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All questions of CBSE New Pattern: Term II Practice Questions for JEE Exam

Direction: Read the following text and answer the following questions on the basis of the same:
The reliability of a COVID PCR test is specified as follows:
Of people having COVID, 90% of the test detects the disease but 10% goes undetected. Of people free of COVID, 99% of the test is judged COVID negative but 1% are diagnosed as showing COVID positive. From a large population of which only 0.1% have COVID, one person is selected at random, given the COVID PCR test, and the pathologist reports him/ her as COVID positive.
Q. What is the probability that the ‘person is actually having COVID given that ‘he is tested as COVID positive’?
a)
0.83
b)
0.0803
c)
0.083
d)
0.089
Correct answer is option 'C'. Can you explain this answer?

Anjali Sharma answered

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Direction: Read the following text and answer the following questions on the basis of the same:
In an office three employees Vinay, Sonia and Iqbal process incoming copies of a certain form. Vinay process 50% of the forms. Sonia processes 20% and Iqbal the remaining 30% of the forms. Vinay has an error rate of 0.06, Sonia has an error rate of 0.04 and Iqbal has an error rate of 0.03.
Q. The probability that Sonia processed the form and committed an error is :
a)
0.005
b)
0.006
c)
0.008
d)
0.68
Correct answer is option 'C'. Can you explain this answer?

Ananya Das answered

P (sonia processed the form and committed an error) = 20% × 0.4

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Direction: Read the following text and answer the following questions on the basis of the same:
A farmer has a square plot of land. Three of its boundaries are x = 0, y = 0 and y = 4. He wants to divide this land among his three sons A, B and C as shown in figure.
Q. Area received by son B is _______ sq. units.
a)
4
b)
16
c)
16/3
d)
8/3
Correct answer is option 'C'. Can you explain this answer?

Anjali Sharma answered

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Direction: Read the following text and answer the following questions on the basis of the same:
Anand, Samanyu and Shah of SHORTCUTS classes were given a problem in Mathematics whose respective probabilities of solving it are 1/2 , 1/3 and 1/4. They were asked to solve it independently.
Based on the above data, answer any four of the following questions.
Q. The probability that Anand alone solves it is _______.
a)
1/4
b)
3/4
c)
11/24
d)
17/24
Correct answer is option 'A'. Can you explain this answer?

Ananya Das answered

Let A → event that Anand solves

B → event that Samanyu solves

C → event that Shah solves

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Direction: Read the following text and answer the following questions on the basis of the same:
A mobile tower stands at the top of a hill. Consider the surface on which the tower stands as a plane having points A(1, 0, 2), B(3, –1, 1) and C(1, 2, 1) on it. The mobile tower is tied with 3 cables from the points A, B and C such that it stands vertically on the ground. The top of the tower is at the point (2, 3, 1) as shown in the figure.
Q. The equation of the plane passing through the points A, B and C is
a)
3x – 2y + 4z = –11
b)
3x + 2y + 4z = 11
c)
3x – 2y – 4z = 11
d)
–3x + 2y + 4z = –11
Correct answer is option 'B'. Can you explain this answer?

Veda Institute answered

A(1, 0, 2), B(3, –1, 1) and C(1, 2, 1)

(x - 1)(1 + 2) -y (-2, 0) + (z - 2) (4 - 0)

3x - 3 + 2y + 4z - 8 = 0

3x + 2y + 4z = 11

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Direction: Read the following text and answer the following questions on the basis of the same:
ABCD is a parallelogram whose adjacent sides are represented by the vectors Three of its vertices are A(1, 2, 3), B(2, 0, 5) and D(–1, 4, 1).
Q. The diagonal
a)
b)
c)
d)
Correct answer is option 'C'. Can you explain this answer?

Krishna Iyer answered

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Direction: Read the following text and answer the following questions on the basis of the same:
A class XII student appearing for a competitive examination was asked to attempt the following questions.
Let be three non zero vectors.
Q. are unit vectors and θ be the angle between them then
a)
sin θ/2
b)
2sin θ/2
c)
2cos θ/2
d)
cos θ/2
Correct answer is option 'B'. Can you explain this answer?

