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Question 1Mathematics

The value of the determinant |2 3; 4 5| is:

  1. 22
  2. -2 Correct answer.
  3. 10
  4. 2

Explanation

For a 2×2 determinant |a b; c d| the value is ad − bc. Here that is (2)(5) − (3)(4) = 10 − 12 = −2.

Question 2Mathematics

The modulus of the complex number z = 3 − 4i is:

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

Explanation

The modulus of a + bi is √(a² + b²). For 3 − 4i this gives √(9 + 16) = √25 = 5.

Question 3Mathematics

The slope of the line passing through the points (1, 2) and (4, 11) is:

  1. 9
  2. 13/5
  3. 1/3
  4. 3 Correct answer.

Explanation

Slope equals the change in y divided by the change in x: (11 − 2)/(4 − 1) = 9/3 = 3.

Question 4Mathematics

The derivative of sin(3x) with respect to x is:

  1. 3sin(3x)
  2. cos(3x)
  3. 3cos(3x) Correct answer.
  4. −3cos(3x)

Explanation

By the chain rule, d/dx sin(u) = cos(u)·du/dx with u = 3x, so du/dx = 3. Hence the derivative is 3cos(3x).

Question 5Mathematics

The value of ∫₀¹ x² dx is:

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

Explanation

An antiderivative of x² is x³/3. Evaluating from 0 to 1 gives 1/3 − 0 = 1/3.

Question 6Mathematics

The principal value of sin⁻¹(1/2) is:

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

Explanation

The principal range of sin⁻¹ is [−π/2, π/2], and sin(π/6) = 1/2. Therefore the principal value is π/6.

Question 7Mathematics

The order and degree of the differential equation (d²y/dx²)³ + (dy/dx)² + y = 0 are respectively:

  1. 3 and 2
  2. 2 and 3 Correct answer.
  3. 2 and 2
  4. 3 and 3

Explanation

Order is the highest derivative present, which is the second derivative, so order = 2. Degree is the power of that highest derivative after the equation is polynomial in derivatives, which is 3.

Question 8Mathematics

The value of the limit lim(x→0) (sin 5x)/x is:

  1. 0
  2. 1
  3. 5 Correct answer.
  4. does not exist

Explanation

Using the standard limit lim(t→0)(sin t)/t = 1 with t = 5x, we get (sin 5x)/x = 5·(sin 5x)/(5x) → 5·1 = 5.

Question 9Mathematics

The general solution of the differential equation dy/dx = y is:

  1. y = x²/2 + C
  2. y = C + x
  3. y = Cx
  4. y = Ceˣ Correct answer.

Explanation

Separating variables gives dy/y = dx, and integrating both sides yields ln|y| = x + k. Exponentiating gives y = Ceˣ.

Question 10Mathematics

If A is a 3×3 matrix with |A| = 4, then |2A| equals:

  1. 32 Correct answer.
  2. 24
  3. 8
  4. 16

Explanation

For an n×n matrix, |kA| = kⁿ|A|. Here n = 3 and k = 2, so |2A| = 2³ × 4 = 8 × 4 = 32.

Question 11Mathematics

The coefficient of x³ in the expansion of (1 + x)⁶ is:

  1. 20 Correct answer.
  2. 120
  3. 6
  4. 15

Explanation

The general term is C(6,r)xʳ, so the coefficient of x³ is C(6,3) = 6!/(3!3!) = 20.

Question 12Mathematics

The sum of the infinite geometric series 1 + 1/3 + 1/9 + 1/27 + … is:

  1. 2/3
  2. 9/4
  3. 3/2 Correct answer.
  4. 3

Explanation

Here the first term a = 1 and the common ratio r = 1/3, with |r| < 1. The sum is a/(1 − r) = 1/(1 − 1/3) = 1/(2/3) = 3/2.

Question 13Mathematics

The eccentricity of the ellipse x²/25 + y²/9 = 1 is:

  1. 16/25
  2. 4/5 Correct answer.
  3. 5/4
  4. 3/5

Explanation

Here a² = 25 and b² = 9, so a = 5, b = 3 and c = √(a² − b²) = √16 = 4. The eccentricity is e = c/a = 4/5.

Question 14Mathematics

The maximum value of the function f(x) = x³ − 3x on the interval [−2, 2] is:

  1. 18
  2. −2
  3. 2 Correct answer.
  4. 0

Explanation

f′(x) = 3x² − 3 = 0 gives x = ±1. Evaluating: f(−1) = 2, f(1) = −2, f(−2) = −2, f(2) = 2. The maximum value over the interval is 2.

