NTA JEE Main 10th January 2019 Shift 1 - Mathematics

Consider the quadratic equation $$(c-5)x^2 - 2cx + (c-4) = 0$$, $$c \neq 5$$. Let $$S$$ be the set of all integral values of $$c$$ for which one root of the equation lies in the interval $$(0, 2)$$ and its other root lies in the interval $$(2, 3)$$. Then the number of elements in $$S$$ is:

Let $$z_1$$ and $$z_2$$ be any two non-zero complex numbers such that $$3|z_1| = 4|z_2|$$. If $$z = \frac{3z_1}{2z_2} + \frac{2z_2}{3z_1}$$ then maximum value of $$|z|$$ is:

If 5, 5$$r$$, 5$$r^2$$ are the lengths of the sides of a triangle, then $$r$$ can not be equal to:

The sum of all two digit positive numbers which when divided by 7 yield 2 or 5 as remainder is:

If $$\sum_{i=1}^{20} \left(\frac{^{20}C_{i-1}}{^{20}C_i + ^{20}C_{i-1}}\right)^3 = \frac{k}{21}$$, then $$k$$ equals:

If the third term in the binomial expansion of $$(1 + x^{\log_2 x})^5$$ equals 2560, then a possible value of $$x$$ is:

The sum of all values of $$\theta \in (0, \frac{\pi}{2})$$ satisfying $$\sin^2 2\theta + \cos^4 2\theta = \frac{3}{4}$$ is:

If the line $$3x + 4y - 24 = 0$$ intersects the $$x$$-axis at the point $$A$$ and the $$y$$-axis at the point $$B$$, then the incentre of the triangle $$OAB$$, where $$O$$ is the origin, is:

A point $$P$$ moves on the line $$2x - 3y + 4 = 0$$. If $$Q(1, 4)$$ and $$R(3, -2)$$ are fixed points, then the locus of the centroid of $$\triangle PQR$$ is a line:

If a circle $$C$$ passing through the point $$(4, 0)$$ touches the circle $$x^2 + y^2 + 4x - 6y = 12$$ externally at the point $$(1, -1)$$, then the radius of $$C$$ is:

If the parabolas $$y^2 = 4b(x-c)$$ and $$y^2 = 8ax$$ have a common normal, then which one of the following is a valid choice for the ordered triad $$(a, b, c)$$:

The equation of a tangent to the hyperbola, $$4x^2 - 5y^2 = 20$$, parallel to the line $$x - y = 2$$, is:

For each $$t \in R$$, let $$[t]$$ be the greatest integer less than or equal to $$t$$. Then, $$\lim_{x \to 1^+} \frac{(1-|x|+\sin|1-x|)\sin\left(\frac{\pi}{2}[1-x]\right)}{|1-x|[1-x]}$$

Consider the statement: "$$P(n): n^2 - n + 41$$ is prime". Then which one of the following is true?

The mean of five observations is 5 and their variance is 9.20. If three of the given five observations are 1, 3 and 8, then a ratio of other two observations is:

Consider a triangular plot $$ABC$$ with sides $$AB = 7$$ m, $$BC = 5$$ m and $$CA = 6$$ m. A vertical lamp-post at the mid-point $$D$$ of $$AC$$ subtends an angle 30$$^{\circ}$$ at $$B$$. The height (in m) of the lamp-post is:

In a class of 140 students numbered 1 to 140, all even numbered students opted Mathematics course, those whose number is divisible by 3 opted Physics course and those whose number is divisible by 5 opted Chemistry course. Then the number of students who did not opt for any of the three courses is:

If the system of equations $$x + y + z = 5$$, $$x + 2y + 3z = 9$$, $$x + 3y + \alpha z = \beta$$ has infinitely many solutions, then $$\beta - \alpha$$ equals:

Let $$d \in R$$, and $$A = \begin{bmatrix} -2 & 4+d & (\sin\theta)-2 \\ 1 & (\sin\theta)+2 & d \\ 5 & (2\sin\theta)-d & (-\sin\theta)+2+2d \end{bmatrix}$$, $$\theta \in [0, 2\pi]$$. If the minimum value of $$\det(A)$$ is 8, then a value of $$d$$ is:

Let $$f(x) = \begin{cases} \max(|x|, x^2), & |x| \leq 2 \\ 8-2|x|, & 2 < |x| \leq 4 \end{cases}$$. Let $$S$$ be the set of points in the interval $$(-4, 4)$$ at which $$f$$ is not differentiable. Then $$S$$:

Let, $$f: R \to R$$ be a function such that $$f(x) = x^3 + x^2f'(1) + xf''(2) + f'''(3)$$, $$\forall x \in R$$. Then $$f(2)$$ equals:

The shortest distance between the point $$\left(\frac{3}{2}, 0\right)$$ and the curve $$y = \sqrt{x}$$, $$(x > 0)$$, is:

Let, $$n \geq 2$$ be a natural number and $$0 \lt \theta \lt \frac{\pi}{2}$$. Then $$\int \frac{(\sin^n\theta - \sin\theta)^{1/n} \cos\theta}{\sin^{n+1}\theta} d\theta$$, is equal to:

Let $$I = \int_a^b (x^4 - 2x^2)dx$$. If $$I$$ is minimum then the ordered pair $$(a, b)$$ is:

If the area enclosed between the curves $$y = kx^2$$ and $$x = ky^2$$, $$(k \gt 0)$$, is 1 sq. unit. Then $$k$$ is:

If $$\frac{dy}{dx} + \frac{3}{\cos^2 x}y = \frac{1}{\cos^2 x}$$, $$x \in \left(-\frac{\pi}{3}, \frac{\pi}{3}\right)$$, and $$y\left(\frac{\pi}{4}\right) = \frac{4}{3}$$, then $$y\left(-\frac{\pi}{4}\right)$$ equals:

Let $$\vec{a} = 2\hat{i} + \lambda_1\hat{j} + 3\hat{k}$$, $$\vec{b} = 4\hat{i} + (3-\lambda_2)\hat{j} + 6\hat{k}$$ and $$\vec{c} = 3\hat{i} + 6\hat{j} + (\lambda_3 - 1)\hat{k}$$ be three vectors such that $$\vec{b} = 2\vec{a}$$ and $$\vec{a}$$ is perpendicular to $$\vec{c}$$. Then a possible value of $$(\lambda_1, \lambda_2, \lambda_3)$$ is:

Let $$A$$ be a point on the line $$\vec{r} = (1-3\mu)\hat{i} + (\mu-1)\hat{j} + (2+5\mu)\hat{k}$$ and $$B(3, 2, 6)$$ be a point in the space. Then the value of $$\mu$$ for which the vector $$\vec{AB}$$ is parallel to the plane $$x - 4y + 3z = 1$$ is:

The plane passing through the point $$(4, -1, 2)$$ and parallel to the lines $$\frac{x+2}{3} = \frac{y-2}{-1} = \frac{z+1}{2}$$ and $$\frac{x-2}{1} = \frac{y-3}{2} = \frac{z-4}{3}$$ also passes through the point:

An unbiased coin is tossed. If the outcome is a head then a pair of unbiased dice is rolled and the sum of the numbers obtained on them is noted. If the toss of the coin results in tail then a card from a well-shuffled pack of nine cards numbered 1, 2, 3, ..., 9 is randomly picked and the number on the card is noted. The probability that the noted number is either 7 or 8 is: