MARCH 2024 - Maharashtra Board STD 12 Science Maths Questions

Q 1MCQ2 Marks

If the mean and variance of a binomial distribution are $18$ and $12$ respectively, then $n = .......$

A
$36$
✓
$54$
C
$18$
D
$27$

Answer: B.

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Q 2MCQ2 Marks

The integrating factor of linear differential equation $x \frac{d y}{d x}+2 y=x^2 \log x$ is $......$

A
$x$
B
$\frac{1}{x}$
✓
$x^2$
D
$\frac{1}{x^2}$

Answer: C.

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Q 3MCQ2 Marks

If $\int_{\frac{-\pi}{4}}^{\frac{\pi}{4}} x^3 \cdot \sin ^4 x d x= k$ then $k = ......$

A
$1$
B
$2$
C
$4$
✓
$0$

Answer: D.

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Q 4MCQ2 Marks

The slope of the tangent to the curve $x=\sin \theta$ and $y=\cos 2 \theta$ at $\theta=\frac{\pi}{6}$ is ________.

A
$-2 \sqrt{3}$
B
$-2 \sqrt{3}$
✓
$-2$
D
$-\frac{1}{2}$

Answer: C.

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Q 5MCQ2 Marks

The perpendicular distance of the plane $\bar{r} \cdot(3 \hat{i}+4 \hat{j}+12 \widehat{k})=78$ from the origin is ________.

A
4
B
5
✓
6
D
8

Answer: C.

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Q 6Answer the following questions in short.1 Mark

A particle is moving along the X-axis. Its acceleration at time t is proportional to its velocity at that time. Find the differential equation of the motion of the particle.

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Q 7Answer the following questions in short.1 Mark

Evaluate: $\int \frac{1}{x^2+25} d x$

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Q 8Answer the following questions in short.1 Mark

If the vectors $2 \hat{i}-3 \hat{j}+4 \hat{k}$ and $p \hat{i}+6 \hat{j}-8 \hat{k}$ are collinear, then find the value of $p$.

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Q 9Answer the following questions in short.1 Mark

Write the following compound statement 'Nagpur is in Maharashtra and Chennai' symbolically.

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Q 10Solve the Following Question.(2 Marks)2 Marks

If two coins are tossed simultaneously, write the probability distribution of the number of heads.

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Q 11Solve the Following Question.(2 Marks)2 Marks

Solve: $1+\frac{d y}{d x}=\operatorname{cosec}(x+y)$; put $x+y=u$.

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Q 12Solve the Following Question.(2 Marks)2 Marks

Find the area of the region bounded by the curve $y=x^2$, and the lines $x=1, x=2$, and $y=0$.

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Q 13Solve the Following Question.(2 Marks)2 Marks

Evaluate the definite integral: $\int_0^{\frac{\pi}{2}} \cos ^2 x d x$

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Q 14Solve the Following Question.(2 Marks)2 Marks

Evaluate: $\int \log x d x$

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Q 15Solve the Following Question.(3 Marks)3 Marks

A die is thrown 6 times. If ‘getting an odd number’ is a success, find the probability of 5 successes.

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Q 16Solve the Following Question.(3 Marks)3 Marks

Find $k$, if
$f(x)=k x^2(1-x)$, for $0 < x < 1$,
$=0$
otherwise
is the p.d.f. of random variable $X$.

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Q 17Solve the Following Question.(3 Marks)3 Marks

Solve the following differential equation:
$x \frac{d y}{d x}-y+x \sin \left(\frac{y}{x}\right)=0$

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Q 18Solve the Following Question.(3 Marks)3 Marks

Prove that $\int \frac{1}{a^2-x^2} d x=\frac{1}{2 a} \log \left|\frac{a+x}{a-x}\right|+c$

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Q 19Solve the Following Question.(3 Marks)3 Marks

Find the approximate value of $\tan ^{-1}$ (1.002). [Given: $\pi=3.1416$ ]

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Q 20Solve the Following Question.(4 Marks)4 Marks

Prove that : $\int_0^{2 a} f(x) d x=\int_0^a f(x) d x+\int_0^a f(2 a-x) dx$
Hence show that : $\int_0^\pi \sin x d x=2 \int_0^{\frac{\pi}{2}} \sin x dx$

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Q 21Solve the Following Question.(4 Marks)4 Marks

Evalution $\int \frac{5 e^x}{\left(e^x+1\right)\left(e^{2 x}+9\right)} d x$

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Q 22Solve the Following Question.(4 Marks)4 Marks

A box with a square base is to have an open top. The surface area of box is 147 sq. cm. What should be its dimensions in order that the volume is largest?

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Q 23Solve the Following Question.(4 Marks)4 Marks

If $x=f(t)$ and $y=g(t)$ are differentiable functions of $t$, so that $y$ is function of $x$ and $\frac{d x}{d t} \neq 0$
then prove that $\frac{ dy }{ d x}=\frac{\frac{ dy }{ dt }}{\frac{ dx }{ dt }}$.
Hence find $\frac{d y}{d x}$, if $x=a t^2, y=2 a t$.

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Q 24Solve the Following Question.(4 Marks)4 Marks

Solve the following L.P.P. by graphical method :
Maximize : $Z=10 x+25 y$
$\begin{aligned} \text { Subject to : } & 0 \leq x \leq 3, \\ 0 & \leq y \leq 3, \\ & x+y \leq 5 .\end{aligned}$

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MARCH 2024 Maths - MARCH 2024

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