This section comprises of **20 multiple choice questions (MCQs)** of 1 mark each.
- If \(A=\begin{bmatrix}-1&0&0\\ 0&1&0\\ 0&0&1\end{bmatrix},\) then \(A^{-1}\) is
- \(\begin{bmatrix}-1&0&0\\ 0&-1&0\\ 0&0&-1\end{bmatrix}\)
- \(\begin{bmatrix}1&0&0\\ 0&-1&0\\ 0&0&-1\end{bmatrix}\)
- \(\begin{bmatrix}-1&0&0\\ 0&-1&0\\ 0&0&1\end{bmatrix}\)
- \(\begin{bmatrix}-1&0&0\\ 0&1&0\\ 0&0&1\end{bmatrix}\)
- If vector \(\vec{a}=3\hat{i}+2\hat{j}-\hat{k}\) and vector \(\vec{b}=\hat{i}-\hat{j}+\hat{k}\) then which of the following is correct?
- \(\vec{a}||\vec{b}\)
- \(\vec{a}\perp\vec{b}\)
- \(|\vec{b}|>|\vec{a}|\)
- \(|\vec{a}|=|\vec{b}|\)
- \(\int_{-1}^{1}\frac{|x|}{x} dx\), \(x\ne0\) is equal to
- -1
- 0
- 1
- 2
- Which of the following is not a homogeneous function of x and y ?
- \(y^{2}-xy\)
- \(x-3y\)
- \(\sin^{2}\frac{y}{x}+\frac{y}{x}\)
- \(\tan x-\sec y\)
- If \(f(x)=|x|+|x-1|\), then which of the following is correct?
- \(f(x)\) is both continuous and differentiable, at \(x=0\) and \(x=1\)
- \(f(x)\) is differentiable but not continuous, at \(x=0\) and \(x=1\)
- \(f(x)\) is continuous but not differentiable, at \(x=0\) and \(x=1.\)
- \(f(x)\) is neither continuous nor differentiable, at \(x=0\) and \(x=1\).
- If A is a square matrix of order 2 such that \(\det (A)=4,\) then \(\det (4 \text{ adj } A)\) is equal to:
- 16
- 64
- 256
- 512
- If E and F are two independent events such that \(P(E)=\frac{2}{3},\) \(P(F)=\frac{3}{7},\) then \(P(E/\overline{F})\) is equal to:
- \(\frac{1}{6}\)
- \(\frac{1}{2}\)
- \(\frac{2}{3}\)
- \(\frac{7}{9}\)
- The absolute maximum value of function \(f(x) = x^{3}-3x + 2\) in \([0, 2]\) is:
- 0
- 2
- 4
- 5
- Let \(A=\begin{bmatrix}1&-2&-1\\ 0&4&-1\\ -3&2&1\end{bmatrix},\) \(B=\begin{bmatrix}-2\\ -5\\ -7\end{bmatrix},\) \(C=[9 \ 8 \ 7]\) which of the following is defined?