Swati Verma answered

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Direction: Read the following text and answer the following questions on the basis of the same:
Consider the plane π₁ : 2x – 3y + 4z + 9 = 0 and the point P(1, –2, 3). π₁ is a plane parallel to π₁ and containing the point P.
Q. Distance between π₁ and π₂ is _______ units.
a)
5
b)
c)
d)
2√3
Correct answer is option 'B'. Can you explain this answer?

Swati Verma answered

Distance between π₁ and π₂

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Direction: Read the following text and answer the following questions on the basis of the same:
The equation of motion of a missile are x = 3t, y = –4t, z = t, where the time ‘t’ is given in seconds, and the distance is measured in kilometres.
Q. At what distance will the rocket be from the starting point (0, 0, 0) in 5 seconds?
a)
b)
c)
d)
Correct answer is option 'B'. Can you explain this answer?

Varun Kapoor answered

Here,

t = 5 seconds

x = 3t = 3 x 5 = 15

y = -4t = -4 x 5 = -20

z = t = 5

(x, y, z) (15, -20, 5)

Distance from starting point(0, 0, 0)

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Read the following text and answer the following questions on the basis of the same:
Q.
a)
1
b)
0
c)
-1
d)
π
Correct answer is option 'B'. Can you explain this answer?

Anjali Sharma answered

since sin³ x is an odd function.

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Direction: Read the following text and answer the following questions on the basis of the same:
The bridge connects two hills 100 feet apart. The arch on the bridge is in a parabolic form. The highest point on the bridge is 10 feet above the road at the middle of the bridge as seen in the figure.
Q. The value of the integral
a)
1000/3
b)
250/3
c)
1200
d)
0
Correct answer is option 'A'. Can you explain this answer?

Anjali Sharma answered

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Direction: Read the following text and answer the following questions on the basis of the same:
In an office three employees Vinay, Sonia and Iqbal process incoming copies of a certain form. Vinay process 50% of the forms. Sonia processes 20% and Iqbal the remaining 30% of the forms. Vinay has an error rate of 0.06, Sonia has an error rate of 0.04 and Iqbal has an error rate of 0.03.
Q. Let A be the event of committing an error in processing the form and let E₁, E₂ and E₃ be the events that Vinay, Sonia and Iqbal processed the form. The value of
a)
0
b)
0.03
c)
0.06
d)
1
Correct answer is option 'D'. Can you explain this answer?

Neha Joshi answered

[∵ sum of all occurrence of an event is equal to 1]

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Direction: Read the following text and answer the following questions on the basis of the same:
A farmer has a square plot of land. Three of its boundaries are x = 0, y = 0 and y = 4. He wants to divide this land among his three sons A, B and C as shown in figure.
Q. Area received by son A is _______ sq. units.
a)
4
b)
16
c)
16/3
d)
8/3
Correct answer is option 'C'. Can you explain this answer?

Anjali Sharma answered

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Direction: Read the following text and answer the following questions on the basis of the same:
The Indian coast guard, while patrolling, saw a suspicious boat with people. They were nowhere looking like fishermen. The coast guard were closely observing the movement of the boat for an opportunity to seize the boat. They observed that the boat is moving along a planar surface. At an instant of time, the coordinates of the position of the coast guard helicopter and the boat is (1, 3, 5) and (2, 5, 3) respectively.
Q. At that given instant of time, the equation of line passing through the positions of the helicopter and boat is
a)
b)
c)
d)
Correct answer is option 'A'. Can you explain this answer?

Varun Kapoor answered

Here, direction cosines are 1, 2, –2.

Equation of line passing through the positions of the helicopter and boat is

⇒

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Read the following text and answer the following questions on the basis of the same:
∫ e^x[f(x) +f′(x)]dx = ∫ e^xf(x)dx + ∫e^x f′(x)dx
= f(x)e^x – ∫ f ′(x)e^x dx + ∫ f ′(x)e^x dx
= e^x f(x) + c
Q. ∫ e^x (sinx + cos x)dx = _______.
a)
e^x cos x + c
b)
e^x sin x + c
c)
e^x + c
d)
e^x (–cos x + sin x) +c
Correct answer is option 'B'. Can you explain this answer?

