Q1
mcq
1 mark
If 7 cos^2 \theta + 3 sin^2 \theta = 4, then the value of \theta is :
- A. 30
- B. 45
- C. 60
- D. 90
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Q2
mcq
1 mark
The probability of drawing an even prime number out of numbers from 1 to 30 is :
- A. 1/30
- B. 1/15
- C. 1/2
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Q3
mcq
1 mark
The quadratic equation whose roots are 7 and 1/7 is :
- A. 7x^2 - 50x + 7 = 0
- B. 7x^2 - 50x + 1 = 0
- C. 7x^2 + 50x - 7 = 0
- D. 7x^2 + 50x - 1 = 0
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Q4
mcq
1 mark
The least number which is a perfect square and is divisible by each of 16, 20 and 50, is :
- A. 1200
- B. 100
- C. 3600
- D. 2400
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Q5
mcq
1 mark
The coordinates of the end points of a diameter of a circle are (5, 2) and (5, -2). The length of the radius of the circle is :
- A. ± 2
- B. ± 4
- C. 4
- D. 2
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Q6
mcq
1 mark
The points (- 5, 0), (5, 0) and (0, 4) are the vertices of a triangle which is a/an :
- A. right-angled triangle
- B. isosceles triangle
- C. equilateral triangle
- D. scalene triangle
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Q7
mcq
1 mark
In the given figure, RS is the tangent to the circle at the point L and MN is the diameter. If NML = 30°, then RLM is :
- A. 30°
- B. 60°
- C. 90°
- D. 120°
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Q8
mcq
1 mark
In the given figure, PQ||BC. If AP/PB = 2/3 and AC = 20.4 cm, then the length of AQ is :
- A. 2.8 cm
- B. 5.8 cm
- C. 3.8 cm
- D. 4.8 cm
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Q9
mcq
1 mark
Which of the following statements is incorrect ?
- A. Two congruent figures are always similar.
- B. A square and a rhombus of the same area are always similar.
- C. Two equilateral triangles are always similar.
- D. Two similar triangles need not be congruent.
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Q10
mcq
The sum of the exponents of prime factors in the prime factorisation of 4004 is :
- A. 5
- B. 4
- C. 3
- D. 2
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Q11
mcq
In a cricket match, a batsman hits the boundary 7 times out of the 42 balls he plays. The probability of his not hitting a boundary is :
- A. 1/6
- B. 1/7
- C. 5/6
- D. 6/7
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Q12
mcq
If a large circular pizza is divided into 5 equal sectors, then the central angle of each sector will be :
- A. 60^{\circ}
- B. 90^{\circ}
- C. 45^{\circ}
- D. 72^{\circ}
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Q13
mcq
If $\sin 30^{\circ} \tan 45^{\circ} = k$, then the value of k is :
- A. 4
- B. 3
- C. 2
- D. 1
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Q14
mcq
The line represented by the equation $x - y = 0$ is :
- A. parallel to x-axis
- B. parallel to y-axis
- C. passing through the origin
- D. passing through the point (3, 2)
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Q15
mcq
The 10th term of the AP 5, 8, 11, 14, ... is :
- A. 30
- B. 32
- C. 35
- D. 38
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Q16
mcq
If 4 is a zero of the polynomial $p(x) = x^2 - x(2 + 2k)$, then the value of k is :
- A. 3
- B. 9
- C. 6
- D. 9
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Q17
mcq
The equation of a line parallel to the x-axis and at a distance of 3 units below x-axis is :
- A. x = 3
- B. x = -3
- C. y = 3
- D. y = -3
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Q18
mcq
The HCF of 40, 110 and 360 is :
- A. 40
- B. 110
- C. 360
- D. 10
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Q19
mcq
Assertion (A) : Common difference of the AP : 5, 1, 3, -1 is -4. Reason (R): Common difference of the AP : $a_1, a_2, a_3, ... a_n$ is obtained by $d = a_n - a_{n-1}$.
- 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
Assertion (A) : The pair of linear equations $px + 3y + 59 = 0$ and $2x + 6y + 118 = 0$ will have infinitely many solutions if $p = 1$. Reason (R): If the pair of linear equations $px + 3y + 19 = 0$ and $2x + 6y + 157 = 0$ has a unique solution, then $p \neq 1$.
- 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
If p and q are zeroes of the polynomial $p(y) = 21y^2 - y - 2$, then find the value of $(1-p)(1-q)$.
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Q22
short answer
2 marks
In the given figure, the shape of the top of a table is that of a sector of a circle with centre O and $\angle AOB = 90^{\circ}$. If $AO = OB = 42$ cm, then find the perimeter of the top of the table.
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Q23
short answer
2 marks
In the given figure, three sectors of a circle of radius 5 cm, making angles 35^{\circ}, 50^{\circ} and 95^{\circ} at the centre are shaded. Find the area of the shaded region. [Use $\pi = 22/7$]
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Q23
short answer
2 marks
If $\tan A = 3/4$; where A is an acute angle, then find the value of $\sin A \cos A$.
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Q25
short answer
2 marks
In the given figure, D is a point on the side BC of $\triangle ABC$ such that $\angle ADC = \angle BAC$. Prove that $CA^2 = CD \cdot CB$.
