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Nust entry test 2020, Exams of Mathematics

Mathematics

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NUST Past Paper – Engineering
Total Time: 3 Hrs Total Question: 200
1. If sin-1x + sin-1 y + sin-1z =3π/2 then the value of x9 + y9 + z9 – 1/ x9 y9 z9 is equal to
a. 0
b. 1
c. 2
d. 3
2. Let p, q, r be the sides opposite to the angle P,Q.R respectively in a triangle PQR. If r2 sin
P sin Q = pq then the triangle is
a. Equilateral
b. Acute angled but not equilateral
c. Obtuse angled if sin
d. Right angled
3. Let p, q, and r be sides opposite to the angles P, Q, R respectively in a triangle PQR. Then
2 prsin (P-Q+R/2) equals
a. p2 + q2 + r2
b. p2 + r2 - q2
c. q2 + r2 - p2
d. p2 + q2 - r2
4. Let P (2,-3), Q (-2, 1) be the vertices of the triangle PQR. If the centroid of ΔPQR lies on the line
2x +3y = 1, then the locus of R is
a. 2x + 3y = 9
b. 2x - 3y = 9
c. 3x + 2y = 5
d. 3x - 2y = 5
5. If n(A) = m, then nP(A) =
a. 2 n
b. 2n
c. 2m
d. 2m
6. If f is a real-valued differentiable function such that f(x) f’(x) < 0 for all real x, then
a. F(x) must be an increasing function
b. F(x) must be an decreasing function
c. |F(x)| must be an increasing function
d. |F(x)| must be an decreasing function
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NUST Past Paper – Engineering

Total Time: 3 Hrs Total Question: 200

  1. If sin-1x + sin-1^ y + sin-1z =3π/2 then the value of x^9 + y^9 + z^9 – 1/ x^9 y^9 z^9 is equal to a. 0 b. 1 c. 2 d. 3
  2. Let p, q, r be the sides opposite to the angle P,Q.R respectively in a triangle PQR. If r^2 sin P sin Q = pq then the triangle is a. Equilateral b. Acute angled but not equilateral c. Obtuse angled if sin d. Right angled
  3. Let p, q, and r be sides opposite to the angles P, Q, R respectively in a triangle PQR. Then 2 prsin (P-Q+R/2) equals a. p^2 + q^2 + r^2 b. p^2 + r^2 - q^2 c. q^2 + r^2 - p^2 d. p^2 + q^2 - r^2
  4. Let P (2,-3), Q (-2, 1) be the vertices of the triangle PQR. If the centroid of ΔPQR lies on the line 2x +3y = 1, then the locus of R is a. 2x + 3y = 9 b. 2x - 3y = 9 c. 3x + 2y = 5 d. 3x - 2y = 5
  5. If n(A) = m, then nP(A) = a. 2 n b. 2n c. 2 m d. 2m
  6. If f is a real-valued differentiable function such that f(x) f’(x) < 0 for all real x, then a. F(x) must be an increasing function b. F(x) must be an decreasing function c. |F(x)| must be an increasing function d. |F(x)| must be an decreasing function
  1. Role’s theorem is applicable in the interval [-2,2] for the function a. F(x) =x^3 b. F(x) =4x^4 c. F(x) =2x^3 + 3 d. F(x) =π|x|
  2. The solution of 25 d^2 y/dx^2 -10dy/dx + y = 0 , y(0) =1y(1) =2e1/5^ is a. y= e5x^ + e-5x b. y=(1 +x) e5x c. y=(1 +x) ex/ d. y=(1 +x) e- x/
  3. Let P be the midpoint of a chord joining the vertex of the parabola y^2 = 8x to another point on it. then the locus of P is a. = 2x b. y^2 = 4x c. x^2 /4 + y^2 = 1 d. x^2 + y^2 /4 = 1
  4. the line x =2y intersects the ellipse x^2 /4 + y^2 =1 at the point P and Q. the equation of the circle with PQ as diameter is a. x^2 + y^2 = 1/ b. x^2 + y^2 = 1 c. x^2 + y^2 = 2 d. x^2 + y^2 = 5/
  5. the eccentric angle in the first quadrant of a point on the ellipse x^2 /10 + y^2 /8= 1 at a distance 3 units from the center of the ellipse is a. π/ b. π/ c. π/ d. π/
  6. The transverse axis of a hyperbola is along the x axis and its length is 2a. The vertex of the hyperbola bisects the line segment joining the center and the focus. The equation of the hyperbola is a. 6x^2 - y^2 = 3a^2 b. x^2 - 3 y^2 = 3a^2 c. x^2 - 6 y^2 = 3a^2 d. 3x^2 - y^2 = 3a^2
  7. A point moves in such a way that the difference of its distance from two point (8, 0) and (-8, 0) always remains 4. Then the locus of the point is a. A circle b. A parabola c. An ellipse d. A hyperbola
  1. The number of the words that can be written using all the letter of the word “irrational” is a. 10! / (2!)^3 b. 10! / (2!)^2 c. 10! /2! d. 10!
  2. Four speakers will address a meeting where speaker Q will always speak after speaker. Then the number of ways in which the order of speakers can be prepared is a. 256 b. 128 c. 24 d. 12
  3. The number of diagonals in a regular polygon of 100 sides is a. 4950 b. 4850 c. 4750 d. 4650
  4. Let the coefficients of powers of x in the 2nd, 3rd^ and 4th terms in the expansion of (1 +x)n^ where is a +ive integer be in arithmetic progression. Then the sum of the coefficients of odd power of x in the expansion is a. 23 b. 64 c. 128 d. 256
  5. The sum 1 x 1! + 2 x 2! + ………..50 x 50! Equal to a. 51! b. 51!- c. 51!+ d. 51! X 2
  6. Six numbers are in AP. Such that their sum is 3 the first term is 4 times the third term. Then the fifth term is a. - b. - c. 9 d. -
  7. The sum of the infinite series 1 + 1/3 + 1.3/1.6 + 1.3.5/3.6.9 + 1.3.5.7/3.6.9.12 + ………………. Is equal to a. (^) √ b. √ c. √3/ d. (^) √1/
  1. The equations x^2 + x+ a = 0 and x^2 + ax+ 1 =0 have a common real root a. For no value of a b. For exactly one value of a c. For exactly two value of a d. For exactly three value of a
  2. If 64, 27, 36, are the Pth , Qth^ and the Rth^ terms of the G.P then P + 2Q is equal to a. R b. 2R c. 3R d. 4R
  3. The equation y^2 + 4x +4y + k = 0 represents a parabola whose lotus rectum is a. 1 b. 2 c. 3 d. 4
  4. If the circles x^2 + y^2 +2x + 2ky + 6 = 0 and x^2 + y^2 + 2ky + k = 0 intersect orthogonally, then k is equal to a. 2 or -3/ b. -2 or-3/ c. 2 or 3/ d. -2 or 3/
  5. If four distinct points(2k,3k),(2,0),(0,3),(0,0) lie on a circle , then a. K< 0 b. 0< K < 1 c. K = 1 d. K > 1
  6. The line joining a( bcos α, bsin) and B( acos β, asin β) , where a ≠ b, is produced to the point M(x,y) so that AM:MB = b:a. then x cos (α + β/2 ) +y sin (α + β/2 ) a. 0 b. 1 c. - d. a^2 + b^2
  7. let the foci of the ellipse x^2 /9 + y^2 = 1 subtend right angle at a point P then the locus of P is a. x^2 + y^2 = 1 b. x^2 + y^2 = 2 c. x^2 + y^2 = 4 d. x^2 + y^2 = 8
  1. A non-empty set on which a binary operation can be defined is called a. Group b. Semi group c. Groupoid d. Ableian group e. Monoid
  2. The value of the integral 2 ∫ (^) -2 (1 +2sinx)e|x|^ dx is equal to a. 0 b. e^2 - c. 2(e^2 – 1) d. 1
  3. If (α +√𝛽) and (α – √𝛽) are the roots of the equation x + px+ q =0 where α , β,p,q are real then the roots of the equation(p^2 -4q) (p^2 x^2 + 4px) – 16q =0 are a. (1/α + 1/√𝛽 )and( 1/α - 1/√𝛽) b. (1/√α + 1/𝛽)and( 1/√α - 1/𝛽 c. (1/√α + 1/√𝛽 )and( 1/√α - 1/√𝛽) d. (√α +√𝛽) and (√α - √𝛽)
  4. The number of solutions of the equation log2(x^2 + 2x -1)=1 is a. 0 b. 1 c. 2 d. 3
  5. The sum of the series 1 + 1 n/2 C1 + 1 n/3 C2 + ……………. + 1 n/n+1Cn. a. 2 n+1^ -1 / n+ b. 3(2n-1)/2n c. 2 n+1/ n+ d. 2 n+1/ 2n
  6. The value of ∑ 1 + 2 + 3 + ⋯ …. (𝑟 − 1) 𝑟!