Anjali Sharma answered

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Direction: In the following questions, A statement of Assertion (A) is followed by a statement of Reason (R). Mark the correct choice as:
a)
Both A and R are true and R is the correct explanation of A
b)
Both A and R are true but R is NOT the correct explanation of A
c)
A is true but R is false
d)
A is false and R is True
Correct answer is option 'D'. Can you explain this answer?

Tanuja Kapoor answered

This is a property of the definite integrals and hence R is true.

Hence A is false.

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Direction: In the following questions, A statement of Assertion (A) is followed by a statement of Reason (R). Mark the correct choice as:

Assertion (A): ∫e^x[sin x - cos x] dx = e^x sin x + C

Reason (R): ∫e^x[f(x) +f′(x)]dx =e^x f(x) + c

a)
Both A and R are true and R is the correct explanation of A

b)
Both A and R are true but R is NOT the correct explanation of A

c)
A is true but R is false

d)
A is false and R is True

Correct answer is option 'D'. Can you explain this answer?

Geetika Shah answered

∫e^x [f(x) + f′(x)]dx = ∫e^xf(x)dx + ∫e^xf′(x)dx

= f(x)e^x - ∫f′(x)e^xdx + ∫f′(x)e^xdx

= e^xf(x) + c

Hence R is true.

∫e^x(sin x - cos x)dx = e^x (-cos) + c

= -e^x cos x + c

Hence A is false.

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Read the following text and answer the following questions on the basis of the same:
Let’s say that we want to evaluate ∫[P(x)/Q(x)] dx, where P(x)/Q(x) is a proper rational fraction. In such cases, it is possible to write the integrand as a sum of simpler rational functions by using partial fraction decomposition. Post this, integration can be carried out easily. The following image indicates some simple partial fractions which can be associated with various rational functions:
In the above table, A, B and C are real numbers to be determined suitably.
Q.
a)
b)
c)
d)
Correct answer is option 'C'. Can you explain this answer?

Krishna Iyer answered

Equating the coefficients of x², x and constant term, we obtain

A + B = 0

C = 0

A = 1

On solving these equations, we obtain

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Direction: In the following questions, A statement of Assertion (A) is followed by a statement of Reason (R). Mark the correct choice as
Assertion (A): The area of parallelogram with diagonals
Reason (R): If represent the adjacent sides of a triangle, then the area of triangle can be obtained by evaluating
a)
Both A and R are true and R is the correct explanation of A
b)
Both A and R are true but R is NOT the correct explanation of A
c)
A is true but R is false
d)
A is false but R is True
Correct answer is option 'C'. Can you explain this answer?

Tanuja Kapoor answered

represent the adjacent sides of a triangle, then the area of triangle can be obtained by evaluating

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Direction: In the following questions, A statement of Assertion (A) is followed by a statement of Reason (R). Mark the correct choice as
Assertion (A) : The probability of an impossible event is 1.
Reason (R) : If A is a perfect subset of B and P(A) < p(b),="" then="" p(b="" –="" a)="" is="" equal="" to="" p(b)="" –="" />
a)
Both A and R are true and R is the correct explanation of A
b)
Both A and R are true but R is NOT the correct explanation of A
c)
A is true but R is false
d)
A is false but R is True
Correct answer is option 'D'. Can you explain this answer?

Anjali Sharma answered

Assertion (A) is wrong.

If the probability of an event is 0, then it is called an impossible event.

But Reason (R) is correct.

From Basic Theorem of Probability, P(B – A) = P(B) – P(A), this is true only if the condition given in the question is true.

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इनमें से कौन-सा क्षेत्र प्रागैतिहासिक शैल कला चित्रों की मेजबानी के लिए प्रसिद्ध हैं?

