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RD Sharma

For the binary operation ×10 set S = {1, 3, 7, 9}, find the inverse of 3.

Binary Operations, CBSE, Class 12, Exercise 3.5, Maths, RD Sharma

Answer: 1 ×10 1 = remainder obtained by dividing 1 × 1 by 10 = 1 3 ×10 7 = remainder obtained by dividing 3 × 7 by 10 = 1 7 ×10 9 = remainder obtained by dividing 7 × 9 by 10 = 3 Composition table:...

Construct the composition table for ×5 on set Z5 = {0, 1, 2, 3, 4}

Binary Operations, CBSE, Class 12, Exercise 3.5, Maths, RD Sharma

Answer: 1 ×5 1 = remainder obtained by dividing 1 × 1 by 5 = 1 3 ×5 4 = remainder obtained by dividing 3 × 4 by 5 = 2 4 ×5 4 = remainder obtained by dividing 4 × 4 by 5 = 1 Composition table: ×5 0 1...

Construct the composition table for ×6 on set S = {0, 1, 2, 3, 4, 5}.

Binary Operations, CBSE, Class 12, Exercise 3.5, Maths, RD Sharma

Answer: 1 ×6 1 = remainder obtained by dividing 1 × 1 by 6 = 1 3 ×6 4 = remainder obtained by dividing 3 × 4 by 6 = 0 4 ×6 5 = remainder obtained by dividing 4 × 5 by 6 = 2 Composition table: ×6 0 1...

Construct the composition table for +5 on set S = {0, 1, 2, 3, 4}

Binary Operations, CBSE, Class 12, Exercise 3.5, Maths, RD Sharma

Answer: 1 +5 1 = remainder obtained by dividing 1 + 1 by 5 = 2 3 +5 1 = remainder obtained by dividing 3 + 1 by 5 = 2 4 +5 1 = remainder obtained by dividing 4 + 1 by 5 = 3 Composition Table: +5 0 1...

Construct the composition table for ×4 on set S = {0, 1, 2, 3}.

Binary Operations, CBSE, Class 12, Exercise 3.5, Maths, RD Sharma

Answer: Given, ×4 on set S = {0, 1, 2, 3} 1 ×4 1 = remainder obtained by dividing 1 × 1 by 4 = 1 0 ×4 1 = remainder obtained by dividing 0 × 1 by 4 = 0 2 ×4 3 = remainder obtained by dividing 2 × 3...

Let * be a binary operation on Z defined by a * b = a + b – 4 for all a, b ∈ Z. (i) Show that * is both commutative and associative. (ii) Find the identity element in Z

Binary Operations, CBSE, Class 12, Exercise 3.4, Maths, RD Sharma

Answers: (i) Consider, a, b ∈ Z a * b = a + b – 4 = b + a – 4 = b * a a * b = b * a, ∀ a, b ∈ Z Then, * is commutative on Z. a * (b * c) = a * (b + c – 4) = a + b + c -4 – 4 = a + b + c – 8 (a * b)...

Let * be a binary operation on Q0 (set of non-zero rational numbers) defined by a * b = (3ab/5) for all a, b ∈ Q0. Show that * is commutative as well as associative. Also, find its identity element, if it exists.

Binary Operations, CBSE, Class 12, Exercise 3.4, Maths, RD Sharma

Answer: Consider, a, b ∈ Q0 a * b = (3ab/5) = (3ba/5) = b * a a * b = b * a, for all a, b ∈ Q0 a * (b * c) = a * (3bc/5) = [a (3 bc/5)] /5 = 3 abc/25 (a * b) * c = (3 ab/5) * c = [(3 ab/5)...

Let * be a binary operation on Q – {-1} defined by a * b = a + b + ab for all a, b ∈ Q – {-1}. Then, (i) Show that * is both commutative and associative on Q – {-1} (ii) Find the identity element in Q – {-1}

Binary Operations, CBSE, Class 12, Exercise 3.4, Maths, RD Sharma

Answers: (i) Consider, a, b ∈ Q – {-1} a * b = a + b + ab = b + a + ba = b * a a * b = b * a, ∀ a, b ∈ Q – {-1} a * (b * c) = a * (b + c + b c) = a + (b + c + b c) + a (b + c + b c) = a + b +...

Let * be a binary operation on Q – {-1} defined by a * b = a + b + ab for all a, b ∈ Q – {-1}. Then, Show that every element of Q – {-1} is invertible. Also, find inverse of an arbitrary element.

Binary Operations, CBSE, Class 12, Exercise 3.4, Maths, RD Sharma

Answer: Consider, a ∈ Q – {-1} and b ∈ Q – {-1} be the inverse of a. a * b = e = b * a a * b = e and b * a = e a + b + ab = 0 and b + a + ba = 0 b (1 + a) = – a Q – {-1} b = -a/1 + a Q – {-1}...

Let A = R0 × R, where R0 denote the set of all non-zero real numbers. A binary operation ‘O’ is defined on A as follows: (a, b) O (c, d) = (ac, bc + d) for all (a, b), (c, d) ∈ R0 × R. (i) Show that ‘O’ is commutative and associative on A (ii) Find the identity element in A

Binary Operations, CBSE, Class 12, Exercise 3.4, Maths, RD Sharma

Answers: (i) Consider, X = (a, b) Y = (c, d) ∈ A, ∀ a, c ∈ R0 b, d ∈ R X O Y = (ac, bc + d) Y O X = (ca, da + b) X O Y = Y O X, ∀ X, Y ∈ A O is not commutative on A. X = (a, b) Y = (c, d) a Z = (e,...

Let A = R0 × R, where R0 denote the set of all non-zero real numbers. A binary operation ‘O’ is defined on A as follows: (a, b) O (c, d) = (ac, bc + d) for all (a, b), (c, d) ∈ R0 × R. Find the invertible element in A.

Binary Operations, CBSE, Class 12, Exercise 3.4, Maths, RD Sharma

Answer: Consider, F = (m, n) be the inverse in A ∀ m ∈ R0 and n ∈ R X O F = E F O X = E (am, bm + n) = (1, 0) and (ma, na + b) = (1, 0) Considering (am, bm + n) = (1, 0) am = 1 m = 1/a And bm + n =...

Let * be a binary operation on Z defined by a * b = a + b – 4 for all a, b ∈ Z. Find the invertible element in Z.

Binary Operations, CBSE, Class 12, Exercise 3.4, Maths, RD Sharma

Answer: Consider, a ∈ Z and b ∈ Z be the inverse of a. a * b = e = b * a a * b = e and b * a = e a + b – 4 = 4 and b + a – 4 = 4 b = 8 – a ∈ Z Hence, 8 – a is the inverse of a ∈...

Find the identity element in the set of all rational numbers except – 1 with respect to * defined by a * b = a + b + ab

Binary Operations, CBSE, Class 12, Exercise 3.3, Maths, RD Sharma

Answer: Consider, e be the identity element in I+ with respect to * such that a * e = a = e * a, ∀ a ∈ Q – {-1} a * e = a and e * a = a, ∀ a ∈ Q – {-1} a + e + ae = a and e + a + ea = a, ∀ a ∈ Q –...

Find the identity element in the set I+ of all positive integers defined by a * b = a + b for all a, b ∈ I+.

Binary Operations, CBSE, Class 12, Exercise 3.3, Maths, RD Sharma

Answer: Consider, e be the identity element in I+ with respect to * a * e = a = e * a, ∀ a ∈ I+ a * e = a and e * a = a, ∀ a ∈ I+ a + e = a and e + a = a, ∀ a ∈ I+ e = 0, ∀ a ∈ I+ Hence, 0 is the...

