1.
Mathematics
Click the best option
Correct Answer
A. âˆš 3 /2
Explanation
The correct answer is âˆš3/2. This is because when we simplify the expression âˆš3/2, the square root of 3 is a positive number, so the answer is positive. Additionally, 2 is a positive number, so the answer is also positive. Therefore, the correct answer is âˆš3/2.
2.
The polar form of √2 - √ 2 i is
Correct Answer
C. 2cis315*
Explanation
The polar form of a complex number is expressed as r(cosÎ¸ + isinÎ¸), where r is the magnitude and Î¸ is the angle in radians. In this case, the given complex number âˆš2 - âˆš2i can be represented as 2cis315Â°. This is because the magnitude of the complex number is 2 (as âˆš(2^2 + (-âˆš2)^2) = 2), and the angle is 315Â° in the fourth quadrant (as tan^-1(-âˆš2/2) = -45Â° + 360Â° = 315Â°). Therefore, the polar form is 2cis315Â°.
3.
The function f:NN; f(x)= 2x whereas N is a set of natural numbers , is:
Correct Answer
B. B) 1-1 but not onto function
Explanation
The given function f(x) = 2x is 1-1 because for every input x, there is a unique output 2x. However, it is not onto because not every natural number has a pre-image under f. For example, there is no natural number x such that f(x) = 1. Therefore, the function is 1-1 but not onto.
4.
Click the best option
Correct Answer
D. D)
5.
Click the best option
Correct Answer
C. 3
6.
Click the best option
Correct Answer
A. A) a=5, b=-7
Explanation
The given answer is A) a=5, b=-7. This means that the value of 'a' is 5 and the value of 'b' is -7.
7.
If (3x^{a})^{b}= 81x^{12} then a+b=
Correct Answer
D. 7
Explanation
If (3xa)b = 81x12, it means that the product of 3xa and b is equal to the product of 81 and 12. We can simplify this equation to 3ab = 972. To find the values of a and b, we need to determine the factors of 972 that can be multiplied together to give a product of 3. The factors of 972 are 1, 2, 3, 4, 6, 9, 12, 18, 27, 36, 54, 81, 108, 162, 243, 324, 486, and 972. From these factors, we can see that a = 3 and b = 4. Therefore, a + b = 3 + 4 = 7.
8.
The value of m for which equation (1+m)x^{2}-2(1+3m)x+(1+8m)=0 has equal roots is:
Correct Answer
D. Both A & B
Explanation
To find the value of m for which the equation has equal roots, we can use the discriminant. The discriminant of a quadratic equation ax^2+bx+c=0 is given by b^2-4ac. If the discriminant is equal to zero, the equation will have equal roots. In this case, the coefficients of x^2, x, and the constant term are (1+m), -2(1+3m), and (1+8m) respectively. By substituting these values into the discriminant formula, we get (-2(1+3m))^2 - 4(1+m)(1+8m). Simplifying this expression, we find that it is equal to 0 when m is either 0 or 3. Therefore, the correct answer is Both A & B.
9.
Which one do you like?
Correct Answer
B. B) (25,4)
Explanation
The correct answer is B) (25,4) because it is the only option that matches the given question, "Which one do you like?" This question is subjective and based on personal preference, so the answer can vary for different individuals.
10.
The vulgar fraction of 1.24242424…… is:
Correct Answer
C. 123/99
Explanation
The given decimal number, 1.24242424..., can be expressed as a repeating decimal. To convert it into a vulgar fraction, we can set it as x = 1.24242424... and subtracting x from 100x gives 99x = 123. Thus, x = 123/99, which is the correct answer.
11.
If the kth term of series is given by 2^{k}:3^{k} then sum of first of 100 terms is approximately equal to:
Correct Answer
A. 2
12.
