Jkbose previous year question paper 2022 set c
A-6-C
[Total No. of Questions : 29]
[Total No. of Printed Pages : 8]
{$$12^{\text {th }}$$ SZARJD22}
6006-C
MATHEMATICS
[Maximum Marks : 100
Time : $$\mathbf{2 . 3 0}$$ Hours]
Section-A \\ (Multiple Choice Questions)
1. Let R be the relation on the set N given by $$\mathrm{R}=\{(a, b) \mid a=b-2$$, $$b>6\}$$. Choose the correct answer.
(A) $$(2,4) \in \mathrm{R}$$
(B) $$(3,8) \in R$$
(C) $$(6,8) \in \mathrm{R}$$
(D) $$(8,7) \in \mathrm{R}$$
2. $$\cos ^{-1} \cos \left(\frac{7 \pi}{6}\right)$$ is equal to :
(A) $$\frac{7 \pi}{6}$$
(B) $$\frac{5 \pi}{6}$$
(C) $$\frac{\pi}{3}$$
(D) $$\frac{\pi}{6}$$
3. If the matrix A is both symmetric and skew-symmetric, then :
(A) A is a diagonal matrix
(C) A is a square matrix
(B) A is a null matrix
(D) None of these
4. If $$\vec{a}$$ is a non-zero vector of magnitude ‘ $$a$$ ‘ and $$\lambda$$ is a non-zero scalar, then $$\lambda \vec{a}$$ is a unit vector if :
(A) $$\lambda=1$$
(B) $$\lambda=-1$$
(C) $$a=|\lambda|$$
(D) $$\quad a=\frac{1}{|\lambda|}$$
\section*{Section-B \\ (Very Short Answer Type Questions) \\ 2 each}
5. Find the value of $$x$$ for which :
$$
\left|\begin{array}{ll}
3 & x \\
x & 1
\end{array}\right|=\left|\begin{array}{ll}
3 & 2 \\
4 & 1
\end{array}\right|
$$
6. Evaluate :
$$
\int(1-x) \sqrt{x} d x
$$
7. Find the rate of change of area of a circle with respect to its radius $$r$$ when $$r=6 \mathrm{~cm}$$.
$$12^{\text {th }}$$ SZARJD22-6006-C
A-6-C
8. Find the general solution of the differential equation :
$$
\frac{d y}{d x}=e^{x+y}
$$
9. Find angle $$\theta$$ between the vectors $$\vec{a}=\hat{i}+\hat{j}-\hat{k}$$ and $$\vec{b}=\hat{i}-\hat{j}+\hat{k}$$.
10. Solve the following L.P.P. graphically :
Maximise :
$$
\mathrm{Z}=3 x+4 y
$$
Subject to the constraints :
$$
\begin{gathered}
x+y \leq 4 \\
x \geq 0, y \geq 0
\end{gathered}
$$
11. If A and B are two events such that $$\mathrm{P}(\mathrm{A})=\frac{1}{4}, \mathrm{P}(\mathrm{B})=\frac{1}{2}$$ and $$\mathrm{P}(\mathrm{A} \cap \mathrm{B})=\frac{1}{8}$$, find $$\mathrm{P}($$ not A and not B$$)$$.
12. If $$\mathrm{P}(\mathrm{A})=\frac{7}{13}, \mathrm{P}(\mathrm{B})=\frac{9}{13}$$ and $$\mathrm{P}(\mathrm{A} \cap \mathrm{B})=\frac{4}{13}$$, evaluate $$\mathrm{P}\left(\frac{\mathrm{A}}{\mathrm{B}}\right)$$.
{Section-C}
(Short Answer Type Questions)
13. If
$$
f(x)=\frac{4 x+3}{6 x-4} \quad x \neq \frac{2}{3}
$$
show that $$f \circ f=x$$ for all $$x \neq \frac{2}{3}$$. What is the inverse of $$f$$ ?
14. Prove that :
$$
\tan ^{-1} x+\tan ^{-1} \frac{2 x}{1-x^{2}}=\tan ^{-1}\left(\frac{3 x-x^{3}}{1-3 x^{2}}\right)
$$
15. If $$A=\left[\begin{array}{rr}3 & 1 \\ -1 & 2\end{array}\right]$$. show that :
$$
A^{2}-5 A+7 I=0
$$
16. Find the relationship between $$a$$ and $$b$$ so that the function $$f$$ defined 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
(b) Strictly decreasing
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 $$|\vec{a} \times \vec{b}|$$ if $$\vec{a}=\hat{i}-7 \hat{j}+7 \hat{k}$$ and $$\vec{b}=3 \hat{i}-2 \hat{j}+2 \hat{k}$$.
22. If $$\vec{a}=5 \hat{i}-\hat{j}-3 \hat{k}$$ and $$\vec{b}=\hat{i}+3 \hat{j}-5 \hat{k}$$, then show that $$\vec{a}+\vec{b}$$ an
$$\vec{a}-\vec{b}$$ are perpendicular.
23. Solve the following problem graphically :
Minimise and maximise: $$\quad \mathrm{Z}=3 x+9 y$$
Subject to the linear constraints
and
$$
\begin{gathered}
x+3 y \leq 60, x \leq y \\
x+y \geq 10, x \geq 0, y \geq 0
\end{gathered}
$$
{Section D}
24. By using properties of determinants prove that :
$$
\left|\begin{array}{ccc}
a & a+b & a+b+c \\
2 a & 3 a+2 b & 4 a+3 b+2 c \\
3 a & 6 a+3 b & 10 a+6 b+3 c
\end{array}\right|=a^{3}
$$
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 $$x^{y}+y^{x}=1$$.
$$
\mathrm{Or}
$$
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-C
a arr.
{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).
Or
Find the angle between the line $$\frac{x+1}{2}=\frac{y}{3}=\frac{z-3}{6}$$ and the plane $$10 x+2 y-11 z=3$$.
29. Find the probability distribution of number of doublets in three throws of a pair of dice.
$$
\mathrm{Or}
$$
Two balls are drawn at random with 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
