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Mathematics - X

The Model Paper - 1

Year : 2024 - 25

Time - 1 Hour 0 minutes

Full Marks - 27

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1 ) If (x - 2) is a common factor of expression $x^2 +ax + b$ and $x^2 +cx + d$, then the value of $\frac{b-d}{c-a}$

2 ) If the slant height of a right circular cone is 15 cm and the length of the base diameter is 16 cm, then the lateral surface area of the cone is -

3 ) Find the coordinates of the points which divides the join of X (-1, 7) and Y (4, -3) in the ratio 7 : 2.

4 ) In an examination, a student gets 32% marks and fails by 20 marks. Another student gets 42% marks and get 30 marks more than the minimum pass marks.Find the maximum marks and pass percentage of marks.

5 ) Lemons are bought for 5 for a rupee. How many for a rupee should these be sold to gain 25% ?

6 ) Which ratio is greater 4 : 5 and 9 : 16 ?

7 ) When $ 4x^3 - 3x^2+7x+k $ is divided by x + 2, the remainder is -40. The value of k is

8 ) A number of two digits is equal to three times the sum of digits. Also the unit digit is one more than three times ten digit. The number is

9 ) If the sum of m terms of an A.P. is to the sum of n terms $m^2$ to $n^2$, then m th term is to n th is

10 ) If $\, x = a(sin \theta + cos \theta ) \, $ and $\, y = b(sin \theta - cos \theta ) \,$, then $\, \theta \,$ eliminant is

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Section-A contains 3 questions of 2 marks each.

Section-B contains 7 questions of 3 marks each.

1 )

If (x - 2) is a common factor of expression $x^2 +ax + b$ and $x^2 +cx + d$, then the value of $\frac{b-d}{c-a}$

1) 8

2) 5

3) 2

4) 3

2 )

If the slant height of a right circular cone is 15 cm and the length of the base diameter is 16 cm, then the lateral surface area of the cone is -

1) $80 \pi \, cm^2$

2) $100 \pi \, cm^2$

3) $130 \pi \, cm^2$

4) $120 \pi \, cm^2$

3 )

Find the coordinates of the points which divides the join of X (-1, 7) and Y (4, -3) in the ratio 7 : 2.

1) (-2, 3)

2) (-2, -3)

3) (2, -3)

4) (2, 3)

4 )

In an examination, a student gets 32% marks and fails by 20 marks. Another student gets 42% marks and get 30 marks more than the minimum pass marks.Find the maximum marks and pass percentage of marks.

1) Max marks = 600 and Pass% = 36

2) Max marks = 500 and Pass% = 34

3) Max marks = 500 and Pass% = 36

4) Max marks = 500 and Pass% = 35

5 )

Lemons are bought for 5 for a rupee. How many for a rupee should these be sold to gain 25% ?

1) 6 for a rupee

2) 3 for a rupee

3) 4 for a rupee

4) 5 for a rupee

6 )

Which ratio is greater 4 : 5 and 9 : 16 ?

1) 9 : 16

2) 4 : 5

3) both are equal

4) none of these

7 )

When $ 4x^3 - 3x^2+7x+k $ is divided by x + 2, the remainder is -40. The value of k is

1) 18

2) 15

3) 24

4) 12

8 )

A number of two digits is equal to three times the sum of digits. Also the unit digit is one more than three times ten digit. The number is

1) 47

2) 27

3) 37

4) 24

9 )

If the sum of m terms of an A.P. is to the sum of n terms $m^2$ to $n^2$, then m th term is to n th is

1) $\frac{2m-1}{2n+1}$

2) $\frac{2m-1}{2n-1}$

3) $\frac{2m+1}{2n-1}$

4) $\frac{2m-1}{-2n+1}$

10 )

If $\, x = a(sin \theta + cos \theta ) \, $ and $\, y = b(sin \theta - cos \theta ) \,$, then $\, \theta \,$ eliminant is

1) $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 2$

2) $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 2$

3) $- \frac{x^2}{a^2} + \frac{y^2}{b^2} = 2$

4) $- \frac{x^2}{a^2} - \frac{y^2}{b^2} = 2$

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1 ) 2 2 ) $120 \pi \, cm^2$ 3 ) (2, -3) 4 ) Max marks = 500 and Pass% = 36 5 ) 4 for a rupee 6 ) 9 : 16 7 ) 18 8 ) 27 9 ) $\frac{2m-1}{2n-1}$ 10 ) $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 2$

Mathematics - X

The Model Paper - 2

Year : 2024 - 25

Time - 1 marks 0 minutes

Full Marks - 27

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1 ) If $\alpha$ and $\beta$ are zeroes of the quadratic polynomial $x^2-6x+8$ then the value of $\alpha - \beta$ $(\alpha \gt \beta)$.

2 ) Keeping the radius of a right circular same, if the height of it is increased twice, the volume of it will be increased by

3 ) Find the ratio in which the point X (-6, h) divides the join of P (-4, 4) and Q (6, -1) and here hence find the value of h.

4 ) In an election, there were only two candidates. The winner obtained 56% votes and won by 8400 votes. Find the total number votes polled.

5 ) A T.V. set was sold at a gain of 15%. Had it been sold for 375 more , the gain would have been 20%. The cost price of set is

6 ) If a : d = 3 : 7, then the value of (5a + b) : (4a + 5b)

7 ) Using factor theorem, factorise the expression $2x^3 - 3x^2 - 11x +6$ .

