Bihar Board 12th Maths Model Papers

## Bihar Board 12th Maths Model Question Paper 1 in English Medium

Time : 3 Hours 15 Min

Full Marks: 100

Instructions for the candidates :

- Candidates are required to give their answers in their own words as far as practicable.
- Figure in the right-hand margin indicates full marks.
- While answering the questions, the candidate should adhere to the word limit as far as practicable.
- 15 Minutes of extra time has been allotted for the candidate to read the questions carefully.
- This question paper is divided into two sections. Section-A and Section-B
- In Section A, there are 1-50 objective type questions which are compulsory, each carrying 1 mark. Darken the circle with blue/black ball pen against the correct option on the OMR Sheet provided to you. Do not use Whitener/Liquid/ Blade/Nail on OMR Sheet otherwise result will be invalid.
- In section-B, there are 25 short answer type questions (each carrying 2 marks), out of which only 15 (fifteen) questions are to be answered.

A part from this there is 08 Long Answer Type questions (each carrying 5 marks), out of which 4 questions are to be answered. - Use of any electronic device is prohibited.

Objective Type Questions

There are 1 to 50 objective type questions with 4 options, choose the correct option which, is to be answered on OMR Sheet. (50 x 1 = 50)

Question 1.

Which of the following has its inverse function one-one and onto?

(a) one-one onto

(b) one-one into

(c) many one onto

(d) many on into

Answer:

(a) one-one onto

Question 2.

A relation R in a set X is an equivalence relation if R is

(a) reflexive

(b) symmetric

(c) transitive

(d) All of above

Answer:

(d) All of above

Question 3.

The number of all one-one functions from set {a, b, c, d} to itself is

(a) 12

(b) 24

(c) 36

(d) None

Answer:

(b) 24

Question 4.

If \(-\frac{\pi}{2}<x<\frac{\pi}{2}\) then tan(tan^{-1} x) =

(a) tan x

(b) cotx

(c) x

(d) -x

Answer:

(c) x

Question 5.

Sin^{-1} x + Cos^{-1} y =

(a) 0

(b) \(\frac{\pi}{4}\)

(c) \(\frac{\pi}{2}\)

(d) π

Answer:

(c) \(\frac{\pi}{2}\)

Question 6.

If x > 0, y > 0, xy < 1, then tan^{-1}x + tan^{-1} y =

(a) tan^{-1} (x+y)

(b) \(\tan ^{-1} \frac{x+y}{1-x y}\)

(c) tan^{-1}

(d) sin^{-1}(x + y)

Answer:

(b) \(\tan ^{-1} \frac{x+y}{1-x y}\)

Question 7.

(a) tan^{-1}2x

(b) \(\tan ^{-1} \frac{2 x}{1-x^{2}} \)

(c) \(\tan ^{-1} \frac{2 x}{1+x^{2}}\)

(d) \(\cot ^{-1} \frac{2}{x}\)

Answer:

(b) \(\tan ^{-1} \frac{2 x}{1-x^{2}} \)

Question 8.

\(2\left[\begin{array}{ll}

x & y \\

1 & m

\end{array}\right]\) =

Answer:

(c) \(\left[\begin{array}{cc}

2 x & 2 y \\

1 & 2 m

\end{array}\right]\)

Question 9.

\(5\left|\begin{array}{ll}

2 & 3 \\

3 & 4

\end{array}\right|=\)

Answer:

\(\left|\begin{array}{cc}

2 & 3 \\

15 & 20

\end{array}\right|\)

Question 10.

If A = \(\left[\begin{array}{cc}

5 & -5 \\

5 & -5

\end{array}\right]\) then A’ =

Answer:

(c) \(\left[\begin{array}{cc}

5 & 5 \\

-5 & -5

\end{array}\right]\)

Question 11.

\(\left|\begin{array}{ll}

1 & 0 \\

0 & 1

\end{array}\right|=\)

(a) 0

(b) 1

(c) -1

(d) 2

Answer:

(b) 1

Question 12.

