Q1
mcq
1 mark
The number of polynomials having zeroes 3 and 5 is :
- A. only one
- B. infinite
- C. exactly two
- D. at most two
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Q2
mcq
1 mark
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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Q3
mcq
1 mark
The common difference of the A.P. whose $n^{th}$ term is given by $a_n = 3n + 7$, is :
- A. 7
- B. 3
- C. 3n
- D. 1
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Q4
mcq
1 mark
In the given figure, DE || BC. The value of x is :
- A. 6
- B. 12·5
- C. 8
- D. 10
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Q5
mcq
1 mark
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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Q6
mcq
1 mark
If tan $\theta$ = 12/5, then the value of (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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Q6
mcq
If $\tan \theta = \frac{12}{5}$, then the value of $\frac{\cos \theta + \sin \theta}{\cos \theta - \sin \theta}$ is :
- A. $\frac{7}{17}$
- B. $\frac{-7}{17}$
- C. $\frac{13}{17}$
- D. $\frac{13}{7}$
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Q7
mcq
1 mark
The distance between the points P (5, 3/11) and Q (5, 3/2) is :
- A. 6 units
- B. 4 units
- C. 2 units
- D. 3 units
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Q8
mcq
1 mark
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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Q9
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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Q10
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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Q11
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. $\sqrt{3}$ cm
- C. 2 cm
- D. $\sqrt{3}$ cm
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Q12
mcq
$\frac{1-\tan^2 30}{1+\tan^2 30}$ is equal to :
- A. sin 60
- B. cos 60
- C. tan 60
- D. sin 30
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Q13
mcq
In triangles ABC and DEF, $\frac{FD}{BC} = \frac{DE}{AB}$. 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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Q14
mcq
The 11th term from the end of the A.P. : 10, 7, 4, ......., 62 is :
- A. 25
- B. 16
- C. 32
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Q15
mcq
Two coins are tossed together. The probability of getting at least one tail is :
- A. 1/4
- B. 1/2
- C. 3/4
- D. 1
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Q16
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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Q17
mcq
Which of the following numbers cannot be the probability of happening of an event?
- B. 0·017
- C. 0·07
- D. 0·73
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Q18
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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Q19
mcq
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.
- 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 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.
- 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
(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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Q22
short answer
2 marks
In the given figure, PT is a tangent to the circle centered at O. OC is perpendicular to chord AB. Prove that PA . PB = PC^2 AC^2.
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Q23
short answer
2 marks
Using prime factorisation, find HCF and LCM of 96 and 120.
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Q24
short answer
2 marks
Find the ratio in which y-axis divides the line segment joining the points (5, 6) and (1, 4).
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Q25
short answer
2 marks
(a) If a cos + b sin = m and a sin b cos = n, then prove that a^2 + b^2 = m^2 + n^2. OR (b) Prove that : 1 Asec/1 Asec + 1 Asec/1 Asec = 2 cosec A
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Q26
short answer
3 marks
(a) Prove that 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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Q27
short answer
3 marks
If p-th term of an A.P. is q and q-th term is p, then prove
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Q27
short answer
If $p^{th}$ term of an A.P. is $q$ and $q^{th}$ term is $p$, then prove that its $n^{th}$ term is $(p + q - n)$.
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Q29
short answer
3 marks
Prove that $3\sqrt{2}$ is an irrational number.
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Q29
short answer
Two people are 16 km apart on a straight road. They start walking at the same time. If they walk towards each other with different speeds, they will meet in 2 hours. Had they walked in the same direction with same speeds as before, they would have met in 8 hours. Find their walking speeds.
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Q30
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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Q31
short answer
In the given figure, CD is the perpendicular bisector of AB. EF is perpendicular to CD. AE intersects CD at G. Prove that $DG/FG = CD/CF$.
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Q31
short answer
Find the mean of the following frequency distribution: Classes 25-30, 30-35, 35-40, 40-45, 45-50, 50-55, 55-60. Frequencies 14, 22, 16, 6, 5, 3, 4.
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Q32
short answer
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
One observer estimates the angle of elevation to the basket of a hot air balloon to be 60, while another observer 100 m away estimates the angle of elevation to be 30. Find: (a) The height of the basket from the ground. (b) The distance of the basket. (c) The horizontal distance of the second observer from the basket.
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Q35
short answer
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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Q37
long answer
5 marks
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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Q38
long answer
5 marks
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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Q39
short answer
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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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.
Q41
short answer
1 mark
Find the surface area of one such dimple.
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Q42
short answer
1 mark
Find the volume of the material dug out to make one dimple.
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Q43
short answer
2 marks
Find the total surface area exposed to the surroundings.
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Q44
short answer
2 marks
Find the volume of the golf ball.
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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.
Q45
short answer
1 mark
List all possible outcomes of the game.
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Q46
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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Q47
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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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 :
Q48
short answer
1 mark
Find zeroes of polynomial p(t) = 20t 16t^2.
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Q49
short answer
1 mark
Which of the following types of graph represents p(t) ?
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Q50
short answer
2 marks
What would be the value of h at t = 2/3 ? Interpret the result.
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Q51
short answer
2 marks
How much distance has the dolphin covered before hitting the water level again ?
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