Jkbose previous year question paper 2022 set B

A-6-B

[Total No. of Questions : 29]

{$$12^{\text {th }}$$ SZARJD22}

6006-B

{MATHEMATICS}

Time : 2.30 Hours]
[Maximum Marks : 100]
Total No. of Printed Pages 8

Section-A

(Multiple Choice Questions) 1 each

1. If $$A$$. $$B$$ are symmetric matrices of same order, then $$A B-B A$$ is a
(A) Symmetric matrix
(B) Skew symmetric matrix
(C) Zero matrix
(D) Identity matrix

2. The principal value of $$\cos ^{-1}\left(-\frac{1}{\sqrt{2}}\right)$$ is :

(A) $$\frac{\pi}{4}$$

(B) $$\frac{\pi}{2}$$

(C) $$\frac{\pi}{3}$$

(D) $$\frac{3 \pi}{4}$$

$$12^{\text {th }} S Z A R J D 22-6006-A$$

3. Two matrices $$A$$ and $$B$$ of the same order are said to be equal, if:

(A) $$a_{i j}=0$$

(B) $$b_{i j}=0$$

(C) $$a_{i j}+b_{i j}=0$$.

(D) $$a_{i j}=b_{i j}$$ for all $$i, j$$

4. The direction cosines of a unit vector along $$x$$-axis are :

(A) (1, 0, 0)

(B) (0,1,0)

(C) (0, 0, 1)

(D) $$(1,1,1)$$

{Section-B}

{(Very Short Answer Type Questions)}
5. Find $$x$$ and $$y$$ if :
$$
2\left[\begin{array}{ll}
1 & 3 \\
0 & x
\end{array}\right]+\left[\begin{array}{ll}
y & 0 \\
1 & 2
\end{array}\right]=\left[\begin{array}{ll}
5 & 6 \\
1 & 8
\end{array}\right]
$$
6. Evaluate :
$$
\int\left(4 e^{3 x}+1\right) d x
$$
7. Verify that the function $$y=x^{2}+2 x+\mathrm{C}$$ is the solution of the differential equation $$\frac{d y}{d x}-2 x-2=0$$.
$$12^{\text {th }}$$ SZARJD22-6006-B
A-6-B
8. Find the rate of change of the area of a circle with respect to its radius $$r$$ well $$r=4 \mathrm{~cm}$$.
9. Find the angle between the vectors $$\hat{i}-2 \hat{j}+3 \hat{k}$$ and $$3 \hat{i}-2 \hat{j}+\hat{k}$$.
10. Solve the following L.P.P. graphically :

Maximise :
$$
Z=3 x+4 y
$$

Subject to the constraints :
$$
\begin{gathered}
x+y \leq 4 \\
x \geq 0, y \geq 0
\end{gathered}
$$
11. If $$P(A)=\frac{6}{11}$$,
$$P(B)=\frac{5}{11}$$ and $$P(A \cup$$ B) $$=\frac{7}{11}$$, find $$P\left(\frac{A}{B}\right)$$.
12. Given two independent events $$A$$ and $$B$$ such that $$P(A)=0\cdot3$$,
$$P(B)=6$$. Find $$P(A$$ or $$B)$$.

{Section-C}
(Short Answer Type Questions)
13. Given $$f: \mathrm{R} \rightarrow \mathrm{R}$$ given by $$f(x)=4 x+3$$. Show that $$f$$ is invertible