Differentiate $\sin ^{2} x$ with respect to $e^{\cos x}$

CBSE, Class 12, Differentiation, Exercise 11.8, Maths, RD Sharma

As per the given question,

Differentiate $\tan ^{-1}\left(\frac{1-x}{1+x}\right)$ with respect to $\sqrt{1-x^{2}}$ , if $-1<x<1$

CBSE, Class 12, Differentiation, Exercise 11.8, Maths, RD Sharma

As per the given question,

Differentiate $\sin ^{-1}\left(2 a x \sqrt{1-a^{2} x^{2}}\right)$ with respect to $\sqrt{1-a^{2} x^{2}}$ , if $-\frac{1}{\sqrt{2}}<a x<\frac{1}{\sqrt{2}}$

CBSE, Class 12, Differentiation, Exercise 11.8, Maths, RD Sharma

As per the given question,

Differentiate $\sin ^{-1} \sqrt{1-x^{2}}$ with respect to $\cot ^{-1}\left(\frac{x}{\sqrt{1-x^{2}}}\right)$ , if $0<x<1$

CBSE, Class 12, Differentiation, Exercise 11.8, Maths, RD Sharma

As per the given question,

Differentiate $\tan ^{-1}\left(\frac{x}{\sqrt{1-x^{2}}}\right)$ with respect to $\sin ^{-1}\left(2 x \sqrt{1-x^{2}}\right)$ , if $-\frac{1}{\sqrt{2}}<x<\frac{1}{\sqrt{2}}$

CBSE, Class 12, Differentiation, Exercise 11.8, Maths, RD Sharma

As per the given question,

Differentiate $\cos ^{-1}\left(4 x^{3}-3 x\right)$ with respect to $\tan ^{-1}\left(\frac{\sqrt{1-x^{2}}}{x}\right)$ , if $\frac{1}{2}<x<1$

CBSE, Class 12, Differentiation, Exercise 11.8, Maths, RD Sharma

As per the given question,

Differentiate $\sin ^{-1}\left(\frac{2 x}{1+x^{2}}\right)$ with respect to $\tan ^{-1}\left(\frac{2 x}{1-x^{2}}\right)$ , if $-1<x<1$

CBSE, Class 12, Differentiation, Exercise 11.8, Maths, RD Sharma

As per the given question,

Differentiate $\tan ^{-1}\left(\frac{\cos x}{1+\sin x}\right)$ with respect to $\sec ^{-1} x$

CBSE, Class 12, Differentiation, Exercise 11.8, Maths, RD Sharma

As per the given question,

Differentiate $\tan ^{-1}\left(\frac{x-1}{x+1}\right)$ with respect to $\sin ^{-1}\left(3 x-4 x^{3}\right)$ , if $-\frac{1}{2}<x<\frac{1}{2}$

CBSE, Class 12, Differentiation, Exercise 11.8, Maths, RD Sharma

As per the given question,

Differentiate $\tan ^{-1}\left(\frac{2 x}{1-x^{2}}\right)$ with respect to $\cos ^{-1}\left(\frac{1-x^{2}}{1+x^{2}}\right)$ , if $0<x<1$

CBSE, Class 12, Differentiation, Exercise 11.8, Maths, RD Sharma

As per the given question,

Differentiate $\sin ^{-1}\left(2 x \sqrt{1-x^{2}}\right)$ with respect to $\tan ^{-1}\left(\frac{x}{\sqrt{1-x^{2}}}\right)$ , if $-\frac{1}{\sqrt{2}}<x<\frac{1}{\sqrt{2}}$

CBSE, Class 12, Differentiation, Exercise 11.8, Maths, RD Sharma

As per the given question,

Differentiate $\tan ^{-1}\left(\frac{1+a x}{1-a x}\right)$ with respect to $\sqrt{1+a^{2} x^{2}}$

CBSE, Class 12, Differentiation, Exercise 11.8, Maths, RD Sharma

As per the given question,

Differentiate $\sin ^{-1}\left(\frac{2 x}{1+x^{2}}\right)$ with respect to $\cos ^{-1}\left(\frac{1-x^{2}}{1+x^{2}}\right)$ , if $0<x<1$

CBSE, Class 12, Differentiation, Exercise 11.8, Maths, RD Sharma

As per the given question,

Differentiate $(\cos x)^{\sin x}$ with respect to $(\sin x)^{\cos x}$ .

CBSE, Class 12, Differentiation, Exercise 11.8, Maths, RD Sharma

As per the given question, ......(i) ........(ii)

Differentiate $\sin ^{-1}\left(2 x \sqrt{1-x^{2}}\right)$ with respect to $\sec ^{-1}\left(\frac{1}{\sqrt{1-x^{2}}}\right)$ , if $x \in\left(\frac{1}{\sqrt{2}}, 1\right)$

CBSE, Class 12, Differentiation, Exercise 11.8, Maths, RD Sharma

As per the given question,

Differentiate $\sin ^{-1}\left(2 x \sqrt{1-x^{2}}\right)$ with respect to $\sec ^{-1}\left(\frac{1}{\sqrt{1-x^{2}}}\right)$ , if $x \in\left(0, \frac{1}{\sqrt{2}}\right)$

CBSE, Class 12, Differentiation, Exercise 11.8, Maths, RD Sharma

As per the given question,

Differentiate $\tan ^{-1}\left(\frac{\sqrt{1+x^{2}}-1}{x}\right)$ with respect to $\sin ^{-1}\left(\frac{2 x}{1+x^{2}}\right)$ , if $-1<x<1, x \neq 0$

CBSE, Class 12, Differentiation, Exercise 11.8, Maths, RD Sharma

As per the given question,

Differentiate $\sin ^{-1}\left(4 x \sqrt{1-4 x^{2}}\right)$ with respect to $\sqrt{1-4 x^{2}}$ , if $x \in\left(-\frac{1}{2},-\frac{1}{2 \sqrt{2}}\right)$

CBSE, Class 12, Differentiation, Exercise 11.8, Maths, RD Sharma

As per the given question,

Differentiate $\sin ^{-1}\left(4 x \sqrt{1-4 x^{2}}\right)$ with respect to $\sqrt{1-4 x^{2}}$ , if $x \in\left(\frac{1}{2 \sqrt{2}}, \frac{1}{2}\right)$

CBSE, Class 12, Differentiation, Exercise 11.8, Maths, RD Sharma

As per the given question,

Differentiate $\sin ^{-1}\left(4 x \sqrt{1-4 x^{2}}\right)$ with respect to $\sqrt{1-4 x^{2}}$ , if $x \in\left(\frac{1}{-2 \sqrt{2}}, \frac{1}{2 \sqrt{2}}\right)$

CBSE, Class 12, Differentiation, Exercise 11.8, Maths, RD Sharma

As per the given question,

Differentiate $\sin ^{-1} \sqrt{1-x^{2}}$ with respect to $\cos ^{-1} x_{1}$ if $x \in(-1,0)$

CBSE, Class 12, Differentiation, Exercise 11.8, Maths, RD Sharma

As per the given question,

Differentiate $\sin ^{-1} \sqrt{1-x^{2}}$ with respect to $\cos ^{-1} x$ , if $x \in(0,1)$

CBSE, Class 12, Differentiation, Exercise 11.8, Maths, RD Sharma

As per the given question,

Differentiate $(\log x)^{x}$ with respect to $\log x$

CBSE, Class 12, Differentiation, Exercise 11.8, Maths, RD Sharma

As per the given question,

Differentiate $\log \left(1+x^{2}\right)$ with respect to $\tan ^{-1} x$

CBSE, Class 12, Differentiation, Exercise 11.8, Maths, RD Sharma

As per the given question,

Differentiate $\sec ^{2}\left(x^{2}\right)$ with respect to $x^{2}$

CBSE, Class 12, Differentiation, Exercise 11.8, Maths, RD Sharma

As per the given question,

If $x=a(2 \theta-\sin 2 \theta)$ and $y=a(1-\cos 2 \theta)$ , find $\frac{d y}{d x}$ when $\theta=\frac{\pi}{3}$

CBSE, Class 12, Differentiation, Exercise 11.7, Maths, RD Sharma

As per the given question,

If $\sin x=\frac{2 t}{1+t^{2}}, \tan y=\frac{2 t}{1-t^{2}}$ find $\frac{d y}{d x} \quad$

CBSE, Class 12, Differentiation, Exercise 11.7, Maths, RD Sharma

As per the given question,

If $x=3 \sin t-\sin 3 t, y=3 \cos t-\cos 3 t$ , find $\frac{d y}{d x}$ at $t=\frac{\pi}{3}$ .

CBSE, Class 12, Differentiation, Exercise 11.7, Maths, RD Sharma

As per the given question,

If $x=\frac{1+\log t}{t^{2}}, y=\frac{3+2 \log t}{t}$ , find $\frac{d y}{d x}$

CBSE, Class 12, Differentiation, Exercise 11.7, Maths, RD Sharma

As per the given question,

If $x=\cos t\left(3-2 \cos ^{2}\right)$ and $y=\sin t\left(3-2 \sin ^{2} t\right)$ find the value of $\frac{d y}{d x}$ at $t=\frac{\pi}{4}$

CBSE, Class 12, Differentiation, Exercise 11.7, Maths, RD Sharma

As per the given question,

If $x=a \sin 2 t(1+\cos 2 t)$ and $y=b \cos 2 t(1-\cos 2 t)$ , show that at $t=\frac{\pi}{4}, \frac{d y}{d x}=\frac{b}{a}$ .

CBSE, Class 12, Differentiation, Exercise 11.7, Maths, RD Sharma

As per the given question,

If $x=a(\theta-\sin \theta)$ and $y=a(1+\cos \theta)$ find $\frac{d y}{d x}$ , at $\theta=\frac{\pi}{3}$ .

