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JEE Main Paper 2B - B.Planning study materials
JEE Main Paper 2B - B.Planning study resources

JEE Main Paper 2B - B.Planning practice questions

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Question 1Mathematics

If A = {1, 2, 3} and B = {3, 4, 5}, then A ∩ B is

  1. {1, 2, 3, 4, 5}
  2. {3, 4}
  3. {3} Correct answer.
  4. {1, 2}

Explanation

The intersection of two sets contains exactly the elements that belong to both sets. The only element common to A and B is 3.

Question 2Mathematics

The value of i^2025, where i^2 = -1, is

  1. 1
  2. i Correct answer.
  3. -1
  4. -i

Explanation

Powers of i repeat in a cycle of length 4: i, -1, -i, 1. Since 2025 leaves remainder 1 on division by 4, i^2025 = i.

Question 3Mathematics

The value of sin(π/6) is

  1. 1/2 Correct answer.
  2. 0
  3. 1
  4. √3/2

Explanation

The angle π/6 radians is 30 degrees, and sin 30° = 1/2. This is a standard trigonometric value.

Question 4Mathematics

The number of distinct arrangements of the letters of the word PLAN is

  1. 48
  2. 24 Correct answer.
  3. 12
  4. 6

Explanation

The word PLAN has 4 distinct letters, so all 4! permutations are distinct. Thus the number of arrangements is 4! = 24.

Question 5Mathematics

The slope of the line joining (1, 2) and (3, 6) is

  1. 4
  2. 1/2
  3. -2
  4. 2 Correct answer.

Explanation

The slope is (y₂ - y₁)/(x₂ - x₁). Substituting gives (6 - 2)/(3 - 1) = 4/2 = 2.

Question 6Mathematics

The derivative of x^3 with respect to x is

  1. x^3/3
  2. 3x^2 Correct answer.
  3. 3x
  4. x^2

Explanation

Using the power rule, d/dx(x^n) = n x^(n-1). For n = 3, this gives 3x^2.

Question 7Mathematics

The number of subsets of a set containing 4 elements is

  1. 16 Correct answer.
  2. 4
  3. 32
  4. 8

Explanation

A set with n elements has 2^n subsets, including the empty set and the whole set. For n = 4, the number is 2^4 = 16.

Question 8Mathematics

The arithmetic mean of 2, 4, 6, 8 is

  1. 7
  2. 6
  3. 4
  4. 5 Correct answer.

Explanation

The arithmetic mean is the sum divided by the number of observations. The sum is 20 and there are 4 observations, so the mean is 5.

Question 9Mathematics

If α and β are the roots of x^2 - 5x + 6 = 0, then α^2 + β^2 equals

  1. 25
  2. 13 Correct answer.
  3. 12
  4. 11

Explanation

For a quadratic, sum of roots = 5 and product of roots = 6. Then α^2 + β^2 = (α+β)^2 - 2αβ = 25 - 12 = 13.

Question 10Mathematics

The coefficient of x^3 in the expansion of (1+x)^6 is

  1. 15
  2. 6
  3. 20 Correct answer.
  4. 30

Explanation

By the binomial theorem, the coefficient of x^r in (1+x)^n is C(n,r). Here r = 3 and n = 6, so the coefficient is C(6,3) = 20.

Question 11Mathematics

The 10th term of the arithmetic progression 3, 7, 11, ... is

  1. 39 Correct answer.
  2. 36
  3. 35
  4. 40

Explanation

Here the first term a = 3 and the common difference d = 4. The 10th term is a + 9d = 3 + 36 = 39.

Question 12Mathematics

The centre of the circle x^2 + y^2 - 4x + 6y - 12 = 0 is

  1. (-2, 3)
  2. (2, -3) Correct answer.
  3. (2, 3)
  4. (-2, -3)

Explanation

For x^2 + y^2 + 2gx + 2fy + c = 0, the centre is (-g, -f). Here 2g = -4 and 2f = 6, so g = -2 and f = 3, giving centre (2, -3).

Question 13Mathematics

lim_{x→0} (sin 3x)/x equals

  1. 1/3
  2. 0
  3. 1
  4. 3 Correct answer.

Explanation

Write (sin 3x)/x = 3 · (sin 3x)/(3x). Since (sin t)/t tends to 1 as t tends to 0, the limit is 3.

Question 14Mathematics

∫_0^1 (2x + 1) dx equals

  1. 2 Correct answer.
  2. 3/2
  3. 5/2
  4. 1

Explanation

An antiderivative of 2x + 1 is x^2 + x. Evaluating from 0 to 1 gives (1 + 1) - 0 = 2.