1. मध्य प्रदेश की विंध्य पर्वतमाला
2. उत्तर प्रदेश का कैमूरियन विस्तार

उपरोक्त में से कौन सा/से सही हैं?

a)
केवल 1

b)
केवल 2

c)
दोनों 1 और 2

d)
इनमे से कोई भी नहीं

Correct answer is option 'C'. Can you explain this answer?

Amit Sharma answered

मध्य प्रदेश के विंध्य पर्वतमाला और उत्तर प्रदेश में उनके कैमूरियन विस्तार से सबसे समृद्ध चित्रों की सूचना मिली है। ये पर्वत श्रृंखलाएं पुरापाषाण और मध्यपाषाणकालीन अवशेषों से परिपूर्ण हैं। 9000 ईसा पूर्व में पाषाण युग की संस्कृति में एक मध्यवर्ती चरण शुरू हुआ, जिसे मेसोलिथिक युग कहा जाता है।
मध्य प्रदेश, उत्तर प्रदेश, आंध्र प्रदेश, कामतका और बिहार के कई जिलों की गुफाओं में शैल चित्रों के अवशेष हैं।
उत्तराखंड में कुमाऊं की पहाड़ियां भी शैल चित्रों के लिए जानी जाती हैं।

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Direction: Read the following text and answer the following questions on the basis of the same:
Suppose the floor of a hotel is made up of mirror polished Salvatore stone. There is a large crystal chandelier attached to the ceiling of the hotel room. Consider the floor of the hotel room as a plane having the equation x – y + z = 4 and the crystal chandelier is suspended at the point (1, 0, 1).
Q. The equation of the perpendicular from the point (1, 0, 1) to the plane x – y + z = 4 is
a)
b)
c)
d)
Correct answer is option 'C'. Can you explain this answer?

Swati Verma answered

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Direction: Read the following text and answer the following questions on the basis of the same:
A cricket match is organized between two Clubs A and B for which a team from each club is chosen. Remaining players of Club A and Club B are respectively sitting on the plane represented by the equation to cheer the team of their own clubs.
Q. The distance of the plane, on which players of Club B are seated, from the origin is
a)
b)
c)
d)
Correct answer is option 'A'. Can you explain this answer?

Geetika Shah answered

We know

∴

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Direction: Read the following text and answer the following questions on the basis of the same:
A coach is training 3 players. He observes that player A can hit a target 4 times in 5 shots, player B can hit 3 times in 4 shots and the player C can hit 2 times in 3 shots.
Q. What is the probability that ‘any two of A, B and C will hit?
a)
1/30
b)
11/30
c)
17/30
d)
13/30
Correct answer is option 'D'. Can you explain this answer?

Varun Kapoor answered

Probability that any two of A, B and C will hit

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Direction: In the following questions, A statement of Assertion (A) is followed by a statement of Reason (R). Mark the correct choice as
Assertion (A): P is a point on the line segment joining the points (3, 2, –1) and (6, –4, –2). If x coordinate of P is 5, then its y coordinate is –2.
Reason (R): The two lines x = ay + b, z = cy + d and x = a’y + b’, z = c’y + d’ will be perpendicular, iff aa’ + bb’ + cc’ = 0.
a)
Both A and R are true and R is the correct explanation of A
b)
Both A and R are true but R is NOT the correct explanation of A
c)
A is true but R is false
d)
A is false but R is True
Correct answer is option 'C'. Can you explain this answer?

Shalini Bose answered

Assertion (A): P is a point on the line segment joining the points (3, 2, –1) and (6, –4, –2). If x coordinate of P is 5, then its y coordinate is –2.

Reason (R): The two lines x = ay b, z = cy d and x = a’y b’, z = c’y d’ will be perpendicular, iff aa’ bb’ cc’ = 0.

The given assertion relates to the coordinates of a point P on a line segment joining two given points. The reason provided relates to the condition for two lines to be perpendicular.

Explanation:

To understand this assertion and reason, let's first find the equation of the line segment joining the points (3, 2, –1) and (6, –4, –2).