8 ) If 2 is added to a number obtained by reversing a given number, then the digits are equal. The face value of each digit of resulting number is

9 ) The sum of p terms of an ap is q and the sum of q terms is p; then the sum of p + q terms is

10 ) If $\, \frac{sin \theta +cos \theta}{sin \theta -cos \theta} = 7 \,$ then the value of $\, tan \theta \, $ is

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Section-A contains 3 questions of 2 marks each.

Section-B contains 7 questions of 3 marks each.

1 )

If $\alpha$ and $\beta$ are zeroes of the quadratic polynomial $x^2-6x+8$ then the value of $\alpha - \beta$ $(\alpha \gt \beta)$.

1) 3

2) 2

3) 4

4) 5

2 )

Keeping the radius of a right circular same, if the height of it is increased twice, the volume of it will be increased by

1) 100%

2) 300%

3) 200%

4) 150%

3 )

Find the ratio in which the point X (-6, h) divides the join of P (-4, 4) and Q (6, -1) and here hence find the value of h.

1) 3 : 4; h = 2

2) 3 : 2; h = 1

3) 3 : 2; h = 2

4) 3 : 2; h = 3

4 )

In an election, there were only two candidates. The winner obtained 56% votes and won by 8400 votes. Find the total number votes polled.

1) 80000

2) 65000

3) 60000

4) 70000

5 )

A T.V. set was sold at a gain of 15%. Had it been sold for 375 more , the gain would have been 20%. The cost price of set is

1) Rs 7500

2) Rs 7000

3) Rs 8500

4) Rs 8000

6 )

If a : d = 3 : 7, then the value of (5a + b) : (4a + 5b)

1) 22 : 47

2) 22 : 53

3) 22 : 15

4) 21 : 47

7 )

Using factor theorem, factorise the expression $2x^3 - 3x^2 - 11x +6$ .

1) (x - 2)(x - 3)(2x - 1)

2) (x + 2)(x - 3)(2x - 1)

3) (x + 2)(x + 3)(2x - 1)

4) (x + 2)(x - 3)(2x + 1)

8 )

If 2 is added to a number obtained by reversing a given number, then the digits are equal. The face value of each digit of resulting number is

1) 8

2) 4

3) 7

4) 5

9 )

The sum of p terms of an ap is q and the sum of q terms is p; then the sum of p + q terms is

1) -p+q

2) -p-q

3) p-q

4) p+q

10 )

If $\, \frac{sin \theta +cos \theta}{sin \theta -cos \theta} = 7 \,$ then the value of $\, tan \theta \, $ is

1) $\frac{2}{3}$

2) $\frac{4}{3}$

3) $\frac{5}{3}$

4) $\frac{1}{3}$

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1 ) 2 2 ) 100% 3 ) 3 : 2; h = 2 4 ) 70000 5 ) Rs 7500 6 ) 22 : 47 7 ) (x + 2)(x - 3)(2x - 1) 8 ) 7 9 ) -p-q 10 ) $\frac{4}{3}$

Mathematics - X

The Model Paper - 3

Year : 2024 - 25

Time - 1 marks 0 minutes

Full Marks - 27

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1 ) Sum of zeros of following of $x^2 + \frac{1}{2} = 3x$ is

2 ) if the length of the radius of cone is $\frac{r}{2}$ unit and slant height of it is 2l unit, then the total surface is -

3 ) Find the ratio in which the line segment PQ, where P (4, -2) and Q (1, 3), is divided by the x-axis.

4 ) On decreasing the price of a tea-set by 8%, it becomes Rs 317.40. What was its original price ?

5 ) Find a single discount equivalent to two successive discounts of 20% and 10%

6 ) Divide Rs 1100 among A, B and C so that A shall receive $\frac{3}{7}$ of what B and C together receive and B may receive $\frac{2}{9}$ of what A and C receive. what are their shares ?

7 ) If (x - 2) and (x + 3) are factors of $2x^3 + mx^2 -11x + n$, then the value of m and n are

8 ) A tank of 10,000 litres capacity is being filled with gasoline by two pumps. The second of which supplies 10 litres less a minute than first. In 10 minutes, the tank is half full. How many litres of gasoline does each pump pour in ?

9 ) If the sum of 7 terms of an A. P. is 49 and the sum of 17 terms of that A. P. is 289, then the sum of n terms is

10 ) The value of $\, \frac{cos A}{1-tan A} + \frac{sin A}{1-cot A}$

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Section-A contains 3 questions of 2 marks each.

Section-B contains 7 questions of 3 marks each.

1 )

Sum of zeros of following of $x^2 + \frac{1}{2} = 3x$ is

1) $\sqrt{2}$

2) $\sqrt{7}$

3) $\sqrt{3}$

4) $\sqrt{5}$

2 )

if the length of the radius of cone is $\frac{r}{2}$ unit and slant height of it is 2l unit, then the total surface is -

1) $2 \pi r (l+\frac{1}{4})$ sq. units

2) $\pi r (l+\frac{1}{3})$ sq. units

3) $\pi r (l+\frac{1}{4})$ sq. units

4) $\pi r (l+r)$ sq. units

3 )

Find the ratio in which the line segment PQ, where P (4, -2) and Q (1, 3), is divided by the x-axis.