Question 15Mathematics

The determinant of the matrix [[2, 3], [1, 4]] is

  1. 7
  2. 5 Correct answer.
  3. 11
  4. 3

Explanation

For a 2×2 matrix [[a,b],[c,d]], the determinant is ad - bc. Thus det = 2·4 - 3·1 = 8 - 3 = 5.

Question 16Mathematics

If a = i + 2j - k and b = 2i + 3k, then a·b equals

  1. 1
  2. 5
  3. -1 Correct answer.
  4. -5

Explanation

The dot product is the sum of the products of corresponding components. Thus a·b = 1·2 + 2·0 + (-1)·3 = 2 - 3 = -1.

Question 17Mathematics

The number of diagonals in a regular hexagon is

  1. 6
  2. 15
  3. 12
  4. 9 Correct answer.

Explanation

A polygon with n sides has n(n-3)/2 diagonals. For n = 6, this gives 6·3/2 = 9.

Question 18Mathematics

Two fair dice are rolled. The probability that the sum of the numbers is 7 is

  1. 1/9
  2. 1/6 Correct answer.
  3. 1/12
  4. 5/36

Explanation

There are 36 equally likely outcomes for two dice. The pairs summing to 7 are (1,6), (2,5), (3,4), (4,3), (5,2), and (6,1), so the probability is 6/36 = 1/6.

Question 19Mathematics

The distance of the point (3, 4) from the origin is

  1. 25
  2. 1
  3. 7
  4. 5 Correct answer.

Explanation

The distance from (x, y) to the origin is √(x² + y²). Thus the distance is √(9 + 16) = √25 = 5.

Question 20Mathematics

The domain of the real function f(x) = 1/√(x - 2) is

  1. x < 2
  2. x ≠ 2
  3. x ≥ 2
  4. x > 2 Correct answer.

Explanation

The expression under the square root must be positive for the square root to be real and for the denominator to be nonzero. Therefore x - 2 > 0, so x > 2.

Question 21Mathematics

If a = i + j, b = j + k, and c = k + i, then the scalar triple product [a b c] equals

  1. 0
  2. -2
  3. 2 Correct answer.
  4. 1

Explanation

The scalar triple product [a b c] is the determinant formed by the components of a, b, and c. Computing it gives 1(1·1 - 1·0) - 1(0·1 - 1·1) + 0 = 2.

Question 22Mathematics

The equation of the tangent to the circle x^2 + y^2 = 25 at the point (3, 4) is

  1. 4x - 3y = 25
  2. 3x + 4y = 25 Correct answer.
  3. 4x + 3y = 25
  4. 3x - 4y = 25

Explanation

For the circle x^2 + y^2 = a^2, the tangent at (x₁, y₁) on the circle is xx₁ + yy₁ = a². Substituting (3,4) gives 3x + 4y = 25.

Question 23Mathematics

The area of the region bounded by y = x^2 and y = x is

  1. 1/6 Correct answer.
  2. 1/3
  3. 1/2
  4. 1/12

Explanation

The curves meet where x^2 = x, so x = 0 or x = 1. Between these points x ≥ x^2, and the area is ∫_0^1 (x - x^2) dx = 1/2 - 1/3 = 1/6.

Question 24Mathematics

The eccentricity of the ellipse 9x^2 + 16y^2 = 144 is

  1. √7/3
  2. √7/4 Correct answer.
  3. 3/4
  4. 4/5

Explanation

Dividing by 144 gives x^2/16 + y^2/9 = 1, so a = 4 and b = 3. Then c^2 = a^2 - b^2 = 7, and the eccentricity is e = c/a = √7/4.

Question 25Mathematics

A fair die is rolled twice. Given that the sum of the outcomes is 8, the probability that the first roll is 3 is

  1. 1/6
  2. 1/4
  3. 5/36
  4. 1/5 Correct answer.

Explanation

The outcomes with sum 8 are (2,6), (3,5), (4,4), (5,3), and (6,2), giving 5 equally likely outcomes under the condition. Only (3,5) has first roll 3, so the conditional probability is 1/5.

Question 26Mathematics

If A = {1, 2, 3} and B = {3, 4}, then the number of elements in the symmetric difference A Δ B is

  1. 3 Correct answer.
  2. 4
  3. 2
  4. 5

Explanation

The symmetric difference consists of elements that belong to exactly one of the two sets. Here A Δ B = {1, 2, 4}, so it has 3 elements.

