Bihar Board 12th Maths Objective Important Questions Part 4

Bihar Board 12th Maths Objective Important Questions Part 4

Question 1.
The order of the differential equation \(2 x^{2} \frac{d^{2} y}{d x^{2}}+3 \frac{d y}{d x}+y=0\) is ’
(a) 3
(b) 2
(c) 1
(d) not defined
Answer:
(a) 3

Question 2.
The number of arbitrary constants in the general solution of a differential equation of fourth order are.
(a) 0
(b) 2
(c) 3
(d) 4
Answer:
(d) 4

Question 3.
The number of arbitrary constants in the particular solution of a differential equation of third order are :
(a) 3
(b) 2
(c) 1
(d) 0
Answer:
(d) 0

Question 4.
Which of the following differential equations hasy = c1e^x + c2e^-x as the general solution?
(a) \(\frac{d^{2} y}{d x^{2}}\) + y = 0
(b) \(\frac{d^{2} y}{d x^{2}}\) – y = 0
(c) \(\frac{d^{2} y}{d x^{2}}\) + 1 = 0
(d) \(\frac{d^{2} y}{d x^{2}}\) – 1 = 0
Answer:
(b) \(\frac{d^{2} y}{d x^{2}}\) – y = 0

Question 5.
Which of the following differential equations has y =x as one crits particular solution ?

Answer:
(c)

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Question 6.
The general solution of the differential equation \(\frac{d y}{d x}\) = e^x+y is ;
(a) e^x + e^-xy = c
(b) e^x + e^y = c
(c) e^-x + e^y = c
(d) e^-x + e^-y = c
Answer:
(a) e^x + e^-xy = c

Question 7.
A homogeneous differential equation of the front \(\frac{d x}{d y}=h\left(\frac{x}{y}\right)\) can be solved by marking the substitution : .
(a) y = Vx
(b) V = yx
(c) x = Vy
(d) x = V
Answer:
(c) x = Vy

Question 8.
Which of the following is a homogeneous differential equation :
(a) (4x + 6y + 5) dy – |3y + 2x + 4| dx = 0
(b) (xy) dx – (x³ + y³) dy = 0
(c) (x³ + 2y²)dx + 2xydy = 0
(d) y²dx + (x² – xy – y³)dy = 0
Answer:
(d) y²dx + (x² – xy – y³)dy = 0

Question 9.
The integrating factor of the differential equation \(x \frac{d y}{d x}-y=2 x^{2}\) is :
(a) e^-x
(b) e^-y
(c) 1/x
(d) x
Answer:
(c) 1/x

Question 10.
The general solution of a differential equation of the type \(\frac{d x}{d y}+P_{1} x=Q_{1}\) is :

Answer:
(c)

Question 11.
The integrating factor of the differential equation
(a-y²)\(\frac{d x}{d y}\)

Answer:
(d) \(\frac{1}{\sqrt{1-y^{2}}}\)

Question 12.
In triangle ABC, which of the following is not true :


Answer:
(c)

Question 13.
If \(\vec{a}\) and \(\vec{b}\) are two colliner vectors, then which of the following are incorrect:
(a) \(\vec{b}\) = λ \(\vec{a}\), for some scalar λ
(b) \(\vec{a}\) = ±\(\vec{b}\)
(c) The respective compounds of \(\vec{a}\) and \(\vec{b}\) are proportional
(d) Both the vectors \(\vec{a}\) and \(\vec{b}\) have some direction, but different magnitudes.
Answer:
(d) Both the vectors \(\vec{a}\) and \(\vec{b}\) have some direction, but different magnitudes.

Question 14.
Let the vectors \(\vec{a}\) and \(\vec{b}\) be such that |\(\vec{a}\)| = 3 and |\(\vec{a}\)|= \(\frac{\sqrt{2}}{3}\), then \(\overrightarrow{a} \times \overrightarrow{b}\) is a unit vector, If the angle between \(\vec{a}\) and \(\vec{b}\) is :
(a) \(\frac{\pi}{6}\)
(b) \(\frac{\pi}{4}\)
(c) \(\frac{\pi}{3}\)
(d) \(\frac{\pi}{2}\)
Answer:
(b) \(\frac{\pi}{4}\)

Question 15.
Area of a rectangle having vertics A, B, C and D with position :
Vectors \(-\hat{i}+\frac{1}{2} \hat{j}+4 \hat{k}, \hat{i}+\frac{1}{2} \hat{j}+4 \hat{k}, \hat{i}-\frac{1}{2} \hat{j}+4 \hat{k}\) and \(-\hat{i}-\frac{1}{2} \hat{j}+4 \hat{k}\) respectively is:
(a) \(\frac { 1 }{ 2 }\)
(b) 1
(c) 2
(d) 4
Answer:
(c) 2