1) 2 : 5

2) 2 : 3

3) 3 : 4

4) 3 : 5

4 )

On decreasing the price of a tea-set by 8%, it becomes Rs 317.40. What was its original price ?

1) Rs 350

2) Rs 345

3) Rs 300

4) Rs 400

5 )

Find a single discount equivalent to two successive discounts of 20% and 10%

1) 20%

2) 30%

3) 24%

4) 28%

6 )

Divide Rs 1100 among A, B and C so that A shall receive $\frac{3}{7}$ of what B and C together receive and B may receive $\frac{2}{9}$ of what A and C receive. what are their shares ?

1) The share of A = Rs 330, The share of B = Rs 250 and The share of C = Rs 570

2) The share of A = Rs 330, The share of B = Rs 200 and The share of C = Rs 560

3) The share of A = Rs 330, The share of B = Rs 200 and The share of C = Rs 570

4) The share of A = Rs 300, The share of B = Rs 200 and The share of C = Rs 570

7 )

If (x - 2) and (x + 3) are factors of $2x^3 + mx^2 -11x + n$, then the value of m and n are

1) m=-3, n=-6

2) m=3, n=6

3) m=-3, n=-6

4) m=3, n=-6

8 )

A tank of 10,000 litres capacity is being filled with gasoline by two pumps. The second of which supplies 10 litres less a minute than first. In 10 minutes, the tank is half full. How many litres of gasoline does each pump pour in ?

1) 2550 l, 2450 l

2) 2550 l, 2440 l

3) 2550 l, 2460 l

4) 2550 l, 2430 l

9 )

If the sum of 7 terms of an A. P. is 49 and the sum of 17 terms of that A. P. is 289, then the sum of n terms is

1) $n^2$

2) $n^3$

3) $2n$

4) $3n$

10 )

The value of $\, \frac{cos A}{1-tan A} + \frac{sin A}{1-cot A}$

1) cos A - sin A

2) cos A + sin A

3) -cos A + sin A

4) -cos A - sin A

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1 ) $\sqrt{5}$ 2 ) $\pi r (l+\frac{1}{4})$ sq. units 3 ) 2 : 3 4 ) Rs 345 5 ) 28% 6 ) The share of A = Rs 330, The share of B = Rs 200 and The share of C = Rs 570 7 ) m=3, n=-6 8 ) 2550 l, 2450 l 9 ) $n^2$ 10 ) cos A + sin A

Mathematics - X

The Model Paper - 4

Year : 2024 - 25

Time - 1 marks 0 minutes

Full Marks - 27

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1 ) If x be real, the maximum value of $7+10x-5x^2$

2 ) If each of radius and height of a cone is increased by twice of its length of its length, then the volume of it will be

3 ) Let M (-3, 5) be the middle point of the line segment XY whose one end has the coordinates (0, 0). Find the coordinates of the other end.

4 ) A number is increased by 20% and then increased number is decreased by 20%. Find the net increase or decrease percent.

5 ) Find the single discount equivalent to two successive discounts of 20% and 5%.

6 ) Arrange the following ratios in ascending order of magnitude : 2 : 3, 11 : 15, 7 : 12 and 9 : 16

7 ) When divided by x - 3, the polynomials $x^3-px^2+x+6 $ and $2x^3-x^2-(p+3)x-6$ leave the same remainder. The value of p is

8 ) If 1 is added to each of of the two given numbers, their ratio becomes 1 : 2 and 5 is subtracted from each, the ratio becomes 5 : 11. The numbers are

9 ) The pth term of an A.P. is q and the q th term is p, then the m th term is

10 ) If $\, (1+4x^2)cos A = 4x \,$ then the value of cosec A + cot A is

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Section-A contains 3 questions of 2 marks each.

Section-B contains 7 questions of 3 marks each.

1 )

If x be real, the maximum value of $7+10x-5x^2$

1) 20

2) 10

3) 8

4) 12

2 )

If each of radius and height of a cone is increased by twice of its length of its length, then the volume of it will be

1) 6 times of the previous one

2) 8 times of the previous one

3) 5 times of the previous one

4) 4 times of the previous one

3 )

Let M (-3, 5) be the middle point of the line segment XY whose one end has the coordinates (0, 0). Find the coordinates of the other end.

1) (6, -10)

2) (-6, -10)

3) (6, 10)

4) (-6, 10)

4 )

A number is increased by 20% and then increased number is decreased by 20%. Find the net increase or decrease percent.

1) 5%

2) 6%

3) 4%

4) none of these

5 )

Find the single discount equivalent to two successive discounts of 20% and 5%.

