Test-12 (2026/27)

Inverse trigo

1) If sin⁻¹(sin⁻¹(1/5) + cos⁻¹x)= 1 then find the value of x.        1/5

2) Solve: tan⁻¹{(1- x)/+1+ x)}= (1/2) tan⁻¹x, x> 0.          1/√3

3) Prove: cot⁻¹7 + cot⁻¹8 + cot⁻¹18= cot⁻¹3.

4) Prove: sin⁻¹(8/17) + sin⁻¹(3/5) = cos⁻¹(36/85).

5) Show that tan⁻¹√x = (1/2) cos⁻¹{(1- x)/(1+ x)}, x ∈ [0,1].

6) Show that cos[tan⁻¹{sin(cot⁻¹x)}]= √{(1+ x²)/(2+ x²)}.

7) Find the value of cot{π/2 - 2 cot⁻¹√3}.        √3

8) Find the value of tan⁻¹[2 sin(2 cos⁻¹(√3/2))].      π/3

9) If sin[cot⁻¹(x +1)]= cos(tan⁻¹x) then find x.      -1/2

10) If tan⁻¹{(x -2)/(x -4)} + tan⁻¹{(x +2)/(x +4)}= π/4 then find the value of x.    ±√2

11) Solve for x : cos⁻¹x + sin⁻¹+x/2)=π/6.       1

12) Show: tan⁻¹(3/4)+ tan⁻¹(3/5) - tan⁻¹(8/19)=π/4.     

13) Show: sin⁻¹(8/17) + sin⁻¹(3/5) = tan⁻¹(77/36).

14) Find the principal value of tan⁻¹√3 - sec⁻¹(-2).       

15) Show: tan⁻¹(1/4) + tan⁻¹(2/9) = (1/2) sin⁻¹(4/5).

16) Evaluate: tan[2 tan⁻¹(1/2) - cot⁻¹3].      9/13

17) If cos⁻¹x + cos⁻¹y + cos⁻¹z =π then show that x²+ y²+ z²+ 2xyz = 1.

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

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

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

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

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

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

a) 0 b) 2 c) - 1 d) 1 

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

a) 2a b) a c) a/2 d) 0 

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

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

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

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

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

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

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

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

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

a) e⁻ʸ b) eˣ c) e⁻ˣ d) eʸ

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

2)a) Find the maximum value of 

1      1          1

1   1+sinx   1

1      1  1 + cosx             (1)

b) Solve the equation for x: 

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

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

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

a) mutually exclusive.

b) independent.                            (2)

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

6) For what value of x is the given matrix

    2x+4            4

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

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

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

9) Prove: 1+ a      1        1

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

                   1         1     1+c 

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

     f(x)= x²,          if x ≤ 2

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

OR

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

f(x)= kx +1,       if x ≤π

          cosx,       if x > π

is continuous at x =π

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

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

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

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

OR

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

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

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

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

OR

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

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

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

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

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

OR

If A= 1       -1       0

         2        5       3

         0        2       1   Find A⁻¹.          

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

OR

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

19) Evaluate 

a) ∫ x² sin⁻¹x dx. (3)

b) ∫ x/(x²+ 4x +3) dx.       (3) 

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

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

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

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

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

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

a) 30° b) 45° c) 60° d) 90° (1)

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

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

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

 (a x b)²= a.a           a.b

                a. b           b.b                 (4)

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

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

OR

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

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

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

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

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

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

i) Find the marginal cost (MC).

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

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

OR

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

X: 1       2        3        4         5

Y: 7       6        5        4         3 

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

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

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

 (6)

REVISION - 1

1) If x= a sin³t and y= a cos³t, find dy/dx. 

2) If y= ₑ(m cos⁻¹x), show that (1- x²) d²y/dx² - x dy/dx = m²y.

3) If y= eˣ log(tan2x), find dy/dx. 

4) If xʸ = eˣ⁻ʸ, show that dy/dx = (logₑx)/(1+ logₑx)².

5) If y= (cosx)ᶜᵒˢˣ, find dy/dx.

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

7) If y= ₑsin²x, find dy/dx. 

8) If x= a(cosθ+ θ sinθ), y= a(sinθ - θ cosθ). Then find dy/x at θ= π/4. 

9) If y= logₑ{x + √(x²+ k²), find dy/dx.

10) If xʸyˣ = 5, then show that dy/dx = |- (log y + (y/x))/(log x + (x/y)).

11) If y= (sin⁻¹x)/√(1- x²), then show that (1- x²) dy/dx - xy = 1.

12) If xᵖ yᑫ = (x + y)ᵖ⁺ᑫ, then show that dy/dx = y/x.

13) If y= xʸ, show that x. dy/dx = y²/(1- y log x).

14) If √(1- x⁴)+ √(1- y⁴)= λ(x² - y²), then show that y √(1- x⁴) dy/dx = x √(1- y⁴).

15) If sin(xy)+ cos(xy)= 1 and tan(xy)≠ 1, then show that dy/dx = -y/x.

