This section comprises of **18 multiple choice questions (MCQs)** (Q1 to Q18) and **2 Assertion-Reason based questions** (Q19, Q20) of 1 mark each.
- If the sum of all the elements of a \(3\times3\) scalar matrix is 9, then the product of all its elements is:
- 0
- 9
- 27
- 729
- Let \(f:R_{+}\rightarrow[-5,\infty)\) be defined as \(f(x)=9x^{2}+6x-5\), where \(R_{+}\) is the set of all non-negative real numbers. Then, f is:
- one-one
- onto
- bijective
- neither one-one nor onto
- If \(\begin{vmatrix}-a&b&c\\ a&-b&c\\ a&b&-c\end{vmatrix}= kabc,\) then the value of k is:
- 0
- 1
- 2
- 4
- The number of points of discontinuity of
\(f(x)=\begin{cases}|x|+3,&if~x\le-3\\ -2x,&if-3\lt x \lt 3\\ 6x+2,&if~x\ge 3\end{cases}\)
- 0
- 1
- 2
- infinite
- The function \(f(x)=x^{3}-3x^{2}+12x-18\) is:
- strictly decreasing on R
- strictly increasing on R
- neither strictly increasing nor strictly decreasing on R
- strictly decreasing on \((-\infty, 0)\)
- \(\int_{0}^{\pi/2}\frac{\sin~x-\cos~x}{1+\sin~x~\cos~x}dx\) is equal to:
- \(\pi\)
- Zero (0)
- \(\int_{0}^{\pi/2}\frac{2~\sin~x}{1+\sin~x~\cos~x}dx\)
- \(\frac{\pi^{2}}{4}\)
- The differential equation \(\frac{dy}{dx}=F(x,y)\) will not be a homogeneous differential equation, if \(F(x,y)\) is :
- \(\cos~x-\sin(\frac{y}{x})\)
- \(\frac{y}{x}\)
- \(\frac{x^{2}+y^{2}}{xy}\)
- \(\cos^{2}(\frac{x}{y})\)
- For any two vectors \(\vec{a}\) and \(\vec{b}\), which of the following statements is always true?
- \(\vec{a}.\vec{b}\ge|\vec{a}||\vec{b}|\)
- \(\vec{a}.\vec{b}=|\vec{a}||\vec{b}|\)
- \(\vec{a}.\vec{b}\le|\vec{a}||\vec{b}|\)
- \(\vec{a}.\vec{b}<|\vec{a}||\vec{b}|\)
- The coordinates of the foot of the perpendicular drawn from the point \((0, 1, 2)\) on the x-axis are given by:
- \((1,0,0)\)
- \((2,0,0)\)
- \((\sqrt{5},0,0)\)
- \((0,0,0)\)
- The common region determined by all the constraints of a linear programming problem is called :
- an unbounded region
- an optimal region
- a bounded region
- a feasible region
- Let E be an event of a sample space S of an experiment, then \(P(S|E)=\)
- \(P(S\cap E)\)
- \(P(E)\)
- 1
- 0
- If \(A=[a_{ij}]\) be a \(3\times3\) matrix, where \(a_{ij}=i-3j\), then which of the following is false ?
- \(a_{11}<0\)
- \(a_{12}+a_{21}=-6\)
- \(a_{13}>a_{31}\)
- \(a_{31}=0\)
- The derivative of \(\tan^{-1}(x^{2})\) w.r.t. x is :
- \(\frac{x}{1+x^{4}}\)
- \(\frac{2x}{1+x^{4}}\)
- \(-\frac{2x}{1+x^{4}}\)
- \(\frac{1}{1+x^{4}}\)
- The degree of the differential equation \((y^{\prime\prime})^{2}+(y^{\prime})^{3}=x~\sin(y^{\prime})\) is:
- 1
- 2
- 3
- not defined
- The unit vector perpendicular to both vectors \(\hat{i}+\hat{k}\) and \(\hat{i}-\hat{k}\) is:
- \(2\hat{j}\)
- \(\hat{j}\)
- \(\frac{\hat{i}-\hat{k}}{\sqrt{2}}\)
- \(\frac{\hat{i}+\hat{k}}{\sqrt{2}}\)
- Direction ratios of a vector parallel to line \(\frac{x-1}{2}=-y=\frac{2z+1}{6}\) are:
- \(2,-1,6\)
- \(2, 1, 6\)
- \(2, 1, 3\)
- \(2,-1, 3\)
- If \(F(x)=[\begin{matrix}\cos~x&-\sin~x&0\\ \sin~x&\cos~x&0\\ 0&0&1\end{matrix}]\) and \([F(x)]^{2}=F(kx)\), then the value of k is :
- 1
- 2
- 0
- -2
- If a line makes an angle of \(30^{\circ}\) with the positive direction of x-axis, \(120^{\circ}\) with the positive direction of y-axis, then the angle which it makes with the positive direction of z-axis is:
- \(90^{\circ}\)
- \(120^{\circ}\)
- \(60^{\circ}\)
- \(0^{\circ}\)
-
Assertion (A): For any symmetric matrix A, B'AB is a skew-symmetric matrix.