1) 20%

2) 25%

3) 15%

4) 24%

6 )

Arrange the following ratios in ascending order of magnitude : 2 : 3, 11 : 15, 7 : 12 and 9 : 16

1) 9 : 16 $\lt$ 2 : 3 $\lt$ 7 : 12 $\lt$ 11 : 15

2) 7 : 12 $\lt$ 9 : 16 $\lt$ 2 : 3 $\lt$ 11 : 15

3) 9 : 16 $\lt$ 7 : 12 $\lt$ 11 : 15 $\lt$ 2 : 3

4) 9 : 16 $\lt$ 7 : 12 $\lt$ 2 : 3 $\lt$ 11 : 15

7 )

When divided by x - 3, the polynomials $x^3-px^2+x+6 $ and $2x^3-x^2-(p+3)x-6$ leave the same remainder. The value of p is

1) 2

2) 1

3) 3

4) -1

8 )

If 1 is added to each of of the two given numbers, their ratio becomes 1 : 2 and 5 is subtracted from each, the ratio becomes 5 : 11. The numbers are

1) 35 and 71

2) 25 and 71

3) 45 and 71

4) 35 and 70

9 )

The pth term of an A.P. is q and the q th term is p, then the m th term is

1) p + q + m

2) p - q - m

3) p - q + m

4) p + q - m

10 )

If $\, (1+4x^2)cos A = 4x \,$ then the value of cosec A + cot A is

1) $\frac{1+2x}{-1-2x}$

2) $\frac{1-2x}{1+2x}$

3) $\frac{-1+2x}{1-2x}$

4) $\frac{1+2x}{1-2x}$

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1 ) 12 2 ) 8 times of the previous one 3 ) (-6, 10) 4 ) 4% 5 ) 24% 6 ) 9 : 16 $\lt$ 7 : 12 $\lt$ 2 : 3 $\lt$ 11 : 15 7 ) 1 8 ) 35 and 71 9 ) p + q - m 10 ) $\frac{1+2x}{1-2x}$

Mathematics - X

The Model Paper - 5

Year : 2024 - 25

Time - 1 marks 0 minutes

Full Marks - 27

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1 ) If the equation $ax^2+bx+c=0$ and $x^2+6x+4=0$ have both roots common, then a : b : c is

2 ) The curved surface area of right circular cone is $\sqrt{5}$ times of its base area. Then, height : radius = ?

3 ) In what ratio is the line segment joining X (2, -3) and Y (5, 6) divides by the x-axis? Also, find the coordinates of the point of division.

4 ) In an election, there were only two candidates. The inner obtained 56% votes and won by 8400 votes. Find the total number of votes polled.

5 ) The cost price of 12 fans is equal to selling price of 16 fans. The gain or loss per cent is

6 ) If a is one forth of b and b is half of c, then a : c is

7 ) If (x + 2) and (x + 4) are factors of $3x^3 +ax^2 -6x-b $, Then the values of a and b are

8 ) At present, the ratio of the ages of two brothers is 2 : 1. After 5 years, the ratio of their ages will be 3 : 2. After how many years from now, their ages will bear a ratio 4 : 3 ?

9 ) The sum of n terms of two arithmetic series are in the ratio of (7n + 1) : (4n + 27), then the ratio of their terms is

10 ) The value of $ \, \frac{1 + cos A - sin A}{1 + cos A + sin A} \,$ is

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Section-A contains 3 questions of 2 marks each.

Section-B contains 7 questions of 3 marks each.

1 )

If the equation $ax^2+bx+c=0$ and $x^2+6x+4=0$ have both roots common, then a : b : c is

1) 1 : 6 : 4

2) 2 : 3 : 4

3) 1 : 3 : 4

4) 1 : 6 : 3

2 )

The curved surface area of right circular cone is $\sqrt{5}$ times of its base area. Then, height : radius = ?

1) 2 : 1

2) 2 : 3

3) 3 : 1

4) none of these

3 )

In what ratio is the line segment joining X (2, -3) and Y (5, 6) divides by the x-axis? Also, find the coordinates of the point of division.

1) 1 : 2; (2, 0)

2) 1 : 2; (3, 0)

3) 1 : 2; (3, -1)

4) 1 : 2; (3, 1)

4 )

In an election, there were only two candidates. The inner obtained 56% votes and won by 8400 votes. Find the total number of votes polled.

1) 65000

2) 60000

3) 70000

4) 75000

5 )

The cost price of 12 fans is equal to selling price of 16 fans. The gain or loss per cent is

1) gain=20%

2) loss=20%

3) gain=25%

4) loss=25%

6 )

If a is one forth of b and b is half of c, then a : c is

1) 4 : 1

2) 1 : 4

3) 8 : 1

4) 1 : 8

7 )

If (x + 2) and (x + 4) are factors of $3x^3 +ax^2 -6x-b $, Then the values of a and b are

1) a = 18, b = 40

2) a = 15, b = 40

3) a = 13, b = 40

4) a = 10, b = 40

8 )

At present, the ratio of the ages of two brothers is 2 : 1. After 5 years, the ratio of their ages will be 3 : 2. After how many years from now, their ages will bear a ratio 4 : 3 ?

1) 6 years

2) 12 years

3) 8 years

4) 10 years

9 )

The sum of n terms of two arithmetic series are in the ratio of (7n + 1) : (4n + 27), then the ratio of their terms is

1) 4 : 3

2) 3 : 4

3) 2 : 3

4) 4 : 5

10 )

The value of $ \, \frac{1 + cos A - sin A}{1 + cos A + sin A} \,$ is

1) $ \, \frac{sin A }{1 - cos A} \,$

2) $ \, \frac{sin A }{1 + cos A} \,$

3) $ \, \frac{cos A }{1 + sin A} \,$

4) $ \, \frac{cos A }{1 - sin A} \,$

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1 ) 1 : 6 : 4 2 ) 2 : 1 3 ) 1 : 2; (3, 0) 4 ) 70000 5 ) loss=25% 6 ) 1 : 8 7 ) a = 13, b = 40 8 ) 10 years 9 ) 4 : 3 10 ) $ \, \frac{cos A }{1 + sin A} \,$

Mathematics - X

The Model Paper - 6

Year : 2024 - 25

Time - 1 marks 0 minutes

Full Marks - 27

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1 ) The zeroes of the polynomial $4x^2 -3x - 1$

2 ) If the volume of right circular cone is V cubic units, base area is A sq. units and height is H units, then the value of $\frac{AH}{V}$.