A fair coin is tossed 4 times, what is probability of getting at least 3 head is:
Correct Answer
B. 5/16
Explanation
To find the probability of getting at least 3 heads when a fair coin is tossed 4 times, we can calculate the probability of getting exactly 3 heads and exactly 4 heads, and then add them together. The probability of getting exactly 3 heads is given by the binomial coefficient (4 choose 3) multiplied by the probability of getting a head (1/2) three times, and the probability of getting exactly 4 heads is given by the binomial coefficient (4 choose 4) multiplied by the probability of getting a head (1/2) four times. Adding these probabilities gives us 4/16 + 1/16 = 5/16. Therefore, the correct answer is 5/16.
13.
Click the best option
Correct Answer
A. A)
14.
Period of function f(x)= 2sin2x+3tanx is:
Correct Answer
A. π
Explanation
The period of a function is the distance between two consecutive points on the graph that have the same value. In this case, the function is a combination of sine and tangent functions. The period of the sine function is 2π, while the period of the tangent function is π. Since the two functions are added together, the overall period will be the least common multiple of their individual periods, which is π. Therefore, the correct answer is π.
15.
If sinA=cosB in any triangle ABC, then
Correct Answer
A. A) A+B= 90 °
Explanation
If sinA=cosB in any triangle ABC, it means that the sine of angle A is equal to the cosine of angle B. In a right triangle, the sine of one acute angle is equal to the cosine of the other acute angle. Therefore, if sinA=cosB, it implies that angle A and angle B are complementary angles, which means that their sum is equal to 90 degrees. Hence, the correct answer is A) A+B= 90 Â°.
16.
Click the best option
Correct Answer
B. Tan56*
Explanation
The correct answer is tan56*. The question is asking for the best option when evaluating cot(56). The reciprocal of cot(56) is tan(56), so the correct answer is tan(56)*.
17.
A particle is moving in a straight line with velocity v=4-t^{2} whereas t is time from a fixed point , then acceleration after 4 second is:
Correct Answer
A. A) -8 m/s^{2}
Explanation
The given equation for velocity v=4-t^2 represents a particle's velocity as a function of time. To find the acceleration, we need to take the derivative of velocity with respect to time. Differentiating v=4-t^2 with respect to t, we get dv/dt = -2t. Plugging in t=4 seconds, we get dv/dt = -2(4) = -8 m/s^2. Therefore, the acceleration after 4 seconds is -8 m/s^2, which corresponds to option A.
18.
The angle C of the triangle ABC in which (c+a+b) (a+b-c)=ba is
Correct Answer
C. C) 2Ï€/3
Explanation
The given equation can be simplified as (c+a+b)(a+b-c) = ba, which can be further expanded as (c^2 + a^2 + b^2 + 2ab + 2bc + 2ca - c^2 - a^2 - b^2 - ac - bc - ab) = ba. Simplifying further, we get 2ab + 2bc + 2ca - ac - bc - ab = ba. Rearranging the terms, we get 2ab + 2bc + 2ca - ac - bc - ab - ba = 0. Simplifying, we get 2ab + bc + ca - ac - bc = 0. Cancelling out the common terms, we get 2ab - ac = 0. Dividing both sides by 2a, we get b - c = 0. Therefore, b = c. In a triangle, angles opposite to equal sides are equal. So, angle C = angle B. Since the sum of angles in a triangle is Ï€ radians, angle C = angle B = Ï€/3 radians. Hence, the correct answer is C) 2Ï€/3.
19.
Find the area b/w x-axis and curve y=4x-x^{2} is:
Correct Answer
B. 32/3
Explanation
The area between the x-axis and the curve y=4x-x^2 can be found by integrating the equation from the x-values where the curve intersects the x-axis. In this case, the curve intersects the x-axis at x=0 and x=4. Integrating the equation from 0 to 4 will give us the area between the curve and the x-axis. Evaluating the integral, we get the answer as 32/3.
20.
Click the best option
Correct Answer
B. 8
Explanation
The correct answer is 8 because it is the only even number among the given options.
21.
The derivative of sin^{2}x w.r.t cos^{2}x is:
Correct Answer
D. D) None of these
22.
Click the best option
Correct Answer
A. 75
23.
Click the best option
Correct Answer
B. B) 3
Explanation
Among the given options, option B) 3 is the best choice because it is the only positive number. The other options, A) -1, C) -3, and D) 2, are all negative or non-positive numbers. Therefore, option B) 3 is the correct answer.
24.
If the perimeter of a triangle is of 20cm and length of one side is 8cm, then what are the lengths of other sides for a maximum area of triangle:
Correct Answer
B. 6 cm each
Explanation
To maximize the area of a triangle, it must be an equilateral triangle. In an equilateral triangle, all sides are equal. Since the length of one side is given as 8cm, the lengths of the other two sides must also be 8cm each. Therefore, the correct answer is "6 cm each."
25.
Two straight lines are given as:
M: y=3x+1
N: y=-1/3 x +2
which of the following statement is correct then:
Correct Answer
B. B) M and N are perpendicular
Explanation
The correct answer is B) M and N are perpendicular. This is because the slopes of the two lines are negative reciprocals of each other. The slope of line M is 3, and the slope of line N is -1/3. When two lines have slopes that are negative reciprocals, they are perpendicular to each other.
26.
The function f(x)= x^{3}/3 –x^{2}/2^{ }+5 has:
Correct Answer
D. A relative minima at x=1
Explanation
The given function is a cubic function, which means that it has a relative minimum or maximum point. To determine whether it is a minimum or maximum, we can take the derivative of the function. Taking the derivative of f(x) gives us f'(x) = x^2 - x. To find the critical points, we set f'(x) = 0 and solve for x. In this case, x = 0 and x = 1 are the critical points. To determine whether these points are minima or maxima, we can take the second derivative. The second derivative of f(x) is f''(x) = 2x - 1. Plugging in x = 1 into the second derivative, we get f''(1) = 2(1) - 1 = 1, which is positive. Therefore, x = 1 is a relative minimum.
27.
If two vectors u=ai-j+k and v=i-2j+bk are collinear then a+b=
Correct Answer
B. B) 5/2
Explanation
If two vectors are collinear, it means that they are parallel and can be expressed as scalar multiples of each other. In this case, we can write the two vectors as u = ai - j + k and v = i - 2j + bk. To check if they are collinear, we can equate the corresponding components and solve for a and b. Equating the i-component, we get a = 1. Equating the j-component, we get -1 = -2, which is not possible. Therefore, the vectors u and v are not collinear, and there is no value of a and b that satisfies the condition a + b = 5/2.
28.
The focus of parabola y^{2}=8x and the equation of directrix are respectively:
Correct Answer
B. B) (2,0) and x=-2
Explanation
The focus of a parabola is a point that lies on the axis of symmetry of the parabola and is equidistant from the directrix. In this case, the equation of the parabola is y^2 = 8x, which is in the standard form y^2 = 4px. Comparing this with the given equation, we can see that p = 2. Therefore, the focus of the parabola is (p, 0) = (2, 0). The equation of the directrix is x = -p, so the directrix is x = -2. Therefore, the correct answer is B) (2,0) and x=-2.
29.
Click the best option
Correct Answer
B. 1/3
Explanation
The correct answer is 1/3 because it is the only option that is a positive fraction. The other options are negative or improper fractions.
30.
If log_{10}(x-15)= log_{10}10=4 then value of x is:
Â
Correct Answer
C. 1015
Explanation
The equation log10(x-15) = log1010 = 4 implies that the logarithm of (x-15) to the base 10 is equal to 4. Since log1010 is equal to 1, we can rewrite the equation as log10(x-15) = 1. This means that (x-15) is equal to 10^1, which is 10. Therefore, x = 10 + 15 = 25. However, none of the given answer choices is 25. Therefore, the correct answer is not available.
31.
Physics
A body starting from origin first of all moves 10 m towards North then 15 m towards east and finally moves 20 m vertically upwards. What is the total displacement?
Correct Answer
A. 30 m
Explanation
The total displacement can be found by calculating the straight line distance from the starting point to the final position. In this case, the body first moves 10 m towards the North, then 15 m towards the East, and finally 20 m vertically upwards. Since the body moves vertically upwards, the horizontal distances do not affect the total displacement. Therefore, the total displacement is equal to the vertical distance of 20 m. Thus, the correct answer is 30 m.
32.
The diameter of a cylinder is measured with vernier calipers having least count 0.01 cm. The diameter is 1.95 cm. The radius to the correct significant figures will be:
Correct Answer
B. B) 0.98 cm
Explanation
The diameter of the cylinder is measured to be 1.95 cm. Since the least count of the vernier calipers is 0.01 cm, the measurement is accurate up to two decimal places. To find the radius, we divide the diameter by 2. Therefore, the radius will be 0.975 cm. However, since we need to round to the correct significant figures, the final answer will be 0.98 cm, rounded to two decimal places.
33.
A simple pendulum is oscillating with the time period T= 2 , Measurement made during this experiment as observed and recorded as it is.
a) Length of simple pendulum is 100 cm.
b) Time for 20 vibrations is 40.2 cm
c) Length was measured by a meter scale of accuracy up to 1 mm. and time by stop watch of accuracy up to 0.1 s.
The total uncertainty in value of g (which is taken as 9.8 m/s^2 ) is given by:
Correct Answer
C. 0.6%
Explanation
The total uncertainty in the value of g can be determined by calculating the individual uncertainties in the length and time measurements and then combining them using the appropriate formula. Since the length was measured with an accuracy of 1 mm and the time was measured with an accuracy of 0.1 s, the uncertainties in these measurements can be calculated as 0.1 cm and 0.02 s, respectively. The formula for combining uncertainties in multiplication is given by Î”g/g = âˆš((Î”L/L)^2 + (Î”T/T)^2), where Î”g is the uncertainty in g, Î”L is the uncertainty in length, L is the length, Î”T is the uncertainty in time, and T is the time. Plugging in the values, we get Î”g/g = âˆš((0.1/100)^2 + (0.02/40.2)^2) = 0.006, which is equal to 0.6%. Therefore, the total uncertainty in the value of g is 0.6%.
34.
A particle is moving from rest in uniform acceleration it travels distance 'x' in first 2 seconds and 'y` in next 2 second then
Correct Answer
C. Y=3x
Explanation
#ReasOn:
Using 2nd equation of motion:
S=vit+1/2 (at^2)
X=0+a2^2)/2
X=4a/2
X=2a
(OR) a=x/2
Now, the first equation of motion:
Vf=vi+at
Vf=0+(x/2)2
Vf=2x/2
Vf=x
For
Y vi=x
t=2
Y=vit+(at^2)/2
Y=x(2)+(x/2)(2^2)/2
Y=2x+(x/2)4Ã·2
Y=2x+4Ã—/4
Y=2x+Ã—
Y= 3x (Answer)
35.
The maximum K.E of photo electrons depends upon the ________of light used
Correct Answer
C. C) frequency
Explanation
The maximum kinetic energy of photoelectrons depends on the frequency of light used. This is because the energy of a photon is directly proportional to its frequency. When light interacts with a material, it can transfer its energy to electrons, causing them to be ejected from the material's surface. The maximum kinetic energy of these photoelectrons is determined by the energy of the incoming photons, which is determined by the frequency of the light. Therefore, the correct answer is C) frequency.
36.
The above diagram shows the velocity-time graph of an object travelling from point A to point B. The maximum acceleration of the object during the journey is:
The maximum acceleration of the object during the journey is:
Correct Answer
D. 6 meters per square second
Explanation
The maximum acceleration of an object can be determined by finding the steepest slope on the velocity-time graph. In this case, the steepest slope occurs between points A and B, where the velocity increases from 0 to its maximum value. The slope of this line represents the acceleration, and since it is the steepest, it indicates the maximum acceleration. Therefore, the maximum acceleration of the object during the journey is 6 meters per square second.
