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Quick Revision Series

Logical Reasoning Concepts for Competitive Exams 2026

Supercharge your preparation with high-yield reasoning tricks, number series, and syllogism concepts. Perfect for quick memory recall before SSC, Banking, Railways, and State exams.

Logical Reasoning

Logical Reasoning one liner - Data Sufficiency

Data Sufficiency

  • When is Statement I alone sufficient to answer a question? - When Statement I yields a single unique answer, while Statement II does not
  • When is Statement II alone sufficient to answer a question? - When Statement II yields a single unique answer, while Statement I does not
  • When is "Either Statement I alone or Statement II alone is sufficient" selected? - When both statements independently give the exact same or unique answer
  • When is "Neither Statement I nor Statement II is sufficient" selected? - When combining both statements still fails to provide a unique, definitive solution
  • When is "Both Statements I and II together are necessary" selected? - When neither statement alone is sufficient, but combining both yields a unique answer
  • Is a definite "No" answer to a yes/no question considered sufficient in Data Sufficiency? - Yes (A definite 'No' is equally sufficient as a definite 'Yes')
  • If a statement yields two possible values for a variable (e.g., x=4 or x=−4), is it sufficient? - No (A sufficient statement must yield a single unique value)
  • Do you need to perform complete numerical calculations to solve Data Sufficiency questions? - No (You only need to determine if a unique solution is mathematically obtainable)
  • Question: "What is the value of integer x?" | (I) x2=36 (II) x>0. Sufficiency? - Both I and II together are necessary (x=6)
  • Question: "What is the value of integer y?" | (I) 2y+6=16 (II) y3=125. Sufficiency? - Either I alone or II alone is sufficient (y=5)
  • Question: "Is n an even integer?" | (I) n is divisible by 4. (II) n is divisible by 6. Sufficiency? - Either I alone or II alone is sufficient (Multiples of 4 or 6 are always even)
  • Question: "Is k an odd integer?" | (I) k+1 is an even integer. (II) k−2 is an odd integer. Sufficiency? - Either I alone or II alone is sufficient
  • Question: "Who is the tallest among A, B, and C?" | (I) A is taller than B. (II) C is taller than A. Sufficiency? - Both I and II together are necessary (C>A>B)
  • Question: "Who is the shortest among P, Q, and R?" | (I) P is shorter than Q. (II) R is shorter than P. Sufficiency? - Both I and II together are necessary (Q>P>R)
  • Question: "What is the rank of Rohit from the top in a class of 40?" | (I) Rohit is 15th from the bottom. (II) Rohit scored 85% marks. Sufficiency? - Statement I alone is sufficient (40−15+1=26th)
  • Question: "How many students are in the row?" | (I) A is 8th from the left. (II) B is 10th from the right. Sufficiency? - Neither I nor II is sufficient (Overlap/gap unknown)
  • Question: "How many students are in the row?" | (I) A is 8th from the left and 14th from the right. (II) B is in the exact middle. Sufficiency? - Statement I alone is sufficient (8+14−1=21)
  • Question: "What day of the week is today?" | (I) Yesterday was Thursday. (II) Tomorrow is Saturday. Sufficiency? - Either I alone or II alone is sufficient (Today is Friday)
  • Question: "In which direction is Point X with respect to Point Y?" | (I) X is North of Z. (II) Y is South of Z. Sufficiency? - Both I and II together are necessary (X is North of Y)
  • Question: "In which direction is Town M with respect to Town N?" | (I) M is East of P. (II) N is West of P. Sufficiency? - Both I and II together are necessary (M is East of N)
  • Question: "How is A related to B?" | (I) A is the brother of C. (II) C is the daughter of B. Sufficiency? - Both I and II together are necessary (A is the son of B)