3 ) The coordinates of the midpoint of the line segment AB are (1, -2). The coordinate of A are (-3, 2). Find the coordinate of B.

4 ) A man gave 35% of his money to his elder son and 40% of the remainder to younger. Now he is son left with Rs 11700. How much money had he ?

5 ) By selling 33 metres of cloth, one gains the selling price of 11 metres. The gain per cent is

6 ) The ratio between two numbers is 3 : 5 and their sum is 40. The larger of two number is

7 ) The expressions $f(x)=x^2 +n^2x+m$, $h(x)=x^2 +m^2x+n$ have x+a as a common factor. Then, find the relation among a, m and n.

8 ) A told B, $^"$ When I was as old as you are now, then your age was four years less than half of my present age$^"$. If the sum of present age of A and B is 61 years. B's present age is

9 ) The sum of four integers in A.P. is 24, and their product is 945, the possible first integer is

10 ) If $\, sin \theta + cosec \theta = 2 \,$ then the value of $\, sin^{10} \theta + cosec^{10} \theta \, $ is

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Section-A contains 3 questions of 2 marks each.

Section-B contains 7 questions of 3 marks each.

1 )

The zeroes of the polynomial $4x^2 -3x - 1$

1) 1, $\frac{1}{4}$

2) 1, $- \frac{1}{4}$

3) 1, $- \frac{3}{4}$

4) 1, $- \frac{1}{3}$

2 )

If the volume of right circular cone is V cubic units, base area is A sq. units and height is H units, then the value of $\frac{AH}{V}$.

1) 2

2) 3

3) 5

4) 4

3 )

The coordinates of the midpoint of the line segment AB are (1, -2). The coordinate of A are (-3, 2). Find the coordinate of B.

1) (-5, 6)

2) (5, -6)

3) (-5, -6)

4) (5, 6)

4 )

A man gave 35% of his money to his elder son and 40% of the remainder to younger. Now he is son left with Rs 11700. How much money had he ?

1) Rs 25000

2) Rs 45000

3) Rs 40000

4) Rs 30000

5 )

By selling 33 metres of cloth, one gains the selling price of 11 metres. The gain per cent is

1) 40%

2) 50%

3) 55%

4) 45%

6 )

The ratio between two numbers is 3 : 5 and their sum is 40. The larger of two number is

1) 30

2) 25

3) 28

4) 35

7 )

The expressions $f(x)=x^2 +n^2x+m$, $h(x)=x^2 +m^2x+n$ have x+a as a common factor. Then, find the relation among a, m and n.

1) a(m+n)+2=0

2) a(m-n)+1=0

3) a(m+n)-1=0

4) a(m+n)+1=0

8 )

A told B, $^"$ When I was as old as you are now, then your age was four years less than half of my present age$^"$. If the sum of present age of A and B is 61 years. B's present age is

1) 30

2) 24

3) 25

4) 22

9 )

The sum of four integers in A.P. is 24, and their product is 945, the possible first integer is

1) 9

2) 5

3) 3

4) 7

10 )

If $\, sin \theta + cosec \theta = 2 \,$ then the value of $\, sin^{10} \theta + cosec^{10} \theta \, $ is

1) 0

2) 1

3) 2

4) 3

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1 ) 1, $- \frac{1}{4}$ 2 ) 3 3 ) (5, -6) 4 ) Rs 30000 5 ) 50% 6 ) 25 7 ) a(m+n)+1=0 8 ) 25 9 ) 3 10 ) 2

Mathematics - X

The Model Paper - 7

Year : 2024 - 25

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Full Marks - 27

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1 ) Find the value of k for which x = 3 is a zero of the quadratic polynomial $f(x) = (k + 2)x^2 -kx + 6$ .

2 ) A conical tent , 6 m in diameter and 4 m high, is made of canvas. How many such tent can be made from 100 m long and 100 cm wide canvas, allowing 10% for eastage ?

3 ) Find the ratio in which the line segment PQ, where P (-5, 2) and Q (2, 3), is divided by the y-axis.

4 ) Rice is costlier than wheat by 20%. By what percent is wheat cheaper than rice ?

5 ) A man sold a toaster at a profit of 10%. had it purchased it for 5% less and sold for Rs 56 more, he could have gained 25%. For how much did he buy it ?

6 ) The numbers are in the ratio 7 : 8. On adding 3 to the first and 8 to the second. their ratio becomes 3 : 4. The numbers are

7 ) Solve : $\frac{x}{x+1} + \frac{x+1}{x} = \frac{34}{15}$

8 ) The sum of two numbers is 1000 and difference between their squares is 256000. The numbers are

9 ) The 4th, 42nd and last terms of an A.P. are 0, -95 and -125 respectively, then the first term and number of terms are

10 ) If $\, x= a sin \theta \,$ and $\, y= b tan \theta \,$ then the value of $\, \frac{a^2}{x^2} - \frac{b^2}{y^2} \, $ is

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Section-A contains 3 questions of 2 marks each.

Section-B contains 7 questions of 3 marks each.

1 )

Find the value of k for which x = 3 is a zero of the quadratic polynomial $f(x) = (k + 2)x^2 -kx + 6$ .

1) -4

2) 4

3) -5

4) 5

2 )

A conical tent , 6 m in diameter and 4 m high, is made of canvas. How many such tent can be made from 100 m long and 100 cm wide canvas, allowing 10% for eastage ?

1) 20

2) 21

3) 24

4) 15

3 )

Find the ratio in which the line segment PQ, where P (-5, 2) and Q (2, 3), is divided by the y-axis.

1) 5 : 4

2) 5 : 1

3) 5 : 3

4) 5 : 2

4 )

Rice is costlier than wheat by 20%. By what percent is wheat cheaper than rice ?

1) $16\frac{1}{3}%$

2) $16\frac{2}{3}%$

3) 15%

4) 20%

5 )

A man sold a toaster at a profit of 10%. had it purchased it for 5% less and sold for Rs 56 more, he could have gained 25%. For how much did he buy it ?

1) Rs 600

2) Rs 640

3) Rs 660

4) Rs 540

6 )

The numbers are in the ratio 7 : 8. On adding 3 to the first and 8 to the second. their ratio becomes 3 : 4. The numbers are

1) 21 and 24

2) 21 and 25

3) 20 and 24

4) 24 and 30

7 )

Solve : $\frac{x}{x+1} + \frac{x+1}{x} = \frac{34}{15}$

1) $ \frac{5}{2}$, $- \frac{3}{2}$

2) $- \frac{5}{2}$, $- \frac{3}{2}$

3) $- \frac{5}{2}$, $ \frac{3}{2}$

4) $ \frac{5}{2}$, $ \frac{3}{2}$

8 )

The sum of two numbers is 1000 and difference between their squares is 256000. The numbers are

1) 628, 372

2) 628, 362

3) 628, 370

4) 638, 372

9 )

The 4th, 42nd and last terms of an A.P. are 0, -95 and -125 respectively, then the first term and number of terms are

1) first term=$5 \frac{1}{2}$ and number of terms=54

2) first term=$7 \frac{1}{2}$ and number of terms=54

3) first term=$3 \frac{1}{2}$ and number of terms=54

4) first term=$9 \frac{1}{2}$ and number of terms=54

10 )

If $\, x= a sin \theta \,$ and $\, y= b tan \theta \,$ then the value of $\, \frac{a^2}{x^2} - \frac{b^2}{y^2} \, $ is

1) 2

2) -1

3) 0

4) 1

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1 ) -4 2 ) 21 3 ) 5 : 2 4 ) $16\frac{2}{3}%$ 5 ) Rs 640 6 ) 21 and 24 7 ) $- \frac{5}{2}$, $ \frac{3}{2}$ 8 ) 628, 372 9 ) first term=$7 \frac{1}{2}$ and number of terms=54 10 ) 1

Mathematics - X

The Model Paper - 8

Year : 2024 - 25

Time - 1 marks 0 minutes

Full Marks - 27

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1 ) If the difference of the roots of the equation $x^2 + kx +7 = 0$ is 6 then possible values of k are

2 ) The height of a right circular cone is 12 cm and its volume is $100 \pi \, cm^3$. the length of the cone is -

3 ) Find the coordinates of the points which divides the join of P (-1, 7) and Q (4, -3) in the ratio 2 : 3.

4 ) The salary of an officer is increased by 20%. By what percent should the new salary be reduced to restore the original salary ?

5 ) A table was sold for Rs 2142 at a gain 5%. At what price should it be sold to gain 10% ?

6 ) The age of husband exceeds that of his wife by six years. 10 years ago, the ratio of their ages was 5 : 4. The present age of the husband is

7 ) Solve : $\frac{x^2-4x+5}{4x-5} = \frac{x^2-7x-10}{7x+10}$, where $x \ne 0$

8 ) A two-digited number is formed by either substracting 17 from nine times the sum of the digits or by adding 21 to 13 times the difference of the digits. The number is

9 ) If the sum of n terms of an A.P. is $ 2n +3n^2$, then the r th term is

10 ) If $\, \frac{sin^4 \alpha}{a} + \frac{cos^4 \alpha}{b} = \frac{1}{a+b} \,$ then the value $ \, sin \alpha =$

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Section-A contains 3 questions of 2 marks each.

Section-B contains 7 questions of 3 marks each.

1 )

If the difference of the roots of the equation $x^2 + kx +7 = 0$ is 6 then possible values of k are

1) $\pm 5$

2) $\pm 8$

3) $\pm 4$

4) $\pm 3$

2 )

The height of a right circular cone is 12 cm and its volume is $100 \pi \, cm^3$. the length of the cone is -

1) 3 cm

2) 6 cm

3) 8 cm

4) 5 cm

3 )

Find the coordinates of the points which divides the join of P (-1, 7) and Q (4, -3) in the ratio 2 : 3.