\(\left[\begin{array}{ll}

\mathbf{a} & \mathbf{b} \\

\mathbf{c} & \mathbf{d}

\end{array}\right]+\left[\begin{array}{ll}

\mathbf{p} & \mathbf{q} \\

\mathbf{r} & \mathbf{s}

\end{array}\right]=\)

Answer:

\(\left[\begin{array}{cc}

a+p & b+q \\

c+r & d+s

\end{array}\right]\)

Question 13.

For a non-invertible matrix A,

(a) | A | = 0

(b)|A|≠ 0

(c) | A | = 1

(d) | A | = 2

Answer:

(b)|A|≠ 0

Question 14.

If 1 is an unit matrix of order 2 × 2 then I^{3} =

(a) 3I^{2}

(b) 3 + I

(c) 3I

(d) I

Answer:

(d) I

Question 15.

If A = \(\left[\begin{array}{ll}

2 & 3 \\

4 & 5

\end{array}\right]\) and \(\left[\begin{array}{ll}

4 & 6 \\

8 & 10

\end{array}\right]\) then 2A – B =

(a) [0]

(b) [0,0]

(c) \(\left[\begin{array}{ll}

0 & 0 \\

0 & 0

\end{array}\right]\)

(d) [3]

Answer:

(c) \(\left[\begin{array}{ll}

0 & 0 \\

0 & 0

\end{array}\right]\)

Question 16.

If the order of two matrices A and B are 2×4 and 3×2 respectively then the order of AB is

(a) 2 x 2

(b) 4 x 3

(c) 2 x 3

(d) it is not possible to find AB

Answer:

(d) it is not possible to find AB

Question 17.

\(\frac{d}{d x}(\tan x)=\) =

(a) cot x

(b) Sec^{2}x

(c) Secx tanx

(d) Secx

Answer:

(b) Sec^{2}x

Question 18.

\(\frac{d}{d x}\left(\sin ^{3} x\right)=\)

(a) 3 cos^{3}

(b) 3 sin^{2}xcosx

(c) 3sin^{2}x

(d) cos^{3}x

Answer:

(b) 3 sin^{2}xcosx

Question 19.

\(\frac{d}{d x}\) [3 (sin^{2} x + cos^{2}x )] =

(a) 3

(b) 1

(c) 0

(d) 6 sin x cos x

Answer:

(c) 0

Question 20.

\(\frac{d}{d x}\) (e^{4x}) =

(a) e^{4x}

(b) e^{x}

(c) \(\frac{e^{4 x}}{4}\)

(d) 4e^{4x}

Answer:

(d) 4e^{4x}

Question 21.

\(\frac{d}{d x}\left(x^{5}\right)=\)

(a) 5x^{5}

(b) 5x^{4}

(c) \(\frac{x^{6}}{6}\)

(d) \(\frac{x^{4}}{4} \)

Answer:

(b) 5x^{4}

Question 22.

\(\frac{d}{d x}\) [log(x^{3})] =

(a) \(\frac{1}{x^{3}}\)

(b) \(\frac{3}{x}\)

(c) 3x

(d) \(\frac{3}{x^{3}}\)

Answer:

(b) \(\frac{3}{x}\)

Question 23.

If x= cosθ,y = sinθ then \(\frac{d y}{d x}\) =

(a) tanθ

(b) sec^{2}θ

(c) cotθ

(d) – cotθ

Answer:

(d) – cotθ

Question 24.

\(\frac{d}{d x}\left(x^{1 / 3}\right)=\)

Answer:

(d) \(\frac{1}{3} x^{2 / 3}\)

Question 25.

∫ sin 2x dx =

(a) K + 2 cos2x

(b) \(\frac{\cos 2 x}{2}+K\)

(c) \(K-\frac{\cos 2 x}{2}\)

(d) \(K-\frac{\cos 2 x}{3}\)

Answer:

(c) \(K-\frac{\cos 2 x}{2}\)

Question 26.

∫x^{4}dx =

(a) K + x^{5}

(b) \(K+\frac{x^{4}}{5}\)

(c) \(K+\frac{x^{5}}{5}\)

(d) \(K+\frac{x^{5}}{4}\)

Answer:

(c) \(K+\frac{x^{5}}{5}\)

Question 27.

