values of p and q.       (3)

Day - 5

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

Graph the solution set on the number line.

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

-3(x -7)≥ 15 - 7x > (x +1)/3, x ∈R

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

-2+ 10x ≤ 13x +10 < 24+ 10x, x ∈ Z

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

11x - 4 < 15x +4 ≤ 13x + 14, x ∈ W

Represent the solution on a real number line.

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

2y - 3 < y +1 ≤ 4y +7, y ∈ R

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

2x -5 ≤ 5x +4 < 11, x ∈I

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

13x -5 < 15x +4 < 7x +12, x∈R

Day -4

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

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

2) If - x ≥ -3 then:

a) x≤ -3 b) x ≥ 3 c) x = 3 d) x ≤ 3

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

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

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

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

5) If a> b, then:

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

6) If x≥ 5 and- ax ≥ 5a, then :

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

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

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

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

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

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

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

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

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

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

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

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

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

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

a) {-3/2≤ x < 2, x ∈R}

b) {-3/2 < x < 2, x ∈R}

c) {-2/3 ≤ x ≤ 1, x ∈R}

d) {-1/3 ≤ x ≤ 2, x ∈R}

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

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

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

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

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

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

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

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

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

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

Day 3

Solve the following:

1) 35x²+ 13x -12=0.

2) mnx²+ (n²- m²)x - mn=0

3) x²- x - 42= 0.

Day- 2

1) Calculate, without actual division, the remainder when 5x³+8x²- 2x-9 is divided by x+2.

a) 6 b) 7 c) 8 d) 9 e) none

2) If f(x)= 24x³+ px²- 5x+ q has 8 factors 2x+1 and 3x-1, then find p and q. Also Factorise f(x) completely.

3) remainder if 2x³-3x²+7x-8 is divided by x-1

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

4) Find the number should be added with the number 2x²+3x+1 to make x-1 is the factor .

A) 6 B)-6 C) none D) none of these

5) If f(x)= 24x³ + px² - 5x +q has two factors 2x+1 and 3x-1, then find p and q.Also Factorise completely. 

6) Factorise completely : x³ + x² - 4x -4. 

7) find remainder when 3x³+ 5x²- 11x -4 is divided by 3x+1. (2)

e) When kx³+ 9x²+ 4x -10 is divided by (x+1), the remainder is 2. Find k. 

8) Factorise completely: x³ - 3x² - 4x +12. (5)

Day- 1

1) When 2x³- x²- 3x +5 is divided by 2x+1, then the remainder is

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

2) If on dividing 4x²- 3kx +5 by x+2, the remainder is -3 then the value of k is

a) 4  b) -4 c) 3 d) -3

3) If on dividing 2x³+ 6x² - (2k -7)x+5 by x+ 3, the remainder is k -1 then the value of k is

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

4) The remainder when x⁵¹+ 51 is divided by x+1, is

a) 51 b) 50 c) -1 d) 0

5) The remainder when x²+ 2x +1 is divided by x +1 is

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

6) The remainder when f(x)= x³+ 4x²- 3x +1 is divided by x -2 is

a) 16 b) 12 c) 17 d) 19

4) If x+1 is a factor of 3x³+ kx²+ 7x +4, then the value of k is

a) -1 b) 0 c) 6 d) 10

MIXED MODEL TEST PAPER - XI AND XII

M. A- R-1

1) Show that, (5+√5)/√{5+ 3√5}= ⁴√20.

2) 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 .

3) If n be the positive integer greater than unity, then show that 49ⁿ - 16n -1 is divisible by 64.

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

5) How many numbers of four digits can be formed from the numbers 1,2,3, 4? Find the sum of all such numbers (digits being used once only).   24, 66660

6) If 9α=π, find the value of sinα sin2α sin3α sin4α.        3/16

7) If tan θ = (tanα - tan β)/(1- tanα tan β), then show that sin2θ = (sin2α - sin2 β)/(1- sin2α sin2β).

8) If 8R²= a²+ b²+ c² (or, cos²A+ cos²B + cos²C= 1), then show that the triangle ABC is right angled.

9) Solve: cos³ θ cos3 θ+ sin³ θsin3 θ = 1/8.        nπ± π/6

10) If {m tan (x- y)}/cos²y= n tan y/cos²(x - y), show that y= (1/2)[x - tan⁻¹{n - m)/(n+ m)} tan x].

11) if lx + m y=0 be the perpendicular bisector of the segment joining the point (a,b) and (c,d). then prove that (c - a)/l = (d - b)/m = 2(la+ mb)/(l²+ m²).