The equation of a line segment joining two points (x₁, y₁, z₁) and (x₂, y₂, z₂) can be written as:
(x - x₁)/(x₂ - x₁) = (y - y₁)/(y₂ - y₁) = (z - z₁)/(z₂ - z₁)

Plugging in the given values, we have:
(x - 3)/(6 - 3) = (y - 2)/(-4 - 2) = (z + 1)/(-2 - (-1))

Simplifying, we get:
(x - 3)/3 = (y - 2)/(-6) = (z + 1)/(-1)

Now, let's consider the x-coordinate of P as 5. Substituting this value into the equation, we get:
(5 - 3)/3 = (y - 2)/(-6) = (z + 1)/(-1)

2/3 = (y - 2)/(-6) = (z + 1)/(-1)

Simplifying further, we get:
(y - 2)/(-6) = (z + 1)/(-1)

Cross-multiplying, we get:
-6(y - 2) = -1(z + 1)

Expanding, we get:
-6y + 12 = -z - 1

Rearranging, we get:
z = -6y + 13

Comparing this equation with the given equations:
x = ay + b
z = cy + d

We can see that a = 0, b = 0, c = -6, and d = 13.

Now, let's consider the given reason. According to the reason, the two lines x = ay + b, z = cy + d and x = a'y + b', z = c'y + d' will be perpendicular if aa' + bb' + cc' = 0.

Plugging in the values, we have:
(0)(0) + (0)(0) + (-6)(0) = 0

Therefore, aa' + bb' + cc' = 0, which means the two lines are perpendicular.

Therefore, both the assertion and the reason are true, and the reason correctly explains the assertion.

Hence, the correct answer is option C.

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Direction: Read the following text and answer the following questions on the basis of the same:
Polio drops are delivered to 50K children in a district. The rate at which polio drops are given is directly proportional to the number of children who have not been administered the drops. By the end of 2nd week half the children have been given the polio drops. How many will have been given the drops by the end of 3rd week can be estimated using the solution to the differential equation dy/dx = k(50 - y) where x denotes the number of weeks and y the number of children who have been given the drops.
Q. The solution of the differential equation dy/dx = k(50 - y) is given by,
a)
log |50 – y| = kx + C
b)
–log |50 – y| = kx + C
c)
log |50 – y| = log|kx|+ C
d)
50 – y = kx + C
Correct answer is option 'B'. Can you explain this answer?

Ananya Das answered

Dy/dx = k(50 - y)

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Direction: Read the following text and answer the following questions on the basis of the same:
The bridge connects two hills 100 feet apart. The arch on the bridge is in a parabolic form. The highest point on the bridge is 10 feet above the road at the middle of the bridge as seen in the figure.
Q. The area formed by the curve x² = 250y, x-axis, y = 0 and y = 10 is
a)
1000√2/3
b)
4/3
c)
1000/3
d)
0
Correct answer is option 'C'. Can you explain this answer?

Anjali Sharma answered

X² = 250 y

at = y = 0 x = 0

at y = 10 x = 50, -50

∴ Area formed by curve

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Direction: Read the following text and answer the following questions on the basis of the same:
Polio drops are delivered to 50K children in a district. The rate at which polio drops are given is directly proportional to the number of children who have not been administered the drops. By the end of 2nd week half the children have been given the polio drops. How many will have been given the drops by the end of 3rd week can be estimated using the solution to the differential equation dy/dx = k(50 - y) where x denotes the number of weeks and y the number of children who have been given the drops.
Q. Which of the following solutions may be used to find the number of children who have been given the polio drops?
a)
y = 50 – e^kx
b)
y = 50 – e^–kx
c)
y = 50(1 – e^–kx)
d)
y = 50(e^–kx – 1)
Correct answer is option 'C'. Can you explain this answer?

Ananya Das answered

We have

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भारतीय उपमहाद्वीप में मेसोलिथिक युग से लौह युग तक के प्रमुख पुरातात्विक स्थलों पर विचार करें। वर्तमान भारत में उनके संबंधित क्षेत्रों के साथ उनका मिलान करें:
1. कोल्डिहवा : मध्य प्रदेश
2. मेहरगढ़: हरियाणा
3. पय्यमपल्ली: तमिलनाडु
नीचे दिए गए कोड का उपयोग करके सही उत्तर चुनें।
a)
केवल 1 और 2
b)
केवल 3
c)
2 और 3 ही
d)
1 ही
Correct answer is option 'B'. Can you explain this answer?

