Jkbose previous year question paper 2021
B-5-C
Roll No
[Total No. of Questions : 29]
[Total No. of Printed Pages : 8]
XIIKDAR21
5005-C MATHEMATICS
Time : 3 Hours ]
{Section-A}
(Multiple Choice Questions) 1 each
1. Let R be the relation in the set N given by R = {(a,b)|a=b-2,(b>6):
(A) $$(2,4) \in R$$
(B) $$(3,8) \in R$$
(C) $$(6,8) \in R$$
(D) $$(8,7) \in R$$
(Choose the correct answer)
2. $$\cos ^{-1}\left(\cos \frac{7 \pi}{6}\right)$$ is equal to :
(A) $$\frac{7 \pi}{6}$$
(B) $$\frac{5 \pi}{6}$$
(C) $$\frac{\pi}{3}$$
(D) $$\frac{\pi}{6}$$
(Choose the correct answer)
3. A and B are symmetric matrices of same order, then $$\mathrm{AB}-\mathrm{BA}$$ is a
(A) Skew symmetric matrix
(B) Symmetric matrix
(C) Zero matrix
(D) Identity matrix
(Choose the correct answer)
4. The value of :
$$
\hat{i} \cdot(\hat{j} \times \hat{k})+\hat{j} \cdot(\hat{k} \times \hat{i})+\hat{k} \cdot(\hat{j} \times \hat{i})
$$
is :
(A) 0
(B) -1
(C) 1
(D) none of these
(Choose the correct answer)
Section-B
(Very Short Answer Type Questions)}
5. Evaluate :
$$
\int \frac{\sin \left(\tan ^{-1} x\right) d x}{1+x^{2}}
$$
6. Evaluate :
$$
\int_{0}^{1} \frac{d x}{\sqrt{1-x^{2}}}
$$
7. Find the values of $$x, y$$ and $$z$$ from the equation
$$
\left[\begin{array}{c}
x+y+z \\
x+z \\
y+z
\end{array}\right]=\left[\begin{array}{l}
9 \\
5 \\
7
\end{array}\right]
$$
8. Find the order and degree of differential equation :
$$
\left(\frac{d s}{d t}\right)^{4}+3 s \frac{d^{2} s}{d t^{2}}=4
$$
9. If a line has direction ratios $$2,-1,2$$, determine its direction cosines.
10. Define objective function and optimal solution of L.P.P.
11. $$P(A)=\frac{6}{11}, P(B)=\frac{5}{11}, P(A \cup B)=\frac{7}{11}$$, find $$P(A \cap B)$$.
12. If $$P(A)=\frac{3}{5}$$ and $$P(B)=\frac{1}{5}$$, find $$(A \cap B)$$ if $$A$$ and $$B$$ are independent events.
{Section-C}
(Short Answer Type Questions) 4 each
13. If :
$$
f(x)=\frac{4 x+3}{6 x-4} \quad x \neq \frac{2}{3} .
$$
Show that $$f \circ f(x)=x$$ for all $$x \neq \frac{2}{3}$$ What is the inverse of $$f$$ ?
14. Write in the simplest form
$$
\tan ^{-1}\left(\frac{\cos x-\sin x}{\cos x+\sin x}\right)
$$
15. Express the matrix $$B=\left[\begin{array}{rrr}2 & -2 & -4 \\ -1 & 3 & 4 \\ 1 & -2 & 3\end{array}\right]$$ as the sum of a symmetric and
a skew symmetric matrix.
16. Find the equation of tangent and normal to the curve at the indicated point $$y=x^{4}-6 x^{3}+13 x^{2}-10 x+5$$ at $$(0,5)$$.
17. Use differentials, find the approximate value up to 3 places of decimal $$(25)^{1 / 3}$$.
18. Integrate the function $$x \tan ^{-1} x$$.
19. Find the relationship between $$a$$ and $$b$$ such that the function $$f$$ defined by :
$$
f(x)= \begin{cases}a x+1 & x \leq 3 \\ b x+3 & x>3\end{cases}
$$
is continuous at $$x=3$$.
20. If $$y=\sin ^{-1}$$ x, show that
$$
\left(1-x^{2}\right) \frac{d^{2} y}{d x^{2}}-x \frac{d y}{d x}=0
$$
21. Find the unit vector perpendicular to each of the vectors $$\vec{a}+\vec{b}$$ and $$\vec{a}-\vec{b}$$ where $$\vec{a}=\hat{i}+\hat{j}+\hat{k}$$ and $$\vec{b}=\hat{i}+2 \hat{j}+3 \hat{k}$$.
22. Find the angle between two planes:
$$
2 x+y-2 z=5
$$
and
$$
3 x-6 y-2 z=7
$$
using vector method.
23. Solve graphically (L.P.P.)
Maximize
$$
Z=4 x+y
$$
Subject to the constraints :
$$
\begin{array}{r}
x+y \leq 50 \\
3 x+y \leq 90 \\
x \geq 0, y \geq 0
\end{array}
$$
{Section-D)}
(Long Answer Type Questions)
24. Using the properties of determinants. Prove that
$$
\left|\begin{array}{ccc}
1 & 1 & 1 \\
a & b & c \\
a^{3} & b^{3} & c^{3}
\end{array}\right|=(a-b)(b-c)(c-a)(a+b+c)
$$
Or
Solve the system of equations by matrix method:
$$
\begin{gathered}
3 x-2 y+3 z=8 \\
2 x+y-z=1 \\
4 x-3 y+2 z=4
\end{gathered}
$$
25. If $$y=x^{\sin x}+(\sin x)^{\cos x}$$. find $$\frac{d y}{d x}$$.
{Or}
Show that of all the rectangles inscribed in a given fixed circle the square has the maximum area.
XIIKDAR21-5005-C
B-5-C
26. Find :
$$
\int \frac{(3 \sin x-2) \cos x d x}{5-\cos ^{2} x-4 \sin x}
$$
Or
Find the area of the region bounded by the ellipse :
$$
\frac{x^{2}}{16}+\frac{y^{2}}{9}=1
$$
27. Find the general solution of the differential equation given by :
$$
\begin{gathered}
x \frac{d y}{d x}+2 y=x^{2} \log x \\
\text { Or }
\end{gathered}
$$
Find the general solution of the differential equation :
$$
\frac{d y}{d x}=\frac{x+1}{2-y}, \quad y \neq 2
$$
28. Find the equation of the plane passing through three points (1, 1, 0) $$(1,2,1),(-2,2,-1)$$.
$$
\mathrm{Or}
$$
Find the shortest distance between the lines whose vector equations are :
$$
\vec{r}=\hat{i}+2 \hat{j}+\hat{k}+\lambda(\hat{i}-\hat{j}+\hat{k})
$$
and
$$
\vec{r}=2 \hat{i}-\hat{j}-\hat{k}+\mu(2 \hat{i}+\hat{j}+2 \hat{k})
$$
XIIKDAR21-5005-C
:-5-C
29. Find the probability distribution of the number of doublets in three throws of a pair of dice.
{Or}
A die is thrown 6 times if “getting an odd number” is a success. What is the probability of :
(i) 5 success
(ii) at least 5 success
(iii) at most 5 success
