Science (English Medium) Class 12 - CBSE Question Bank Solutions for Mathematics

Let d₁, d₂, d₃ 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 d₂, and others had disease d₃. 1500 patients with disease d_1, 1200 patients with disease d₂, and 900 patients with disease d₃ showed the symptom. Which of the diseases is the patient most likely to have?

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

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?

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

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

There are three categories of students in a class of 60 students:
A : Very hardworking ; B : Regular but not so hardworking; C : Careless and irregular 10 students are in category A, 30 in category B and the rest in category C. It is found that the probability of students of category A, unable to get good marks in the final year examination is 0.002, of category B it is 0.02 and of category C, this probability is 0.20. A student selected at random was found to be one who could not get good marks in the examination. Find the probability that this student is category C.

Find : ` int  (sin 2x ) /((sin^2 x + 1) ( sin^2 x + 3 ) ) dx`

If tan^-1 x - cot^-1 x = tan^-1 `(1/sqrt(3)),`x> 0 then find the value of x and hence find the value of sec^-1 `(2/x)`.

A company manufactures two types of novelty souvenirs made of plywood. Souvenirs of types A require 5 minutes each for cutting and 10 minutes each for assembling. Souvenirs of type B require 8 minutes each for cutting and 4 hours available for assembling. The profit is ₹ 50 each for type A and ₹60 each for type B souvenirs. How many souvenirs of each type should the company manufacture in order to maximize profit? Formulate the above  LPP and solve it graphically and find the maximum profit.

Find: ∫ sin x · log cos x dx

Solve: tan^-1 4 x + tan^-1 6x `= π/(4)`.

If `"A" = [(1,1,1),(1,0,2),(3,1,1)]`, find A^-1. Hence, solve the system of equations x + y + z = 6, x + 2z = 7, 3x + y + z = 12.

Solve for x : `tan^-1 ((2-"x")/(2+"x")) = (1)/(2)tan^-1  ("x")/(2), "x">0.`

Show that height of the cylinder of greatest volume which can be inscribed in a right circular cone of height h and semi-vertical angle α is one-third that of the cone and the greatest volume of the cylinder is `(4)/(27) pi"h"^3 tan^2 α`.

Let A = R − (2) and B = R − (1). If f: A ⟶ B is a function defined by`"f(x)"=("x"-1)/("x"-2),` how that f is one-one and onto. Hence, find f⁻¹. 

Integrate the following functions w.r.t. x : `(e^(2x) + 1)/(e^(2x) - 1)`

Evaluate: `int log ("x"^2 + "x")` dx

Let the function f: R → R be defined by f(x) = 4x – 1, ∀ x ∈ R. Then, show that f is one-one.

Let f: R → R be the function defined by f(x) = 4x – 3 ∀ x ∈ R. Then write f^–1 

If A = {a, b, c, d} and f = {a, b), (b, d), (c, a), (d, c)}, show that f is one-one from A onto A. Find f^–1