Previous Year Papers

Elementary Mathematics 2020 (II)

Set A

View solutions

100Questions

120Minutes

100Marks

Question index

Jump to any question in this paper.

Q1 $x^{3} + x^{2} + 16$ is exactly divisible by x, where x is a positive integer. The number of all such possible values of x isDivisibilityEasy
Q2The number of $(a, b, c)$ , where a, b, c are positive integers such that $ab c = 30$ , isFactors and DivisorsModerate
Q3If the roots of the quadratic equation $x^{2} - 4 x - lo g_{10} N = 0$ are real, then what is the minimum value of N ?RootsEasy
Q4The number of different solutions of the equation $x + y + z = 12$ , where each of x, y and z is a positive integer, isLinear EquationsModerate
Q5If $I = a^{2} + b^{2} + c^{2}$ , where a and b are consecutive integers and $c = ab$ , then I isAlgebraic IdentitiesModerate
Q6If the number $23 P 62971335$ is divisible by the smallest odd composite number, then what is the value of P ?DivisibilityModerate
Q7What is the remainder when the sum $1^{5} + 2^{5} + 3^{5} + 4^{5} + 5^{5}$ is divided by $4$ ?DivisibilityEasy
Q8What is the digit in the unit place of $3^{99}$ ?Unit DigitEasy
Q9LCM of two numbers is $28$ times their HCF. The sum of the HCF and the LCM is $1740$ . If one of these numbers is $240$ , then what is the other number ?LCM and HCFEasy
Q10 $(x^{n} - a^{n})$ is divisible by $(x - a)$ , where $x \neq = a$ , for everyFactor and Remainder TheoremEasy
Q11If $1 7^{2020}$ is divided by $18$ , then what is the remainder ?DivisibilityEasy
Q12What is the value of $\frac{1}{1 + 2} + \frac{1}{2 + 3} + \frac{1}{3 + 4} + ... + \frac{1}{99 + 100}$ ?SimplificationModerate
Q13If $x^{m} = 14 x x x$ , then what is the value of m ?SimplificationModerate
Q14The sum of all possible products taken two at a time out of the numbers $\pm 1, \pm 2, \pm 3, \pm 4, \pm 5$ isAlgebraic IdentitiesModerate
Q15A train of length $110$ m is moving at a uniform speed of $132$ km/hr. The time required to cross a bridge of length $165$ m isTrainsEasy
Q16The simple interest on a certain sum is one-fourth of the sum. If the number of years and the rate of annual interest are numerically equal, then the number of years isSimple InterestEasy
Q17A 60-page book has n lines per page. If the number of lines were reduced by $3$ in each page, the number of pages would have to be increased by $10$ to give the same writing space. What is the value of n ?Linear EquationsEasy
Q18If x men working x hours per day can do x units of work in x days, then y men working y hours per day in y days would be able to do k units of work. What is the value of k ?Time and WorkModerate
Q19Let d(n) denote the number of positive divisors of a positive integer n. Which of the following are correct ?Factors and DivisorsModerate
Q20If $A_{n} = P_{n} + 1$ , where $P_{n}$ is the product of the first n prime numbers, then consider the following statements :Prime NumbersModerate
Q21A shopkeeper sells his articles at their cost price but uses a faulty balance which reads $1000$ gm for $800$ gm. What is the actual profit percentage ?False WeightsModerate
Q22A river $3$ m deep and $40$ m wide is flowing at the rate of $2$ km/hr and falls into the sea. What is the amount of water in litres that will fall into the sea from this river in a minute ?CuboidModerate
Q23If a television set is sold at Rs. x, a loss of $28%$ would be incurred. If it is sold at Rs. y, a profit of $12%$ would be incurred. What is the ratio of y to x ?Profit and LossEasy
Q24By increasing the speed of his car by $15$ km/hr, a person covers a distance of $300$ km by taking an hour less than before. What was the original speed of the car ?Time, Speed and DistanceModerate
Q25Three persons start a business with capitals in the ratio $\frac{1}{3} : \frac{1}{4} : \frac{1}{5}$ . The first person withdraws half his capital after $4$ months. What is his share of profit if the business fetches an annual profit of Rs. $96, 800$ ?PartnershipHard
Q26If x varies as y, then which of the following is/are correct ?Direct and Inverse VariationHard
Q27Ena was born $4$ years after her parents marriage. Her mother is $3$ years younger than her father and $24$ years older than Ena, who is $13$ years old. At what age did Ena's father get married ?Age ProblemsEasy
Q28Mahesh is $60$ years old. Ram is $5$ years younger to Mahesh and $4$ years elder to Raju. Babu is a younger brother of Raju and he is $6$ years younger. What is the age difference between Mahesh and Babu?Age ProblemsEasy
Q29The number of items in a booklet is N. In the first year there is an increase of $x %$ in this number and in the subsequent year there is a decrease of $x %$ . At the end of the two years, what will be the number of items in the booklet?Successive Increase and DecreaseEasy
Q30If $ab + x y - x b = 0$ and $b c + y z - cy = 0$ , then what is $\frac{x}{a} + \frac{c}{z}$ equal to?Algebraic IdentitiesModerate
Q31What is the HCF of the polynomials $x^{6} - 3 x^{4} + 3 x^{2} - 1$ and $x^{3} + 3 x^{2} + 3 x + 1$ ?HCF and LCMModerate
Q32The HCF and the LCM of two polynomials are $3 x + 1$ and $30 x^{3} + 7 x^{2} - 10 x - 3$ respectively. If one polynomial is $6 x^{2} + 5 x + 1$ , then what is the other polynomial?HCF and LCMModerate
Q33If $(p + 2) (2 q - 1) = 2 pq - 10$ and $(p - 2) (2 q - 1) = 2 pq - 10$ , then what is $pq$ equal to?Linear EquationsModerate
Q34What is the value of $\frac{a ^{2} + a c}{a ^{2} c - c ^{3}} - \frac{a ^{2} - c ^{2}}{a ^{2} c + 2 a c ^{2} + c ^{3}} - \frac{2 c}{a ^{2} - c ^{2}} + \frac{3}{a + c}$ ?SimplificationHard
Q35What is the square root of $4 x^{4} + 8 x^{3} - 4 x + 1$ ?FactorisationModerate
Q36The sum of the digits of a two digit number is $13$ and the difference between the number and that formed by reversing the digits is $27$ . What is the product of the digits of the number?Linear EquationsEasy
Q37If $\frac{x}{b + c} = \frac{y}{c + a} = \frac{z}{b - a}$ , then which one of the following is correct?ProportionEasy
Q38X, Y and Z travel from the same place with uniform speeds $4$ km/hr, $5$ km/hr and $6$ km/hr respectively. Y starts $2$ hours after X. How long after Y must Z start in order that they overtake X at the same instant?Relative SpeedHard
Q39 $1 - x - x^{n} + x^{n + 1}$ , where n is a natural number, is divisible byFactorisationEasy
Q40A person sold an article for Rs. $75$ which cost him Rs. x. He finds that he realised $x %$ profit on his outlay. What is x equal to?Word ProblemsModerate
Q41A car did a journey in t hours. Had the average speed been x kmph greater, the journey would have taken y hours less. How long was the journey?Time, Speed and DistanceModerate
Q42When a ball is allowed to fall, the time it takes to fall any distance varies as the square root of the distance and it takes $4$ seconds to fall $78.40$ m. How long would it take to fall $122.50$ m?Direct and Inverse VariationEasy
Q43If $6^{3 - 4 x} 4^{x + 5} = 8$ (Given $lo g_{10} 2 = 0.301$ and $lo g_{10} 3 = 0.477$ ), then which one of the following is correct?EquationsModerate
Q44The Euclidean algorithm is used to calculate theLCM and HCFVery easy
Q45If radius of a sphere is rational, then which of the following is/are correct?Rational NumbersModerate
Q46If $cosec θ - sin θ = m$ and $sec θ - cos θ = n$ , then what is $m^{\frac{4}{3}} n^{\frac{2}{3}} + m^{\frac{2}{3}} n^{\frac{4}{3}}$ equal to?Elimination of the AngleHard
Q47If $cos θ + sec θ = k$ , then what is the value of $sin^{2} θ - tan^{2} θ$ ?Trigonometric IdentityModerate
Q48ABC is a triangle inscribed in a semicircle of diameter AB. What is $cos (A + B) + sin (A + B)$ equal to?Complementary AnglesVery easy
Q49Consider the following statements:Maximum and Minimum ValuesEasy
Q50What is the magnitude (in radian) of the interior angle of a regular pentagon?Interior and Exterior AnglesEasy
Q51The difference between two angles is $1 5^{\circ}$ and the sum of the angles in radian is $\frac{5 π}{12}$ . The bigger angle is k times the smaller angle. What is k equal to?Degree and RadianEasy
Q52Consider the following statements:Trigonometric EquationModerate
Q53A road curve is to be laid out on a circle. What radius should be used if the track is to change direction by $4 2^{\circ}$ in distance of $44$ m? (Assume $π = \frac{22}{7}$ )CircleEasy
Q54What is the maximum value of $3 sin θ - 4$ ?Maximum and Minimum ValuesVery easy
Q55If $sin θ + cos θ = 2$ , then what is $sin^{6} θ + cos^{6} θ + 6 sin^{2} θ cos^{2} θ$ equal to ?Trigonometric IdentityModerate
Q56What is the least value of $9 sin^{2} θ + 16 cos^{2} θ$ ?Maximum and Minimum ValuesEasy
Q57If $cos 4 7^{\circ} + sin 4 7^{\circ} = k$ , then what is the value of $cos^{2} 4 7^{\circ} - sin^{2} 4 7^{\circ}$ ?Trigonometric IdentityHard
Q58If $cosec θ - sin θ = p^{3}$ and $sec θ - cos θ = q^{3}$ , then what is the value of $tan θ$ ?Elimination of the AngleHard
Q59If $0 \leq α, β \leq 9 0^{\circ}$ such that $cos (α - β) = 1$ , then what is $sin α - sin β + cos α - cos β$ equal to ?Trigonometric EquationVery easy
Q60Consider the following statements :Complementary AnglesModerate
Q61In a quadrilateral ABCD, $∠ B = 9 0^{\circ}$ and $A B^{2} + B C^{2} + C D^{2} - A D^{2} = 0$ , then what is $∠ A C D$ equal to ?Pythagoras TheoremModerate
Q62In a $△ A B C$ , $A C = 12$ cm, $A B = 16$ cm and AD is the bisector of $∠ A$ . If $B D = 4$ cm, then what is DC equal to ?Angle BisectorEasy
Q63ABCD is a cyclic quadrilateral. The bisectors of the angles A, B, C and D cut the circle at P, Q, R and S respectively. What is $∠ P QR + ∠ R S P$ equal to ?Arcs and AnglesHard
Q64ABC is an equilateral triangle. The side BC is trisected at D such that $B C = 3 B D$ . What is the ratio of $A D^{2}$ to $A B^{2}$ ?Pythagoras TheoremModerate
Q65Consider the following statements :TrapeziumEasy
Q66If H, C and V are respectively the height, curved surface area and volume of a cone, then what is $3 π V H^{3} + 9 V^{2}$ equal to ?ConeModerate
Q67How many solid lead balls each of diameter $2$ mm can be made from a solid lead ball of radius $8$ cm ?Recasting and MeltingEasy
Q68The two sides of a triangle are $40$ cm and $41$ cm. If the perimeter of the triangle is $90$ cm, what is its area ?TriangleEasy
Q69The diagonals of a rhombus differ by $2$ units and its perimeter exceeds the sum of the diagonals by $6$ units. What is the area of the rhombus ?QuadrilateralHard
Q70What is the area of a right-angled triangle, if the radius of the circumcircle is $5$ cm and altitude drawn to the hypotenuse is $4$ cm ?TriangleEasy
Q71In a triangle, values of all the angles are integers (in degree measure). Which one of the following cannot be the proportion of their measures ?Angles of a TriangleEasy
Q72The length of a rectangle is increased by $10%$ and breadth is decreased by $10%$ . Then the area of the new rectangle isQuadrilateralEasy
Q73The surface areas of two spheres are in the ratio $1 : 4$ . What is the ratio of their volumes ?SphereEasy
Q74The length, breadth and height of a brick are $20$ cm, $15$ cm and $10$ cm respectively. The number of bricks required to construct a wall with dimensions $45$ m length, $0.15$ m breadth and $3$ m height isCuboidModerate
Q75If the sum of all interior angles of a regular polygon is twice the sum of all its exterior angles, then the polygon isInterior and Exterior AnglesVery easy
Q76A bicycle wheel makes $5000$ revolutions in moving $11$ km. What is the radius of the wheel ? (Assume $π = \frac{22}{7}$ )CircleEasy
Q77The volumes of two cones are in the ratio $1 : 4$ and their diameters are in the ratio $4 : 5$ . What is the ratio of their heights ?ConeModerate
Q78In a triangle ABC, if $2∠ A = 3∠ B = 6∠ C$ , then what is $∠ A + ∠ C$ equal to ?Angles of a TriangleEasy
Q79If the perimeter of a circle and a square are equal, then what is the ratio of the area of the circle to that of the square ?Plane FiguresEasy
Q80The lengths of the sides of a right-angled triangle are consecutive even integers (in cm). What is the product of these integers ?Pythagoras TheoremModerate
Q81A circle is inscribed in a triangle ABC. It touches the sides AB and AC at M and N respectively. If O is the centre of the circle and $∠ A = 7 0^{\circ}$ , then what is $∠ M O N$ equal to?Inscribed FiguresEasy
Q82The sum of the squares of sides of a right-angled triangle is $8, 450$ square units. What is the length of its hypotenuse?Pythagoras TheoremEasy
Q83A triangle and a parallelogram have equal areas and equal bases. If the altitude of the triangle is k times the altitude of the parallelogram, then what is the value of k?Plane FiguresEasy
Q84Areas of two squares are in the ratio $m^{2} : n^{4}$ . What is the ratio of their perimeters?QuadrilateralEasy
Q85AD is the median of the triangle ABC. If P is any point on AD, then which one of the following is correct?Medians and CentroidModerate
Q86What is the area of a segment of a circle of radius r subtending an angle $θ$ at the centre?CircleModerate
Q87ABC is a triangle right-angled at C. Let P be any point on AC and Q be any point on BC. Which of the following statements is/are correct?Pythagoras TheoremModerate
Q88Four circular coins of equal radius are placed with their centres coinciding with four vertices of a square. Each coin touches two other coins. If the uncovered area of the square is $42 cm^{2}$ , then what is the radius of each coin? (Assume $π = \frac{22}{7}$ )Composite and Shaded RegionsModerate
Q89The radii of the flat circular faces of a bucket are x and $2 x$ . If the height of the bucket is $3 x$ , what is the capacity of the bucket? (Assume $π = \frac{22}{7}$ )FrustumModerate
Q90If p, q, r, s and t represent length, breadth, height, surface area and volume of a cuboid respectively, then what is $\frac{1}{p} + \frac{1}{q} + \frac{1}{r}$ equal to?CuboidEasy
Q91Fifteen candidates appeared in an examination. The marks of the candidates who passed in the examination are $9, 6, 7, 8, 8, 9, 6, 5, 4$ and $7$ . What is the median of marks of all the fifteen candidates?MedianHard
Q92If the yield (in gm) of barley from $7$ plots of size one square yard each, were found to be $180, 191, 175, 111, 154, 141$ and $176$ , then what is the median yield?MedianVery easy
Q93Which one of the following measures of central tendency will be used to determine the average size of the shoe sold in the shop?ModeVery easy
Q94When the class intervals have equal width, the height of a rectangle in a histogram representsFrequency DistributionVery easy
Q95The ages of $7$ family members are $2, 5, 12, 18, 38, 40$ and $60$ years respectively. After $5$ years a new member aged x years is added. If the mean age of the family now goes up by $1.5$ years, then what is the value of x?MeanModerate
Q96The mean weight of $100$ students in a class is $46 kg$ . The mean weight of boys is $50 kg$ and that of girls is $40 kg$ . The number of boys exceeds the number of girls byWeighted MeanEasy
Q97What is the algebraic sum of the deviations from the mean of a set of values $25, 65, 73, 75, 83, 76, 17, 15, 7, 14$ ?MeanEasy
Q98The mean of five observations x, x + 2, x + 4, x + 6, x + 8 is m. What is the mean of the first three observations?MeanEasy
Q99What is the median of $2, 4, 6$ , ..., $100$ ?MedianEasy
Q100The harmonic mean and the geometric mean of two numbers are $10$ and $12$ respectively. What is their arithmetic mean?MeanEasy

Topic mix

Arithmetic26
Mensuration19
Algebra18
Geometry14
Trigonometry13
Statistics10

Total questions100

Difficulty

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

Very easy8
Easy44
Moderate38
Hard10
Very hard0

Marking scheme

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

Options per question: 4Correct options per question: 1