CONTINUITY
1) 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.
2) The number of points of discontinuity of f defined by f(x)= |x | - |x +1| is
a) 1 b) 2 c) 0 d) 5.
3) Show that the function
f(x)= (sinx)/x + cosx, if x≠ 0
2, if x=0 is continuous at x=0.
4) If f(x)= {√(4 +x 0) -2}/x, x ≠ 0 be continuous at x= 0, then 4f(0) is equal to
a) 1/2 b) 1/4 c) 1 d) 3/2.
5) If the function f defined as
f(x)= (x²-9)/(x-3) , x≠ 3
k x=3 is continuous at x= 3, then the value of k is
a) 1 b) 2 c) 6 d) 5
6) The function
f(x)= (k cosx)/(π- 2x), if x≠ π/2
3. if x=π/2 is continuous at x=π/2, when k equals
a) -6 b) 6 c) 5 d) -5.
7) Determine f(0), so that the function f(x) defined by
f(x)= (4ˣ -1)³/{sin(x/4) log(1+ x²/3)} becomes continuous at x=0.
8) Find the value of a, for which the function
f(x)= {√(1+ ax) - √(1- ax)}/x, if -1≤x <0
(2x+1)/(x-1), if 0≤ x < 1 is continuous at x=0.
9) The function f(x)= xⁿ, where n is a positive integer, is
a) continuous at x = n
b) limit of f at x= n does not exist
c) limit of f at x= n exists and not equal to f(x)
d) none of the above.
10) If f(x)= λ(x²-2x), if x≤ 0
4x +1, if x> 0 then which one of the following is correct.
a) f(x) is continuous at x= 0 for any value of λ
b) f(x) is discontinuous at x= 0 for any value of λ
c) f(x) is continuous at x= 1 for any value of λ
d) none.
11) The function= (4- x²)/(4x - x³) is
a) discontinuous at only one point
b) discontinuous at exactly two points
c) discontinuous at exactly three points
d) none.
12) The function defined by
f(x)= x+5, if x≤ 1
x -5, if x> 1 is discontinuous at
a) x= 0 b) x=1 c) x=2 d) none
13) The function= (9- x²)/(9x - x³) is
a) discontinuous at only one point
b) discontinuous at exactly two points
c) discontinuous at exactly three points
d) none.
14) Examine the continuity of
f(x)=(logx - log2)/(x -2), x> 2
1/2, x= 2 at x=2
2{(x -2)/(x²-4)}, x < 2
15) f(x)= 1, if x≠ 0
2, if x= 0 is not continuous at
a) x= 0 b) x= 1 c) x= -1 d) none
16) If f(x)= 2x and g(x)= (x²/2) +1, then which of the following can be discontinuous
a) f(x) + g(x)
b) f(x) - g(x)
c) f(x) . g(x)
d) g(x)/f(x).
17) Find the value/s of k. So that the following functions is continuous at x=0
f(x)= (1- cos kx)/(x sinx), if x≠ 0
1/2, if x=0.
18) All the points of discontinuity of the function f defined by
f(x)= 3, if 0≤ x ≤ 1
4, if 1< x<3
5, if 3≤ x ≤10 are
a) 1,3 b) 3,10 c) 1,3,10 d) 0,1,3.
19) (1- cos4x)/x², x < 0
If f(x)= a, x= 0
√x/√{16+ √x -4}, x> 0 and f is continuous at x= 0, then find the value of a.
20) The function
f(x)= (sin3x)/x, x≠ 0
k/2, x=0 continuous at x=0, then the value of k is
a) 2 b) 4 c) 6 d) 8
21) Examine the continuity of a function
f(x)= |x| cos(1/x), if x≠ 0
0, if x=0
at x=0.
LINEAR PROGRAMMING
1) The feasible region corresponding to the linear constraints of a Linear Programming Problem is given below.
Which of the following is not a constraint to the given Linear Programming Problem?
a) x+ y ≥ 2 b) x+ 2y ≤ 10 c) x- y ≥ 1 d) x1 y ≤ 1.
2) The corner points of the bounded feasible region determined by a system of linear constraints are (0, 3), (1,1 0) and (3,0). Let Z= px + qy, where p,q > 0. The condition on p and q, so that the minimum of Z occurs at (3,0) and (1,1).
a) p=2q b) p= q/2 c) p= 3q d) p= q.
3) Solve the following Linear Programming Problems graphically,
Minimise Z = x + 2y
Subject to constraints: x + 2y≥ 100; 2x - y≤ 0, 2x + y ≤ 200, x, y≥ 0.
4) Solve the following Linear Programming Problems graphically,
Maximise Z = - x + 2y
Subject to constraints: x ≥ 3, x + y≥ 5, x + y ≥ 6, y≥ 0.
5) The number of feasible solutions of the linear programming problem given as Maximize Z= 15 x + 30y, subject to constraints : 3x + y≤ 12, x + 2y≤ 10, x≥ 0, y≥ 0 is
a) 1 b) 2 c) 3 d) infinity.
6) The feasible region of a linear programming problem is shown in the figure below
Which of the following are the possible constraints?
a) x+ 2y≥ 4, x + 2y≤ 3, x ≥ 0, y≥ 0.
b) x+ 2y≤ 4, x + y≤ 3, x ≥ 0, y≥ 0.
c) x+ 2y≥ 4, x + 2y≥ 3, x ≥ 0, y≥ 0.
d) x+ 2y≥ 4, x + 2y≥ 3, x ≤ 0, y≤ 0.
7) Solve the following Linear Programming Problems graphically,
Minimise Z = 500x + 409y
Subject to constraints: x + y≤ 200; x ≥ 20, y ≥ 4x, y≥ 0.
8) The graph of the ineuality 2x + 3y > 6 is
a) half plane that contains the origin.
b) half plane that neither contains the origin not the points of the line 2x + 3=6
c) Whole XOY plane excluding the points on the line 2x + 3y = 6
d) entire XOY plane.
9) In an LPP , if the objective function has Z= ax+ by has the same maximum value on two corner points of the feasible region, then the number of points at which Zₘₐₓ occurs is
a) 0 b) 2 c) finite d) infinite.
10) If Z= 2x + 3y, subject to constraints x. + 2y ≤ 10, 2x + y≤ 14, x, y≥ 0, then find the corner points of feasible region.
11) Solve the following LPP graphically:
Maximize Z= 4x + 6y,
Subject to constraints x + y≤ 8, x, y ≥0.
12) Find the minimum value of Z, where Z= 2x + 3y, subject to constraints 2x + y≥ 23; x + y≤ 24, x, y≥ 0.
13) The feasible solution for LPP is shown below and the objective function is Z= 15 x - 4y.
Based on the above information, answer the following questions.
a) Find the value of (n -1)², where n is number of corners points.
b) Find Z₆ ,₁/₂ + Z₀,₂.
c) Find the co-ordinate point of D.
d) find the maximum of Z.
14) The feasible region for an LPP shown in the following figure.
Let F= 3x - 4y be the objective function. Maximum value of F is
a) 0 b) 8 c) 12 d) -18.
15) if feasible solution of a LPP is given as follows :
And the objective function is Z= 10500x + 9000y.
On the basis of above information, answer the following question.
a) if n is the number of Corner Points. then find the value of (n + 2)³.
b) find the value of Z₀ , ₁.
c) Find the point where objective function is maximum.
d) Evaluate Z₂₀ , ₂₀ - Z₀₀ , ₁₀.
16) Solve the following Linear Programming Problems graphically,
Minimise Z = -50x + 20y
Subject to constraints: 2x - y≥ -5; 3x +y≥ 3, 2x -3y ≤ 12, x, y≥ 0.
17) Solve the following Linear Programming Problems graphically,
Minimise Z = 2x + 5y
Subject to constraints: 2x + 4y≤ 8; 3x + y≤ 6, 2x + y ≤ 4, x, y≥ 0.
18) Solve the following Linear Programming Problems graphically,
Minimise Z = 5x + 10y
Subject to constraints: x + 2y≤ 120; x +y≥ 60, x - 2y ≥ 0, x, y≥ 0.
19) Solve the following Linear Programming Problems graphically,
Minimise Z = 3x + 5y
Subject to constraints: 3x -4y≥ -12; 2x - y≥ -2, 2x + 3y ≥ 12, 0 ≤ x ≤ 4, y≥ 2.
INCREASING AND DECREASING
1) Find the intervals in which the function f: R---> R defined by f(x)= xeˣ is increasing.
2) If f(x)= a(tanx - cotx), where a> 0, then find f(x) is increasing or decreasing function in its domain.
3) Show that the function f defined by f(x)= (x -1) eˣ +1 is an increasing function for all x > 0.
4) The function f given by f(x)= 3x +17, is
a) strictly increasing on R
b) strictly decreasing on R
c) decreasing on R
d) both (b) and (c).
5) Find the intervals on which the function f(x)= (x -1)³(x -2)² is strictly increasing and strictly decreasing.
6) Which of the following statements is true for f(x)= 4x³- 6x²- 72x + 30?
I) f is increasing in the interval of (-∞,-2)
II) f is strictly increasing in the interval (3, ∞).
III) f is strictly decreasing in the interval (-2,3)
IV) f is neither increasing nor decreasing in R.
a) I and II are true
b) II and III are true
c) II and IV are true
d) All are true
7) Show that the function f(x)= (x³- 6x²+ 12x -18) is an increasing function on R.
8) Show that the function f given by f(x)= log cosx is strictly decreasing function for x ∈ (0, π/2).
9) If y= x(x -3)² decreases for the values of x given by
a) 1< x <3 b) x< 0 c) x>0 d) 0< x < 3/2.
10) The interval in which y= x² e⁻ˣ is
a) (-∞, ∞) b) (-2, 0) c) (2, ∞) d) (0,2).
11) Find the intervals in which the function f(x)= 20- 9x + 6x²- x³ is
a) Strictly increasing.
b) strictly decreasing.
12) If I be any interval disjoint from [-1,1], then the function f given by f(x)=x + 1/x is
a) strictly decreasing on I
b) strictly increasing on I
c) decreasing on I
d) Only (a) and (c) are true.
13) The least value of a, such that the function f given by f(x)=x²+ ax + 1 is strictly increasing on (1,2) is
a) -1 b) -2 c) 0 d) 1.
14) The function f(x)= 4 sin³x - 6 sin²x +12 sinx +100 is strictly
a) increasing in {π, 3π/2)
b) decreasing in (π/2,π)
c) decreasing in [-π/2, π/2]
d) decreasing in [0, π/2].
15) Show that the function f(x)= x/3 + 3/x decreases in the intervals (-3,0) U (0,3).
16) The function f(x)= sinx + cosx x belongs to [0, π/4] is
a) increasing function b) strictly increasing function c) decreasing function d) strictly decreasing function.
17) Show that y= log(1+x) - 2x/(2+ x), x > -1 is an increasing function of x, throughout its domain.
18) Find the interval in which the function f given by f(x)= tanx - 4x, x belongs to (0,π/2) is
a) strictly increasing.
b) strictly decreasing.
METRIX
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
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
3) The value of |A|, if
A= 0 2x -1 √x
1-2x -2√x 2 √x
-√x -2√x 0 where x ∈ R⁺ , is
a) (2x+1)² b) 0 c) (2x +1)³ d) none
4) 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
5) Solve: 2/x + 3/y + 10/z =4; 4/x - 6/y + 5/z =1, 6/x + 9/y - 20/z =2.
6) 1 4 x
If A= z 2 y
-3 -1 3 is a symmetric matrix, then the value of x + y + z is
a) 10 b) 6 c) 8 d) 0
7) If A. (adj A)= 3 0 0
0 3 0
0 0 3 then the value of|A|+ |adj A| is equal to
a) 12 b) 9 c) 3 d) 27
8) A and B are skew-symmetric matrices of same order. AB is symmetric, if
a) AB=0 b) AB= - BA c) AB= BA d) AB=0
9) 1 0 2
If A= 0 2 1
2 0 3 then show that A³- 6A²+ 7A + 2I =0.
10) If A= 3 2
5 -7 then find A⁻¹.
11) If A is a square matrix of order 3, such that A(adj A)= 10I, then|adj A| is equal to
a) 1 b) 10 c) 100 d) 101
12) If A is a 3x3 Matrix such that |A|= 8, then |3A| equals
a) 8 b) 24 c) 72 d) 216
13) 2 -3 5
If A= 3 2 -4
1 1 -2 then find A⁻¹. Using A⁻¹, solve the following system of equations 2x - 3y + 5z = 11; 3x + 2y - 4z = -5; x +y -2z = -3.
14) x+3 z+4 2y-7 0 6 3y-2
If -6 a-1 0 = -6 -3 2c+2
b-3 -21 0 2b+4 -21 0
Then values of x,y,z,a,b,c are
a) x=-3, y=-5, z=2, a= -2, b= -7, c= -1
b) x=-2, y=-7, z=-1, a= -3, b= -5, c= 2
c) x=-3, y=-5, z=2, a= 2, b= 7, c= 1
d) x=3, y=5, z=2, a= 2, b= 7, c= 1.
15) If matrix A given by
A= 1. -1
0 3
2 5 then the order of the matrix A is
a) 1x2 b) 2x3 c) 3x2 d) 2x2
16) If A= 4 2
-1 1 show that (A'- 2I)(A - 3I)= O
17) 2 0 1
If A= 2 1 3
1 -1 0 then find the value of A²- 5A.
18) In the following questions, a statement of Assertion(A) is followed by a statement of Reason (R). Choose the correct answer out of the following choices.
a) Both A and R are true and R is the correct explanation of A
b) both A and R are true but R is not the correct explanation of A
c) A is true but R is false
d) A is false but R is true
Assertion (A) The matrix
A= 3 -1 0
3/2 3 √2. 1
4 3 -1 is rectangular Matrix of order 3.
Reason (R): If A=[aᵢⱼ]ₘₓ₁ , then A is column matrix.
19) Solve the following system of equations by matrix method when x≠ 0, y≠ 0 and z≠ 0.
2/x - 3/y + 3/z = 10
1/x + 1/y + 1/z = 10
3/x - 1/y + 2/z = 13.
20) The sum of 3 number is 6. Twice the third number when added to the first number gives 7. On adding the sum of the second and third numbers to thrice the first number, we get 12. Find the numbers, using Matrix method.
21) if A=[aᵢⱼ] be a man matrix, then the matrix obtained by interchanging the rows and columns of A is called the transpose of A.
A square matrix A=[aᵢⱼ] is said to be symmetric , If Aᵀ= A for all possible values of i and j.
A square metrix A=[aᵢⱼ] is said to be skew-symmetric, if Aᵀ = - A for all possible values of i and j.
Based on the above information, answer the following questions.
i) find the transpose of [1 -2 -5].
ii) find the transpose a matrix (ABC). iii) Evaluate (A+ B)ᵀ - A, where
A= 0 1 & B= 1 2
2 -1 3 4
22) Evaluate (AB)ᵀ, where
A= 1 1 & B= 3 2
0 1 1 4
23) 5 0 4 1 3 3
A= 2 3 2 1 4 3
1 2 1 1 3 4 Compute (AB)⁻¹
24) -4 4 4 1 -1 1
A= -7 1 3 & B= 1 -2 -2
5 -3 -1 2 1 3 Determine the product of AB and then use to solve the system of equations x - y + z= 4, x - 2y - 2z =9 and 2x + y +3z =1.
