Madhyamik Practice Test Paper

Prime Maths | Vinod Singh

Madhyamik Test Examination – Practice Paper | Subject: MATHEMATICS

Time: 3 Hours 15 Minutes

Full Marks: 90

Type: Free Practice Test Paper

General Instructions:

• The figures in the margin indicate full marks.
• Use of calculators is not permitted.
• Graph papers will be supplied if required.

1. Multiple Choice Questions [1 × 6 = 6]

Select the correct answer in each of the following cases:

iIf the mean of a frequency distribution is 8.1, \(\sum f_i x_i = 132 + 5k\), and \(\sum f_i = 20\), then the value of \(k\) is:

(a) 4

(b) 2

(c) 3

(d) 6

iiIf \(\cos\theta + \sec\theta = 2\), then the value of \(\cos^5\theta + \sin^5\theta\) is:

(a) -1

(b) 1

(c) 0

(d) -2

iiiThe volumes of two right circular cones are equal. If the radii of their circular bases are in the ratio 1:2, their heights are in the ratio:

(a) 2:1

(b) 3:1

(c) 4:1

(d) 8:1

ivIn the right-angled triangle ABC, \(\angle A\) is a right angle, AB = 12 cm, AC = 5 cm, and BC = 13 cm. AD is perpendicular to BC. The length of AD is:

(a) \(\frac{65}{12}\) cm

(b) \(\frac{60}{13}\) cm

(c) \(\frac{13}{60}\) cm

(d) \(\frac{5}{15}\) cm

vX and Y are two points on sides PQ and PR of triangle PQR respectively. If PX = 2 cm, XQ = 3.5 cm, YR = 7 cm, and PY = 4.25 cm, then:

(a) XY || QR

(b) XY = QR

(c) XY is not parallel to QR

(d) None of these

viIf \(x \propto y\), then:

(a) \(x^{420} \propto y^{202}\)

(b) \(x^{2025} \propto y^{3025}\)

(c) \(x^{2024} \propto y^{2025}\)

(d) \(x^{2025} \propto y^{2025}\)

2. Fill in the blanks (any five) [1 × 5 = 5]

iIf the difference between the roots of the equation \(x^2 - px + q = 0\) is 1, then \(p^2 - 4q = \) __________.

iiIf \(x = 2 - \sqrt{3}\) and \(y = 4\sqrt{2}-2\), then \((x + y)^4 = \) __________.

iiiThe tangents drawn at the two ends of a diameter of a circle are __________ to each other.

ivThe ratio of simple interest on a fixed amount of money at the same annual rate for 4 years and 8 years is __________.

vIf the compound ratio of 2:3, 4:5, and 6:\(x\) is 2:5, then the value of \(x = \) __________.

viA tangent drawn from an external point M to a circle with center O touches the circle at point N. If ON = 9 cm and MO = 15 cm, then the length of MN = __________.

3. Write True or False (any five) [1 × 5 = 5]

iA starts a business with ₹10,000, and 6 months later B invests ₹20,000. At the end of the year, their profit shares will be equal.

iiThe ratio of the volumes of a right circular cone and a right circular cylinder with the same base and same height is 1:3.

iiiThe arithmetic mean and mode of the numbers 4, 3, 2, 5, 3, 4, 5, 1, 7, 3, 2, 1 are equal.

ivIn trapezium ABCD, AB || CD; diagonals AC and BD intersect at point O, then \(AO \cdot OD = BO \cdot OC\).

vAB is a diameter of a circle with center O. P is any point on the circumference. If \(\angle POA = 120^\circ\), then \(\angle PBO + \angle POA = 140^\circ\).

viThe current price of a mobile phone is ₹20,000. If the price depreciates by 15% every year, the price of the mobile phone after 2 years will be ₹14,450.

