Jkbose sample paper 2024-2025 Maths

Issue by jkbose 

General Instructions:

1. This Question paper contains – Four sections A, B, C and D. Each section is compulsory.
2. Section A-Question 1 to 10 comprises of $$\mathbf{1 0}$$ Very Very Short Answer type questions of 1 mark each
3. Section B-Question 11 to $$\mathbf{2 0}$$ comprises of $$\mathbf{1 0}$$ Very Short Answer (VSA)-type questions of 2 marks each.
4. Section C-Question 21 to 28 comprises of 8 Short Answer (SA)-type questions of 4 marks each.
5. Section D-Question 29 to 31 comprises of Long Answer (LA)-type questions of 6 marks each.

{SECTION A}
Q.1) If $$\left|\begin{array}{ll}2 & 4 \\ 5 & 1\end{array}\right|=\left|\begin{array}{cc}2 x & 4 \\ 6 & x\end{array}\right|$$, then the possible value(s) of ‘ $$x$$ ‘ is/are
(a) 3
(b) $$\sqrt{3}$$
(c) $$-\sqrt{3}$$
(d) $$\sqrt{3},-\sqrt{3}$$
Q.2) If $$\mathrm{A}, \mathrm{B}$$ are non-singular square matrices of the same order, then $$\left(A B^{-1}\right)^{-1}=$$
(a) $$A^{-1} B$$
(b) $$A^{-1} B^{-1}$$
(c) $$B A^{-1}$$
(d) $$A B$$
Q.3)

The degree of the differential equation $$1+\left(\frac{d y}{d x}\right)^{2}=x$$ is $$\qquad$$ .
Q.4) If $$y=A e^{5 x}+B e^{-5 x}$$, then $$\frac{d^{2} y}{d x^{2}}$$ is equal to
(a) 25 y
(b) 5 y
(c) -25 y
(d) 15 y
Q.5) If $$f^{\prime}(x)=x+\frac{1}{x}$$, then $$f(x)$$ is
(a) $$x^{2}+\log |x|+C$$
(b) $$\frac{x^{2}}{2}+\log |x|+C$$
(c) $$)_{2}^{x}+\log |x|+C$$
(d) $$\frac{x}{2}-\log |x|+C$$
Q.6) If the radius of the circle is increasing at the rate of $$0.5 \mathrm{~cm} / \mathrm{s}$$, then the rate of increase of its circumference is $$\qquad$$ ….

Q.7)
$$\int \frac{e^{x}(1+x)}{\cos ^{2}\left(\mathrm{xe}^{\mathrm{x}}\right)} \mathrm{dx}$$ is equal to
(A) $$\tan \left(\mathrm{xe}^{\mathrm{x}}\right)+\mathrm{c}$$
(B) $$\quad \cot \left(\mathrm{xe}^{\mathrm{x}}\right)+\mathrm{c}$$
(C) $$\quad \cot \left(\mathrm{e}^{\mathrm{x}}\right)+\mathrm{c}$$
(D) $$\tan \left|\mathrm{e}^{\mathrm{x}}(1+\mathrm{x})\right|+\mathrm{c}$$
Q.8) The value of $$p$$ for which $$p(\hat{i}+\hat{j}+\hat{k})$$ is a unit vector is
(A) 0
(B) $$\frac{1}{\sqrt{3}}$$
(C) 1
(D) $$\sqrt{3}$$
Q.9) The vector equation of XY-plane is
(A) $$\vec{r} \cdot \hat{k}=0$$
(B) $$\quad \vec{r} \cdot \hat{j}=0$$
(C) $$\vec{r} \cdot \hat{\mathrm{i}}=0$$
(D) None of these
Q.10) The feasible region for an LPP is shown below :