Question 27Mathematics

The value of log_2 8 + log_3 9 is

  1. 7
  2. 4
  3. 5 Correct answer.
  4. 6

Explanation

Using the definition of logarithm, log_2 8 = 3 because 2^3 = 8, and log_3 9 = 2 because 3^2 = 9. Their sum is 3 + 2 = 5.

Question 28Mathematics

If f(x) = 2x + 5, then f^{-1}(x) equals

  1. (x+5)/2
  2. x/2+5
  3. (x-5)/2 Correct answer.
  4. 2x-5

Explanation

To find the inverse, set y = 2x + 5 and solve for x. This gives x = (y - 5)/2, so f^{-1}(x) = (x - 5)/2.

Question 29Mathematics

The sum of the first 20 natural numbers is

  1. 200
  2. 190
  3. 210 Correct answer.
  4. 205

Explanation

The sum of the first n natural numbers is n(n+1)/2. For n = 20, this is 20 × 21 / 2 = 210.

Question 30Mathematics

The value of sin^2 30° + cos^2 60° is

  1. 1/4
  2. 1
  3. 1/2 Correct answer.
  4. 3/4

Explanation

We have sin 30° = 1/2 and cos 60° = 1/2. Therefore sin^2 30° + cos^2 60° = 1/4 + 1/4 = 1/2.

Question 31Mathematics

If A is a matrix of order 2 × 3 and B is a matrix of order 3 × 4, then the order of AB is

  1. 2 × 4 Correct answer.
  2. 4 × 2
  3. 3 × 3
  4. 2 × 3

Explanation

For the product AB to be defined, the number of columns of A must equal the number of rows of B. The product then has order (rows of A) × (columns of B), which is 2 × 4.

Question 32Mathematics

The slope of the line 3x - 4y + 7 = 0 is

  1. 3/4 Correct answer.
  2. 4/3
  3. -3/4
  4. -4/3

Explanation

Rewrite the line as 4y = 3x + 7, so y = (3/4)x + 7/4. The coefficient of x is the slope, which is 3/4.

Question 33Mathematics

The number of ways of selecting 2 students from a group of 5 students is

  1. 20
  2. 10 Correct answer.
  3. 5
  4. 15

Explanation

Selecting students without regard to order uses combinations. The number is C(5, 2) = 5!/(2!3!) = 10.

Question 34Mathematics

If z = (1 + i)/(1 - i), then z^2 equals

  1. -i
  2. 1
  3. -1 Correct answer.
  4. i

Explanation

Multiply numerator and denominator by 1 + i: z = (1 + i)^2/(1^2 + 1^2) = (2i)/2 = i. Therefore z^2 = i^2 = -1.

Question 35Mathematics

The number of distinct arrangements of the letters of the word LEVEL is

  1. 30 Correct answer.
  2. 20
  3. 60
  4. 120

Explanation

The word LEVEL has 5 letters with L repeated twice and E repeated twice. The number of distinct arrangements is 5!/(2!2!) = 30.

Question 36Mathematics

If A and B are independent events with P(A) = 0.4 and P(B) = 0.5, then P(A ∪ B) equals

  1. 0.2
  2. 0.7 Correct answer.
  3. 0.5
  4. 0.9

Explanation

For independent events, P(A ∩ B) = P(A)P(B) = 0.4 × 0.5 = 0.2. Then P(A ∪ B) = P(A) + P(B) - P(A ∩ B) = 0.4 + 0.5 - 0.2 = 0.7.

Question 37Mathematics

The distance between the parallel planes 2x - y + 2z = 3 and 2x - y + 2z = 9 is

  1. 1
  2. 6
  3. 3
  4. 2 Correct answer.

Explanation

For parallel planes ax + by + cz = d1 and ax + by + cz = d2, the distance is |d1 - d2|/√(a^2 + b^2 + c^2). Here it is |3 - 9|/√(4 + 1 + 4) = 6/3 = 2.

Question 38Mathematics

The equation of the normal to the curve y = x^2 at the point (1, 1) is

  1. x - 2y + 1 = 0
  2. 2x + y - 3 = 0
  3. 2x - y - 1 = 0
  4. x + 2y - 3 = 0 Correct answer.

Explanation

The derivative is dy/dx = 2x, so at x = 1 the tangent slope is 2. The normal slope is -1/2, and through (1, 1): y - 1 = -1/2(x - 1), which simplifies to x + 2y - 3 = 0.

Question 39Mathematics

The value of ∫_1^2 (1/x) dx is

  1. ln(1/2)
  2. ln 2 Correct answer.
  3. 1
  4. 1/2

Explanation

An antiderivative of 1/x is ln x. Evaluating from 1 to 2 gives ln 2 - ln 1 = ln 2.

