RBSE Class 12 Maths Board Paper 2018 English Medium

RBSE Class 12 Maths Board Paper 2018 English Medium are part of RBSE Class 12 Maths Board Model Papers. Here we have given  Rajasthan RBSE Class 12 Maths Board Paper 2018 English Medium.

Board	RBSE
Textbook	SIERT, Rajasthan
Class	Class 12
Subject	Maths
Paper Set	Board Paper 2018
Category	RBSE Model Papers

Rajasthan RBSE Class 12 Maths Board Paper 2018 English Medium

Time : 3 ¼ Hours
Maximum Marks: 80

General instructions to the examines

1. Candidate must write first his/her Roll No. on the question paper compulsorily.
   
2. All the questions are compulsory.
   
3. Write the answer to each question in the given answer book only.
   
4. For questions having more than one part the answers to those parts are to be written together in continuity.

SECTION – A

Question 1.
If \(f : \mathrm{R} \rightarrow \mathrm{R}, f(x)=x^{2}-5 x+7\), then find the value of f^-1(1). [1]

Question 2.
Find the value of \(\sin ^{-1}\left(\frac{1}{2}\right)+2 \cos ^{-1}\left(\frac{1}{2}\right)\) [1]

Question 3.
[1]

Question 4.
[1]

Question 5.
Find \(\int x e^{x} d x\) [1]

Question 6.
Find a vector of magnitude 5 units along the vector \(\hat{i}-2 \hat{j}+2 \hat{k}\) [1]

Question 7.
Find the projection of the vector \(\hat{i}-\hat{j}\) on the vector \(\hat{i}+\hat{j}\). [1]

Question 8.
Find the direction cosines of the line \(\frac{x-2}{2}=\frac{y+1}{-2}=\frac{z-1}{1}\) [1]

Question 9.
Show the region of feasible solution under the following constraints 2x + y ≤ 6; x ≥ 0; y ≥ 0. [1]

Question 10.
If A and B are two independent events with P(A) = 0.2 and P(B) = 0.5 then find the value of P(AUB). [1]

SECTION – B 

Question 11.
If \(f : \mathrm{R} \rightarrow \mathrm{R} \text { and } g : \mathrm{R} \rightarrow \mathrm{R}\), are defined such that \(f(x)=x^{2}+3 ; g(x)=1-\frac{1}{(1-x)}\) then find gof(x) and fog(x) [2]

Question 12.
[2]

Question 13.
[2]

Question 14.
Find \(\int \frac{d x}{\sqrt{1+x}-\sqrt{x}}\) [2]

Question 15.
Find the vector product of the vectors \(2 \hat{i}-\hat{j}+\hat{k}\) and \(3 \hat{i}+\hat{j}-2 \hat{k}\). [2]

SECTION – C

Question 16.
Solve the equation \(\cos ^{-1} x+\cos ^{-1} 2 x=\frac{2 \pi}{3}\) [3]
OR

Question 17.
[3]

Question 18.
Solve the following system of equations by using Cramer’s rule. [3]
5x – 4y= 7
x + 3y=9

Question 19.
Find the intervals in which the function f given by f(x) = sin x + cos x; 0 ≤ x ≤ 2π is [3]

1. Strictly increasing.
2. Strictly decreasing.

Question 20.
Prove that the value of function \(\frac{x}{1+x \tan x}\) is maximum at x = cos x. [3]

Question 21.
Find \(\int \frac{1}{\sqrt{\left(5 x-6-x^{2}\right)}} d x\) [3]
OR
Find \(\int \frac{d x}{x\left[6(\log x)^{2}+7 \log x+2\right]}\)

Question 22.
Find the area bounded by curves x² + y² = 1 and y= [x]. [3]

Question 23.
Find the area of the region bounded by the parabolas y² = 4x and x² = 4y. [3]

Question 24.
For any vector \(\vec{a}\), prove that \(|\vec{a} \times \hat{i}|^{2}+|\vec{a} \times \hat{j}|^{2}+|\vec{a} \times \hat{k}|^{2}=2|a|^{2}\). [3]
OR
For any vector \(\vec{a}\), prove that \(\vec{a}=(\vec{a} \cdot \hat{i}) \hat{i}+(\vec{a} \cdot \hat{j}) \hat{j}+(\vec{a} . \hat{k}) \hat{k}\).

Question 25.
By graphical method solve the following linear programming problem for [3]
Minimum z = 8000 x + 12000 y
Constraints 3x + 4y ≤ 60
x + 3y ≤ 30
x ≥ 0, y ≥ 0.

SECTION -D

Question 26.
Differentiate (log x)²+x^{log x}  with respect to x. [6]

Question 27.
Prove that \(\int_{0}^{\pi} \log _{e}(1+\cos x) d x=\pi \log _{e}\left(\frac{1}{2}\right)\) [6]

Question 28.
Solve the differential equation \(\frac{d y}{d x}=\frac{x+y+1}{2 x+2 y+3}\)
OR
Find the particular solution of the differential equation (tan^-1y- x)dy = (1 + y²)dx if x = 0 and y= 0. [6]

Question 29.
Prove that the lines \(\vec{r}=(\hat{i}+\hat{j}-\hat{k})+\lambda(3 \hat{i}-\hat{j}) \text { and } \vec{r}=(4 \hat{i}-\hat{k})+\mu(2 \hat{i}+3 \hat{j})\) are intersecting, also find the point of intersection. [6]

Question 30.
Bag A contains 3 red and 4 black balls and bag B contains 4 red and 5 black balls. One ball
transferred from bag A to bag B and then a ball is drawn from bag B. The ball so drawn is found to be red in colour. Find the probability that the transferred ball is black. [6]
OR
Two cards are drawn successively with replacement from a well-shuffled deck of 52 cards. Find the probability distribution and mean of the number of aces.

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