(A) $$(0,0)$$
(B) $$(0,8)$$
(C) $$(5,0)$$
(D) $$(4,10)$$

{SECTION B}
Q.11) Check if the relation $$R$$ on the set $$A=\{1,2,3,4,5,6 ;$$ defined as $$\mathrm{R}=\{(x, y): y$$ is divisible by $$x\}$$ is (i) symmetric (ii) transitive
Q.12) Find the value of $$\sin ^{-1}\left[\sin \left(\frac{13 \pi}{7}\right)\right]$$
Q.13) Show that the function $$f$$ defined by $$f(x)=(x-1) \mathrm{e}^{x}+1$$ is an increasing function for all $$x>0$$.
Q.14) Find the unit vector perpendicular to each of the vectors $$\vec{a}=4 \hat{i}+3 \hat{j}+\hat{k}$$ and $$\vec{b}=2 \hat{i}-\hat{j}+2 \hat{k}$$.
Q.15) Find $$|\vec{x}|$$ if $$(\vec{x}-\vec{a}) \cdot(\vec{x}+\vec{a})=12$$, where $$\vec{a}$$ is a unit vector.
Q.16) Evaluate the integral $$\int(2 x+3) d x$$
Q.17) Evaluate $$\int_{0}^{2 \pi}|\sin x| d x$$
Q.18) Compute $$P(A \cap B)$$, Where $$P(A)=0.8, P(B)=0.5$$ and $$P(A / B)=0.4$$
Q.19) If $$P(A)=0.25$$ then find $$P(not A)$$
Q.20) If A is a symmetric Matric, then show that $$\mathrm{A}-A^{\prime}$$ is a skew symmetric matrix.

{SECTION C}
Q.21) Solve the differential equation: $$y d x+\left(x-y^{2}\right) d y=0$$
Q.22) Evaluate: $$\int_{0}^{4}|x-1| d x$$

Q.23) If $$y=a \cos (\log x)+b \sin (\log x)$$, show that $$x^{2} \frac{d^{2} y}{d x^{2}}+x \frac{d y}{d x}+y=0$$,
Q.24) Find the shortest distance between the two lines:
$$
\begin{aligned}
\bar{r} & =6 \hat{\imath}+2 \hat{\jmath}+2 \hat{k}+\lambda(\hat{\imath}-2 \hat{\jmath}+2 \hat{k}) \\
\text { and } \quad \bar{r} & =-4 \hat{\imath}-\hat{k}+\mu(3 \hat{\imath}-2 \hat{\jmath}-2 \hat{k})
\end{aligned}
$$
Q.25) If $$\vec{a}=\hat{\imath}-\hat{j}+7 \hat{k}$$ and $$\vec{b}=5 \hat{\imath}-\hat{\jmath}+\lambda \hat{k}$$, then find the value of $$\lambda$$ so that the vectors $$\vec{a}+\vec{b}$$ and $$\vec{a}-\vec{b}$$ are orthogonal.
Q.26

Solve the following Linear Programming Problem graphically:
Maximize $$Z=400 \mathrm{x}+300 \mathrm{y}$$ subject to $$x+y \leq 200, x \leq 40, x \geq 20, y \geq 0$$
Q.27) Show that the function $$f: \mathrm{R} \rightarrow \mathrm{R}$$ defined by $$f(x)=\frac{x}{x^{2}+1}, \forall x \in \mathrm{R}$$ is neither one-one nor onto.
Q.28) The probability distribution of a random variable X , where k is a constant is given below:
$$
\mathrm{P}(\mathrm{X}=\mathrm{x})=\left\{\begin{array}{ccc}
0 \cdot 1, & \text { if } & \mathrm{x}=0 \\
\mathrm{kx}, & \text { if } & \mathrm{x}=1 \\
\mathrm{kx}, & \text { if } & \mathrm{x}=2 \text { or } 3 \\
0, & \text { otherwise } &
\end{array}\right.
$$

Determine
(a) the value of k
(b) $$\quad \mathrm{P}(\mathrm{x} \leq 2)$$

{SECTION D}
Q.29) Solve the following system of equations by matrix method:
$$
\begin{aligned}
& x-y+2 z=7 \\
& 2 x-y+3 z=12 \\
& 3 x+2 y-z=5
\end{aligned}
$$

OR

If $$A=\left[\begin{array}{ccc}1 & 3 & 2 \\ 2 & 0 & -1 \\ 1 & 2 & 3\end{array}\right]$$, then show that $$A^{3}-4 A^{2}-3 A+11 I=0$$. Hence find $$A^{4}$$.
Q. 30) Evaluate: $$\int \frac{\mathrm{dx}}{\sqrt{5-4 \mathrm{x}-\mathrm{x}^{2}}}$$

OR
$$
\text { Find } \int \frac{\left(x^{3}+x+1\right)}{\left(x^{2}-1\right)} d x
$$
Q. 31) If $$\mathrm{y}=\left(\tan ^{-1} x\right)^{2}$$, show that $$\left(x^{2}+1\right)^{2} y_{2}+2 x\left(x^{2}+1\right) y_{1}=2$$

OR

Find $$\frac{d y}{d x}$$, if $$x^{y} \cdot y^{x}=x^{x}$$.