Kuldeep Rajput answered

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Direction: Read the following text and answer the following questions on the basis of the same:
A cricket match is organized between two Clubs A and B for which a team from each club is chosen. Remaining players of Club A and Club B are respectively sitting on the plane represented by the equation to cheer the team of their own clubs.
Q. The Cartesian equation of the plane on which players of Club A are seated is
a)
2x − y + z = 3
b)
2x − y + 2z = 3
c)
2x − y + z =−3
d)
x − t + z = 3
Correct answer is option 'A'. Can you explain this answer?

Defence Exams answered

2x - y + z = 3

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Read the following text and answer the following questions on the basis of the same:
Q.
a)
0
b)
1
c)
-1
d)
2
Correct answer is option 'A'. Can you explain this answer?

Krishna Iyer answered

since x⁹⁹ is an odd function.

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Direction: Read the following text and answer the following questions on the basis of the same:
The reliability of a COVID PCR test is specified as follows:
Of people having COVID, 90% of the test detects the disease but 10% goes undetected. Of people free of COVID, 99% of the test is judged COVID negative but 1% are diagnosed as showing COVID positive. From a large population of which only 0.1% have COVID, one person is selected at random, given the COVID PCR test, and the pathologist reports him/ her as COVID positive.
Q. What is the probability of the ‘person to be tested as COVID positive’ given that ‘he is actually having COVID’?
a)
0.001
b)
0.1
c)
0.8
d)
0.9
Correct answer is option 'D'. Can you explain this answer?

Anjali Sharma answered

E = Person selected has Covid

F = Does not have Covid

G = Test judge Covid positive

Probability of the person to be tested as Covid positive given that he is actually having Covid

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Direction: Read the following text and answer the following questions on the basis of the same:
Solar Panels have to be installed carefully so that the tilt of the roof, and the direction to the sun, produce the largest possible electrical power in the solar panels.
A surveyor uses his instrument to determine the coordinates of the four corners of a roof where solar panels are to be mounted. In the picture , suppose the points are labelled counter clockwise from the roof corner nearest to the camera in units of meters P₁(6, 8, 4), P₂(21, 8, 4), P₃(21, 16, 10) and P₄(6, 16, 10)
Q. What is the magnitude of and in what units?
a)
100
b)
150
c)
50
d)
90
Correct answer is option 'B'. Can you explain this answer?

Anjali Sharma answered

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Direction: Read the following text and answer the following questions on the basis of the same:
Polio drops are delivered to 50K children in a district. The rate at which polio drops are given is directly proportional to the number of children who have not been administered the drops. By the end of 2nd week half the children have been given the polio drops. How many will have been given the drops by the end of 3rd week can be estimated using the solution to the differential equation dy/dx = k(50 - y) where x denotes the number of weeks and y the number of children who have been given the drops.
Q. The value of C in the particular solution given that y(0) = 0 and k = 0.049 is
a)
log 50
b)
log 1/50
c)
50
d)
-50
Correct answer is option 'B'. Can you explain this answer?

Ananya Das answered

Given,

y(0) = and k = 0.049

We have, -log|50 - y| = Kx + C

log |50 - y| = -Kx - C

log |50 - 0| = 0 - C

[∵ x = 0, K = 0.049, y(0) = 0]

log 50 = - C

C = log 1/50

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Direction: In the following questions, A statement of Assertion (A) is followed by a statement of Reason (R). Mark the correct choice as
Assertion (A): The position vector of a point say and its magnitude is
Reason (R): If then coefficient of are called the direction ratios of vector
a)
Both A and R are true and R is the correct explanation of A
b)
Both A and R are true but R is NOT the correct explanation of A
c)
A is true but R is false
d)
A is false but R is True
Correct answer is option 'B'. Can you explain this answer?