- Only AB
- Only AC
- Only BA
- All AB, AC and BA
- If \(\int\frac{2^{\frac{1}{x}}}{x^{2}}dx=k\cdot2^{\frac{1}{x}}+C,\) then k is equal to
- \(\frac{-1}{\log 2}\)
- \(-\log 2\)
- -1
- \(\frac{1}{2}\)
- If \(\vec{a}+\vec{b}+\vec{c}=0,\) \(|\vec{a}|=\sqrt{37},\) \(|\vec{b}|=3\) and \(|\vec{c}|=4,\) then angle between \(\vec{b}\) and \(\vec{c}\) is
- \(\frac{\pi}{6}\)
- \(\frac{\pi}{4}\)
- \(\frac{\pi}{3}\)
- \(\frac{\pi}{2}\)
- The integrating factor of differential equation \((x+2y^{3})\frac{dy}{dx}=2y\) is
- \(e^{\frac{y^{2}}{2}}\)
- \(\frac{1}{\sqrt{y}}\)
- \(\frac{1}{y^{2}}\)
- \(e^{-\frac{1}{y^{2}}}\)
- If \(A=\begin{bmatrix}7&0&x\\ 0&7&0\\ 0&0&y\end{bmatrix}.\) is a scalar matrix, then \(y^{x}\) is equal to
- 0
- 1
- 7
- \(\pm7\)
- The corner points of the feasible region in graphical representation of a L.P.P. are \((2, 72)\), \((15, 20)\) and \((40, 15)\). If \(Z=18x+9y\) be the objective function, then
- Z is maximum at \((2, 72)\), minimum at \((15, 20)\)
- Z is maximum at \((15, 20)\) minimum at \((40, 15)\)
- Z is maximum at \((40, 15)\), minimum at \((15, 20)\)
- Z is maximum at \((40, 15)\), minimum at \((2, 72)\)
- If A and B are invertible matrices, then which of the following is not correct?
- \((A+B)^{-1}=B^{-1}+A^{-1}\)
- \((AB)^{-1}=B^{-1}A^{-1}\)
- \(\text{adj} (A)=|A|A^{-1}\)
- \(|A|^{-1}=|A^{-1}|\)
- If the feasible region of a linear programming problem with objective function \(Z=ax+by\) is bounded, then which of the following is correct?
- It will only have a maximum value.
- It will only have a minimum value.
- It will have both maximum and minimum values.
- It will have neither maximum nor minimum value.
- The area of the shaded region bounded by the curves \(y^{2}=x\), \(x=4\) and the x-axis is given by
- \(\int_{0}^{4} x dx\)
- \(\int_{0}^{2} y^{2} dy\)
- \(2\int_{0}^{4}\sqrt{x}dx\)
- \(\int_{0}^{x}\sqrt{x}dx\)
- The graph of a trigonometric function is as shown in the original document (figure showing a \(\sin x\) or \(\cos x\) type wave from \(-\pi/2\) to \(\pi/2\)). Which of the following will represent graph of its inverse?
- [Graph A]
- [Graph B]
- [Graph C]
- [Graph D]
-
Assertion (A): Let Z be the set of integers. A function \(f:Z\rightarrow Z\) defined as \(f(x)=3x-5,\) \(\forall x\in Z\) is a bijective.
Reason (R): A function is a bijective if it is both surjective and injective.
- Both Assertion (A) and Reason (R) are true and the Reason (R) is the correct explanation of the Assertion (A).
- Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of the Assertion (A).
- Assertion (A) is true, but Reason (R) is false.
- Assertion (A) is false, but Reason (R) is true.
-
Assertion (A): \(f(x)=\begin{cases}3x-8,&x\le5\\ 2k&x>5\end{cases}\) is continuous at \(x=5\) for \(k=\frac{5}{2}\).
Reason (R): For a function \(f\) to be continuous at \(x=a\), \(\lim_{x\rightarrow a^{-}}f(x)=\lim_{x\rightarrow a^{+}}f(x)=f(a).\)
- Both Assertion (A) and Reason (R) are true and the Reason (R) is the correct explanation of the Assertion (A).
- Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of the Assertion (A).
- Assertion (A) is true, but Reason (R) is false.
- Assertion (A) is false, but Reason (R) is true.
This section comprises of **5 Very Short Answer (VSA)** type questions of 2 marks each.
-
(a) Differentiate \(2^{\cos^{2}x}\) w.r.t \(\cos^{2}x.\)
OR
(b) If \(\tan^{-1}(x^{2}+y^{2})=a^{2}\), then find \(\frac{dy}{dx}.\)
- Evaluate : \(\tan^{-1}[2\sin(2\cos^{-1}\frac{\sqrt{3}}{2})]\)
- The diagonals of a parallelogram are given by \(\vec{a}=2\hat{i}-\hat{j}+\hat{k}\) and \(\vec{b}=\hat{i}+3\hat{j}-\hat{k}\). Find the area of the parallelogram.
