Arts (English Medium)
Science (English Medium)
Commerce (English Medium)
Academic Year: 20222023
Date: March 2023
Duration: 3h
General Instructions :
 This Question paper contains  five sections A, B, C, D, and E. Each section is compulsory. However, there are internal choices in some questions.
 Section A has 18 MCQs and 02 AssertionReason based questions of 1 mark each.
 Section B has 5 Very Short Answer (VSA)type questions of 2 marks each.
 Section C has 6 Short Answer (SA)type questions of 3 marks each.
 Section D has 4 Long Answer (LA)type questions of 5 marks each.
 Section E has 3 source based/case based/passage based/integrated units of assessment (4 marks each) with subparts.
If A = [a_{ij}] is a skewsymmetric matrix of order n, then ______.
`a_(ij) = 1/(a_(ji)) ∀ i, j`
`a_(ij) ≠ 0 ∀ i, j`
`a_(ij) = 0, where i = j`
`a_(ij) ≠ 0 where i = j`
Chapter: [0.03] Matrices
If A is a square matrix of order 3, A′ = −3, then AA′ = ______.
9
−9
3
−3
Chapter: [0.04] Determinants
The area of a triangle with vertices A, B, C is given by ______.
`vec("AB") xx vec("AC")`
`1/2vec("AB") xx vec("AC")`
`1/4vec("AC") xx vec("AB")`
`1/8vec("AC") xx vec("AB")`
Chapter: [0.04] Determinants
The value of ‘k’ for which the function f(x) = `{{:((1  cos4x)/(8x^2)",", if x ≠ 0),(k",", if x = 0):}` is continuous at x = 0 is ______.
0
–1
1
2
Chapter: [0.05] Continuity and Differentiability
If f'(x) = `x + 1/x`, then f(x) is ______.
`x^2 + log x + C`
`x^2/2 + log x + C`
`x/2 + log x + C`
`x/2  log x + C`
Chapter: [0.07] Integrals
If m and n, respectively, are the order and the degree of the differential equation `d/(dx) [((dy)/(dx))]^4` = 0, then m + n = ______.
1
2
3
4
Chapter: [0.09] Differential Equations
The solution set of the inequality 3x + 5y < 4 is ______.
an open halfplane not containing the origin.
an open halfplane containing the origin.
the whole XYplane not containing the line 3x + 5y = 4.
a closed halfplane containing the origin.
Chapter: [0.12] Linear Programming
The scalar projection of the vector `3hati  hatj  2hatk` on the vector `hati + 2hatj  3hatk` is ______.
`7/sqrt(14)`
`7/14`
`6/13`
`7/2`
Chapter: [0.1] Vectors
The value of `int_2^3 x/(x^2 + 1)`dx is ______.
`log 4`
`log 3/2`
`1/2 log2`
`log 9/4`
Chapter: [0.07] Integrals
If A, B are nonsingular square matrices of the same order, then (AB^{–1})^{–1} = ______.
A^{–1}B
A^{–1}B^{–1}
BA^{–1}
AB
Chapter: [0.03] Matrices
The corner points of the shaded unbounded feasible region of an LPP are (0, 4), (0.6, 1.6) and (3, 0) as shown in the figure. The minimum value of the objective function Z = 4x + 6y occurs at ______.
(0.6, 1.6) only
(3, 0) only
(0.6, 1.6) and (3, 0) only
at every point of the linesegment joining the points (0.6, 1.6) and (3, 0)
Chapter: [0.12] Linear Programming
If `(2, 4),(5, 1) = (2x, 4),(6, x)`, then the possible value(s) of ‘x’ is/are ______.
3
`sqrt(3)`
`sqrt(3)`
`sqrt(3), sqrt(3)`
Chapter: [0.04] Determinants
If A is a square matrix of order 3 and A = 5, then adj A = ______.
5
25
125
`1/5`
Chapter: [0.04] Determinants
Given two independent events A and B such that P(A) = 0.3, P(B) = 0.6 and P(A' ∩ B') is ______.
0.9
0.18
0.28
0.1
Chapter: [0.13] Probability
The general solution of the differential equation y dx – x dy = 0 is ______.
xy = C
x = Cy^{2}
y = Cx
y = Cx^{2}
Chapter: [0.09] Differential Equations
If y = sin^{–1}x, then (1 – x^{2})y_{2} is equal to ______.
xy_{1}
xy
xy_{2}
x^{2}
Chapter: [0.05] Continuity and Differentiability
If two vectors `veca` and `vecb` are such that `veca` = 2, `vecb` = 3 and `veca.vecb` = 4, then `veca  2vecb` is equal to ______.
`sqrt(2)`
`2sqrt(6)`
24
`2sqrt(2)`
Chapter: [0.1] Vectors
P is a point on the line joining the points A(0, 5, −2) and B(3, −1, 2). If the xcoordinate of P is 6, then its zcoordinate is ______.
10
6
–6
–10
Chapter: [0.11] Three  Dimensional Geometry
Assertion (A): The domain of the function sec^{–1}2x is `(∞,  1/2] ∪ pi/2, ∞)`
Reason (R): sec^{–1}(–2) = ` pi/4`
Both A and R are true and R is the correct explanation of A.
Both A and R are true but R is not the correct explanation of A.
A is true but R is false.
A is false but R is true.
Chapter: [0.02] Inverse Trigonometric Functions
Assertion (A): The acute angle between the line `barr = hati + hatj + 2hatk + λ(hati  hatj)` and the xaxis is `π/4`
Reason(R): The acute angle 𝜃 between the lines `barr = x_1hati + y_1hatj + z_1hatk + λ(a_1hati + b_1hatj + c_1hatk)` and `barr = x_2hati + y_2hatj + z_2hatk + μ(a_2hati + b_2hatj + c_2hatk)` is given by cosθ = `(a_1a_2 + b_1b_2 + c_1c_2)/sqrt(a_1^2 + b_1^2 + c_1^2 sqrt(a_2^2 + b_2^2 + c_2^2)`
Both A and R are true and R is the correct explanation of A.
Both A and R are true but R is not the correct explanation of A.
A is true but R is false.
A is false but R is true.
Chapter: [0.11] Three  Dimensional Geometry
Find the value of `sin^1 [sin((13π)/7)]`
Chapter: [0.02] Inverse Trigonometric Functions
Prove that the function f is surjective, where f: N → N such that `f(n) = {{:((n + 1)/2",", if "n is odd"),(n/2",", if "n is even"):}` Is the function injective? Justify your answer.
Chapter: [0.01] Relations and Functions
A man 1.6 m tall walks at the rate of 0.3 m/sec away from a street light that is 4 m above the ground. At what rate is the tip of his shadow moving? At what rate is his shadow lengthening?
Chapter: [0.06] Applications of Derivatives
If `veca = hati  hatj + 7hatk` and `vecb = 5hati  hatj + λhatk`, then find the value of λ so that the vectors `veca + vecb` and `veca  vecb` are orthogonal.
Chapter: [0.1] Vectors
Find the direction ratio and direction cosines of a line parallel to the line whose equations are 6x − 12 = 3y + 9 = 2z − 2
Chapter: [0.1] Vectors
If `ysqrt(1  x^2) + xsqrt(1  y^2)` = 1, then prove that `(dy)/(dx) =  sqrt((1  y^2)/(1  x^2))`
Chapter: [0.05] Continuity and Differentiability
Find `vecx` if `(vecx  veca).(vecx + veca)` = 12, where `veca` is a unit vector.
Chapter: [0.1] Vectors
Find: `int (dx)/sqrt(3  2x  x^2)`
Chapter: [0.07] Integrals
Three friends go for coffee. They decide who will pay the bill, by each tossing a coin and then letting the “odd person” pay. There is no odd person if all three tosses produce the same result. If there is no odd person in the first round, they make a second round of tosses and they continue to do so until there is an odd person. What is the probability that exactly three rounds of tosses are made?
Chapter: [0.13] Probability
Find the mean number of defective items in a sample of two items drawn onebyone without replacement from an urn containing 6 items, which include 2 defective items. Assume that the items are identical in shape and size.