Question 15Mathematics

The value of ∫ x·eˣ dx is:

  1. x·eˣ + eˣ + C
  2. x²eˣ/2 + C
  3. x·eˣ − eˣ + C Correct answer.
  4. eˣ + C

Explanation

Integrating by parts with u = x and dv = eˣ dx gives x·eˣ − ∫eˣ dx = x·eˣ − eˣ + C.

Question 16Mathematics

If the vectors a = 2i + 3j − k and b = i − j + 2k, then a·b equals:

  1. 7
  2. 3
  3. 1
  4. −3 Correct answer.

Explanation

The dot product is the sum of products of corresponding components: (2)(1) + (3)(−1) + (−1)(2) = 2 − 3 − 2 = −3.

Question 17Mathematics

The mean of the binomial distribution with n = 10 and p = 0.3 is:

  1. 2.1
  2. 3 Correct answer.
  3. 7
  4. 0.3

Explanation

For a binomial distribution the mean is np. Here np = 10 × 0.3 = 3.

Question 18Mathematics

The value of the determinant |1 2 3; 0 1 4; 0 0 1| is:

  1. 24
  2. 0
  3. 6
  4. 1 Correct answer.

Explanation

For an upper triangular matrix the determinant equals the product of the diagonal entries. The product 1 × 1 × 1 = 1.

Question 19Mathematics

The equation of the tangent to the curve y = x² at the point (2, 4) is:

  1. y = 2x
  2. y = 4x − 4 Correct answer.
  3. y = x + 2
  4. y = 4x + 4

Explanation

The slope dy/dx = 2x equals 4 at x = 2. Using point-slope form: y − 4 = 4(x − 2), which simplifies to y = 4x − 4.

Question 20Mathematics

The value of ∫₀^{π/2} sin x dx is:

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

Explanation

An antiderivative of sin x is −cos x. Evaluating −cos x from 0 to π/2 gives (−cos(π/2)) − (−cos 0) = 0 + 1 = 1.

Question 21Mathematics

A fair die is rolled twice. The probability that the sum of the two outcomes is 7 is:

  1. 7/36
  2. 1/36
  3. 1/6 Correct answer.
  4. 1/12

Explanation

There are 36 equally likely ordered outcomes. The pairs summing to 7 are (1,6),(2,5),(3,4),(4,3),(5,2),(6,1), six in all, so the probability is 6/36 = 1/6.

Question 22Mathematics

If A = [1 2; 3 4], then A² equals:

  1. [7 10; 15 16]
  2. [5 6; 7 8]
  3. [1 4; 9 16]
  4. [7 10; 15 22] Correct answer.

Explanation

Matrix multiplication sums products of rows of A with columns of A. Row 1 of A² is (1×1+2×3, 1×2+2×4) = (7, 10); row 2 is (3×1+4×3, 3×2+4×4) = (15, 22).

Question 23Mathematics

The value of lim(x→0) (1 − cos 2x)/x² is:

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

Explanation

Using 1 − cos 2x = 2sin²x, the quotient becomes 2(sin x/x)². Since sin x/x → 1, the limit is 2 × 1² = 2.

Question 24Mathematics

The area enclosed between the curve y = x² and the line y = x is:

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

Explanation

The curves meet where x² = x, i.e. x = 0 and x = 1. The area is ∫₀¹ (x − x²) dx = [x²/2 − x³/3]₀¹ = 1/2 − 1/3 = 1/6.

Question 25Mathematics

The number of distinct real roots of the equation x³ − 3x + 1 = 0 is:

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

Explanation

Let f(x) = x³ − 3x + 1. Its derivative 3x² − 3 vanishes at x = ±1, where f(−1) = 3 and f(1) = −1. Since the local maximum is positive and the local minimum is negative, the cubic crosses the x-axis three times, giving three distinct real roots.

Question 26Mathematics

For the matrix A = [[2, 1], [4, 3]], the value of the determinant |A| is:

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

Explanation

For a 2×2 matrix [[a, b], [c, d]] the determinant is ad − bc. Here that gives (2)(3) − (1)(4) = 6 − 4 = 2.

Question 27Mathematics

If z = 3 + 4i, then the modulus |z| and the argument (in the first quadrant) are respectively:

  1. 5 and tan⁻¹(3/4)
  2. 7 and tan⁻¹(3/4)
  3. 5 and tan⁻¹(4/3) Correct answer.
  4. 25 and tan⁻¹(4/3)

Explanation

The modulus is √(3² + 4²) = √25 = 5. Since both real and imaginary parts are positive, the argument is tan⁻¹(4/3), measured from the positive real axis.