- Find the intervals in which function \(f(x)=5x^{\frac{3}{2}}-3x^{\frac{5}{2}}\) is (i) increasing (ii) decreasing.
-
(a) Two friends while flying kites from different locations, find the strings of their kites crossing each other. The strings can be represented by vectors \(\vec{a}=3\hat{i}+\hat{j}+2\hat{k}\) and \(\vec{b}=2\hat{i}-2\hat{j}+4\hat{k}.\) Determine the angle formed between the kite strings. Assume there is no slack in the strings.
OR
(b) Find a vector of magnitude 21 units in the direction opposite to that of \(\vec{AB}\) where A and B are the points \(A(2,1,3)\) and \(B(8,-1,0)\) respectively.
This section comprises of **6 Short Answer (SA)** type questions of 3 marks each.
- The side of an equilateral triangle is increasing at the rate of \(3\text{ cm/s}.\) At what rate its area increasing when the side of the triangle is \(15\text{ cm}\)?
-
Solve the following **linear programming problem graphically**:
Minimise \(Z=x+2y\)
subject to the constraints:
\begin{align*} x-y &\ge 0 \\ x-2y &\ge -2 \\ x &\ge 0 \\ y &\ge 0 \end{align*}
-
(a) Find: \(\int\frac{x+\sin x}{1+\cos x}dx\)
OR
(b) Evaluate : \(\int_{0}^{\frac{\pi}{4}}\frac{dx}{\cos^{3}x\sqrt{2\sin 2x}}\)
-
(a) Verify that lines given by \(\vec{r}=(1-\lambda)\hat{i}+(\lambda-2)\hat{j}+(3-2\lambda)\hat{k}\) and \(\vec{r}=(\mu+1)\hat{i}+(2\mu-1)\hat{j}-(2\mu+1)\hat{k}\) are skew lines. Hence, find shortest distance between the lines.
OR
(b) During a cricket match, the position of the bowler, the wicket keeper and the leg slip fielder are in a line given by \(\vec{B}=2\hat{i}+8\hat{j}\), \(\vec{W}=6\hat{i}+12\hat{j}\) and \(\vec{F}=12\hat{i}+18\hat{j}\) respectively. Calculate the ratio in which the wicketkeeper divides the line segment joining the bowler and the leg slip fielder.
-
(a) The probability distribution for the number of students being absent in a class on a Saturday is as follows:
| \(X\) | 0 | 2 | 4 | 5 |
|---|
| \(P(X)\) | \(p\) | \(2p\) | \(3p\) | \(p\) |
Where X is the number of students absent.
- Calculate \(p\). (1 mark)
- Calculate the mean of the number of absent students on Saturday. (2 marks)
OR
(b) For the vacancy advertised in the newspaper, 3000 candidates submitted their applications. From the data it was revealed that two third of the total applicants were females and other were males. The selection for the job was done through a written test. The performance of the applicants indicates that the probability of a male getting a distinction in written test is 0.4 and that a female getting a distinction is 0.35. Find the probability that the candidate chosen at random will have a distinction in the written test.
- Sketch the graph of \(y=|x+3|\) and find the area of the region enclosed by the curve, x-axis, between \(x=-6\) and \(x=0\), using integration.
This section comprises of **4 Long Answer (LA)** type questions of 5 marks each.
-
(a) If \(\sqrt{1-x^{2}}+\sqrt{1-y^{2}}=a(x-y).\) then prove that \(\frac{dy}{dx}=\sqrt{\frac{1-y^{2}}{1-x^{2}}}.\)
OR
(b) If \(x=a(\cos\theta+\log\tan\frac{\theta}{2})\) and \(y=\sin\theta\), then find \(\frac{d^{2}y}{dx^{2}}\) at \(\theta=\frac{\pi}{4}.\)
- Find the absolute maximum and absolute minimum of function \(f(x)=2x^{3}-15x^{2}+36x+1\) on \([1, 5]\).