CBSE, Class 12, Differentiation, Exercise 11.7, Maths, RD Sharma

As per the given question,

Find $dy/dx$ , if $y\;=\;12(1-cos t),$ $x\;=\;10(1-sin t)$

CBSE, Class 12, Differentiation, Exercise 11.7, Maths, RD Sharma

As per the given question,

If $x=a\left(\frac{1+t^{2}}{1-t^{2}}\right)$ and $y=\frac{2 t}{1-t^{2}}$ , find $\frac{d y}{d x} \quad$

CBSE, Class 12, Differentiation, Exercise 11.7, Maths, RD Sharma

As per the given question,

If $x=\left(t+\frac{1}{t}\right)^{2}$ and $y=a^{1+\frac{1}{t}}$ , find $\frac{d y}{d x}$

CBSE, Class 12, Differentiation, Exercise 11.7, Maths, RD Sharma

As per the given question,

If $x$ and $y$ are connected parametrically by the equation, without eliminating the parameter, find $\frac{d y}{d x}$ . $x=\frac{\sin ^{3} t}{\sqrt{\cos 2 t}}, y=\frac{\cos ^{3} t}{\sqrt{\cos 2 t}}$

CBSE, Class 12, Differentiation, Exercise 11.7, Maths, RD Sharma

As per the given question,

If $x=\sin ^{-1}\left(\frac{2 t}{1+t^{2}}\right)$ and $y=\tan ^{-1}\left(\frac{2 t}{1+t^{2}}\right) .-1<t<1$ , prove that $\frac{d y}{d x}=1$

CBSE, Class 12, Differentiation, Exercise 11.7, Maths, RD Sharma

As per the given question,

If $x=a\left(t+\frac{1}{t}\right)$ and $y=a\left(t-\frac{1}{t}\right)$ , prove that $\frac{d y}{d x}=\frac{x}{y}$ .

CBSE, Class 12, Differentiation, Exercise 11.7, Maths, RD Sharma

As per the given question,

If $x=\cos t$ and $y=\sin t$ , prove that $\frac{d y}{d x}=\frac{1}{\sqrt{3}}$ at $t=\frac{2 \pi}{3}$

CBSE, Class 12, Differentiation, Exercise 11.7, Maths, RD Sharma

As per the given question,

If $x=x=e^{\cos 2 t}$ and $y=e^{\sin 2 t}$ , prove that $\frac{d y}{d x}=-\frac{y \log x}{x \log y}$

CBSE, Class 12, Differentiation, Exercise 11.7, Maths, RD Sharma

As per the given question,

If $x=2 \cos \theta-\infty \sin \theta$ and $y=2 \sin \theta-\sin 2 \theta$ , prove that $\frac{d y}{d x}=\tan \left(\frac{3 \theta}{2}\right)$

CBSE, Class 12, Differentiation, Exercise 11.7, Maths, RD Sharma

As per the given question,

Find $\frac{d y}{d x}$ , when ${x}=\frac{1-t^{2}}{1+t^{2}}$ and $y=\frac{2 t}{1+t^{2}}$

CBSE, Class 12, Differentiation, Exercise 11.7, Maths, RD Sharma

As per the given question,

Find $\frac{d y}{d x}$ , when $x=\cos ^{-1}\left(\frac{1}{\sqrt{1+t^{2}}}\right)$ and $y=\sin ^{-1} \frac{t}{\sqrt{1+t^{2}}}, t \in R$

CBSE, Class 12, Differentiation, Exercise 11.7, Maths, RD Sharma

As per the given question,

Find $\frac{d y}{d x}$ , when $x=\frac{2 t}{1+t^{2}}$ and $y=\frac{1-t^{2}}{1+t^{2}}$

CBSE, Class 12, Differentiation, Exercise 11.7, Maths, RD Sharma

As per the given question,

Find $\frac{d y}{d x}$ , when Find $\frac{d y}{d x}$ , when $x=e^{\theta}\left(\theta+\frac{1}{\theta}\right)$ and $y=e^{-\theta}\left(\theta-\frac{1}{\theta}\right)$

CBSE, Class 12, Differentiation, Exercise 11.7, Maths, RD Sharma

As per the given question,

If $x$ and $y$ are connected parametrically by the equation, without eliminating the parameter, find $\frac{d y}{d x}$ . $x=a(\cos \theta+\theta \sin \theta), y=a(\sin \theta-\theta \cos \theta)$

CBSE, Class 12, Differentiation, Exercise 11.7, Maths, RD Sharma

As per the given question,

Find $\frac{d y}{d x}$ , when $x=\frac{3 a t}{1+t^{2}}$ and $y=\frac{3 a t^{2}}{1+t^{2}}$

CBSE, Class 12, Differentiation, Exercise 11.7, Maths, RD Sharma

As per the given question,

Find $\frac{d y}{d x}$ , when $\quad x=\frac{e^{t}+e^{-t}}{2}$ and $y=\frac{e^{t}-e^{-t}}{2}$

CBSE, Class 12, Differentiation, Exercise 11.7, Maths, RD Sharma

As per the given question,

Find $\frac{d y}{d x}$ , when $x=a(1-\cos \theta)$ and $y=a(\theta+\sin \theta) \mathrm{at} \theta=\frac{\pi}{2} .$

CBSE, Class 12, Differentiation, Exercise 11.7, Maths, RD Sharma

As per the given question,

Find $\frac{d y}{d x}$ , when $\quad x=b \sin ^{2} \theta$ and $y=a \cos ^{2} \theta$

CBSE, Class 12, Differentiation, Exercise 11.7, Maths, RD Sharma

As per the given question,

Find $\frac{d y}{d x}$ , when $x=a e^{\theta}(\sin \theta-\cos \theta), y=a e^{\theta}(\sin \theta+\cos \theta)$

CBSE, Class 12, Differentiation, Exercise 11.7, Maths, RD Sharma

As per the given question,

Find $\frac{d y}{d x}$ , when $\quad x=a \cos \theta, y=b \sin \theta$

CBSE, Class 12, Differentiation, Exercise 11.7, Maths, RD Sharma

As per the given question,

Find $\frac{d y}{d x}$ , when $x=a(\theta+\sin \theta)$ and $y=a(1-\cos \theta)$

CBSE, Class 12, Differentiation, Exercise 11.7, Maths, RD Sharma

As per the given question,

Find $dy/dx$ when: $x\;=\;at^2$ and $y\;=\;2at$

CBSE, Class 12, Differentiation, Exercise 11.7, Maths, RD Sharma

We have, $y=2 a t$ $\frac{d y}{d t}=2 a \frac{d}{d t}(t)=2 a(1)=2 a \\ \text { also } x=a t^{2} \\ \frac{d x}{d x}=a \frac{d}{d t}\left(t^{2}\right)=a(2 t)=2 a t \\ \text { now } \frac{d y}{d...

A laboratory blood test is $99 \%$ effective in detecting a certain disease when it is in fact, present. However, the test also yields a false positive result for $0.5 \%$ of the healthy person tested (that is, if a healthy person is tested, then, with probability $0.005$ , the test will imply he has the disease). If $0.1$ percent of the population actually has the disease, what is the probability that a person has the disease given that his test result is positive?

CBSE, Class 12, Exercise 31.7, Maths, Probability, RD Sharma

As per the given question,

In answering a question on a multiple choice test a student either knows the answer or guesses. Let 3/4 be the probability that he knows the answer and 1/4 be the probability that he guesses. Assuming that a student who guesses at the answer will be correct with probability 1/4. What is the probability that a student knows the answer given that he answered it correctly?

CBSE, Class 12, Exercise 31.7, Maths, Probability, RD Sharma

As per the given question,

A speaks the truth 8 times out of 10 times. A die is tossed. He reports that it was 5 What is the probability that it was actually $5 ?$

CBSE, Class 12, Exercise 31.7, Maths, Probability, RD Sharma

As per the given question,

$A$ is known to speak truth 3 times out of 5 times. He throws a die and reports that it is $1 .$ Find the probability that it is actually $1 .$

CBSE, Class 12, Exercise 31.7, Maths, Probability, RD Sharma

As per the given question,

Let $d_{1}, d_{2}, d_{3}$ be three mutually exclusive diseases. Let $S$ be the set of observable symptoms of these diseases. A doctor has the following information from a random sample of 5000 patients: 1800 had disease $d_{1}, 2100$ has disease $d_{2}$ and the others had disease $d_{3}$ 1500 patients with disease $d_{1}, 1200$ patients with disease $d_{2}$ and 900 patients with disease $d_{3}$ showed the symptom. Which of the diseases is the patient most likely to have?

CBSE, Class 12, Exercise 31.7, Maths, Probability, RD Sharma

As per the given question,

A test for detection of a particular disease is not fool proof. The test will correctly detect the disease $90 \%$ of the time, but will incorrectly detect the disease $1 \%$ of the time. For a large population of which an estimated $0.2 \%$ have the disease, a person is selected at random, given the test, and told that he has the disease. What are the chances that the person actually have the disease?