Test paper - XII maths

Part- A 

1) Choose the correct option (1x 10=10)

a) If P is a linear on the set A={1,2,3,5,6} defined by P={(a, b) belongs to A x A, a - b=15} then P is 

i) reflexive ii) symmetric iii) transitive iv) void relation 

b) If A and B are arbitrary square matrix 

i) (AB)'= A' B' ii) (A')'(B')'= B'A' iii) (A+ B)' = A'B'. d) (AB)'= B'A'

c) The number of points of discontinuous of f(x)= (x²+4)/(4x - x³) is 

i) 1 ii) 2 iii) 3 iv) 4

d) If a and b are two vectors such that 

a, b< 0 and|a. b|= |a x b| then the angle between a and b is

i) π ii) 7π/4 iii) 3π/4 d) π/2

e) i. (j x k)+ j. (k x i)+ k. (i x j)

i) 1  b) 2 c) 3 d) -3

f) f(x)= logx , then d/dx(f(logx))

i) (1/x) logx ii) x/logx iii) 1/(x logx) iv)  x log x

g) Two integers are selected at random from 1 to 15. If the sum is even, then the probability that both the number will be odd is

i) 3/7 ii) 5/7 iii) 4/7 iv) 6/7

h) if f(x)={3x -4, 0≤ x < 2

                  2x + k, 2 < x ≤ 3 

is continuous at x= 2. Find the value of k.

i) 0 ii) 2 iii) -2 iv) -1

i) Two vectors are α = 2i - 5j - 3k and β = -4i + 10j + 6k then α and β are 

i) like ii) collinear iii) equal iv) none

j) The tangent to the curve y= x³ at (0,0) is 

i) x axis ii) y-axis iii) x= y iv) none

Type- B

2) answer any one question (2 x2= 2)

a) i) Show that, the function f: IR --> IR, f(x) = |x| is neither oneone nor onto 

ii) Differentiate log(logx).

b) i)Answer any one question.      2x 2= 2

Without expanding prove:

a    a²    bc  = 1     a²     a³

b.   b²    ca     1     b²     b³

c    c²    ab     1     c²     c³

ii) Find real values of k for which the following system of operations has non trivial solution 

x- ky - z=0; Kx - y - z= 0, x+ y - z= 0.

c) Answer any three questions.  (2cpx3=6)

i) Show that f(x)= |x|/x, x≠ 0

                               0,       x= 0

is discontinuous at x= 0

ii) Evaluate ∫ xeˣ dx.

iii) Find order and degree of differential equation d³y/dx³+ y = ³√(1+ dy/dx).

iv) Show that f(x)= logx has neither a maximum value nor a minimum value.

v) Find dy/dx when eˣʸ - 4xy = 4.

d) Answer any one (2x2= 2)

i) Find direction cosine of the straight line in YZ plane which makes π/6 with zaxis.

ii) If the Cartesian equation of a line AB is (3- x)/1= (y+2)/-2= (z -5)/4. Find the direction ratio of a line parallel to AB.

e) Answer any one (2x2=2)

i) P(A' UB')= 5/6, P(A)= 1/2, P(B')= 2/3, show A and B are two independent event.

ii) If a random variable x has the following probability distribution 

X:       4      5     6       8

P(x). 0.1   0.3  0.4   0.2

Find expectation of random variable of x.

Type- 3

3) a) Answer any one.     (4x 1= 4

i) Show that, the relation R is reflexive, transitive not symmetric on W where 

R={(x,y): x belongs to IN, y belongs to IN, x is multiple of y}

ii) If two function f: IR ---> IR  g: IR--> IR Defined by 

f(x)= 2x+ 3 and g(x)= 4x -1. Show that (fog)⁻¹= g⁻¹ o f⁻¹.

b) Answer the following:    (4x2= 8

i) If matrix 

A= 1  -2   2 & B= 1    2    0

      0   1  -1          2    3   -1

      0   0   1          0    -1  -2 show that (A'B)A is diagonal matrix.

OR

Prove that, the inverse of a given square matrix if it exists, is unique.

ii) Prove the determinants 

x     y      z 

x²   y²     z²

yz   zx    xy

= (x - y)(y- x)= (z - x)(xy + yz + zx)

OR

Solve by Cramer's Rule 

2/x + 3/y - 4/z= -3,

1/x + 2/y - 6/z= 2

3/x - 1/y - 2/z= 5

c) Answer the following:      (4x 3= 12)

i) If y= eᵘ and u= f(x) show that d²y/dx²= eᵘ [d²u/dx²+ (du/dx)²]

OR

i) If y= 2 sin⁻¹{(x -2)/√6}  - √(2+ 4x - x²), then show that dy/dx |ₓ₌₂ = 2/√6

ii) Evaluate , ∫ (x -1)dx/{(x -3)(x+2)}

OR

Find , ∫ eˣ(x log x +1)/x    dx.

iii) Solve, (1+ e²ˣ) dy + eˣ(1+ y²) dx = 0, given y= 1, x= 0

OR 

Solve, cos⁻¹(dy/dx)= x + y.

d) Answer any one.   4x1= 4

i) Let a= i + 4j + 2k, b = 3i - 2j + 7k and c= 2i - j + 4k. Find a vector d whris perpendicular to both a and b and c.d= 18

ii) ABCD is a quadrilateral. M and N are midpoint of AB and CD. O is the midpoint of MN, show that OA+ OB+ OC + OD= 0.

e) Answer any one.    4x4= 4

i) Evaluate ¹₀∫ (x +2) dx from defination of definite integral.

ii) ²₁∫  dx/{√(x +2). + √(x -2)}

f) Answer any one.    4x 1= 4

i) What is probability of getting 53 Sundays in a leap year?

ii) There are 3 black balls and 5 white balls in a bag A and 5 black balls and 4 white balls in bag B. One ball is chosen at random from any bag. What is the getting the ball from bag B when the ball is white?

Type- 4

 4) a) Answer any one.     5 x 1= 5

i) A furniture manufacturing rplana to make two products, chairs and tables from its available resources of 400 board feet of mahogany lumber and 10 hours. A chair requires 5 board feet of lumber and 10 manhours and gives a profit of Rs 45 while each table uses 20 board feet of lumber and 15 man hours and a profit of Rs 80. Formulate the LPP to maximize the profit.

ii) Solve graphically the LPP given 

Min: z= 3x + 2y

2x + y≥ 14; 2x + 3y≥ 22; x+ y≥ 5, x≥ 0, y≥ 0

b) Answer any two (5x.2=10)

i) Prove that the sum of the x and y intercepts made by the tangents to the curve √x + √y at any point on its a constant.

ii) Find the area between two curves y= x² and y²= x.

iii) Show that, a conical tent of given capacity will require the least amount of canvas, when the height is √2 times of its radius.

iv) If the straight line lx + my = n is a normal to the hyperbola a²/l² - b²/m²= (a²+ b²)²/n².

c) Answer any one.    (5x 1=5)

i) Find the vector equation of a line which passes through the point with position vector i- 2j + 4k and is in the direction of i + 2j - k. Also reduce it to Cartesian form.

ii) Show that, the lines (x -1)/3= (y+1)/2= (z -1)/5 and (x +2)/4 = (y -1)/3= (z +1)/-2 do not intersect.