4. Answer the following questions (any ten) [2 × 10 = 20]

iIf the simple interest on ₹8,000 for 2 years is ₹720, find the annual rate of interest.

iiIf the compound interest on a sum of money for 2 years is ₹512 and the simple interest for the same time and at the same rate is ₹500, what is the rate of interest?

iiiIn a partnership business, the ratio of the capitals of three persons is 3 : 5 : 8. If the profit of the first person is ₹60 less than the profit of the third person, what was the total profit in the business?

ivIf 2 is a root of the equations \(x^2 + bx + 12 = 0\) and \(x^2 + bx + q = 0\), find the value of \(q\).

vIf \(a+b=2\sqrt{5}\) and \(a-b=\sqrt{2}\), what is the value of \(a^2 + b^2\)?

viThe length of each of two parallel chords AB and CD is 16 cm. If the radius of the circle is 10 cm, find the distance between the two chords.

viiIn a circle, two chords PQ and PR are perpendicular to each other. If the length of the radius of the circle is \(r\) cm, find the length of the chord QR.

viiiIf \((4a+5b):(4c+5d) = (4a-5b):(4c-5d)\), prove that a, b, c, and d are in proportion.

ixIf the volume of a right circular cone is \(V\), base area is \(A\), and height is \(H\), find the value of \(\frac{AH}{V}\).

xIf \(\theta = 30^\circ\), what is the value of \(\sin 2\theta + \cos 2\theta\)?

xiFind the median and arithmetic mean of the numbers: 2, 2, 3, 4, 6, 5, 5, 8, 9, 10, 11.

xiiThe length, breadth, and height of a rectangular cuboid room are 5 m, 4 m, and 3 m respectively. Calculate the length of the longest rod that can be kept in that room.

5. Answer any one question [5 × 1 = 5]

iA person borrowed ₹500 at 6% simple interest per annum, and exactly two years after the first loan, borrowed a second loan of ₹900 at 4% simple interest per annum. How many years after taking the first loan will the interest on these two loans be equal, and what will be the total interest on the two loans at that time?

iiA and B started a joint business with ₹6,200 and ₹10,000 respectively. They decided that A would receive 20% of the profit for managing the business, and 10% of the remaining profit would be deposited as savings. The rest of the profit would be divided in proportion to their capitals. If the total profit at the end of the year is ₹45,000, how much money will A receive in total?

6. Answer any one question [3 × 1 = 3]

iSolve: \(\frac{2x+1}{2x-1} + \frac{2x-1}{2x+1} = 1 + \frac{7}{2x-1}, x \neq \frac{1}{2}\)

iiIf \(x \propto y\) and \(y \propto z\), prove that \(x+y+z \propto \sqrt{yz}+\sqrt{zx}+\sqrt{xy}\).

7. Solve any one question [3 × 1 = 3]

iFind the simplest value: \( (\sqrt{5}+\sqrt{3})\big( \frac{3\sqrt{3}}{\sqrt{5}+\sqrt{2}}-\frac{\sqrt{5}}{\sqrt{3}+\sqrt{2}}\big)\)

iiThe volume of a cone is in joint variation with the square of its base radius and its height. If the ratio of the base radii of two cones is 3:4 and the ratio of their heights is 6:5, find the ratio of their volumes.

8. Answer any one question [3 × 1 = 3]

iIf \(a, b, c, d\) are in continued proportion, prove that \((b-c)^2 + (c-a)^2 + (b-d)^2 = (a-d)^2\).

iiIf \(\frac{x}{b+c} = \frac{y}{c+a} = \frac{z}{a+b}\), then prove that \(\frac{a}{y+z-x} = \frac{b}{z+x-y} = \frac{c}{x+y-z}\).

9. Answer any one question [5 × 1 = 5]

iProve that the opposite angles of any cyclic quadrilateral are supplementary.

iiProve that if two tangents are drawn to a circle from an external point, the line segments joining the points of contact to the external point are equal in length and subtend equal angles at the center.

10. Answer any one question [3 × 1 = 3]

iIn an isosceles triangle ABC, AB = AC. A straight line parallel to BC intersects AB and AC at points P and Q respectively. Prove that BCQP is a cyclic quadrilateral.

iiA right-angled triangle ABC is drawn where \(\angle A\) is a right angle. P and Q are two points on sides AB and AC respectively. B, Q and C, P are joined. Prove that \(BQ^2 + PC^2 = BC^2 + PQ^2\).

