JKBOSE 2024 MATHS PAPER SET-X
{B-7-X}
ROLLNo.
[Total No. of Questions:31]
[Total No. of Printed Pages : 8]
$$12^{\text {th }}$$ ARM(SZ)JKUT2024
{1107-X}
MATHEMATICS
[Time : 3 Hours]
[Maximum Marks : 80]
SECTION-A
(OBJECTIVE TYPE QUESTIONS
MULTIPLE CHOICE QUESTIONS) 1 each
1. In determinant
$$
\left|\begin{array}{rr}
3 & 6 \\
-2 & 5
\end{array}\right|
$$
cofactor of -2 is:
(A) 6
(B) 3
(C) 5
(D) -6
2. For a square matrix $$A \cdot A({adj} A)=({adj} A) A=$$
12ARM(SZ)IKUT2024-1107-x
B-7-X
3. Order of differential equation :
$$
\left[1+\left(\frac{d y}{d x}\right)^{2}\right]^{3 / 2}=\frac{d^{2} y}{d x^{2}}
$$
is:
(A) 3
(B) 2
(C) 1
(D) 4
4. Derivative of $$e^{\log \sin x}$$ is :
(A) $$\sin x$$
(B) $$\cos x$$
(C) $$\sec x$$
(D) $$\tan x$$
5. Number of points of discontinuity of the function:
$$
f(x)=\frac{1}{x-5}, x \neq 5
$$
is:
(A) 1
(B) 2
(C) 3
(D) 5
6. $$\int x e^{2} d x$$ is equal to $$\frac{e^{x^{2}}}{x}+c$$.
7. The function $$f(x)=3 x+5$$ is increasing for……..
8. Direction cosines of $$x$$-axis are :
(A) (1,0,0)
(B) (0,1,0)
(C) (0,0,1)
(D) ($$0,0,0$$)
9. Magnitude of Vector $$\hat{i}+2 \hat{j}+3 \hat{k}$$ is :
(A) $$\frac{1}{\sqrt{14}}$$
(D) $$\sqrt{14}$$
(C) 1
(D) 10
10. Define objective function.
12ARM(SZ)JKUT2024-1107-X
{section-B (VERY SHORT ANSWER TYPE QUESTIONS) 2 each}
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11. Check the injectivity and subjectivity of the function $$\quad f:=\rightarrow$$ : given by $$f(x)=x^{2}$$.
12. Find the principal value of
$$
\sin ^{-1}\left(-\frac{1}{2}\right)
$$
13. Prove that the logarithmic function is strictly increasing on $$(0, \infty)$$.
14. Find the projection of the vector $$\hat{i}+3 \hat{j}+7 \hat{k}$$ on the vector $$7 \hat{i}-\hat{j}+8 \hat{k}$$.
15. Find a unit vector perpendicular to each of the vector $$\vec{a}+\vec{b}$$ and $$\vec{a}-\vec{b}$$, where $$\vec{a}=3 \hat{i}+2 \hat{j}+2 \hat{k}$$ and $$\vec{b}=\hat{i}+2 \hat{j}-2 \hat{k}$$.
16. Find:
$$
\int \frac{(\log x)^{2}}{x} d x
$$
17. Evaluate:
$$
\int_{0}^{1} \frac{d x}{\sqrt{1-x^{2}}}
$$
12ARM(SZ)JKUT2024-1107-X
B-7-X
18.
$$
P(A)=\frac{3}{5}
$$
find $$P(A \cap B)$$ if $$A$$ and $$B$$ are independent events.
$$
\begin{aligned}
& P(B)=\frac{1}{5} \\
& B \text { are independent events. }
\end{aligned}
$$
19. Evaluate $$P(A \cup B)$$ if $$2 \underline{P}(A)=P(B)=\frac{5}{13}$$ and $$P(A / B)=\frac{2}{5}$$.
20. Show that the matrix :
$$
A=\left[\begin{array}{rrr}
1 & -1 & 5 \\
-1 & 2 & 1 \\
5 & 1 & 3
\end{array}\right]
$$
is a symmetric matrix.
(SHORT ANSWER TYPE QUESTIONS) 4 each
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21. Find the general solution of the dierential equation:
$$
x \log x \frac{d y}{d x}+y=\frac{2}{x} \log x
$$
22. Evaluate:
$$
\int_{0}^{\pi / 2} \frac{\sqrt{\sin x}}{\sqrt{\sin x}+\sqrt{\cos x}} d x
$$
23. If
$$
x=\sqrt{a^{\sin ^{-t \prime}}}, y=\sqrt{a^{\cos ^{-t \prime}}}
$$
show that:
$$
\frac{d y}{d x}=-\frac{y}{x}
$$
24. Find the shortest distance between the lines:
$$
\vec{r}=(\hat{i}+2 \hat{j}+\hat{k})+\lambda(\hat{i}-\hat{j}+\hat{k})
$$
and
$$
\vec{r}=(2 \hat{i}-\hat{j}-\hat{k})+\mu(2 \hat{i}+\hat{j}+2 \hat{k})
$$
25. If
$$
\vec{a}=2 \hat{i}+2 \hat{j}+3 \hat{k}, \vec{b}=-\hat{i}+2 \hat{j}+\hat{k}
$$
and
$$
\vec{c}=3 \hat{i}+\hat{j}
$$
are such that:
$$
\vec{a}+\lambda \vec{b}
$$
is perpendicular to $$\vec{c}$$. then find the value of $$\lambda$$.
Jkboseoldpapers.com
26. Solvethe linearprogramming problem graphically
Minimize $$:=-3 x+4 y$$
subject to:
$$
\begin{gathered}
x+2 y \leq 8 \\
3 x+2 y \leq 12 \\
x \geq 0, y \geq 0
\end{gathered}
$$
27. Show that the relation $$R$$ in the set $$\mathbf{Z}$$ of integers given by $$R=\{(a, b): 2$$ divides $$a-b\}$$ is an equivalence relation.
28. A bag contains 4 red and 4 black balls. another bag contains 2 red and 6 black balls. One of the two bags is selected at random and a hall is drawn from the bag which is found to be red. Find the probability that the ball is drawn from the first bag.
{SECTION-D}
(LONG ANSWER TYPE QUESTIONS) 6 each
29. If :
$$
A=\left[\begin{array}{rrr}
1 & -1 & 2 \\
3 & 0 & -2 \\
1 & 0 & 3
\end{array}\right]
$$
verify $$A(\operatorname{adj} A)=(\operatorname{adj} A) A=|A| I$$.
12ARM(SZ)JKUT2024-1107-X
Or
Let :
$$
A=\left[\begin{array}{ll}
3 & 7 \\
2 & 5
\end{array}\right] \text { and } B=\left[\begin{array}{ll}
6 & 8 \\
7 & 9
\end{array}\right]
$$
verify $$(A B)^{-1}=B^{-1} \cdot A^{-1}$$.
30. Evaluate: Jkboseoldpapers.com $$
\int_{-1}^{1} 5 x^{4} \sqrt{x^{5}+1} d x
$$}
Or Evaluate:
$$
\int \frac{(3 x-1) d x}{(x-1)(x-2)(x-3)}
$$
31. If $$y=A e^{m x}+B e^{n x}$$, show that:
$$
\frac{d^{2} y}{d x^{2}}-(m+n) \frac{d y}{d x}+m n y=0
$$
Or
Differentiate w.r.t. $$x$$ :
$$
(\log x)^{x}+x^{\log x}
$$