CBSE, Class 12, Exercise 31.7, Maths, Probability, RD Sharma

As per the given question,

By examining the chest $X$ -ray, probability that T.B is detected when a person is actually suffering is $0.99$ . The probability that the doctor diagnoses incorrectly that a person has T.B. on the basis of $X$ -ray is $0.001$ . In a certain city 1 in 100 persons suffers from T.B. A person is selected at random is diagnosed to have T.B. What is the chance that he actually has T.B.?

CBSE, Class 12, Exercise 31.7, Maths, Probability, RD Sharma

As per the given question,

If a machine is correctly set up, it produces $90 \%$ acceptable items. If it is incorrectly set up, it produces only $40 \%$ acceptable item s. Past experience shows that $80 \%$ of the set ups are correctly done. If after a certain set up, the machine produces 2 acceptable items, find the probability that the machine is correctly setup.

CBSE, Class 12, Exercise 31.7, Maths, Probability, RD Sharma

As per the given question,

Coloured balls are distributed in four boxes as shown in the following table
$\begin{tabular}{cc|c|c|c} \hline Box & \multicolumn{3}{c}{ Colour } \\ & Black & White & Red & Blue \\ \hline I & 3 & 4 & 5 & 6 \\ \hline II & 2 & 2 & 2 \\ \hline III & 1 & 2 & 1 \\ \hline IV & 4 & 3 & 5 \\ \hline \end{tabular}$
A box is selected at random and then a ball is randomly drawn from the selected box. The colour of the ball is black, what is the probability that ball drawn is from the box III.

CBSE, Class 12, Exercise 31.7, Maths, Probability, RD Sharma

As per the given question,

Assume that the chances of the patient having a heart attack are $40 \%$ . It is also assumed that a meditation and yoga course reduce the risk of heart attack by $30 \%$ and prescription of certain drug reduces its chances by $25 \%$ . At a time a patient can choose any one of the two options with equal probabilities. It is given that after going through one of the two options the patient selected at random suffers a heart attack. Find the probability that the patient followed a course of meditation and yoga?

CBSE, Class 12, Exercise 31.7, Maths, Probability, RD Sharma

As per the given question,

There are three coins. One is two headed coin (having head on both faces), another is a biased coin that comes up heads $75 \%$ of the time and third is an unbiased coin. One of the three coins is chosen at random and tossed, it shows heads, what is the probability that it was the two headed coin?

CBSE, Class 12, Exercise 31.7, Maths, Probability, RD Sharma

As per the given question,

Of the students in a college, it is known that $60 \%$ reside in a hostel and $40 \%$ do not reside in hostel. Previous year results report that $30 \%$ of students residing in hostel attain $A$ grade and $20 \%$ of ones not residing in hostel attain $A$ grade in their annual examination. At the end of the year, one students is chosen at random from the college and he has an A grade. What is the probability that the selected student is a hosteller?

CBSE, Class 12, Exercise 31.7, Maths, Probability, RD Sharma

As per the given question,

An insurance company insured 2000 scooters and 300 motorcycles. The probability of an accident involving a scooter is $0.01$ and that of a motorcycle is $0.02$ . An insured vehicle met whith an accident. Find the probability that the accident vehicle was a motorcycle

CBSE, Class 12, Exercise 31.7, Maths, Probability, RD Sharma

As per the given question,

For $A, B$ and $C$ the chances of being selected as the manager of a firm are in the ratio $4: 1: 2$ respectively. The respective probabilities for them to introduce a radical change in marketing startegy are $0.3,0.8$ and $0.5$ . If the change does take place, find the probability that it is due to the appointment of $B$ or $C$ .

CBSE, Class 12, Exercise 31.7, Maths, Probability, RD Sharma

As per the given question,

In a certain college, $4 \%$ of boys and $1 \%$ of girls are taller than $1.75$ meters. Further more, $60 \%$ of the students in the college are girls. A student selected at random from the college is found to be taller than $1.75$ meters. Find the probability that the selected student is girl.

CBSE, Class 12, Exercise 31.7, Maths, Probability, RD Sharma

As per the given question,

A bag contains 1 white and 6 red balls, and a second bag contains 4 white and 3 red balls. One of the bags is picked up at random and a ball is randomly drawn from it, and is found to be white in colour. Find the probability that the drawn ball was from the first bag.

CBSE, Class 12, Exercise 31.7, Maths, Probability, RD Sharma

As per the given question,

A factory has three machines $A, B$ and $C$ , which produce 100,200 and 300 items of a particular type daily. The machines produœe $2 \%, 3 \%$ and $5 \%$ defective items respectively. One day when the production was over, an item was picked up randomly and it was found to be defective. Find the probability that it was produced by machine $A$ .

CBSE, Class 12, Exercise 31.7, Maths, Probability, RD Sharma

As per the given question,

In a group of 400 people, 160 are smokers and non-vegetarian, 100 are smokers and vegetarian and the remaining are nonsmokers and veqetarian. The probabilities of aettina a special chest disease are $35 \%, 20 \%$ and $10 \%$ respectively. A person is chosen from the group at random and is found to be suffering from the disease. What is the probability that the selected person is a smoker and non-vegetarian?

CBSE, Class 12, Exercise 31.7, Maths, Probability, RD Sharma

As per the given question,

Three urns $A, B$ and $C$ contain 6 red and 4 white; 2 red and 6 white; and 1 red and 5 white balls redpectively. an urn is chosen at random and a ball is drawn. If the ball drawn is found to be red, find the probability that the ball was drawn from urn $A$ .

CBSE, Class 12, Exercise 31.7, Maths, Probability, RD Sharma

As per the given question,

A company has two plants to manufacture bicycles. The first plant manufactures $60 \%$ of the bicycles and the second plant $40 \%$ . Out of that $80 \%$ of the bicycles are rated of standard quality at the first plant and $90 \%$ of standard quality at the second plant. A bicycle is picked up at random and found to be standard quality. Find the probability that it comes from the second plant.

CBSE, Class 12, Exercise 31.7, Maths, Probability, RD Sharma

As per the given question,

In a factory, machine $A$ produces $30 \%$ of the total output, machine $B$ produces $25 \%$ and the machine $C$ produces the remaining output. If defective items produced by machines $A, B$ and $C$ are $1 \%, 1.2 \%$ respectively. Three machines working output and found to be defective. Find the probability that it was produced by machine $B$ ?

CBSE, Class 12, Exercise 31.7, Maths, Probability, RD Sharma

As per the given question,

There are three coins. One is two-headed coin (having head on both faces), another is biased coin that comes up heads $75 \%$ of the times and third is also a biased coin that comes up tail $40 \%$ of the times. One of the three coins is chosen at random and tossed, and it shows heads. What is the probability that it was the two-headed coin?

CBSE, Class 12, Exercise 31.7, Maths, Probability, RD Sharma

As per the given question,

An item is manufactured by three machine A, B and C. out of the total number of items manufactured durina a specified period. $50 \%$ are manufacture on machine A $30 \%$ on $\mathrm{B}$ and $20 \%$ on C. $2 \%$ of the items produced on $\mathrm{A}$ and $2 \%$ of items produced on $\mathrm{B}$ are defective and $3 \%$ of these produced on $\mathrm{C}$ are defective. All the items stored at one godown. One items is drawn at random and is found to be defective. What is the probability that it was manufactured on machine A?

CBSE, Class 12, Exercise 31.7, Maths, Probability, RD Sharma

As per the given question,

A manufacturer has three machine operators $A, B$ and $C$ . The first operator A produces $1 \%$ defective items, where as the other two operators $B$ and $C$ produce $5 \%$ and $7 \%$ defective items respectively. $A$ is on the job for $50 \%$ of the time, $B$ is on the job for $30 \%$ of the time and $\mathrm{C}$ is on the job for $20 \%$ of the time. A defective item is produced, what is the probability that was produced by $A$ ?

CBSE, Class 12, Exercise 31.7, Maths, Probability, RD Sharma

As per the given question,

Suppose we have four boxes $A, B, C$ and $D$ containing coloured marbles as given below:
$\begin{tabular}{|c|c|c|c|} \hline Box & \multicolumn{3}{|c|}{ Colour } \\ \cline { 2 - 4 } & Red & White & Black \\ \hline A & 1 & 6 & 3 \\ \hline B & 6 & 2 & 2 \\ \hline C & 8 & 1 & 1 \\ \hline D & 0 & 6 & 4 \\ \hline \end{tabular}$
one of the boxes has been selected at random and a single marble is drawn from it. If the marble is red, what is the probability that it was drawn from that box $A$ ? Box B? Box C?

CBSE, Class 12, Exercise 31.7, Maths, Probability, RD Sharma

As per the given question,

An insurance company issued 3000 scooters, 4000 cars and 5000 trucks. The probabilities of the accident involving a scooter, a car and a truck are $0.02$ , $0.03$ and $0.04$ respectively. One of the insured vehicles meet with an accident Find the probability that it is a truck.