∫e^{3x} dx =

(a) e^{3x} + K

(b) K + 3e^{3x}

(c) \(\frac{e^{3 x}}{3}+K\)

(d) \(\frac{e^{3 x}}{4}+K\)

Answer:

(c) \(\frac{e^{3 x}}{3}+K\)

Question 28.

\(\int \frac{3}{x} d x=\)

(a) K + 3x^{2}

(b) \(K-\frac{3}{x^{2}}\)

(c) 3x + K

(d) K + 3 log|x|

Answer:

(d) K + 3 log|x|

Question 29.

∫3dx =

(a) 3 + K

(b) x + K

(c) 3x + K

(d) 3K

Answer:

(c) 3x + K

Question 30.

∫√x.dx =

Answer:

(c) \(\frac{2}{3} x^{3 / 2}+K\)

Question 31.

\(\int_{0}^{\pi / 2} \sin x d x=\)

(a) 0

(b) 1

(c) -1

(d) \(\frac{\pi}{2}\)

Answer:

(b) 1

Question 32.

\(\int_{0}^{1} e^{x} d x=\)

(a) e

(b) e + 1

(c) e – 1

(d) 2e

Answer:

(c) e – 1

Question 33.

The solution of the equation \(\frac{d x}{x}=\frac{d y}{y}\) is

(a) x = Ky

(b) xy = K

(c) x + y = K

(d) x – y = K

Answer:

(a) x = Ky

Question 34.

The solution of the differential equation cosx dx + cosy dy = 0 is

(a) sinx + cosy = K

(b) sinx + siny = K

(c) cosx + cosy = K

(d) None of these

Answer:

(b) sinx + siny = K

Question 35.

The solution of e^{x}dx + e^{y}dy = 0

(a) e^{x} + e^{y}= K

(b) e^{x} – e^{y} = K

(c) e^{x+y} = K

(d) None of these

Answer:

(a) e^{x} + e^{y}= K

Question 36.

The integrating factor of the linear differential equation \(\frac{d y}{d x}+x y=x^{3}\) is

(a) e^{x}

(b) \(e^{\frac{x^{2}}{2}}\)

(c) x

(d) None of these

Answre:

(b) \(e^{\frac{x^{2}}{2}}\)

Question 37.

∫ logdx =

(a) x log x – x + K

(b) x log x + x + K

(c) \(\frac{1}{x}+K\)

(d) \(\frac{1}{2}(\log x)^{2}+K\)

Answer:

(a) x log x – x + K

Question 38.

\(\int \frac{d x}{1+x^{2}}=\)

(a) tan^{-1} x + K

(b) sin^{-1} x + K

(c) cos^{-1} x + K

(d) cot^{-1} x + K

Answer:

(a) tan^{-1} x + K

Question 39.

\(\overrightarrow{| \vec{i}} |=\)

(a) 0

(b) 1

(c) 2

(d) 3

Answer:

(b) 1

Question 40.

\(\vec{i} \cdot \vec{i}=\)

(a) 0

(b) 1

(c) \(\vec{j}\)

(d) \(\vec{k}\)

Answer:

(b) 1

Question 41.

\(\vec{i} \times \vec{i}=\)

(a) \(\vec{i}\)

(b) \(\vec{0}\)

(c) \(\vec{j}\)

(d) \(\vec{k}\)

Answer:

(b) \(\vec{0}\)

Question 42.

If O is the origin and the position vector of a point(2,3,4) then \(\overrightarrow{\mathrm{OA}}=\)

Answer:

(b) \(2 \vec{i}+3 \vec{j}+3 \vec{k}\)

Question 43.

\(\overrightarrow{| i}+2 \vec{j}+3 \vec{K} |=\)

(a) 14

(b) 6

(c) 1

(d) √14

Answer:

(d) √14

Question 44.

\((2 \vec{i}-3 \vec{j}+4 \vec{k}) \cdot(3 \vec{i}+4 \vec{j}-5 \vec{k})=\)

(a) 14

(b) -14

(c) 26

(d) -26

Answer:

(d) -26

Question 45.

The direction cosines of the y-axis are

(a) 0, 0, 0

(b) 1, 0, 0

(c) 0, 1, 0

(d) 0, 0, 1

Answer:

(c) 0, 1, 0

Question 46.

