Q1
mcq
1 mark
If $n$ is a natural number, then which of the following numbers ends with zero?
- A. $(3 \times 2)^n$
- B. $(2 \times 5)^n$
- C. $(6 \times 2)^n$
- D. $(5 \times 3)^n$
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Q2
mcq
1 mark
In a lottery, there are 5 prizes and 20 blanks. The probability of getting a prize is:
- A. 1/4
- B. 1/20
- C. 1/25
- D. 1/5
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Q3
mcq
1 mark
If $2x + 3y = 15$ and $3x + 2y = 25$, then the value of $x - y$ is:
- A. 10
- B. 8
- C. -10
- D. -8
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Q4
mcq
1 mark
In the given figure, $\angle A = \angle C$, $AB = 6$ cm, $AP = 12$ cm, $CP = 4$ cm. Then length of $CD$ is:
- A. 2 cm
- B. 6 cm
- C. 8 cm
- D. 18 cm
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Q4
mcq
In the given figure, $A = C$, $AB = 6\text{ cm}$, $AP = 12\text{ cm}$, $CP = 4\text{ cm}$. Then length of $CD$ is :
- A. 2 cm
- B. 6 cm
- C. 8 cm
- D. 18 cm
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Q5
mcq
1 mark
The sum of zeroes of the polynomial $2x^2 - 17$ are given as:
- A. $\frac{17}{2}$
- B. $-\frac{17}{2}$
- D. 1
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Q6
mcq
1 mark
If the area of the base of a cone is $51$ cm$^2$ and its volume is $85$ cm$^3$, then the vertical height of the cone is given as:
- A. $\frac{6}{5}$ cm
- B. $\frac{3}{5}$ cm
- C. $\frac{2}{5}$ cm
- D. 5 cm
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Q7
mcq
1 mark
What is the length of the arc of the sector of a circle with radius $14$ cm and of central angle $90^{\circ}$?
- A. 22 cm
- B. 44 cm
- C. 88 cm
- D. 11 cm
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Q8
mcq
The coordinates of the vertex A of a rectangle ABCD whose three vertices are given as B(0, 0), C(3, 0) and D(0, 4) are :
- A. (4, 0)
- B. (0, 3)
- C. (3, 4)
- D. (4, 3)
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Q9
mcq
The area of the triangle formed by the line $\frac{x}{a} + \frac{y}{b} = 1$ with the coordinate axes is :
- A. ab
- B. $\frac{1}{2}ab$
- C. $\frac{1}{4}ab$
- D. 2ab
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Q10
mcq
The hour-hand of a clock is $6\text{ cm}$ long. The angle swept by it between 7:20 a.m. and 7:55 a.m. is :
- A. $\frac{35}{4}$
- B. $\frac{35}{2}$
- C. 35
- D. 70
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Q11
mcq
The zeroes of the polynomial $p(x) = x^2 + 4x + 3$ are given by :
- A. 1, 3
- B. -1, 3
- C. 1, -3
- D. -1, -3
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Q12
mcq
In the given figure, $AB \parallel PQ$. If $AB = 6\text{ cm}$, $PQ = 2\text{ cm}$ and $OB = 3\text{ cm}$, then the length of $OP$ is :
- A. 9 cm
- B. 3 cm
- C. 4 cm
- D. 1 cm
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Q13
mcq
In the given figure, the quadrilateral $PQRS$ circumscribes a circle. Here $PA + CS$ is equal to :
- A. QR
- B. PR
- C. PS
- D. PQ
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Q14
mcq
If one zero of the polynomial $6x^2 + 37x - (k - 2)$ is reciprocal of the other, then what is the value of $k$?
- A. 4
- B. 6
- C. -6
- D. -4
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Q15
mcq
If three coins are tossed simultaneously, what is the probability of getting at most one tail?
- A. $\frac{3}{8}$
- B. $\frac{4}{8}$
- C. $\frac{5}{8}$
- D. $\frac{7}{8}$
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Q16
mcq
If the pair of equations $3x - y + 8 = 0$ and $6x - ry + 16 = 0$ represent coincident lines, the $r$ value is :
- A. $\frac{1}{2}$
- B. $-\frac{1}{2}$
- C. 2
- D. -2
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Q17
mcq
If $\triangle ABC \sim \triangle PQR$, $A = 32^\circ$ and $R = 65^\circ$, then the measure of $B$ is :
- A. 32
- B. 65
- C. 83
- D. 97
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Q17
mcq
If $\angle ABC = 32^\circ$ and $\angle R = 65^\circ$, then the measure of $\angle B$ is :
- A. 32^\circ
- B. 65^\circ
- C. 83^\circ
- D. 97^\circ
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Q18
mcq
Which of the following quadratic equations has sum of its roots as 4?
- A. $2x^2 - 4x + 8 = 0$
- B. $x^2 + 4x + 4 = 0$
- C. $2x^2 - 8x + 1 = 0$
- D. $4x^2 - 4x + 4 = 0$
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Q19
mcq
1 mark
Assertion (A): The tangent at any point of a circle is perpendicular to the radius through the point of contact. Reason (R): The lengths of tangents drawn from an external point to a circle are equal.
- A. Both (A) and (R) are true and (R) is the correct explanation of (A).
- B. Both (A) and (R) are true but (R) is not the correct explanation of (A).
- C. (A) is true but (R) is false.
- D. (A) is false but (R) is true.
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Q20
mcq
1 mark
Assertion (A): The polynomial $p(x) = x^2 + 3x + 3$ has two real zeroes. Reason (R): A quadratic polynomial can have at most two real zeroes.
- A. Both (A) and (R) are true and (R) is the correct explanation of (A).