25) 1 -1 0 2 2 -4
A= 2 3 4 & B=-4 2 -4
0 1 2 2 -1 5 then find AB. Use this to solve the system of equations x - y = 3, 2 x +3y +4z =17 and y +2z =7.
26) 1 0 -2
If A= -2 -1 2
3 4 1 show that A³- A²- 3A - I= O.
27) If a Matrix has 8 elements , then which of the following will not be a possible order of the matrix ?
a) 1x8 b) 2x4 c) 4x2 d) 4x4
28) For what values of K, The Matrix
2- K 4
-5 1 is not invertible ?
a) 8 b) 17 c) 22 d) 25
29) If A= 2 3
-4 -6 then which of the following is true?
a) A(adj A)≠ |A| I
b) A(adj A)≠ (adjA)A
c) A(adj A)= (adj A)A=|A| I = O
d) none
30) Let A= 2 3
-1 2 and f(x)= x²- 4x +7.
Show that f(A)= O. Use this result to find A⁵.
31) If A= (2 -3 4) & B= 3
2
2
& X= (1 2 3) and Y= 2
3
4
Then AB+ XY is equal to
a) [28] b) [24] c) 28 d) 24
32) Find the inverse of the matrix
1 0 0
0 cosx sinx
0 sinx - cosx
33) If x +y 7 = 2 7
9 x- y 9 4 then xy is equal to
a) 1 b) 2 c) -3 d) -5
34) If A+ B= 1 0 & A - 2B= -1 1
1 1 0 -1 then matrix A is
a) 1/3 1/3 b) 1/3 2/3
2/3 1/3 1/3 1/3
c) 1/2 3/2 d) 1 1
5/2 5/2 1/3 2/3
35) If A= 4 6 & B= 3 -6
-3 7 5 -8 with the relation X+ A= B, then matrix X is
a) -1 -12 b) 1 -12 c) -1 -1 d) 7 0
8 -15 -8 -15 0 0 2 -1
36) A= 3 -4
1 -1 then A⁻¹ is equal
a) -1 4 b) 1 4 c) 1 -4 d) 1 4
-1 3 -1 3 1 -3 3 -1
37) In the following questions , a statement Assertion (A) is followed by the statement of Reason (R). Choose the correct answer out of the following choices.
a) Both A and R are true and R is the correct explanation of A
b) both A and R are true but R is not the correct explanation of A
c) A is true but R is false
d) A is false but R is true
Assertion (A)
If A= 3 1
-5 x then (-A) is given by
-3 -1
5 -x
Reason (R): The negative of a Matrix is given by - A and is defined as -A = (-1) A.
38) if A is a matrix of order 3 such that A(adj A)= 10I. Then the value of |adj A| is
a) 10 b) 100 c) 110 d) 5
39) if A is skew-symmetric matrix, than A² is
a) symmetric matrix
b) skew-symmetric matrix
c) null matrix d) none.
40) Find A⁻¹ if
A= 0 1 1
1 0 1
1 1 0 and show that A⁻¹ = (A²- 3I)/2
41) if a matrix A is both symmetric and skew-symmetric, then A is
a) null matrix
b) identity Matrix
c) diagonal Matrix
d) none of these
42) If A is symmetric matrix, then B'A is
a) symmetric matrix
b) skew-symmetric matrix
c) scalar matrix d) none
43) If A= 2 3
-1 2 then find A²- 4A + I
44) If A= 1 0 & B= 1 0
-1 7 0 1 then find K so that A²= 8A + KI.
45) If A⁻¹ = 2 3 & B= -1 0
1 2 1 2, then value of [A+ 2B]⁻¹ is
a) -4 1 b) -4 5 c) -4 5 d) -4 1
5 6 1 6 6 1 2 4
46) The non-zero values of x satisfying the matrix equation xA+ 2B = 2C where
A= 2x 2 & B= 8 5x & C= x²+8 24
3 x 4. 4x 10 6x is
a) 1 b) 2 c) 3 d) 4