Question 16.
If \(\vec{a}\) is a nonzero vector of magnitude ‘a’ and λ nonzero scalar, then λ\(\vec{a}\) is unit vector If,
(a) λ = 1
(b) λ = -1
(c) a = |λ|
(d) a = \(\frac{1}{|\lambda|}\)
Answer:
(d) a = \(\frac{1}{|\lambda|}\)

Question 17.
The general solution of the differential equation
e^xdy + (ye^x +2x)dx = 0 is :
(a) xe^x + x² = c
(b) xe^y + y² = c
(c) ye^x + x² = c
(d) ye^y + x² = c
Answer:
(c) ye^x + x² = c

Question 18.
If θ is the angle between two vectors \(\vec{a}\) and \(\vec{b}\), the \(\vec{b}\), \(\vec{a}\), \(\vec{b}\) ≥ 0 only when :
(a) 0 < θ < \(\frac{\pi}{2}\)
(b) 0 ≤ θ ≤ \(\frac{\pi}{2}\)
(c) 0 < θ < π
(d) 0 ≤ θ ≤ π
Answer:
(b) 0 ≤ θ ≤ \(\frac{\pi}{2}\)

Question 19.
Let \(\vec{a}\) and \(\vec{b}\) be two unit vectors and θ is the angle between them; Then \(\vec{a}\) + \(\vec{b}\) is a unit vector if:
(a) θ = \(\frac{\pi}{4}\)
(b) θ = \(\frac{\pi}{3}\)
(c) θ = \(\frac{\pi}{2}\)
(d) θ = \(\frac{2\pi}{3}\)
Answer:
(d) θ = \(\frac{2\pi}{3}\)

Question 20.
The value of \(\hat{i}(\hat{j} \times \hat{k})+\hat{j}(\hat{i}+\hat{k})+\hat{k}(\hat{i}+\hat{j})\) is :
(a) 0
(b) -1
(c) 1
(d) 3
Answer:
(c) 1

Question 21.
If θ is the angle between any two vectors \(\vec{a}\) and \(\vec{b}\) the \(|\vec{a} \cdot \vec{b}| \vec{a} \cdot \vec{b}|=| \vec{a} \times \vec{b} \mid\) when θ, is equal to
(a) 0
(b) \(\frac{\pi}{4}\)
(c) \(\frac{\pi}{2}\)
(d) π
Answer:
(b) \(\frac{\pi}{4}\)

Question 22.
The planes 2x – y + 4z = 5 and Sx – 2.5y + 10z = 6 are :
(a) perpendicular
(b) parallel
(c) intersect y-axis
(d) passes through (0, 0, \(\frac{5}{4}\)
Answer:
(b) parallel

Question 23.
Distance between the two planes 2x + 3 y +-4z = 4 and 4x + 6y + 8x = 12 is
(a) 2 units
(b) 4 units
(c) 8 units
(d) 9 units
Answer:
(d) 9 units

Question 24.
The solution set of the in equation 2x + y > 5 is :
(a) half plane that contains the origin
(b) open half plane not containing the origin
(c) when xy-plane except the reints lying on the live 2x + y = 5
(d) none of these
Answer:
(b) open half plane not containing the origin

Question 25.
Which of the following is not a convex set ?
(a) {(x, y)/2x + 5y < 7}
(b) {(x,y)/x² + y² ≤ 4} .
(c) {x/ |x| = 5}
(d) {(x,y)/3x² + 2y ≤ b
Answer:
(c) {x/ |x| = 5}

Question 26.
The reints of which the maximum value of x + y. Subject to the constraints x + 2y ≤ 70, 2x + y ≤ 95,x,y ≥ 0 is obtained is :
(a) (30,25)
(b) (20,35)
(c) (35,20)
(d) (40,15)
Answer:
(d) (40,15)

Question 27.
Objective function of a LPP is :
(a) a constraint
(b) a function to be optimized
(c) a relation b/w the variables
(d) none of these
Answer:
(b) a function to be optimized

Question 28.
The Probability of obtaining an even prime number on each die, when a pair of dice is rolled is :
(a) 0
(b) \(\frac { 1 }{ 3 }\)
(c) \(\frac { 1 }{ 12 }\)
(d) \(\frac { 1 }{ 36 }\)
Answer:
(d) \(\frac { 1 }{ 36 }\)

Question 29.
If P (A) = \(\frac { 1 }{ 2 }\) P(B) = 0 then P(A/B) is :
(a) 0
(b) \(\frac { 1 }{ 2 }\)
(c) not defined
(d) 1
Answer:
(c) not defined

