Q1
mcq
1 mark
If $\tan 3\theta = 1$, then $\theta$ equals :
- A. 60^\circ
- B. 30^\circ
- C. 20^\circ
- D. 10^\circ
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Q2
mcq
1 mark
If $x$ is the LCM of 4, 6, 8 and $y$ is the LCM of 3, 5, 7 and $p$ is the LCM of $x$ and $y$, then which of the following is true ?
- A. p = 35x
- B. p = 4y
- C. p = 8x
- D. p = 16y
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Q3
mcq
1 mark
The system of equations $6x + y = 3$ and $36x + 6y = 3$ have infinitely many solutions is :
- A. 6
- B. Not readable
- C. Not readable
- D. Not readable
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Q4
mcq
1 mark
If $\alpha$ and $\beta$ are the zeroes of the polynomial $p(x) = x^2 - ax - b$, then the value of $(\alpha + \beta)$ is equal to :
- A. a + b
- B. a - b
- C. a * b
- D. a + b
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Q5
mcq
1 mark
If equation is given, then the values of $x$ are :
- A. 6
- B. 4
- C. 12
- D. 3
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Q6
mcq
1 mark
The line represented by $\frac{x}{a} + \frac{y}{b} = 1$, intersects x-axis and y-axis respectively at P and Q. The coordinates of the mid-point of line segment PQ are :
- A. (2, 3)
- B. (3, 2)
- C. (2, 0)
- D. (0, 3)
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Q7
mcq
1 mark
Two of the vertices of PQR are P(1, 5) and Q(5, 2). The coordinates of a point which divides PQ in the ratio 2 : 1 are :
- A. (3, 3)
- B. (5, 5)
- C. (3, 3)
- D. (5, 1)
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Q8
mcq
1 mark
If tangents PA and PB drawn from an external point P to the circle with centre O are inclined to each other at an angle of 80 degrees as shown in the figure, then the measure of POA is :
- A. 40
- B. 50
- C. 60
- D. 80
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Q9
mcq
1 mark
(cot $\theta$ + tan $\theta$) equals :
- A. cosec $\theta$ sec $\theta$
- B. sin $\theta$ sec $\theta$
- C. cos $\theta$ tan $\theta$
- D. sin $\theta$ cos $\theta$
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Q10
mcq
1 mark
If in two triangles DEF and PQR, D = Q and R = E, then which of the following is not true?
- A. Not readable
- B. Not readable
- C. Not readable
- D. Not readable
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Q11
mcq
1 mark
The measurements of LMN and ABC are shown in the figure. The length of side AC is :
- A. 16 cm
- B. 7 cm
- C. 8 cm
- D. 4 cm
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Q12
mcq
If the volumes of two cubes are in the ratio 8 : 125, then the ratio of their surface areas is :
- A. 8 : 125
- B. 4 : 25
- C. 2 : 5
- D. 16 : 25
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Q13
mcq
If the area of a sector of circle of radius 36 cm is 54 \text{ cm}^2, then the length of the corresponding arc of the sector is :
- A. 8 \text{ cm}
- B. 6 \text{ cm}
- C. 4 \text{ cm}
- D. 3 \text{ cm}
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Q14
mcq
A die is thrown once. The probability of getting a number which is not a factor of 36, is :
- A. 1/3
- B. 2/3
- C. 1/6
- D. 5/6
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Q15
mcq
If the mean of 2, 9, x+6, 2x+3, 5, 10, 5 is 7, then the value of x is :
- A. 9
- B. 6
- C. 5
- D. 3
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Q16
mcq
AOBC is a rectangle whose three vertices are A(0, 2), O(0, 0) and B(4, 0). The square of the length of its diagonal is equal to :
- A. 36
- B. 20
- C. 16
- D. 4
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Q17
mcq
Zeroes of the polynomial p(x) = x^2 - 3x + 4 are :
- A. 2, 1
- B. 2, -1
- C. 4, -1
- D. -4, 2
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Q18
mcq
In the given figure, in ABC, AD \perp BC and BAC = 90^\circ. If BC = 16 cm and DC = 4 cm, then the value of x is :
- A. 4 \text{ cm}
- B. 5 \text{ cm}
- C. 8 \text{ cm}
- D. 3 \text{ cm}
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Q19
mcq
Assertion (A) : A ladder leaning against a wall, stands at a horizontal distance of 6 m from the wall. If the height of the wall up to which the ladder reaches is 8 m, then the length of the ladder is 10 m.
Reason (R): The ladder makes an angle of 60^\circ with the ground.
- 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) : If two tangents are drawn to a circle from an external point, then they subtend equal angles at the centre of the circle.
Reason (R): A parallelogram circumscribing a circle is a rhombus.
- 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 4k = \tan^2 60^\circ - 2 \text{cosec}^2 30^\circ - 2 \tan^2 30^\circ, then find the value of k.
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Q22
short answer
2 marks
The probability of guessing the correct answer of a certain test question is x/12. If the probability of not guessing the correct answer is 2/3, then find the value of x.
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Q23
short answer
2 marks
Find the smallest number which is divisible by both 644 and 462.
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Q24
short answer
2 marks
Two numbers are in the ratio 4 : 5 and their HCF is 11. Find the LCM of these numbers.
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Q25
short answer
2 marks
If $4x^2 + kx + 1 = 0$ has real and equal roots, find the value of k.