  • Question: "How is P related to Q?" | (I) P is the mother of R. (II) R is the wife of Q. Sufficiency? - Both I and II together are necessary (P is the mother-in-law of Q)
  • Question: "What is the code for 'apple'?" | (I) 'apple sweet' is coded as 'ka la'. (II) 'sweet mango' is coded as 'la pa'. Sufficiency? - Both I and II together are necessary ('sweet' = 'la' → 'apple' = 'ka')
  • Question: "What is the code for 'good'?" | (I) 'is good' is 'ma ta'. (II) 'he is' is 'ta pa'. Sufficiency? - Both I and II together are necessary ('is' = 'ta' → 'good' = 'ma')
  • Question: "What is the area of a rectangle?" | (I) Length is 12 cm. (II) Breadth is 8 cm. Sufficiency? - Both I and II together are necessary (Area=12×8=96 cm2)
  • Question: "What is the perimeter of a square?" | (I) Side length is 5 cm. (II) Area is 25 cm2. Sufficiency? - Either I alone or II alone is sufficient (Perimeter=20 cm)
  • Question: "What is the radius of a circle?" | (I) Circumference is 44 cm. (II) Area is 154 cm2. Sufficiency? - Either I alone or II alone is sufficient (r=7 cm)
  • Question: "What is the speed of a train?" | (I) Train crosses a 200m platform in 20 seconds. (II) Train crosses a signal pole in 10 seconds. Sufficiency? - Both I and II together are necessary (Speed=20 m/s)
  • Question: "What is the length of a train?" | (I) Speed is 72 km/h. (II) It crosses a standing pole in 15 seconds. Sufficiency? - Both I and II together are necessary (Length=20×15=300 m)
  • Question: "What is the age of Ram?" | (I) Ram is 4 years older than Shyam. (II) Shyam is 18 years old. Sufficiency? - Both I and II together are necessary (Ram=22 years)
  • Question: "Is A older than B?" | (I) A was born in 1995. (II) B was born in 1998. Sufficiency? - Both I and II together are necessary (Yes, A is older)
  • Question: "How many days are in month X?" | (I) Month X immediately follows July. (II) Month X is August. Sufficiency? - Either I alone or II alone is sufficient (31 days)
  • Question: "What is the two-digit number?" | (I) Sum of the digits is 9. (II) Ten's digit is twice the unit digit. Sufficiency? - Both I and II together are necessary (The number is 63)
  • Question: "Is x>y?" | (I) x−y=5. (II) x>0. Sufficiency? - Statement I alone is sufficient (x−y>0⟹x>y)
  • Question: "Is m greater than n?" | (I) m+n=20. (II) m−n=4. Sufficiency? - Both I and II together are necessary (Yes, m=12>n=8)
  • Question: "What is the average of three numbers a,b,c?" | (I) a+b+c=60. (II) a=20. Sufficiency? - Statement I alone is sufficient (Average=60/3=20)
  • Question: "What is the value of a+b?" | (I) a2−b2=40. (II) a−b=4. Sufficiency? - Both I and II together are necessary (a+b=40/4=10)
  • Question: "Is triangle ABC equilateral?" | (I) All three angles are equal. (II) Side AB=BC=CA. Sufficiency? - Either I alone or II alone is sufficient
  • Question: "What is the simple interest on ₹1,000?" | (I) Rate is 10% per annum. (II) Time period is 2 years. Sufficiency? - Both I and II together are necessary (SI=₹200)
  • Question: "What is the compound interest on ₹5,000?" | (I) Rate of interest is 10% compounded annually. (II) Time is 2 years. Sufficiency? - Both I and II together are necessary (CI=₹1,050)
  • Question: "Who arrived first among X, Y, and Z?" | (I) X arrived before Y. (II) Z arrived after Y. Sufficiency? - Both I and II together are necessary (X arrived first)
  • Question: "On which floor does Amit live in a 5-story building?" | (I) Amit lives on an odd floor above floor 2. (II) Amit lives on a floor below floor 4. Sufficiency? - Both I and II together are necessary (Floor 3)
  • Question: "On which floor does Priya live in a 6-story building?" | (I) Priya lives on an even floor below floor 4. (II) Priya does not live on floor 2. Sufficiency? - Both I and II together are necessary (None/Invalid or Floor 2/4 boundary)