MATRIX (1)

1) if A= 2   -1

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

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

2) The multiplicative inverse of matrix 

2    1

7    4 is

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

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

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

a) AB= AC doesn't imply B= C

b) A+ B = B+ A

c) (AB)'= B'A'

d) AB= 0 implies A= O or B= O

4) If A= 1  0  2 & Adj A= 5 a -2

             -1  1 -2                 1 1  0

               0 2  1                -2 -2 b

Then the values of a and b are 

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

5) If A= -1   0

               0   2 then the value of A³- A² is equal to 

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

6) If A= - x   - y

                z     t then the transpose of adj A is 

a) t     z b) t    y c) t -z d) none 

    -y - x     -z  -x     y -x

Hy

7) If A=3  5 & B= 1  17

             2  0          0  -10 then|AB| is equal to 

a) 80 b) 100 c) -110 d) 92

8) If A= 5.  -2

              3    1 find the inverse of A

9) If A is singular matrix of order n then A. (Adj A) is equal to 

a) a null matrix 

b) a row matrix 

c) a column matrix 

d) none 

10) If A= 3   -5

               -4    2 then find the value of A²- 5A is equal to 

a) I b) 14I c) O d) none 

11) If A= 1  2 & B= 1   2

                2  3          2   1 

                3  4 

Then

a) both AB and BA exist 

b) neither AB nor BA exist 

c) AB exists but BA does not exist 

d) AB does not exist but BA exist 

12) If A= 2  -1 & B= 1 0

                0   1         -1 -1 then+A+ B)² is not equal to 

a) A²+ AB+ BA+ B²

b) A²+ AB+ BA+ B²I

c) A²I + AB+ BA+ B²

d) A²+ 2AB+ B²

13) If A be an n × n matrix and k any scalar then det. kA is equal to 

a) k detA b) nᵏdetA c) kⁿ detA d) kn detA

14) If A= 1    2 

                3   -5 find inverse of A 

15) If A= -1   2 & B= 5

                 2  -1          7 and AX= B, then X is equal to 

a) 19  17 b) 19/3  c)19/3 17/3 d) 19

                     17/3                            17

16) If A≠ O and B≠ O are two 2 x 2 matrices such that AB= O, then which of the following is correct?

a) detA= 0 or detB= 0

b) detA= 0 and detB= 0

c) detA= 0 = detB≠ 0 d) none 

17) If A is a square matrix 3x3 and k is a scalar, then adj(kA) is equal to which of the following?

a) k adj A B) k² adj A c) k³ adj A d) 2k adj A

18) If A= 0   1   2 

                1   2   3 

                3   1   1 and its inverse B= [bᵢⱼ] , then the element b₂₃ of matrix B is 

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

19) If A=a₁₁ a₁₂ a₁₃ & B=1  2  3 

               a₂₁ a₂₂ a₂₃         2. 3. 4

               a₃₁ a₃₂ a₃₃         3  4  5 

C= -1  -2 & D= -4  -5  -6

       3   0           0   0   1

       0  -4 

With the relation A= BCD, then the value of a₂₂ is 

a) 40 b) -40 c) -20 d) 20

20) If A= 1  2 & B= 3   8

                3  4          7   2 with the relation 2X+ A= B, then the matrix X is equal to 

a) 2 6 b) 1 -3 c) 1 3 d) 2  -6

    4 -2     2 -1     2 -1     4  -2

21) If A= a  2

                2  a and |A³|= 125, then the value of a is

a) ±2 b) ±2 c) ±5 d) 0

22) If A= |aᵢⱼ| and Aᵢⱼ denotes the Cofactor of aᵢⱼ, then which of the following is not equal to zero ?

a) a₃₁A₁₁ + a₃₂A₁₂+ a₃₃A₁₃

b) a₁₁A₃₁+ a₁₂A₃₂+ a₁₃A₃₃

c) a₂₁A₂₁+ a₂₂A₂₂+ a₂₃A₂₃

d) a₃₁A₂₁+ a₃₂A₂₂+ a₃₃A₂₃

23) A= 1   0    0 

             a   1    0

             b   c    1 find inverse of A 

24) The minors of (-4) and 9 and the Cofactor of (-4) and 9 in matrix 

-1   -2    3

-4   -5   -6

-7    8    9 are respectively 

a) 42,3; -42,3 b) -42,-3; 42,-3  c) 42,3; -42,-3 d) 42,3; 42,3 

25) If A= 0   3 & kA= 0   4a

                4   5          3b  60

Then the values of k, a and b are respectively 

a) 12,9,16 b) 9,12,16 c) 12,9,12 d) 16,12,9

26) B= 1  3 & C= 1   1

             0  1          0  -1 

If the matrix A satisfies the equation BA= C, then which one of the following represents A?

a) 1 4  b) 1 4  c) 1 -4  d) 1 -2

    -1 0      0 -1     1   0      0  -1

27) For any matrix A, if A⁻¹ exists then which of the following is not true?

