JEE Physics Basic Mathematics Used in Physics MCQs Set 01

Answer: (b) decreases

Explanation:
1. cos 0° = 1 and cos 90° = 0, and cos θ falls steadily in between.
2. So cos θ decreases.

Teacher's Note:
1. Remember that sin θ increases and cos θ decreases in the first quadrant.
2. Check with 30° and 60°.

Answer: (b) 13

Explanation:
1. The maximum of \( a\sin\theta+b\cos\theta \) is \( \sqrt{a^2+b^2} \).
2. Here \( \sqrt{25+144}=13 \).

Teacher's Note:
1. Write the expression as R cos(θ + α) to see the maximum.
2. The sign of a does not affect the maximum.

Answer: (a) \( \sqrt{\frac56} \)

Explanation:
1. Take a right triangle with perpendicular 1 and base \( \sqrt5 \), so the hypotenuse is \( \sqrt6 \).
2. \( \cos\theta=\frac{\sqrt5}{\sqrt6}=\sqrt{\frac56} \), positive in the first quadrant.

Teacher's Note:
1. Draw the right triangle from the given tangent.
2. cos is positive in the first quadrant.

Answer: (c) parabola

Explanation:
1. From y = 2t we get t = y/2.
2. Then x = 4(y/2)² = y², which is a parabola.

Teacher's Note:
1. Eliminate the time parameter to get the path.
2. An equation like x = y² is a parabola.

Answer: (b) m₁ < m₂ < m₃

Explanation:
1. The curve gets steeper as x increases (it bends upwards).
2. So the slope increases from point 1 to 2 to 3: m₁ < m₂ < m₃.

Teacher's Note:
1. The slope of a graph is the steepness of the tangent.
2. An upward-bending curve has an increasing slope.

Answer: (b) y-coordinate

Explanation:
1. Differentiating with time: dy/dt = 3 dx/dt.
2. So y changes three times as fast as x.

Teacher's Note:
1. The slope 3 gives the ratio of the rates.
2. Use the chain rule with respect to time.

Answer: (b) first decreases then increases

Explanation:
1. On the rising part the slope is positive and decreases to zero at the top.
2. On the falling part the slope is negative and its magnitude increases again.
3. So the magnitude of the slope first decreases and then increases.

Teacher's Note:
1. Zero slope occurs at the top of the arch.
2. Magnitude ignores the sign of the slope.

Answer: (c) both (1, 0) and (2, 0)

Explanation:
1. Put y = 0: x² − 3x + 2 = 0, i.e. (x − 1)(x − 2) = 0.
2. So x = 1 and x = 2.

Teacher's Note:
1. Intersection with the x-axis means y = 0.
2. Factorise the quadratic.

Answer: (a) 5

Explanation:
1. At t = 1: A = (3, 1) and B = (6, 5).
2. Distance = √[(6 − 3)² + (5 − 1)²] = √25 = 5.

Teacher's Note:
1. Substitute t = 1 first.
2. Use the distance formula.

Answer: (a) \( y=\frac x2 \)

Explanation:
1. The point is (2k, k), so the slope is k/2k = 1/2.
2. The line through the origin is y = x/2.

Teacher's Note:
1. Abscissa means the x-coordinate and ordinate means the y-coordinate.
2. The slope is ordinate divided by abscissa.

Answer: (d) 0.8 cm/s

Explanation:
1. Perimeter P = 4a.
2. dP/dt = 4 da/dt = 4 × 0.2 = 0.8 cm/s.

Teacher's Note:
1. Differentiate the perimeter formula with time.
2. The constant 4 multiplies the rate.

Answer: (d) f² on the ordinate and 1/ℓ on the abscissa

Explanation:
1. Squaring gives \( f^2=\frac{g}{4\pi^2}\cdot\frac1\ell \).
2. This has the form y = m x with y = f² and x = 1/ℓ, a straight line through the origin.

Teacher's Note:
1. Convert the relation to y = mx form.
2. Square to remove the root.

Answer: (d) \( \frac43 \)

Explanation:
1. This is an infinite GP with a = 1 and r = 1/4.
2. Sum = \( \frac{a}{1-r}=\frac{1}{3/4}=\frac43 \).

Teacher's Note:
1. Use S = a/(1 − r) for
2. r
3. less than 1.
4. Check that the ratio is constant.

Answer: (c) \( z=\sqrt{2x+3} \)

Explanation:
1. After the first machine the value is 2x + 3.
2. The second machine takes the square root, so z = √(2x + 3).

Teacher's Note:
1. Follow the input through the machines in order.
2. Doubling then adding three is 2x + 3, not 2(x + 3).