The condition for two lines having direction cosines I_{1} m_{1},n_{1} and I_{2}, m_{2}, n_{2} being parallel is

(a) l_{1}l_{2} + m_{1}m_{2} + n_{1}n_{2} = 0

(b) \(\frac{I_{1}}{I_{2}}+\frac{m_{1}}{m_{2}}+\frac{n_{1}}{n_{2}}=0\)

(c) \(\frac{I_{1}}{I_{2}}=\frac{m_{1}}{m_{2}}=\frac{n_{1}}{n_{2}}\)

(d) None of these

Answer:

(c) \(\frac{I_{1}}{I_{2}}=\frac{m_{1}}{m_{2}}=\frac{n_{1}}{n_{2}}\)

Question 47.

The equation of a plane parallel to the plane x + 2y + 3z + 5 = 0 is

(a) x + 2y + 3z + 5 = 0

(b) x – 2y + 3z + 5 = 0

(c) x + 2y – 3z + 5 = 0

(d) None of these

Answer:

(d) None of these

Question 48.

The equation of a plane parallel to yz-plane is

(a) x + K

(b) y = K

(c) z = K

(d) None of these

Answer:

(a) x + K

Question 49.

If A and B be two arbitrary events where A ≠ φ then P (A∩B) =

(a) P (A). P (B/A)

(b) P (A) + P (B/A)

(c) P (A) – P (B/A)

(d) None of these

Answer:

(a) P (A). P (B/A)

Question 50.

The function in a linear programming problem whose maximum or minimum value has to be determined is called

(a) Objective function

(b) Constraint

(c) Both (a) and (b)

(d) None of these

Answer:

(a) Objective function

Non-Objective Type Questions

Short Answer Type Questions

Question No. 1 to 25 are short answer type questions. Each question of this category carries 2 marks. Answer any 15 questions. (15 x 2 = 30)

Question 1.

Examine whether the function f: R → R is one-one or many-one where f (x) = | x |, x ∈ R

Answer:

We have f(-1) = |- 1| = 1 and f(-1) = | 1 | = 1

Thus two different elements in R have the same Image,

∴ f is not one-one function, f is many one function.

Question 2.

Prove that 2 \(\tan ^{-1} \frac{1}{5}+\tan ^{-1} \frac{1}{4}=\tan ^{-1} \frac{32}{43}\)

Answer:

Question 3.

Solve for x : cot^{-1} x + sin ^{-1}\(\frac{1}{\sqrt{5}}\) = \(\frac{\pi}{4}\)

Answer:

Question 4.

Find the value of x from the following

\(\left[\begin{array}{cc}

2 x-y & 5 \\

3 & y

\end{array}\right]=\left[\begin{array}{cc}

6 & 5 \\

3 & -2

\end{array}\right]\)

Answer:

Given that \(\left[\begin{array}{cc}

2 x-y & 5 \\

3 & y

\end{array}\right]=\left[\begin{array}{cc}

6 & 5 \\

3 & -2

\end{array}\right]\)

∴ 2x – y = 6 ………..(1)

y = -2………..(2)

from (i) 2x + 2 = 6 ⇒ 2x = 4. ∴ x = 2

∴x = 2, y = 2

Question 5.

Evaluate:

\(\left|\begin{array}{ccc}

16 & 9 & 7 \\

23 & 16 & 7 \\

32 & 19 & 13

\end{array}\right|\)

Answer:

Question 6.

When Evaluate x : \(\left|\begin{array}{ll}

x & 4 \\

2 & 2 x

\end{array}\right|=0\)

Answer:

Given that \(\left|\begin{array}{ll}

x & 4 \\

2 & 2 x

\end{array}\right|=0\)

⇒ 2x^{2} – 8 = 0

⇒ 2x^{2} = 8

⇒ x^{2} = 4

∴ x = ±2

Question 7.

If A = \(\left[\begin{array}{c}

2 \\

-4 \\

3

\end{array}\right]\) and B = [ 2 3 4] then find B’A’

Answer:

Question 8.

If y + x = sin (y +x) then find dy/dx

Answer:

Question 9.