Question 28Mathematics

The 5th term of the arithmetic progression 3, 7, 11, 15, … is:

  1. 21
  2. 23
  3. 17
  4. 19 Correct answer.

Explanation

The first term is a = 3 and the common difference is d = 4. The nth term is a + (n − 1)d, so the 5th term is 3 + 4·4 = 19.

Question 29Mathematics

The slope of the normal to the parabola y² = 8x at the point (2, 4) is:

  1. 2
  2. −2
  3. −1 Correct answer.
  4. 1

Explanation

Differentiating gives 2y·dy/dx = 8, so dy/dx = 4/y, which equals 1 at (2, 4). The normal is perpendicular to the tangent, so its slope is the negative reciprocal, −1.

Question 30Mathematics

The value of ∫ (2x + 3)⁴ dx is:

  1. (2x + 3)⁵ + C
  2. 4(2x + 3)³ + C
  3. (2x + 3)⁵/10 + C Correct answer.
  4. (2x + 3)⁵/5 + C

Explanation

Substituting u = 2x + 3 gives du = 2 dx, so the integral becomes ½∫u⁴ du = u⁵/10 + C. Replacing u restores (2x + 3)⁵/10 + C.

Question 31Mathematics

If the vectors p = i + 2j + 2k and q = 2i + j − 2k are given, then the angle between them is:

  1. 60°
  2. 45°
  3. 90° Correct answer.
  4. 0°

Explanation

The dot product is (1)(2) + (2)(1) + (2)(−2) = 0. Since neither vector is zero, a zero dot product means the vectors are perpendicular, so the angle is 90°.

Question 32Mathematics

The probability that a randomly chosen two-digit number is a multiple of 7 is:

  1. 13/90 Correct answer.
  2. 1/7
  3. 14/90
  4. 12/90

Explanation

Two-digit numbers run from 10 to 99, giving 90 in total. The multiples of 7 in this range are 14, 21, …, 98, which number 13. The probability is therefore 13/90.

Question 33Mathematics

The standard deviation of the data 2, 4, 6, 8, 10 is:

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

Explanation

The mean is 6, and the squared deviations are 16, 4, 0, 4, 16, whose sum is 40. The variance is 40/5 = 8, so the standard deviation is √8 = 2√2.

Question 34Mathematics

The value of tan⁻¹(1) + tan⁻¹(2) + tan⁻¹(3) is:

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

Explanation

Using tan⁻¹a + tan⁻¹b = tan⁻¹((a + b)/(1 − ab)) for ab < 1 gives tan⁻¹(1) + tan⁻¹(2) = tan⁻¹(−3), taken as 3π/4 in the principal range. Adding tan⁻¹(3) then gives 3π/4 + tan⁻¹(3) = π.

Question 35Mathematics

If the roots of x² − px + q = 0 are consecutive integers, then p² − 4q equals:

  1. 0
  2. p
  3. 4q
  4. 1 Correct answer.

Explanation

Consecutive integer roots differ by 1, so the discriminant (difference of roots)² equals 1. Since the discriminant is p² − 4q, its value is 1.

Question 36Mathematics

The equation of the circle with centre (2, −3) that passes through the origin is:

  1. x² + y² − 4x + 6y = 13
  2. x² + y² − 2x + 3y = 0
  3. x² + y² + 4x − 6y = 0
  4. x² + y² − 4x + 6y = 0 Correct answer.

Explanation

The radius is the distance from (2, −3) to (0, 0), namely √13. Expanding (x − 2)² + (y + 3)² = 13 and simplifying gives x² + y² − 4x + 6y = 0, consistent with passing through the origin.

Question 37Mathematics

The derivative of sin(log x) with respect to x is:

  1. cos(log x)/x Correct answer.
  2. −cos(log x)/x
  3. cos(log x)
  4. sin(log x)/x

Explanation

By the chain rule, differentiate the outer sine to get cos(log x) and multiply by the derivative of log x, which is 1/x. The result is cos(log x)/x.

Question 38Mathematics

The value of ∫₁^e (1/x) dx is:

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

Explanation

The antiderivative of 1/x is log x, so the definite integral equals log e − log 1 = 1 − 0 = 1.

Question 39Mathematics

The number of ways of arranging the letters of the word 'LEVEL' so that the two L's are always together is:

  1. 6
  2. 24
  3. 12 Correct answer.
  4. 60

Explanation

Treat the two L's as one block, giving 4 units: the block, E, V, E. These can be arranged in 4!/2! = 12 ways because the two E's are identical, and the L's inside the block have only one order.

