Q1
mcq
1 mark
Using SAS criterion of similarity, Devansh proved that $\Delta ABC \sim \Delta PQR$. If $\angle C = \angle R$, then which of the following statements was found to be true?
- A. $\frac{AC}{AB} = \frac{PR}{PQ}$
- B. $\frac{BC}{AC} = \frac{PR}{QR}$
- C. $\frac{AC}{BC} = \frac{PR}{PQ}$
- D. $\frac{AC}{BC} = \frac{PR}{QR}$
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Q1
mcq
1 mark
Devansh proved that $\Delta ABC \sim \Delta PQR$ using SAS similarity criteria. If he found $\angle C = \angle R$, then which of the following was proved true ?
- A. $\frac{AC}{AB} = \frac{PR}{PQ}$
- B. $\frac{BC}{AC} = \frac{PR}{QR}$
- C. $\frac{AC}{BC} = \frac{PR}{PQ}$
- D. $\frac{AC}{BC} = \frac{PR}{QR}$
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Q2
mcq
1 mark
In the given figure, PQ is a tangent to a circle with centre O. Point S lies on the circle such that $\angle SQT = 55^{\circ}$. The measure of $\angle QPS$ is
- A. $55^{\circ}$
- B. $20^{\circ}$
- C. $35^{\circ}$
- D. $70^{\circ}$
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Q3
mcq
1 mark
In an A.P., if $a_{14} - a_{8} = 24$, then the common difference of the A.P. is
- A. 6
- B. 4
- C. $\pm 4$
- D. 3
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Q4
mcq
1 mark
The roots of the quadratic equation $x^2 - px + 6 = 0$ are rational. Which of the following is a value of p?
- A. 1
- B. $-5$
- C. 25
- D. 5
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Q4
mcq
1 mark
The value of p for which roots of the quadratic equation $x^2 - px + 6 = 0$ are rational, is
- A. 1
- B. -5
- C. 25
- D. 5
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Q5
mcq
1 mark
In the given figure, PQ || YZ and XP : PY = 2 : 3. If PQ = 5 cm, then YZ is equal to
- A. 12.5 cm
- B. 10 cm
- C. 15 cm
- D. 7.5 cm
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Q5
mcq
1 mark
In the given figure, $PQ \parallel YZ$ such that $XP : PY = 2 : 3$. If $PQ = 5 \text{ cm}$, then $YZ$ equals
- A. 12.5 cm
- B. 10 cm
- C. 15 cm
- D. 7.5 cm
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Q6
mcq
1 mark
For an acute angle $\theta$, if $\cos \theta = \frac{1}{8}$, then $\frac{8 \sec \theta + 1}{8 \sec \theta - 1}$ equals
- A. \frac{64}{63}
- C. \frac{65}{63}
- D. 1
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Q7
mcq
1 mark
A card is drawn at random from a well shuffled deck of 52 playing cards. The probability that it is either a ten or a king is
- A. \frac{1}{26}
- B. \frac{2}{13}
- C. \frac{1}{13}
- D. \frac{8}{26}
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Q8
mcq
1 mark
The line segment joining the points $P(-4, -2)$ and $Q(10, 4)$ is divided by y-axis in the ratio
- A. 2 : 5
- B. 1 : 2
- C. 2 : 1
- D. 5 : 2
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Q9
mcq
1 mark
Simplest form of $\frac{\sec A}{\sec^2 A - 1}$ is
- A. \sin A
- B. \tan A
- C. \csc A
- D. \cos A
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Q10
mcq
1 mark
A wire is attached from a point A on the ground to the top of a pole BC, making an angle of elevation as $60^\circ$. If $AB = 5\sqrt{3} \text{ m}$, then length of the wire is
- A. 10 m
- B. 10\sqrt{3} m
- C. 15 m
- D. \frac{5}{2}\sqrt{3} m
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Q11
mcq
1 mark
If sum and product of zeroes of a polynomial are (-3) and (-2) respectively, then a polynomial is
- A. x^2 - 3x - 2
- B. -x^2 - 3x + 2
- C. -x^2 + 3x - 2
- D. x^2 + 3x + 2
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Q12
mcq
Meena calculates that the probability of her winning the first prize in a lottery is 0.08. If total 800 tickets were sold, the number of tickets bought by her, is
- A. 64
- B. 640
- C. 100
- D. 10
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Q13
mcq
A conical cavity of maximum volume is carved out from a wooden solid hemisphere of radius 10 cm. Curved surface area of the cavity carved out is (use $\pi$ = 3.14)
- A. 314 $\sqrt{2}$ $\text{cm}^2$
- B. 314 $\text{cm}^2$