CBSE, Class 12, Exercise 31.7, Maths, Probability, RD Sharma

As per the given question,

An insurance company issued 3000 scooters, 4000 cars and 5000 trucks. The probabilities of the accident involving a scooter, a car and a truck are $0.02$ , $0.03$ and $0.04$ respectively. One of the insured vehicles meet with an accident Find the probability that it is a (i) scooter (ii) car

CBSE, Class 12, Exercise 31.7, Maths, Probability, RD Sharma

As per the given question,

A factory has three machines X, Y, and Z producing 1000, 2000 and 3000 bolts per day respectively. The machine X produces 1% defective bolts, Y produces 1.5% and Z produces 2% defective bolts. At the end of the day, a bolt is drawn at random and is found to be defective. What is the probability that this defective bolt has been produced by machine?

CBSE, Class 12, Exercise 31.7, Maths, Probability, RD Sharma

Total number of bolts produced in day =(1000+2000+3000) =6000 Let E1, E2 and E3 be the events of drawing a bolt produced by machines X, Y and Z respectively. Then, P(E)=1000/6000=1/6,...

In a class, $5 \%$ of the boys and $10 \%$ of the girls have an IQ of more than $50 .$ In this class, $60 \%$ of the students are boys. If a student is selected at random and found to have an IQ of more than 150 , find the probability that the student is a boy.

CBSE, Class 12, Exercise 31.7, Maths, Probability, RD Sharma

As per the given question,

A letter is known to have come either from LONDON or CLIFTON. On the envelope just two consecutive letters $\mathrm{ON}$ are visible. What is the probability that the letter has come from (i) LONDON (ii) CLIFTON?

CBSE, Class 12, Exercise 31.7, Maths, Probability, RD Sharma

As per the given question,

Suppose 5 men out of 100 and 25 women out of 1000 are good orators. An orator is chosen at random. Find the probability that a male person is selected. Assume that there are equal number of men and women.

CBSE, Class 12, Exercise 31.7, Maths, Probability, RD Sharma

As per the given question,

Two groups are competing for the positions of the Board of Directors of a Corporation. The probabilities that the first and the second groups will win are $0.6$ and $0.4$ respectively. Further, if the first group wins, the probability of introducing a new product is $0.7$ and the correspond ing probability is $0.3$ if the second group wins. Find the probability that the new product introduced was by the second group.

CBSE, Class 12, Exercise 31.7, Maths, Probability, RD Sharma

As per the given question,

Suppose a girl throws a die. If she gets 1 or 2 , she tosses a coin three times and notes the number of tails. If she gets $3,4,5$ or 6 , she tosses a coin once and notes whether a ‘head’ or ‘tail’ is obtained. If she obtained exactly one ‘tail’, what is the probability that she threw $3,4,5$ or 6 with the die?

CBSE, Class 12, Exercise 31.7, Maths, Probability, RD Sharma

As per the given question,

The contents of three urns are as follows: Urn 1: 7 white, 3 black balls, Urn 2: 4 white, 6 black balls, and Urn 3: 2 white, 8 black balls. One of these urns is chosen at random with probabilities $0.20,0.60$ and $0.20$ respectively. From the chosen urn two balls are drawn at random without replacement. If both these balls are white, what is the probability that these came from urn 3 ?

CBSE, Class 12, Exercise 31.7, Maths, Probability, RD Sharma

As per the given question,

Three urns contain 2 white and 3 black balls; 3 white and 2 black balls and 4 white and 1 black ball respectively. One ball is drawn from an urn chosen at random and it was found to be white. Find the probability that it was drawn from the first urn.

CBSE, Class 12, Exercise 31.7, Maths, Probability, RD Sharma

As per the given question,

A bag $A$ contains 2 white and 3 red balls and a bag $B$ contains 4 white and 5 red balls. One ball is drawn at random from one of the bags and is found to be red. Find the probability that it was drawn from bag $B$ .

CBSE, Class 12, Exercise 31.7, Maths, Probability, RD Sharma

As per the given question,

The contents of urns, I, II, III are as follows: Urn I: 1 white, 2 black and 3 red balls Urn II: 2 white, 1 black and 1 red balls Urn III: 4 white, 5 black and 3 red balls. One urn is chosen at random and two balls are drawn. They happen to be white and red. What is the probability that they come from Urns, I, II, III?

CBSE, Class 12, Exercise 31.7, Maths, Probability, RD Sharma

As per the given question,

Three machines E 1,E 2,E 3 in a certain factory produce 50%, 25% and 25% respectively, of the total daily output of electric bulbs. It is known that 4% of the tubes produced one each of machines E 1 and E 2 are defective, and that 5% of those produced on E 3 are defective. If one tube is picked up at random from a day’s production, calculate the probability that it is defective.

CBSE, Class 12, Exercise 31.6, Maths, Probability, RD Sharma

Let A1: Event that the bulb is produced by machine E1 A2: Event that the bulb is produced by machine E2 A3: Event that the bulb is produced by machine E3 A: Event that the picked up bulb is...

A bag contains 6 red and 8 black balls and another bag contains 8 red and 6 black balls. A ball is drawn from the first bag and without noticing its colour is put in the second bag. A ball is drawn from the second bag. Find the probability that the ball drawn is red in colur.

CBSE, Class 12, Exercise 31.6, Maths, Probability, RD Sharma

As per the given question,

An urn contains 10 white and 3 black balls. Another urn contains 3 white and 5 black balls. Two are drawn from first urn and put into the second urn and then a ball is drawn from the latter. Find the probability that it is a white ball.

CBSE, Class 12, Exercise 31.6, Maths, Probability, RD Sharma

As per the given question,

One bag contains 4 white and 5 black balls. Another bag contains 6 white and 7 black balls. A ball is transferred from first bag to the second bag and then a ball is drawn from the second bag. Find the probability that the ball drawn is white.

CBSE, Class 12, Exercise 31.6, Maths, Probability, RD Sharma

As per the given question,

A bag contains 4 white and 5 black balls and another bag contains 3 white and 4 black balls. A ball is taken out from the first bag and without seeing its colour is put in the second bag. A ball is taken out from the later. Find the probability that the ball drawn is white.

CBSE, Class 12, Exercise 31.6, Maths, Probability, RD Sharma

As per the given question,

The bag $A$ contains 8 white and 7 black balls while the bag $B$ contains 5 white and 4 black balls. One ball is randomly picked up from the bag $A$ and mixed up with the balls in baq $B$ . Then a ball is randomly drawn out from it. Find the probability that ball drawn is white.

CBSE, Class 12, Exercise 31.6, Maths, Probability, RD Sharma

As per the given question,

A factory has two machines $A$ and $B$ . Past records show that the machine $A$ produced $60 \%$ of the items of output and machine $B$ produced $40 \%$ of the items. Further $2 \%$ of the items produced by machine $A$ were defective and $1 \%$ produced by machine $B$ were defective. If an item is drawn at random, what is the probability that it is defective?

CBSE, Class 12, Exercise 31.6, Maths, Probability, RD Sharma

As per the given question,

An unbiased coin is tossed. If the result is a head, a pair of unbiased dice is rolled and the sum of the numbers obtained is noted. If the result is a tail, a card from a well shuffled pack of eleven cards numbered $2,3,4, \ldots, 12$ is picked and the number on the card is noted. What is the probability that the noted number is either 7 or 8 ?

CBSE, Class 12, Exercise 31.6, Maths, Probability, RD Sharma

An unbiased coin is tossed, then I:- If head occurs, a pair of dice is rolled and sum on them is either $7$ or $8.$ II:- If tail occurs, a card is drawn from cards number $2,3,....12$ and is $7$ or...

The contents of three bags I, II and III are as follows: Bag I: 1 white, 2 black and 3 red balls, Bag II: 2 white, 1 black and 1 red, and Bag III: 4 white, 5 black and 3 red balls. A bag is chosen at random and two balls are drawn. What is the probability that the balls are white and red?

CBSE, Class 12, Exercise 31.6, Maths, Probability, RD Sharma

As per the given question,

A bag contains 3 white and 2 black balls and another bag contains 2 white and 4 black balls. One bag is chosen at random. From the selected bag, one ball is drawn. Find the probability that the ball drawn is white.

CBSE, Class 12, Exercise 31.6, Maths, Probability, RD Sharma

As per the given question,

One bag contains $4$ yellow and $5$ red balls. Another bag contains $6$ yellow and $3$ red balls. A ball is transferred from the first bag to the second bag and then a ball is drawn from the second bag. Find the probability that the ball drawn is yellow.

CBSE, Class 12, Exercise 31.6, Maths, Probability, RD Sharma

Given, $Bag\;I$ contains $4$ yellow and $5$ red balls $Bag\;II$ contains $6$ yellow and $3$ red balls Now, there are two ways of transferring a ball from $bag\;I\;to\;bag\;II$ $Way – 1$ By...