∞ 𝑟=

I sequal to a. e b. 2e c. e/ d. 3e/

  1. If P = Q=PPt^ , then the value of the determinant of Q is equal to a. 2 b. - c. 1 d. 0
  2. The remainder obtained when 1! +2! +………………. +95! Is divided by 15 is a. 14 b. 3 c. 1 d. 0
  3. If P, Q R, are angles of triangle PQR then the value of is equal to a. - b. 0 c. ½ d. 1
  4. The number of real values of α for which the system of equations x +3y +5z =αx, 5x +y+3z =αy, 3x + 5y + z = αz has infinite number of solutions is a. 1 b. 2 c. 4 d. 6
  5. The total number of injections(one – one into mappings) from {a1,a2,a3,a4} to {b1,b2,b3,b4,b5,b6,b7} is a. 400 b. 420 c. 800 d. 840
  6. It the set G = {1, ω, ω2} is an abelian group w.r.t multiplication then inverse of ω is? a. 1 b. ω c. ω^2 d. does not contain an inverse
  7. Two decks of playing cards are well shuffled and 26 cards are randomly distributed to a player. Then the probability that the player gets all distinct cards o s a. 52C 26 / 104C 26 b. 2 x 52C 26 / 104C 26 c. 213 x 52C 26 / 104C 26 d. 2 26 x 52C 26 / 104C 26