12) Show that the two circles x²+ y²+ 2gx + 2fy=0 and x²+ y²+ 2g'x + 2f'y=0 will touch each other if f'g= g'f.

13) Find the equation and the latus rectum on the parabola whose focus is (5,3) and vertex is (3,1).       x²+ y²- 2xy - 20x -12y+ 68=0; 8√2

14) if α and β be the eccentric angles of the extremities of a focal chord of the ellipse  x²/a²+ y²/b²=1.

Show that tan(α/2) tan(β/2) = (e -1)/(e+1) or (e +1)/(e-1).

15)  Find dy/dx when y= ₓx²+ ₐx².      ₓx²+1+ (1+ 2 logx)+ ₐx² log a

16) Evaluate lim ₓ→₀ (tan2x - x)/(3x - sinx).       1/2

17) if y= xⁿ{a cos(logx)+ b sin(logx)}, show that x² d²y/dx²+ (1- 2n) x dy/dx + (1+ n²)y= 0.

18) Show that eˢᶦⁿˣ  - e⁻ˢᶦⁿˣ = 4 has no real solution.    

19) find the derivative of x² cosx.      2x cosx - x² sinx

20) prove ᵗᵃⁿˣ₁/ₑ ∫ t dt/(1+ t²)+ ᶜᵒᵗˣ₁/ₑ ∫ dt/t(1+ t²)= 1.

21) solve: (x + y)² dy/dx = 2x + 2y+5.     y= log{(x + y+1)²+4} + (3/2) tan⁻¹{(x+ y+1)/2}+ c

22) Evaluate lim ₙ→∞ (1/n)[sec²(π/4n) + sec²(2π/4n + .....+ sec²(nπ/4n)].    4/π

23) integrate: ∫ (cosx - sinx)(2+ 2 sin2x)/(cosx + sinx) dx.     Sin2x + c

24) solve: d²y/dx² - 2a dy/dx + a²y= 0, given y= a and dy/dx = 0 when x= 0.    y= a(1- ax)eᵃˣ

25) Shade the region above the x-axis , included between the parabola y²= 4x and the circle (x -4)= 4 cos θ , y= 4 sin θ. Find the area of the region by integration.    (4π - 32/3) Square.unit

26) Show that the maximum value of the function x + 1/x is less than its minimum value.

27) Show that the line lx + my = n is a normal to the ellipse x²/a²+ y²/b² = 1, if a²/l²+ b²/m² = (a²- b²)²/n².

28) Show that log(1 + x)> (tan⁻¹x)/(1+ x) for all x > 0.

29) prove that ³√(2+ √5)+ ³√(2 - √5)= 1.

30) determine the sign of the expression 

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

31) If cot x = 2 and cot y=3, then find (x - y).    π/4

32) Find when the solution of the equation acos x + b sin x = c is possible.    c≤ √(a²+ b²)

33) Find the square root of 4ab - 2i(a²- b²).     ±(a+ b) i(a- b)

34) if  θ (x)= (x -1)eˣ + 1, Show that θ (x) is positive for all the values of x> 0.    

35) If y= f(x)= (x+1)/(x+2) show that, f(y)= (2x+3)/(3x +5).   

36) Evaluate: ¹₋₁∫ sin³x cos²x dx.     0

37) Is it possible to draw a tangent from the point (-2,-1) to the x²+ y²- 4x + 6y - 12= 0 ? give reasons .          No

38) If f(x)= tan(x - π/4), find f(x) x f(-x).     1

M. A- R-2

1) If x= {q - √(p²- 4q)}/{q+ √(p²- 4q)} , show that (q²- p²+ 4q)(x²+1) - 2(p²+ q²- 4q)x = 0.

2) show that log₂₀3 lies between 1/2 and 1/3.

3) In the expansion of (√x - √k/x²)¹⁰, the term independent of the x be 405; find the value of k.      9

4) if the equation x²+ bx + ca = 0 and x²+ cx + ab = 0 have a common root, prove that their other roots will satisfy the equation x²+ ax + bc = 0.    

5) if x - 1/x = 2i sin θ, Show that x⁴- 1/x⁴ = 2i sin4θ.

6) prove that tan 6° tan 42° tan 66° tan 78°=1.

7) Show that time 20 tan 40 tan18 = tan60.

8) if a cosθ + b sinθ = a cos α+ b sinα, show that sin(α +θ)= 2ab/(a²+ b²).

9) Solve : sin⁸x + cos⁸x = 17/32.     (2n +1)π/8

10) If a²(1- sinα) + b²(1+ sinα)= 2ab cosα , show that a/b - b/a = 2 tanα.