A purse contains $2$ silver and $4$ copper coins. A second purse contains $4$ silver and $3$ copper coins. If a coin is pulled at random from one of the two purses, what is the probability that it is a silver coin?

CBSE, Class 12, Exercise 31.6, Maths, Probability, RD Sharma

As per the given question,

A bag A contains 5 white and 6 black balls. Another bag B contains 4 white and 3 black balls. A ball is transferred from bag A to the bag B and then a ball is taken out of the second bag. Find the probability of this ball being black.

CBSE, Class 12, Exercise 31.6, Maths, Probability, RD Sharma

A black ball can be drawn in two mutually exclusive ways: (I) By transferring a white ball from bag A to bag B, then drawing a black ball (II) By transferring a black ball from bag A to bag B, then...

In a hockey match, both teams $A$ and $B$ scored same number of goals upto the end of the game, so to decide the winner, the refree asked both the captains to throw a die alternately and decide that the team, whose captain gets a first six, will be declared the winner. If the captain of team $A$ was asked to start, find their respective probabilities of winning the match and state whether the decision of the refree was fair or not.

CBSE, Class 12, Exercise 31.5, Maths, Probability, RD Sharma

As per the given question,

Out of $100$ students, two sections of $40$ and $60$ are formed. If you and your friend are among 100 students, what is the probability that: (i) you both enter the same section? (ii) you both enter the different section?

CBSE, Class 12, Exercise 31.5, Maths, Probability, RD Sharma

As per the given question,

A card is drawn from a well-shuffled deck of $52\;cards.$ The outcome is noted, the card is replaced and the deck reshuffled. Another card is then drawn from the deck. (i) What is the probability that both the cards are of the same suit? (ii) What is the probability that the first card is an ace and the second card is a red queen?

CBSE, Class 12, Exercise 31.5, Maths, Probability, RD Sharma

As per the given question,

An urn contains 7 red and 4 blue balls. Two balls are drawn at random with replacement. Find the probability of getting one red and one blue ball.

CBSE, Class 12, Exercise 31.5, Maths, Probability, RD Sharma

As per the given question,

An urn contains 7 red and 4 blue balls. Two balls are drawn at random with replacement. Find the probability of getting (i) 2 red balls (ii) 2blue balls

CBSE, Class 12, Exercise 31.5, Maths, Probability, RD Sharma

As per the given question,

A bag contains $8$ marbles of which $3$ are blue and $5$ are red. one marble is drawn at random, its colour is noted and the marble is replaced in the bag. A marble is again drawn from the bag and its colour is noted. Find the probability that the marble will be Of the same colour.

CBSE, Class 12, Exercise 31.5, Maths, Probability, RD Sharma

Bag contains blue, red marbles. One marble is drawn, its colour noted and replaced, then again a marble drawn and its colour noted.

A bag contains $8$ marbles of which $3$ are blue and $5$ are red. one marble is drawn at random, its colour is noted and the marble is replaced in the bag. A marble is again drawn from the bag and its colour is noted. Find the probability that the marble will be (i) blue followed by red. (ii) blue and red in any order.

CBSE, Class 12, Exercise 31.5, Maths, Probability, RD Sharma

Bag contains $3$ blue, $5$ red marbles. One marble is drawn, its colour noted and replaced, then again a marble drawn and its colour noted.

Fatima and John appear in an interview for two vacancies in the same post. The probability of Fatima’s selection is $\frac{1}{7}$ and that of John’s selection is $\frac{1}{5}$ . What is the probability that none of them will be selected?

CBSE, Class 12, Exercise 31.5, Maths, Probability, RD Sharma

As per the given question,

Fatima and John appear in an interview for two vacancies in the same post. The probability of Fatima’s selection is $\frac{1}{7}$ and that of John’s selection is $\frac{1}{5}$ . What is the probability that (i) both of them will be selected? (ii) only one of them will be selected?

CBSE, Class 12, Exercise 31.5, Maths, Probability, RD Sharma

As per the given question,

There are $3$ red and $5$ black balls in bag ‘ $A^{\prime}$ and $2$ red and $3$ black balls in bag ‘ $B^{\prime}$ . one ball is drawn from bag ‘ $A^{\prime}$ and two from bag ‘ $B^{\prime}$ . Find the probability that out of the $3$ balls drawn one is red and $2$ are black.

CBSE, Class 12, Exercise 31.5, Maths, Probability, RD Sharma

As per the given question,

$A$ and $B$ take turns in throwing two dice, the first to throw 10 being awarded the prize, show that if $A$ has the first throw, their chance of winning are in the ratio $12: 11$ .

CBSE, Class 12, Exercise 31.5, Maths, Probability, RD Sharma

As per the given question,

Three persons $A, B, C$ throw a die in succession till one gets a ‘six’ and wins the game. Find their respective probabilities of winning.

CBSE, Class 12, Exercise 31.5, Maths, Probability, RD Sharma

As per the given question,

$A,$ $B$ and $C$ in order toss a coin. The one to throw a head wins. What are their respective chances of winning assuming that the game may continue indefinitely?

CBSE, Class 12, Exercise 31.5, Maths, Probability, RD Sharma

As per the given question,

$A$ and $B$ take tums in throwing two dice, the first to throw $9$ being awarded the prize. Show that their chance of winning are in the ratio $9: 8$ .

CBSE, Class 12, Exercise 31.5, Maths, Probability, RD Sharma

As per the given question,

$X$ is taking up subjects – Mathematics, Physics and Chemistry in the examination. His probabilities of getting grade $A$ in these subjects are $0.2,0.3$ and $0.5$ respectively. Find the probability that he gets. Grade $A$ in two subjects.

CBSE, Class 12, Exercise 31.5, Maths, Probability, RD Sharma

As per the given question,

$X$ is taking up subjects – Mathematics, Physics and Chemistry in the examination. His probabilities of getting grade $A$ in these subjects are $0.2,0.3$ and $0.5$ respectively. Find the probability that he gets. (i) Grade $A$ in all subjects (ii) Grade $A$ in no subject

CBSE, Class 12, Exercise 31.5, Maths, Probability, RD Sharma

As per the given question,

There are three urns $A, B$ and $C$ . Urn $A$ contains $4$ red balls and $3$ black balls. Urn $B$ contains $5$ red balls and $4$ black balls. Urn $C$ contains $4$ red and $4$ black balls. One ball is drawn from each of these urns. What is the probability that $3$ balls drawn consist of $2$ red balls and a black ball?

CBSE, Class 12, Exercise 31.5, Maths, Probability, RD Sharma

As per the given question,

The probability of student $A$ passing an examination is $\frac{2}{9}$ and of student $B$ passing is $\frac{5}{9}$ . Assuming the two events: ‘ $A$ passes’, ‘ $B$ passes as independent, find the probability of : (i) only A passing the examination (ii) only one of them passing the examination.

CBSE, Class 12, Exercise 31.5, Maths, Probability, RD Sharma

As per the given question,

A can hit a target $3$ times in $6$ shots, $B: 2$ times in $6$ shots and $C: 4$ times in $4$ shots. They fix a volley. What is the probability that at least 2 shots hit?

CBSE, Class 12, Exercise 31.5, Maths, Probability, RD Sharma

As per the given question,

A bag contains $4$ red and $5$ black balls, a second bag contains $3$ red and $7$ black balls. One ball is drawn at random from each bag; find the probability that the (i) balls are of different colours (ii) balls are of the same colour.

CBSE, Class 12, Exercise 31.5, Maths, Probability, RD Sharma

As per the given question,

Three cards are drawn with replacement from a well shuffled pack of cards. Find the probability that the cards are a king, a queen and a jack.

CBSE, Class 12, Exercise 31.5, Maths, Probability, RD Sharma

As per the given question,

A bag contains $4$ white, $7$ black, and $5$ red balls. 4 balls are drawn with replacement What is the probability that at least two are white?

CBSE, Class 12, Exercise 31.5, Maths, Probability, RD Sharma

As per the given question,

A bag contains $4$ white balls and $2$ black balls. Another contains 3 white balls and 5 black balls. If one ball is drawn from each bag, find the probability that One is white and one is black.

CBSE, Class 12, Exercise 31.5, Maths, Probability, RD Sharma

As per the given question,

A bag contains $4$ white balls and $2$ black balls. Another contains 3 white balls and 5 black balls. If one ball is drawn from each bag, find the probability that (i) Both are white (ii) Both are black

CBSE, Class 12, Exercise 31.5, Maths, Probability, RD Sharma

As per the given question,

$A,\;B$ , and $C$ are independent witness of an event which is known to have occurred. $A$ speaks the truth three times out of four, $B$ four times out of five and $C$ five times out of six. What is the probability that the occurrence will be reported truthfully by majority of three witnesses?

