Date: March 2019

Duration: 3h

Two positive integers a and b can be written as a = x^{3}y^{2 }and b = xy^{3}. x, y are prime numbers. Find LCM (a, b).

Chapter: [1.01] Real Numbers

How many two digits numbers are divisible by 3?

Chapter: [2.01] Pair of Linear Equations in Two Variables

In Fig. 1, DE || BC, AD = 1 cm and BD = 2 cm. What is the ratio of the ar (Δ ABC) to the ar (Δ ADE)?

Chapter: [3.02] Triangles

Find the coordinates of a point A, where AB is the diameter of circle whose centre is (2, − 3) and B is (1, 4)

Chapter: [6.01] Lines (In Two-dimensions)

For what value of k, the roots of the equation x^{2 }+ 4x + k = 0 are real?

Chapter: [2.03] Quadratic Equations

Find the value of k for which the roots of the equation 3x^{2 }- 10x + k = 0 are reciprocal of each other.

Chapter: [2.03] Quadratic Equations

Find A if tan 2A = cot (A-24°).

Chapter: [4.02] Trigonometric Identities [4.03] Introduction to Trigonometry

Find the value of (sin^{2} 33 + sin^{2} 57°)

Chapter: [3.02] Triangles

Find, how many two digit natural numbers are divisible by 7.

Chapter: [1.01] Real Numbers

If the sum of first n terms of an AP is n^{2}, then find its 10th term.

Chapter: [2.02] Arithmetic Progressions

A game consist of tossing a one-rupee coin 3 times and noting the outcome each time. Ramesh will win the game if all the tosses show the same result, (i.e. either all thee heads or all three tails) and loses the game otherwise. Find the probability that Ramesh will lose the game.

Chapter: [5.01] Probability [5.01] Probability

Find the ratio in which the segment joining the points (1, –3) and (4, 5) is divided by the *x*-axis? Also, find the coordinates of this point on the *x*-axis.

Chapter: [3.03] Constructions

A die is thrown once. Find the probability of getting a number which (i) is a prime number (ii) lies between 2 and 6.

Chapter: [5.01] Probability [5.01] Probability

Find c if the system of equations cx + 3y + (3 – c) = 0; 12x + cy – c = 0 has infinitely many solutions?

Chapter: [2.03] Quadratic Equations

Find the HCF of 1260 and 7344 using Euclid's algorithm.

Chapter: [1.01] Real Numbers

Show that every positive odd integer is of the form (4q + 1) or (4q + 3), where q is some integer.

Chapter: [1.01] Real Numbers

Find all zeros of the polynomial 3x^{3} + 10x^{2} − 9x − 4 if one of its zero is 1.

Chapter: [2.04] Polynomials

PQ is a chord of length 8 cm of a circle of radius 5 cm. The tangents at P and Q intersect at a point T. Find the lengths of TP and TQ.

Chapter: [3.01] Circles [3.01] Circles

Prove that `(2+sqrt3)/5` is an irrational number, given that `sqrt 3` is an irrational number.

Chapter: [1.01] Real Numbers

Prove that (sin θ + cosec θ)^{2} + (cos θ + sec θ)^{2} = 7 + tan^{2 }θ + cot^{2 }θ.

Chapter: [4.02] Trigonometric Identities [4.03] Introduction to Trigonometry

Prove that (1 + cot A - cosec A ) (1 + tan A + sec A) = 2

Chapter: [3.02] Triangles

A father's age is three times the sum of the ages of his two children. After 5 years his age will be two times the sum of their ages. Find the present age of the father.

Chapter: [2.01] Pair of Linear Equations in Two Variables

A fraction becomes `(1)/(3)` when 2 is subtracted from the numerator and it becomes `(1)/(2)` when 1 is subtracted from the denominator. Find the fraction.

Chapter: [2.01] Pair of Linear Equations in Two Variables

Find the point on the y-axis which is equidistant from the points (5, −2) and (−3, 2).