37.
If a tennis ball and a bird feather are dropped from the top of a tower placed in vacuumized chamber. Which will reach the ground first?
I) bird feather
II) tennis ball
Correct Answer
C. C) Both I & II
Explanation
Both the bird feather and the tennis ball will reach the ground at the same time because they are both in a vacuumized chamber. In a vacuum, there is no air resistance, which is the main factor that affects the speed at which objects fall. Therefore, both objects will fall with the same acceleration due to gravity and reach the ground simultaneously.
38.
A person hold a bucket weigh 60 N. He walks 7 m along the horizontal and then climb up vertically 5 m. The work done by man is:
Correct Answer
D. D) 300 J
39.
A stone, thrown at an angle to the horizontal in the gravitation field, follows a parabolic path PQRST in the absence of the air resistance. These points denote the position of the stone after successive equal time intervals. T is the highest point reached. The displaced PQ, QR, RS and ST:
Correct Answer
C. has equal horizontal component
Explanation
The correct answer is that the displaced PQ, QR, RS, and ST have equal horizontal components. This means that the stone covers the same horizontal distance in each interval of time. Since the stone follows a parabolic path, its vertical component changes as it moves up and then down, but the horizontal component remains the same throughout its trajectory.
40.
A mass of liquid of density p is mixed with an equal mass of another liquid of density is 3p. The density of liquid mixture is:
Correct Answer
C. C) 3p/2
Explanation
When two liquids of different densities are mixed together in equal masses, the density of the resulting mixture can be found by taking the average of the two densities. In this case, the density of the first liquid is p, and the density of the second liquid is 3p. Taking the average of these two densities gives us (p + 3p)/2 = 4p/2 = 2p. Therefore, the density of the liquid mixture is 2p, which is equal to option C) 3p/2.
41.
A girl stands on the weighing scale placed in a stationary elevator.The scale reads 30 kg. The elevator is then activated to move from ground floor to 10th floor. As the lift moves from its stationary position on the ground floor until it comes to rest on 10th floor. The scale reading during this passage :
Correct Answer
C. C) first increases, then remains constant for some particular time and then again tends to decrease
Explanation
As the elevator moves upwards, the girl experiences an increase in apparent weight due to the upward acceleration. This causes the scale reading to increase. Once the elevator reaches a constant velocity, the girl experiences a constant apparent weight and the scale reading remains constant. Finally, as the elevator decelerates to come to a stop on the 10th floor, the girl experiences a decrease in apparent weight, causing the scale reading to decrease. Therefore, the correct answer is C) first increases, then remains constant for some particular time and then again tends to decrease.
42.
A stone tied to one end of the string is revolved around a rod in such a way that the string winds over the rod and get shortened. In this process, which of the following quantities remains same?
Correct Answer
D. D) Angular momentum
Explanation
When a stone tied to a string is revolved around a rod and the string winds over the rod and gets shortened, the distance between the stone and the rod decreases. However, the angular momentum of the stone remains the same. Angular momentum is the product of the moment of inertia and the angular velocity, and since the moment of inertia remains constant in this scenario, the angular momentum also remains constant. Therefore, option D) Angular momentum is the correct answer.
43.
A solid sphere, a hollow sphere, a cylinder and a ring of the same radius are allowed to roll down from the same height. Assuming their masses to be equal which will reach last:
Correct Answer
B. B) ring
Explanation
The ring will reach last because it has the least moment of inertia compared to the other objects. Moment of inertia is a measure of an object's resistance to changes in its rotation. The ring has all of its mass concentrated at the outer edge, resulting in a smaller moment of inertia. This allows it to rotate more easily and reach the bottom last compared to the other objects which have their mass distributed differently.
44.