CBSE, Class 12, Exercise 31.5, Maths, Probability, RD Sharma

As per the given question,

A bag contains $7$ white, $5$ black and $4$ red balls. Four balls are drawn without replacement. Find the probability that at least three balls are black.

CBSE, Class 12, Exercise 31.5, Maths, Probability, RD Sharma

As per the given question,

A husband and wife appear in an interview for two vacancies in the same post. The probability of husband’s selection is $\frac{1}{7}$ and that of wife’s selection is $\frac{1}{5}$ . What is the probability that none of them will be selected?

CBSE, Class 12, Exercise 31.5, Maths, Probability, RD Sharma

As per the given question,

A husband and wife appear in an interview for two vacancies in the same post. The probability of husband’s selection is $\frac{1}{7}$ and that of wife’s selection is $\frac{1}{5}$ . What is the probability that (a) both of them will be selected? (b) only one of them will be selected?

CBSE, Class 12, Exercise 31.5, Maths, Probability, RD Sharma

As per the given question,

In a family, the husband tells a lie in $30 \%$ cases and the wife in $35 \%$ cases. Find the probability that both contradict each other on the same fact.

CBSE, Class 12, Exercise 31.5, Maths, Probability, RD Sharma

As per the given question,

Tickets are numbered from $1\;to\;10\;.$ Two tickets are drawn one after the other at random. Find the probability that the number on one of the tickets is a multiple of $5$ and on the other a multiple of $4 .$

CBSE, Class 12, Exercise 31.5, Maths, Probability, RD Sharma

As per the given question,

Two cards are drawn from a well shuffled pack of $52$ cards, one after another without replacement. Find the probability that one of these is red card and the other a black card?

CBSE, Class 12, Exercise 31.5, Maths, Probability, RD Sharma

As per the given question,

$A$ and $B$ toss a coin alternately till one of them gets a head and wins the game, If $A$ starts the game, find the probability that $B$ will win the game.

CBSE, Class 12, Exercise 31.5, Maths, Probability, RD Sharma

As per the given question,

Arun and Tarun appeared for an interview for two vacancies. The probability of Arun’s selection is $\frac{1}{4}$ and that of Tarun’s rejection is $\frac{2}{3}$ . Find the probability that at least and of them will be selected.

CBSE, Class 12, Exercise 31.5, Maths, Probability, RD Sharma

As per the given question,

A bag contains $8$ red and $6$ green balls. Three balls are drawn one after another without replacement. Find the probability that at least two balls drawn are green.

CBSE, Class 12, Exercise 31.5, Maths, Probability, RD Sharma

As per the given question,

A bag contains $3$ white, $4$ red and $5$ black balls. Two balls are drawn one after the other, without replacement. What is the probability that one is white and the other is black?

CBSE, Class 12, Exercise 31.5, Maths, Probability, RD Sharma

As per the given question,

Kamal and Monica appeared for an interview for two vacancies. The probability of Kamal’s selection is $\frac{1}{3}$ and that of Monika’s selection is $\frac{1}{5}$ . Find the probability that (i) at least one of them will be selected (ii) only one of them will be selected.

CBSE, Class 12, Exercise 31.5, Maths, Probability, RD Sharma

As per the given question, (i) (ii)

Kamal and Monica appeared for an interview for two vacancies. The probability of Kamal’s selection is $\frac{1}{3}$ and that of Monika’s selection is $\frac{1}{5}$ . Find the probability that $\begin{array}{ll}\text { (i) both of them will be selected } & \text { (ii) none of them will be selected }\end{array}$

CBSE, Class 12, Exercise 31.5, Maths, Probability, RD Sharma

As per the given question,

A speaks truth in $75 \%$ and $B$ in $80 \%$ of the cases. In what percentage of cases are they likely to contradict each other in narrating the same incident?

CBSE, Class 12, Exercise 31.5, Maths, Probability, RD Sharma

As per the given question,

Two cards are drawn successively without replacement from a well-shuffled deck of cards. Find the probability of exacty one ace.

CBSE, Class 12, Exercise 31.5, Maths, Probability, RD Sharma

As per the given question,

Two balls are drawn at random with replacement from a box containing $10$ black and $8$ red balls. Find the probability that one of them is black and other is red.

CBSE, Class 12, Exercise 31.5, Maths, Probability, RD Sharma

As per the given question,

Two balls are drawn at random with replacement from a box containing 10 black and 8 red balls. Find the probability that (i) both the ball are red (ii) first ball is black and second is red.

CBSE, Class 12, Exercise 31.5, Maths, Probability, RD Sharma

As per the given question,

A bag contains $3$ red and $5$ black balls and a second bag contains $6$ red and $4$ black balls. A ball is drawn from each bag. Find the probability that one is red and the other is black.

CBSE, Class 12, Exercise 31.5, Maths, Probability, RD Sharma

As per the given question,

A bag contains $6$ black and $3$ white balls. Another bag contains $5$ black and $4$ white balls. If one ball is drawn from each bag, find the probability that these two balls are of the same color.

CBSE, Class 12, Exercise 31.5, Maths, Probability, RD Sharma

As per the given question,

Let $A$ and $B$ be two independent events such that $P(A)=D_{1}$ and $P(B)=p_{2}$ . Describe in words the events whose probabilities are $p_{1}+p_{2}=2 p_{1} p_{2}$

CBSE, Class 12, Exercise 31.4, Maths, Probability, RD Sharma

Events are said to be independent, if the occurrence or non – occurrence of one does not affect the probability of the occurrence or non – occurrence of the other.

Let $A$ and $B$ be two independent events such that $P(A)=D_{1}$ and $P(B)=p_{2}$ . Describe in words the events whose probabilities are $1-\left(1-p_{1}\right)\left(1-p_{2}\right)$

CBSE, Class 12, Exercise 31.4, Maths, Probability, RD Sharma

Events are said to be independent, if the occurrence or non – occurrence of one does not affect the probability of the occurrence or non – occurrence of the other.

Let $A$ and $B$ be two independent events such that $P(A)=D_{1}$ and $P(B)=p_{2}$ . Describe in words the events whose probabilities are (i) $p_{1} p_{2}$ (ii) $\left(1-p_{1}\right) p_{2}$

CBSE, Class 12, Exercise 31.4, Maths, Probability, RD Sharma

Events are said to be independent, if the occurrence or non - occurrence of one does not affect the probability of the occurrence or non - occurrence of the other.

Two dice are thrown together and the total score is noted. The event $E_{1} F$ and $G$ are “a total $4$ “, “a total of $9$ or more” and “a total divisible by $5^{\text {" }}$ , respectively. Calculate $\mathrm{P}(\mathrm{E}), \mathrm{P}(\mathrm{F})$ and $\mathrm{P}(\mathrm{G})$ and decide which pairs of events, if any, are independent.

CBSE, Class 12, Exercise 31.4, Maths, Probability, RD Sharma

As per the given question,

The probabilities of two students $\mathrm{A}$ and $\mathrm{B}$ coming to the school in time are $\frac{3}{7}$ and $\frac{5}{7}$ respectively. Assuming that the events, ‘A coming in time’ and ‘B coming in time’ are independent, find the probability of only one of them coming to the school in time. Write at least one advantage of coming to school in time.

CBSE, Class 12, Exercise 31.4, Maths, Probability, RD Sharma

Given that the events 'A coming in time' and 'B coming in time' are independent. The advantage of coming to school in time is that you will not miss any part of the lecture and will be able to learn...

An urn contains $4\;red\;and\;7\;black\;balls.$ Two balls are drawn at random with replacement. Find the probability of getting one red and one blue ball.

CBSE, Class 12, Exercise 31.4, Maths, Probability, RD Sharma

As per the given question,

An urn contains $4\;red\;and\;7\;black\;balls.$ Two balls are drawn at random with replacement. Find the probability of getting $\begin{array}{ll}\text { (i) } 2 \text { red balls } & \text { (ii) } 2 \text { blue balls }\end{array}$

CBSE, Class 12, Exercise 31.4, Maths, Probability, RD Sharma

As per the given question,

Two balls are drawn at random with replacement from a box containing $10$ black and $8$ red balls. Find the probability that one of them is black and other is red

CBSE, Class 12, Exercise 31.4, Maths, Probability, RD Sharma

As per the given question,

Two balls are drawn at random with replacement from a box containing $10$ black and $8$ red balls. Find the probability that (i) Both balls are red (ii) First ball is black and second is red.

CBSE, Class 12, Exercise 31.4, Maths, Probability, RD Sharma

As per the given question,

A die is thrown thrice. Find the probability of getting an odd number at least once.

CBSE, Class 12, Exercise 31.4, Maths, Probability, RD Sharma

As per the given question,

The odds against a certain event are $5$ to $2$ and the odds in favour of another event, independent to the former are 6 to 5 . Find the probability that (a) at least one of the events will occur, and (b) none of the events will occur.

