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Maths MCQ

MARCH 2024 · Maharashtra Board STD 12 Science (MSBSHSE - English Medium). 8 questions.

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8 questions · timed · auto-graded

MCQ 12 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
Correct option: B.
$54$
$(B)$ If the mean and variance of a binomial distribution are $18$ and $12$ respectively, then $n =\underline{ 5 4 }$.
Explanation:
$np = 18$ and $npq = 12$
$\therefore \frac{ npq }{ np }=\frac{12}{18}$
$\therefore q =\frac{2}{3}$
$\therefore p =1- q =1-\frac{2}{3}=\frac{1}{3}$
$\therefore n \left(\frac{1}{3}\right)=18$
$\therefore n =54$
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MCQ 22 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
Correct option: C.
$x^2$
$e^{\int \frac{2}{x} d x}$
$=e^{2 \log x}$
$\text { I.F }=x^2$
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MCQ 32 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
Correct option: D.
$0$
Let $I =\int_{\frac{-\pi}{4}}^{\frac{\pi}{4}} x^3 \sin ^4 x \cdot d x$
Let $f ( x )=x^3 \sin ^4 x$
$\therefore f(-x)=(-x)^3 \sin ^4(-x)$
$=-x^3 \sin ^4 x$
$=-f(x)$
$\therefore f$ is an odd function.
$\therefore \int_{\frac{-\pi}{4}}^{\frac{\pi}{4}} f(x) \cdot d x=0$,
i.e. $\int_{\frac{-\pi}{4}}^{\frac{\pi}{4}} x^3 \sin ^4 x \cdot d x=0$.
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MCQ 42 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
Correct option: C.
$-2$
(C) The slope of the tangent to the curve $x =\sin \theta$ and $y =\cos 2 \theta$ at $\theta=\frac{\pi}{6}$ is $\underline{ - 2 }$.
Explanation:
$x=\sin \theta, y=\cos \theta$, at $\theta=\frac{\pi}{6}$
$\frac{d y}{d x}=\frac{\frac{d y}{d \theta}}{\frac{d x}{d \theta}}$
$=\frac{-\sin \theta \times 2}{\cos \theta}$
$=\frac{-2 \times 2 \sin \theta \cdot \cos \theta}{\cos \theta}$
$=-4 \sin \theta$
$=-4 \sin \frac{\pi}{6}$
$=-4 \times \frac{1}{2}$
$=-2$
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MCQ 52 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
Correct option: C.
6
(C) The perpendicular distance of the plane $\bar{r} .(3 \hat{i}+4 \hat{j}+12 \widehat{k})=78$ from the origin is $\underline{6}$.
Explanation:
Given plane is $\bar{r} .(3 \hat{i}+4 \hat{j}+12 \widehat{k})=78$
$\bar{n}=3 \hat{i}+4 \hat{j}+12 \widehat{k}$
$|\bar{n}|=\sqrt{(3)^2+(4)^2+(12)^2}$
$|\bar{n}|=\sqrt{9+16+144}$
$|\bar{n}|=\sqrt{13}$
$\therefore \bar{r} . \hat{i}=\frac{78}{|\bar{n}|}$
$=\frac{7}{13}$
=6
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MCQ 62 Marks
If α, β, γ are direction angles of a line and α = 60°, β = 45°, γ = ______.
  • A
    $30^{\circ}$ or $90^{\circ}$
  • B
    $45^{\circ}$ or $60^{\circ}$
  • C
    $90^{\circ}$ or $30^{\circ}$
  • ✓
    $60^{\circ}$ or $120^{\circ}$
Answer
Correct option: D.
$60^{\circ}$ or $120^{\circ}$
(D) If α, β, γ are direction angles of a line and α =$60^{\circ}$ , β = $45^{\circ}$, γ = $60^{\circ}$ or $120^{\circ}$.
Explanation:
By applying the relation $\cos ^2 \alpha+\cos ^2 \beta+\cos ^2 \gamma$= 1 with α = $60^{\circ}$ and β = $45^{\circ}$, the calculation yields γ as $60^{\circ}$ or $120^{\circ}$, conforming to the condition for direction angles.
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MCQ 72 Marks
The principle solutions of the equation $\cos \theta=\frac{1}{2}$ are ________.
  • A
    $\frac{\pi}{6}, \frac{5 \pi}{6}$
  • ✓
    $\frac{\pi}{3}, \frac{5 \pi}{3}$
  • C
    $\frac{\pi}{6}, \frac{7 \pi}{6}$
  • D
    $\frac{\pi}{3}, \frac{2 \pi}{3}$
Answer
Correct option: B.
$\frac{\pi}{3}, \frac{5 \pi}{3}$
(B) The principle solutions of the equation $\cos \theta=\frac{1}{2}$ are $\frac{ \pi }{ 3 }, \frac{ 5 \pi }{ 3 }$.
Explanation:
$\cos \theta=\frac{1}{2}$
$\cos \theta=\cos \frac{\pi}{3}$
$\cos \theta=\cos \left(2 \pi-\frac{\pi}{3}\right)$
$\frac{\pi}{3}, \frac{5 \pi}{3}$
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MCQ - Maths STD 12 Science Questions - Vidyadip