-1 cosR cosQ cosR -1 cosP cosQ cosP -

  1. If f (x) and g(X) are twice differentiable functions on (0,3) satisfying f”(x) =g”(x), f(1) =4 g(1)= f(2) =3 g(2) =9 then f(1)-g(1) is a. 4 b. - c. 0 d. -
  2. Let (x) denote the greater integer less than or equal to x, then the value of the integral 1 ∫ (^) - [|x| -2[x]]dx is equal to a. 3 b. 2 c. - d. -
  3. The points representing the complex number z for which arg(z-2/z+2) =π/3 lies on a. A circle b. A straight line c. An ellipse d. A parabola
  4. Let a, b, c, p, q, r be positive real numbers such that a, b ,c are in G.P and ap^ =bq^ =cr^ then A,B,C a. p, q rare in G.P b. p, q rare in A.P c. p, q rare in H.P d. p^2 ,q^2 and r^2 rare in A.P
  5. a compound statement at the form “If p then q ” is called a. implication b. hypothesis c. tautology d. contingency
  6. The quadratic equation 2x^2 (a^3 +8a -1) x a^2 -4a =0 possesses roots of opposite sign. then a. a ≤ 0 b. 0<a< c. 4 ≤ a < 8 d. a≥ 8
  7. if log (x2 -16 ) ≤ log(4𝑥 − 11) , 𝑡ℎ𝑒𝑛 a. 4<x≤ 5 b. X< -4 0r x> c. -1≤ 𝑥 ≤ 5 d. X<-1 0r x>
  1. The coefficient of x 10 in the expansion of 1+ (1+x) +………………………+(1+x)^10 is a. 19 C 9 b. 20 C 10 c. 21 C 11 d. 22 C 12
  2. The system of linear equation λx+ y+ z =3, x-y-2z=6, -x + y +z =𝜇 a. Infinite number of solutions for λ ≠-1 and all 𝜇 b. Infinite number of solutions for λ =-1 and all 𝜇 = c. No solution for λ ≠- d. Unique solution for λ =-1 and all 𝜇 =
  3. Let A and B be two events with P(Ac) =0.3, P (B)=0.4 and P(A ∩B’) =0.5 Then P(B/(AUB’)) is equal to a. ¼ b. 1/ c. ½ d. 2/
  4. The set of real number is a subset of a. Set at natural number b. Set of whole number c. Set of……….. d. Set of complex number
  5. Let C 1 and C 2 denote the cents of the circles x^2 + y^2 =4 and (x-2)^2 + y^2 =1 respectively and let P and Q be their Points of intersection. The n the area of triangle C 1 PQ and C 2 PQ are in ration a. 3: b. 5: c. 7: d. 9:
  6. A Straight line through the point of intersection of the lines x +2y =4 and 2x +y =4 meet the coordinates axes at A and B the locus of the midpoint of AB is a. 3(x + y) =2xy b. 2(x + y) =3xy c. 2(x + y) =xy d. (x y) =3xy
  7. Let P and Q be the points on the parabola y^2 =4x so that the line segment PQ subtends right angle at the vertex. If PQ intersects the axis of the parabola at R then the distance of the vertex from R is a. 1 b. 2 c. 4 d. 6
  1. Which one of the following groups has quantities that do not have the same dimensions a. Velocity, speed b. Pressure, stress c. Force, impulse d. Work , energy
  2. The %age errors in the measurement of mass and speed are 3% and 4% respectively. The maximum error in the measurement of K.E is a. 11% b. 10 5 c. 8% d. 9%
  3. The vector product of two vectors is zero, when a. They are parallel to each other b. They are equal vectors c. They are perpendicular to each other d. They are inclined at angle of 60^0
  4. In right hand rule, the direction of the product vector will be a. Along the thumb erect b. Perpendicular to the erect thumb c. Along the rotation of fingers d. None
  5. When an object slides at constant speed down an inclined plane, the coefficient of friction may be approximately be a. sinӨ b. cos Ө c. tan Ө d. cot Ө
  6. Two forces 3N and 2N are at an angle Ө such that the resultant is R the first force is now increased to 6N and the resultant becomes 2R. the value of Ө is a. 300 b. 600 c. 900 d. 1200
  7. Torque acting on a body determines a. Acceleration b. Linear acceleration c. Angular acceleration d. Direction of motion of the body
  1. If the velocity of a body is uniform the velocity – time graph is a straight line which is a. Parallel to x axis b. Parallel c. At an angle of 45^0 with the x-axis d. Along the y-axis
  2. At what angle of projection the horizontal range of a projectile is max? a. 300 b. 450 c. 600 d. 900
  3. What will be the ratio of the distance moved by a freely falling body from rest in 4th^ and 5th second of journey a. 4: 5 b. 7: c. 16: d. 1:
  4. According to the postulates of the theory of relativity, a fourth dimension has been added to the three dimensions already associated with a Cartesian frame of reference. Which is the fourth dimension? a. Space b. Inertial frame of reference c. Speed of light d. Time
  5. If the water fall from a dam to into a turbine wheel 19.6m below, then the velocity of water at the turbine is (Take g =9.8m/s^2 ) a. 9.8m/s b. 19.6 m/s c. 39.2 m/s d. 98.0 m/s