Chapter: [0.13] Probability
Evaluate: `int_(pi/6)^(pi/3) (dx)/(1 + sqrt(tanx)`
Chapter: [0.07] Integrals
Evaluate the following integral:
Chapter: [0.07] Integrals
Solve the differential equation: y dx + (x – y^{2})dy = 0
Chapter: [0.09] Differential Equations
Solve the differential equation: xdy – ydx = `sqrt(x^2 + y^2)dx`
Chapter: [0.09] Differential Equations
Solve the following Linear Programming Problem graphically:
Maximize Z = 400x + 300y subject to x + y ≤ 200, x ≤ 40, x ≥ 20, y ≥ 0
Chapter: [0.12] Linear Programming
Evaluate the following integral:
Chapter: [0.07] Integrals
Make a rough sketch of the region {(x, y): 0 ≤ y ≤ x^{2}, 0 ≤ y ≤ x, 0 ≤ x ≤ 2} and find the area of the region using integration.
Chapter: [0.08] Applications of the Integrals
Define the relation R in the set N × N as follows:
For (a, b), (c, d) ∈ N × N, (a, b) R (c, d) if ad = bc. Prove that R is an equivalence relation in N × N.
Chapter: [0.01] Relations and Functions
Given a nonempty set X, define the relation R in P(X) as follows:
For A, B ∈ P(X), (4, B) ∈ R iff A ⊂ B. Prove that R is reflexive, transitive and not symmetric.
Chapter: [0.01] Relations and Functions
An insect is crawling along the line `barr = 6hati + 2hatj + 2hatk + λ(hati  2hatj + 2hatk)` and another insect is crawling along the line `barr =  4hati  hatk + μ(3hati  2hatj  2hatk)`. At what points on the lines should they reach so that the distance between them s the shortest? Find the shortest possible distance between them.
Chapter: [0.11] Three  Dimensional Geometry
The equations of motion of a rocket are:
x = 2t,y = –4t, z = 4t, where the time t is given in seconds, and the coordinates of a ‘moving point in km. What is the path of the rocket? At what distances will the rocket be from the starting point O(0, 0, 0) and from the following line in 10 seconds? `vecr = 20hati  10hatj + 40hatk + μ(10hati  20hatj + 10hatk)`
Chapter: [0.11] Three  Dimensional Geometry
If A = `[(2, 3, 5),(3, 2, 4),(1, 1, 2)]`, find A^{–1}. Use A^{–1} to solve the following system of equations 2x − 3y + 5z = 11, 3x + 2y – 4z = –5, x + y – 2z = –3
Chapter: [0.04] Determinants
Read the following passage and answer the questions given below.

 Is the function differentiable in the interval (0, 12)? Justify your answer.
 If 6 is the critical point of the function, then find the value of the constant m.
 Find the intervals in which the function is strictly increasing/strictly decreasing.
OR
Find the points of local maximum/local minimum, if any, in the interval (0, 12) as well as the points of absolute maximum/absolute minimum in the interval [0, 12]. Also, find the corresponding local maximum/local minimum and the absolute ‘maximum/absolute minimum values of the function.
Chapter: [0.06] Applications of Derivatives
Read the following passage and answer the questions given below.
In an elliptical sport field the authority wants to design a rectangular soccer field with the maximum possible area. The sport field is given by the graph of `x^2/a^2 + y^2/b^2` = 1. 
 If the length and the breadth of the rectangular field be 2x and 2y respectively, then find the area function in terms of x.
 Find the critical point of the function.
 Use First derivative Test to find the length 2x and width 2y of the soccer field (in terms of a and b) that maximize its area.
OR
Use Second Derivative Test to find the length 2x and width 2y of the soccer field (in terms of a and b) that maximize its area.
Chapter: [0.06] Applications of Derivatives
Read the following passage and answer the questions given below.
There are two antiaircraft guns, named as A and B. The probabilities that the shell fired from them hits an airplane are 0.3 and 0.2 respectively. Both of them fired one shell at an airplane at the same time. 
 What is the probability that the shell fired from exactly one of them hit the plane?
 If it is known that the shell fired from exactly one of them hit the plane, then what is the probability that it was fired from B?
Chapter: [0.13] Probability
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