Q1
mcq
$\frac{30 \tan 1}{1 + 30 \tan^2 1}$ is equal to :
- A. sin 60
- B. cos 60
- C. tan 60
- D. cos 30
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Q2
mcq
$\frac{FD}{BC} = \frac{DE}{AB}$. Which of the following makes the two triangles similar ?
- A. A = D
- B. B = D
- C. B = E
- D. A = F
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Q3
mcq
The 13th term from the end of the A.P. : 20, 13, 6, 1, ....., 148 is :
- A. 57
- B. -57
- C. 64
- D. -64
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Q4
mcq
Two dice are rolled together. What is the probability of getting a sum greater than 10 ?
- A. 1/9
- B. 1/6
- C. 1/12
- D. 5/18
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Q5
mcq
In the given figure, AB = BC = 10 cm. If AC = 7 cm, then the length of BP is :
- A. 3.5 cm
- B. 7 cm
- C. 6.5 cm
- D. 5 cm
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Q6
mcq
Water in a river which is 3 m deep and 40 m wide is flowing at the rate of 2 km/h. How much water will fall into the sea in 2 minutes ?
- A. 800 m^3
- B. 4000 m^3
- C. 8000 m^3
- D. 2000 m^3
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Q7
mcq
If the mean and the median of a data are 12 and 15 respectively, then its mode is :
- A. 13.5
- B. 21
- C. 6
- D. 14
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Q8
mcq
In the given figure, AB is a tangent to the circle centered at O. If OA = 6 cm and OAB = 30, then the radius of the circle is :
- A. 3 cm
- B. 3$\sqrt{3}$ cm
- C. 2 cm
- D. \sqrt{3}$ cm
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Q9
mcq
In the given figure, AC and AB are tangents to a circle centered at O. If COD = 120, then BAO is equal to :
- A. 30
- B. 60
- C. 45
- D. 90
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Q10
mcq
Which of the following numbers cannot be the probability of happening of an event ?
- B. 0.17
- C. 0.07
- D. 3.07
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Q11
mcq
If every term of the statistical data consisting of n terms is decreased by 2, then the mean of the data :
- A. decreases by 2
- B. remains unchanged
- C. decreases by 2n
- D. decreases by 1
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Q12
mcq
In the given figure, DE || BC. The value of x is :
- A. 6
- B. 12.5
- C. 8
- D. 10
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Q13
mcq
A quadratic equation whose roots are $(2 + \sqrt{3})$ and $(2 - \sqrt{3})$ is :
- A. $x^2 - 4x + 1 = 0$
- B. $x^2 + 4x + 1 = 0$
- C. $4x^2 - 3 = 0$
- D. $x^2 - 1 = 0$
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Q14
mcq
If $\tan \theta = \frac{12}{5}$, then the value of $\frac{\cos \theta - \sin \theta}{\cos \theta + \sin \theta}$ is :
- A. 7/17
- B. -7/17
- C. 13/17
- D. 13/7
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Q15
mcq
If end points of a diameter of a circle are (5, 4) and (1, 0), then the radius of the circle is :
- A. \sqrt{32} units
- B. \sqrt{13} units
- C. 24 units
- D. 22 units
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Q16
mcq
The number of polynomials having zeroes 1 and 2 is :
- A. exactly 2
- B. only 1
- C. at most 2
- D. infinite
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Q17
mcq
The pair of equations $ax + 2y = 9$ and $3x + by = 18$ represent parallel lines, where a, b are integers, if :
- A. a = b
- B. 3a = 2b
- C. 2a = 3b
- D. ab = 6
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Q18
mcq
The common difference of the A.P. whose $n^{th}$ term is given by $a_n = 5n - 7$ is :
- A. 7
- B. -7
- C. 5
- D. 2
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Q19
short answer
1 mark
Assertion (A) : The number $5^n$ cannot end with the digit 0, where n is a natural number. Reason (R): Prime factorisation of 5 has only two factors, 1 and 5.
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Q20
short answer
1 mark
Assertion (A) : If the points A(4, 3) and B(x, 5) lie on a circle with centre O(2, 3), then the value of x is 2. Reason (R) : Centre of a circle is the mid-point of each chord of the circle.
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Q21
short answer
2 marks
Find the HCF and LCM of 96 and 120 using prime factorization.
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Q21
short answer
2 marks
Using prime factorisation, find HCF and LCM of 96 and 120.
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Q22
short answer
2 marks
Find the ratio in which the line $y = x$ divides the line segment joining the points (6, 3) and (1, 6).
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Q23
short answer
2 marks
If $a \cos \theta + b \sin \theta = m$ and $a \sin \theta - b \cos \theta = n$, then prove that $a^2 + b^2 = m^2 + n^2$.
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Q23
short answer
2 marks
(a) If $a \cos \theta + b \sin \theta = m$ and $a \sin \theta - b \cos \theta = n$, then prove that $a^2 + b^2 = m^2 + n^2$. OR (b) Prove that : $1/(1-sin A) + 1/(1+sin A) = 2 sec^2 A$
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Q24
short answer
2 marks
(a) The line segment joining the points A(4, 5) and B(4, 5) is divided by the point P such that AP : AB = 2 : 5. Find the coordinates of P. OR (b) Point P(x, y) is equidistant from points A(5, 1) and B(1, 5). Prove that x = y.