CBSE, Class 12, Exercise 31.4, Maths, Probability, RD Sharma

As per the given question,

An anti-aircraft gun can take a maximum of $4\;shots$ at an enemy plane moving away from it. The probabilities of hitting the plane at the first, second, third and fourth shot are $0.4,0.3,0.2$ and $0.1$ respectively. What is the probability that the gun hits the plane?

CBSE, Class 12, Exercise 31.4, Maths, Probability, RD Sharma

As per the given question,

The probability that $A$ hits a target is $\frac{1}{3}$ and the probability that $B$ hits it, is $\frac{2}{5}$ What is the probability that the target will be hit, if each one of $A$ and $B$ shoots at the target?

CBSE, Class 12, Exercise 31.4, Maths, Probability, RD Sharma

As per the given question,

An article manufactured by a company consists of two parts $X$ and $Y$ . In the process of manufacture of the part $x, 9$ out of $100$ parts may be defective. Similarly, $5$ out of $100$ are likely to be defective in the manufacture of part $Y$ . Calculate the probability that the assembled product will not be defective.

CBSE, Class 12, Exercise 31.4, Maths, Probability, RD Sharma

As per the given question,

Three cards are drawn with replacement from a well shuffled pack of cards. Find the probability that the cards drawn are king, queen and jack.

CBSE, Class 12, Exercise 31.4, Maths, Probability, RD Sharma

As per the given question,

A bag contains $3\;red$ and $2\;black$ balls. One ball is drawn from it at random. Its color is noted and then it is put back in the bag. A second draw is made and the same procedure is repeated. Find the probability of drawing first red and second black ball.

CBSE, Class 12, Exercise 31.4, Maths, Probability, RD Sharma

As per the given question,

A bag contains $3\;red$ and $2\;black$ balls. One ball is drawn from it at random. Its color is noted and then it is put back in the bag. A second draw is made and the same procedure is repeated. Find the probability of drawing (i) two red balls, (ii) two black balls

CBSE, Class 12, Exercise 31.4, Maths, Probability, RD Sharma

As per the given question,

An unbiased die is tossed twice. Find the probability of getting $4,\;5,\;or\;6$ on the first toss and $1,2,3$ or 4 on the second toss.

CBSE, Class 12, Exercise 31.4, Maths, Probability, RD Sharma

As per the given question,

Given the probability that $A$ can solve a problem is $\frac{2}{3}$ and the probability that $B$ can solve the same problem is $\frac{3}{5}$ . Find the probability that none of the two will be able to solve the problem.

CBSE, Class 12, Exercise 31.4, Maths, Probability, RD Sharma

As per the given question,

A die is tossed twice. Find the probability of getting a number greater than $3$ on each toss.

CBSE, Class 12, Exercise 31.4, Maths, Probability, RD Sharma

As per the given question,

If $A$ and $B$ are two independent events such that $P(A \cup B)=0.60$ and $P(A)=0.2$ find $P(B)$

CBSE, Class 12, Exercise 31.4, Maths, Probability, RD Sharma

As per the given question,

$A$ and $B$ are two independent events. The probability that $A$ and $B$ occur is $\frac{1}{6}$ and the probability that neither of them oocurs is $\frac{1}{3}$ . Find the probability of occurrence of two events.

CBSE, Class 12, Exercise 31.4, Maths, Probability, RD Sharma

As per the given question, ......(i)

If $A$ and $B$ are two independent events such that $p(\bar{A} \cap B)=\frac{2}{15}$ and $P(A \cap \bar{B})=\frac{1}{6}$ , then find $P(B)$ .

CBSE, Class 12, Exercise 31.4, Maths, Probability, RD Sharma

As per the given question,

If $P(not B)=0.65, P(A \cup B)=0.85$ , and $A$ and $B$ are independent events, then find $P(A)$ .

CBSE, Class 12, Exercise 31.4, Maths, Probability, RD Sharma

As per the given question,

Given two independent events $A$ and $B$ such that $P(A)=0.3$ and $P(B)=0.6$ , Find $P\left(\frac{B}{A}\right)$

CBSE, Class 12, Exercise 31.4, Maths, Probability, RD Sharma

Given that $A\;and\;B$ are independent events and $P(A)\;=\;0.3,\; P(B)\;=\;0.6$

Given two independent events $A$ and $B$ such that $P(A)=0.3$ and $P(B)=0.6$ , Find (i) $P(A \cup B)$ (ii) $P\left(\frac{A}{B}\right) \quad$

CBSE, Class 12, Exercise 31.4, Maths, Probability, RD Sharma

Given that $A\;and\;B$ are independent events and $P(A)\;=\;0.3,\; P(B)\;=\;0.6$ (i) (ii)

Given two independent events $A$ and $B$ such that $P(A)=0.3$ and $P(B)=0.6$ , Find $\text { (i) } P(\bar{A} \cap B) \text { (ii) } P(\bar{A} \cap \bar{B})$

CBSE, Class 12, Exercise 31.4, Maths, Probability, RD Sharma

Given that $A\;and\;B$ are independent events and $P(A)\;=\;0.3,\; P(B)\;=\;0.6$ (i) (ii)

Given two independent events $A$ and $B$ such that $P(A)=0.3$ and $P(B)=0.6$ , Find $\text { (i) } P(A \cap B) \text { (ii) } P(A \cap \bar{B})$

CBSE, Class 12, Exercise 31.4, Maths, Probability, RD Sharma

Given that $A\;and\;B$ are independent events and $P(A)\;=\;0.3,\; P(B)\;=\;0.6$

If $A$ and $B$ be two events such that $P(A)=\frac{1}{4}, P(B)=\frac{1}{3}$ and $P(A \cup B)=\frac{1}{2}$ , show that $A$ and $B$ are independent events.

CBSE, Class 12, Exercise 31.4, Maths, Probability, RD Sharma

As per the given question,

A coin is tossed three times. Let the events $A, B$ and $C$ be defined as follows: $A=$ first toss is head, $B=$ second toss is head, and $C=$ exactly two heads are tossed in a row. $\begin{array}{lll}\text { Check the independence of } \; C \text { and } A\end{array}$

CBSE, Class 12, Exercise 31.4, Maths, Probability, RD Sharma

As per the given question,

A coin is tossed three times. Let the events $A, B$ and $C$ be defined as follows: $A=$ first toss is head, $B=$ second toss is head, and $C=$ exactly two heads are tossed in a row. $\begin{array}{lll}\text { Check the independence of (i) } A \text { and } B & \text { (ii) } B \text { and } C \end{array}$

CBSE, Class 12, Exercise 31.4, Maths, Probability, RD Sharma

As per the given question,

A card is drawn from a pack of $52\;cards$ so that each card is equally likely to be selected. State whether events $A$ and $B$ are independent if, $A=$ the card drawn is spade, $B=$ the card drawn in an ace

CBSE, Class 12, Exercise 31.4, Maths, Probability, RD Sharma

As per the given question,

(i) A card is drawn from a pack of 52 cards so that each card is equally likely to be selected. State whether events $A$ and $B$ are independent if, $A=$ the card drawn is a king or queen $B=$ the card drawn is a queen or jack (ii) A card is drawn from a pack of 52 cards so that each card is equally likely to be selected. State whether events $A$ and $B$ are independent if, $A=$ the card drawn is black, $B=$ the card drawn is a king

CBSE, Class 12, Exercise 31.4, Maths, Probability, RD Sharma

As per the given question, (i) (ii)

Prove that in throwing a pair of dice, the occurrence of the number $4$ on the first die is independent of the occurrence of $5$ on the second die.

CBSE, Class 12, Exercise 31.4, Maths, Probability, RD Sharma

As per the given question,

A coin is tossed thrice and all the eight outcomes are assumed equally likely. State whether events $A$ and $B$ are independent if, $A=$ the number of heads is two, $B=$ the last throw results in head

CBSE, Class 12, Exercise 31.4, Maths, Probability, RD Sharma

As per the given question,

(i) A coin is tossed thrice and all the eight outcomes are assumed equally likely. State whether events $A$ and $B$ are independent if, $A=$ the first throw results in head, $B=$ the last throw results in tail (ii) A coin is tossed thrice and all the eight outcomes are assumed equally likely. State whether events $A$ and $B$ are independent if, $A=$ the number of heads is odd, $B=$ the number of tails is odd

CBSE, Class 12, Exercise 31.4, Maths, Probability, RD Sharma

As per the given question, So, $A\;and\;B$ are independent events. (ii)

Assume that each born child is equally likely to be a boy or a girl. If a family has two children, what is the conditional probability that both are girls? Given that (i) the youngest is a girl (ii) at least one is girl.

CBSE, Class 12, Exercise 31.3, Maths, Probability, RD Sharma

As per the given question, (i) Let $'A'$ be the event that both the children born are girls. Let $'B'$ be the event that the youngest is a girl. We have to find conditional probability $P(A/B).$...