Class 12 maths previous year question paper 2023
Set x

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B-12-X

Total No. of Questions : 29]
[Total No. of Printed Pages : 8
{XIIARJKUT23}

9112-X MATHEMATICS
[Maximum Marks : 100

Time : 3 Hours]

{SECTION-A}

1. Range of the function $$f: \mathrm{R} \rightarrow \mathrm{R}$$ defined as $$f(x)=x^{2}$$ is :
(A) $$(0, x)$$
(B) $$(-\infty, 0)$$
(C) $$[0, x)$$
(D) $$(-\infty, 0]$$
2. $$\sin ^{-1} x+\cos ^{-1} x, x \in[-1,1]$$ is equal to :
(x) $$\frac{\pi}{2}$$
(B) $$\frac{\pi}{3}$$
(C) $$\frac{\pi}{4}$$
(D) $$\pi$$
3. If A is a square matrix of order $$n$$, then $$\mathrm{A}({adj} \mathrm{A})$$……..
(A) $$|\mathrm{A}|$$
(B) I
(C) $$|A| I$$
(D) None of these
4. If $$\vec{a}$$ and $$\vec{b}$$ are two unit vectors, then $$\vec{a} \cdot \vec{b}=\ldots \ldots \ldots \ldots$$
(A) $$\cos \theta$$
(B) $$\sin \theta$$
(C) $$a b \cos \theta$$
(D) $$a b \sin \theta$$
[ARJKUT23-9112-X-12-X

{SECTION-B}
(VERY SHORT ANSWER TYPE QUESTIONS)

2 each
5. If $$\left|\begin{array}{cc}x & 2 \\ 18 & x\end{array}\right|=\left|\begin{array}{cc}6 & 2 \\ 18 & 6\end{array}\right|$$. find the values of $$x$$.
6. Examine the continuity of the function $$f(x)=2 x^{2}-1$$ at $$x=3$$.
7. Differentiate $$\sin \left(x^{2}+5\right)$$ with respect to $$x$$.
8. Find :
$$
\int \frac{x^{3}+5 x^{2}-4}{x^{2}} d x
$$
9. A coin is tossed three times. Find $$P(E / F)$$. where $$E$$ is the event “head on third toss” and $$F$$ is the event “heads on first two tosses”.
10. Compute $$\mathrm{P}(\mathrm{A} \cap \mathrm{B}$$ ), where $$\mathrm{P}(\mathrm{A})=0.8$$. $$\mathrm{P}(\mathrm{B})=0.5$$ and $$\mathrm{P}(\mathrm{B} / \mathrm{A})=0.4$$.
11. Find the vector in the direction of vector $$5 \hat{i}-\hat{j}+2 \hat{k}$$ and having magnitude of 8 units.
12. Define Linear Constraints.

XIIARJKUT23-9112-X B-12-X

{SECTION-C}

{(SHORT ANSWER TYPE QUESTIONS)}

4 each
13. Find $$g \circ f$$ and $$f \circ g$$ if $$f(x)=|x|$$ and $$g(x)=\mid 5 x-21|$$.
14. If $$\sin \left(\sin ^{-1} \frac{1}{5}+\cos ^{-1} x\right)=1$$, find the value of $$x$$.
15. If :
$$
A=\left[\begin{array}{lll}
1 & 0 & 2 \\
0 & 2 & 1 \\
2 & 0 & 3
\end{array}\right]
$$
prove that $$\mathrm{A}^{3}-6 \mathrm{~A}^{2}+7 \mathrm{~A}+2 \mathrm{I}=0$$.
16. Find local maxima and local minima if any of the function :
$$
f(x)=x^{3}-6 x^{2}+9 x+15
$$
17. Find general solution of differential equation:
$$
x \cdot \frac{d y}{d x}-y+x \sin \left(\frac{y}{x}\right)=0
$$
18. Find $$\frac{d y}{d x}$$ if $$x^{3}+x^{2} y+x y^{2}+y^{3}=81$$.

IIARJKUT23-9112-X
– $$\boldsymbol{n} \boldsymbol{N}$$

19. Prove that:
$$
\int \sqrt{a^{2}-x^{2}} d x=
\frac{x \sqrt{a^{2}-x^{2}}}{2}+\frac{a^{2}}{2} \sin ^{-1}\left(\frac{x}{a}\right)+C
$$
20. (ff $$y=\left(\tan ^{-1} x\right)^{2}$$. show that :
$$
\left(x^{2}+1\right)^{2} y_{2}+2 x\left(x^{2}+1\right) y_{1}=2
$$
21. If $$\vec{a}, \vec{b}, \vec{c}$$ are unit vectors such that $$\vec{a}+\vec{b}+\vec{c}=\overrightarrow{0}$$, find the value of $$\vec{a} \cdot \vec{b}+\vec{b} \cdot \vec{c}+\vec{c} \cdot \vec{a}$$.
22. Find the. Vector and Cartesian equations of the line that passes through the points $$(3,-2,-5)$$ and $$(3,-2,6)$$.
23. Solve the following graphically :

Minimise :
$$
\mathbf{Z}=x+2 y
$$

Subject to the constraints:
$$
\begin{array}{r}
2 x+y \geq 3 \\
x+2 y \geq 6 \\
x, y \geq 0
\end{array}
$$

{SECTION -D)}
(LONG ANSWER TYPE QUESTIONS)

24. Using properties of determinants show that :
$$
\left|\begin{array}{lll}
a & a^{2} & b c \\
b & b^{2} & c a \\
c & c^{2} & a b
\end{array}\right|=
(a-b)(b-c)(c-a)(a b+b c+c a)
$$

Or

Solve the system of linear equations using matrix method :
$$
\begin{aligned}
2 x+y+z & =1 \\
x-2 y-z & =\frac{3}{2} \\
3 y-5 z & =9
\end{aligned}
$$

✅ For Answer Click Here

25. Find $$\frac{d y}{d x}$$ if $$x=a\left(\cos t+\log \tan \frac{t}{2}\right), y=a \sin t$$.

Find $$\frac{d y}{d x}$$ if $$(\cos x)^{y}=(\cos y)^{x}$$.

XIIARJKUT23-9112-X B-12-X

26. Find $$\int_{0}^{\pi / 2} \sqrt{\sin \phi} \cos ^{5} \phi d \phi$$.

{Or}

Using integration, find the area of region bounded by triangle whose vertices are $$(-1,0),(1,3),(3,2)$$.
27. If $$\vec{a}=\hat{i}+4 \hat{j}+2 \hat{k}, \vec{b}=3 \hat{i}-2 \hat{j}+7 \hat{k},
\vec{c}=2 \hat{i}-\hat{j}+4 \hat{k}$$, find the vector $$\vec{d}$$ which is perpendicular to both $$\vec{a}$$ and $$\vec{b}$$ and $$\vec{c} \cdot \vec{d}=15$$.
Or

Find the shortest distance between the lines $$\frac{x+1}{7}=\frac{y+1}{-6}=\frac{z+1}{1}$$ and $$\frac{x-3}{1}=\frac{y-5}{-2}=\frac{z-7}{1}$$.
28. Find the intervals in which the function $$f(x)=-2 x^{3}-9 x^{2}-12 x+1$$ is strictly increasing or decreasing.
Or

Evaluate :
$$
\int_{0}^{\pi} \frac{x \sin x}{1+\cos ^{2} x} d x
$$

29. There are $$5 \%$$ defective items in a large bulk of items. What is the probability that a sample of 10 items will include not more defective item.
Or

Find the mean and variance of the number obtained on a throw of an unbiased die.