  • Question: "How many red marbles are in the bag?" | (I) Total marbles is 20. (II) Probability of drawing red is 1/4. Sufficiency? - Both I and II together are necessary (20×1/4=5)
  • Question: "What is the cost price of the article?" | (I) Selling price is ₹120. (II) Profit percentage is 20%. Sufficiency? - Both I and II together are necessary (CP=₹100)
  • Question: "What is the profit earned on an item?" | (I) Cost price is ₹500. (II) Selling price is ₹650. Sufficiency? - Both I and II together are necessary (Profit=₹150)
  • Question: "Is P the grandmother of Q?" | (I) P is the mother of R. (II) R is the father of Q. Sufficiency? - Both I and II together are necessary (Yes, paternal grandmother)
  • Question: "How many children does Mohan have?" | (I) Mohan has two sons. (II) Mohan has one daughter. Sufficiency? - Both I and II together are necessary (3 children)
  • Question: "What is the value of x/y?" | (I) 3x=4y. (II) x=8. Sufficiency? - Statement I alone is sufficient (x/y=4/3)
  • Question: "Is a×b>0?" | (I) a>0 and b>0. (II) a<0 and b<0. Sufficiency? - Either I alone or II alone is sufficient (Both yield positive product)
  • Question: "Is integer z positive?" | (I) z3=27. (II) z2=9. Sufficiency? - Statement I alone is sufficient (z=3>0)
  • Question: "What is the median of a dataset?" | (I) Dataset has numbers 5, 8, 12, 16, 20. (II) Mean is 12.2. Sufficiency? - Statement I alone is sufficient (Median=12)
  • Question: "What is the mode of numbers 2, 4, 4, 6, 8, x?" | (I) x=4. (II) x=2. Sufficiency? - Either I alone or II alone is sufficient (If x=4, mode is 4; if x=2, bi-modal)
  • Question: "What is the distance between Town A and Town B?" | (I) A car travels at 60 km/h. (II) The car takes 3 hours. Sufficiency? - Both I and II together are necessary (60×3=180 km)
  • Question: "What is the time taken to fill a tank?" | (I) Pipe A fills in 10 hrs. (II) Pipe B fills in 15 hrs. Sufficiency? - Both I and II together are necessary (1/10+1/15→6 hrs)
  • Question: "Is quadrilateral ABCD a rectangle?" | (I) All interior angles are 90°. (II) Diagonals are equal and bisect each other. Sufficiency? - Either I alone or II alone is sufficient
  • Question: "What is the slope of line L?" | (I) Line passes through (0,0) and (2,4). (II) Equation is y=2x. Sufficiency? - Either I alone or II alone is sufficient (Slope=2)
  • Question: "In which year was Tina born?" | (I) Tina is 25 years old in 2026. (II) Tina's sister was born in 2005. Sufficiency? - Statement I alone is sufficient (2026−25=2001)
  • Question: "What is the ratio of boys to girls?" | (I) Total students is 50. (II) Number of boys is 30. Sufficiency? - Both I and II together are necessary (Ratio=30:20=3:2)
  • Question: "Is line M parallel to line N?" | (I) Corresponding angles are equal. (II) Alternate interior angles are equal. Sufficiency? - Either I alone or II alone is sufficient
  • Question: "What is the total salary of A and B?" | (I) A earns ₹30,000. (II) B earns ₹25,000. Sufficiency? - Both I and II together are necessary (₹55,000)
  • Question: "Who sits to the immediate left of P in a row facing North?" | (I) Q sits to the immediate left of P. (II) R sits to the immediate right of P. Sufficiency? - Statement I alone is sufficient (Q sits to the immediate left)
  • Question: "What is the 3rd term of an Arithmetic Progression?" | (I) First term is 5. (II) Common difference is 3. Sufficiency? - Both I and II together are necessary (a3=5+2(3)=11)
  • Question: "What is the common ratio of a GP?" | (I) First term is 2. (II) Second term is 6. Sufficiency? - Both I and II together are necessary (r=6/2=3)