If \(y=\log \left(x^{2} \sqrt{x^{2}+1}\right)\) then find \(\frac{d y}{d x}\)

Answer:

Question 10.

If x = a cosθ, y = b sinθ, then find dy/dx

Answer:

Given that x = acosθ

D.W.R. to θ \(\frac{d x}{d \theta}\) = -asinθ…(i)

and y = bsinθ

D.w.r. to θ ; \(\frac{d y}{d \theta}\) = bcosθ …(ii)

\(\frac{(\mathrm{ii})}{(\mathrm{i})} \frac{d y}{d x}=-\frac{b}{a} \cot \theta\)

Question 11.

Integrate ∫ (sin x +cos x)^{2}dx.

Answer:

Let I = ∫ (sin x + cos x)^{2} dx

= ∫ (sin^{2}x + cos^{2}x + 2sinx-cosx)dlv

= ∫ (l + sin 2x)clx = ∫ dx + ∫ sin 2x dx

= \(x-\frac{\cos 2 x}{2}+c\)

Question 12.

Evaluate: \(\int_{0}^{\pi / 2} \frac{d x}{1+\sin x}\)

Answer:

Question 13.

Evaluate : \(\int_{0}^{\pi / 2} \frac{\sin x d x}{\sin x+\cos x}\)

Answer:

Question 14.

Solve \(\frac{d y}{d x}\) – y tanx =-ysec^{2}x.

Answer:

Given differentia] Equation is \(\frac{d y}{d x}\) – y tanx =-ysec^{2}x.

This is L.D.E. of the form \(\frac{d y}{d x}\) + Py = Q

Here p = – tan x Q = sec^{2}x

∴ Solution of given diff. eqn. is

y x l.F.= ∫ Q-(lF)dx + C

⇒ ycosx = ∫sec^{2}x.cosxdx + C

⇒ ycosx = ∫ secxdx + C

⇒ ycosx = log |secx + tanx| + C

This is requried solution of given diff. eqn.

Question 15.

Integrate : ∫x^{2}e^{x}dx

Answer:

Let I = ∫x^{2}e^{x}dx

Question 16.

Find the scalar product of \(\overrightarrow{5} \vec{i}+\vec{j}-3 \vec{k}\) and \(3 \vec{i}-\overrightarrow{4 j}+7 \vec{K}\)

Answer:

Let \(\vec{a}=5 \hat{l}+\hat{j}-3 \hat{k}\) and \(\vec{b}=3 \hat{\imath}-4 \hat{j}-7 \hat{k}\)

∴ \(\vec{a} \cdot \vec{b}\) = (5 x 3) + (1 x -4)÷(- 3 x 7)

15 – 4 – 21 = – 10

Question 17.

If \(\vec{a}=3 \vec{i}+4 \vec{j}-5 \vec{k}\) and b = \(7 \vec{i}-3 \vec{j}+6 \vec{K}\) then find \((\vec{a}+\vec{b}) \times(\vec{a}-\vec{b})\)

Answer:

Question 18.

Find the acute angle between two straight lines whose direction ratios are (1,1,0) and (2,1,2).

Answer:

Direction ratios of the first line are 1,1,0

∴Its direction cosines are

Direction ratios of the second line are 2,1,2

its direction cosines are

Question 19.

Find the values of p so that the lines \(\frac{11-x}{p}=\frac{3 y-3}{2}=\frac{17-z}{5}\) and

\(\frac{x-22}{3 p}=\frac{2 y-7}{27 p}=\frac{z-100}{6 / 5}\) are perpendicular to each other.

Answer:

⇒ -3p^{2} + 9p – 6 = 0

⇒ 3p^{2} – 9p + 6 = 0

⇒ p^{2} – 3p + 2 = 0

⇒ p^{2} – 2p – p + 2 = 0

⇒ p(p – 2) -1 (p – 2) =0

⇒ (p – 2) (p – 1) = 0

∴ p = 2, 1

Question 20.

Prove that the two planes, 3x – 4y + 5z = 0 and 2x – y – 2z = 5 are mutually perpendicular.