11. Answer any one question [5 × 1 = 5]

iConstruct an isosceles triangle whose one angle is \(120^\circ\) and the length of each of the equal sides is 4 cm. Draw the circumcircle of the triangle.

iiConstruct a square equal in area to a rectangle with sides of length 6 cm and 3 cm. (Only traces of construction are required).

12. Solve any two questions [3 × 2 = 6]

iIf \(m = \tan\theta + \sin\theta\) and \(n = \tan\theta - \sin\theta\), prove that \(m^2 - n^2 = 4\sqrt{mn}\) (\(0^\circ < \theta < 90^\circ\)).

iiIf \(\angle A + \angle B = 90^\circ\), show that \(1 + \frac{\tan A}{\tan B} = \sec^2 A\).

iiiOne angle of a triangle is 60° and another is \(\frac{\pi}{6}\) radians. What is the measurement of the third angle in degrees?

13. Answer any one question [5 × 1 = 5]

iFrom the roof and the foot of a 16-meter high house, the angles of elevation of the top of a temple are observed to be 45° and 60° respectively. Find the height of the temple and its horizontal distance from the house.

iiThe angle of elevation of the top of a tower from a point A on the ground is 30°. After moving 20 meters from point A towards the foot of the tower to point B, the angle of elevation increases to 60°. Find the height of the tower and the distance of the tower from point A.

14. Answer any two questions [4 × 2 = 8]

iA large solid sphere is made by melting three solid copper spheres with radii of lengths 3 cm, 4 cm, and 5 cm. Calculate the length of the radius of the large sphere.

ii77 square meters of tarpaulin were required to make a right circular cone-shaped tent. If the slant height of the tent is 7 meters, calculate the base area of the tent.

iiiDetermine the ratio of the volumes of a solid cone, a solid hemisphere, and a solid cylinder having equal base diameters and equal heights.

15. Answer any two questions [4 × 2 = 8]

iCalculate the arithmetic mean from the following frequency distribution:

Class Limit	0-10	10-20	20-30	30-40	40-50	50-60	60-70
Frequency	4	7	10	15	10	8	5

iiFind the median of the data below:

Class Limit	1-5	6-10	11-15	16-20	21-25	26-30	31-35
No. of Workers	2	3	6	7	5	4	3

iiiDetermine the mode of the following frequency distribution:

Class Limit	3-6	6-9	9-12	12-15	15-18	18-21	21-24
Frequency	2	6	12	24	21	12	3

Prime Maths - Vinod Singh, 9038126497

Subject: Mathematics | Total Marks: 90 | Time: 3 Hours 15 Minutes

Madhyamik Practice Test Paper

Mathematics Test Examination Paper

Class X (Madhyamik)

3 Hours 15 Minutes

Full Marks: 90

West Bengal Board Format | Test Paper I - Vinod Singh

About This Examination Paper

This is a Mathematics Test Examination paper from Prime Maths, designed for Class X (Madhyamik) students. It follows the standard West Bengal Board format with a total of 90 marks and a duration of 3 hours and 15 minutes.

Arithmetic

Problems on simple interest, compound interest, and partnership business

Algebra

Quadratic equations, ratio and proportion, variation, and surds

Geometry

Theorems related to circles, tangents, circumcenters, and construction

Mensuration

Problems involving cubes, cylinders, cones, spheres, and cuboids

Trigonometry

Solving identities, trigonometric ratios, heights and distances problems

Statistics

Calculation of Mean, Median, and Mode from frequency distribution tables

This paper serves as excellent practice material for students preparing for their final Madhyamik examination, testing their understanding of both objective (MCQ, True/False, Fill in the blanks) and subjective (long answer) type questions.