  • Question: "Is p divisible by 5?" | (I) Last digit of p is 0. (II) Last digit of p is 5. Sufficiency? - Either I alone or II alone is sufficient
  • Question: "What is the units digit of integer N?" | (I) N is an even number. (II) N is divisible by 10. Sufficiency? - Statement II alone is sufficient (Units digit=0)
  • Question: "What is the total surface area of a cylinder?" | (I) Radius is 7 cm. (II) Height is 10 cm. Sufficiency? - Both I and II together are necessary
  • Question: "What is the volume of a sphere?" | (I) Radius is 3 cm. (II) Diameter is 6 cm. Sufficiency? - Either I alone or II alone is sufficient
  • Question: "Is x+y even?" | (I) Both x and y are odd. (II) Both x and y are even. Sufficiency? - Either I alone or II alone is sufficient (Sum is always even)
  • Question: "What is the remainder when N is divided by 5?" | (I) N ends in digit 7. (II) N=27. Sufficiency? - Either I alone or II alone is sufficient (Remainder=2)
  • Question: "What is the code for 'sun'?" | (I) 'sun bright' is '1 2'. (II) 'moon bright' is '2 3'. Sufficiency? - Both I and II together are necessary ('sun' = 1)
  • Question: "How many books are on the shelf?" | (I) 10 math books and 15 science books. (II) No other books are present. Sufficiency? - Both I and II together are necessary (25 books)
  • Question: "Who is the youngest in the family?" | (I) A is younger than B and C. (II) D is younger than A. Sufficiency? - Both I and II together are necessary (D is youngest)
  • Question: "What is the angle between clock hands at 4:00?" | (I) Hour hand is at 4. (II) Minute hand is at 12. Sufficiency? - Either I alone or II alone is sufficient (Angle is 120°)
  • Question: "Is year Y a leap year?" | (I) Y is divisible by 4 but not 100. (II) Y is 2024. Sufficiency? - Either I alone or II alone is sufficient
  • Question: "What is the discount percentage?" | (I) Marked price is ₹200. (II) Selling price is ₹160. Sufficiency? - Both I and II together are necessary (Discount=20%)
  • Question: "What is the marked price?" | (I) Cost price is ₹400. (II) Markup is 25%. Sufficiency? - Both I and II together are necessary (MP=₹500)
  • Question: "What is the perimeter of a right triangle?" | (I) Sides are 3 cm, 4 cm, 5 cm. (II) Area is 6 cm2. Sufficiency? - Statement I alone is sufficient (Perimeter=12 cm)
  • Question: "What is the speed of stream?" | (I) Downstream speed is 12 km/h. (II) Upstream speed is 8 km/h. Sufficiency? - Both I and II together are necessary (Speed=(12−8)/2=2 km/h)
  • Question: "What is the speed of boat in still water?" | (I) Downstream speed is 15 km/h. (II) Upstream speed is 9 km/h. Sufficiency? - Both I and II together are necessary (Speed=(15+9)/2=12 km/h)
  • Question: "What is the probability of picking an ace?" | (I) Standard deck of 52 cards. (II) Deck has 4 aces. Sufficiency? - Either I alone or II alone is sufficient (4/52=1/13)
  • Question: "What is the sum of angles of polygon P?" | (I) Polygon P is a hexagon (n=6). (II) Perimeter is 30 cm. Sufficiency? - Statement I alone is sufficient ((6−2)×180∘=720∘)
  • Question: "How many diagonals does polygon Q have?" | (I) Q has 5 sides (pentagon). (II) Sum of angles is 540°. Sufficiency? - Either I alone or II alone is sufficient (n(n−3)/2=5)
  • Question: "Is x a prime number?" | (I) x is an even number greater than 2. (II) x=10. Sufficiency? - Either I alone or II alone is sufficient (Definite 'No')
  • Question: "What is the HCF of 12 and y?" | (I) y=18. (II) LCM of 12 and y is 36. Sufficiency? - Either I alone or II alone is sufficient (HCF=6)
  • Question: "What is the LCM of 8 and 12?" | (I) HCF is 4. (II) Product of numbers is 96. Sufficiency? - Either I alone or II alone is sufficient (LCM=24)
  • Question: "Is a/b>1?" | (I) a>b>0. (II) a=10,b=5. Sufficiency? - Either I alone or II alone is sufficient (Yes)