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Q25
short answer
3 marks
At point A on the diameter AB of a circle of radius 10 cm, tangent XAY is drawn to the circle. Find the length of the chord CD parallel to XY at a distance of 16 cm from A.
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Q26
short answer
2 marks
In the given figure, $OA \cdot OB = OC \cdot OD$. Show that $\angle A = \angle C$ and $\angle B = \angle D$.
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Q28
short answer
3 marks
Prove that the parallelogram circumscribing a circle is a rhombus.
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Q28
short answer
3 marks
If the mid-point of the line segment joining the points A(3, 4) and B(k, 6) is P(x, y) and $x + y - 10 = 0$, then find the value of k.
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Q29
short answer
3 marks
Prove that the angle between the two tangents drawn from an external point to a circle is supplementary to the angle subtended by the line-segment joining the points of contact at the centre.
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Q29
short answer
3 marks
The length of the hour hand of a clock is 10 cm. Find the area of the minor sector swept by the hour hand of the clock between 5 a.m. to 8 a.m. Also, find the area of the major sector.
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Q30
short answer
3 marks
Prove that : $\frac{1}{1+\sin A} + \frac{1}{1-\sin A} = 2\sec^2 A$.
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Q30
short answer
3 marks
Prove that $\sqrt{2}$ is an irrational number.
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Q31
short answer
3 marks
Prove that : $\frac{1}{\text{cosec}A-1} + \frac{1}{\text{cosec}A+1} = 2\sec^2 A$.
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Q31
short answer
3 marks
A sum of 2,000 is invested at 7% per annum simple interest. Calculate the interests at the end of 1st, 2nd and 3rd year. Do these interests form an AP? If so, find the interest at the end of the 27th year.
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Q34
long answer
5 marks
Prove that a line drawn parallel to one side of a triangle to intersect the other two sides in distinct points divides the other two sides in the same ratio. Hence, in the figure given below, prove that $\frac{AM}{AB} = \frac{AN}{AD}$ where $LM \parallel CB$ and $LN \parallel CD$.
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Q35
long answer
5 marks
Find the Mean and Mode of the following frequency distribution:
Class Frequency
0-10 8
10-20 7
20-30 15
30-40 20
40-50 12
50-60 8
60-70 10
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Q36
long answer
5 marks
Two ships are sailing in the sea on either side of a lighthouse. The angles of depression to the two ships as observed from the top of the lighthouse are $60^{\circ}$ and $45^{\circ}$, respectively. If the distance between the ships is 100 m, then find the height of the lighthouse.
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Q37
long answer
5 marks
The angles of depression of the top and the bottom of an 8 m tall building from the top of another multistoried building are $30^{\circ}$ and $45^{\circ}$, respectively. Find the height of the multistoried building and the distance between the two buildings.
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Q38
long answer
5 marks
The sum of the areas of two squares is 52 $cm^2$ and difference of their perimeters is 8 cm. Find the lengths of the sides of the two squares.
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Q39
long answer
5 marks
The time taken by a person to travel an upward distance of 150 km was 2 hours more than the time taken in the downward return journey. If he returned at a speed of 10 km/h more than the speed while going up, find the speeds in each direction.
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A school is organizing a grand cultural event to show the talent of its students. To accommodate the guests, the school plans to rent chairs and tables from a local supplier. It finds that rent for each chair is 50 and for each table is 200. The school spends 30,000 for renting the chairs and tables. Also, the total number of items (chairs and tables) rented are 300.
Q42
short answer
1 mark
Write down the pair of linear equations representing the given information.
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Q43
short answer
2 marks
Find the number of chairs and number of tables rented by the school.
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Q44
short answer
2 marks
If the school wants to spend a maximum of 27,000 on 300 items (tables and chairs), then find the number of chairs and tables it can rent.
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Q45
short answer
1 mark
What is maximum number of tables that can be rented in 30,000 if no chairs are rented?
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Rahul is a lucky charm for his cricket team. He has a jar of cards with numbers from 10 to 74. Before each match, he draws a card from the jar. If the card bears an even number, the team wins. If the number is even and divisible by 5, they win by a big margin. If the number is an odd number less than 30, they win by a small margin. And if the number is a prime number between 50 and 74, they lose.
Q46
short answer
1 mark
What is the probability that Rahul draws a card with an even number?
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Q47
short answer
2 marks
What is the probability that Rahul draws a card with a prime number between 50 and 74?
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A skilled carpenter decided to craft a special rolling pin for the local baker. He carefully joined three cylindrical pieces of wood two small ones on the ends and one larger in the centre to create a perfect tool. The baker loved the rolling pin, as it rolled out the smoothest dough for breads and pastries. The length of the bigger cylindrical part is 12 cm and diameter is 7 cm and the length of each smaller cylindrical part is 5 cm and diameter is 2.1 cm.
Q48
short answer
1 mark
Find the volume of the bigger cylindrical part.
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Q49
short answer
1 mark
Find the curved surface area of the bigger cylindrical part.
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Q50
short answer
2 marks
Find the ratio of the volume of the bigger cylindrical part to the total volume of the two smaller (identical) cylindrical parts.
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Q51
short answer
2 marks
Find the sum of the curved surface areas of the two identical smaller cylindrical parts.
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