MADHYAMIK PRACTICE TEST PAPER

Time: 3 Hours 15 Minutes  |  Full Marks: 90

1. Select the correct answer in each of the following cases: [1 × 6 = 6]

1. If the difference between compound interest and simple interest on a certain principal for 3 years at 20% compound interest rate per annum is ₹48, the principal will be:

   • a) ₹350
   • b) ₹375
   • c) ₹400
   • d) ₹425

2. For what value(s) of \(k\) will the quadratic equation \(2x^2 - kx + k = 0\) have equal roots?

   • a) Only 0
   • b) Only 8
   • c) 4
   • d) 0 and 8

3. If \(A:B=2:3\), \(B:C=5:8\), and \(C:D=6:7\), what is the value of \(A:D\)?

   • a) 2:7
   • b) 5:8
   • c) 7:2
   • d) 5:14

4. \(O\) is the circumcenter of \(\triangle ABC\). Given \(\angle BAC=85^\circ\) and \(\angle BCA=55^\circ\), the value of \(\angle AOC\) will be:

   • a) \(65^\circ\)
   • b) \(45^\circ\)
   • c) \(50^\circ\)
   • d) \(100^\circ\)

5. When two circles touch each other internally, what is the maximum number of common tangents that can be drawn?

   • a) 1
   • b) 3
   • c) 4
   • d) 2

6. The diameters of two solid cylinders are 8 cm and 12 cm respectively, and their heights are \(3(n + 1)\) cm and \((n + 3)\) cm respectively. If the volumes of the two cylinders are equal, what is the value of \(n\)?

   • a) 13
   • b) 7
   • c) 8
   • d) 5

2. Fill in the blanks (Any five): [1 × 5 = 5]

1. If \(P_1: P_2: P_3\) is the ratio of profits and \(t_1:t_2:t_3\) is the ratio of time, then the ratio of invested capital is ______________.
2. \(7\sqrt{\frac{101}{404}}\) is a ______________ number.
3. If \(x \propto z\) and \(y \propto z\), then \(x+y \propto\) ______________.
4. The value of \(\cos 28^\circ \csc 62^\circ + \tan 1^\circ \cot 89^\circ\) = ______________.
5. If the ratio of three consecutive angles of a cyclic quadrilateral is \(1 : 3 : 4\), the measure of the fourth angle = ______________.
6. If the volume of a cube is \(V\), the total surface area of the cube = ______________ square units.

3. Write True or False (Any five): [1 × 5 = 5]

1. A partnership business requires at least 4 people.
2. If the annual simple interest rate is 12%, the interest for \(8\frac{1}{3}\) years will be equal to the principal.
3. The two roots of the equation \(x^2 = -2024\) are real numbers.
4. In two triangles ABC and DEF, if \(\angle A= \angle D\), \(\angle B= \angle F\), and \(\angle C = \angle E\), then \(\frac{AB}{DE} = \frac{AC}{DF} = \frac{BC}{EF}\).
5. Two concentric circles of different radius will have one common tangent.
6. If the radius of the base of a right circular cone is halved and the height is doubled, the volume of the cone remains the same.

4. Answer the following questions (Any ten): [2 × 10 = 20]

1. If the simple interest on a principal for \(n\) years at \(r\%\) per annum is \(\frac{pnr}{25}\), show that the Principal = \(4p\).
2. If ₹8,000 is invested for 3 years at 10% compound interest per annum, how much interest will be earned at the end of 2 years?
3. In a business, A invested ₹1800 and B invested ₹1000 for 9 months. If both their profit shares are equal, for how long was A's money invested?
4. What is the ratio of the sum and product of the roots of the equation \(-x^2 - 6x + 2 = 0\)?
5. Rationalize the denominator: \(\frac{2\sqrt{2}-3}{\sqrt{2}}\).
6. AB and AC are two chords of a circle that are perpendicular to each other. If AB = 5 cm and AC = 12 cm, find the length of the radius of the circle.
7. AD is perpendicular to the hypotenuse BC of a right-angled triangle ABC. If BC = 16 cm and BD = 9 cm, find the length of AB.
8. The curved surface area of a right circular cone is \(\sqrt{5}\) times its base area. What is the ratio of the height to the radius of the cone?
9. If \(x = a \cos \theta + b \sin \theta\) and \(y = a \sin \theta - b \cos \theta\), what is the value of \(x^2 + y^2\)?
10. Find the median and arithmetic mean of the numbers: 1, 2, 3, 4, 6, 5, 7, 8, 9, 10, 11.
11. The length of the diagonal of a cube is \(4\sqrt{3}\) cm. Find the total surface area of the cube.

5. Answer any one question: [5 × 1 = 5]

1. The simple interest and annual compound interest on a certain capital for 2 years at the same rate are ₹4,000 and ₹4,100 respectively. Find the capital and the rate of interest.
2. The current number of students in all secondary education centers in a district is 3993. If the number of students increases by 10% compared to the previous year every year, determine what the number of students was in all secondary education centers of that district 3 years ago.

6. Solve any one question: [3 × 1 = 3]

1. \(\frac{1}{x} - \frac{1}{x+b} = \frac{1}{a} - \frac{1}{a+b}, \quad x \neq 0, -b.\)
2. If \(x = \frac{\sqrt{a+2} + \sqrt{a-2}}{\sqrt{a+2} - \sqrt{a-2}}\), where \(a > 2\), find the simplest value of \(x + \frac{1}{x}\).

7. Solve any one question: [3 × 1 = 3]

1. Find the simplest value: \(\frac{2(\sqrt{2}+\sqrt{3})}{\sqrt{3}+1} - \frac{2(\sqrt{2}-\sqrt{3})}{\sqrt{3}-1}\).
2. If it takes 9 days for 5 men to cultivate 10 acres of land, use the theory of variation to find how many days it will take for 25 men to cultivate 30 acres of land.

8. Answer any one question: [3 × 1 = 3]

1. If \(a:b = b:c\), prove that \(\frac{(a+b)}{(b+c)^2} = \frac{a^2+b^2}{b^2+c^2}\).
2. If \(\frac{a+b-c}{a+b} = \frac{b+c-a}{b+c} = \frac{c+a-b}{c+a}\) and \(a+b+c \neq 0\), prove that \(a = b = c\).

9. Answer any one question: [5 × 1 = 5]

1. Prove that the angle subtended by an arc of a circle at the center is double the angle subtended by it at any point on the remaining part of the circle.
2. Prove that if a perpendicular is drawn from the center of a circle to a chord which is not a diameter, the perpendicular bisects the chord.

10. Answer any one question: [3 × 1 = 3]

1. Prove that a cyclic trapezium is an isosceles trapezium and the lengths of its diagonals are equal.
2. In an isosceles triangle ABC, AB = AC and BE is perpendicular to AC from B. Prove that \(BC^2 = 2AC \times CE\).

11. Answer any one question: [5 × 1 = 5]

1. Draw a square figure with a side length of 14 cm.
2. Draw a right-angled triangle whose sides adjacent to the right angle have lengths 4 cm and 8 cm. Draw the incircle of the triangle. (Only traces of construction are required).

12. Solve any two questions: [3 × 2 = 6]

1. If \(m = \cos \theta - \sin \theta\) and \(n = \cos \theta + \sin \theta\), prove that \(\sqrt{\frac{m}{n}} + \sqrt{\frac{n}{m}} = \frac{2}{\sqrt{1-\tan^2 \theta}}\).
2. Find the value of \(x\): \(x \sin 60^\circ \cos^2 30^\circ = \frac{\tan^2 45^\circ \sec 60^\circ}{\csc 60^\circ}\).
3. The measures of three angles of a quadrilateral are \(\frac{\pi}{3}\), \(\frac{5\pi}{6}\), and \(90^\circ\). Find the circular measure (in radians) of the fourth angle.

13. Answer any one question: [5 × 1 = 5]

1. From an object on the roof of a house \(30\sqrt{3}\) meters high, the angles of depression of the top and bottom of a lamp post are \(30^\circ\) and \(60^\circ\) respectively. Find the height of the lamp post.
2. From the roof of a five-story building 18 meters high, the angle of elevation of the top of a monument is observed to be \(45^\circ\) and the angle of depression of the foot of the monument is observed to be \(60^\circ\). What is the height of the monument?

14. Answer any two questions: [4 × 2 = 8]

1. If the radius of a solid sphere and a solid right circular cylinder are equal and their volumes are also equal, calculate the ratio of the radius and height of the cylinder.
2. The volume of a right circular cone is \(100\pi\) cubic cm and its height is 12 cm. Calculate and write the slant height of the cone.
3. A rectangular pond is 20 m long and 18.5 m wide, and has water 3.2 m deep. How long will it take to irrigate the entire water of the pond using a pump that can irrigate 160 kiloliters of water per hour?