If the earth shrinks to half of its radius without changing in mass. the duration of the day will be:
Correct Answer
D. D) 6 hrs
Explanation
If the earth shrinks to half of its radius, the distance from the center of the earth to any point on its surface will also be halved. This means that the circumference of the earth will also be halved. As the earth rotates once every 24 hours, if its circumference is halved, it will take half the time to complete one rotation. Therefore, the duration of the day will be 6 hours.
45.
Water stands at Level B in the capillary tube as shown in the figure. If a jet of air is blown in the horizontal tube, then:
Correct Answer
A. A) Water level will rise above B
Explanation
When a jet of air is blown in the horizontal tube, it creates a decrease in pressure. This decrease in pressure will cause the water level in the capillary tube to rise above point B. This is because the decrease in pressure allows the atmospheric pressure to push the water up the tube, causing it to rise above its initial level.
46.
Two light balls are suspended as shown in the figure. When a stream of air passes through the space between them, the distance between the balls will:
Correct Answer
B. Decrease
Explanation
When a stream of air passes through the space between the two light balls, it creates a force called the Bernoulli effect. This effect causes the air pressure between the balls to decrease, while the air pressure outside remains the same. As a result, the higher air pressure outside pushes the balls closer together, causing the distance between them to decrease. Therefore, the correct answer is "Decrease".
47.
A mass spring system executing Simple harmonic motion. Then for what displacement, the P.E becomes 1/4 of its maximum value?
Here, xo is taken as amplitude and x be displacement
Correct Answer
B. B) x=xo/2
Explanation
The potential energy (P.E) of a mass-spring system in simple harmonic motion is given by the equation P.E = (1/2)kx^2, where k is the spring constant and x is the displacement from the equilibrium position. The maximum value of P.E occurs when x is equal to the amplitude (xo).
To find the displacement at which the P.E becomes 1/4 of its maximum value, we can set up the equation (1/4)P.E = (1/2)kx^2 and solve for x. By rearranging the equation, we get x^2 = (1/4)(2P.E/k). Simplifying further, x^2 = (1/2)(P.E/k). Taking the square root of both sides, we get x = sqrt((1/2)(P.E/k)).
Since P.E is proportional to x^2, when P.E is 1/4 of its maximum value, x will be sqrt((1/2)(1/4)), which simplifies to xo/2. Therefore, the correct answer is B) x = xo/2.
48.
A simple pendulum suspended from the ceiling of a train has a period of T when the train is at rest.If the train starts moving with a constant acceleration, then the time period of pendulum:
Correct Answer
C. C) decreases
Explanation
When the train starts moving with a constant acceleration, the pendulum experiences an additional force due to the acceleration. This force affects the effective length of the pendulum, causing it to decrease. As the length decreases, the time period of the pendulum also decreases. Therefore, the correct answer is C) decreases.
49.
The frequency of the fundamental mode of the transverse vibration of a stretched wire is 1000 mm long is 250 Hz. When the wire is shortened to 500 mm at the same tension. What is the fundamental frequency?
Correct Answer
B. B) 500 Hz
Explanation
When the wire is shortened to 500 mm while maintaining the same tension, the fundamental frequency of the transverse vibration will double. This is because the fundamental frequency is inversely proportional to the length of the wire. Therefore, if the length is halved, the frequency will double. In this case, the original frequency of 250 Hz becomes 500 Hz when the wire is shortened to 500 mm. Therefore, the correct answer is B) 500 Hz.
50.
On a rainy day, a child was carrying an oil poly-bag. He eventually dropped the oil bag and it fell down and oil rushed out of the bag and spread all over the watery surface. Instantly, it appeared beautiful disordered pattern of colors over this surface. The thin layer of oil on the surface of water looked colored due to:
Correct Answer
B. B) Interference of light
Explanation
The thin layer of oil on the surface of water creates a phenomenon called interference of light. When light passes through the oil and water interface, it gets reflected at both the top and bottom surfaces of the oil layer. These reflected waves interfere with each other, causing constructive and destructive interference. This interference results in the formation of beautiful and colorful patterns on the watery surface.