  • Question: "What is the value of (a+b)2?" | (I) a2+b2=25. (II) ab=12. Sufficiency? - Both I and II together are necessary (25+2(12)=49)
  • Question: "What is the value of (a−b)2?" | (I) a2+b2=50. (II) ab=20. Sufficiency? - Both I and II together are necessary (50−2(20)=10)
  • Question: "Is point (2,3) on line L?" | (I) Equation of line L is 2x+y=7. (II) Line L passes through (1,5) and (3,1). Sufficiency? - Either I alone or II alone is sufficient (2(2)+3=7, Yes)
  • Question: "What is the y-intercept of line K?" | (I) Equation is y=3x+5. (II) Line passes through (0,5). Sufficiency? - Either I alone or II alone is sufficient (y-intercept=5)
  • Question: "What is the x-intercept of line K?" | (I) Equation is 2x+4y=8. (II) Line passes through (4,0). Sufficiency? - Either I alone or II alone is sufficient (x-intercept=4)
  • Question: "Is triangle PQR right-angled?" | (I) Sides are 5, 12, 13. (II) 52+122=132. Sufficiency? - Either I alone or II alone is sufficient (Yes)
  • Question: "What is the supplementary angle of θ?" | (I) θ=70∘. (II) Complementary angle of θ is 20°. Sufficiency? - Either I alone or II alone is sufficient (180∘−70∘=110∘)
  • Question: "What is the complementary angle of α?" | (I) α=40∘. (II) Supplementary angle of α is 140°. Sufficiency? - Either I alone or II alone is sufficient (90∘−40∘=50∘)
  • Question: "What is the value of angle x in a straight line?" | (I) Adjacent angle is 60°. (II) Angles form a linear pair. Sufficiency? - Both I and II together are necessary (x=180∘−60∘=120∘)
  • Question: "Is ABCD a rhombus?" | (I) All 4 sides are equal. (II) Diagonals bisect at 90°. Sufficiency? - Either I alone or II alone is sufficient
  • Question: "Is quadrilateral XYZW a square?" | (I) All sides are equal and all angles are 90°. (II) Diagonals are equal and perpendicular. Sufficiency? - Either I alone or II alone is sufficient
  • Question: "What is the volume of a cone?" | (I) Base radius is 3 cm. (II) Height is 4 cm. Sufficiency? - Both I and II together are necessary (V=31π(9)(4)=12π cm3)
  • Question: "What is the capacity of a hemisphere?" | (I) Radius is 6 cm. (II) Diameter is 12 cm. Sufficiency? - Either I alone or II alone is sufficient (V=32πr3)
  • Question: "What is the total surface area of a hemisphere?" | (I) Radius is 7 cm. (II) Curved surface area is 308 cm2. Sufficiency? - Either I alone or II alone is sufficient (3πr2)
  • Question: "Is polygon A regular?" | (I) All sides are equal. (II) All interior angles are equal. Sufficiency? - Both I and II together are necessary (Both conditions required)
  • Question: "What is the interior angle of a regular octagon?" | (I) Octagon has 8 sides. (II) Sum of interior angles is 1080°. Sufficiency? - Either I alone or II alone is sufficient (1080∘/8=135∘)
  • Question: "What is the exterior angle of a regular pentagon?" | (I) Pentagon has 5 sides. (II) Sum of exterior angles is 360°. Sufficiency? - Either I alone or II alone is sufficient (360∘/5=72∘)
  • Question: "What is the mean of numbers 10, 20, 30, 40, x?" | (I) x=50. (II) Sum of numbers is 150. Sufficiency? - Either I alone or II alone is sufficient (Mean=30)
  • Question: "What is the range of numbers 4, 8, 15, 23, 42?" | (I) Lowest is 4, highest is 42. (II) Dataset is complete. Sufficiency? - Either I alone or II alone is sufficient (Range=42−4=38)
  • Question: "What is the variance of a dataset?" | (I) Standard deviation is 5. (II) Mean is 20. Sufficiency? - Statement I alone is sufficient (Variance=52=25)
  • Question: "What is log10(100)?" | (I) Base is 10. (II) 102=100. Sufficiency? - Either I alone or II alone is sufficient (2)
  • Question: "What is log2(8)?" | (I) 23=8. (II) Base is 2. Sufficiency? - Either I alone or II alone is sufficient (3)
  • Question: "Is x=y?" | (I) x−y=0. (II) x/y=1 (where y=0). Sufficiency? - Either I alone or II alone is sufficient