Raghav Bansal answered

Assertion (A) and Reason (R) both are individually correct.

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Direction: Read the following text and answer the following questions on the basis of the same:
A class XII student appearing for a competitive examination was asked to attempt the following questions.
Let be three non zero vectors.
Q. then evaluate
a)
0
b)
4
c)
3
d)
2
Correct answer is option 'A'. Can you explain this answer?

Raghav Bansal answered

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Direction: Read the following text and answer the following questions on the basis of the same:
A cricket match is organized between two Clubs A and B for which a team from each club is chosen. Remaining players of Club A and Club B are respectively sitting on the plane represented by the equation to cheer the team of their own clubs.
Q. The intercept form of the equation of the plane on which players of Club B are seated is
a)
b)
c)
d)
Correct answer is option 'C'. Can you explain this answer?

Hansa Sharma answered

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Read the following text and answer the following questions on the basis of the same:
Let’s say that we want to evaluate ∫[P(x)/Q(x)] dx, where P(x)/Q(x) is a proper rational fraction. In such cases, it is possible to write the integrand as a sum of simpler rational functions by using partial fraction decomposition. Post this, integration can be carried out easily. The following image indicates some simple partial fractions which can be associated with various rational functions:
In the above table, A, B and C are real numbers to be determined suitably.
Q.
a)
b)
c)
d)
Correct answer is option 'D'. Can you explain this answer?

Hansa Sharma answered

Let ex = t

⇒ e^x dx = dt

1 = A(t - 1) + Bt …..(i)

Substituting t = 1 and t = 0 in equation (i), we obtain

A = –1 and B = 1

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Read the following text and answer the following questions on the basis of the same:
Let’s say that we want to evaluate ∫[P(x)/Q(x)] dx, where P(x)/Q(x) is a proper rational fraction. In such cases, it is possible to write the integrand as a sum of simpler rational functions by using partial fraction decomposition. Post this, integration can be carried out easily. The following image indicates some simple partial fractions which can be associated with various rational functions:
In the above table, A, B and C are real numbers to be determined suitably.
Q. Integration of
a)
b)
c)
d)
Correct answer is option 'B'. Can you explain this answer?

Varun Kapoor answered

Equating the coefficients of x and constant term, we obtain

A + B = 1

2A + B = 0

On solving, we obtain A = -1 and B = 2

= -log|x+1|+2log|x+2|+C

= log(x +2)² - log|x+1|+C

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Read the following text and answer the following questions on the basis of the same:
Q.
a)
π
b)
0
c)
2π
d)
π/2
Correct answer is option 'C'. Can you explain this answer?

Neha Joshi answered

Since, xsin x is an even function

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Read the following text and answer the following questions on the basis of the same:
∫ e^x[f(x) +f′(x)]dx = ∫ e^xf(x)dx + ∫e^x f′(x)dx
= f(x)e^x – ∫ f ′(x)e^x dx + ∫ f ′(x)e^x dx
= e^x f(x) + c
Q.
a)
0
b)
1
c)
-1
d)
-e^π
Correct answer is option 'A'. Can you explain this answer?

Raghav Bansal answered

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Direction: Read the following text and answer the following questions on the basis of the same:
The rate of increase in the number of bacteria in a certain bacteria culture is proportional to the number present. Given that the number triples in 5 hours.
Q. If N₀ is the initial count of bacteria, after 10 hours the count is _______.
a)
1/5 log3
b)
3log N₀
c)
9N₀
d)
2N₀
Correct answer is option 'C'. Can you explain this answer?

Ananya Das answered

Given when t = 0, N = N₀.

From (i), log|N₀| = C

∴ (i) ⇒ log|N| = Kt + log|N₀|

⇒

….(ii)

Given when t = 5, N = 3N₀.

From (ii), log|3| = 5K

⇒ K = 1/5 log3

∴ The particular solution is

When t = 10,

N/N₀ = 9

⇒ N = 9N₀

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Direction: Read the following text and answer the following questions on the basis of the same:
A differential equation of the form where P and Q are functions of x alone, is called a first order linear differential equation. It has many applications in physics including RL circuits.
Consider the linear differential equation
Q. The value of ∫ P dx = ______.
a)
b)
log x² + C
c)
log|x| + C
d)
Correct answer is option 'B'. Can you explain this answer?