11) In a triangle ABC, prove that (bc - r₂r₃)/r₁ = (ca - r₃r₁)/r₂ = (ab - r₁r₂)/r₃.

12) y= mx is the equation of a chord of the circle x²+ y²- 2ax =0. Prove that the equation of the circle on this chord as diameter is (1+ m²)(x²+ y²) - 2a(x + my)= 0.

13) The point (1,3) and (5,1) are two opposite vertices of a rectangle. The other two vertices lie on the line y= 2x + c. Find c and the remaining vertices.    -4, (2,0),(4,4)

14) P is a variable point on the hyperbola x²- y²= a² and A is the fixed point (2 a, 0). Show that the locus of the midpoint of the line segment AP is another hyperbola.    

15) Given the ellipse 4x²+ 9y²= 36, find the equation of the chord which is bisected at (2,1).       8x - 9y=25

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

f(x)= 2 - |x|/x, when  -2≤ x ≤ 2.    

a) Draw the graph of f(x) and discuss the continuity at x=0.        discontinuous at x= 0.

b) Does the limit of f(x) exist when x= 0? Explain.    

17) Find derivative of tan(√x).    (Sec²√x)/2√x.

18) If cosy = x cos(a+ y), show that dy/dx = (cos²(a+ y)/sina.

19) Evaluate: lim ₓ→₀ (x log(1+ x))/(1- cosx).      2

20)  lim ₓ→₁ (1+ cosπx)/(1- x)².         π²/2

21) If x= eᶻ, then show that x² d²y/dx² = d²y/dz² - dy/dz.

22) Evaluate: lim ₙ→∞ [1/(n +1) + 1/(n +2)+ 1/(n +3)+ .....+ 1/6n].    Log6

23) ∫ dx/[(x - a)√{(x - a)(b - x)}].    2/(a - b) √{(b - x)/(x - a)}+ c

24) ∫ (x cosx)/sin³x dx.      (-1/2) (x cosec²x + cotx)

25) Show that: π⁾⁴₀∫  sin²x cos²x/(sun³x + cos³x)² dx = 1/6

26) Solve: xy dy/dx = (1+ y²)(1+ x + x²)/(1+ x²).       (1/2) (1+ y²)= Logx + tan⁻¹x + c

27) find the value of ᵇₐ∫ e⁻ˣ dx.        e⁻ᵃ - e⁻ᵇ

28) Find the area bounded by the upper half of the ellipse x²/25 + y²/9= 1, the x-axis and the straight lines x=3 and x= 4.      (15/2) (Sin⁻¹ - sin⁻¹(3/5)) Sq. Unit

29) Prove that the normal chord of the parabola y²= ax which is normal at the point (a/2, a/√2) subtends a right angle at the vertex.        

30) The eccentric angles of two points on the ellipse x²/a²+ y²/b²= 1 are  α and β. If the tangents at those two points and intersect at (h,k), show that (h²/a²+ k²/b²)² sin²(α - β)= 4(h²/a²+ k²/b² -1).

31) Prove that the extremum of u(x)/v(x) is given by u've v - uv'= 0 and the extremum is a maximum or a minimum according as u" v - uv" is < or > 0.

32) which term of the following two series is equal?

  2+ 7+12+...... And 101+97 +93 +....12th

33) The roots of (a²+ b²)x²+ 2(a+ b)x +2= 0 are always:

a) real  b) imaginary c) positive  d) equal 

34) If tan200 = a find (sin110 - cos 250)/(cosec 160+ sec 340) in terms of a.  a/(1+ a²)

35) evaluate: tan⁻¹(a/b) -  tan⁻¹{(a-b)/(a+ b)}.      π/4

36) Find the minimum value of ligₐx + logₓ a for 0< a < x.    2

37) If y= c₁x⁻¹ + c₂x², the value of x² d²y/dx² is:

a) x b) y c) 2x d) 2y

38) The centre of a circle is (3,4) and the length of a tangent drawn from (-2,-2) to the circle is 6. Find the radius of the circle.     5

39) π₀∫ f(x) dx = 0, then f(π - x) is equals to:

a) f(x) b) f(-x)  c) - f(x) d) none of these

40) Find the differential equation of the straight lines which pass through the origin.     y= x dy/dx

41) If f(x)= x⁹- 6x⁸ - 2x⁷+ 12x⁶+ x⁴ - 7x³+ 6x²+ x -3, then find the value of f(6).     3

M. A - R- 3

α 

β λ θ γ δ ⁻¹