dakshana question paper-MATHEMATICS question paper

Mathematics Questions with SVGs

1. A square ABCD of side 10 cm has a circle inscribed inside it touching all four sides. What is the area of the shaded region at one of the corner gaps?

(a) 25 −

25π4

(b)

254

(4 − π)

(d) None of these

2. In the right circular cone shown below, the base radius is r = 6 cm and the vertical height is h = 8 cm. What is the total surface area of this cone?

(a) 60π cm²

(b) 96π cm²

(d) 108π cm²

3. If tan α =

x y

, then what is the value of

x sin α + y cos α x sin α − y cos α

(a)

x² − y² x² + y²

(b)

x² + y² x² − y²

(c)

x + y x − y

(d)

x − y x + y

4. यदि sin β + cos β = °2 sin(90° − β), तो cot β का मान क्या होगा?

(a) °2 − 1

(b) °2 + 1

(c)

1°2 − 1

(d) −(°2 + 1)

5. एक निष्पक्ष पासे (unbiased dice) को दो बार फेंका जाता है। दोनों बार आने वाले अंकों का योगफल कम से कम 10 होने की प्रायिकता (probability) क्या होगी?

(a) 1/6

(b) 1/4

(d) 5/36

6. ताश की एक गड्डी में से एक पत्ता निकाला जाता है। निकाले गए पत्ते के या तो हुकुम (spade) का पत्ता या फिर एक इक्का (ace) होने की प्रायिकता क्या है?

(a) 17/52

(b) 4/13

(d) 1/4

7. यदि किसी द्विघात समीकरण (quadratic equation) के मूलों का योग 6 है और उनके वर्गों का योग 20 है, तो वह समीकरण क्या होगा?

(a) x² − 6x + 8 = 0

(b) x² + 6x + 8 = 0

(d) x² − 20x + 6 = 0

Mathematics Questions 8-11

8. In the triangle PQR shown below, S is a point on PQ such that PS : SQ = 2 : 1, and T is a point on PR such that PT : TR = 1 : 3. If QT and RS intersect at point O, find the ratio RO : OS.

(a) 3 : 1

(b) 4 : 1

(d) 2 : 1

9. An equilateral triangle ABC has side length 14 cm. Three identical circular sectors of radius 7 cm are drawn centered at vertices A, B, and C. Find the area of the remaining unshaded region inside the triangle.

(a) 49°3 − 24.5π

(b) 49°3 − 49π

(d) None of these

10. A circle of radius R is inscribed inside an isosceles right-angled triangle. A smaller circle is then placed in the remaining gap near the right-angled corner, touching both legs and the primary circle. Find the radius of this smaller circle.

(a) R(°2 − 1)

(b) R(°2 − 1)²

(c)

R(°2 − 1) 2

(d)

R 3

11. In the figure shown, two circles with radii 9cm and 4cm touch each other externally. A common external tangent line touches the circles at points P and Q respectively. Find the length of the direct common tangent segment PQ.

(a) 12 cm

(b) 13 cm

(d) 6 cm

Mathematics Questions 12-18

12. For what value of k will the expression x⁴ − 6x³ + kx² − 12x + 4 become a perfect square of a quadratic polynomial?

(a) 11

(b) 13

(d) 15

13. If 25^x − 25^x−1 = 1200, then what is the value of (3x)^x?

(a) 15°15

(b) 27

(d) None of these

14. If 3x + 2y = 12 and xy = 6, then what is the value of 27x³ + 8y³?

(a) 360

(b) 432

(d) 648

15. The vertices of a triangle ABC are A(2, 3), B(4, −1), and C(8, 7). If AD is the median from vertex A to the side BC, what are the coordinates of the point D?

(a) (6, 3)

(b) (3, 1)

(d) (6, 2)

16. If the centroid and the circumcenter of a triangle are located at (4, 3) and (2, 2) respectively, then what will be the coordinates of its orthocenter?

(a) (8, 5)

(b) (6, 4)

(d) (5, 4)

17. Shweta has ₹50 and ₹100 currency notes with her. If the total number of notes she has is 35 and the total value of all the money is ₹2,500, how many ₹50 notes does she have?

