Jkbose previous year question paper 2022 set A

A-6-A

Total No. of Questions : 29

[Total No. of Printed Pages : 8]

$$12^{\text {th }}$$ SZARJD22 6006-A

{MATHEMATICS}

Time : 2.30 Hours]

[Maximum Marks : 100

Section-A

(Multiple Choice Questions) 1 each

1. The relation R in the set $$\{1,2,3\}$$ given by $$\mathrm{R}=\{(x, y) \mid x<y$$,

$$
x, y \in A\} \text { is : }
$$

(A) Reflexive

(B) Symmetric

(C) Transitive

(D) Anti-symmetric

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 the values of $$x, y$$ and $$z$$ from the following equation :
$$
\left[\begin{array}{c}
x+y+z \\
x+z \\
y+z
\end{array}\right]=\left[\begin{array}{l}
9 \\
5 \\
7
\end{array}\right]
$$
6. Evaluate :
$$
\int\left(a x^{2}+b x+c\right) d x
$$
7. Find the rate of change of area of a circle with respect to its radius when $$r=5 \mathrm{~cm}$$.
8. Form the differential equation representing the family of curve $$y=m x^{2}$$, where $$m$$ is arbitrary constant.
9. Find the projection of the vector $$\hat{i}+3 \hat{j}+7 \hat{k}$$ on the vector $$7 \hat{i}-\hat{j}+8 \hat{k}$$.
10. Solve the following L.P.P. graphically :

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

Subject to constraints :
$$
\begin{gathered}
x+y \leq 4 \\
x \geq 0, y \geq 0
\end{gathered}
$$
11. Two cards are drawn at random and without replacement from a pack of 52 playing cards. Find the probability that both the cards are black.
12. If $$P(A)=\frac{3}{5}$$ and $$P(B)=\frac{1}{5}$$, find $$P(A \cap B)$$ if $$A$$ and $$B$$ are independent events.

{Section-C \\ (Short Answer Type Questions)}
13. Show that $$f:\left(\begin{array}{ll}-1 & 1\end{array}\right) \rightarrow \mathrm{R}$$ given by $$f(x)=\frac{x}{x+2}$$ is one-one. Find the inverse of the function $$f:\left(\begin{array}{ll}-1 & 1\end{array}\right) \rightarrow$$ Range of $$f$$.
14. Solve :
$$
\tan ^{-1} 2 x+\tan ^{-1} 3 x=\frac{\pi}{4}
$$
15.For the matrix A and B verify that $$(\mathrm{AB})^{\prime}=\mathrm{B}^{\prime} \mathrm{A}^{\prime}, \mathrm{A}=\left[\begin{array}{r}1 \\ -4 \\ 3\end{array}\right]. B=\left[\begin{array}{lll}-1 & 2 & 1\end{array}\right]$$.

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 oven 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 $$\vec{a}+\vec{b}$$ 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 :
and
$$
\begin{gathered}
x+3 y \leq 60, x \leq y \\
x+y \geq 10, x \geq 0, y \geq 0
\end{gathered}
$$
th $$\operatorname{cosAR} 1 D 22$$-6006-A

{(Section-D)}

{(long Answer Type Questions)}

24 By using properties of determinants prove that
$$
\left|\begin{array}{ccc}
a & a+b & a+b+c \\
2 a & 3 a+2 b & 4 a+3 b+2 c \\
3 a & 6 a+3 b & 10 a+6 b+3 c
\end{array}\right|=a^{3}
$$

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 :
$$
\frac{2 x-3}{\left(x^{2}-1\right)(2 x+3)}
$$

{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.
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