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Q25
short answer
2 marks
Prove that the tangents drawn at the ends of a diameter of a circle are parallel.
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Q26
short answer
2 marks
If $p(y) = y^2 - 5y + 3$, then find the value of $\alpha/\beta + \beta/\alpha$.
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Q27
short answer
3 marks
Prove that $\sqrt{3}$ is an irrational number given that $\sqrt{2}$ is an irrational number.
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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, find the value of k.
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Q29
short answer
3 marks
Find the coordinates of the points which divide the line segment joining A(-2, 2) and B(2, 8) into four equal parts.
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Q29
short answer
3 marks
A chord of a circle of radius 10 cm subtends a right angle at the centre of the circle. Find the area of the corresponding minor segment. [Use $\pi = 3.14$]
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Q30
short answer
3 marks
Three unbiased coins are tossed simultaneously. Find the probability of getting: (a) exactly two tails (b) at least one head (c) at most two heads
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Q31
short answer
3 marks
Prove that: $\frac{\cos A}{1 - \sin A} + \frac{\cos A}{1 + \sin A} = 2 \text{cosec} A$.
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Q31
short answer
3 marks
In the given figure, PC is a tangent to the circle at C. AOB is the diameter which when extended meets the tangent at P. Find $\angle CBA$ and $\angle BCO$, if $\angle PCA = 110^{\circ}$.
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Q32
short answer
3 marks
Prove that: $\frac{\tan A + \sec A - 1}{\tan A - \sec A + 1} = \frac{1 + \sin A}{\cos A}$.
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Q32
long answer
5 marks
The perimeter of an isosceles triangle is 32 cm. If each equal side is 5/6 times of the base, find the area of the triangle.
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Q35
long answer
5 marks
A vessel is in the form of an inverted cone. Its height is 8 cm and the radius of its top, which is open, is 5 cm. It is filled with water up to the brim. When lead shots, each of which is a sphere of radius 0.5 cm, are dropped into the vessel, one-fourth of the water flows out. Find the number of lead shots dropped in the vessel.
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A garden designer is planning a rectangular lawn that is to be surrounded by a uniform walkway. The total area of the lawn and the walkway is 360 square metres. The width of the walkway is same on all sides. The dimensions of the lawn itself are 12 metres by 10 metres.
Q36
long answer
4 marks
A garden designer is planning a rectangular lawn that is to be surrounded by a uniform walkway. The total area of the lawn and the walkway is 360 square metres. The width of the walkway is same on all sides. The dimensions of the lawn itself are 12 metres by 10 metres. Based on the information given above, answer the following questions: (i) Formulate the quadratic equation representing the total area of the lawn and the walkway, taking width of walkway = x m. (ii) Solve the quadratic equation to find the width of the walkway.
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Q37
long answer
5 marks
The sum of the third term and the seventh term of an AP is 6 and their product is 8. Find the sum of the first sixteen terms of the AP.
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Q38
long answer
5 marks
The minimum age of children eligible to participate in a painting competition is 8 years. It is observed that the age of the youngest boy was 8 years and the ages of the participants, when seated in order of age, have a common difference of 4 months. If the sum of the ages of all the participants is 168 years, find the age of the eldest participant in the painting competition.
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Q39
long answer
5 marks
In the given figure, PA, QB and RC are perpendicular to AC. If PA = x units, QB = y units and RC = z units, prove that $1/x + 1/z = 1/y$.
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Q40
long answer
5 marks
Sides AB and BC and median AD of triangle ABC are respectively proportional to sides PQ and QR and median PM of PQR. Show that $\triangle ABC \sim \triangle PQR$.
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Q43
short answer
1 mark
Formulate the quadratic equation representing the total area of the lawn and the walkway, taking width of walkway = x m.
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Q44
short answer
2 marks
Solve the quadratic equation to find the width of the walkway.
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Q45
short answer
2 marks
If the cost of paving the walkway at the rate of 50 per square metre is 12,000, calculate the area of the walkway.
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A lighthouse stands tall on a cliff by the sea, watching over ships that pass by. One day a ship is seen approaching the shore and from the top of the lighthouse, the angles of depression of the ship are observed to be 30° and 45° as it moves from point P to point Q. The height of the lighthouse is 50 metres.
Q47
short answer
1 mark
Find the distance of the ship from the base of the lighthouse when it is at point Q, where the angle of depression is 45°.
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Q48
short answer
1 mark
Find the measures of $\angle PBA$ and $\angle QBA$.
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Q49
short answer
2 marks
Find the distance travelled by the ship between points P and Q.
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Q50
short answer
2 marks
If the ship continues moving towards the shore and takes 10 minutes to travel from Q to A, calculate the speed of the ship in km/h, from Q to A.
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The India Meteorological Department observes seasonal and annual rainfall every year in different sub-divisions of our country. It helps them to compare and analyse the results. The table below shows sub-divisions wise seasonal (monsoon) rainfall (in mm) in 2023. Rainfall (mm) | No. of Sub-divisions 200-400 | 3 400-600 | 4 600-800 | 7 800-1000 | 4 1000-1200 | 3 1200-1400 | 3
Q52
short answer
2 marks
Find the median of the given data.
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Q53
short answer
2 marks
Find the mean rainfall in the season.
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Q54
short answer
1 mark
If a sub-division having at least 800 mm rainfall during monsoon season is considered a good rainfall sub-division, then how many sub-divisions had good rainfall?
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