1) (1, 3)

2) (1, 2)

3) (2, 3)

4) (2, 1)

4 )

The salary of an officer is increased by 20%. By what percent should the new salary be reduced to restore the original salary ?

1) $16%$

2) $18\frac{2}{3}%$

3) $14\frac{2}{3}%$

4) $16\frac{2}{3}%$

5 )

A table was sold for Rs 2142 at a gain 5%. At what price should it be sold to gain 10% ?

1) Rs 2250

2) Rs 2254

3) Rs 2244

4) Rs 2240

6 )

The age of husband exceeds that of his wife by six years. 10 years ago, the ratio of their ages was 5 : 4. The present age of the husband is

1) 45 years

2) 35 years

3) 40 years

4) 30 years

7 )

Solve : $\frac{x^2-4x+5}{4x-5} = \frac{x^2-7x-10}{7x+10}$, where $x \ne 0$

1) x = -5

2) x = -4

3) x = -8

4) x = 5

8 )

A two-digited number is formed by either substracting 17 from nine times the sum of the digits or by adding 21 to 13 times the difference of the digits. The number is

1) 27

2) 73

3) 67

4) 76

9 )

If the sum of n terms of an A.P. is $ 2n +3n^2$, then the r th term is

1) 6r - 1

2) 6r + 1

3) 5r - 1

4) 5r + 1

10 )

If $\, \frac{sin^4 \alpha}{a} + \frac{cos^4 \alpha}{b} = \frac{1}{a+b} \,$ then the value $ \, sin \alpha =$

1) $\sqrt{\frac{b}{a+b}}$

2) $\sqrt{\frac{a}{a+b}}$

3) $\sqrt{\frac{a}{b}}$

4) $\sqrt{\frac{b}{a}}$

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1 ) $\pm 8$ 2 ) 5 cm 3 ) (1, 3) 4 ) $16\frac{2}{3}%$ 5 ) Rs 2244 6 ) 40 years 7 ) x = -5 8 ) 73 9 ) 6r - 1 10 ) $\sqrt{\frac{a}{a+b}}$

Mathematics - X

The Model Paper - 9

Year : 2024 - 25

Time - 1 marks 0 minutes

Full Marks - 27

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1 ) Find the value(s) of k so that quadratic polynomial $kx^2 + x + k$ has equal zeros.

2 ) Find the total surface areaof a cone of 6 cm radius and 8 cm slant height.

3 ) If the point (p, q) is the middle point of the line segment joining the points P (7, -4) and Q (-1, 2) then find p and q.

4 ) If the price of sugar is raised by 25%, by how much percent a housewife should reduce the consumption of sugar so as not to increase the expenditure ?

5 ) Apples are bought at 2 for a rupee and sold at 5 for Rs 3. The gain per cent is

6 ) A bag contains 50P, 25P and 10P coins in the ratio 5 : 9 : 4 amounting to Rs 206. The numbers of coins of each type are

7 )

8 ) Ramesh has some goats and hens in his shed. Upon counting Ramesh found that the total number of legs is 112 and the total number of heads is 40. The number of hens in his shed is

9 ) The 2nd 31st and last terms of an A.P. are $7 \frac{3}{4}$ , $ \frac{1}{2}$ and $-6 \frac{1}{2}$ respectively, then the first term and number of terms are

10 ) If $\, cos \theta - sin \theta = sqrt{2} sin \theta \,$ then the value of $\, cos \theta + sin \theta \, $ is

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Section-A contains 3 questions of 2 marks each.

Section-B contains 7 questions of 3 marks each.

1 )

Find the value(s) of k so that quadratic polynomial $kx^2 + x + k$ has equal zeros.

1) $\pm \frac{3}{4}$

2) $\pm \frac{1}{3}$

3) $\pm \frac{1}{4}$

4) $\pm \frac{1}{2}$

2 )

Find the total surface areaof a cone of 6 cm radius and 8 cm slant height.

1) $264 cm^2$

2) $260 cm^2$

3) $254 cm^2$

4) $272 cm^2$

3 )

If the point (p, q) is the middle point of the line segment joining the points P (7, -4) and Q (-1, 2) then find p and q.

1) p = 3, q = -1

2) p = -3, q = -1

3) p = -3, q = -1

4) p = 3, q = 1

4 )

If the price of sugar is raised by 25%, by how much percent a housewife should reduce the consumption of sugar so as not to increase the expenditure ?

1) 20%

2) 25%

3) 15%

4) none of these

5 )

Apples are bought at 2 for a rupee and sold at 5 for Rs 3. The gain per cent is

1) 20%

2) 25%

3) 15%

4) none of these

6 )

A bag contains 50P, 25P and 10P coins in the ratio 5 : 9 : 4 amounting to Rs 206. The numbers of coins of each type are

1) No. of 50P coins = 220, No. of 25P coins = 360 and No. of 10P coins = 160

2) No. of 50P coins = 200, No. of 25P coins = 360 and No. of 10P coins = 160

3) No. of 50P coins = 200, No. of 25P coins = 340 and No. of 10P coins = 160

4) No. of 50P coins = 200, No. of 25P coins = 360 and No. of 10P coins = 180

7 )

8 )

Ramesh has some goats and hens in his shed. Upon counting Ramesh found that the total number of legs is 112 and the total number of heads is 40. The number of hens in his shed is