Answer:

Given equations of plane are 3x – 4y + 5z = 0 …(i)

and 2x-y-2z = 5 …(ii)

Here a_{1}= 3 b_{1}= – 4, c_{1} = 5; a_{2} = 2, b_{2} = -1 c_{2}=- 2

a_{1}a_{2} + b_{1}b_{2} + c_{1}c_{2}; 6 + 4 – 10 = 0

Since product of Direction’ratios at two planes are zero, plane (i) & (ii) are perpendicular.

Question 21.

What is the probability of occurrence of a number greater than 2 if it is known that only even num¬bers can occur ?

Answer:

Let S = {1,2,3,4,5,6}; A = {2,4,6}

B= {3, 4, 5,6}

Question 22.

A person tosses a coin 3 times. Find the probability of occurrence of exactly one head.

Answer:

Let p = Probability of getting a head in one trial

Question 23.

If y = sin x + cosx, then find \(\frac{d^{2} y}{d x^{2}}\)

Answre:

Given that y = sinx + cosx; D.w.r. to x both sides

∴ \(\frac{d y}{d x}\) = cps x – sin x, Again D.w.r.to x both sides

⇒ \(\frac{d^{2} y}{d x^{2}}\) = – sin x – cosx (sinx + cosx)

Question 24.

Find the values of \(\left|\begin{array}{lll}

a & a^{2} & a^{3} \\

b & b^{2} & b^{3} \\

c & c^{2} & c^{3}

\end{array}\right|\)

Answer:

Question 25.

If A and B be two events and 2P (A) = P (B) = 6/13 and P (A/B) = 1/3, then find (P(A ∪ B)

Answer:

Long Answer Type Questions

Question no. 26 to 33 are long answer type questions. Each question carries 5 marks.

Answer any 4 questions out of these. (4 × 5 = 20)

Question 26.

If y = \(e^{x^{x}}\) then find \(\frac{d y}{d x}\)

Answer:

Question 27.

Prove that sinθ (1 + cosθ)has maximum value at θ = \(\frac{\pi}{3}\)

Answer:

Let y = sinθ(1 + cosθ)

D .w.r. to 0 both sides

Question 28.

Evaluate : \(\int_{0}^{\pi} \frac{x}{1+\sin x} d x\)

Answer:

Question 29.

Solve: (x^{2} + y2)\(\frac{d y}{d x}\) = 2xy

Answer:

Given differential equ. is (x^{2} + y2)\(\frac{d y}{d x}\) = 2xy

This is Homogeneons diff. equation put y = vx

Question 30.

Maximize : Z = 50x + 15y

subject to : 5A + y ≤ 5, x + y ≤ 3 and x, y ≥ 0

Answer:

Its corresponding equation

5x + y = 5 …… (i)

x + y = 3 ……..(ii)

Question 31.

A speaks the truth in 75% casses and B in 80% of the cases. In what percentage of cases are they likely to contradict each other in stating the same fact ?

Answer:

Let E = The Event of A speaking the truth and F = The Event of B speaking the truth.

Then P(E) = \(\frac{75}{100}=\frac{3}{4}\) and P(F) = \(\frac{80}{100}=\frac{4}{5}\)

Required probability P(A & B contradicting each other)

= P(EF̄) or ĒF) = P(EF̄+ĒF)= P(E)- P(F̄) + P(Ē).P(F)

= P(E) .[1 – P(F)]+[1 – P(E). P(F)]

∴ A &.B are likely to contradict each other in 35% cases.

Question 32.

Find the acute angle between the straight line \(\frac{x}{1}=\frac{y}{3}=\frac{z}{0}\) and plane 2x + y = 5

Answer:

Given that \(\frac{x}{1}=\frac{y}{3}=\frac{z}{0}\) and 2x + y = 5

Here a_{1}, b_{1}, c_{1} & a_{2} = 2, b_{2}= 1, c_{2} = 0

∴ Acute angle between the given line and plane is

Question 33.

Factorize. \(\left|\begin{array}{ccc}

(b+c)^{2} & a^{2} & a^{2} \\

b^{2} & (c+a)^{2} & b^{2} \\

c^{2} & c^{2} & (a+b)^{2}

\end{array}\right|\)

Answer: