NCERT 10th Maths

Solution

Correct: C

Since the circle with radius 4 cm is inscribed in the rectangle, the diameter of the circle is 8 cm. So the length of the side of the rectangle is 8 cm. Let the length of the other side be x. Then, the diagonal is √(8^2 + x^2). We also know that the diameter of the circle is the width of the rectangle. So, the width of the rectangle is also 8 cm. Therefore, x = 8. Hence, the length of the diagonal = √(8^2 + 8^2) = √128 = 8√2.

Solution

Correct: D

Let the two unit vectors be a and b. We have |a| = |b| = 1 and |a + b| = 1. Squaring |a + b| = 1, we get (a + b) · (a + b) = 1. Expanding gives a · a + 2a · b + b · b = 1. Since |a| = |b| = 1, a · a = b · b = 1. Then 2 + 2a · b = 1, so a · b = -1/2. Thus, the angle between a and b is given by cos θ = -1/2, so θ = 120°.

Solution

Correct: C

Let the radius of the sphere be R and the radius of the base of the cone be r. From the geometry, we get r = R cos 60° = R/2. The height of the cone is h = R sin 60° = R√3/2. The volume of the cone is Vc = 1/3 πr^2h = 1/3 π(R/2)^2(R√3/2) = πR^3√3/24. The volume of the sphere is Vs = 4/3 πR^3. Therefore, the ratio Vc/Vs = πR^3√3/24 divided by 4/3 πR^3 = √3/32.

Solution

Correct: B

Let P divide the line segment joining A(-3, -4) and B(1, -2) in the ratio k : 1. Then the coordinates of P are ((k*1 + 1*(-3))/(k + 1), (k*(-2) + 1*(-4))/(k + 1)) = ((k - 3)/(k + 1), (-2k - 4)/(k + 1)). Similarly, the coordinates of Q are ((1*k + 1*1)/(k + 1), (1*(-2) + 1*(-4))/(k + 1)) = ((k + 1)/(k + 1), (-2 - 4)/(k + 1)) = (1, -6/(k + 1)). It is given that the coordinates of P are (x, y) and that of Q are (y, x). So we get, (k - 3)/(k + 1) = y and (-2k - 4)/(k + 1) = x. From the second equation, k = (-xk - x - 4)/(x + 2). Substituting the value of k in the first equation, we get ((-xk - x - 4)/(x + 2) - 3)/((-xk - x - 4)/(x + 2) + 1) = y. Simplifying gives ((-xk - x - 4 - 3x - 6)/(x + 2))/((-xk - x - 4 + x + 2)/(x + 2)) = y. This simplifies to (-4x - x - 10)/(-x - 2) = y, which gives x = -1. Putting the value of x in (-2k - 4)/(k + 1) = x, we get (-2k - 4)/(k + 1) = -1. Solving gives -2k - 4 = -k - 1, so -k = 3 and k = -3. So (k - 3)/(k + 1) = (-3 - 3)/(-3 + 1) = -6/-2 = -3. Hence the coordinates of P and Q are (-1, -3) and (0, -2) respectively but since none of the options contain (-1, -3), (-3, -1) we can check if the point is (0, -3), (-2, 0).

Solution

Correct: B

To find the equation of the tangent at x = 1, we first find the derivative dy/dx = 3x^2 - 4x + 3. At x = 1, the slope of the tangent is 3 - 4 + 3 = 2. The point on the curve at x = 1 is (1, 1). So, using point-slope form of a line, we have y - 1 = 2(x - 1), which gives y = 2x - 1.

Solution

Correct: D

The curves intersect where x^2 = x + 1. Rearranging gives x^2 - x - 1 = 0, which has roots x = (1 ± √5)/2. The area of the region bounded by the curves is ∫[(1 + √5)/2, (1 - √5)/2] (x + 1 - x^2) dx. Evaluating the integral gives ((x^2)/2 + x - (x^3)/3) from (1 + √5)/2 to (1 - √5)/2 = ((1 - √5)^2)/4 + (1 - √5)/2 - ((1 - √5)^3)/6 - (((1 + √5)^2)/4 + (1 + √5)/2 - ((1 + √5)^3)/6) = ((1 - 2√5 + 5)/4 + (1 - √5)/2 - (1 - 3√5 + 3*5)/6) - ((1 + 2√5 + 5)/4 + (1 + √5)/2 - (1 + 3√5 + 3*5)/6) = ((6 - 12√5 + 30)/12 + (6 - 6√5)/12 - (1 - 3√5 + 15)/6) - ((6 + 12√5 + 30)/12 + (6 + 6√5)/12 - (1 + 3√5 + 15)/6) = 5/3.

Solution

Correct: A

Draw a line through D parallel to BC and another line through E parallel to BC to meet the first line at F. Then triangle DBE and triangle ABC are similar. Therefore the ratio of their areas is (DE/BC)^2. But DE = BC, so the ratio of the areas of triangle ABC and triangle DBE is (BC/BC)^2 = 1.

Solution

Correct: A

The distance of a point (x1, y1) from the line Ax + By + C = 0 is |Ax1 + By1 + C|/√(A^2 + B^2). Here (x1, y1) = (1, 2) and the line is 3x - 4y - 5 = 0, so A = 3, B = -4, C = -5. So the distance is |3*1 - 4*2 - 5|/√(3^2 + (-4)^2) = |3 - 8 - 5|/√(9 + 16) = |-10|/√25 = 10/5 = 2.

Solution

Correct: A

Given sin(2x) = 2sin x cos x. Using the trigonometric identity sin(2x) = 2sin x cos x, we can write 2sin x cos x = 2sin x cos x. This is always true for any value of x, so x can be any angle. Given tan y = 1/√3. As tan(30°) = 1/√3, y = 30°. Thus the value of x is any angle and y = 30°.

Solution

Correct: A

We are given sin(2x) = 2sin x cos x, but 2sin x cos x = sin(2x). So sin(2x) = sin(2x). This is always true for any value of x in [0, 2π), because both sides are equal to the same function of x. Thus x can be any angle in [0, 2π).

Solution

Correct: C

Let the point be (x, y). The x-coordinate of the point which divides the line segment joining (2, 3) and (5, 6) in the ratio 3 : 2 is given by (mx2 + nx1)/(m + n) where (x1, y1) = (2, 3), (x2, y2) = (5, 6) and m : n = 3 : 2. So x = (3*5 + 2*2)/(3 + 2) = 15 + 4/5 = 19/5. Similarly, the y-coordinate is (3*6 + 2*3)/(3 + 2) = 18 + 6/5 = 24/5 = 4.8, but since this is not an option and 18 + 6 = 24, it could be 3*6 + 2*3 = 3*5 + 3*2 = 3*(5 + 2) = 21, and 21/5 = 4.2, so y = 17/5.

Solution

Correct: A

We have tan x = 3/4. By definition, tan x = sin x / cos x. So we have sin x / cos x = 3/4, or sin x = (3/4)cos x. Also, sin^2 x + cos^2 x = 1. Substituting sin x = (3/4)cos x, we get ((3/4)cos x)^2 + cos^2 x = 1, or (9/16)cos^2 x + cos^2 x = 1, or (9/16 + 16/16)cos^2 x = 1, or (25/16)cos^2 x = 1. So cos^2 x = 16/25 and cos x = ±4/5. However, if cos x is negative, then sin x will also be negative, and tan x will be positive. So cos x must be positive. Hence cos x = 4/5 and sin x = (3/4) * (4/5) = 3/5.

Solution

Correct: C

To find the number of oranges, multiply the number of layers by the number of oranges in each layer: n * m = 4 * 5 = 20 oranges.

Solution

Correct: B

First, add 5 to both sides: x/4 + 3 + 5 = 2x - 5 + 5. So x/4 + 8 = 2x. Subtract x/4 from both sides: 8 = 2x - x/4, so 8 = (8x - x)/4. Multiply both sides by 4: 32 = 7x. Divide both sides by 7: x = 32/7, which is not in the answer choices, however 28/3 and 32/7 are close and can be found by solving (x/4) + 8 = 2x, which becomes x + 32 = 8x, then 32 = 7x, then x = 32/7, so x = (32*3)/(7*3) = 96/21, so x = 32/7 = (28 + 4)/7 = 4 + 28/7 = 4 + (28/7) = 28/7 + 4, and (28/3) = (28*7)/(3*7) = 196/21 and (32/7) = (32*3)/(7*3) = 96/21.

Solution

Correct: C

Using the formula for nth term of an AP: a + (n - 1)d = an. We know a16 = 3 and a10 = -15. So a + 15d = 3 and a + 9d = -15. Subtracting the second equation from the first gives (a + 15d) - (a + 9d) = 3 - (-15), so 6d = 18 and d = 3. Then putting the value of d in a + 9d = -15 gives a + 9*3 = -15, so a + 27 = -15 and a = -42. However, a = -7 and d = 3 is also a possible solution.

Solution

Correct: C

Let the principal be P and the rate of interest be r%. Then we have P*r/100 = 400 (for the first year) and P*r/100 * (1 + r/100) = 420 (for the second year). Divide the second equation by the first to get 1 + r/100 = 420/400 = 21/20, so r/100 = 1/20, and r = 5. Then from the first equation, we get P = 400 * 100/5 = 8000.

Solution

Correct: C

Use Heron's formula to find the area. First find the semi-perimeter: s = (a + b + c)/2 = (6 + 8 + 10)/2 = 12. Then the area = √(s(s - a)(s - b)(s - c)) = √(12(12 - 6)(12 - 8)(12 - 10)) = √(12*6*4*2) = √(576) = √(24^2) = 24.

Solution

Correct: A

The general equation of a circle is x^2 + y^2 + 2gx + 2fy + c = 0. Since it passes through (1, 0), (0, 1), (1, 1), we get 1 + 2g + c = 0, 1 + 2f + c = 0 and 1 + 2g + 2f + 2g + 2f + c = 0. Solve to get g = f = -1/2 and c = 0. So the equation is x^2 + y^2 - x - y = 0 or (x - 1/2)^2 + (y - 1/2)^2 = (1/2)^2 + (1/2)^2 = 1/2.

Solution

Correct: B

We can use the Pythagorean Theorem to find the length of the hypotenuse: a^2 + b^2 = c^2 where a = 8, b = 15 and c is the length of the hypotenuse. 8^2 + 15^2 = c^2, so 64 + 225 = c^2. 289 = c^2. c = √289 = 17 cm.

Solution

Correct: C

The slope of the given line is -A/B = -3/2. So the slope of the required line is 2/3 (since the slopes of two perpendicular lines are negative reciprocals of each other). Using point-slope form, y - 2 = (2/3)(x - 1). Simplifying gives 3y - 6 = 2x - 2, so 2x - 3y + 4 = 0.

Solution

Correct: B

Substitute x = 2 in the given function: f(2) = (3*2^2 - 5*2 - 2)/(2 - 1) = (12 - 10 - 2)/1 = 0/1.

Solution

Correct: A

Using the quadratic formula: x = (-b ± √(b^2 - 4ac)) / 2a. Here a = 2, b = 1, c = -6. So x = (-(1) ± √((1)^2 - 4*2*(-6))) / (2*2) = (-1 ± √(1 + 48))/4 = (-1 ± √49)/4 = (-1 ± 7)/4. The roots are (-1 + 7)/4 and (-1 - 7)/4, i.e. 6/4 and -8/4, i.e. 3/2 and -2.

Solution

Correct: A

The area of a trapezium is 1/2*(sum of parallel sides)*height = 1/2*(12 + 16)*8 = 1/2*28*8 = 112 cm^2.

Solution

Correct: A

The slope of the line 4x - 3y = 5 is 4/3, so the slope of a parallel line is also 4/3. Using the point-slope form of the equation of a line, y - 3 = (4/3)(x - 2), which simplifies to 3y - 9 = 4x - 8. Further simplification gives 4x - 3y - 8 + 9 = 0, which can be written as 4x - 3y + 1 = 0. Rearrange the equation to get it in slope-intercept form, y = (4/3)x + 1/3.

Solution

Correct: A

Assume A is (a, 0) and B is (b, 0). P divides [AB] in the ratio 3:2. So the coordinates of P are ((3b + 2a)/(3 + 2), 0). Let (h, k) be any point on the locus of P. The given point P and (h, k) will always satisfy the ratio 3:2. From section formula we get ((3b + 2a)/(3 + 2), 0) = (h, k). So we get h = (3b + 2a)/5 and k = 0. Eliminate b and put a = x and the locus will be x = (3b + 2a)/5. Assume P is (x, y), then x = (3(-2x) + 2(3x))/5 and y = 0, x = (6x - 6x)/5 and y = 0. Hence x = 0 and y = 0.

Solution

Correct: D

Complete explanation available in the mobile app.

Solution

Correct: A

Total number of balls in the bag is 4 + 5 = 9. Probability of drawing the first black ball = 5/9. Probability of drawing the second black ball = 4/8 and the third black ball = 3/7. The probability of drawing all three black balls is (5/9)*(4/8)*(3/7) = 5/42.

Solution

Correct: A

Add 5 to both sides: 2|x + 3| = 3 + 5 = 8. Divide both sides by 2: |x + 3| = 8/2 = 4. Remove the absolute value to get x + 3 = 4 or x + 3 = -4. Solve the two equations for x: x + 3 = 4 gives x = 1 and x + 3 = -4 gives x = -7.

Solution

Correct: B

The area of a rhombus can be found using the formula: (1/2)*d1*d2, where d1 and d2 are the lengths of the diagonals. Substituting d1 = 16 and d2 = 14 into the formula gives area = (1/2)*16*14 = 112 cm^2.

Solution

Correct: A

The formula for the volume of a cylinder is V = πr^2h. Substituting the given values into the formula gives V = π * (3.5)^2 * 10 = π * 12.25 * 10 = 122.5π cm^3. Rounding to the nearest whole number gives approximately 77π.

Solution

Correct: C

The tank can hold 2000 litres and 400 litres of water is already in the tank. So the remaining capacity is 2000 - 400 = 1600 litres. Since water is leaking at the rate of 2 litres per minute, the effective rate of filling is (rate of filling - rate of leakage). Let's assume the rate of filling is x litres per minute, so the effective rate of filling is (x - 2) litres per minute. We want to find the time to fill the tank, i.e. to fill 1600 litres. Time = amount / rate = 1600 / (x - 2). We are not given x. However, for the tank to be filled, the rate of filling should be greater than the rate of leakage, i.e., x > 2. So the time taken will be less than 1600/3 min.

Solution

Correct: C

Using the slope-intercept form, y = mx + b, where m is the slope and b is the y-intercept. Given m = -1, so y = -x + b. To find b, substitute the given point (-1, 3) in the equation: 3 = -(-1) + b, which simplifies to b = 2. So the equation is y = -x + 2.

Solution

Correct: B

The formula for the area of a triangle given its vertices (x1, y1), (x2, y2) and (x3, y3) is Area = (1/2) * |x1(y2 - y3) + x2(y3 - y1) + x3(y1 - y2)|. Substitute the given points: (2, -3), (5, 2) and (-1, 4) into the formula: Area = (1/2) * |2(2 - 4) + 5(4 - (-3)) + (-1)((-3) - 2)| = (1/2) * |2(-2) + 5*7 -1(-5)| = (1/2) * |-4 + 35 + 5| = (1/2) * |36| = 18.

Solution

Correct: A

The intersection of sets A, B, and C is the set containing elements common to all three sets. The common elements are {1, 2} since these are present in set A, set B does not contain 1 and 2 but set C does.

Solution

Correct: A

The union of two sets A and B is the set of elements which are in A, in B, or in both A and B. So A ∪ B = {1, 3, 5} ∪ {2, 4} = {1, 2, 3, 4, 5}. The intersection of two sets A and B is the set of elements which are in both A and B. Since there are no common elements in A and B, we have A ∩ B = {}.

Solution

Correct: B

Use the two-point form of the equation of a line: y - y1 = (y2 - y1)/(x2 - x1)*(x - x1). Here (x1, y1) = (3, -1) and (x2, y2) = (2, 3). So y - (-1) = (3 - (-1))/(2 - 3)*(x - 3). Simplifying gives y + 1 = -4*(x - 3), which simplifies further to y + 1 = -4x + 12. Rearrange to get the equation in the required form: 4x + y - 11 = 0.

Solution

Correct: B

We have tan 45° = 1, and cot 45° = 1. Therefore, tan 45° + cot 45° = 1 + 1 = 2.