a) (A⁻¹)⁻¹= A

b) (Aᵀ)⁻¹= (A⁻¹)ᵀ

c) (A²)⁻¹ = (A⁻¹)²

d) |A⁻¹|= |A|⁻¹

28) If A and B are two square matrices of the same order, then (A - B)² is equal to 

a) A²- 2AB+ B²

b) A²- AB - BA + B²

c) A²- 2BA+ B²

d) A²+ 2AB+ B²

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

3   -1+ x   2

3     -1   x+2

x+3 -1     2 

is singular 

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

30) A= 7  1  2 & B= 3 & C= 4

            9   2  1          4          2

                                  5

find the relation AB+ 2C

a) 43 b) 43 c) 45 d) 44

    44      45      44     45

31) If A= 3   4 

                5   7

Then the value of A(adj A) is equal to 

a) I b) |A| c) |A| I d) none

32) If x+ y  2x + z = 4   7

          x -y   2z+ w    0   10 then the values of x, y, z and w are

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

33) The matrix 

2   k.   -4

-1  3     4 

1  -2    -3 is non-singular if

a) k≠ 2 b) k≠ 3 c) k≠ -3 d) k≠ -2

34) If matrix A= 3    2

                            4    5 and

AC= 19     24

         37    46

Then the matrix C is equal to 

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

    5 6      4 6     2  6     6 4

35) Let A= [5] be a matrix of order 1x1. Then adj A is equal to 

a) [1] b) [5]  c) [0] d) 1

                                   5

36) If A²- A + I= 0 then the inverse of matrix A is 

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

37) 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

38) If A= 4  2

               -1  1 then the value of (A - 2I)(A- 3I) is 

a) A b) I c) O d) 4I

39) If A= 1  -1 & B= a  1

                2  -1          b  -1 and (A+ B)²= A²+ B², then the values of a and b are 

a) a=4, b= 1 b) a=1, b= 4 c)  a=9, b= 4 d) a=2, b= 4

40) Let A= a  0 & B= 1 0

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

41) The matrix 0  7  4

                          -7  0 -5

                         -4   5  0 is

a) symmetric b) skew symmetric c) nonsingular d) orthogonal 

42) Let A= 1  -1  1 & B= 4 2 2

                   2   1 -3         -5 0 a

                   1   1 1           1 -2 3

If B is the inverse of the matrix A, then the value of a is

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

43) If A= 0   0   -1

                0  -1    0

               -1   0    0

Then the only correct statement about the matrix A is 

a) inverse of A does not exist 

b) A= (-1) I

c) A is a zero matrix 

d) A²= I

         SECTION : A (80 Marks)

Question 1) (10x2= 20 Marks)

i) If A= 2     3

             4     5 , find Inverse of A.

ii) Show that the function f: R-> R, given by f(x) = | x | is neither one one or onto.

iii) Show sin⁻¹cos sin⁻¹x+ cos⁻¹sin cos⁻¹ x = π/2

iv) If y= tan⁻¹(secx+ tanx), find d²y/dx²

v) ∫ x eˣ dx    

vi) 

vii) Prove without expanding:

a - b     1     a            a      1      b

b - c     1     b    =      b      1      c

c - a     1     c            c       1      a

viii) If x > 1/2, show that the function f(x)= x(4x²-3) is strictly increasing.

ix) Solve 2ˣ⁻ʸ dx + 2ʸ ⁻ˣ dy = 0

x) A and B are two Independent events with P(A)= 2/5 and P(B)= 1/3, Evaluate P(AUB).

Question 2).                                 (4)

Prove: 1+a²-b²       2ab            - 2b

                2ab      1 - a²+b²         2a

                 2b           - 2a       1 - a² - b²

 = (1+a²+b²)³.                              

Question 3).                                   (4)

If tan⁻¹(yz/xr) + tan⁻¹(zx/yr) + tan⁻¹(xy/zr) = π/2 then Prove that, x² + y² + z² = r². 

Question 4).                                    (4)

A bag contains 5 white, 7 red and 3 black balls. If three balls are drawn one by one without replacement. Find the probability that none is red.                       

Question 5).                                 (4)

If y= (tan⁻¹x)², show that (1+x²) d²y/dx² + 2x(1+x²) dy/dx - 2= 0.  

Question 6).                                   (4)

Evaluate ∫ 2⁴ˣ sin 3x dx. 

                   OR

Evaluate:

 ∫ (cosx + x sinx)/{x(x+ cosx) dx

Question 7)                                    (4)

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

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

b) Perpendicular to 5y-15x= 13

                       OR

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

                  

Question 8).                                  (4)

Solve by matrix inversion method

x+2y+z= 7; x + 3z= 11; 2x - 3y=1

                 OR

Show that

a + b+ 2c         a                  b

      c           b+c + 2a           b

      c                 a.            c+a+2b          = 2(a+b+c)³. 

Question 9).                                   (6)

Given x+y= 3, find the maximum and minimum values of 9/x + 36/y 

                 OR

A closed right circular cylinder is has a volume of 2156cm³. What will be the radius of the base so that total surface area is minimum.

Question 10).                                  (4)

 Show that the function f in A= R - {2/3} defined as f(x)=(4x-3)/(6x-4) is one-one and onto .

Question 11).                           2+2=4

 Solve:

a) dy/dx + y secx = tan x. 

b) tan x dy/dx= 1+y² where x= π/2 and y= 1. 

Question 12).                   3+3= 6

a) Evaluate ∫ |sin x| dx at (π/2,-π/2). 

b) Prove ∫ {log(1+x)}/(1+x²) dx at (1,0) = π/8 . Log 2. 

Question 13).                      (3x2= 6)

 a) If x= sint and y= cos pt, p is constant, then find the value of (1-x²) d²y/dx² - x dy/dx.            

b) If m² = p² cos² t + q²sin²t, then show that m+ d²m/dt² = p²q²/m²

Question 14).                  3+3=6

A) It is known that 5 men out of 100 and 25 women out of 1000 are colour blind. A colour blind person is chosen at random. Assuming that males and females are in equal proportion, find the probability of the person to be male.

B) Rajiv and Robin play 12 games of chess. Rajiv wins 6 games, Robin wins 4 games and and 2 games end in a draw. They agree to play 3 more games. Calculate the probability that out of these 3 games, two games end in a draw.

    

                      Or

Evaluate:

A) ∫ x² sin⁻¹x dx

B) ∫x² eᵃˣ dx at (a,0)

            SECTION C.        (20 Marks)

Question 15).                    2+2+2

A) Given demand function x= 50- 0.5 P and cost function C=50+40x, find price for break-even price.

B) 4x+y-10= 0, 2x + 5y -14= 0 are two regression lines. Find the correlation coefficient between variables x and y.

C) The total cost C(x) of a firm is C(x)= 0.0005x³ - 0.7x² - 30x + 3000 where x is the output. Determine:

    a) average cost (AC)

    b) Marginal cost (MC)

Question 16).                               (4)

The two lines of Regression for a distribution (x,y) are 3x+2y= 7 and x+4y= 9. Find the regression coefficient X on Y and Y on X.

                         OR

Treating x as an independent variable. Find the line of best fit for the following date:

X: 15    12       11        14          13

Y: 25    28       24        22          30

Hence, predict the value of y when x= 10.

Question 17)                               (4)

The marginal cost function of manufacturing x units of a commodity is 6+10x - 6x². The total cost of producing one unit of the commodity is ₹ 12. Find the total and average cost functions.

                       OR

If c= 2x{(x+4)/(x+1)} + 6 is the total cost of production of x units of a commodity, show that marginal cost falls continuously as a x increases.

Question 18).                              (6)

A small firm manufacturers gold rings and chains. The combined number of rings and chains manufactured per day is almost 24. It takes one hour to make a ring and half an hour for a chain. The maximum number of hours available per day is 16. If the profit on a ring is ₹300 and on a chain is ₹190, how many of each should be manufactured daily so as to maximize the profit?

MATRIX (2)

1) If A= 1   0   0

              0   1   0

              a    b  -1

Then A² is equal to 

a) a null matrix b) a unit matrix c) - A d) A

2) If A and B are symmetric matrices of the same order, then ABᵀ - BAᵀ is a 

a) skew symmetric matrix 

b) null matrix 

c) symmetric matrix  d) none 

3) If A and B are symmetric matrices of the same order, then AB= A and -BA = B, then B² is equal to 

a) B b) A c) 1 d) 0

4) If AB= A and BA= B, where A and B are square matrices, then 

a) B²= B b) B²≠ B and A²= A c) A²≠ A, B²= B d) A²≠ A, B²≠ B

5) If A and B are two matrices such that AB= B and BA= A, then A²+ B² is equal to 

a) 2AB b) 2BA c) A + B d) AB

6) A= 1  0  0

           0 2  0

           0 0  4 is 

a) Identity matrix 

b) symmetric matrix 

c) Skew symmetric matrix 

d) diagonal matrix 

7) If the matrix AB is zero, then 

a) it is not necessary that either A= O or B= O

b) A= O or B= O

c) A= O and B= O

d) all the above statements are wrong.

8) A= 0  -5  8

          5  0  12

         -8  -12 0  is a

a) diagonal matrix 

b) symmetric matrix 

c) skew symmetric matrix 

d) scalar matrix 

9) If A, B are square matrices of order 3, A is a non singular and AB= O, then B is a 

a) null matrix 

b) singular matrix 

c) unit matrix 

d) non singular matrix 

10) A= n 0 0 & B= a₁ a₂ a₃

            0 n 0          b₁ b₂ b₃

            0 0 n          c₁ c₂ c₃

Then AB is equal to 

a) B b) nB c) Bⁿ d) A+ B

11) If A=2  -1  3 & B= 2. 3

             -4   5   1          4  2

                                     1  5

Then 

a) only AB is defined 

b) only BA is defined 

c) AB and BA both are defined 

d) AB and BA both are not defined 

12) If A=1 2 x & B= 1 -2 y

               0 1 0          0  1 0

               0 0 1          0. 0 1 and AB= I₃, then x+ y is equal to 

a) 0 b) -1 c) 2 d) none 

13) If A= 1  -1 & B= a  1

                2  -1          b -1 and (A+ B)²= A²+ B², then values of a and b are 

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

14) If A= a  b

                c  d is such that A²= I, then 

a) 1+ a²+ bc= 0

b) 1- a²+ bc= 0

c) 1- a²- bc= 0

d) 1+ a²- bc= 0

15) If S= [sᵢⱼ] is a scalar matrix such that sᵢⱼ= k and A is a square matrix of the same order, then AS= SA= ?

a) Aᵏ b) k+ A c) kA d) kS

16) If A is a square matrix such that A²= A, then (I+ A)³- 7A is equal to 

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

17) If a matrix A is both symmetric and skew symmetric, then 

a) A is a diagonal matrix 

b) A is a zero matrix 

c) A is a scalar matrix 

d) A is a square matrix 

18) The matrix 

0  5  -7

-5 0 11

7 -11 0 

is

a) a skew symmetric matrix 

b) a symmetric matrix 

c) a diagonal matrix 

d) an upper triangular matrix 

19) If A is a square matrix, then AA is a 

a) skew symmetric matrix 

b) symmetric matrix 

c) diagonal matrix 

d) none of these 

20) If A and B are symmetric matrices, then ABA is 

a) symmetric matrix 

b) skew symmetric matrix 

c) diagonal matrix 

d) scalar matrix 

21) If A= 5  x

                y  0 and A= Aᵀ, then

a) x=0, y= 5

b) x + y= 5

c) x=y  d) none 

22) If A is 3x4 matrix and B is a matrix such that AᵀB and BAᵀ are both defined. Then, B is of the type 

a) 3x4 b) 3x3 c) 4x4 d) 4x3

23) If A= [aᵢⱼ] is a square matrix of even order that aᵢⱼ= i²- j², then

a) A is a skew symmetric matrix and |A|= 0

b) A is symmetric matrix and |A| is a square 

c) A is symmetric matrix and |A|= 0 d) none 

24) If A and B are square matrices of the same order, then (A+ B)(A- B) is equal to 

a) A²- B² b) A²- BA - AB - B² c) A²- B²+ BA - AB d) A²- BA+ B²+ AB

25) If A= 2   0   -3

               4    3    1

              -5    7    2

is expressed as the sum of a symmetric and skew symmetric matrix, then the symmetric matrix is 

a) 2 2 -4 b) 2 4 -5

    2 3   4     0 3  7

   -4 4   2    -3 1 2 

c) 4 4 -8 d) 1 0 0

    4 6  8      0 1 0

   -8 8 4       0 0 1 

26) Out of the following matrices, choose that matrix which is scalar matrix 

a) 0 0 b) 0 0 0 c) 0 0 d) 0

    0 0      0 0 0     0 0      0 

                             0 0      0 

27) The number of all possible matrices of order 3 x 3 with each entry 0 or 1 is

a) 27 b) 18 c) 81 d) 512

28) Which of the given values of x and y make the following pairs of matrices equal?

a) x=-1/3, y = 7

b) x=-2/3, y = 7

c) x=-1/3, y = -2/5

d) not possible to find 

29) If A=0  2 & kA= 0  3a

               3 -4           2b 24

Then the values of k,a,b are respectively 

a) -6,-12,-18 b) -6, -4,9 c) -6,-4,-9 dx) -6,12,18

30) If matrix A= [aᵢⱼ] 2x2, 

where aᵢⱼ={ 1, if i≠ j

                    0, if i+ j

Then A² is equal to 

a) I b) A c) O d) -I

31) The trace of the matrix 

1  -5  7

0   7  9

11 8  9 is

a) 17 b) 25 x) 3 d) 12

32) If A= [aᵢⱼ] is a scalar matrix of order n x n such that aᵢⱼ= k for all i, then trace of A is equal to 

a) nk b) n+ k c) n/k d) none 

33) A= 0  0  4

            0  4  0

            4  0  0 is a

a) square matrix 

b) diagonal matrix 

c) unit matrix 

d) none 

34) The number of possible matrices of order 3x3 with each entry 2 or 0 is

a) 9 b) 27 c) 81 d) none 

35) If 2x+ y   4x = 7   7y-13

          5x-7     4x    y    x+ 6

Then the value of x, y is

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

36) If A is a square matrix such that A²= I, then (A- I)³+ (A+ I)³- 7A is equal to 

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

37) If A and B are two matrices of order 3x m and 3x n respectively and m= n, then the order of 5A - 2B is

a) mx3 b) 3x3 c) m x n d) 3 x n

38) If A is a matrix of order m x n and B is matrix such that ABᵀ and BᵀA are both defined, then the order of matrix B is 

a) mx n b) nx n c) nx n d) mx n

39) If A and B are matrices of the same order, then ABᵀ - BᵀA is a

a) skew symmetric matrix 

b) null matrix 

c) unit matrix 

d) symmetric matrix 

MATRIX (1)

1) if A= 2   -1

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

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

2) The multiplicative inverse of matrix 

2    1

7    4 is

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

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

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

a) AB= AC doesn't imply B= C

b) A+ B = B+ A

c) (AB)'= B'A'

d) AB= 0 implies A= O or B= O

4) If A= 1  0  2 & Adj A= 5 a -2

             -1  1 -2                 1 1  0

               0 2  1                -2 -2 b

Then the values of a and b are 

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

5) If A= -1   0

               0   2 then the value of A³- A² is equal to 

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

6) If A= - x   - y

                z     t then the transpose of adj A is 

a) t     z b) t    y c) t -z d) none 

    -y - x     -z  -x     y -x

7) If A=3  5 & B= 1  17

             2  0          0  -10 then|AB| is equal to 

a) 80 b) 100 c) -110 d) 92

8) If A= 5.  -2

              3    1 find the inverse of A

9) If A is singular matrix of order n then A. (Adj A) is equal to 

a) a null matrix 

b) a row matrix 

c) a column matrix 

d) none 

10) If A= 3   -5

               -4    2 then find the value of A²- 5A is equal to 

a) I b) 14I c) O d) none 

11) If A= 1  2 & B= 1   2

                2  3          2   1 

                3  4 

Then

a) both AB and BA exist 

b) neither AB nor BA exist 

c) AB exists but BA does not exist 

d) AB does not exist but BA exist 

12) If A= 2  -1 & B= 1 0

                0   1         -1 -1 then+A+ B)² is not equal to 

a) A²+ AB+ BA+ B²

b) A²+ AB+ BA+ B²I

c) A²I + AB+ BA+ B²

d) A²+ 2AB+ B²

13) If A be an n × n matrix and k any scalar then det. kA is equal to 

a) k detA b) nᵏdetA c) kⁿ detA d) kn detA

14) If A= 1    2 

                3   -5 find inverse of A 

15) If A= -1   2 & B= 5

                 2  -1          7 and AX= B, then X is equal to 

a) 19  17 b) 19/3  c)19/3 17/3 d) 19

                     17/3                            17

16) If A≠ O and B≠ O are two 2 x 2 matrices such that AB= O, then which of the following is correct?

a) detA= 0 or detB= 0

b) detA= 0 and detB= 0

c) detA= 0 = detB≠ 0 d) none 

17) If A is a square matrix 3x3 and k is a scalar, then adj(kA) is equal to which of the following?

a) k adj A B) k² adj A c) k³ adj A d) 2k adj A

18) If A= 0   1   2 

                1   2   3 

                3   1   1 and its inverse B= [bᵢⱼ] , then the element b₂₃ of matrix B is 

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

19) If A=a₁₁ a₁₂ a₁₃ & B=1  2  3 

               a₂₁ a₂₂ a₂₃         2. 3. 4

               a₃₁ a₃₂ a₃₃         3  4  5 

C= -1  -2 & D= -4  -5  -6

       3   0           0   0   1

       0  -4 

With the relation A= BCD, then the value of a₂₂ is 

a) 40 b) -40 c) -20 d) 20

20) If A= 1  2 & B= 3   8

                3  4          7   2 with the relation 2X+ A= B, then the matrix X is equal to 

a) 2 6 b) 1 -3 c) 1 3 d) 2  -6

    4 -2     2 -1     2 -1     4  -2

21) If A= a  2

                2  a and |A³|= 125, then the value of a is

a) ±2 b) ±2 c) ±5 d) 0

22) If A= |aᵢⱼ| and Aᵢⱼ denotes the Cofactor of aᵢⱼ, then which of the following is not equal to zero ?

a) a₃₁A₁₁ + a₃₂A₁₂+ a₃₃A₁₃

b) a₁₁A₃₁+ a₁₂A₃₂+ a₁₃A₃₃

c) a₂₁A₂₁+ a₂₂A₂₂+ a₂₃A₂₃

d) a₃₁A₂₁+ a₃₂A₂₂+ a₃₃A₂₃

23) A= 1   0    0 

             a   1    0

             b   c    1 find inverse of A 

24) The minors of (-4) and 9 and the Cofactor of (-4) and 9 in matrix 

-1   -2    3

-4   -5   -6

-7    8    9 are respectively 

a) 42,3; -42,3 b) -42,-3; 42,-3  c) 42,3; -42,-3 d) 42,3; 42,3 

25) If A= 0   3 & kA= 0   4a

                4   5          3b  60

Then the values of k, a and b are respectively 

a) 12,9,16 b) 9,12,16 c) 12,9,12 d) 16,12,9

26) B= 1  3 & C= 1   1

             0  1          0  -1 

If the matrix A satisfies the equation BA= C, then which one of the following represents A?

a) 1 4  b) 1 4  c) 1 -4  d) 1 -2

    -1 0      0 -1     1   0      0  -1

27) For any matrix A, if A⁻¹ exists then which of the following is not true?

a) (A⁻¹)⁻¹= A

b) (Aᵀ)⁻¹= (A⁻¹)ᵀ

c) (A²)⁻¹ = (A⁻¹)²

d) |A⁻¹|= |A|⁻¹

28) If A and B are two square matrices of the same order, then (A - B)² is equal to 

a) A²- 2AB+ B²

b) A²- AB - BA + B²

c) A²- 2BA+ B²

d) A²+ 2AB+ B²

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

3   -1+ x   2

3     -1   x+2

x+3 -1     2 

is singular 

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

30) A= 7  1  2 & B= 3 & C= 4

            9   2  1          4          2

                                  5

find the relation AB+ 2C

a) 43 b) 43 c) 45 d) 44

    44      45      44     45

31) If A= 3   4 

                5   7

Then the value of A(adj A) is equal to 

a) I b) |A| c) |A| I d) none

32) If x+ y  2x + z = 4   7

          x -y   2z+ w    0   10 then the values of x, y, z and w are

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

33) The matrix 

2   k.   -4

-1  3     4 

1  -2    -3 is non-singular if

a) k≠ 2 b) k≠ 3 c) k≠ -3 d) k≠ -2

34) If matrix A= 3    2

                            4    5 and

AC= 19     24

         37    46

Then the matrix C is equal to 

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

    5 6      4 6     2  6     6 4

35) Let A= [5] be a matrix of order 1x1. Then adj A is equal to 

a) [1] b) [5]  c) [0] d) 1

                                   5

36) If A²- A + I= 0 then the inverse of matrix A is 

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

37) 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

38) If A= 4  2

               -1  1 then the value of (A - 2I)(A- 3I) is 

a) A b) I c) O d) 4I

39) If A= 1  -1 & B= a  1

                2  -1          b  -1 and (A+ B)²= A²+ B², then the values of a and b are 

a) a=4, b= 1 b) a=1, b= 4 c)  a=9, b= 4 d) a=2, b= 4

40) Let A= a  0 & B= 1 0

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

41) The matrix 0  7  4

                          -7  0 -5

                         -4   5  0 is

a) symmetric b) skew symmetric c) nonsingular d) orthogonal 

42) Let A= 1  -1  1 & B= 4 2 2

                   2   1 -3         -5 0 a

                   1   1 1           1 -2 3

If B is the inverse of the matrix A, then the value of a is

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

43) If A= 0   0   -1

                0  -1    0

               -1   0    0

Then the only correct statement about the matrix A is 

a) inverse of A does not exist 

b) A= (-1) I

c) A is a zero matrix 

d) A²= I

TEST- 10/7/26

1) Prove: cot⁻¹(1/2) - (1/2) cot⁻¹(4/3) =π/4.

2) y= 2 sin{(x -2)/√6 - √(2+ 4x - x²)}, find dy/dx at x= 2 

3) Let y= (sin⁻x)²+ (cosx)², find dy/dx  

4) If (a+ bx)eʸ/ˣ= x, find the value of (x dy/dx - y)².

5) Three normals are drawn from the point (14,7) to the curve y² - 16x - 8y =0. Find the coordinates of the feet of the normal.

6) For what values of x the function y= 2 sinx + cos2x(0< x ≤2π) attains maximum and minimum values.

7) d/dx (logₑ(ax)ˣ), where a is a constant), is equal to 

a) 1 b) logₑ(ax) c) 1/a d) logₑ(ax) +1

8) Find the value of a if f(x)= a |sin x|+ 2x is continuous at x=0.