Question 40Mathematics

The general solution of the differential equation dy/dx = y/x, for x ≠ 0, is

  1. y = Cx^2
  2. y = C + x
  3. y = C/x
  4. y = Cx Correct answer.

Explanation

Separate variables: dy/y = dx/x. Integrating gives ln|y| = ln|x| + c, so y = Cx.

Question 41Mathematics

The eccentricity of the hyperbola x^2/9 - y^2/16 = 1 is

  1. 5/4
  2. 3/5
  3. 4/3
  4. 5/3 Correct answer.

Explanation

For x^2/a^2 - y^2/b^2 = 1, the eccentricity is e = √(1 + b^2/a^2). Here a^2 = 9 and b^2 = 16, so e = √(1 + 16/9) = √(25/9) = 5/3.

Question 42Mathematics

The variance of the data 2, 4, 6, 8, 10 is

  1. 4
  2. 6
  3. 8 Correct answer.
  4. 10

Explanation

The mean is (2+4+6+8+10)/5 = 6. The squared deviations are 16, 4, 0, 4, 16, whose sum is 40; dividing by 5 gives variance 8.

Question 43Mathematics

If sin θ = 3/5 and θ lies in the second quadrant, then tan θ equals

  1. -4/3
  2. 4/3
  3. -3/4 Correct answer.
  4. 3/4

Explanation

In the second quadrant, sine is positive and cosine is negative. Since sin θ = 3/5, we get cos θ = -4/5, so tan θ = sin θ / cos θ = -3/4.

Question 44Mathematics

The term independent of x in the expansion of (x + 1/x)^8 is

  1. 28
  2. 56
  3. 112
  4. 70 Correct answer.

Explanation

The general term is C(8, r) x^(8-r)(1/x)^r = C(8, r) x^(8-2r). For the term independent of x, 8 - 2r = 0, so r = 4, giving C(8, 4) = 70.

Question 45Mathematics

The projection of the vector a = 2i + 3j on the vector b = i + j is

  1. 5√2
  2. 5/√2 Correct answer.
  3. √5
  4. 5

Explanation

The scalar projection of a on b is (a·b)/|b|. Here a·b = 2 + 3 = 5 and |b| = √2, so the projection is 5/√2.

Question 46Mathematics

If A = [[1, 2], [3, 4]], then A^2 - 5A + 2I equals

  1. [[2, 0], [0, 2]]
  2. [[1, 0], [0, 1]]
  3. [[0, 0], [0, 0]]
  4. [[4, 0], [0, 4]] Correct answer.

Explanation

Compute A^2 = [[7, 10], [15, 22]], 5A = [[5, 10], [15, 20]], and 2I = [[2, 0], [0, 2]]. Thus A^2 - 5A + 2I = [[4, 0], [0, 4]].

Question 47Mathematics

lim_{x→0} (tan x - sin x)/x^3 equals

  1. 0
  2. 1/2 Correct answer.
  3. 1
  4. 1/6

Explanation

Use tan x - sin x = sin x(1 - cos x)/cos x. As x → 0, sin x ~ x and 1 - cos x ~ x^2/2, so the numerator behaves like x^3/2 and the limit is 1/2.

Question 48Mathematics

Box I contains 2 white and 3 black balls; Box II contains 3 white and 2 black balls. A box is chosen at random and a white ball is drawn. The probability that it came from Box I is

  1. 1/5
  2. 1/2
  3. 2/5 Correct answer.
  4. 3/5

Explanation

By Bayes' theorem, P(I | W) = (1/2 × 2/5)/[(1/2 × 2/5) + (1/2 × 3/5)] = (1/5)/(1/2) = 2/5.

Question 49Mathematics

The distance of the point (1, 2, 3) from the plane 2x - 2y + z = 5 is

  1. 4/3 Correct answer.
  2. 5/3
  3. 2
  4. 1

Explanation

For a point (x1, y1, z1) and plane ax + by + cz = d, the distance is |ax1 + by1 + cz1 - d|/√(a^2 + b^2 + c^2). Here it is |2 - 4 + 3 - 5|/√(4 + 4 + 1) = 4/3.

Question 50Mathematics

The number of 4-digit numbers that can be formed using the digits 1, 2, 3, 4, 5 without repetition such that the number is even is

  1. 24
  2. 48 Correct answer.
  3. 36
  4. 60

Explanation

An even number must end in 2 or 4, giving 2 choices for the last digit. The remaining 3 positions are filled from the other 4 digits in 4P3 = 24 ways, so the total is 2 × 24 = 48.