Question 40Mathematics

The value of the determinant |2 3; 1 4| is:

  1. 8
  2. 5 Correct answer.
  3. −5
  4. 11

Explanation

For [[a, b], [c, d]] the determinant is ad − bc = (2)(4) − (3)(1) = 8 − 3 = 5.

Question 41Mathematics

If f(x) = x² − 4x + 7, the point of minimum of the curve y = f(x) is:

  1. (4, 7)
  2. (−2, 19)
  3. (2, 7)
  4. (2, 3) Correct answer.

Explanation

Setting f′(x) = 2x − 4 = 0 gives x = 2, and f″(x) = 2 > 0 confirms a minimum. Substituting gives f(2) = 4 − 8 + 7 = 3, so the minimum point is (2, 3).

Question 42Mathematics

The value of the limit lim(x→∞) (3x² + 2x + 1)/(x² − 5) is:

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

Explanation

Divide numerator and denominator by x² to get (3 + 2/x + 1/x²)/(1 − 5/x²). As x → ∞, the terms with x in the denominator vanish, leaving 3/1 = 3.

Question 43Mathematics

The value of the integral ∫₀¹ x·e^(x²) dx is:

  1. (e − 1)/2 Correct answer.
  2. 1/2
  3. e − 1
  4. (e + 1)/2

Explanation

Substitute u = x², so du = 2x dx and x dx = du/2. The integral becomes ½∫₀¹ e^u du = ½(e − 1).

Question 44Mathematics

If the position vectors of A and B are 2i − j + k and i + j + 2k, then the vector AB is:

  1. −i − 2j + k
  2. −i + 2j + k Correct answer.
  3. i − 2j − k
  4. 3i + 0j + 3k

Explanation

The vector AB equals the position vector of B minus that of A: (1 − 2)i + (1 − (−1))j + (2 − 1)k = −i + 2j + k.

Question 45Mathematics

If the mean of the binomial distribution is 4 and the variance is 3, then the number of trials n is:

  1. 16 Correct answer.
  2. 12
  3. 8
  4. 4

Explanation

For a binomial distribution, mean = np = 4 and variance = np(1 − p) = 3. Dividing gives 1 − p = 3/4, so p = 1/4, and then n = 4/(1/4) = 16.

Question 46Mathematics

The value of sin⁻¹(sin 5π/6) is:

  1. π/3
  2. π/6 Correct answer.
  3. 5π/6
  4. −π/6

Explanation

The principal value of sin⁻¹ lies in [−π/2, π/2]. Since sin(5π/6) = sin(π/6) = ½, the principal value is π/6.

Question 47Mathematics

The general solution of the differential equation d²y/dx² − 5(dy/dx) + 6y = 0 is:

  1. y = C₁e^(2x) + C₂e^(−3x)
  2. y = (C₁ + C₂x)e^(2x)
  3. y = C₁e^(2x) + C₂e^(3x) Correct answer.
  4. y = C₁e^(−2x) + C₂e^(−3x)

Explanation

The auxiliary equation is m² − 5m + 6 = 0, with roots m = 2 and m = 3. Distinct real roots give the solution y = C₁e^(2x) + C₂e^(3x).

Question 48Mathematics

The equation of the line passing through (1, 2) and perpendicular to the line 2x + 3y = 5 is:

  1. 3x − 2y + 1 = 0 Correct answer.
  2. 2x − 3y + 4 = 0
  3. 3x + 2y − 7 = 0
  4. 2x + 3y − 8 = 0

Explanation

The given line has slope −2/3, so a perpendicular line has slope 3/2. Using point-slope form with (1, 2): y − 2 = (3/2)(x − 1), which simplifies to 3x − 2y + 1 = 0.

Question 49Mathematics

The coefficient of x⁵ in the expansion of (x + 2)⁷ is:

  1. 84 Correct answer.
  2. 21
  3. 168
  4. 42

Explanation

The general term is C(7, r)x^(7−r)2^r. For the power x⁵ take r = 2, giving C(7, 2)·2² = 21·4 = 84.

Question 50Mathematics

If A = [[1, 2], [3, 4]], then the inverse A⁻¹ is:

  1. [[4, −2], [−3, 1]]
  2. [[4, 2], [3, 1]]
  3. [[−2, 1], [1, −1/2]]
  4. [[−2, 1], [3/2, −1/2]] Correct answer.

Explanation

The determinant is 1·4 − 2·3 = −2, which is nonzero, so A is invertible. The inverse is (1/−2)·[[4, −2], [−3, 1]] = [[−2, 1], [3/2, −1/2]].