Answer: (d) A vector has only magnitude whereas a scalar has both magnitude and direction

Explanation:
1. A scalar has magnitude only and a vector has both magnitude and direction.
2. So statement 4 has them reversed and is false.

Teacher's Note:
1. Scalars: mass, speed, energy, distance.
2. Vectors: momentum, force, torque, displacement.

Answer: (a) \( \hat n=\frac{\vec A}{|\vec A|} \)

Explanation:
1. A unit vector has magnitude 1 and the same direction as \( \vec A \).
2. Dividing \( \vec A \) by its magnitude gives @\hat n=\vec A/
3. \vec A
4. @.

Teacher's Note:
1. Divide a vector by its own magnitude to get its unit vector.
2. The other options do not have unit magnitude.

Answer: (b) may be a vector

Explanation:
1. Vectors have direction, but some quantities with a direction, such as electric current, are scalars.
2. So it may be a vector but need not be.

Teacher's Note:
1. A direction alone does not make a quantity a vector; it must also obey the vector addition rule.
2. Current is the standard counter-example.

Answer: (d) torque

Explanation:
1. Axial vectors act along an axis of rotation: angular velocity, torque and angular momentum.
2. Displacement, force and velocity are polar vectors.

Teacher's Note:
1. Axial vectors come from a cross product.
2. Torque is r × F.

Answer: (c) non-coplanar and concurrent forces

Explanation:
1. Meeting at one point means concurrent.
2. Not lying in one plane means non-coplanar.

Teacher's Note:
1. Concurrent means passing through a common point.
2. Coplanar means in one plane.

Answer: (d) the axis of rotation

Explanation:
1. By the right-hand rule, angular velocity points along the axis of rotation.
2. It is perpendicular to the plane of the circular motion.

Teacher's Note:
1. Curl the fingers of the right hand in the direction of rotation.
2. The thumb gives the direction.

Answer: (b) cannot be zero

Explanation:
1. \( \vec A+\vec B \) lies in the plane, while \( \vec C \) is out of it.
2. The sum of an in-plane vector and an out-of-plane vector cannot be zero.

Teacher's Note:
1. Zero resultant needs equal and opposite vectors.
2. The out-of-plane component of C cannot be cancelled.

Answer: (b) 45°

Explanation:
1. R is perpendicular to A, so R is the component of B perpendicular to A: \( R=B\sin\theta \).
2. \( B\sin\theta=\frac{B}{\sqrt2} \) gives \( \sin\theta=\frac1{\sqrt2} \), so θ = 45°.

Teacher's Note:
1. Use the right triangle in the figure.
2. R is the side opposite to θ and B is the hypotenuse.

Answer: (c) α < β if A > B

Explanation:
1. The resultant lies closer to the larger vector.
2. If A > B, it makes a smaller angle with A, so α < β.

Teacher's Note:
1. The larger vector pulls the resultant towards itself.
2. When A = B the angles are equal.

Answer: (c) \( \vec A \) is perpendicular to \( \vec B \)

Explanation:
1. \( C^2=A^2+B^2+2AB\cos\theta \).
2. Comparing with \( C^2=A^2+B^2 \) gives \( \cos\theta=0 \), so θ = 90°.

Teacher's Note:
1. This is the Pythagoras condition for vectors.
2. Perpendicular vectors satisfy it.

Answer: (a) 2

Explanation:
1. Two vectors of equal magnitude and opposite direction give a zero resultant.

Teacher's Note:
1. Equal and opposite vectors cancel.
2. Three or more are not required.

Answer: (b) 3

Explanation:
1. Two vectors of different magnitudes cannot cancel.
2. Three coplanar vectors of different magnitudes can form a closed triangle.

Teacher's Note:
1. A closed triangle means zero resultant.
2. The sides of a triangle can all differ.

Answer: (c) 4

Explanation:
1. Vectors in different planes cannot form a closed triangle.
2. Four vectors can form a closed (skew) polygon, so at least 4 are needed.

Teacher's Note:
1. Three vectors that sum to zero must be coplanar.
2. Four vectors can close a skew quadrilateral.

Answer: (d) infinite

Explanation:
1. A vector can be resolved into any number of components along different directions.
2. So the number of possible components is infinite.

Teacher's Note:
1. There is no limit on the number of components.
2. Rectangular components are limited, but components in general are not.

Answer: (a) 2

Explanation:
1. In a plane there are only two mutually perpendicular directions (x and y).
2. So at most 2 rectangular components.

Teacher's Note:
1. Rectangular means mutually perpendicular.
2. A plane has two independent perpendicular axes.