Motion in plane practice · Mixed difficulty

Option A ($\cos^{-1}(\frac{1}{3})$) is correct because it is the angle between the diagonals calculated using the dot product formula.

Option D (zero) is correct because if the magnitude of the resultant vector is equal to the sum of the magnitudes of the individual vectors, the vectors must be parallel, implying an angle of 0 degrees between them.

Option D ($\sin ^{-1}(\frac{\sqrt{5}}{3})$) is correct because the angle between the vectors is found using the dot product formula, and then converting cosine to sine.

Option C (Both I and II) is correct because a unit vector is dimensionless and when multiplied by a scalar, it results in a vector.

Option B (If both assertion and reason are true but reason is not the correct explanation of assertion) is correct because the resultant magnitude can be less than either vector if the angle between them is greater than 90 degrees, and the law of parallelogram is used to calculate the resultant, but the latter doesn't explain the former.

Option A (0) is correct because the dot product of a unit vector and its derivative with respect to time is always zero, as the magnitude of the unit vector is constant.

Option B (3) is correct because by using the given condition $OG_1 \cdot BG_2 = 0$ and the properties of centroids, we can derive the relationship $\frac{{a^2 + c^2}}{{b^2}} = 3$.

Option D (Potential energy) is correct because potential energy is a scalar quantity, while weight, nuclear spin, and momentum are vector quantities.

Option A (If both Assertion & Reason are True & the Reason is a correct explanation of the Assertion) is correct because the vertical velocity decreases due to the constant downward acceleration due to gravity.

Option D (It falls 4.9 m) is correct because it is calculated using the equation of motion for vertical displacement under gravity.

Option D (Zero) is correct because the vector projection onto the y-axis depends only on the y-component of the vector, which is zero in this case.

Option B (If both assertion and reason are true but reason is not the correct explanation of assertion) is correct because while both statements are true, the reason doesn't directly explain why we can treat the sum of two vectors as the difference of two vectors.

Option B ($7.5 \mathrm{~m}$) is correct because it is the height at which the ball hits the wall, calculated using projectile motion equations.

Option A (Statement-1 and Statement-2 are true, Statement-2 is the correct explanation for Statement-1) is correct because the constant acceleration due to gravity results in the same change in velocity over equal time intervals.

Option A (If both Assertion & Reason are True & the Reason is a correct explanation of the Assertion) is correct because the given position vector leads to a parabolic trajectory, and the reason correctly derives the parabolic equation.

Option C (${A}$ and ${B}$ acts in the opposite direction) is correct because the dot product is negative when the angle between the vectors is 180 degrees.

Option B ($3 / 4$) is correct because it is the value of $\lambda$ that makes the dot product of the two vectors zero, indicating perpendicularity.

Option B (at each point of flight horizontal component of momentum remains constant) is correct because in projectile motion, the horizontal component of velocity remains constant, and therefore the horizontal component of momentum also remains constant.

Option D (Zero) is correct because at the highest point of a projectile launched at 90 degrees, the horizontal velocity is zero, resulting in zero kinetic energy.

Option D (If Assertion is false but Reason is true) is correct because the angle between $({A} \times {B})$ and $({B} \times {A})$ is $180^{\circ}$, making the cross product zero, and the reason correctly states that they are anti-parallel.

Option A ($2(\vec{b} \times \vec{a})$) is correct because expanding the cross product results in $2(\vec{b} \times \vec{a})$.

Option C (At the top most point) is correct because at the highest point, the vertical velocity is zero, and the acceleration due to gravity is vertically downwards, making them perpendicular.

Option D (Adding a scalar to a vector of the same dimensions) is correct because scalars and vectors are fundamentally different mathematical objects and cannot be directly added.

Option A is correct because the time it takes for an object to fall vertically depends only on the height and gravity, and both particles are dropped from the same height with the same gravitational acceleration.

Option C ($\sqrt{\frac{8 H}{g}}$) is correct because the time of flight is calculated using the maximum height and the acceleration due to gravity.

Option B ($\hat{i}-\hat{j}$) is correct because it represents the direction of the net acceleration (and therefore the force) at t=3s, considering both x and y components.

Option C ($\alpha > \frac{\pi}{2}$) is correct because by finding the vector c and its components, we determine the angle it makes with the x-axis, which is less than $\frac{\pi}{2}$.

Option C is correct. By resolving the vectors and using trigonometric relationships, the magnitudes of the two vectors are found to be 6 units and 10 units.

Option D ($0^{\circ}$) is correct because the dot product magnitude equals the product of the magnitudes only when the angle between the vectors is 0 degrees.

Option C ($\frac{u^2}{2 g}$) is correct because the sum of maximum heights for complementary angles of projection equals $\frac{u^2}{2g}$.