Previous Year Papers

Elementary Mathematics 2016 (II)

Set A

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100Questions

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Q1What is the solution of $lo g_{10} [1 - {1 - (1 - x^{2})^{- 1}}^{- 1}]^{- \frac{1}{2}} = 1$ ?EquationsModerate
Q2If $λ$ is an integer and $α, β$ are the roots of $4 x^{2} - 16 x + \frac{λ}{4} = 0$ such that $1 < α < 2$ and $2 < β < 3$ , then how many values can $λ$ take?RootsHard
Q3What is $\frac{6 ^{2} + 7 ^{2} + 8 ^{2} + 9 ^{2} + 1 0 ^{2}}{7 + 4 3 - 4 + 2 3}$ equal to?SimplificationModerate
Q4If $x^{2} = y + z$ , $y^{2} = z + x$ and $z^{2} = x + y$ , then what is the value of $\frac{1}{x + 1} + \frac{1}{y + 1} + \frac{1}{z + 1}$ ?Symmetric EquationsModerate
Q5What would be the maximum value of $Q$ in the equation $5 P 9 + 3 R 7 + 2 Q 8 = 1114$ ?Digit ProblemsEasy
Q6 $A$ is a set of positive integers such that when divided by $2, 3, 4, 5$ and $6$ leaves the remainder $1, 2, 3, 4$ and $5$ respectively. How many integers between $0$ and $100$ belong to the set $A$ ?DivisibilityModerate
Q7In an examination, a student was asked to divide a certain number by $8$ . By mistake he multiplied it by $8$ and got the answer $2016$ more than the correct answer. What was the number?Number SystemEasy
Q8In the quadratic equation $x^{2} + a x + b = 0$ , $a$ and $b$ can take any value from the set ${1, 2, 3, 4}$ . How many pairs of values of $a$ and $b$ are possible in order that the quadratic equation has real roots?RootsModerate
Q9The sum of the squares of two positive integers is $208$ . If the square of the larger number is $18$ times the smaller number, then what is the difference of the larger and smaller numbers?Word ProblemsEasy
Q10Let $A = {7, 8, 9, 10, 11, 12}$ and $B = {7, 10, 14, 15}$ . What is the number of elements in $(A - B)$ and $(B - A)$ respectively?OperationsEasy
Q11A boy saves Rs. $4.65$ daily. What is the least number of days in which he will be able to save an exact number of rupees?LCM and HCFEasy
Q12Two men, $A$ and $B$ run a $4$ km race on a course $0.25$ km round. If their speeds are in the ratio $5 : 4$ , how often does the winner pass the other?RacesModerate
Q13Which one of the following rational numbers has non-terminating and repeating decimal expansion?Rational NumbersEasy
Q14If $α$ and $β$ are the two zeros of the polynomial $25 x^{2} - 15 x + 2$ , then what is a quadratic polynomial whose zeros are $(2 α)^{- 1}$ and $(2 β)^{- 1}$ ?RootsModerate
Q15What is the remainder when $2^{100}$ is divided by $101$ ?DivisibilityHard
Q16In an office, one-third of the workers are women, half of the women are married and one-third of the married women have children. If three-fourth of the men are married and one-third of the married men have children, then what is the ratio of married women to married men?Ratio and ProportionEasy
Q17 $A$ can do $50%$ more work than $B$ in the same time. $B$ alone can do a piece of work in $30$ hours. $B$ starts working and had already worked for $12$ hours when $A$ joins him. How many hours should $B$ and $A$ work together to complete the remaining work?Time and WorkModerate
Q18When the speed of a train is increased by $20%$ , it takes $20$ minutes less to cover the same distance. What is the time taken to cover the same distance with the original speed?Time, Speed and DistanceEasy
Q19A candidate scoring $x %$ marks in an examination fails by $a$ marks, while another candidate who scores $y %$ marks gets $b$ marks more than the minimum required pass marks. What is the maximum marks for the examination?PercentageModerate
Q20If $p$ and $q$ are the roots of $x^{2} + p x + q = 0$ , then which one of the following is correct?RootsModerate
Q21The cost of $2.5$ kg rice is Rs. $125$ . The cost of $9$ kg rice is equal to that of $4$ kg pulses. The cost of $14$ kg pulses is equal to that of $1.5$ kg tea. The cost of $2$ kg tea is equal to that of $5$ kg nuts. What is the cost of $11$ kg nuts?Ratio and ProportionModerate
Q22Which one of the following is correct in respect of the number $1729$ ?Number SystemModerate
Q23A shopkeeper increases the cost price of an item by $20%$ and offers a discount of $10%$ on this marked price. What is his percentage gain?DiscountEasy
Q24There are two numbers $p$ and $q$ such that their HCF is $1$ . Which of the following statements are correct?LCM and HCFModerate
