TEST PAPER -5 (2025)
Time: 3 hrs: Max. Marks: 80
GENERAL INSTRUCTIONS
1. This question papers contains 5 section A, B, C, D and E. Each Section is compulsory. However, there are internal choices in some questions .
2. Section A has 18 MCQ's and 02 Assertion -Reason based questions of 1 mark each .
3. Section B has 5 very Short Answer (VSA) type questions of 2 marks each.
4. Section C has 6 Short Answer(SA) type questions of 3 marks each.
5. Section D has 4 Long Answer (LA) type questions of 5 marks each.
6. Section E has 3 source based/case/passage based/integrated units of assessment (4 marks each) with sub-parts.
• SECTION - A
(Multiple Choice Questions) Each question carries 1 mark
1) The direction angles of line l are
a) α, β, γ
b) -α, -β, -γ
c) π -α, π-β, π-γ d) none
2) ∫ log tanx dx at (π/2,0) is equals to
a) 1 b) -1 c) 0 d) 2
3) If tan⁻¹(x²+ y²)= a, then dy/dx is equals to
a) x/y b) -x/y c) y/x d) -y/x
4) If sin⁻¹x = y, then
a) 0≤ y ≤ x
b) -π/2 ≤ y ≤ π/2
c) 0< y < π
d) -π/2 < y < π/2
5) ∫ eᵅˣ {af(x)+ f'(x)} dx is equals to
a) eᵅˣ f(x)+ C
b) aeᵅˣ f(x)+ C
c) eᵅˣ f'(x)+ C
d) eˣ f(ax)+ C
6) The degree of the differential equation (d²y/dx²)² + (dy/dx)²= x sin(dy/dx) is
a) 1 b) 2 c) not defined d) 3
7) If A= a b
c d then determinants of A is written as
a) a b b) a c c) a c d) a b
c d d b b d d c
8) The matrix 2x +y 4x= 7 7y - 13
5x - 7 4x y x + 6 then the value of x + y is
a) 1 b) 2 c) 4 d) 5
9) The area bounded by y= - x²+ 2x +3 and y=0 is
a) 32 sq units
b) 32/3 sq units
c) 1/32 sq units
d)1/3 sq units
10) The function f(x)= x³- 6x²+ 12x - 18 is
a) strictly increasing function
b) increasing function
c) decreasing functions
d) strictly decreasing function
11) if the relation R defined on the set A={1, 2, 3,4, 5, 6} is R={(a,b); b= a+1}, then R is
a) reflexive
b) reflexible and symmetric
c) not reflexible
d) reflexive but not run transitive
12) For Matrix A= 1 -2
3 5 , (A') A is equals to
a)10 13 b) 10 13 c) 13 29 d) 1 10
13 29 29 13 10 13 1 10
13) The projection of the vector 7i + j - 4k on 2i + 6j + 3k is
a) 7/8 b) 8/7 c) 1/7 d) 1/8
14) If A is a matrix of order 3 x 3 such that |A|= 5, then |A (adj A)| is equal to
a) 25 b) 125 c) 5 d) 1/125
15) Solve (2y -1)dx - (2x +3) dy =0
a) (2x +3)(2y -1)= C
b) (2y -1)(2x +3)= C
c) (y -1)(2x +3)= C
d) (2y -1)(x +3)= C
16) If y= (1+ x¹⁾⁶)(1+ x¹⁾³)(1- x¹⁾⁶), then dy/dx at x=1 is equal to
a) 2/3 b) -2/3 c) 3 d) -4/3
17) ∫ sin²(x/2) dx is equals to
a) (x - sinx)/2+ C
b) (x + sinx)/2+ C
c) (sinx -x)/2+ C
d) (x/2 + sinx) + C
18) If a= i + j + 2k and b = 3i + 2j - k, then the value of (a+ 3b). (2a - b) is
a) 15 b) 5 c) -15 d) 10
Assertion -Reason Based Questions
In the following questions , a statement of Assertion (A) is followed by 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 A.
c) A is true but R is false.
d) A is false but R is true.
19) If R is the relation in the set A ={1, 2, 3, 4, 5 } given by R={(a,b): |a - b| is even }.
Assertion (A): R is an equivalence relation.
Reason (R): All elements of {1,3,5} are related to all elements of {2,4}.
20) Assertion (A): If A= 2 3 -1
1 4 2
and B= 2 3
4 5
2 1 then AB and BA both are defined.
Reason (R): For the two matrices A and B, the product AB is defined , if number of columns in A is equal to the number of rows in B.
SECTION B
This Section compromises of very short answer type questions (VSA) of 2 marks each)
21) Find the cartesian and vector equation for the line passing through the points A(-1,1,2) and B(2,4,5).
OR
The x-cordinate of a point on the line joining the points P(2,2,1) and Q(5,1,-2) is 4. Find its x-cordinate.
22) Find the domain of the function f(x)= cos⁻¹x + sin⁻¹2x.
23) If y= x¹⁾ˣ, find dy/dx
OR
If x = a(θ + sinθ), y= a(1- cosθ), find dy/dx.
24) Evaluate ∫ (x⅖-1)/(x²+4) dx.
25) Find the values of λ and μ, for which (2i + 6j + 27k) x (i + λj + μk)= 0.
SECTION C
(This Section compromises of short answer type questions (SA) of 3 marks each)
26) Show that for a≥ 1,
f(x)= √3 sinx - cosx - 2ax + b is decreasing in R.
27) Find the particular solution of the differential equation
eˣ tany dx + (2- eˣ) sec²y dy = 0, given that y=π/4 when x=0.
OR
Find the particular solution of the differential equation
(3xy + y²) dx + (x²+ xy) dy =0, for x= 1, y= 1.
28) Evaluate ∫ (1+ x²)/(1+ x⁴) dx.
OR
Evaluate ∫ x(logx)² dx.
29) If A={1, 2,3,.....9} and R is the relation in Ax A defined by (a,b) R(c,d), if a + d = b + c for (a,b), (c,d) in A x A.
Prove that R is an equivalence relation. Also, obtain the equivalence class [(2,5)].
OR
If f: X --> Y is a function. Define a relation R on X given by R ={(a,b): f(a)= f(b)}. Show that R is an equivalence relation on X.
30) Find the shortest distance between the lines whose vector equations are
r= i + j + λ(2i - j + k) and 2i + j - k + μ(3i - 5j + 2k).
31) Find the area of the region bounded by the line y= 3x +2, the x-axis and the ordinates x =-1 and x =1.
SECTION D
(This Section compromises of long answer type questions (LA) of 5 marks each)
32) For any two vectors a and b, show that (1+ |a|²) (1+ |b|²)= {(1- a.b)}²+ |a+ b + (ax b)|².
33) Solve the following system of equations by metrix method, where x ≠ 0, y≠ 0 and z≠ 0.
2/x - 3/y + 3/z =10, 1/x + 1/y + 1/z =10, and 3/x - 1/y + 2/z =13.
OR
Determine the product of
-4 4 4 & 1 -1 1
-7 1 3 1 -2 -2
5 -3 -1 2 1 3 and then use to solve the system equations x - y + z = 4, x - 2y - 2z = 9 and 2x + y + 3z =1.
34) Solve the following LPP maximize
Z= 12x + 16y subject to constraints,
x + y ≤ 1200,
y≤ x/2,
x ≤ 3y + 600, x, y ≥ 0
OR
Solve the LPP maximize Z= 40x + 50y
Subject to constraints
3x + y ≤9, x + 2y ≤ 8, x, y ≥ 0
35) Evaluate ∫ (x +1)/{x(1+ xeˣ)²} dx.
SECTION E
(This Section compromises of 3 case study/passage based questions of 4 marks each)
36) Consider the given equation dy/dx + Py = Q.
The above equation is known as linear differential equation, Here, IF = ₑ∫P dx and solution is given by y. IF = ∫ (Q. IF) dx + C. Now, consider the given equation
(1+ sinx) dy/dx + y cosx + x =0.
On the basis of above information, answer the following questions .
a) Find the value of P and Q.
b) Find IF.
c) Find the general solution of the given equation.
OR
If y(1+ sinx)= -x²/2 + C and y(0)=1, then find y and y(π/2).
37) A random variable X has the following probability distribution
x: 0 1 2 3 4 5 6 7 8
P(x): a 3a 5a 7a 9a 11a 13a 15a 17a
On the basis of above information, answer the following questions.
a) Find the value of a.
b) Find P(X= 4).
c) Find P(X>5) and P(0≤ X ≤ 2).
OR
Find (1≤ x ≤4) and P(3<x ≤6).
38) An electronic assembly consists of two subsystem say A and B as shown below.
From previous testing procedures , the following probabilities are assumed to be known P(A fails)= 0.2, P(B fails alone)= 0.15, P(A and B fail)= 0.15.
On the basis of above information, answer the following questions .
a) Find the probability P(B fails) and probability P(A fails alone).
b) Find the probability P(whole system fail) and the probability of P(A fails/B has failed ).
1c 2c 3b 4b 5a 6c 7a 8d 9b 10b 11c 12a 13b 14b 15b 16b 17a 18c 19c 20a
21) (x +1)/1= (y -1)/1= (z -2)/1; r= - i + j + 2k + λ(i + j+ k) or -1
22) (-1/2,1/2)
23) x¹⁾ˣ{(1- logx)/x²} or tan(θ/2)
24) x - (5/2) tan⁻¹(x/2)+ C
25) λ = 3 and μ = 27/2
27) tany = 2- eˣ or |y²+ 2xy|= 3/x²
28) (1/√2) tan⁻¹{(x²-1)/√(2x)}+ C or (x²/2) (logx)² - (x²/2) logx + x²/4+ C
29) {(1,4),(2,5),(3,6),(4,7),(5,8),(6,9)} 30) 10/√59
31) 13/3 sq units
33) x=1/2, y = 1/3 and z = 1/5 or
8 0 0
0 8 0
0 0 8 x=3, y=-2, z= -1
34) max= 16000 at (800,400) or min= 230 at (2,3)
35) log(xeˣ) - log(1+ xeˣ) + 1/(1+ xeˣ) + C
36) P= cosx/(1+ sinx) , Q= -x/(1+ sinx)
1+ sinx
y(1+ sinx)= -x²/2 + C or (8- π²)/16
37) 1/81
1/9
5/9, 1/9 or 8/27, 11/27
38) 0.30 and 0.05
0.35 and 0.5
μ λ μ ⁻¹
TEST PAPER -4
θ μ λ μ
TEST PAPER -4
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) lim ₓ→₀ (log cos x)/sin²x
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?
TEST PAPER - 1
Time : 3 hours. Full Marks: 100
Group A:
a)
b) Fill in the gaps: The value of the determinant
3 1975 1978
4 1982 1986 is ______
5 1995 2000
c) State whether the following statement is true or false:
" The product of two non-zero matrices be a non-zero matrix.
d) If y= logₑlogₑx, x > 1 which one is true
i) x dy/dx = 1
ii) (xlogₑx) dy/dx = 1
iii) (logₑx) dy/dx= 1
iv) (logₑx) dy/dx = x
e) If x = a(θ - sinθ) and y= a(1+ cosθ) then which one of the following is the value of dy/dx.
i) - cot(θ/2) ii) - cotθ iii) -tan(θ/2) iv) cot(θ/2)
f) If y= cos²x, then which one of the following is the value of d²y/dx² ?
i) - 2cos2x ii) 2 cos2x iii) - 2sin2x iv) cos2x
g) ∫ tan²dx = ______
h) ²₀∫ dx/(x²+4) =π/16. T/F
i) ∫ sin⁵x dx at (π/2, -π/2)= _____
j) The gradient of the tangent at the point (8,-4) to the parabola y²= 8(x - 6) is -1. T/F
2) a) Examine whether AB= BA for the two matrices A= 1 5 & B= 0 0
1 3 0 1. (2)
b) If y= (Secx)ᵗᵃⁿˣ , then find dy/dx. (2)
3) If x³+ y³= 2xy, then find the value of dy/dx at the point (1,1). (2)
4) Evaluate: ∫ 2sinx/(5+ 3 cosx) dx
Or
Evaluate: ∫ xe²ˣ dx
5) What are the order and degree of the following differential equation?(d²y/dx²)² + 5(dy/dx)³+ 2y =0.
) An integer is chosen at random from 100 integers 1,2,3,.....,100. What is the probability that the selected integer is divisible by 5 or 7? (4)
) ᵗᵗ⁻¹ˣʸ³⁺²ˣ⁻¹ˣˣ⁻ˣˣ³₂
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