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
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