- C. $\frac{3140}{3}$ $\text{cm}^2$
- D. 3140 $\sqrt{2}$ $\text{cm}^2$
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Q14
mcq
While calculating mean of a grouped frequency distribution, step deviation method was used $\left[\frac{x - a}{h} = u\right]$. It was found that $\bar{x} = 64, h = 5$ and $a = 62.5$. The value of $\bar{u}$ is
- A. 0.5
- B. 1.5
- C. 0.3
- D. 7.5
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Q15
mcq
The area of a sector of a circle of radius 10 cm is $\frac{55}{3}$ $\text{cm}^2$. The value of central angle is
- A. $\frac{21^\circ}{2}$
- B. 42 $^\circ$
- C. 105 $^\circ$
- D. 21 $^\circ$
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Q16
mcq
A camping tent in hemispherical shape of radius 1.4 m, has a door opening of area 0.50 $\text{m}^2$. Outer surface area of the tent is
- A. 11.78 $\text{m}^2$
- B. 12.32 $\text{m}^2$
- C. 11.82 $\text{m}^2$
- D. 12.86 $\text{m}^2$
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Q17
mcq
Which of the following can not be the probability of an event ?
- A. $\frac{39}{100}$
- B. $\frac{0.001}{20}$
- C. $\frac{10}{0.2}$
- D. 10%
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Q18
mcq
The value of k for which the equation $kx^2 - 6x - 4 = 0$ has real and equal roots, is
- A. $\frac{9}{4}$
- B. -4
- C. $-\frac{9}{4}$
- D. -2
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Q19
mcq
Assertion (A) : The system of linear equations $3x - 5y + 7 = 0$ and $-6x + 10y + 14 = 0$ is inconsistent.
Reason (R) : When two linear equations don’t have unique solution, they always represent parallel lines.
- 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) : H.C.F. $(36 m^2, 18 m) = 18 m$, where m is a prime number.
Reason (R) : H.C.F. of two numbers is always less than or equal to the smaller number.
- 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
In the given figure, AB || DE and AC || DF. Show that $\Delta ABC \sim \Delta DEF$. If BC = 10 cm, EB = CF = 5 cm and AB = 7 cm, then find the length DE.
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Q23
short answer
2 marks
Prove that $4 - \sqrt{25}$ is an irrational number given that $\sqrt{5}$ is irrational.
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Q24
short answer
2 marks
A bag contains 25 balls. Some of them are yellow and others are green. One ball is drawn at random. If probability of getting a green ball is $3/5$, then find the number of yellow balls.
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Q25
short answer
2 marks
For acute angles A and B, if $\sec (2A - B) = 2$ and $\csc (A + B) = 2$, then find the values of A and B.
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Q26
short answer
2 marks
Evaluate : $\frac{2 \cos 30^\circ - \cot^3 60^\circ}{\tan 30^\circ}$
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Q27
short answer
3 marks
The dimensions of a window are 156 cm x 216 cm. Arjun wants to put grill on the window creating complete squares of maximum size. Determine the side length of the square and hence find the number of squares formed.
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Q28
long answer
Use graphical method to solve the system of linear equations : $y = -3$ and $x + 2y = 4$.
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Q29
short answer
2 marks
Vertices of a right triangle ABC with $\angle B = 90^\circ$ are A(3, 4), B(1, 1) and C(-8, 7). Find the value of $\tan A$.
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Q30
short answer
2 marks
Using distance formula, prove that the points A(2, 3), B(-7, 0) and C(-1, 2) are collinear.
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Q30
long answer
A circle of diameter 20 cm is equally divided into five sectors. Find the area and perimeter of one of the sectors.
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Q31
short answer
3 marks
A circle centered at (2, 1) passes through the points A(5, 6) and B(-3, K). Find the value(s) of K. Hence find length of chord AB.