  • Question: "What is the value of x0 (for x=0)?" | (I) x=5. (II) x>0. Sufficiency? - Either I alone or II alone is sufficient (Value is always 1)
  • Question: "What is the factorial of 5 (5!)?" | (I) 5!=5×4×3×2×1. (II) 4!=24. Sufficiency? - Either I alone or II alone is sufficient (120)
  • Question: "What is the value of (25)?" | (I) Formula 2!3!5!. (II) (25)=(35). Sufficiency? - Statement I alone is sufficient (10)
  • Question: "Is integer A divisible by 2?" | (I) A is divisible by 6. (II) A is an even number. Sufficiency? - Either I alone or II alone is sufficient
  • Question: "Is integer B divisible by 3?" | (I) Sum of digits of B is 12. (II) B=237. Sufficiency? - Either I alone or II alone is sufficient
  • Question: "Is integer C divisible by 9?" | (I) Sum of digits of C is 18. (II) C=999. Sufficiency? - Either I alone or II alone is sufficient
  • Question: "Is integer D divisible by 11?" | (I) Difference between sum of odd and even place digits is 0. (II) D=121. Sufficiency? - Either I alone or II alone is sufficient
  • Question: "What is the absolute value of m (∣m∣)|?" | (I) m=−7. (II) m=7. Sufficiency? - Either I alone or II alone is sufficient (∣m∣=7)
  • Question: "What is the cube root of 216?" | (I) 63=216. (II) 216=23×33. Sufficiency? - Either I alone or II alone is sufficient (6)
  • Question: "Is line segment AB equal to CD?" | (I) Length of AB is 8 cm. (II) Length of CD is 8 cm. Sufficiency? - Both I and II together are necessary (Yes, both 8 cm)
  • Question: "What is the midpoint of segment from (2,4) to (6,8)?" | (I) Coordinates are given. (II) Midpoint formula is (2x1+x2,2y1+y2). Sufficiency? - Statement I alone is sufficient ((4,6))
  • Question: "What is the centroid of a triangle with vertices (0,0), (3,0), (0,3)?" | (I) Vertices are given. (II) Centroid formula is (3∑x,3∑y). Sufficiency? - Statement I alone is sufficient ((1,1))
  • Question: "Is polygon M convex?" | (I) All interior angles are less than 180°. (II) All diagonals lie entirely inside M. Sufficiency? - Either I alone or II alone is sufficient (Yes)
  • Question: "Is polygon N concave?" | (I) At least one interior angle is greater than 180°. (II) At least one diagonal passes outside N. Sufficiency? - Either I alone or II alone is sufficient (Yes)
  • Question: "What is the order of rotational symmetry of a square?" | (I) Square has 4 equal sides and angles. (II) It maps onto itself at 90°, 180°, 270°, 360°. Sufficiency? - Either I alone or II alone is sufficient (Order 4)
  • Question: "What is the order of rotational symmetry of an equilateral triangle?" | (I) Triangle has 3 equal sides. (II) Rotates at 120°, 240°, 360°. Sufficiency? - Either I alone or II alone is sufficient (Order 3)
  • Question: "How many lines of symmetry does a regular hexagon have?" | (I) It has 6 equal sides. (II) It is a regular 6-gon. Sufficiency? - Either I alone or II alone is sufficient (6 lines)
  • Question: "How many lines of symmetry does a circle have?" | (I) Circle is symmetric about any diameter. (II) Diameter count is infinite. Sufficiency? - Either I alone or II alone is sufficient (Infinite)
  • Question: "What is the probability of getting heads on a fair coin?" | (I) Coin has 2 outcomes: Head and Tail. (II) Outcomes are equally likely. Sufficiency? - Both I and II together are necessary (1/2)
  • Question: "What is the probability of rolling a 6 on a fair dice?" | (I) Dice has 6 faces numbered 1 to 6. (II) Faces are equally likely. Sufficiency? - Both I and II together are necessary (1/6)
  • Question: "What is the probability of rolling an even number on a dice?" | (I) Even numbers are 2, 4, 6. (II) Total outcomes is 6. Sufficiency? - Both I and II together are necessary (3/6=1/2)