Reason (R): A square matrix P is skew-symmetric if \(P^{\prime}=-P\)
(Choose the correct option from the standard codes A, B, C, D)
-
Assertion (A): For two non-zero vectors \(\vec{a}\) and \(\vec{b}\), \(\vec{a} \cdot \vec{b} = \vec{b} \cdot \vec{a}\).
Reason (R): For two non-zero vectors \(\vec{a}\) and \(\vec{b}\), \(\vec{a} \times \vec{b} = \vec{b} \times \vec{a}\).
(Choose the correct option from the standard codes A, B, C, D)
This section comprises of **5 Very Short Answer (VSA)** type questions of 2 marks each. (Internal choice has been provided in 2 questions)
-
(a) Find the value of \(\tan^{-1}(-\frac{1}{\sqrt{3}})+\cot^{-1}(\frac{1}{\sqrt{3}})+\tan^{-1}[\sin(-\frac{\pi}{2})].\)
OR
(b) Find the domain of the function \(f(x)=\sin^{-1}(x^{2}-4).\) Also, find its range. (2)
-
(a) If \(f(x)=|\tan~2x|\), then find the value of \(f^{\prime}(x)\) at \(x=\frac{\pi}{3}\)
OR
(b) If \(y=\operatorname{cosec}(\cot^{-1}x)\), then prove that \(\sqrt{1+x^{2}}\frac{dy}{dx}-x=0\) (2)
- If M and m denote the local maximum and local minimum values of the function \(f(x)=x+\frac{1}{x}(x\ne0)\) respectively, find the value of \((M-m)\) (2)
- Find: \(\int\frac{e^{4x}-1}{e^{4x}+1}dx\) (2)
- Show that \(f(x)=e^{x}-e^{-x}+x-\tan^{-1}x\) is strictly increasing in its domain. (2)
This section comprises of **6 Short Answer (SA)** type questions of 3 marks each. (Internal choice has been provided in 3 questions)
-
(a) If \(x=e^{\cos~3t}\) and \(y=e^{\sin~3t}\), prove that \(\frac{dy}{dx}=-\frac{y~\log~x}{x~\log~y}\)
OR
(b) Show that: \(\frac{d}{dx}(|x|)=\frac{x}{|x|},x\ne0\) (3)
-
(a) Evaluate : \(\int_{-2}^{2}\sqrt{\frac{2-x}{2+x}}dx\)
OR
(b) Find: \(\int\frac{1}{x[(\log~x)^{2}-3~\log~x-4]}dx\) (3)
-
(a) Find the particular solution of the differential equation given by \(2xy+y^{2}-2x^{2}\frac{dy}{dx}=0\) \(y=2\), when \(x=1.\)
OR
(b) Find the general solution of the differential equation : \(y~dx=(x+2y^{2})~dy\) (3)
- The position vectors of vertices of \(\Delta\) ABC are \(A(2\hat{i}-\hat{j}+\hat{k}),\) \(B(\hat{i}-3\hat{j}-5\hat{k})\) and \(C(3\hat{i}-4\hat{j}-4\hat{k})\) Find all the angles of \(\Delta\) AВС. (3)
- A pair of dice is thrown simultaneously. If X denotes the absolute difference of the numbers appearing on top of the dice, then find the probability distribution of X. (3)
- Find: \(\int x^{2}\cdot \sin^{-1}(x^{3/2})dx\) (3)
This section comprises of **4 Long Answer (LA)** type questions of 5 marks each. (Internal choice has been provided in 2 questions)
-
(a) Show that a function \(f:R\rightarrow R\) defined by \(f(x)=\frac{2x}{1+x^{2}}\) is neither one-one nor onto. Further, find set A so that the given function \(f:R\rightarrow A\) becomes an onto function.
OR
(b) A relation R is defined on \(N\times N\) (where N is the set of natural numbers) as: \((a, b)~R~(c,d)\Leftrightarrow a-c=b-d\) Show that R is an equivalence relation. (5)
- Find the equation of the line which bisects the line segment joining points \(A(2,3,4)\) and \(B(4,5,8)\) and is perpendicular to the lines \(\frac{x-8}{3}=\frac{y+19}{-16}=\frac{z-10}{7}\) and \(\frac{x-15}{3}=\frac{y-29}{8}=\frac{z-5}{-5}\) (5)
-
(a) Solve the following system of equations, using matrices: \(\frac{2}{x}+\frac{3}{y}+\frac{10}{z}=4\), \(\frac{4}{x}-\frac{6}{y}+\frac{5}{z}=1\), \(\frac{6}{x}+\frac{9}{y}-\frac{20}{z}=2\) where x, y, \(z\ne0\)
OR
(b) If \(A=[\begin{matrix}1&\cot~x\\ -\cot~x&1\end{matrix}]\) show that \(A^{\prime}A^{-1}=[\begin{matrix}-\cos~2x&-\sin~2x\\ \sin~2x&-\cos~2x\end{matrix}]\) (5)
- If \(A_{1}\) denotes the area of region bounded by \(y^{2}=4x,\) \(x=1\) and x-axis in the first quadrant and \(A_{2}\) denotes the area of region bounded by \(y^{2}=4x,\) \(x=4\), find \(A_{1}:A_{2}\). (5)
This section comprises of **3 case study based questions** of 4 marks each. (Internal choice has been provided in 2 questions)
-
Case Study 1: Overspeeding and Fuel Consumption
Overspeeding increases fuel consumption and decreases fuel economy as a result of tyre rolling friction and air resistance. While vehicles reach optimal fuel economy at different speeds, fuel mileage usually decreases rapidly at speeds above \(80~km/h\). The relation between fuel consumption \(F(l/100~km)\) and speed \(V(km/h)\) under some constraints is given as \(F=\frac{V^{2}}{500}-\frac{V}{4}+14.\)
On the basis of the above information, answer the following questions :
- Find F, when \(V=40~km/h\). (1 mark)
- Find \(\frac{dF}{dV}\) (1 mark)
-
(a) Find the speed V for which fuel consumption F is minimum. (2 marks)
OR
(b) Find the quantity of fuel required to travel 600 km at the speed V at which \(\frac{dF}{dV}=-0\cdot01.\) (2 marks)
(4)
-
Case Study 2: Diet and Linear Programming
The month of September is celebrated as the Rashtriya Poshan Maah across the country. Following a healthy and well-balanced diet is crucial in order to supply the body with the proper nutrients it needs. A balanced diet also keeps us mentally fit and promotes improved level of energy. A dietician wishes to minimize the cost of a diet involving two types of foods, food \(X(x~kg)\) and food \(Y(y~kg)\) which are available at the rate of \(\text{₹}16/kg\) and \(\text{₹}20/kg\) respectively. The feasible region satisfying the constraints is shown in Figure-2.
On the basis of the above information, answer the following questions :
- Identify and write all the constraints which determine the given feasible region in Figure-2. (2 marks)
- If the objective is to minimize cost \(Z=16x+20y\), find the values of x and y at which cost is minimum. Also, find minimum cost assuming that minimum cost is possible for the given unbounded region. (2 marks)
(4)
-
Case Study 3: [Title is missing from source document, appears to be related to transport]
(The preamble and questions for this case study are missing from the source document.)
- [Part 1 of Q38, 1 mark - Missing content]
- [Part 2 of Q38, 1 mark - Missing content]
-
(a) [Part 3(a) of Q38, 2 marks - Missing content]
OR
(b) [Part 3(b) of Q38, 2 marks - Missing content]
(4)