15. Answer any two questions: [4 × 2 = 8]

1. Determine the Arithmetic Mean from the following frequency distribution:

   Value:	Below 10	Below 20	Below 30	Below 40	Below 50	Below 60	Below 70	Below 80
   Frequency:	4	16	40	76	96	112	126	125

2. The frequency distribution of daily wages of 100 workers is as follows. Find the median of this distribution.

   Daily Wage (in ₹):	460-470	470-480	480-490	490-500	500-510	510-520	520-530
   No. of Workers:	6	4	29	23	16	10	2

3. Determine the Mode of the frequency distribution below:

   Class Limit:	0-5	5-10	10-15	15-20	20-25	25-30	30-35
   Frequency:	5	12	18	28	17	12	8

Problems on Algebra : ICSE and CBSE

Algebraic Identities Problem Set

Master Algebra through Practice Problems. Prime Maths

About This Problem Set

This collection of problems is designed for Class IX students following CBSE, ICSE, and State Board curricula. The problems cover fundamental algebraic identities, inequalities, factorization, and polynomial expansions. Work through these problems systematically to strengthen your understanding of algebraic concepts and their applications.

Problem 1

Problem 1 of 22

Click to reveal hint

Select a Problem

Practice these problems to master algebraic identities. Remember to attempt each problem before checking the hint!

Designed for Class IX Mathematics

Designed by Vinod Singh, 9038126497

Indices Interactive Chapter Test: ICSE and CBSE

Prime Maths

25 Question MCQ Chapter on Indices & Exponents for ICSE, CBSE, WBBSE and other State Boards

👨‍🏫 Author: Vinod Singh

📞 WA: +91-9038126497

Prime Maths

Indices & Exponents Quiz

Test your understanding of mathematical indices

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• This quiz contains 25 multiple choice questions.
• Select only one correct answer per question.
• Use the navigator to jump between questions.
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Question 1

\( \big( 125^{-4}\times 256^{-3/2}\big)^{-1/6} \times \big( (x^{2}y^{-4})^{1/3} \times \sqrt{y^{3}x^{-5}}\big)^{12}\times (x^{2})^{11}\)

A\( 10y^{2}\) B\( 100y^{2}\) C\( 10x^{2}\) D\( 100x^{2}\)

Question 2

The value of \( (5^{0} + 3^{0} + 2^{0}) ÷\bigg( \frac{2^{n}(2^{n-1})^n}{2^{n+1}.2^{n-1}} \times \bigg( \frac{\sqrt [3] {8^{n}}}{4}\bigg)^{-n} \bigg) \) is:

A0 B1 C2 D3

Question 3

Simplify: \( \big(\frac{1728}{729}\big)^{\frac{2}{3}}\times \frac{3^{2}}{\sqrt{144}}\times \sqrt{5} \)

A\( \frac{4\sqrt{5}}{3} \) B\( \frac{4\sqrt{5}}{7} \) C\( \frac{\sqrt{5}}{3} \) D\( \frac{7\sqrt{5}}{3} \)

Question 4

If \(5^{10x}=4900 \) and \( 2^{ \sqrt{y}}=25 \), the value of \(4^ { \sqrt{y}}\times 5^{5x-5} \) is:

A12 B16 C14 D13

Question 5

If \(\big( \frac{2a}{b}\big)^{2x-4} \) = \(\big( \frac{b}{2a}\big)^{2x-4} \), then the value of \( x \) is:

A2 B4 C3 D-2

Question 6

\( \big[1-\big(1-\big(1-n\big)^{-1}\big)^{-1} \big]^{-1} = \) where \(n \neq 0,1 \)

A\( n \) B\( 2n \) C\( -n \) D\( \frac{1}{n} \)

Question 7

If \( 2016 = 2^{a}\times 3^{b} \times 5^{c}, \) then the value of \(3^{a}\times 2^{-b} \times 5^{-c} \) is

A\( \frac{243}{4} \) B\( \frac{247}{4} \) C\( \frac{241}{4} \) D\( \frac{234}{4} \)

Question 8

If \( 2^{x}=3^{y}=6^{-z} \) then the value of \(\frac{1}{x}+\frac{1}{y}+\frac{1}{z} =\)