-
(a) Find the image A' of the point \(A(1,6,3)\) in the line \(\frac{x+5}{1}=\frac{y+3}{4}=\frac{z-6}{-9}\). Also, find the equation of the line joining A and A'.
OR
(b) Find a point P on the line \(\frac{x}{1}=\frac{y-1}{2}=\frac{z-2}{3}\) such that its distance from point \(Q(2,4,-1)\) is 7 units. Also, find the equation of line joining P and Q.
-
A school wants to allocate students into three clubs **Sports (x), Music (y) and Drama (z)**, under following conditions:
- The number of students in Sports club should be equal to the sum of the number of students in Music and Drama club. (\(x = y + z\))
- The number of students in Music club should be 20 more than half the number of students in Sports club. (\(y = \frac{1}{2}x + 20\))
- The total number of students to be allocated in all three clubs are 180. (\(x + y + z = 180\))
Find the number of students allocated to different clubs, using **matrix method**.
This section comprises of **3 case study based questions** of 4 marks each.
-
Case Study 1: Solar Panel Installation ☀️
A technical company is designing a rectangular solar panel installation on a roof using 300 metres of boundary material. The design includes a partition running parallel to one of the sides dividing the area (roof) into two sections. Let the length of the side perpendicular to the partition be \(x\) metres and with parallel to the partition be \(y\) metres.
- Write the equation for the total boundary material used in the boundary and parallel to the partition in terms of \(x\) and \(y\). (1 mark)
- Write the area of the solar panel as a function of \(x\). (1 mark)
-
(a) Find the critical points of the area function. Use second derivative test to determine critical points at the maximum area. Also, find the maximum area. (2 marks)
OR
(b) Using first derivative test, calculate the maximum area the company can enclose with the 300 metres of boundary material, considering the parallel partition. (2 marks)
-
Case Study 2: Relations in a Classroom 🧑🏫
A class-room teacher is keen to assess the learning of her students the concept of "relations" taught to them. She writes the following five relations each defined on the set \(A=\{1,2,3\}\):
- \(R_{1}=\{(2,3),(3,2)\}\)
- \(R_{2}=\{(1,2),(1,3),(3,2)\}\)
- \(R_{3}=\{(1,2),(2,1),(1,1)\}\)
- \(R_{4}=\{(1, 1), (1, 2), (3, 3), (2, 2)\}\)
- \(R_{5}=\{(1, 1), (1, 2), (3, 3), (2, 2), (2, 1), (2, 3), (3, 2)\}\)
The students are asked to answer the following questions about the above relations:
- Identify the relation which is reflexive, transitive but not symmetric. (1 mark)
- Identify the relation which is reflexive and symmetric but not transitive. (1 mark)
-
(a) Identify the relations which are symmetric but neither reflexive nor transitive. (2 marks)
OR
(b) What pairs should be added to the relation \(R_{2}\) to make it an equivalence relation? (2 marks)
-
Case Study 3: Traffic Flow Analysis 🚦
An engineer is modeling traffic flow at a major intersection. The number of vehicles passing the intersection at time \(t\) minutes after 8:00 AM is modeled by a function, say \(V(t)\).
- Define the general problem of finding the rate of change of traffic flow. (1 mark)
- If the rate of flow is given by a differential equation \(\frac{dV}{dt} = k(C - V)\), where C is the maximum capacity and k is a constant, identify the order and degree of this differential equation. (1 mark)
-
(a) If the differential equation of traffic flow is \(\frac{dy}{dx} = \frac{x^2+y^2}{xy}\) (homogeneous), find the solution. (2 marks)
OR
(b) Find the general solution of the differential equation \(\frac{dy}{dx} + \frac{y}{x} = x^2\). (2 marks)