1) 20

2) 24

3) 30

4) 28

9 )

The 2nd 31st and last terms of an A.P. are $7 \frac{3}{4}$ , $ \frac{1}{2}$ and $-6 \frac{1}{2}$ respectively, then the first term and number of terms are

1) 1st term=8, number of terms=60

2) 1st term=8, number of terms=49

3) 1st term=6, number of terms=59

4) 1st term=8, number of terms=59

10 )

If $\, cos \theta - sin \theta = sqrt{2} sin \theta \,$ then the value of $\, cos \theta + sin \theta \, $ is

1) $sqrt{2} cos \theta \,$

2) $sqrt{2} sin \theta \,$

3) $sqrt{3} cos \theta \,$

4) $- sqrt{2} cos \theta \,$

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1 ) $\pm \frac{1}{4}$ 2 ) $264 cm^2$ 3 ) p = 3, q = -1 4 ) 20% 5 ) 20% 6 ) No. of 50P coins = 200, No. of 25P coins = 360 and No. of 10P coins = 160 8 ) 24 9 ) 1st term=8, number of terms=59 10 ) $sqrt{2} cos \theta \,$

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The Model Paper - 10

Year : 2024 - 25

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Full Marks - 27

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1 ) If (x + 2)(x + 5) is H. C. F. of polynomials $(x + 2)(x^2 + 6x + a)$ and $(x + 5)(x^2 + 8x + b)$. Then the values of a and b are

2 ) If the ratio of the volumes of two right circular cones is 1 : 4 and the ratio of their radii of the bases is 4 : 5, then the ratio of the heights is -

3 ) Find the coordinates of points of trisection of the line segment joining the point (6, -9) and the origin.

4 ) Wheat is cheaper than rice by 20%. By what percent is rice costlier than wheat ?

5 ) A sells a bicycle to Bat a profit of 20% and B sells to C at a profit of 5%. If C pays Rs 1890, what did A pay for it ?

6 ) There are two classes A and B. If 10 students leaves class A and join B, then the ratio of the number of students in class A and in class B would reverse. Then difference between the number of students in class A and class B is

7 )

8 ) One man and two boys can do a piece of work in 12 days, which could be done in 6 days by three men and one boy. How long would it take one man to do it ?

9 ) If s = n(5n - 3), then the pth term is

10 ) If $\, tan x = \frac{sin y - cos y}{sin y + cos y}$, then the value of sin y + cos y is

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Section-A contains 3 questions of 2 marks each.

Section-B contains 7 questions of 3 marks each.

1 )

If (x + 2)(x + 5) is H. C. F. of polynomials $(x + 2)(x^2 + 6x + a)$ and $(x + 5)(x^2 + 8x + b)$. Then the values of a and b are

1) a=5, b=12

2) a=5, b=10

3) a=5, b=15

4) a=5, b=20

2 )

If the ratio of the volumes of two right circular cones is 1 : 4 and the ratio of their radii of the bases is 4 : 5, then the ratio of the heights is -

1) 1 : 5

2) 5 : 9

3) 25 : 64

4) 25 : 16

3 )

Find the coordinates of points of trisection of the line segment joining the point (6, -9) and the origin.

1) $(\frac{4}{3}, \, - \frac{4}{3} )$, $(\frac{8}{3}, \, - \frac{8}{3} )$

2) $(\frac{4}{3}, \, \frac{4}{3} )$, $(\frac{8}{3}, \, - \frac{8}{3} )$

3) $(\frac{4}{3}, \, - \frac{4}{3} )$, $(\frac{8}{3}, \, \frac{8}{3} )$

4) $(- \frac{4}{3}, \, - \frac{4}{3} )$, $(\frac{8}{3}, \, - \frac{8}{3} )$

4 )

Wheat is cheaper than rice by 20%. By what percent is rice costlier than wheat ?

1) 25%

2) 30%

3) 28%

4) 20%

5 )

A sells a bicycle to Bat a profit of 20% and B sells to C at a profit of 5%. If C pays Rs 1890, what did A pay for it ?

1) Rs 1400

2) Rs 1500

3) Rs 2000

4) Rs 1800

6 )

There are two classes A and B. If 10 students leaves class A and join B, then the ratio of the number of students in class A and in class B would reverse. Then difference between the number of students in class A and class B is

1) 15

2) 10

3) 20

4) 25

7 )

8 )

One man and two boys can do a piece of work in 12 days, which could be done in 6 days by three men and one boy. How long would it take one man to do it ?

1) 15

2) 24

3) 18

4) 20

9 )

If s = n(5n - 3), then the pth term is

1) 10p - 6

2) 10p - 4

3) 10p - 8

4) 10p - 2

10 )

If $\, tan x = \frac{sin y - cos y}{sin y + cos y}$, then the value of sin y + cos y is

1) $\sqrt{2} cos x$

2) $\sqrt{2} sin x$

3) $\sqrt{3} cos x$

4) $- \sqrt{3} cos x$

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1 ) a=5, b=12 2 ) 25 : 64 3 ) $(\frac{4}{3}, \, - \frac{4}{3} )$, $(\frac{8}{3}, \, - \frac{8}{3} )$ 4 ) 25% 5 ) Rs 1500 6 ) 10 8 ) 20 9 ) 10p - 8 10 ) $\sqrt{2} cos x$

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