A0 B1 C-1 D\( \frac{1}{2} \)

Question 9

If \( a^{x}=b^{y}=c^{z} \) and \(b^{2} = ac \) then the value of \( \frac{2xz}{z+x}\) is

Ay B2x Cx D-x

Question 10

The simplest value of \( \frac{(x^{a+b})^{2} . (x^{b+c})^{2} . (x^{c+a})^{2}}{(x^{a}.x^{b}.x^{c})^{4}} \) is

A0 B1 C2 D-1

Question 11

Simplest value of \( \big( \frac{2^{-1}\times 3^{2}}{2^{2}\times 3^{-4}}\big)^{7/2} \times \big( \frac{2^{-2}\times 3^{3}}{2^{3}\times 3^{-5}}\big)^{-5/2} \) is

A12 B22 C2 D11

Question 12

The value of \(x\) satisfying \( 9^{x}-9^{x-1}=648 \) is

A3 B\( \frac{1}{3} \) C2 D-3

Question 13

The simplest value of \( \big[ 5\big(8^{1/3}+27^{1/3}\big)^{3}\big]^{\frac{1}{4}} \) is

A5 B6 C8 D12

Question 14

If \( y= x^{1/3}-x^{-1/3}, \) then \( y^{3}+3y = \)

A\( x-x^{-1} \) B\( x^{2} - x^{-2} \) C\( x^{2}+x^{-2} \) D\( x+x^{-1} \)

Question 15

If \( (10^{11}+25)^{2}-(10^{11}-25)^{2} = 10^{n}\) then the value of \( n \) is

A11 B13 C17 D7

Question 16

The least value of the expression \( 4^{x}+4^{1-x} \) is

A2 B4 C8 D16

Question 17

Value of \(x\) and \(y\) satisfying the equations \( 5^{x}-3^{y}=16; \quad 5^{x-1}+3^{y+1}=32\) are

A\( x=2 \quad y = 2 \) B\( x=2 \quad y = -2 \) C\( x=-2 \quad y = 2 \) D\( x=-2 \quad y = -2 \)

Question 18

If \(\big( x^{n^{2}}\big)^{n}= \big( x^{2^{n}}\big)^{2}\), then the value of \( \sqrt[n+1]{n^{3}} \) is

A2 B4 C1 D\( 2^{4} \)

Question 19

If \( x = 3+3^{2/3}+3^{1/3}\) then the value of \( x^3-9x^2+18x-12 \) is

A0 B1 C3 D-3

Question 20

If \( a^{x}=bc, \quad b^{y}= ca, \quad c^{z}=ab\), then the value of \(\frac{x}{x+1}+\frac{y}{y+1}+\frac{z}{z+1} \quad \) is

A1 B2 C0 D-1

Question 21

The value of \(x\) satisfying \( 4^{x}-3^{x-1/2}=3^{x+1/2}-2^{2x-1}\) is

A\( \frac{3}{2} \) B2 C\( \frac{1}{2} \) D\( \frac{2}{3} \)

Question 22

The value of \( \frac{\big(2^{2n}-3.2^{2n-2}\big)\big(3^{n}-2.3^{n-2}\big)}{3^{n-4}\big( 4^{n+3}-2^{2n}\big)} \quad \) is

A\( \frac{1}{4} \) B\( \frac{3}{4} \) C\( \frac{-3}{4}\) D\( \frac{1}{3} \)

Question 23

If \( a^{x}=b^{y}\) and \( b^{x}=a^{y} \quad ( ab \neq 1)\) then which of the following relation is true

A\( a=b^{2} \) B\( a=b \) C\( b=a^{2} \) D\( ab=2 \)

Question 24

Solve for \(x \) and \( y: \quad 2^{x}+2^{y}=12; \quad x+y= 5\)

A\( x=3, \quad y = 2 \) B\( x=2, \quad y = 3 \) C\( x=4, \quad y = 1\) D\( x=1, \quad y = 4 \)

Question 25

The value of \( \bigg( \frac{9^{n+1/4}.\sqrt{3.3^{n}}}{3.\sqrt{3^{-n}}}\bigg)^{1/n}\) is

A3 B9 C27 D81

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