Q25What is $1 + \frac{1}{1 ^{2}} + \frac{1}{2 ^{2}} + 1 + \frac{1}{2 ^{2}} + \frac{1}{3 ^{2}} + \dots + 1 + \frac{1}{200 7 ^{2}} + \frac{1}{200 8 ^{2}}$ equal to?SummationHard
Q26A person can row downstream $20$ km in $2$ hours and upstream $4$ km in $2$ hours. What is the speed of the current?Boats and StreamsEasy
Q27Consider the following statements in respect of positive odd integers $x$ and $y$ :DivisibilityEasy
Q28What are the roots of the equation $2^{x + 2} \cdot 2 7^{\frac{x}{x - 1}} = 9$ ?EquationsHard
Q29There are twelve friends $A, B, C, D, E, F, G, H, I, J, K$ and $L$ who invested money in some business in the ratio of $1 : 2 : 3 : 4 : 5 : 6 : 7 : 8 : 9 : 10 : 11 : 12$ and the duration for which they invested the money is in the ratio of $12 : 11 : 10 : 9 : 8 : 7 : 6 : 5 : 4 : 3 : 2 : 1$ respectively. Who will get the maximum profit at the end of the year?PartnershipModerate
Q30 $2^{122} + 4^{62} + 8^{42} + 4^{64} + 2^{130}$ is divisible by which one of the following integers?DivisibilityModerate
Q31If $2 p + 3 q = 12$ and $4 p^{2} + 4 pq - 3 q^{2} = 126$ , then what is the value of $p + 2 q$ ?FactorisationModerate
Q32The number of digits in $3^{30}$ is $n$ and it is given that $lo g_{10} 3 = 0.4771$ . What is the value of $n$ ?Characteristic and MantissaEasy
Q33What is the unit digit of $7^{139}$ ?Unit DigitEasy
Q34If $4 x + 3 a = 0$ , then what is the value of $\frac{x ^{2} + a x + a ^{2}}{x ^{3} - a ^{3}} - \frac{x ^{2} - a x + a ^{2}}{x ^{3} + a ^{3}}$ ?SimplificationModerate
Q35Incomes of Mahesh and Kamal are in the ratio $1 : 2$ and their expenses are in the ratio $1 : 3$ . Which one of the following statements is correct?Ratio and ProportionHard
Q36 $X$ and $Y$ entered into partnership with Rs. $700$ and Rs. $600$ respectively. After $3$ months $X$ withdrew $2/7$ of his stock but after $3$ months, he puts back $3/5$ of what he had withdrawn. The profit at the end of the year is Rs. $726$ . How much of this should $X$ receive?PartnershipModerate
Q37A tank can be filled by pipe $X$ in $2$ hours and pipe $Y$ in $6$ hours. At 10 a.m. pipe $X$ was opened. At what time will the tank be filled if pipe $Y$ is opened at 11 a.m.?Pipes and CisternsEasy
Q38A train is travelling at $48$ km/hour completely crosses another train having half its length and travelling in opposite direction at $42$ km/hour in $12$ s. It also passes a railway platform in $45$ s. What is the length of the platform?TrainsModerate
Q39The speeds of three cars are in the ratio $2 : 3 : 4$ . What is the ratio between the times taken by these cars to travel the same distance?Time, Speed and DistanceEasy
Q40If the mean age of combined group of boys and girls is $18$ years and the mean of age of boys is $20$ and that of girls is $16$ , then what is the percentage of boys in the group?Weighted MeanEasy
Q41The area of four walls of a room is $120$ m $^{2}$ . The length of the room is twice its breadth. If the height of the room is $4$ m, what is area of the floor?CuboidEasy
Q42If a square of side $x$ and an equilateral triangle of side $y$ are inscribed in a circle, then what is the ratio of $x$ to $y$ ?Inscribed FiguresModerate
Q43The ratio of the curved surface area to the total surface area of a right circular cylinder is $1 : 2$ . If the total surface area is $616$ cm $^{2}$ , what is the volume of the cylinder?CylinderModerate
Q44A cubic metre of copper weighing $9000$ kg is rolled into a square bar $9$ m long. An exact cube is cut off from the bar. How much does the cube weigh?Recasting and MeltingModerate
Q45Into a conical tent of radius $8.4$ m and vertical height $3.5$ m, how many full bags of wheat can be emptied, if space required for the wheat in each bag is $1.96$ m $^{3}$ ?ConeModerate
Q46Let $A B C$ be a right angled triangle with $B C = 5$ cm and $A C = 12$ cm. Let $D$ be a point on the hypotenuse $A B$ such that $∠ B C D = 3 0^{\circ}$ . What is length of $C D$ ?TriangleHard
Q47Two circles touch externally and sum of their areas is $130 π$ cm $^{2}$ and the distance between their centres is $14$ cm. What is the difference in the radii of the circles?CircleModerate
Q48In a circle of radius $3$ units, a diameter $A B$ intersects a chord of length $2$ units perpendicularly at $P$ . If $A P > B P$ , then what is the ratio of $A P$ to $B P$ ?ChordsModerate