Find the inverse of $$f$$.
14. Write the following function in the sumplest form:
$$
\tan ^{-1}\left[\frac{\cos x-\sin x}{\cos x+\sin x}\right] 0<x<\pi $$
15. Express the matrix $$A=\left[\begin{array}{rrr}2 & -2 & -4 \\ -1 & 3 & 4 \\ 1 & -2 & -3\end{array}\right]$$ as the sum of symmetric and a skew-symmetric matrix.
16. Find the relationship hetween $$a$$ and $$b$$ so that the function $$I$$ detined by : $$ f(x)=\left\{\begin{array}{lll} a x+1 & \text { if } & x \leq 3 \\ b x+3 & \text { if } & x>3
\end{array}\right.
$$
is continuous at $$x=3$$.
17. Find the intervals in which the function $$f$$ given by $$f(x)=4 x^{3}-6 x^{2}$$
– $$72 x+30$$ is :
(a) Strictly increasing
(h) Strictly decreasing
$$12^{\text {th }}$$ SZA.RJD22-6006-B
A-6-B
18. Find the equations of the tangent and normal to the given curve at the indicated point $$y=x^{4}-6 x^{3}+13 x^{2}-10 x+5$$ at ( 0.5 ).
19. Find the general solution of the differential equation :
$$
\frac{d y}{d x}=\frac{1+y^{2}}{1+x^{2}}
$$
20. Find the area of the region bounded by the ellipse :
$$
\frac{x^{2}}{16}+\frac{y^{2}}{9}=1
$$
21. Find $$|\vec{a} \times \vec{b}|$$ f $$\vec{a}=\hat{i}-7 \hat{j}+7 \hat{k}$$ and $$\vec{b}=3 \hat{i}-2 \hat{j}+2 \hat{k}$$.
22. If $$\vec{a}=5 \hat{i}-j-3 \hat{k}$$ and $$\vec{b}=\hat{i}+3 \hat{j}-5 \hat{k}$$. then show that $$\vec{a}+\vec{b}$$ and $$\vec{a}-\vec{b}$$ are perpendicular.
23. Solve the following problem graphically :

Minimise and maximise : $$\quad \mathrm{Z}=3 x+9 y$$
Subject to the linear constraints :
$$
x+3 y \leq 60 . x \leq y
$$
and
$$
x+y \geq 10 . x \geq 0 . y \geq 0
$$
$$12^{\text {th }} \operatorname{sZARJ} 22-620 E \cdot B$$
$$A-6-B$$

section D{(Long Answer Type Questions)}
24. By using properties of determinants prove that :
$$
\left|\begin{array}{ccc}
a & a+b & a+b+c \\
2 a & 3 a+2 b & 4 a+3 b+2 c \\
3 a & 6 a+3 b & 10 a+6 b+3 c
\end{array}\right|=a^{3}
$$
$$
\mathrm{Or}
$$

If $$A=\left[\begin{array}{lll}1 & 3 & 3 \\ 1 & 4 & 3 \\ 1 & 3 & 4\end{array}\right]$$, then verify that $$A(\operatorname{adj} A)=|A| I$$. Also find $$A^{-1}$$
25. Find $$\frac{d y}{d x}$$ of the function $$x^{y}+y^{x}=1$$.

{Or}

If $$y=3 \cos (\log x)+4 \sin (\log x)$$, show that :
$$
x^{2} y_{2}+x y_{1}+y=0
$$
26. Integrate the rational fraction :
$$
\frac{2 x-3}{\left(x^{2}-1\right)(2 x+3)}
$$
$$12^{\text {th }}$$ SZARJD22-6006-B
A-6-B

or

Using the properties of definite integrals evaluate
$$
\int_{-5}^{5}|x+2| d x
$$
27. Find the general solution of the differential equation :
$$
x \frac{d y}{d x}+2 y=x^{2} \log x
$$

Or
Show that the differential equation $$\left(x^{2}-y^{2}\right) d x+2 x y d y=0$$ is homogeneous and solve it.
28. Find the equation of the plane through the intersection of the planes $$3 x-y+2 z-4=0$$ and $$x+y+z-2=0$$ and the point (2. 2, 1).
$$
\mathrm{Or}
$$

Find the angle between the line $$\frac{x+1}{2}=\frac{y}{3}=\frac{z-3}{6}$$ and the plane $$10 x+2 y-11 z=3$$.
$$
\left(\begin{array}{l}
1
\end{array}\right)
$$
29. Find the probability distribution at member of doublets in three throws of a pair of dice
$$
Or
$$

Two balls are drawn at random will replacement from a box containing 10 black balls and 8 red balls. Find the probability that
a) Both balls are red
b) First ball is black and second is red
c) One of them is black and Other is red