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Sample Mock Test JEE
15 questions · Physics, Chemistry, Maths · tap an option to check your answer
Physics
Q1. ABC is a right angled triangular plate of uniform thickness. The sides are such that AB BC> as shown in figure. I1, I2, I3 are moments of inertia about AB, BC, AC, and respectively. Then, which of the following relations is correct?
Answer: B
The moment of inertia of a body about an axis depends on the mass of the body and the distribution of mass from the axis. For a given body, the mass remains constant, so the distribution matters. Since the mass is farthest from axis BC, I2 is maximum, whereas the mass is closest to axis AC, making I3 minimum. Thus, the correct relation is I2 > I1 > I3.
The moment of inertia of a body about an axis depends on the mass of the body and the distribution of mass from the axis. For a given body, the mass remains constant, so the distribution matters. Since the mass is farthest from axis BC, I2 is maximum, whereas the mass is closest to axis AC, making I3 minimum. Thus, the correct relation is I2 > I1 > I3.
Q2. The second overtone of an open organ pipe has the same frequency as the first overtone of a closed pipe L metre long. What is the length of the open pipe?
Answer: B
For an open organ pipe, \( \nu_n = \frac{n v}{2L} \) where n = 1, 2, 3, ... For the second overtone, n=3, \( \nu_3 = \frac{3 v}{2L_1} \)\nFor closed organ pipe, \( \nu_n = \frac{(n+1)v}{4L} \)\nFor the 1st overtone, n=1, \( \nu_1 = \frac{3 v}{4L} \)\nSetting \( \nu_3 = \nu_1 \) leads to \( \frac{3v}{2L_1} = \frac{3v}{4L} \)\nThus, \( L_1 = 2L \).
For an open organ pipe, \( \nu_n = \frac{n v}{2L} \) where n = 1, 2, 3, ... For the second overtone, n=3, \( \nu_3 = \frac{3 v}{2L_1} \)\nFor closed organ pipe, \( \nu_n = \frac{(n+1)v}{4L} \)\nFor the 1st overtone, n=1, \( \nu_1 = \frac{3 v}{4L} \)\nSetting \( \nu_3 = \nu_1 \) leads to \( \frac{3v}{2L_1} = \frac{3v}{4L} \)\nThus, \( L_1 = 2L \).
Q3. The ratio of the specific heats \( \frac{C_p}{C_V} \) in terms of degrees of freedom \( n \) is given by
Answer: A
The ratio of specific heats \( \gamma \) is derived from the relationship between the specific heats at constant pressure and constant volume: \( C_p = C_V + R \). For a gas with \( n \) degrees of freedom, we can express this ratio using statistical mechanics as \( \gamma = \frac{C_p}{C_V} = 1 + \frac{1}{n}. \) Therefore, the correct answer is \( 1 + \frac{1}{n} \).
The ratio of specific heats \( \gamma \) is derived from the relationship between the specific heats at constant pressure and constant volume: \( C_p = C_V + R \). For a gas with \( n \) degrees of freedom, we can express this ratio using statistical mechanics as \( \gamma = \frac{C_p}{C_V} = 1 + \frac{1}{n}. \) Therefore, the correct answer is \( 1 + \frac{1}{n} \).
Q4. The given graph represents V-I characteristic for a semiconductor device. Which of the following statement is correct?
Answer: A
V-I characteristics of a solar cell is shown. Where, A represents open circuit voltage (i.e. I = 0, V = emf) and B shows short circuit voltage (i.e. I = I_s, V = 0).
V-I characteristics of a solar cell is shown. Where, A represents open circuit voltage (i.e. I = 0, V = emf) and B shows short circuit voltage (i.e. I = I_s, V = 0).
Q5. A convex lens is dipped in a liquid whose refractive index is equal to the refractive index of the lens. Then its focal length will
Answer: D
When the convex lens is dipped in a liquid with the same refractive index, the lens maker's formula indicates that the focal length approaches infinity as the refractive indices match, causing the lens to effectively become flat.
When the convex lens is dipped in a liquid with the same refractive index, the lens maker's formula indicates that the focal length approaches infinity as the refractive indices match, causing the lens to effectively become flat.
Chemistry
Q6. For an ideal solution, the correct option is [NEET (National) 2019]
Answer: B
Ideal solutions are those which obey Raoult’s law over all concentration ranges at a given temperature, e.g. benzene-toluene, n-hexane-n-heptane, etc. For an ideal solution, \Delta V_{mix} = 0, \Delta H_{mix} = 0, \Delta G_{mix} < 0, \Delta S_{mix} > 0. Hence, option (b) is correct.
Ideal solutions are those which obey Raoult’s law over all concentration ranges at a given temperature, e.g. benzene-toluene, n-hexane-n-heptane, etc. For an ideal solution, \Delta V_{mix} = 0, \Delta H_{mix} = 0, \Delta G_{mix} < 0, \Delta S_{mix} > 0. Hence, option (b) is correct.
Q7. Oxygen will directly react with each of the following elements except?
Answer: B
Chlorine does not react directly with oxygen, while phosphorus (P), sodium (Na), and sulfur (S) do. The reactions are: P + 5O2 → P4O10, S + O2 → SO2, 4Na + O2 → 2Na2O.
Chlorine does not react directly with oxygen, while phosphorus (P), sodium (Na), and sulfur (S) do. The reactions are: P + 5O2 → P4O10, S + O2 → SO2, 4Na + O2 → 2Na2O.
Q8. The angular shape of ozone molecule (O₃) consists of
Answer: D
In ozone (O₃), there are 2 sigma bonds and 1 pi bond, resulting in an angular shape. Therefore, the answer is 2 sigma and 1 pi-bonds.
In ozone (O₃), there are 2 sigma bonds and 1 pi bond, resulting in an angular shape. Therefore, the answer is 2 sigma and 1 pi-bonds.
Q9. If the E_cell° for a given reaction has a negative value, which of the following gives correct relationships for the values of ∆G° and K_eq?
Answer: A
The standard Gibbs energy change (∆G°) is related to the standard cell potential (E_cell°) by the equation ∆G° = -nFE_cell°. If E_cell° is negative, then ∆G° must be positive (∆G° > 0). The relationship between ∆G° and K_eq is given by the equation ∆G° = -RT ln(K_eq). If ∆G° is positive, then K_eq must be less than 1 (K_eq < 1). Therefore, the correct answer is ∆G° > 0; K_eq < 1.
The standard Gibbs energy change (∆G°) is related to the standard cell potential (E_cell°) by the equation ∆G° = -nFE_cell°. If E_cell° is negative, then ∆G° must be positive (∆G° > 0). The relationship between ∆G° and K_eq is given by the equation ∆G° = -RT ln(K_eq). If ∆G° is positive, then K_eq must be less than 1 (K_eq < 1). Therefore, the correct answer is ∆G° > 0; K_eq < 1.
Q10. What is the pH of pure water at 25°C?
Answer: A
The pH of pure water at 25°C is 7, which is considered neutral. This value indicates that the concentration of hydrogen ions is equal to that of hydroxide ions in pure water.
The pH of pure water at 25°C is 7, which is considered neutral. This value indicates that the concentration of hydrogen ions is equal to that of hydroxide ions in pure water.
Maths
Q11. In a potato race, 20 potatoes are placed in a line at intervals of 4m with the first potato 24m from the starting point. A contestant is required to bring back the potatoes one at a time. How far would he run in bringing back all the potatoes?
Answer: B
According to the given condition, the sequence becomes 24, 28, 32, ..., where the first term (a) is 24m and the common difference (d) is 4m. The number of terms (n) is 20. The total distance run by the contestant is given by the formula for the sum of an arithmetic series: \( S_n = \frac{n}{2} (2a + (n-1)d) \). Substituting the values, we get: \( S_{20} = \frac{20}{2} (2 \times 24 + (20-1) \times 4) = 10 (48 + 76) = 10 \times 124 = 1240 \). Since the contestant has to bring the potatoes back to the starting point, the total distance run is \( 2S_n = 2 \times 1240 = 2480 m \).
According to the given condition, the sequence becomes 24, 28, 32, ..., where the first term (a) is 24m and the common difference (d) is 4m. The number of terms (n) is 20. The total distance run by the contestant is given by the formula for the sum of an arithmetic series: \( S_n = \frac{n}{2} (2a + (n-1)d) \). Substituting the values, we get: \( S_{20} = \frac{20}{2} (2 \times 24 + (20-1) \times 4) = 10 (48 + 76) = 10 \times 124 = 1240 \). Since the contestant has to bring the potatoes back to the starting point, the total distance run is \( 2S_n = 2 \times 1240 = 2480 m \).
Q12. What is the semi-vertical angle of the cone of maximum volume when the slant height is given?
Answer: A
For maxima, we put \( \frac{dV}{dq} = 0 \), leading to \( \tan(q) = 2 \). Hence, the semi-vertical angle \( q \) of the cone of maximum volume is \( \tan^{-1}(1/2) \).
For maxima, we put \( \frac{dV}{dq} = 0 \), leading to \( \tan(q) = 2 \). Hence, the semi-vertical angle \( q \) of the cone of maximum volume is \( \tan^{-1}(1/2) \).
Q13. If the lines are parallel to unit vectors b1 = (1, 1, 1) and b2 = (2, 2, 2), what are the direction cosines of the line perpendicular to both?
Answer: D
To find the direction cosines of a line perpendicular to both b1 and b2, we compute the cross product of the two vectors. This gives us the direction cosines which turn out to be (1, 1, -1).
To find the direction cosines of a line perpendicular to both b1 and b2, we compute the cross product of the two vectors. This gives us the direction cosines which turn out to be (1, 1, -1).
Q14. If \( m \) is the AM of two distinct real numbers \( l \) and \( n \) with \( l < n \) and \( G_1, G_2, G_3 \) are three geometric means between \( l \) and \( n \), then \( G_1^4 + G_2^4 + G_3^4 \) equals?
Answer: B
Given that \( m = \frac{l+n}{2} \) and \( G_1, G_2, G_3 \) are in GP with common ratio \( r \), we find that each geometric mean can be expressed in terms of \( l \) and \( n \). The calculations lead to the result that \( G_1^4 + G_2^4 + G_3^4 = 4^2 lm n \).
Given that \( m = \frac{l+n}{2} \) and \( G_1, G_2, G_3 \) are in GP with common ratio \( r \), we find that each geometric mean can be expressed in terms of \( l \) and \( n \). The calculations lead to the result that \( G_1^4 + G_2^4 + G_3^4 = 4^2 lm n \).
Q15. If a man is not happy, then he is not rich.
Answer: A
The statement can be expressed as a contrapositive. Given p: A man is happy and q: A man is rich, the contrapositive of pq -> is that if p is false (not happy), then q is also false (not rich). Therefore, the correct answer is option A.
The statement can be expressed as a contrapositive. Given p: A man is happy and q: A man is rich, the contrapositive of pq -> is that if p is false (not happy), then q is also false (not rich). Therefore, the correct answer is option A.
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