  6. The escape velocity of earth in Km/s a. 9. b. 11. c. 12. d. 15.
  7. Which is constant for a satellite in orbit? a. Velocity b. K.E c. Angular momentum d. P.E
  1. In which case application of angular velocity is useful? a. When body is rotating b. When velocity of body is in a straight line c. When velocity is in a straight line d. None
  2. If the area of a circle is equal to its circumference the radius of this circle is a. 1 b. 2 c. 3 d. 4
  3. Rotational K.E of a disc is a. K.Erot =1/2 mv^2 b. K.Erot =1/3 mv^2 c. K.Erot =1/4 mv^2 d. None
  4. Which of these statements is not correct a. Moment of inertia is independent of shape and size of the body b. Moment of inertia depends on choice of axes c. Momentum of inertia does not depend on the mass of body d. None
  5. A particle is moving in a vertical circle. The tensions the string when passing through two positions at angles 30^0 and 600 from vertical (lowest positions) are T 1 and T 2 respectively. Then a. T 1 = T 2 b. T 2 > T 1 c. T 1 > T 2 d. Tension in the string always remains the same
  6. At terminal velocity, fluid friction is a. Maximum b. Minimum c. Zero d. Decreasing
  7. 𝑣 = √2g(h1 − h2)^ 𝑠ℎ𝑜𝑤𝑠 𝑡ℎ𝑒 a. Equation of continuity b. Bernoulli’s theorem c. Torricelli’s theorem d. Equation for compressible fluids
  1. With the increase of temperature viscosity a. Increase b. Decrease c. Remain constant d. Doubles
  2. In case of streamed lined flow of liquid the loss of energy is a. Maximum b. Minimum c. Infinite d. Equal to what is in turbulent flow
  3. A car engine is based on the principle of a. Bernoulli’s equation b. Ventura relation c. Torricelli’s theorem d. None
  4. When a beam of light traveling in a rare medium is reflected from a denser medium it a. Suffers no phase change b. Undergoes a phase change of 180^0 c. Undergoes a phase change of 270^0 d. Undergoes a phase change of 90^0
  5. Two water pipes of diameters 4 cm and 8 cm are connected with a supply line. The velocity of flow of water in the pipe 4 cm diameter is a. ¼ times b. 4 times c. Twice d. ½ of 8 cm diameter pipe
  6. The density of water in F.P.S system is a. 50lb/ft^2 b. 50ft/lb c. 50ft/lb^3 d. 50lb/ft^3
  7. Total pressure on 1 m x 1 m gate immersed vertically at a depth of 2 m below the free water surface will be a. 1000 kg b. 2000kg c. 4000kg d. 8000 kg
  1. Which if electromagnetic radiation has the longest wavelength? a. γ rays b. UV c. Microwaves d. X rays
  2. The length of a spring is α when a force of 4 N is applied on it the length is β when 5 N forces is applied then the length of spring when 9N force is applied is a. 5β - 4α b. β - α c. 5α - 4β d. 9(β – α)
  3. Two springs of spring constant k1 and K2 are joined in series. The effective spring constant of combination is given by a. (k1 +k2)/ b. K1+ K c. K1k2/(k1 + k2) d. (^) √𝑘1𝑘
  4. The various features of wave phenomenon can be very conveniently studies by an apparatus called a. Sonometer b. Ripple tank c. hydrometer d. barometer
  5. A highly directional beam of ultrasonic wave can be made to travel in water in a. many meters b. many kilometers c. several kilometers d. none
  6. Applications of the result of scientific studies of sound in the designs of building etc. is called a. Optics b. Wave mechanics c. Acoustics d. Statics
  7. Laplace formula is derived from a. Isothermal; change b. Adiabatic change c. Isobaric change d. Isochoric change
  1. In the absence of an external torque the angular momentum of a rotating body is a. Constant b. Variable c. Unstable d. Zero
  2. Progressive waves of frequency 300 Hz are superimposed to produce a system of stationary waves in which adjacent nodes are 1.5m apart. What is the speed of the progressive waves? a. 100m/s b. 200m/s c. 450 m/s d. 900 m/s
  3. Which one of the following could be the frequency of ultraviolet radiation a. 1.0 x 10^6 Hz b. 1.0 x 10^9 Hz c. 1.0 x 10^12 Hz d. 1.0 x 10^15 Hz
  4. To hear a clear echo, the reflecting surface must be at a minimum distance of a. 10m b. 16.5m c. 33m d. 66m
  5. Which one is not a produced by sound wave in air a. Polarization b. Diffraction c. Refraction d. Reflection
  6. The conduction due to charges produced by pair generation in a semi-conductor is called a. Polarity b. Intrinsic conduction c. Electrostatic d. Amplitude modulation
  7. Ever point of a wave front may be considered as a a. Source b. Source of wave front c. Source of secondary wave front d. None