(a) 15

(b) 20

(d) 25

18. If the sum of the roots of the quadratic equation x² − (2k + 1)x + 3(k − 2) = 0 is equal to twice the product of its roots, what is the value of k?

(a) 13/4

(b) −13/4

(d) −11/4

Mathematics Questions 19-25

19. For what values of m, the roots of the quadratic equation x² + x(10 − m) − 10m + 1 = 0 are equal integers?

(a) −9, −11

(b) −8, −12

(d) −9, −12

20. Two inlet pipes can fill a swimming pool in 4 h and 6 h respectively, while an outlet pipe can empty it in 8 h. If the pool is initially empty and all three pipes are opened at the same time, how long will it take to fill the pool completely?

(a) 2 h 15 min

(b) 2 h 24 min

(d) 3 h 12 min

21. Driving at 25% faster than his standard speed, Amit arrives at his destination 8 minutes early. What is the usual time taken by Amit to cover this distance?

(a) 32 min

(b) 40 min

(d) 48 min

22. A merchant bought wheat at ₹1600 per quintal. 10% of the wheat got ruined due to dampness. He mixes affordable alternative materials to restore the total weight back to its original quantity. To make a net profit of 20%, at what rate should he sell the mixture?

(a) ₹ 19.2 per kg

(b) ₹ 18.5 per kg

(d) ₹ 17.6 per kg

23. An Arithmetic Progression (AP) consists of 30 terms. The first term is 7 and the final term is 122. What is the combined sum of its 11^th and 20^th terms?

(a) 135

(b) 129

(d) Cannot be determined

24. If a₁, a₂, a₃, ……… a_n are in Arithmetic Progression (AP) with a common difference d, then the value of

1a₁a₂

1a₂a₃

1a₃a₄

+ ……… +

1a_n−1a_n

is equal to

(a)

n − 1 a₁a_n

(b)

n a₁a_n

(c)

n − 1 d · a₁a_n

(d) None of the above

25. In the triangle ABC shown below, AD is perpendicular to BC, and M is the absolute mid-point of BC. If BC = w, AC = u, AB = v, AD = p, and DM = z, then find the correct expression for z.

(a)

v² − u² 2w

(b)

u² − v² 2w

(c)

v² − p² −

w²4

(d)

u² + v² − w² 4

Reasoning Questions 1-3

In the question (Q no 1-2), there are statements followed by four conclusions. You have to take the given statements to be true even, if they seem to be at variance from commonly known facts and then decide, which of the given conclusions logically follows from the given statements.

Statements

All books are notes.

All diaries are notes.

Some papers are diaries.

Some pens are notes.

Conclusions –

I. Some books are papers.

II. All papers are notes.

III. Some pens are books.

IV. No book is a paper.

(a) None Follows

(b) Only II follows

(d) Either I or IV follows

Statements

All mobile phones are laptops.

Some mobile phones are not tablets.

No tablet is a desktop computer.

All desktop computers are servers.

Conclusions –

I. Some servers are not laptops.

II. Some servers are not tablets.

III. No laptop is a server.

IV. Some laptops are not tablets.

(a) Only IV follows

(b) II and IV follow

(d) None of the above

3. One statement is given followed by two conclusions I and II. Take the statement to be true and then decide which of the conclusions logically follows.

Statements- Now you don’t need a formal prescription from a practitioner to buy this wellness supplement.

Conclusions –
I. This wellness supplement is now produced locally inside the country.
II. This wellness supplement is now safely allowed to be purchased over-the-counter.

(a) if only Conclusion I follows

(b) if only Conclusion II follows

(d) if neither Conclusion I nor II follows

Reasoning Questions 4-9

4. If 6 * 3 = 100, then what is the value of 9 * 4?

(a) 144

(b) 169

(d) 196

5. Seven students P, Q, R, S, T, U, and V are standing in a straight queue facing forward. Each of them is wearing a distinct colored cap from the VIBGYOR spectrum (Violet, Indigo, Blue, Green, Yellow, Orange, Red). Q can see Green and Red caps in front of him, but not Violet. U can see Violet and Blue caps but not Orange. V can see all colored caps except Yellow. If T is wearing a red cap, then find the color of the cap worn by U.

(a) Yellow

(b) Violet

(d) Orange

6. P, Q, R, S, T, and U are seated symmetrically around a circular dining table. P sits directly between T and U. T is seated exactly opposite to S. R is not a neighbor of T. Who is seated directly opposite to Q?

(a) U

(b) R

(d) None of these

7. On what specific dates of March 2003 did Sunday occur?

(a) 2, 9, 16, 23, and 30

(b) 4, 11, 18, and 25

(d) 1, 8, 15, 22, and 29

8. At what exact time, in minutes, between 4 O’clock and 5 O’clock will both the hands of a clock coincide with each other?

(a) 16

411

(b) 21

911

711

(d) 25

511

The following question is based on the diagram below in which the circle stands for educated, the square stands for hardworking, the triangle stands for urban, and the rectangle stands for honest. Study the diagram carefully and select the correct option.

9. Educated hardworking and honest urban people are represented by which of the following numbered regions?

(a) 1

(b) 2

(d) 4

Reasoning Questions 13-19

13. In a certain code ‘345’ means ‘mangoes are sweet’, ‘674’ means ‘apple is sweet’ and ‘589’ means ‘buy good mangoes’. What code stands for ‘are’ in that language?

(a) 3

(b) 4

(d) 7

14. In the question below, three of the following pairs have a similar mathematical/alphabetical relationship and form a group. Find the one that does not belong to the group.

(a) COMFORT : ZNLENQS

(b) JOURNEY : GPXOKBV

(d) PATIENT : MXQFSBO

15. In the question below, three words are given which share a specific sequential relationship. Identify the option that shares the exact same logical relationship.
Melting : Water : Iceberg

(a) Combustion : Heat : Fuel

(b) Condensation : Dew : Vapor

(d) Seed : Sapling : Plant

16. If a single meaningful English word can be formed using the sixth, tenth, fourteenth, and eighteenth letters from the left in the English alphabetical series, what will be the third letter of that word? If no such word can be formed, mark your answer as ‘X’, and if more than one such word can be formed, mark your answer as ‘P’.

(a) N

(b) R

(d) P

17. Find the missing term to complete the letter series.
BDFH, IKMO, NPRT, ? , ZBDF

(a) UXWY

(b) TVXZ

(d) RTVX

18. In the given number sequence, how many times does an ‘8’ appear such that it is immediately preceded by an odd number and immediately followed by an even number?

3 8 4 7 8 2 5 6 9 8 6 1 8 3 5 8 4 2 7 8 6 3 9 8 4 1 8

(a) two

(b) three

(d) five

19. Arrange the following administrative ranks in a logical hierarchical sequence (either ascending or descending) and select the most appropriate option.

(1) Governor

(2) President

(3) Chief Minister

(4) Mayor

(5) Prime Minister

(a) (4), (3), (1), (5), (2)

(b) (2), (5), (1), (3), (4)

(d) Both (a) and (b) are correct

Reasoning Questions 20-25

Study the following information carefully to answer this question.

20. Rajesh Verma has three children – Anita, Deepak, and Sunil. Sunil married Kavita, the eldest daughter of Mr and Mrs Gupta. The Guptas married their youngest daughter to the eldest son of Mr and Mrs Kapoor, and they had two children, Rahul and Priya. Mr Gupta has two more children, Amit and Kiran, both elder to Sunita. Rohan and Nitin are sons of Sunil and Kavita. Sneha is the daughter of Rahul.

Que. – What is the surname of Rohan?

(a) Gupta

(b) Kapoor

(d) Deepak

21. If ‘North-East’ is called ‘South’, ‘South-East’ is called ‘West’, ‘South-West’ is called ‘North’ and so on, what will ‘East’ be called?

(a) South-West

(b) North-West

(d) South-East

22. Six athletes P, Q, R, S, T, and U compete in a high jump event. It is observed that ‘S’ jumped higher than ‘Q’ but not as high as ‘T’. ‘R’ jumped to a position not higher than ‘S’ and not lower than ‘P’, who jumped not lower than ‘U’ and not higher than ‘Q’. Who achieved the highest jump position?

(a) S

(b) T

(d) Cannot be determined

23. Vikram’s age is 6 years more than four times the sum of the ages of his two twin grandsons. After 6 years, Vikram’s age will be 12 years more than twice the sum of the ages of his grandsons. What was Vikram’s age when his grandsons were born?

(a) 54 years

(b) 60 years

(d) 72 years

24. Divide ₹ 450 among P, Q, and R such that P receives ₹ 40 more than Q, and R receives ₹ 20 more than P.

(a) ₹ 150, ₹ 110, and ₹ 190

(b) ₹ 170, ₹ 130, and ₹ 150

(d) ₹ 140, ₹ 100, and ₹ 210

Read the below information carefully and answer the question

φ(x, y) = (x + y)², if x < y; x − y if x ≥ y

What is the value of

φ[−(−4 + 1), −(−1 + 4)] −(−5 + 3)

(a) 0

(b) 1

(d) 3

Physics Questions 1-7

1. A container of absolute height 60 cm is completely filled with a transparent liquid. If the bottom of the container appears to be shifted upward such that it seems to be 45 cm below the top surface, what is the refractive index of this liquid?

(a)

(b)

(c)

(d)

2. The total equivalent resistance obtained by combining three identical 6 Ω resistors in any possible configuration cannot be equal to which of the following values?

(a) 18 Ω

(b) 2 Ω

(d) 6 Ω

3. What is the total equivalent resistance of the given circuit network measured across the terminal points P and Q?

(a) 12 Ω

(b) 9 Ω

(d) 3 Ω

4. Cylindrical electrical wires fabricated from the exact same material are given with various dimensional modifications. Which specification will demonstrate the minimum electrical resistance?

(a) Length = 2L, Area = A

(b) Length = L, Area = 2A

, Area = 3A

(d) Length = 3L, Area =

5. An electrical hair dryer and an electric oven are rated at 200 W, 220 V and 2000 W, 220 V respectively. Comparing their values, the internal electrical resistance of the oven is:

(a) significantly smaller than that of the hair dryer

(b) significantly larger than that of the hair dryer

(d) dependent on the operating temperature limit

6. A frictionally rubbed glass rod acquires a net positive electrical charge equal to 4.8 × 10⁻¹⁰ C. Determine the number of deficient electrons missing from its surface.

(a) 3 × 10⁹

(b) 3 × 10⁻⁹

(d) 4.8 × 10⁹

7. What is the numerical ratio of the electrostatic Coulomb force acting between two isolated electrons to that acting between two isolated protons placed at identical separation distances inside a vacuum?

(a) 1 : 1

(b) 1 : 1836

(d) 1 : 4

Physics Questions 8-13

8. The periodic time of a tuning fork is 0.005 sec. Its operational wave frequency will be:

(a) 50.0 Hz

(b) 200.0 Hz

(d) 1000.0 Hz

9. Due to persistence of hearing, a human ear can clearly distinguish a distinct echo only if the reflected sound wave reaches back after a minimum time interval of:

(a) 1.0 s

(b) 0.5 s

(d) 0.01 s

10. Rear-view mirrors installed in commercial vehicles to view a wide range of traffic behind are:

(a) concave mirrors

(b) convex mirrors

(d) parabolic concave mirrors

11. Which of the following ray diagrams correctly illustrates geometric specular reflection across two intersecting plane mirrors inclined at an angle of 50°?

(a)

(c)

(b)

(d)

12. A concave mirror produces a magnified real image that is exactly two times as large as an object situated 30 cm away from its vertex. Determine the focal length of this mirror:

(a) 10 cm

(b) 20 cm

(d) 15 cm

13. A thin glass equiconvex lens having a focal length f inside air is completely submerged in an organic oil with a matching refractive index. Its modified focal length under these conditions will turn into:

(a) Zero

(b) Infinity

(c)

(d) 2f

Physics Questions 14-19

14. Which of the following figures correctly illustrates the typical path and relative sequence of chromatic deviation and spectral dispersion when white light undergoes transmission through a triangular glass prism?

(a)

(b)

(c)

(d)

15. A container stores 4 × 10⁻³ kg of compressed gas. A micro-rupture is created on its surface, causing the gas molecules to jet outwards horizontally with a constant uniform velocity of 5 ms⁻¹. If the container completely vents all its contents over a total span of 2.0 s, the average thrust force exerted back onto the container structure is:

(a) 0.010 N

(b) 0.020 N

(d) 4.0 N

16. Two heavy blocks with masses of 5 kg and 15 kg respectively are aligned on a frictionless path and linked together by an ideal massless structural spring. A constant horizontal pulling force of 120 N is applied straight to the 15 kg block. At the precise instant under consideration, the 5 kg block exhibits an acceleration of 6 ms⁻². Determine the corresponding acceleration of the 15 kg block at this same split second:

(a) 6 ms⁻²

(b) 8 ms⁻²

(d) 2 ms⁻²

17. An object is released from rest at the opening of a deep vertical crevice where the steady baseline water surface is located at a depth of d meters directly beneath the top edge. If the speed of acoustic propagation through the air column is c, the total cumulative time interval that elapses from the release to when the impact splash is detected at the top edge is:

(a)

2dc

(b) °

2dg

(c)

d2g

2dc

(d) °

2dc

18. A motor vehicle travels along a straight path. It maintains a constant speed of 60 km/h for the first half of its total travel time duration, and then switches to a constant speed of 90 km/h for the remaining half of its duration. Find the average speed of the vehicle over its entire trip:

(a) 75 km/h

(b) 72 km/h

(d) 65 km/h

19. Which graphic curve below perfectly models the structural variations of gravitational acceleration (g) as a function of the radial distance (r) from the center of a uniform solid planetary body of radius R?

^-2

(a)

^-2

(b)

^-2

(c)

^-2

(d)

Physics Questions 20-25

20. If the total mass of a target planet drops by 25% while its nominal radius decreases by 50%, the fundamental acceleration due to gravity measured at its structural surface would:

(a) increase by 200%

(b) increase by 300%

(d) remain completely identical

21. A stone block with a mass of 500 g is dropped from a cliff height of 10 m. Determine its total Kinetic Energy (K.E.) right at the precise instance it makes contact with the baseline floor, given that active atmospheric air resistance must be factored in:

(a) exactly 49 J

(b) exactly 98 J

(d) more than 98 J

22. An absolute thermodynamic temperature level reading of 233 K is perfectly equivalent to:

(a) −40° F

(b) −4° F

(d) −172° F

23. A solid block of 100 g of pure ice at a thermal state of 0°C is combined inside an insulated vessel with 100 g of liquid water at 80°C. Find the final uniform equilibrium temperature level of this combined mixture (Take L_f = 80 cal/g, S_w = 1 cal/g/°C):

(a) 0°C

(b) 20°C

(d) 60°C

24. The propagation velocity speed of an acoustic sound wave moving through a steady uniform gaseous medium is:

(a) directly proportional to its operational frequency

(b) inversely proportional to its operational frequency

(d) proportional to the square root of its frequency

25. The diagram below illustrates the geometric shape and momentary displacement profiles of specific particles along a string experiencing steady transverse harmonic wave propagation. Which specific pair of labeled points are moving in identical phase with each other?

(a) A and I

(b) B and H

(d) D and J

Chemistry Questions 1-5

1. Which of the following minerals serves as a prominent commercial example of a carbonate ore?

(a) Calamine

(b) Galena

(d) Fluorite

2. The precise molecular mass of a certain gaseous hydrocarbon containing only carbon and hydrogen atoms is measured to be 42. What is its simplest empirical formula?

(a) CH₂

(b) CH₃

(d) C₃H₆

3. Distinct molecular compounds belonging to the exact same organic homologous series will never possess:

(a) identical chemical functional groups

(b) identical molecular weight values

(d) highly similar chemical reactivities

4. Identify the standardized systematic IUPAC name for the organic skeletal structure illustrated below:

(a) 2, 2, 4, 6, 6 – pentamethylheptane

(b) 2, 2, 4, 4, 6 – pentamethylheptane

(d) 1, 1, 3, 5, 5 – pentamethylhexane

5. Which of the following alcohols yields propene as the primary organic product when subjected to acid-catalyzed dehydration?

(a) Methanol

(b) Ethanol

(d) Butan-2-ol

Chemistry Questions 6-11

6. A molar solution (1 M) is defined as a solution that contains exactly 1 mole of a dissolved solute dissolved in:

(a) 1000 g of solvent

(b) 1 L of solution

(d) 22.4 L of solvent

7. What is the calculated molarity of an aqueous chemical solution containing 2 g of NaOH (molar mass = 40 g/mol) dissolved in exactly 200 mL of its total solution?

(a) 0.5 M

(b) 0.25 M

(d) 1.0 M

8. At standard room temperature (25°C), which of the following aqueous conditions represents a strongly alkaline (basic) solution?

(a) when pH = 0

(b) when pOH = 0

(d) Both B and C

9. According to the Brønsted-Lowry acid-base theory, the conjugate base of H₂O is:

(a) H₃O⁺

(b) OH⁻

(d) H₂

10. Spontaneous nuclear decay and radioactivity are fundamentally observed in atomic species that possess:

(a) a stable proton-to-neutron ratio

(b) an unstable nucleus

(d) fully filled electronic valence shells

11. In high-temperature metallurgical smelting processes, an added chemical flux is primarily utilized to:

(a) convert metal oxides into free gases

(b) react with infusible impurities to form a fluid slag

(d) increase the combustion temperature of the furnace fuel

Chemistry Questions 12-18

12. John was examining an unknown liquid mixture in the laboratory. He observed that no visible particles settled down at the bottom of the flask even after standing undisturbed for a week. However, when he passed a laser beam through it, the path of light became clearly visible. This liquid mixture is classified as a:

(a) Solution

(b) Suspension

(d) Pure compound

13. Determine the correct chemical symbols for the species composed of the following distinct sets of subatomic particles:

1. 16p, 16n, 18e 2. 3p, 4n, 2e

(a) S, Li

(b) S²⁻, Li⁺

(d) S²⁻, He

14. An element has an atomic number of 38. Identify the specific period of the modern periodic table to which this element belongs.

(a) 4

(b) 5

(d) 3

15. Which of the following chemical oxides displays the most pronounced acidic character?

(a) CO₂

(b) SO₃

(d) Al₂O₃

16. Which of the following atomic combinations can react together to establish a predominantly covalent chemical bond?

(a) Na and Cl

(b) Mg and O

(d) Ca and F

17. In which of the following stable molecular species is the central atom found to have an expanded octet (more than 8 valence electrons)?

(a) SF₆

(b) CH₄

(d) BF₃

18. A naturally occurring sample of chlorine gas consists of 75.8% of ³⁵Cl (isotopic mass = 34.969 u) and 24.2% of ³⁷Cl (isotopic mass = 36.966 u). Calculate the relative average atomic mass of this chlorine sample.

(a) 35.000 u

(b) 36.000 u

(d) 36.453 u

Chemistry Questions 19-25

19. The chemical formula of a metal chloride is MCl₂. What will be the correct chemical formula of its corresponding phosphate compound?

(a) MPO₄

(b) M₂(PO₄)₃

(d) M₃PO₄

20. The biological metabolic pathway in which glucose is broken down to release energy along with CO₂ is fundamentally a class of:

(a) reduction reaction

(b) oxidation reaction

(d) None of these

21. Choose the correctly balanced stoichiometry for the chemical reaction: C₃H₈ + O₂ → CO₂ + H₂O

(a) C₃H₈ + 3O₂ → 3CO₂ + 4H₂O

(b) C₃H₈ + 5O₂ → 3CO₂ + 4H₂O

(d) C₃H₈ + 4O₂ → 3CO₂ + 4H₂O

22. Consider the gas-phase chemical reaction: CO(g) + 2H₂(g) → CH₃OH(g). If a mixture containing exactly 2 moles of CO and 2 moles of H₂ is reacted to maximum completion:

(a) All carbon monoxide gas will be consumed

(b) Exactly 2 moles of CH₃OH will be produced

(d) Exactly 1 mole of CO will remain unreacted

23. The actual mass quantified in grams contained within exactly 1 gram-atom of magnesium is equal to:

(a) 1 g

(b) 12 g

(d)

124

24. A sample of an ideal gas occupies a volume of 4 L at a specific temperature and pressure. If its absolute temperature is doubled while its pressure is simultaneously halved, the new volume of the gas will be:

(a) 2 L

(b) 4 L

(d) 16 L

25. Which of the following substances exhibits the weakest intermolecular attractive forces in its liquid phase?

(a) HF

(b) Ne

(d) CH₃OH

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