  • Question: "Is event E certain?" | (I) P(E)=1. (II) Event E is guaranteed to occur. Sufficiency? - Either I alone or II alone is sufficient (Yes)
  • Question: "Is event F impossible?" | (I) P(F)=0. (II) Event F cannot occur. Sufficiency? - Either I alone or II alone is sufficient (Yes)
  • Question: "What is P(A∪B) for mutually exclusive events?" | (I) P(A)=0.3,P(B)=0.4. (II) P(A∩B)=0. Sufficiency? - Both I and II together are necessary (0.3+0.4=0.7)
  • Question: "What is P(A∩B) for independent events?" | (I) P(A)=0.5. (II) P(B)=0.6. Sufficiency? - Both I and II together are necessary (0.5×0.6=0.3)
  • Question: "What is the sum of roots of x2−5x+6=0?" | (I) Quadratic equation is x2−5x+6=0. (II) Formula is −b/a. Sufficiency? - Statement I alone is sufficient (Sum=5)
  • Question: "What is the product of roots of x2−5x+6=0?" | (I) Quadratic equation is given. (II) Formula is c/a. Sufficiency? - Statement I alone is sufficient (Product=6)
  • Question: "What are the roots of x2−4=0?" | (I) x2=4. (II) (x−2)(x+2)=0. Sufficiency? - Either I alone or II alone is sufficient (x=2,−2)
  • Question: "Is the discriminant of ax2+bx+c=0 positive?" | (I) Equation has two distinct real roots. (II) b2−4ac>0. Sufficiency? - Either I alone or II alone is sufficient (Yes)
  • Question: "Does ax2+bx+c=0 have equal roots?" | (I) b2−4ac=0. (II) Equation is a perfect square (x−k)2=0. Sufficiency? - Either I alone or II alone is sufficient (Yes)
  • Question: "What is the value of 25?" | (I) 25=2×2×2×2×2. (II) 24=16. Sufficiency? - Either I alone or II alone is sufficient (32)
  • Question: "What is the value of 34?" | (I) 34=81. (II) 32=9. Sufficiency? - Either I alone or II alone is sufficient (81)
  • Question: "What is the reciprocal of 5/8?" | (I) Fraction is 5/8. (II) Product with reciprocal is 1. Sufficiency? - Statement I alone is sufficient (8/5)
  • Question: "What is the multiplicative inverse of −4?" | (I) Number is −4. (II) Definition is 1/x. Sufficiency? - Statement I alone is sufficient (−1/4)
  • Question: "What is the additive inverse of 15?" | (I) Number is 15. (II) Sum with additive inverse is 0. Sufficiency? - Statement I alone is sufficient (−15)
  • Question: "What is the additive identity of numbers?" | (I) Adding 0 leaves number unchanged. (II) x+0=x. Sufficiency? - Either I alone or II alone is sufficient (0)
  • Question: "What is the multiplicative identity of numbers?" | (I) Multiplying by 1 leaves number unchanged. (II) x×1=x. Sufficiency? - Either I alone or II alone is sufficient (1)
  • Question: "Is matrix M invertible?" | (I) det(M)=0. (II) Determinant of M is 5. Sufficiency? - Either I alone or II alone is sufficient (Yes)
  • Question: "Is matrix N singular?" | (I) det(N)=0. (II) N has no inverse. Sufficiency? - Either I alone or II alone is sufficient (Yes)
  • Question: "What is the trace of matrix A?" | (I) Diagonal elements are 2, 4, 6. (II) Trace is sum of diagonals. Sufficiency? - Both I and II together are necessary (12)
  • Question: "What is the transpose of a symmetric matrix S?" | (I) Matrix S is symmetric (ST=S). (II) Matrix S is given. Sufficiency? - Either I alone or II alone is sufficient (Equal to S itself)
  • Question: "What is the dot product of perpendicular vectors?" | (I) Vectors are orthogonal (θ=90∘). (II) cos(90∘)=0. Sufficiency? - Either I alone or II alone is sufficient (0)
  • Question: "What is the cross product of parallel vectors?" | (I) Vectors are collinear (θ=0∘). (II) sin(0∘)=0. Sufficiency? - Either I alone or II alone is sufficient (Zero vector)
  • Question: "What is the magnitude of unit vector u^?" | (I) Vector is a unit vector. (II) ∣u^∣=1. Sufficiency? - Either I alone or II alone is sufficient (1)
  • Question: "Is triangle XYZ isosceles?" | (I) Two sides are equal. (II) Two base angles are equal. Sufficiency? - Either I alone or II alone is sufficient (Yes)