Q49What is the number of rounds that a wheel of diameter $\frac{5}{11}$ m will make in traversing $7$ km?CircleEasy
Q50A building is in the form of a cylinder surmounted by a hemispherical dome on the diameter of the cylinder. The height of the building is three times the radius of the base of the cylinder. The building contains $67 \frac{1}{21}$ m $^{3}$ of air. What is the height of the building?Combination of SolidsModerate
Q51The radius of the base and the height of a solid right circular cylinder are in the ratio $2 : 3$ and its volume is $1617$ cm $^{3}$ . What is the total surface area of the cylinder?CylinderModerate
Q52The cost of painting a spherical vessel of diameter $14$ cm is Rs. $8008$ . What is the cost of painting per square centimetre?SphereEasy
Q53If a quadrilateral has an inscribed circle, then the sum of a pair of opposite sides equalsTangentialModerate
Q54A circle and a square have the same perimeter. Which one of the following is correct?CircleEasy
Q55A drinking glass of height $24$ cm is in the shape of frustum of a cone and diameters of its bottom and top circular ends are $4$ cm and $18$ cm respectively. If we take capacity of the glass as $π x$ cm $^{3}$ , then what is the value of $x$ ?FrustumModerate
Q56In an equilateral triangle another equilateral triangle is drawn inside joining the mid-points of the sides of given equilateral triangle and the process is continued up to $7$ times. What is the ratio of area of fourth triangle to that of seventh triangle?SimilarityModerate
Q57Rain water from a roof $22$ m $\times$ $20$ m drains into a cylindrical vessel having diameter of base $2$ m and height $3.5$ m. If the vessel is just full, what is the rainfall?CylinderEasy
Q58The height of a cone is $60$ cm. A small cone is cut off at the top by a plane parallel to the base and its volume is $\frac{1}{64}$ the volume of original cone. What is the height from the base at which the section is made?FrustumModerate
Q59If two tangents inclined at an angle $6 0^{\circ}$ are drawn to a circle of radius $3$ cm, then what is the length of each tangent?TangentsEasy
Q60What is the volume of a sphere of radius $3$ cm?SphereVery easy
Q61The sides of a triangle are given by $a^{2} + b^{2}$ , $c^{2} + a^{2}$ and $(b + c)$ where $a, b, c$ are positive. What is the area of the triangle equal to?TriangleHard
Q62What is area of largest triangle inscribed in a semi-circle of radius $r$ units?TriangleEasy
Q63The diameter of the front wheel of an engine is $2 x$ cm and that of rear wheel is $2 y$ cm. To cover the same distance, what is the number of times the rear wheel revolves when the front wheel revolves $n$ times?CircleEasy
Q64What is the slant height of the cone?ConeEasy
Q65What is the curved surface area of the cylinder?CylinderEasy
Q66How many square metres of canvas are used?Combination of SolidsModerate
Q67The wheels of a car are of diameter $80$ cm each. The car is travelling at a speed of $66$ km/hour. What is the number of complete revolutions each wheel makes in $10$ minutes?CircleModerate
Q68What is the area of a triangle with sides of length $12$ cm, $13$ cm and $5$ cm?TriangleVery easy
Q69If the perimeter of a circle is equal to that of a square, then what is the ratio of area of circle to that of square?CircleEasy
Q70 $A B C D A$ is a con-cyclic quadrilateral of a circle $A B C D$ with radius $r$ and centre at $O$ . If $A B$ is the diameter and $C D$ is parallel and half of $A B$ and if the circle completes one rotation about the centre $O$ , then the locus of the middle point of $C D$ is a circle of radiusArcs and AnglesModerate
Q71If $sin θ + cos θ = \frac{7}{2}$ , then what is $sin θ - cos θ$ equal to?Trigonometric IdentityEasy
Q72If $sin x + sin^{2} x = 1$ , then what is the value of $cos^{8} x + 2 cos^{6} x + cos^{4} x$ ?Trigonometric IdentityModerate
Q73A man from the top of a $100$ m high tower sees a car moving towards the tower at an angle of depression $3 0^{\circ}$ . After some time, the angle of depression becomes $6 0^{\circ}$ . What is the distance travelled by the car during this time?Heights and DistancesEasy
Q74Two men on either side of a tower $75$ m high observe the angle of elevation of the top of the tower to be $3 0^{\circ}$ and $6 0^{\circ}$ . What is the distance between the two men?Heights and DistancesEasy
Q75What is the value of $cosec^{2} 6 8^{\circ} + sec^{2} 5 6^{\circ} - cot^{2} 3 4^{\circ} - tan^{2} 2 2^{\circ}$ ?Complementary AnglesModerate
Q76If $2 y cos θ = x sin θ$ and $2 x sec θ - y cosec θ = 3$ , then what is $x^{2} + 4 y^{2}$ equal to?Elimination of the AngleModerate
Q77If $sin θ + cos θ = \frac{1 + 3}{2}$ where $0 < θ < \frac{π}{2}$ , then what is $tan θ + cot θ$ equal to?Trigonometric IdentityModerate
Q78If $A = sin^{2} θ + cos^{4} θ$ where $0 \leq θ < \frac{π}{2}$ , then which one of the following is correct?Maximum and Minimum ValuesHard
Q79What is $\frac{c o t A + cosec A - 1}{c o t A - cosec A + 1}$ equal to?Trigonometric IdentityModerate
Q80Consider the following:Degree and RadianModerate
Q81If $tan^{2} x + \frac{1}{t a n ^{2} x} = 2$ and $0^{\circ} < x < 9 0^{\circ}$ , then what is the value of $x$ ?Trigonometric EquationEasy
Q82If the length of the shadow of a tower is equal to its height, then what is the Sun's altitude at that time?Heights and DistancesVery easy
Q83Consider the following:Complementary AnglesModerate
Q84A pole stands vertically inside a triangular park $A B C$ . If the angle of elevation of the top of the pole from each corner of the park is same, then in the triangle $A B C$ , the foot of the pole is at theHeights and DistancesEasy
Q85What is the value of $tan 1^{\circ} tan 2^{\circ} tan 3^{\circ} tan 4^{\circ} \dots tan 8 9^{\circ}$ ?Complementary AnglesEasy
Q86The average score of class $X$ is $83$ . The average score of class $Y$ is $76$ . The average score of class $Z$ is $85$ . The average score of class $X$ and $Y$ is $79$ and average score of class $Y$ and $Z$ is $81$ . What is the average score of $X$ , $Y$ and $Z$ ?Weighted MeanModerate
Q87For $x > 0$ , if a variable takes discrete values $x + 4$ , $x - 3.5$ , $x - 2.5$ , $x - 3$ , $x - 2$ , $x + 0.5$ , $x - 0.5$ , $x + 5$ , then what is the median?MedianEasy
Q88The median of set of $9$ distinct observations is $20.5$ . If each of the largest $4$ observations of the set is increased by $2$ , then the median of the new setMedianEasy
Q89A cricketer has a certain average of $10$ innings. In the eleventh inning he scored $108$ runs, thereby increasing his average by $6$ runs. What is his new average?MeanEasy
Q90The mean of $20$ observations is $17$ . On checking it was found that the two observations were wrongly copied as $3$ and $6$ . If wrong observations are replaced by correct values $8$ and $9$ , then what is the correct mean?MeanEasy
Q91Number of credit cards held by an individual can be treated asData ClassificationEasy
Q92In a class of $100$ students, there are $70$ boys whose average marks in a subject are $75$ . If the average marks of the complete class is $72$ , then what is the average marks of the girls?Weighted MeanEasy
Q93A pie chart depicts the classification of total funds of an organization according to different sources of funds. A particular sector of pie chart for corporate tax has $10 8^{\circ}$ angle at the centre. What is the percentage of income from corporate tax to total funds?Pie ChartVery easy
Q94Consider the following statements:Class IntervalModerate
Q95Suppose the class interval $10 - 15$ has frequency $30$ , then what is the frequency density of this class interval?Frequency DensityEasy
Q96Let $S$ be a set of first ten natural numbers. What is the possible number of pairs $(a, b)$ where $a, b \in S$ and $a \neq = b$ such that the product $ab$ $(> 12)$ leaves remainder $4$ when divided by $12$ ?DivisibilityHard
Q97What is the remainder when $1 3^{5} + 1 4^{5} + 1 5^{5} + 1 6^{5}$ is divided by $29$ ?DivisibilityModerate
Q98What is the difference between the sum of the cubes and that of squares of first ten natural numbers?SummationEasy
Q99A motorist travels to a place $150$ km away at an average speed of $50$ km/hour and returns at $30$ km/hour. What is the average speed for the whole journey?Average SpeedEasy
Q100If $x + \frac{1}{1 + \frac{1}{2 + \frac{1}{3}}} = 2$ , then what is $x$ equal to?SimplificationEasy

Topic mix

Arithmetic28
Mensuration23
Algebra16
Trigonometry16
Statistics11
Geometry6

Total questions100

Difficulty

P(a prepared candidate answers correctly in ~90 seconds)

Very easy4
Easy43
Moderate44
Hard9
Very hard0

Marking scheme

Correct answer+1
Wrong answer−1/3 × 1
Unattempted0

Options per question: 4Correct options per question: 1