Varun Kapoor answered

= 2log |x| + C

= log x² + C

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Direction: Read the following text and answer the following questions on the basis of the same:
The cross product of two vectors gives a vector perpendicular to the plane containing the vectors.
Q. The angle between
a)
0
b)
π
c)
π/2
d)
π/4
Correct answer is option 'B'. Can you explain this answer?

Varun Kapoor answered

The angle between

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Direction: Read the following text and answer the following questions on the basis of the same:
In the figure O(0, 0) is the centre of the circle. The line y = x meets the circle in the first quadrant at the point B.
Q. Area of the shaded region is _______ sq. units.
a)
32π
b)
4π
c)
8
d)
4π - 8
Correct answer is option 'B'. Can you explain this answer?

Anjali Sharma answered

Area of shaded region

= Ar (ΔOBM) + Ar (BAMB)

= 8 + 4π – 8

= 4π sq. units

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Direction: Read the following text and answer the following questions on the basis of the same:
Consider the plane π₁ : 2x – 3y + 4z + 9 = 0 and the point P(1, –2, 3). π₁ is a plane parallel to π₁ and containing the point P.
Q. The image of π on π₁ is _______.
a)
(–1, 1, –1)
b)
(–1, 2, –3)
c)
(–3, 4, –5)
d)
(0, 0, 0)
Correct answer is option 'C'. Can you explain this answer?

Geetika Shah answered

Let P′(α, β, γ) be the image A is mid-point of PP′.

⇒ α = –3, β = 4, γ = –5

∴ P′ = (–3, 4, –5)

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Direction: Read the following text and answer the following questions on the basis of the same:
A group of people start playing cards. And as we know a well shuffled pack of cards contains a total of 52 cards. Then 2 cards are drawn simultaneously (or successively without replacement).
Q. If x = no. of kings = 0, 1, 2. Then P(x = 1) = ?
a)
32/229
b)
32/227
c)
32/221
d)
32/219
Correct answer is option 'C'. Can you explain this answer?

Neha Sharma answered

= 188/221

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Direction: Read the following text and answer the following questions on the basis of the same:
A group of people start playing cards. And as we know a well shuffled pack of cards contains a total of 52 cards. Then 2 cards are drawn simultaneously (or successively without replacement).
Q. If x = no. of kings = 0, 1, 2. Then P(x = 0) = ?
a)
188/221
b)
198/223
c)
197/290
d)
187/221
Correct answer is option 'A'. Can you explain this answer?

Raghav Bansal answered

= 188/221

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Direction: Read the following text and answer the following questions on the basis of the same:
A group of people start playing cards. And as we know a well shuffled pack of cards contains a total of 52 cards. Then 2 cards are drawn simultaneously (or successively without replacement).
Q. If x = no. of kings = 0, 1, 2. Then P(x = 2) = ?
a)
2/219
b)
1/221
c)
3/209
d)
1/209
Correct answer is option 'B'. Can you explain this answer?

Preeti Iyer answered

= 1/221

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Direction: Read the following text and answer the following questions on the basis of the same:
The reliability of a COVID PCR test is specified as follows:
Of people having COVID, 90% of the test detects the disease but 10% goes undetected. Of people free of COVID, 99% of the test is judged COVID negative but 1% are diagnosed as showing COVID positive. From a large population of which only 0.1% have COVID, one person is selected at random, given the COVID PCR test, and the pathologist reports him/ her as COVID positive.
Q. What is the probability of the ‘person to be tested as COVID positive’ given that ‘he is actually not having COVID’?
a)
0.01
b)
0.99
c)
0.1
d)
0.001
Correct answer is option 'A'. Can you explain this answer?

Ananya Das answered

Probability of person to be tested as Covid positive given that he is actually not having Covid = P(G/E)= 1% = 1/100 = 0.01

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