- B. Both (A) and (R) are true but (R) is not the correct explanation of (A).
- C. (A) is true but (R) is false.
- D. (A) is false but (R) is true.
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Q22
short answer
2 marks
Show that the points (-2, 3), (8, 3) and (6, 7) are the vertices of a right-angled triangle.
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Q23
short answer
2 marks
The length of the shadow of a tower on the plane ground is $\sqrt{3}$ times the height of the tower. Find the angle of elevation of the sun.
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Q23
short answer
2 marks
In the given figure, O is the centre of the circle. AB and AC are tangents drawn to the circle from point A. If $\angle BAC = 65^\circ$, then find the measure of $\angle BOC$.
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Q24
short answer
2 marks
The angle of elevation of the top of a tower from a point on the ground which is 30 m away from the foot of the tower, is 30^\circ. Find the height of the tower.
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Q25
short answer
2 marks
Prove that $6 + \sqrt{7}$ is irrational number, given that $\sqrt{7}$ is an irrational number.
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Q26
long answer
3 marks
Prove that : $\frac{\tan \theta}{1 - \cos^2 \theta} + \frac{\cos \theta}{\sin \theta} = 1 + \sin \theta \cos \theta$
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Q27
short answer
2 marks
If $4 \cot^2 45^\circ - \sec^2 60^\circ + \sin^2 60^\circ + p = \frac{4}{3}$, then find the value of p.
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Q27
long answer
3 marks
A chord of a circle of radius 14 cm subtends an angle of 60^\circ at the centre. Find the area of the corresponding minor segment of the circle. (Use $\pi = 3.14$ and $\sqrt{3} = 1.73$)
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Q28
short answer
2 marks
If $\cos A + \cos^2 A = 1$, then find the value of $\sin^2 A + \sin^4 A$.
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Q29
long answer
3 marks
In the given figure, O is the centre of the circle and QPR is the tangent to the circle at point P. Prove that $\angle QAP + \angle APR = 90^\circ$.
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Q30
long answer
3 marks
If point Q(0, 1) is equidistant from points P(5, 3) and R(x, 6), then find the value of x.
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Q32
long answer
5 marks
The mode of the following frequency distribution is 55. Find the missing frequencies. (Table: Class Interval 0-15, 15-30, 30-45, 45-60, 60-75, 75-90; Frequency 6, 7, a, 15, 10, b; Total 51)
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Q33
long answer
5 marks
Prerna saves ₹32 during the first month, ₹36 in the second month and ₹40 in the third month. If she continues to save in this manner, in how many months will she save ₹2,000 ?
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Q34
short answer
Find by prime factorisation the LCM of the numbers 18180 and 7575. Also, find the HCF of the two numbers.
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Q35
short answer
Three bells ring at intervals of 6, 12 and 18 minutes. If all the three bells rang at 6 a.m., when will they ring together again ?
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Q36
short answer
If the system of linear equations $2x + 3y = 7$ and $2ax + (a + b)y = 28$ have infinitely many solutions, then find the values of a and b.
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Q37
short answer
If $217x + 131y = 913$ and $131x + 217y = 827$, then solve the equations for the values of x and y.
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Q40
long answer
5 marks
Sides AB and BC and median AD of a triangle ABC are respectively proportional to sides PQ and QR and median PM of a triangle PQR. Show that $\triangle ABC \sim \triangle PQR$.
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Q41
long answer
5 marks
Through the mid-point M of the side CD of a parallelogram ABCD, the line BM is drawn intersecting AC in L and AD (produced) in E. Prove that EL = 2BL.
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Q42
long answer
5 marks
As observed from the top of a 75 m high lighthouse from the sea-level, the angles of depression of two ships are $30^\circ$ and $60^\circ$. If one ship is exactly behind the other on the same side of the lighthouse, find the distance between the two ships. (Use $\sqrt{3} = 1.73$)
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Q43
long answer
5 marks
From a point on the ground, the angle of elevation of the bottom and top of a transmission tower fixed at the top of 30 m high building are $30^\circ$ and $60^\circ$, respectively. Find the height of the transmission tower. (Use $\sqrt{3} = 1.73$)
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A coffee shop sells coffee in two types of cups. One cup is cylindrical with a diameter of 7 cm and a height of 14 cm and the second cup is hemispherical with a diameter of 21 cm.
Q44
short answer
1 mark
Find the area of the base of the cylindrical cup.
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Q45
short answer
2 marks
Find the capacity of the cylindrical cup.
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Q46
short answer
1 mark
Find the curved surface area of the cylindrical cup.
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In a coffee shop, coffee is served in two types of cups. One is cylindrical in shape with diameter 7 cm and height 14 cm and the other is hemispherical with diameter 21 cm. Based on the above, answer the following questions :
Q47
short answer
2 marks
What is the capacity of the hemispherical cup ?
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Computer-based learning (CBL) refers to any teaching methodology that makes use of computers for information transmission. At an elementary school level, computer applications can be used to display multimedia lesson plans. A survey was done on 1000 elementary and secondary schools of Assam and they were classified by the number of computers they had. [Number of computers table: 1-10 (250), 11-20 (200), 21-50 (290), 51-100 (180), 101 and more (80)]
Q48
short answer
1 mark
Find the probability that the school chosen at random has more than 100 computers.
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In an annual day function of a school, the organizers wanted to give a cash prize along with a memento to their best students. Each memento is made as shown in the figure and its base ABCD is shown from the front side. The rate of silver plating is 20 per cm^2.
Q49
short answer
1 mark
What is the area of the quadrant ODCO ?
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Q51
short answer
2 marks
What is the total cost of silver plating the shaded part ABCD ?
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