Differentiation (Part-38) | S N De

MATHEMATICS (New Syllabus) – 2022

PART-B

1. Choose the correct answer from the given alternatives (Alternatives are to be noted): $1 \times 10 = 10$

(i) An unbiased coin is tossed for $3$ times. Then the probability of getting only one head is:

(a) $\frac{1}{2}$ (b) $\frac{5}{8}$ (c) $\frac{3}{4}$ (d) $\frac{3}{8}$

OR

A coin is tossed $10$ times. The probability of getting head $6$ times is:

(a) ${}^{10}C_5 \cdot \frac{1}{2^{10}}$ (b) ${}^{10}C_3 \cdot \frac{1}{2^{10}}$ (c) ${}^{10}C_4 \cdot \frac{1}{2^{10}}$ (d) ${}^{10}C_8 \cdot \frac{1}{2^{10}}$

(ii) $A = (1, 0, 2)$ and $B = (0, 1, 1)$, then direction cosines of the line $AB$ are:

(a) $1, -1, 1$ (b) $\frac{1}{\sqrt{3}}, -\frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}$ (c) $-\frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}$ (d) $\frac{1}{\sqrt{2}}, -\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}}$

OR

$\vec{a} = \hat{i} + 3\hat{j} - \hat{k}$, and $\vec{b} = 2\hat{i} + 6\hat{j} + \lambda\hat{k}$. If $\vec{a}$ and $\vec{b}$ vectors are parallel, then the value of $\lambda$ is:

(a) $3$ (b) $-6$ (c) $-3$ (d) $-2$

(iii) $f(x) = \log(3x + 1)$, then the value of $f''(1)$ is:

(a) $\frac{9}{16}$ (b) $-\frac{9}{16}$ (c) $\frac{9}{4}$ (d) $-\frac{9}{4}$

OR

If $f(x) = \frac{\sin x}{x} \; (x \neq 0)$ is continuous at $x = 0$, then the value of $f(0)$ will be:

(a) $0$ (b) $1$ (c) $\pi$ (d) $\frac{\pi}{2}$

(iv) $A = \begin{pmatrix} a \\ 6 \end{pmatrix}$ and $B = \begin{pmatrix} 3 \\ b \end{pmatrix}$ and $A = B$, then $(a, b)$ is equal to:

(a) $(3, 6)$ (b) $(6, 3)$ (c) $(6, 6)$ (d) $(3, 3)$

OR

If $a + b + c = 0$, then the value of $\begin{vmatrix} a & b & c \\ b & c & a \\ c & a & b \end{vmatrix}$ is:

(a) $1$ (b) $a$ (c) $0$ (d) $-1$

(v) The domain in which the functions $f(x) = 3x^2 - 2x$ and $g(x) = 3(3x - 2)$ will be equal to:

(a) $\{1, \frac{2}{3}\}$ (b) $\{1, 3\}$ (c) $\{\frac{2}{3}, 3\}$ (d) $\{\frac{2}{3}, 0\}$

OR

Let $A = \{1, 2, 3\}$ and $R$ be a relation defined on $A$, such that $R = \{(1, 1), (1, 2), (2, 1)\}$; then the relation $R$ will be:

(a) Reflexive (b) Symmetric (c) Transitive (d) None of these

(vi) $P(B) = \frac{9}{13}$, $P(A \cap B) = \frac{4}{13}$, then value of $P(A/B)$ is:

(a) $\frac{2}{9}$ (b) $\frac{4}{13}$ (c) $\frac{9}{13}$ (d) $\frac{4}{9}$

(vii) The equation of the plane with intercepts $2, 3, 4$ units on the $x$-axis, $y$-axis and $z$-axis respectively is:

(a) $6x + 4y + 3z = 12$ (b) $4x + 3y + 6z = 12$ (c) $3x + 6y + 4z = 12$ (d) $6x + 2y + 3z = 12$

(viii) The value of the function $f(x) = 4x - x^2 - 3$ will be maximum when:

(a) $x = 3$ (b) $x = 2$ (c) $x = -2$ (d) $x = -3$

(ix) The degree of the differential equation $\frac{d^3y}{dx^3} + y = \sqrt[3]{1 + \frac{dy}{dx}}$ is:

(a) $1$ (b) $2$ (c) $3$ (d) $4$

OR

The value of $\int e^{a \log_e x} \, dx$ will be $(a \neq -1)$:

(a) $\frac{1}{a} e^{a \log_e x} + c$ (b) $\frac{1}{x} + c$ (c) $a x^{a-1} + c$ (d) $\frac{x^{a+1}}{a+1} + c$

(x) The value of $\tan\left(\frac{\pi}{2} - \tan^{-1}\frac{1}{3}\right)$ is equal to:

(a) $\frac{1}{3}$ (b) $3$ (c) $\frac{1}{2}$ (d) $\frac{2}{3}$

HIGHER SECONDARY MATHS COMPLETE SOLUTIONS (2015-2019) [WBCHSE]

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