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Q31
long answer
Prove that : $(1 + \cot \theta - \operatorname{cosec} \theta) (1 + \tan \theta + \sec \theta) = 2$
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Q32
short answer
3 marks
Prove that the point P dividing the line segment joining the points A(-1, 7) and B(4, -3) in the ratio 3 : 2, lies on the line x - 3y = -1. Also find length of PA and PB.
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Q32
long answer
5 marks
By selling an article for ` 48, a trader loses as much percent as half of the cost price of the article. Calculate the cost price and loss amount of the article.
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Q33
long answer
5 marks
PQ and PR are two tangents to a circle with centre O and radius 5 cm. AB is another tangent to the circle at C which lies on OP. If OP = 13 cm, then find the length AB and PA.
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Q35
long answer
In an A.P., 15th term exceeds the 8th term by 21. If sum of first 10 terms is 55, then form the A.P.
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Q36
long answer
The sum of first n terms of an A.P. is $2n^2 + 13n$. Find its nth term and hence 10th term.
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D is the mid-point of side BC of $\Delta ABC$. CE and BF intersect at O, a point on AD. AD is produced to G such that OD = DG. Prove that
Q41
long answer
D is the mid-point of side BC of $\Delta ABC$. CE and BF intersect at O, a point on AD. AD is produced to G such that OD = DG. Prove that (i) OBGC is a parallelogram.
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Through the mid-point Q of side CD of a parallelogram ABCD, the line AR is drawn which intersects BD at P and produced BC at R. Prove that
Q42
long answer
Through the mid-point Q of side CD of a parallelogram ABCD, the line AR is drawn which intersects BD at P and produced BC at R. Prove that (i) AQ = QR
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Q43
long answer
Find the mean and mode of the following frequency distribution: Class Interval: 400-450, 450-500, 500-550, 550-600, 600-650, 650-700; Frequency: 15, 18, 20, 23, 22, 12
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Q44
long answer
If the median of the following frequency distribution is 32.5 and sum of all frequencies is 40, then find the values of $f_1$ and $f_2$: Class Interval: 0-10, 10-20, 20-30, 30-40, 40-50, 50-60, 60-70; Frequency: 3, $f_1$, 9, 12, 6, $f_2$, 2
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Elevated water storage tanks are built to store and supply water to nearby colonies. In the diagram given above, AB is an elevated water tank and CD is a nearby multistorey building. The building is 54 metres away from the water tank. From a window (W) of the building, the angle of elevation of top of the tank is 45$^{\circ}$ and angle of depression of its foot is 30$^{\circ}$.
Q45
short answer
1 mark
Write a relation between d (the height of window) and y.
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Q47
short answer
2 marks
Find the height of the water tank.
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Q48
short answer
2 marks
Find the value of x and the height of the window from the ground level.
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Elevated water storage tanks are built to store and supply water to nearby colonies. In the diagram given above, AB is an elevated water tank and CD is a nearby multistorey building. The building is 54 metres away from the water tank. From a window (W) of the building, the angle of elevation of top of the tank is 45° and angle of depression of its foot is 30°.
Q50
short answer
2 marks
Determine height of the water tank.
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An arch of a railway bridge, built on Chenab riverbed, is shown in the above diagram. It is a parabolic arch connecting two hills at P and Q. If the parabolic curve is represented by the polynomial $p(x) = -0.0025x^2 - 0.025x + 136$. Observe the diagram and based on above information, answer the following questions :
Q51
short answer
1 mark
Write the co-ordinates of point A.
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Q53
short answer
2 marks
Write the zeroes of the polynomial using diagram and verify the relationship between sum of zeroes and polynomials.
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Q54
short answer
2 marks
Find the values of p(x) at x = 100 and x = –100. Are they same ?
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A wall mounted lamp, made of fabric, is shown below. Lamp has cuboidal shape, open from top and bottom. A spherical bulb of diameter 7 cm is latched with a very thin rod. (Ignore the rod while making calculations.) Dimensions of the cuboid are 24 cm $\times$ 12 cm $\times$ 17 cm.
Q55
short answer
1 mark
Find the surface area of the bulb.
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Q56
short answer
1 mark
What could be the maximum diameter of the bulb if at least 1 cm space is left from each side ?
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Q57
short answer
2 marks
Find the area of the fabric used if there is a fold of 2 cm on top and bottom edges.
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Q58
short answer
2 marks
Find the space available inside the lamp.
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