  • Question: "Is triangle DEF scalene?" | (I) All three sides have different lengths. (II) All three angles are different. Sufficiency? - Either I alone or II alone is sufficient (Yes)
  • Question: "What is the sum of interior angles of a decagon (n=10)?" | (I) Decagon has 10 sides. (II) Formula is (n−2)×180∘. Sufficiency? - Statement I alone is sufficient (1440∘)
  • Question: "What is the exterior angle of a regular dodecagon (n=12)?" | (I) Dodecagon has 12 sides. (II) Formula is 360∘/n. Sufficiency? - Statement I alone is sufficient (30∘)
  • Question: "What is the total number of subsets of set S?" | (I) Set S has 4 elements. (II) Formula is 2n. Sufficiency? - Statement I alone is sufficient (24=16)
  • Question: "What is the number of proper subsets of set S?" | (I) Set S has 3 elements. (II) Formula is 2n−1. Sufficiency? - Statement I alone is sufficient (23−1=7)
  • Question: "Is set A equal to set B?" | (I) A⊆B and B⊆A. (II) Both sets contain identical elements. Sufficiency? - Either I alone or II alone is sufficient (Yes)
  • Question: "Is set P disjoint from set Q?" | (I) P∩Q=∅. (II) No elements are common. Sufficiency? - Either I alone or II alone is sufficient (Yes)
  • Question: "What is the power set cardinality of empty set ∅?" | (I) Empty set has 0 elements. (II) Formula 20=1. Sufficiency? - Either I alone or II alone is sufficient (1)
  • Question: "What is the intersection of set A with universal set U?" | (I) Set A is a subset of U. (II) A∩U=A. Sufficiency? - Either I alone or II alone is sufficient (Set A)
  • Question: "What is the union of set A with empty set ∅?" | (I) Set A is given. (II) A∪∅=A. Sufficiency? - Either I alone or II alone is sufficient (Set A)
  • Question: "What is the complement of universal set Uc?" | (I) U is the universal set. (II) Definition Uc=∅. Sufficiency? - Either I alone or II alone is sufficient (Empty set ∅)
  • Question: "What is the complement of empty set ∅c?" | (I) ∅ is empty set. (II) Definition ∅c=U. Sufficiency? - Either I alone or II alone is sufficient (Universal set U)
  • Question: "What is the value of loga(1) for a>0,a=1?" | (I) a0=1. (II) Logarithm property. Sufficiency? - Either I alone or II alone is sufficient (0)
  • Question: "What is the value of loga(a)?" | (I) a1=a. (II) Base and argument are equal. Sufficiency? - Either I alone or II alone is sufficient (1)
  • Question: "What is the expansion of (a+b)(a−b)?" | (I) Algebraic identity. (II) a2−b2. Sufficiency? - Either I alone or II alone is sufficient (a2−b2)
  • Question: "What is (a+b)3?" | (I) Identity formula. (II) a3+b3+3ab(a+b). Sufficiency? - Either I alone or II alone is sufficient (a3+b3+3ab(a+b))
  • Question: "What is (a−b)3?" | (I) Identity formula. (II) a3−b3−3ab(a−b). Sufficiency? - Either I alone or II alone is sufficient (a3−b3−3ab(a−b))
  • Question: "What is a3+b3?" | (I) Identity (a+b)(a2−ab+b2). (II) Values of a,b given. Sufficiency? - Either I alone or II alone is sufficient
  • Question: "What is a3−b3?" | (I) Identity (a−b)(a2+ab+b2). (II) Values of a,b given. Sufficiency? - Either I alone or II alone is sufficient
  • Question: "What is the arithmetic mean of x and y?" | (I) x+y=30. (II) x=10,y=20. Sufficiency? - Either I alone or II alone is sufficient (15)
  • Question: "What is the harmonic mean of 2 and 6?" | (I) Numbers are 2 and 6. (II) Formula is a+b2ab=824. Sufficiency? - Statement I alone is sufficient (3)
  • Question: "What is the golden rule of Data Sufficiency questions?" | (I) Check adequacy of data rather than solving unnecessary arithmetic. (II) Eliminate options systematically based on uniqueness. Sufficiency? - Determine whether a single unique solution is definitively guaranteed
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