Chapter: [6.01] Lines (In Two-dimensions) [6.01] Lines (In Two-dimensions)

The line segment joining the points A(2, 1) and B(5, −8) is trisected at the points P and Q such that P is nearer to A. If P also lies on the line given by 2x − y + k = 0, find the value of k.

Chapter: [3.03] Constructions

Find the mode of the following frequency distribution.

Class | 0-10 | 10-20 | 20-30 | 30-40 | 40-50 | 50-60 | 60-70 |

Frequency | 8 | 10 | 10 | 16 | 12 | 6 | 7 |

Chapter: [5.02] Statistics

Water in a canal, 6 m wide and 1.5 m deep, is flowing with a speed of 10 km/hour. How much area will it irrigate in 30 minutes; if 8 cm standing water is needed?

Chapter: [7.02] Surface Areas and Volumes

In Fig. 3, ∠ACB = 90° and CD ⊥ AB, prove that CD^{2} = BD x AD.

Chapter: [3.02] Triangles

If P and Q are the points on side CA and CB respectively of ΔABC, right angled at C, prove that (AQ^{2} + BP^{2} ) = (AB^{2} + PQ^{2})

Chapter: [3.02] Triangles

Find the area of the shaded region in the given figure, if ABCD is a rectangle with sides 8 cm and 6 cm and O is the centre of the circle.

Chapter: [7.01] Areas Related to Circles

If sec θ = x + `1/(4"x"), x ≠ 0,` find (sec θ + tan θ)

Chapter: [4.02] Trigonometric Identities [4.03] Introduction to Trigonometry

Prove that “That ratio of areas of two similar triangles is equal to the square of the ratio of their corresponding sides.”

Chapter: [6.01] Lines (In Two-dimensions)

The following distribution gives the daily income of 50 workers of a factory.

Daily income (in ₹) | 200-220 | 220-240 | 240-260 | 260-280 | 280-300 |

Number of workers | 12 | 14 | 8 | 6 | 10 |

Convert the distribution above to a 'less than type' cumulative frequency distribution and draw its ogive.

Chapter: [6.01] Lines (In Two-dimensions)

Construct a ΔABC in which CA = 6 cm, AB = 5 cm and ∠BAC = 45°. Then construct a triangle whose sides are `3/5` of the corresponding sides of ΔABC.

Chapter: [3.02] Triangles

A bucket open at the top is in the form of a frustum of a cone with a capacity of 12308.8 cm^{3}. The radii of the top and bottom circular ends are 20 cm and 12 cm, respectively. Find the height of the bucket and the area of metal sheet used in making the bucket. (use *π* = 3.14)

Chapter: [4.01] Heights and Distances

A man in a boat rowing away from a lighthouse 100 m high takes 2 minutes to change the angle of elevation of the top of the lighthouse from 60° to 30°. Find the speed of the boat in metres per minute [Use `sqrt3` = 1.732]

Chapter: [4.03] Introduction to Trigonometry [4.03] Introduction to Trigonometry

Two poles of equal heights are standing opposite each other on either side of the road, which is 80 m wide. From a point between them on the road, the angles of elevation of the top of the poles are 60° and 30° respectively. Find the height of the poles and the distances of the point from the poles.

Chapter: [4.01] Heights and Distances

Two water taps together can fill a tank in `1(7)/(8)` hours. The tap with longer diameter takes 2 hours less than the tap with a smaller one to fill the tank separately. Find the time in which each tap can fill the tank separately.

Chapter: [2.03] Quadratic Equations

A boat goes 30 km upstream and 44 km downstream in 10 hours. In 13 hours, it can go 40 km upstream and 55 km downstream. Determine the speed of the stream and that of the boat in still water.

Chapter: [2.03] Quadratic Equations

If the sum of the first four terms of an AP is 40 and that of the first 14 terms is 280. Find the sum of its first n terms.

Chapter: [2.02] Arithmetic Progressions

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