Q1
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 - 4x + 1 = 0
- D. 4x^2 - 4x + 4 = 0
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Q2
mcq
What is the length of the arc of the sector of a circle with radius 14 cm and of central angle 90 ?
- A. 22 cm
- B. 44 cm
- C. 88 cm
- D. 11 cm
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Q3
mcq
If ABC ~ PQR, A = 32 and R = 65, then the measure of B is :
- A. 32
- B. 65
- C. 83
- D. 97
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Q3
mcq
If $\angle ABC = 32^\circ$ and $R = 65^\circ$, then the measure of $\angle B$ is :
- A. $32^\circ$
- B. $65^\circ$
- C. $83^\circ$
- D. $97^\circ$
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Q5
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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Q6
mcq
If the pair of equations 3x - y + 8 = 0 and 6x - ry + 16 = 0 represent coincident lines, then r =
- A. 1/2
- B. -1/2
- C. 2
- D. -2
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Q7
mcq
A bag contains 100 cards numbered 1 to 100. A card is drawn at random from the bag. What is the probability that the number on the card is a perfect cube ?
- A. 1/20
- B. 3/50
- C. 1/25
- D. 7/100
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Q8
mcq
The pair of equations $x = a$ and $y = b$ graphically represents lines which are :
- A. parallel
- B. intersecting at (b, a)
- C. coincident
- D. intersecting at (a, b)
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Q9
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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Q11
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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Q12
mcq
In the given figure, DE || BC. If AD = 2 units, DB = AE = 3 units and EC = x units, then the value of x is :
- A. 2
- B. 3
- C. 5
- D. $\frac{9}{2}$
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Q13
mcq
The hour-hand of a clock is 6 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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Q14
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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Q15
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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Q16
mcq
If $\alpha$ and $\beta$ are the zeroes of the quadratic polynomial $p(x) = x^2 - ax - b$, then the value of $\alpha^2 + \beta^2$ is :
- A. $a^2 - 2b$
- B. $a^2 + 2b$
- C. $b^2 - 2a$
- D. $b^2 + 2a$
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Q17
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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Q18
mcq
In the given figure, AB || PQ. If AB = 6 cm, PQ = 2 cm and OB = 3 cm, then the length of OP is :
- A. 9 cm
- B. 3 cm
- C. 4 cm
- D. 1 cm
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Q19
mcq
1 mark
Assertion (A) : A tangent to 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 Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of the Assertion (A).
- B. Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of the Assertion (A).
- C. Assertion (A) is true, but Reason (R) is false.
- D. Assertion (A) is false, but Reason (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 Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of the Assertion (A).
- B. Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of the Assertion (A).
- C. Assertion (A) is true, but Reason (R) is false.
- D. Assertion (A) is false, but Reason (R) is true.
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Q21
short answer
2 marks
Prove that 2 + 3 is an irrational number, given that 3 is an irrational number.
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Q21
short answer
2 marks
If 4 cot^2 45 sec^2 60 + sin^2 60 + p = 4/3, then find the value of p.
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Q22
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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Q23
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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Q24
short answer
2 marks
The length of the shadow of a tower on the plane ground is 3 times the height of the tower. Find the angle of elevation of the sun.
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Q25
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. Find the height of the tower.
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Q25
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 BAC = 65, then find the measure of BOC.
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Q26
short answer
3 marks
(a) Find by prime factorisation the LCM of the numbers 18180 and 7575. Also, find the HCF of the two numbers. OR (b) 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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Q27
short answer
3 marks
Prove that: $\frac{\cos \theta}{1 - \sin \theta} + \frac{\cos \theta}{1 + \sin \theta} = 2\sec \theta$
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Q28
short answer
3 marks
If $Q(0, 1)$ is equidistant from $P(5, 3)$ and $R(x, 6)$, find the values of $x$.
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Q29
short answer
3 marks
A car has two wipers which do not overlap. Each wiper has a blade of length $21 \text{ cm}$ sweeping through an angle of $120^\circ$. Find the total area cleaned at each sweep of the two blades.
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Q30
short answer
3 marks
(a) If the system of linear equations $2x + 3y = 7$ and $2ax + (a + b)y = 28$ have infinitely many solutions, find the values of $a$ and $b$. OR (b) If $217x + 131y = 913$ and $131x + 217y = 827$, then solve the equations for the values of $x$ and $y$.
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Q31
long answer
5 marks
In the given figure, O is the centre of the circle and QPR is a tangent to it at P. Prove that $\angle QAP + \angle APR = 90^\circ$.
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Q32
long answer
5 marks
How many terms of the arithmetic progression 45, 39, 33, ........ must be taken so that their sum is 180? Explain the double answer.
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Q33
long answer
5 marks
(a) 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 $\sqrt{3} = 1.73$) OR (b) 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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Q34
long answer
5 marks
A student noted the number of cars passing through a spot on a road for 100 periods each of 3 minutes and summarised it in the table given below. Find the mean and median of the following data: Intervals: 0-10, 10-20, 20-30, 30-40, 40-50, 50-60, 60-70, 70-80. Frequencies: 7, 14, 13, 12, 20, 11, 15, 8.
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Q36
long answer
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. Prove that $\triangle ABC \sim \triangle PQR$.
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Q37
long answer
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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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$^2$.
Q38
short answer
1 mark
What is the area of the quadrant ODCO ?
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Q40
short answer
2 marks
What is the total cost of silver plating the shaded part ABCD ?
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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.
Q42
short answer
1 mark
Find the area of the base of the cylindrical cup.
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Q43
short answer
2 marks
What is the capacity of the hemispherical cup ?
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Q44
short answer
2 marks
Find the capacity of the cylindrical cup.
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Q45
short answer
1 mark
What is the curved surface area of the cylindrical 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: 1-10, 11-20, 21-50, 51-100, 101 and more Number of Schools: 250, 200, 290, 180, 80
Q46
short answer
1 mark
Find the probability that the school chosen at random has more than 100 computers.
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