Question 30.
If Aand B are two events such that P(A) ≠ 0 and P(B/A) = then :
(a ) A⊂B
(b ) B⊂A
(c) B = Φ
(d) A = Φ
Answer”
(a ) A⊂B

Question 31.
If P(B/A)>P(A) then which of the following is correct:
(a) P(B / A) < P(B)
(b) P(AnB)<P(A) P(B) (c) P(B/ A) > P(B)
(d) P(B/A) = P(B)
Answer:
(c) P(B/ A) > P(B)

Question 32.
Two events A and B will be independent, if:
(a) A and B are mutually exclusive
(b) P(A’B’)=[1-P(A)][1-P(B)]
(c )P(A) = P(B)
(d) P(A) + P(B) = 1
Answer:
(b) P(A’B’)=[1-P(A)][1-P(B)]

Question 33.
If A and B are any two events such that P(A) + P(B) – P (A and B) = P(A), the ;
(a) P(B/A) = 1
(b) P(A/B) = 1
(c) P(B/A) = 0
(d) P(A/B) = 0
Answer:
(b) P(A/B) = 1

Question 34.
Suppose that two cards are drown at random from a deck of cards. Let X be the number of aces obtained. Then the. value of E(X) is :
(a) \(\frac{37}{221}\)
(b) \(\frac{9}{13}\)
(c) \(\frac{1}{13}\)
(d) \(\frac{2}{13}\)
Answer:
(d) \(\frac{2}{13}\)

Question 35.
In a box containing 100 bulbs. 10 are defective. The probability that out of a sample of 5 bulbs, none in defective is :
(a) 10^-1
(b) \(\left(\frac{1}{2}\right)^{5}\)
(c) \(\left(\frac{9}{10}\right)^{5}\)
(d) \(\frac{9}{10}\)
Answer:
(c) \(\left(\frac{9}{10}\right)^{5}\)

Question 36.
The probability that a student Is not a swimmer is \(\frac{1}{5}\) then the probability that mat of five students four are swimmers is:
(a) \({ }^{5} C_{4}\left(\frac{4}{5}\right)^{4} \frac{1}{5}\)
(b) \(\left(\frac{4}{5}\right)^{4} \frac{1}{5}\)
(c) \({ }^{5} C_{1} \frac{1}{5}\left(\frac{4}{5}\right)^{4}\)
(d) None of these
Answer:
(a) \({ }^{5} C_{4}\left(\frac{4}{5}\right)^{4} \frac{1}{5}\)

Question 37.
The mean of the numbers of obtained on throwing a die having written. I on three facems, 2 on two faces and 5 on one face is :
(a) 1
(b) 2
(c) 5
(d) \(\frac { 8 }{ 3 }\)
Answer:
(b) 2

Question 38.
If y = log x then \(\frac{d^{2} y}{d x^{2}}\) =
(a) log x
(b) \(\frac{1}{x}\)
(c) \(-\frac{1}{x^{2}}\)
(d) \(\frac{1}{x^{2}}\)
Answer:
(c) \(-\frac{1}{x^{2}}\)

Question 39.
\(\frac{d}{d x}\left[\frac{d}{d x}(\sin x)\right]=\)
(a) sin x
(b) cos x
(c) -sin x
(d) -cos x
Answer:
(c) -sin x

Question 40.
\(\int_{a}^{b}\left[\frac{d}{d x}(\log x)\right]=\)
(a) log \(\frac{b}{a}\)
(b) log \(\frac{a}{b}\)
(c) log(a-b)
(d) log(b-a)
Answer:
(b) log \(\frac{a}{b}\)

Question 41.
dy – dx = y – x hindi
(a) y + x = k
(b) y – x = k
(c) y/x = k
(d) xy = k
Answer:
(b) y – x = k

Question 42.
\(|2 \vec{i}+3 \vec{j}+5 \vec{k}|\)
(a) 10
(b) \(\sqrt{10}\)
(c) 38
(d) \(\sqrt{38}\)
Answer:
(d) 0

Question 43.
\((5 \vec{i}+2 \vec{j}-\vec{k}) \cdot(\vec{i}-2 \vec{j}-\vec{k})=\)
(a) 13
(b) 8
(c) 6
(d) 0
Answer:
(d) 0

Question 44.
\(\frac{d}{d x}\) [(sin^-1x + cos^-1x)] =
(a) \(\frac{\pi}{2}\)
(b) 0
(c) \(\frac{\pi^{2}}{2}\)
(d) 1
Answer:
(b) 0

Bihar Board 12th Maths Important Questions