H. W -1

SAP- 1

1) If A= 2   0   3 & B= 1   2    3 

              1   2   0          2   1    4 find A+ B

2) If A= 1  2   3 & B= 1   2

              4  5   6          3   4

              6  8   9          5   6 find A+ B

3) If A= 0    2   3 & B= 7    8    3

              2    1  4           1    4    3 find 

a) 2A + 3B

b) 3A - B 

c) AB

4) If A= 1   3 

              3   4 and A²- kA - 5=0, then find k.

5) If A= x  y  z & B= a  h  g & C= x

                                  h  b  f          y

                                  g  f   c         z

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

6) If A= 1   0 & B= 0     1 

              0   1         -1     0

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

7) If A= 1   0   0 & B= x₁  y₁  z₁

              0   1   0          x₂  y₂  z₂

              0   0   1          x₃  y₃  z₃

Then show that AB= BA = B

Sap-2 (H. W)

1) If A= 1  -1 & B= a   1 

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

2) If A= 1    -1

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

3) Find the inverse of A 

A= 1   -3     2

      2   5     -1

     -3   1      4

4) A= 20    10

          10     20 find inverse of A 

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

SAP-1

1) If A= 2    -1

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

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

2) The multiplicative inverse of 

2     1

7     4 is

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

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

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

a) AB= AC does not imply B= C

b) A+ B= B+ A

c) (AB)'= B'A'

d) AB= O implies A= O or B= O.     

4) If A=1. 0  2 & Adj A= 5  a  -2

            -1  1 -2                 1  1  0

             0   2  1                -2 -2  b

Then the values of a and b are 

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

5) If A= -1   0

               0   2 then the value of A³- A² is 

a) I b) A c) 2A d) 2I.     

6) If A= -x   - y

               z     t, then transpose of adj A

a.) t  z b) t  y c) t  -z d) none 

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

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

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

8) The inverse of 5   -2

                                 3    1

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

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

       

Mixed b/c/p/m

MIXED CHEMISTRY 

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

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

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

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

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

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

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

3) Fill up the blanks with appropriate words.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

b) State the factor which affect

a) electro affinity 

b) electro negative of elements in a periodic table.

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

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

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

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

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

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

a) NaCl

b) HCl

c) NH₃ gas conduct electricity.

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

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

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

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

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

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

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

15) state reason why zinc is used in used in 

a) galvanization 

b) dry cells

c) alloys 

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

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

18) convert hydrochloric acid to nascent chlorine.

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

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

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

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

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

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

25) State the colour of 

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

f) Zinc blend is roasted in air. Calculate 

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

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

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

2C₆H₆ + 7O₂ --> 4CO₂ + 6H₂O.        350 lits

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

₂₂₂₃₂₂₂₄₂₂₄₄₄₄₂₄₂₄₃₂₃₃₂₂₄₂₂₄₂₂₂₂₃₂₄ ⁻₂₅₂₂₃₂₂₂₂₃₂₂₂₃₂₂₄₄₂₄₄₂₂₂₂₇₂₄₂₂₃₂₅₂₂₇₂₂₇₃ ₆ ₄₂₂₂₆₆₅₁₀₃₆₆₂₂₂₄₂₂₂₂

MATHS

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

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

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

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

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

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

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

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

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

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

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

a) Find his dividend income from the shares.

b) Find his total investment in the shares.

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

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

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

13) A={ x: -1< x ≤ 5, x belongs to R}

       B={x: -4≤ x < 3, x belongs to R}

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

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

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

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

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

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

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

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

21) Solve: √(6x -5) - √(3x -2)= 2.

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

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

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

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

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

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

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

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

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

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

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

a) plot the points on the graph paper.

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

c) Are the two triangles congruent?

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

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

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

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

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

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

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

36) Using properties of proportion, solve for x:

(2x²+ 6x)/(12x²+ 4)= 682/364.

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

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

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

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

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

42) If A= (k    3 & B= (2  & C= 1

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

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

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

44) If A= 4     1

                -1    2 show that A²- 6A + 9I= 0.

45) If A= -3    2 & B= 1     a

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

46) If A=2    -1 & B= 2 

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

47) If A= 1     1

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

48) If A= 2   5 & B= 4     -1

                0   1           2     0 and equation 

 5A+ 3B + 2X= 0      42

                          4       9, find the matrix X.

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

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

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

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

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

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

55) find the y-intercept and slope of 

a) 3y - 6x= 2

b) 2x - 3y = 6.

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

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

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

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

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

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

a) the midpoint of AC 

b) the slope of AC 

c) the equation of the diagonal BD.

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

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

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

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

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

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

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

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

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

66) From the adjoining diagram 

a) Show BM/AL = MC/LQ

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

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

a) QM= QR

b) ∆ LSR= ∆ ||gm PQRS 

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

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

a) QX/XS

b) QX/QS

c) QY/YS

d) QY/QS

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

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

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

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

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

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

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

76) Prove:

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

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

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

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

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

f) If 6 sin²x + 2 cos²x = 3 find x.

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

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

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

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

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

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

Mass (kg)       frequency 

44-47               13

48-51                27

52-55                43

56-59                36

60-63                23

80) Calculate the mean for the following distribution:

Class         frequency 

10-16           1

16-22           10

22-28            5

28-34            3

35-40            6

81) Find mean for the following distribution 

Marks       students 

05-10         5

10-15         6

15-20         15

20-25         10

25-30          5

30-35          4

35-40          3

40-45          2

82) Following are scores made by a cricketer:

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

Calculate a) the mean b) the median score 

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

Class       frequency 

0-2            17

3-5            22

6-8            29

9-11          18

12-14         9

15-17         5

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

84) Following are the marks of 100 candidates 

Marks    candidate

00-10       5

10-20      10

20-30      22

30-40      40

40-50      15

50-60       8

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

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

What percentage of candidates get less than 20 marks.

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

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

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

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