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Q25
short answer
2 marks
In the given figure, PQ is a chord of the circle centered at O. PT is a tangent to the circle at P. If QPT = 55, then find PRQ.
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Q26
short answer
3 marks
Find the mean of the following distribution : Classes 0-15, 15-30, 30-45, 45-60, 60-75, 75-90. Frequency 17, 20, 18, 21, 15, 9
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Q27
short answer
3 marks
A 2-digit number is seven times the sum of its digits. The number formed by reversing the digits is 18 less than the given number. Find the given number.
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Q28
short answer
3 marks
Prove that : $cot \theta / (1-tan \theta) + tan \theta / (1-cot \theta) = 1 + sec \theta cosec \theta$
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Q28
short answer
Prove that: $\frac{\cot \theta}{1-\tan \theta} + \frac{\tan \theta}{1-\cot \theta} = 1 + \sec \theta \csc \theta$
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Q29
short answer
(a) Prove that $\sqrt{3}$ is an irrational number. OR (b) The traffic lights at three different road crossings change after every 48 seconds, 72 seconds and 108 seconds respectively. If they change simultaneously at 7 a.m., at what time will they change together next ?
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Q30
short answer
In an A.P., the sum of the first n terms is given by $S_n = 6n - n^2$. Find its 30th term.
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Q31
short answer
(a) In the given figure, CD is the perpendicular bisector of AB. EF is perpendicular to CD. AE intersects CD at G. Prove that DG/FG = CD/CF. OR (b) In the given figure, ABCD is a parallelogram. BE bisects CD at M and intersects AC at L. Prove that EL = 2BL.
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Q32
long answer
5 marks
(a) A triangle ABC is drawn to circumscribe a circle of radius 4 cm such that the segments BD and DC are of lengths 10 cm and 8 cm respectively. Find the lengths of the sides AB and AC, if it is given that area of triangle ABC = 90 cm^2. OR (b) Two circles with centres O and O' of radii 6 cm and 8 cm, respectively intersect at two points P and Q such that OP and O'P are tangents to the two circles. Find the length of the common chord PQ.
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Q33
long answer
5 marks
A horse is tied to a peg at one corner of a square shaped grass field of side 15 m by means of a 5 m long rope. Find the area of that part of the field in which the horse can graze. Also, find the increase in grazing area if length of rope is increased to 10 m. (Use $\pi = 3.14$)
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Q34
long answer
5 marks
The angle of elevation of the top of a vertical tower from a point P on the ground is 60. From another point Q, 10 m vertically above the first point P, its angle of elevation is 30. Find : (a) The height of the tower. (b) The distance of the point P from the foot of the tower. (c) The distance of the point P from the top of the tower.
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Q35
long answer
5 marks
(a) A train travels at a certain average speed for a distance of 54 km and then travels a distance of 63 km at an average speed of 6 km/h more than the first speed. If it takes 3 hours to complete the journey, what was its first average speed ? OR (b) Two pipes together can fill a tank in 8/15 hours. The pipe with larger diameter takes 2 hours less than the pipe with smaller diameter to fill the tank separately. Find the time in which each pipe can fill the tank separately.
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Q39
long answer
A train travels at a certain average speed for a distance of 54 km and then travels a distance of 63 km at an average speed of 6 km/h more than the first speed. If it takes 3 hours to complete the journey, what was its first average speed ?
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Q40
long answer
Two pipes together can fill a tank in $\frac{8}{15}$ hours. The pipe with larger diameter takes 2 hours less than the pipe with smaller diameter to fill the tank separately. Find the time in which each pipe can fill the tank separately.
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A middle school decided to run the following spinner game as a fund-raiser on Christmas Carnival. Making Purple : Spin each spinner once. Blue and red make purple. So, if Based on the above, answer the following questions :
Q41
short answer
1 mark
List all possible outcomes of the game.
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Q42
short answer
1 mark
Find the probability of making purple.
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Q43
short answer
2 marks
For each win, a participant gets ₹ 10, but if he/she loses, he/she has to pay ₹ 5 to the school. If 99 participants played, calculate how much fund could the school have collected.
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Q44
short answer
2 marks
If the same amount of ₹ 5 has been decided for winning or losing the game, then how much fund had been collected by school? (Number of participants = 99)
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A golf ball is spherical with about 300 500 dimples that help increase its velocity while in play. Golf balls are traditionally white but available in colours also. In the given figure, a golf ball has diameter 4.2 cm and the surface has 315 dimples (hemi-spherical) of radius 2 mm. Based on the above, answer the following questions :
Q45
short answer
1 mark
Find the surface area of one such dimple.
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Q46
short answer
1 mark
Find the volume of the material dug out to make one dimple.
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Q47
short answer
2 marks
Find the total surface area exposed to the surroundings.
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Q48
short answer
2 marks
Find the volume of the golf ball.
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38. In a pool at an aquarium, a dolphin jumps out of the water travelling at 20 cm per second. Its height above water level after t seconds is given by h = 20t - 16t^2. Based on the above, answer the following questions :
Q49
short answer
1 mark
Find zeroes of polynomial p(t) = 20t - 16t^2.
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Q50
short answer
1 mark
Which of the following types of graph represents p(t) ?
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Q51
short answer
2 marks
What would be the value of h at t = 2/3 ? Interpret the result.
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Q52
short answer
2 marks
How much distance has the dolphin covered before hitting the water level again ?
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